Mathematical Methods for Engineering Applications: ICMASE 2022, Bucharest, Romania, July 4–7 3031216997, 9783031216992

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Springer Proceedings in Mathematics & Statistics

Fatih Yilmaz · Araceli Queiruga-Dios · Jesús Martín Vaquero · Ion Mierluş-Mazilu · Deolinda Rasteiro · Víctor Gayoso Martínez   Editors

Mathematical Methods for Engineering Applications ICMASE 2022, Bucharest, Romania, July 4–7

Springer Proceedings in Mathematics & Statistics Volume 414

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Fatih Yilmaz · Araceli Queiruga-Dios · Jesús Martín Vaquero · Ion Mierlu¸s-Mazilu · Deolinda Rasteiro · Víctor Gayoso Martínez Editors

Mathematical Methods for Engineering Applications ICMASE 2022, Bucharest, Romania, July 4–7

Editors Fatih Yilmaz Faculty of Arts and Sciences Ankara Hacı Bayram Veli University Polatli, Ankara, Turkey Jesús Martín Vaquero Department of Applied Mathematics University of Salamanca Salamanca, Spain Deolinda Rasteiro Department of Physics and Mathematics Instituto Superior de Engenharia de Coimbra Coimbra, Portugal

Araceli Queiruga-Dios Department of Applied Mathematics University of Salamanca Salamanca, Spain Ion Mierlu¸s-Mazilu Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest, Romania Víctor Gayoso Martínez Centro Universitario de Tecnología y Arte Digital (U-tad) Las Rozas de Madrid, Spain

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-031-21699-2 ISBN 978-3-031-21700-5 (eBook) https://doi.org/10.1007/978-3-031-21700-5 Mathematics Subject Classification: 08Axx, 11-xx, 65-xx, 97Uxx © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Statistical Analysis of Car Data Using Analysis of Covariance (ANCOVA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thaer Syam, Mahmoud M. Syam, Adnan Khan, Mahmoud I. Syam, and Muhammad I. Syam RBF-FD Solution of Natural Convection Flow of a Nanofluid in a Right Isosceles Triangle Under the Effect of Inclined Periodic Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bengisen Pekmen Geridonmez

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13

Quantum Graph Realization of Transmission Problems . . . . . . . . . . . . . . . Gökhan Mutlu

23

A Monge-Kantorovich—Type Norm on a Vector Measures Space . . . . . . Ion Mierlus-Mazilu and Lucian Nita

33

Further Fixed Point Results for Rational Suzuki F-Contractions in b-Metric-Like Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kastriot Zoto and Ilir Vardhami Mutual Generation of the Choice and Majority Functions . . . . . . . . . . . . . Elmira Yu Kalimulina Dynamical Germ-Grain Models with Ellipsoidal Shape of the Grains for Some Particular Phase Transformations in Materials Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paulo R. Rios, Harison S. Ventura, and Elena Villa Social Interactions and Mathematical Competencies Development . . . . . Daniela Richtarikova Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear Spinor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oktay K. Pashaev

39 49

59 75

81

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Contents

Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s Theorem for Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oktay K. Pashaev

93

Derivative-Free Finite-Difference Homeier Method for Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Yanal Al-Shorman, Obadah Said Solaiman, and Ishak Hashim The Effect (Impact) of Project-Based Learning Through Augmented Reality on Higher Math Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Cristina M. R. Caridade Performance of Machine Learning Methods Using Tweets . . . . . . . . . . . . . 123 ˙Ilkay Tu˘g and Betül Kan-Kilinç A Note on Special Matrices Involving k-Bronze Fibonacci Numbers . . . . 135 Paula Catarino and Sandra Ricardo Influence of the Collaboration Among Predators and the Weak Allee Effect on Prey in a Modified Leslie-Gower Predation Model . . . . . . 147 Alejandro Rojas-Palma and Eduardo González-Olivares Experience in Teaching Mathematics to Engineers: Students Versus Teacher Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Cristina M. R. Caridade On Some Q-Dual Bicomplex Jacobsthal Numbers . . . . . . . . . . . . . . . . . . . . 175 Serpil Halıcı and Sule Curuk The Moore-Penrose Inverse in Rickart ∗-Rings . . . . . . . . . . . . . . . . . . . . . . . 191 Mehsin Jabel Atteya k-Oresme Polynomials and Their Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 201 Serpil Halıcı, Zehra Betül Gür, and Elifcan Sayın k-Oresme Numbers and k-Oresme Numbers with Negative Indices . . . . . 211 Serpil Halıcı, Elifcan Sayın, and Zehra Betül Gür A Note on k-Telephone and Incomplete k-Telephone Numbers . . . . . . . . . 225 Paula Catarino, Eva Morais, and Helena Campos Extended Exponential-Weibull Mixture Cure Model for the Analysis of Cancer Clinical Trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Adam Braima Mastor, Oscar Ngesa, Joseph Mung’atu, Ahmed Z. Afify, and Abdisalam Hassan Muse On the Statistical Properties of the Deformed Algebras on the Jackson q-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Mehmet Niyazi Çankaya

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Quaternion Algebras and the Role of Quadratic Forms in Their Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Nechifor Ana-Gabriela Multicovariance and Multicorrelation for p-variables . . . . . . . . . . . . . . . . . 273 Mehmet Niyazi Çankaya An Individual Work Plan to Influence Educational Learning Paths in Engineering Undergraduate Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 M. E. Bigotte de Almeida, J. R. Branco, L. Margalho, M. J. Cáceres, and A. Queiruga-Dios

Statistical Analysis of Car Data Using Analysis of Covariance (ANCOVA) Thaer Syam, Mahmoud M. Syam, Adnan Khan, Mahmoud I. Syam, and Muhammad I. Syam

1 Introduction Analysis of covariance (ANCOVA) is a technique that combines features of analysis of variance and regression. It can be used for either observational studies or designed experiments. The basic idea is to augment the analysis of variance model containing the factor effects with one or more additional quantitative (covariates) variables that are related to the response variable. This augmentation is intended to reduce the variance of the error terms in the model; to make the analysis more precise. Thus, covariance models are just a special type of regression model [1]. According to [2], data revealed that brand name (of cars especially) has strong influence on purchase decision, so this is probably a major factor that increases prices. According to [3], the factors that influence car price include buying source, car model, rarity of models, their stylish specifications, mileage, age, fuel efficiency/economy, stability. Obviously, these factors will all influence the car desirability and therefore price, T. Syam (B) Department of Mechanical Engineering, Texas A&M University, College Station, TX 3123 TAMU, USA e-mail: [email protected] M. M. Syam School for Engineering of Matter, Arizona State University, Transport & Energy, Tempe, AZ 85287-6106, USA A. Khan Department of Material Science and Engineering, Texas A&M University, College Station, TX 3003 TAMU, USA M. I. Syam Department of Mathematics, Foundation Program, Qatar University, P.O. 2713, Doha, Qatar M. I. Syam Department of Mathematical Sciences College of Science, UAE University Al-Ain, Al Ain, UAE © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_1

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2

T. Syam et al.

either positively or negatively, to some extent. To know which of these many factors are the most influential on car prices is very useful and makes predictions simpler. Regression analysis was carried out on a sample of 30 Japanese used cars (Toyota, Honda, Nissan, and Suzuki) to see which factors were correlated and not correlated with their car prices [4]. It was found that overall mileage (miles already driven by the used car), did not correlate significantly with car prices. However, the maximum power of the car showed strong positive correlation with car prices. A guided statistical investigation by [5] used Multiple Regression Analysis techniques on a data set from the Kelley Blue Book in 2005, about several used American General Motors (GM) cars, including the make, model, equipment, mileage, and the Kelley Blue Book suggested retail price. Their results indicated that an obvious strong positive correlation between the liter volume (of the tank) and cylinder displacement (size of engine). Mileage: had a negative correlation with car price, as expected, which was mitigated to some extent by a combination of liter size and cylinder number. Also, the estimated impact of changing cylinder size did not depend on mileage. Thus, for any given number of miles, when the number of cylinders changed from four to eight, they expected an increase in Price of 4 times. Moreover, the Mileage coefficient stated that holding Cylinder number constant; they expected the Price to decrease by $0.20 for each additional mile on the car (milometer). An additional mile impacted Price differently depending on the second variable, Cylinder number. For example, when there were four cylinders, Price was reduced by $0.058 with each additional mile. When there were eight cylinders, Price was reduced by $0.456 with each additional mile. In this article, Multiple Linear Regression (MLR) will be used to analyze the data to understand and identify the main factors that affects the prices of cars in different countries. Also, a model selection analysis will be done to reduce the model for more significant results. Finally, ANCOVA will be used to analyze these quantitative factors to visualize how they affect the car prices in different countries producers. This work is about analyzing statically a set of measurements of 13 factors of a sample of 66 cars collected by [6] (Chambers, Cleveland, Kleiner and Tukey, 1983). The contribution is to investigate the most significant factors that affect the car price. Also, to understand more comprehensively the relationships between different variables of car data, and finally, to classify and predict the car data based on the country producer.

2 Data Set The car data set consists of 13 variables measured for 66 car types. This data was collected by (Chambers, Cleveland, Kleiner and Tukey, 1983) [6]. This data set can be obtained from the car manufacturers (catalogue); it illustrates most of the cars features that most likely affect their price. This data set is the selected data to study each feature as a variable (covariate) using ANCOVA analysis to see the difference between the three countries producers. This data set is shown in Table 1.

–

–

20

18

15

20

25

23

17

17

22

M

b

–

3

3

4

3

4

3

5

3

3

R1978

c

–

3

4

4

3

4

3

2

1

2

R1977

d

e

–

2.0

4.0

4.0

4.5

2.5

2.5

3.0

3.0

2.5

H

f

–

28.5

30.5

31.5

29.0

26.0

28.0

27.0

25.5

27.5

SC

g

–

16

21

20

16

12

11

15

11

11

T

h

–

3280

3670

4080

3250

2650

2070

2830

3350

2930

W

i

–

200

218

222

196

177

174

189

173

186

L

–

42

43

43

40

34

36

37

40

40

Dj

–

196

231

350

196

121

97

131

258

121

Disp k

b

Price (USD) Mileage (in miles per gallon) c Repair record_1978 (rated on a 5-point scale; 5 best, 1 worst)d Repair record_1977 (rated on a 5-point scale; 5 best, 1 worst) e Headroom (in inches)f Rear seat clearance (distance from front seat back to rear seat, in inches) g Trunk space (in cubic feet)h Weight (in pound) i Length (in inches)j Turning diameter (clearance required to make a U-turn, in feet) k Engine Displacement (in cubic inches)l Gear ratio for high gear m Company headquarter (1 for USA, 2 for Japan, 3 for Europe)

a

Up to…. n = 66

4816

Buick_Century

5189

9735

BMW_320i

Buick_Regal

6295

Audi_Fox

7827

9690

Audi_5000

5788

4749

AMC_Pacer

Buick_Le_Sabre

4099

AMC_Concord

Buick_Electra

P

Car mark

a

Table 1 Car data obtained from [6] l

–

2.93

2.73

2.41

2.93

3.64

3.70

3.20

2.53

3.58

GR

–

1

1

1

1

3

3

3

1

1

Country m

Statistical Analysis of Car Data Using Analysis of Covariance (ANCOVA) 3

4

T. Syam et al.

3 Analysis and Results 3.1 Multiple Linear Regression (MLR) Before start analysing these data using ANCOVA, MLR is applied. The target of the MLR is to study the car price dependency over the other quantitative variables to investigate which factors affect the car price. This task is achieved using the Rstudio software by constructing a linear model. The coefficient of determination of this model is R 2 =0.6282. Indeed, the linear relation among the quantitative variables of the data set can be enhanced by adding more quantitative variables to the existed ones such as weight 2 . This task was achieved firstly by calculating the Pearson correlation coefficient r X,Y between the Price (dependent variable Y) and the other quantitative variables one at a time (independent variable X) to investigate the linear association between Y and X using Excel. Table 2 illustrates the computed correlation coefficient of the data set of Table 1. Some of the variables are linearly related to the price of cars and some are not. For, example displacement and weight have a linear relation the price. However, headroom has not. Secondly, the T03methodology for enhancing the linear relation among the data set is by handling each independent variable and apply some possible functions (log(x), 1/x, ex p(x), and x 2 ) where the Pearson correlation coefficient was computed for each possible function and choosing the one resulted in higher r X,Y . Repair_1978, Repair_1977 and Headroom variables were not analyzed since the linear relation existed between price and these variables are too weak. The result of the analysis concluded that the new variables that could be added are the ones resulted in higher r X,Y than the ones shown in Table 2. Table 3 shows the new added variables with corresponding r X,Y . Therefore, the new modified linear full model of the car price will be constructed based on 17 qualitative independent variables described by Tables 2 and 3. The model is as following:

Table 2 r X,Y between price and other quantitative variables Price, M

Price, R1978

Price, R1977

Price, H

Price, SC

Price, T

r = - 0.462706

r = 0.000816

r = 0.059735

r = 0.124710

r = 0.407124

r = 0.341569

Price, W

Price, L

Price, D

Price, Disp

Price, G R

r = 0.561146

r = 0.447593

r = 0.320233

r = 0.559165

r = −0.404920

Table 3 r X,Y between price and added quantitative variables Price,

1 M

Price, D2

Price, W2

Price, Disp2

Price, L2

Price,

1 GR

r = 0.5651561 r = 0.3278167 r = 0.5993527 r = 0.6182344 r = 0.4534153 r = 0.4788554

Statistical Analysis of Car Data Using Analysis of Covariance (ANCOVA)

5

Price = −105735.8 + (−59.51001 × M) + (30367.62 × M −1 ) + (81.20402 × R1978 ) + (−137.2307 × R1977 ) + (−361.8804 × H ) + (91.48332 × S_C) + (78.18014 × T ) + (−13.60103 × W ) + (0.003123287 × W 2 ) + (1099.126 × L) + (−3.018339 × L 2 )

(1)

+ (−516.3384 × D) + (0.4972458 × D 2 ) + (−23.33155 × Disp) + (0.01728158 × Disp 2 ) + (7079.082 × G R ) + (70344.79 × G −1 R )

The coefficient of determination R 2 = 0.7716 which is better than the previous linear model with = R 2 0.6282.

3.2 Model Selection/Reduction/Analysis of Variance (ANOVA) The strategy that has been followed in the analysis is the search strategy by looking for the variables in the full model that has the highest probabilities from t-distribution Pr (> |t|). Then, the reduced linear model could be constructed by considering all variables except the ones with highest Pr (> |t|) which indicates that they are not significant and investigate the Analysis of Variance (ANOVA) between the initial full model and the reduced one. If the probability from F-distribution Pr (> F) < 0.05 which is the assumption, then the full and the reduced models are significantly differs from each other which concludes accepting the longer model as the optimal model. The first attempt is to construct the reduced model 1 by removing D 2 , R1978 and Disp 2 from the full model which have Pr (> |t|) of 0.97513, 0.82998 and 0.74096 respectively. The coefficient of determination of the reduced model 1 is R 2 = 0.7708. Then, performing the ANOVA analysis between the full model and the reduced model 1 yields: the F-ratio = 0.0594 and Pr (> F) = 0.9808 which indicates 98.08% that both models do not significantly differ and D 2 , R1978 and Disp 2 can be removed from the model. The second attempt is to construct the reduced model 2 by removing Repair_1977 and Mileage from the reduced model 1 which have Pr (> |t|) of 0.765947 and 0.723549 respectively. The R 2 of the reduced model 2 = 0.7696. Then, performing the ANOVA analysis between the full model and the reduced model 2 yields: the F-ratio = 0.0829 and Pr (> F) = 0.9946 which indicates 99.46% that both models do not significantly differ and R1977 and M can be removed from the model. The third attempt is to construct the reduced model 3 by removing T , SC , and H from the reduced model 2 which have Pr (> |t|) of 0.417661, 0.290939 and 0.275480 respectively. The R 2 of the reduced model 3 = 0.7581. Then, performing the ANOVA analysis between the full model and the reduced model 3 yields: the F-ratio = 0.355 and Pr (> F) = 0.9388 which indicates 93.88% that both models do not significantly differ, and T , SC , and H can be removed from the model. All variables in reduced model 3 have very small Pr (> |t|) ≤ 0.032963 which even

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less that a significant level (0.05) and reduced model 3 can be considered as the significant optimal linear model of price. The conclusion is that car price is significantly depends on 9 variables which are M −1 , W , W 2 , L, L 2 , D, Disp, G R , andG −1 R . Price = −130639.9 + (69242.67 × M −1 ) + (−14.78885 × W ) + (0.003223441 × W 2 ) + (1153.893 × L) + (−3.099709 × L 2 ) + (−457.7132 × D) + (−17.49719 × Disp) + (9834.481 × G R )

(2)

+ (96953.52 × G −1 R ).

3.3 Analysis of Covariance (ANCOVA) After reducing the model into a significant model. The target of the ANCOVA is to study the car price for 3 groups of car country producers (USA, Japan, Europe) considering the effect of the quantitative variables (covariates). ANCOVA is investigated for two cases: (1) car price for 3 group countries taking into consideration all quantitative variables (covariates) and (2) car price for 3 group countries taking into consideration one quantitative variable (covariate) at a time.

Car Price Considering All Covariates The ANCOVA model of car price for three group countries taking the effect of all covariates is constructed It can be noticed from the coefficients of the ANCOVA model that at a reference point of the covariates there is a big difference of car price between group countries. Since this model is a multivariable model of 9 independent variables (covariates), then it can’t be plotted as normal 2D or 3D plot. Therefore, a special type of plotting called “scatter plot matrix” using the R command “splom” has been used instead to visualize the correlation between the covariates of the three group countries. It can be noticed from Fig. 1 that grouping is clearly shown between group countries where USA is always grouped from the other two countries. Also, the overall relations between all covariates are fine. Some of them are strong correlated and others are not. This figure was only a visualization of considering all variables as covariates.

Car Price Considering One Covariate at a Time The ANCOVA model for this case is constructed for the 9 different variables taken as a covariant at a time, and this results in the following plot as in Fig. 2.

Fig. 1 Scatter plot matrix shows the correlation between covariates for three countries

Statistical Analysis of Car Data Using Analysis of Covariance (ANCOVA) 7

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T. Syam et al.

Fig. 2 Car price versus Mileage−1 as covariate for three car producing countries

It can be indicated that the models (prices in $) for the three countries producers are as follows with Eq. (3) for USA, Eq. (4) for Japan and Eq. (5) for Europe: YU S A = Bo + Bz ∗ Z

(3)

Y J apan = Bo + B1 + Bz ∗ Z

(4)

Y Eur ope = Bo + B2 + Bz ∗ Z

(5)

where Bo is the intercept, Bz is the covariate coefficient, B1 and B2 are the intercepts for Japan and Europe models, Z is the mean value of the covariate (Mileage−1 ) which equals to 21.18 miles per gallon and by substituting it was found that the car price for USA, JAPAN, and EUROPE is $3,307,896.5, $3,309,040.3, and $3,310,298.471, respectively. These values are the adjusted means for the three countries producers by taking the effect of Mileage−1 as a covariate. Based on the ANCOVA model taking the effect of the Mileage−1 , Europe cars price is more expensive than Japanese and American cars. The differences between mean values between Europe and Japan, Japan and USA, and Europe and USA are $1258.2, $1143.8, and $2401.971. These differences are statistically acceptable and significant. The interpretation of these results can be that from Pr (|| > t) values, it can be concluded whether to reject the null hypothesis (coefficients are equal to zero) if this value is less than 0.05 which is the original assumption. Therefore, we reject the null hypothesis for the Mileage−1 (covariate) coefficient Bz and B2 which means they are significant values and important for this linear regression model. To contrast, we cannot reject the fact that B1 and Bo is different from zero, so they are not significant as in Fig. 2.

Statistical Analysis of Car Data Using Analysis of Covariance (ANCOVA)

9

Figure 3a–h show the other 8 factors as covariates. Based on the ANCOVA model taking the effect of the different parameters, countries were ranked based on cheapest to most expensive. Also, a rejection of the null hypothesis is a space of decision based on the Pr (|| > t) values for each coefficient (t) less than 0.05 significance level. This means that this parameter is very important for this linear regression model. Based on the ANCOVA model, length, gear ratio and gear ratio−1 affect all countries significantly, all coefficients are significant, and these variables are extremely important and significant for this linear regression model. Mileage−1 is a significant factor in Europe only. Weight, weight2 , diameter, and displacement have significant factors in Japan and Europe only, opposite to USA (US) which needs more statistical analysis. Length2 is a significant factor in USA and Europe only.

Fig. 3 Car Price Vs all other 8 variables as covariates for three car producing countries, a Weight, b Weight2 , c Length, d Length−1 , e Diameter, f Displacement, g Gear_ratio, and f Gear_ratio−1

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Table 4 Significance of all covariate’s coefficients and their interpretation Covariate

Ranking of producer’s prices

Significance [Pr(| |>t)] < 0.05 ? Bz

Bo

Mileage−1

E > J > US

✓

Weight

E > J > US

✓

Weight2

E > J > US

✓

Length

E > J > US

✓

✓

Length2

E > J > US

✓

Diameter

E > J > US

✓

Displacement

E > J > US

✓

Gear ratio

E > J > US

✓

Gear ratio−1

E > J > US

✓

B1

Interpretation B2 ✓

Significant factor in Europe only

✓

✓

Significant factor in Japan and Europe only

✓

✓

Significant factor in Japan and Europe only

✓

✓

Significant factor in all countries

✓

✓

Significant factor in USA and Europe only

✓

✓

Significant factor in Japan and Europe only

✓

✓

Significant factor in Japan and Europe only

✓

✓

✓

Significant factor in all countries

✓

✓

✓

Significant factor in all countries

4 Conclusion The article data set was going through some levels before performing ANCOVA analysis directly to get the best possible multiple linear models. Firstly, a MLR analysis was done to construct a full model for this data and how variables affect the car price for different country producers. Then, some analysis to see which factors significantly affect the car price to keep them and get rid of the rest by performing a single linear regression model between each price and each variable to see the value of R 2 . These are ways for enhancement for the multiple linear regression model. The conclusion from the Regression Analysis is that car price is significantly depends on 9 variables. After, that, a model selection/reduction analysis was done using ANCOVA. Three models were suggested, and it was compared with each other, models showed that there is no significant difference between them and hence enhancement and reduction to obtain the best model to perform ANCOVA analysis. It can be concluded from the ANCOVA results that for this data sample, for each covariate Europe cars have higher prices than Japanese and USA cars. Also, all covariates were correlated with each other, and it is obtained that most of them have a relation with each. Results also showed that most of the factors affect the car prices for three countries producers except some of them as obtained in the previous section such as displacement which is not an important factor that affect the car price in the

Statistical Analysis of Car Data Using Analysis of Covariance (ANCOVA)

11

USA. The difference between the new (adjusted) mean values were analyzed and results showed that it is statistically correct and significant. Finally, the intensive analysis which was done answers the two research questions that were mentioned earlier. The most significant factors which affect the car price were identified and how these qualitative factors influenced the car prices in different counties Producers. The statistical analyses which were used were quite useful. It would be good to check different statistical analysis methods on the same data, to ensure that they both come up with the same answers, which they should. Also, the analysis can be done on a wider sample size, a larger number of cars like all cars that were manufactured per year for each country. This will help to get better results. Also, R-studio software was used to implement this analysis. It is coding software; other statistical tools might be used to verify and support the results. In addition, ANCOVA will be best to use with smaller number of covariates. In this project, 9 covariates were analyzed. Future research can aim to reduce these variables more and more for better interpretation. Covariates are independent from the dependent variable (car price); this will show better correlation and hence better analysis. Acknowledgements This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References 1. Heflin L. B, Zuiker N. J, Calkins G. E, Putnam Z. R, and Whitten D 2022 AIAA Sci. Technol. Forum Expo. AIAA SciTech Forum. 2. Alamgir M 2010 The Annals of The “¸Stefan cel Mare. https://ideas.repec.org/a/scm/ausvfe/v10 y2010i2(12)p142-153.html. 3. “How to Determine Price of Used Car | Factors Affecting Used Car Pricing.” https://www.pro kerala.com/automobile/articles/determining-price-of-used-cars.htm. 4. “Statistical Investigation on Car Prices - 1938 Words | 123 Help Me.” https://www.123helpme. com/essay/Statistical-Investigation-on-Car-Prices-149491. 5. Kuiper & Sklar 2012 Ch.3 Multiple regression- How much is your car worth? Practicing Statistics: Guided Investigations for the Second Course, Pearson. 6. Chambers J, Cleveland W, Kleiner B, and Tukey P 1983 Graph. Methods Data Anal, pp 1–395.

RBF-FD Solution of Natural Convection Flow of a Nanofluid in a Right Isosceles Triangle Under the Effect of Inclined Periodic Magnetic Field Bengisen Pekmen Geridonmez

1 Introduction and Problem Definition In the last decade, numerical studies on nanofluids are in a rapid progress due to the improvement in thermal conductivity in the presence of nanoparticles inside a host fluid [1]. Finite difference method [2, 3], finite element method [4, 5] and finite volume method [6, 7] are frequently encountered numerical methods in these problems. In the current study, a local radial basis function method is presented as an alternative numerical method for numerical solutions of these type of problems. In the considered problem, laminar, incompressible, 2D, steady flow of a Newtonian nanofluid exists inside a right triangle as demonstrated in Fig. 1. A heater of length  = 0.5 is located at the center of the left wall. Radiation, viscous dissipation and Brownian motion are ignored. Further, Joule heating effect, Hall effect and induced magnetic field are also assumed as negligible. Single phase nanofluid model is adopted. Thermal and physical properties of water and copper are given in Table 1. Physical relations of nanofluid may be written as ρn f = (1 − φ)ρ f + ρs φ (ρC p )n f = (1 − φ)(ρC p ) f + (ρC p )s φ

(1a) (1b)

(ρβ)n f = (1 − φ)(ρβ) f + (ρβ)s φ 1 μn f = (Brinkman’s model [9]) (1 − φ)2.5   ks + k f ks 1 − φ + 2φ ln ks − k f 2k f  (Xue’s model [10])  kn f = k f kf ks + k f 1 − φ + 2φ ln ks − k f 2k f

(1c) (1d)

(1e)

B. Pekmen Geridonmez (B) Department of Mathematics, TED University, 06420 Ankara, Turkey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_2

13

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B. Pekmen Geridonmez

periodic B

1

Tc Cu-water

Th

B0

θm

θ

 = 0.5

v

· O

u

g

1

Fig. 1 Geometrical description of the problem. θm = θ + 90◦ Table 1 Physical properties of base fluid and Cu nanoparticles [8] Property Water Specific Heat Thermal conductivity Thermal expansion coefficient Density Dynamic viscosity Electrical Conductivity

 σn f = σ f

Cu

C p (J/(kg K)) k (W/(m K) ) β (1/K)

4179 0.613 21 × 10−5

385 401 1.67 × 10−5

ρ (kg/m3 ) μ(kg/(m s)) σ (m)−1

997.1 8.9 × 10−4 0.05

8933 – 5.96 × 107

3(σs /σ f − 1)φ 1+ (σs /σ f + 2) − (σs /σ f − 1)φ

 (Maxwell’s model [11])

(1f)

RBF-FD Solution of Natural Convection Flow of a Nanofluid in a Right Isosceles …

15

where φ is the concentration of nanoparticles, and subindices f, s, n f refer to the base fluid (water), solid (Cu) and nanofluid, respectively, ρ is the density, (ρC p ) is the heat capacitance, (ρβ) is the thermal expansion coefficient, μ is the dynamic viscosity, k is the thermal conductivity and σ is the electrical conductivity. The non-dimensional governing equations in stream function-vorticity formulation is extracted as [12] (2a) ∇ 2 ψ = −w αn f 2 ∂T ∂T ∇ T =u +v , (2b) αf ∂x ∂y   (ρβ)n f ν f ν f ρn f ∂w ∂w ∂T 2 u +v − Ra Pr Pr ∇ w = μn f ∂x ∂y β f μn f ∂x    σn f μ f π ∂u ∂u ∂v 2 2 2 2 A sin θm − cos θm + 2 sin θm cos θm , − H a Pr sin σ f μn f Λ ∂y ∂x ∂x (2c) where αn f = kn f /(ρC p )n f , ν = μ/ρ is the kinematic viscosity, A = x cos θ + y sin θ , Λ is the period of the periodic magnetic field and θm is the angle of the magnetic field. Note that θm = 180◦ is the same as θm = 0◦ . In these dimensionless equations, Prandtl (Pr ), Rayleigh (Ra) and Hartmann (H a) numbers have the following definitions Pr =

gβ f (Th − Tc )L 3 νf , Ra = , H a = B0 L αf νfαf



σf . μf

(3)

Note also that w = ∇ × u, u = u, v, 0, u = ∂ψ/∂ y, v = −∂ψ/∂ x. Velocity (u) and stream function (ψ) have no slip boundary conditions, namely, u = v = ψ = 0. Temperature is Th = 1 on heater and Tc = 0 on hypothenuse and ∂ T /∂n = 0 on jagged walls. Vorticity boundary conditions are set by definition of vorticity w = ∂v/∂ x − ∂u/∂ y on each boundaries.

2 Numerical Procedure Radial basis functions have taken great interest in the last decade. Novel books [13, 14] include many details about RBFs. RBF-FD is a local method. The localization is constructed around each point having a neighborhood with nodes. The progress of the method is very well explained in Flyer et al. [15]. Let i be a stencil centered at xi involving N number of points around xi . In this stencil, the conventional RBF interpolation would be

16

B. Pekmen Geridonmez

u(x) ≈ s(x) =

N 

χi (||x − xi ||) +

m 

i=1

ηk pk (x),

(4)

i=1

in which  is a radial basis function depending on the radial distance r = ||x − xi ||2 with x = (x, y) and center xi = (xi , yi ). There are some constraints defined by N 

χi pk (xi ) = 0, k = 1, 2, . . . , m

(5)

i=1

where m is the total number of polynomial terms taken into account. Equation (4) with linear polynomials 1, x, y may be written as ⎛

φ(||x1 − x1c ||) φ(||x1 − x2c ||) ⎜ φ(||x2 − x1c ||) φ(||x2 − x2c ||) ⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ φ(||x N − xc ||) φ(||x N − x2c ||) 1 ⎜ ⎜− − − − − − −− − − − − − − −− ⎜ ⎜ 1 1 ⎜ ⎝ x2 x1 y2 y1

... ... .. .

φ(||x1 − xcN ||) φ(||x2 − xcN ||) .. .

. . . φ(||x N − xcN ||) −− − − − − − − −− ... 1 ... xN ... yN

| |

1 1

x1 x2

| −− | | |

1 −− 0 0 0

xN −− 0 0 0

⎞⎛ ⎞ ⎛ ⎞ χ1 u(x1 ) y1 ⎜ χ2 ⎟ ⎜ u(x2 ) ⎟ y2 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎟⎜ . ⎟ ⎜ . ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ yN ⎟ ⎟ ⎜ χ N ⎟ = ⎜u(x N )⎟ ⎜−−⎟ ⎜ −− ⎟ −−⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎟ ⎟ ⎜ η1 ⎟ ⎜ 0 ⎟ 0 ⎠ ⎝ η2 ⎠ ⎝ 0 ⎠ 0 η3 0

(6)

where note that x1 = (x1 , y1 ) and superscript ‘c’ refers to center. Equivalently, Eq. (6) may be expressed in matrix-vector form as (Anew )Υ = u i (x) =⇒ Υ = (Anew )−1 u i (x),

(7)

in which Υ is the vector involving the terms χ1 , . . . , χ N , η1 , . . . , η3 . 1 if i = j By using the cardinal basis function ψ j (xi ) = , j = 1, . . . , N , 0 if i = j interpolant may also be expressed as si (x) = Ψ (x)u i (x),

(8)

where Ψ = [ψ1 (x), ψ2 (x), . . . ψ N (x), ψ N +1 (x), ψ N +2 (x), ψ N +3 (x)]. Note that at a point x, if Υ is known, interpolation would be si (x) = Φ(x)Υ,

or

si (x) = Φ(x)(Anew )−1 u i (x),

(9)

where Φ(x) = [(||x − x1c ||), (||x − x2c ||), . . . , (||x − xcN ||), 1, x, y]. With the help of Eqs. (8) and (9), Ψ (x) is found as Ψ (x)u i (x) = Φ(x)(Anew )−1 u i (x) =⇒ Ψ = Φ(x)(Anew )−1 ,

(10)

note that Ψ is a vector of size 1 × (N + 3), but the last three terms are not taken into account.

RBF-FD Solution of Natural Convection Flow of a Nanofluid in a Right Isosceles …

17

Implementation works in the following steps: 1. Around a center node xi , find nodes inside a stencil. 2. Once these local nodes (say, xiloc ) are found, nodes are shifted and centered at origin as described in [15]. 3. A scaling on nodes is done multiplying all nodes inside stencil with this scaling scale = 1/max(abs(xiloc )) as is used by Tominec [16]. Another scaling which also works is also mentioned in [17]. 4. Matrix Anew with xiloc utilizing a polyharmonic spline RBF is constructed. The right hand side of Eq. (6) may also involve x−, y− and x x + yy derivatives, too, as can be noted in [15]. 5. Ψ (or Ψx (for x-derivatives) or Ψ y (for y-derivatives) or Ψ2 (for Laplacian)) is found and the its first N terms are saved. 6. Each row of the obtained large sparse matrix corresponds to a center point’s index, and entries of that row correspond to the indices of stencil nodes with that center point. After finding the differentiation matrices, say Dx , D y and D2 for the x-, y- and the Laplacian, respectively, iterative solution of the dimensionless nonlinear governing equations is performed as follows D2 ψ n+1 = −w n

(11a)

u = u n+1 = D y ψ n+1 , v = v n+1 = −Dx ψ n+1 (11b)   αf D2 − M T n+1 = 0 (11c) αn f   ν f ρn f (ρβ)n f ν f M w n+1 = − Ra Pr Dx T n+1 Pr D2 − μn f β f μn f    σn f μ f π − H a 2 Pr sin2 A (D y u) sin2 θm − (Dx v) cos2 θm + 2 sin θm cos θm (Dx u) , σ f μn f Λ

(11d) where M is the matrix equal to diag(u)Dx + diag(v)D y and n is the iteration level. A relation between convective and conductive heat transfer is interpreted by average Nusselt number defined by kn f 1 Nu = − kf 

 0

∂T d x. ∂y

(12)

This integral is computed by composite Simpson’s rule numerically. A relaxation parameter τ is used after solving the vorticity equation as wn+1 ← τ w n+1 + (1 − τ )w n , where τ ∈ (0, 1). τ is taken between τ = 0.01 (for large H a = 100) and τ = 0.05 in the current executions.

18

B. Pekmen Geridonmez

3 Numerical Computations All numerical computations are done in 2.3 GHz Quad-Core i7 MacOS computer using MATLAB R2022a. Prandtl and Rayleigh numbers are held at 6.2 and 105 . Fluid flow and heat transfer are visualized in streamlines and isotherms in variation of Hartmann number, period (Λ) and the angle (θm ). Regarding to the single phase model, φ is kept at 0.04. Polyharmonic spline RBF, f = r 7 , is adopted with cubic polynomial terms (1, x, y, x 2 , x y, y 2 , x 3 , x 2 y, x y 2 , y 3 ). In the following figure results, the contours in ψ and T rows correspond to the streamlines and isotherms, respectively. Uniform node distribution is settled as in Fig. 2. As an observation, RBF-FD method seems working well only with uniform nodes or nodes close to the uniform mode. On this figure, two stencil examples are also shown. Red point is the center of the stencil while the blue points around the red one are the nodes inside the stencil. The following results are obtained by Nb = 192 number of boundary and Ni = 1953 number of interior nodes, respectively. In Fig. 3, fluid flow and heat transfer in variation of H a number are demonstrated when θm and Λ are kept at 0◦ and 1, respectively. As H a rises, core vortex in streamlines is pushed to the bottom of the triangle cavity and in turn the central streamline values decrease. Convective behavior of streamlines at H a = 10 almost become stabilized at H a = 100 exhibiting conductive behavior due to the retardation of Lorentz force. In Fig. 4, variation in period of periodic magnetic field is illustrated when H a = 30, θm = 0◦ . The most reduction in central fluid velocity occurs at Λ = 1. This is 1 0.9

0.8 0.7 0.6 0.5

0.4 0.3 0.2

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 2 Node setup and stencil examples (Stencil size : N = 20)

0.7

0.8

0.9

1

RBF-FD Solution of Natural Convection Flow of a Nanofluid in a Right Isosceles …

19

-6.65 -2.41

-1.26

Fig. 3 Variation in H a when θm = 0, Λ = 1

-5.97

-4.86

-4.41 -3.79

Fig. 4 Variation in Λ when θm = 0, H a = 30

due to push by one big wave of periodic magnetic field. Isotherms at Λ = 1 are also a little bit seen as stabilized. In Fig. 5, the influence of different angles of periodic magnetic field is observed and the most reducing effect appears at θm = 180◦ (or θm = 0◦ ). This points to the inhibitive effect of Lorentz force coming along the wall having a heater.

4 Conclusion In this study, RBF-FD solution of natural convection flow of Cu-water nanofluid in a right triangle under the effect of periodic magnetic field is presented. Inclination angles between 0◦ and 90◦ are also conducted.

20

B. Pekmen Geridonmez

-4.65

-4.79

-5.68 -3.79

Fig. 5 Variation in θm when H a = 30, Λ = 1

Some results may be listed as follows: • While H a = 30 and Λ = 1, if θm changes from 90◦ to 0◦ , 36.5% reduction in ψmax value and 14.6% reduction in N u number is obtained. That is, periodic magnetic field coming through the wall having the heater has more reducing effect on both fluid flow and convective heat transfer. • While Λ = 1, if H a changes from 10 to 100, – 56% reduction in ψmax value and 29.5% reduction in N u number in case of θm = 90◦ is noted. – 81% reduction in ψmax value and 39.5% reduction in N u number in case of θm = 0◦ is exhibited. That is, Hartmann variation significantly affects the fluid flow and heat transfer in horizontal periodic magnetic field. This result also verifies the first result. • In case of H a = 30, θm = 0◦ , the variation in period is in a decreasing trend. 18.7% reduction in ψmax value and 7.31% reduction in N u number occur if Λ is altered from Λ = 0.1 to 1.

References 1. Choi, S.U.S., Eastman, J.A.: Enhancement thermal conductivity of fluids with nanoparticles, Proceedings of the 1995 ASME international mechanical engineering congress and exposition, 66, 99–105 (1995). 2. Ghalambaz, M., Sheremet, M.A., Pop, I.: Free Convection in a Parallelogrammic Porous Cavity Filled with a Nanofluid Using Tiwari and Das’ Nanofluid Model. Plos One 1–17 (2015). 3. Sheremet, M.A., Oztop, H.F., Pop, I.: MHD natural convection in an inclined wavy cavity with corner heater filled with a nanofluid. J. Magn. Magn. Mater. 416, 37–47 (2016).

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4. Izadi, M., Hoghoughi, G., Mohebbi, R., Sheremet, M.: Nanoparticle migration and natural convection heat transfer of Cu-water nanofluid inside a porous undulant-wall enclosure using LTNE and two-phase model. J. Mol. Liq. 261, 357–372 (2018). 5. Dutta, S., Goswami, N., Biswas, A.K., Pati, S.: Numerical investigation of magnetohydrodynamic natural convection heat transfer and entropy generation in a rhombic enclosure filled with Cu-water nanofluid. Int. J. Heat Mass Transf. 136, 777–798 (2019). 6. Sheikzadeh, G.A., Arefmanesh, A., Kheirkhah, M.H., Abdollahi, R.: Natural convection of Cu-water nanofluid in a cavity with partially active side walls. Eur. J. Mech. B Fluids 30, 166–176 (2011). 7. Hussein, A.K., Mustafa, A.W.: Natural convection in a parabolic enclosure with an internal vertical heat source filled with Cu-water nanofluid. Heat Transfer - Asian Res. 47, 320–336 (2018). 8. Ghalambaz, M., Mehryan, S.A.M., Izadpanahi, E., Chamkha, A.J., Wen, D.: MHD natural convection of Cu-Al2 O3 water hybrid nanofluids in a cavity equally divided into two parts by a vertical flexible partition membrane. J. Therm. Anal. Calorim. 138, 1723–1743 (2019). 9. Brinkman, H.C.: The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 3, 571-581 (1952). 10. Xue, Q. Z.: Model for thermal conductivity of carbon nanotube-based composites. Physica B 368, 302–307 (2005). 11. Maxwell-Garnett, J.C.: Colors in metal glasses and in metallic films. Phil. Trans. R. Soc. A 203, 385-420 (1904). 12. Pekmen Geridonmez, B., Oztop, H.F.: The effect of inclined periodic magnetic field on natural convection flow of Al2O3-Cu/water nanofluid inside right isosceles triangular closed spaces, Eng. Anal. Bound. Elem. 141, 222–234 (2022). 13. Fasshauer, G.E.: Meshfree Approximation Methods with Matlab. World Scientific Publications, Singapore (2007). 14. Fasshauer, G.E. and McCourt, M.: Kernel-based Approximation Methods using MATLAB. World Scientific Publications, Singapore (2015). 15. Flyer, N., Barnett, G. A., Wicker, L.J.: Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations, J. Comput. Phys. 316, 39–62 (2016). 16. An Rbf-Fd implementation example : https://github.com/IgorTo/rbf-fd. 17. Shahane, S., Radhakrishnan, A., Vanka, S.P.: A high-order accurate meshless method for solution of incompressible fluid flow problems. J. Comput. Phys. 445, 110623 (2021).

Quantum Graph Realization of Transmission Problems Gökhan Mutlu

1 Introduction A metric graph is obtained by identifying each edge with an interval of the real line. Hence a metric graph does not just consist of vertices and edges (as relations between vertices), but it consists of all intermediate points on the edges as well. This identification gives rise to a local coordinate xe for each point on edge e on the graph. Moreover, it induces a natural direction starting from xe = 0. Therefore a metric graph is directed. One can endow a metric graph with a natural metric which makes it a topological space and explains the name. An interval can be considered as the simplest example of a metric graph. Quantum graphs are referred to self-adjoint differential operators acting on metric graphs. Namely, a quantum graph consists of a metric graph, a differential operator acting on the edges and suitable vertex conditions imposed at the vertices. These graphs have been used for a long time in order to model real-world problems taking place in a graph-like structure (network). First problem of this type was observed in chemistry [5]. Other models can be observed in quantum mechanics, quantum chaos, quantum wires, dynamical systems, photonic crystals, scattering theory, nanotechnology, mesoscopic physics, etc. [4, 7]. Most of the physical problems can be described by boundary value problems. Usually these boundary value problems involve differential equations and some boundary conditions at the endpoints of the considered interval. However there exist some discontinuous boundary value problems in which there are singularities inside the interval called points of interaction. At these points, transmission conditions (also known as jump conditions or impulsive conditions) are imposed to relate the values of the function and its derivatives on both sides of these points. These type of problems are called boundary value transmission problems. Since these problems have G. Mutlu (B) Faculty of Science, Department of Mathematics, Gazi University, 06560 Ankara, Turkey e-mail: [email protected] URL: https://avesis.gazi.edu.tr/gmutlu University of Nottingham, School of Mathematical Sciences, University Park, NG7 2RD Nottingham, UK © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_3

23

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G. Mutlu

singularities at interior points of the interval the spectral analysis of these problems needs to be handled with new methods. The most effective way to deal with these problems is to transfer them to eigenvalue problems in appropriate Hilbert spaces. It has been shown that these problems are self-adjoint in special Hilbert spaces defined in terms of the transmission conditions [1, 2, 10, 11]. In this study we are interested in the quantum graph realization of boundary value transmission problems and relations between them. Firstly, we show that transmission problem with one discontinuous point can be equivalently considered as a quantum two-star graph. Secondly, we treat a quantum two-star graph with self-adjoint vertex conditions as a boundary value transmission problem. Using the approach in [1, 2, 10, 11] we define a special Hilbert space and transfer the quantum graph Hamiltonian into this new Hilbert space. We prove that this Hamiltonian is self-adjoint in this new Hilbert space. Moreover, we show that the well-known self-adjointness criteria of quantum two-star graph Hamiltonian can be obtained as a special case. The rest of the paper is organized as follows. In Sect. 2, we present some basic notions on quantum graphs and transmission problems which are necessary for our study. In Sect. 3, we state our main results.

2 Preliminaries 2.1 Quantum Graphs in a Nutshell In this subsection we briefly summarize necessary notions about quantum graphs. All of the material presented below can be found in [4, 6, 8]. Let G = (V, E) be a graph with finite set V of vertices and finite set E of edges. An edge e can be represented by e = {u, v} and in this case two vertices u and v are said to be incident in e and also e is adjacent to u and v. The number of adjacent edges to a vertex v is called the degree of v. Let us denote the set of adjacent edges to a given vertex v by E v . Clearly the degree of v is |E v |. We assume for our purposes that each vertex has a finite positive degree (hence no loops). Assigning a finite positive length L e to each edge e, we obtain a metric graph. In this metric graph, each edge e is identified with an interval [0, L e ] and the vertices correspond to the endpoints of these intervals. If a metric graph has a finite number of edges with finite lengths then it is called a compact metric graph and otherwise a non-compact metric graph. A metric graph has a rich structure. Namely, it is a metric space when the distance between two points is defined as the length of the shortest path between these points along the graph. This makes a metric graph a topological space. Moreover, in a compact metric graph, each point on edge e has a local coordinate xe ∈ [0, L e ]. This gives rise to a natural direction on the metric graph. Explicitly, the edge e is directed in the direction starting from the point which has local coordinate xe = 0. Considering a metric graph as a collection of intervals in which the vertices corresponding to the same endpoint are identified, one can naturally define the stan-

Quantum Graph Realization of Transmission Problems

25

dard function spaces as the direct sum of spaces on the individual edges. First, note that a function f on a metric graph is a vector of functions on individual edges f = ( f e )e∈E where f e : [0, L e ] → C. Now we can define the Hilbert space L 2 (G) of square-integrable Lebesgue measurable functions on the metric graph as the direct sum |E|    L 2 0, L ei , L 2 (G) := i=1

with the inner product |E| |E|     f, g :=  f i , gi  = f i (x)gi (x)d x. L ei

i=1

(1)

i=1 0

In order to obtain a quantum graph, we need to consider a differential operator (Hamiltonian) acting on the metric graph. Namely, we should consider a differential operator acting on the edges with appropriate boundary conditions imposed at the vertices (called vertex conditions). The most common differential operator is the Laplacian  2  d2 d . f = ( f e )e∈E → − 2 f = − 2 f e x xe e∈E One can consider a more general Schrödinger operator f = ( f e )e∈E → −

  2 d2 d f + q(x) f = − f + q (x ) f e e e e x2 xe2 e∈E

(2)

with a potential q(x) = (qe (xe ))e∈E. We can write the vertex conditions at a vertex v with degree d in the matrix form 

Av F(v) + Bv F (v) = 0,

(3) 

T d fe where Av , Bv are d × d matrices, F(v) = ( and F (v) = (v) d xe e∈E v are d dimensional vectors of values of the functions and their outgoing derivatives at the vertex v, respectively. Here outgoing derivatives are assumed to be taken in the direction away from the vertex v. It is well-known [6] that the compact quantum graph G = (V, E) with the differential operator (2) and vertex conditions (3) is self-adjoint if and only if for every vertex v ∈ V the following conditions hold T f e (v))e∈E v

(i) rank(Av Bv ) = |E v |, (ii) Av Bv∗ = Bv A∗v .



26

G. Mutlu

There are several examples of vertex conditions. The most frequent vertex conditions are called Neumann-Kirchoff or Kirchoff conditions which are given by (i) f = ( f e )e∈E is continuous at v, i.e., f e1 (v) = f e2 (v) whenever edges e1 and e2 are to v, adjacent d fe (ii) e∈E v d xe (v) = 0, where the derivatives are taken in outgoing directions. There are also Dirichlet conditions which can be stated as f = ( f e )e∈E is continuous at v and f (v) = 0. Generalized δ-type vertex conditions cover Kirchoff (αv = 0) and Dirichlet conditions (αv → ∞) and they can be given by (i) f = ( f e )e∈E is continuous at v, d fe (ii) e∈E v d xe (v) = αv f (v), where αv is a fixed number. As a summary, a quantum graph consists of a metric graph, a differential operator acting on the edges and vertex conditions.

2.2 Basic Notions on Boundary Value Transmission Problems In this subsection, we summarize some basic features of transmission problems with one point of interaction, for simplicity. All of the below information in this subsection can be found in [1, 2, 10, 11]. A boundary value transmission problem on a finite interval with one discontinuous interior point consists of Sturm-Liouville equation L y := −y  (x) + q(x)y(x) = λy(x), x ∈ [a, c) ∪ (c, b] , and boundary conditions

(4)

α1 y(a) + α2 y  (a) = 0,

(5)

β1 y(b) + β2 y  (b) = 0,

(6)

and transmission conditions at the point of interaction γ11 y(c+) + γ12 y  (c+) = y(c−),

(7)

γ21 y(c+) + γ22 y  (c+) = y  (c−),

(8)

where the following conditions hold: (i) q is a real-valued continuous function in the intervals [a, c) and (c, b] and the limits q(c±) := lim →0 q(c ± +) are finite,

Quantum Graph Realization of Transmission Problems

27

(ii) λ is a complex eigenparameter, (iii) α1 , α2 , β1 , β2 , γ11 , γ12 , γ21 , γ22 ∈ R such that |α1 | + |α2 | = 0, |β1 | + |β2 | = 0 and δ := γ11 γ22 − γ12 γ21 > 0. Let us define a new inner product in the direct sum space H := L 2 [a, c) ⊕ L 2 (c, b] as c b f 1 (x)g1 (x)d x + δ f 2 (x)g2 (x)d x, (9)  f, g := a



where f (x) =

c

f 1 (x), x ∈ [a, c) , g(x) = f 2 (x), x ∈ (c, b]



g1 (x), x ∈ [a, c) . g2 (x), x ∈ (c, b]

Let D(L) be the subspace of H which consists of f ∈ H such that (i) L f ∈ H (ii) f and f  are absolutely continuous in the intervals [a, c) and (c, b] with finite limits f (c±) := lim →0 f (c ± +), f  (c±) := lim →0 f  (c ± +), (iii) f satisfies the boundary conditions (5)–(6) and transmission conditions (7)–(8). Note that the boundary value transmission problem is equivalent to the eigenvalue problem L y = λy in the Hilbert space H . It has been shown [3, 9] that the operator L is self-adjoint in the Hilbert space H .

3 Main Results Firstly, we shall show that boundary value transmission problem (4)–(8) can be realized as a quantum two-star graph. Consider the two-star graph Γ (see Fig. 1) with central vertex v at the position x = c and two pendant vertices v1 and v2 at the positions x = a and x = b, respectively in (4)–(8). In this quantum graph, the vertices v1 and v2 have the local coordinates x1 = 0 and x2 = L 2 , respectively and the central vertex has the local coordinates x1 = L 1 and x2 = 0 on edges e1 and e2 , respectively, where ei denotes the edge connecting the vertices vi and v for i = 1, 2. We consider the Schrödinger operator L y = −y  + q(x)y acting on the edges as a Hamiltonian where q = (q1 , q2 ) and qi are real-valued continuous functions on [0, L i ]. Then one can write the boundary conditions (5)–(6) as vertex conditions at v1 and v2 (10) α1 y1 (0) + α2 y1 (0) = 0, β1 y2 (L 2 ) + β2 y2 (L 2 ) = 0,

(11)

respectively, and transmission conditions at the point of interaction as vertex conditions at v (12) γ11 y2 (0) + γ12 y2 (0) = y1 (L 1 ),

28

G. Mutlu

v1

v

v2

Fig. 1 Two-star graph Γ

γ21 y2 (0) + γ22 y2 (0) = y  (L 1 ).

(13)

It is easily seen that transmission problem (4)–(8) can be realized as the quantum two-star graph Γ with vertex conditions (10)–(13). Further, let us show that the self-adjointness of this quantum graph can be obtained by the self-adjointness of the transmission problem (4)–(8). First, we shall write the vertex conditions (12)–(13) in the matrix form as in (3). It can be easily shown that Av =

 −1 0

 γ11 , γ21

Bv =

 0 1

 γ12 . γ22

As stated in Sect. 2, quantum graph Γ is self-adjoint iff rank(Av Bv ) = |E v | and Av Bv∗ = Bv A∗v for every vertex. It easily follows that at the pendant vertices these conditions are satisfied. As for the central vertex, taking δ = 1 in the inner product (9) one obtains the usual inner product (1) in L 2 (Γ ). Since δ = 1 > 0, it follows that the resulting operator is self-adjoint. Therefore we have proven the self-adjointness of quantum graph by using the modified inner product (9) associated with transmission problem (4)–(8). Furthermore, it can be shown by direct calculation that the conditions rank(Av Bv ) = 2, and Av Bv∗ = Bv A∗v are equivalent to the condition δ = 1. This is indeed in accordance with the well-known criteria of self-adjointness of two-star graph [6]. Conversely, one can use the approach in transmission problems for quantum twostar graph. We will justify this in the rest of the paper. Let us consider the quantum graph Γ (see Fig. 1) with vertex conditions given by (10)–(11) for vertices v1 and v2 , respectively and (14) AF(v) + B F  (v) = 0, for the central vertex. Here A = [ai j ] and B = [bi j ] are 2 × 2 matrices with real  T entries, F(v) = ( f 1 (L 1 ), f 2 (0))T and F  (v) = − f 1 (L 1 ), f 2 (0) . Here we consider the Schrödinger operator L y = −y  + q(x)y acting on the edges as a Hamiltonian where q = (q1 , q2 ) and qi are real-valued continuous functions on [0, L i ]. Let us introduce the matrices     b12 a12 −b11 a11 , D= . C= −b21 a21 b22 a22 Let us assume δ1 := −detC > 0 and δ2 := −det D > 0. Following the approach in transmission problems, we can define a new inner product on L 2 (Γ ) = L 2 [0, L 1 ] ⊕ L 2 [0, L 2 ] as

Quantum Graph Realization of Transmission Problems

29

 f, g1 := δ1  f 1 , g1  L 2 [0,L 1 ] + δ2  f 2 , g2  L 2 [0,L 2 ] .

(15)

We will show that the quantum two-star graph Hamiltonian is symmetric with respect to the modified inner product (15). Suppose that f, g ∈ L 2 (Γ ) such that L f, Lg ∈ L 2 (Γ ) and f, g satisfy the vertex conditions (10)–(11) and (14). Then L f, g1 = − f  , g1 +  f, q(x)g1 . Using integration by parts and the fact that f, g satisfy the vertex conditions (10)– (11), we can easily find

 − f  , g1 =  f, −g  1 + δ1 f 1 (L 1 )g1 (L 1 ) − f 1 (L 1 )g1 (L 1 )

 +δ2 f 2 (0)g2 (0) − f 2 (0)g2 (0) . Since δ1 = 0 the inverse matrix C −1 exists. Multiplying the equation (14) from left with C −1 one gets 

  0 −det A δ F(v) + 1 r −δ1 0

 s F  (v) = 0, −det B

(16)

where r := a12 b21 − b11 a22 and s := a21 b12 − a11 b22 . Similarly, taking the complex conjugate of (14) for g and then multiplying it from the left with D −1 we have  det A s

0 −δ2



  g1 (L 1 ) −r + −det B g2 (0)

−δ2 0

   −g1 (L 1 ) = 0. g2 (0)

(17)

Equations (16) and (17) are equivalent to the system − (det A) f 2 (0) − δ1 f 1 (L 1 ) + s f 2 (0) = 0,

(18)

− δ1 f 1 (L 1 ) + r f 2 (0) − (det B) f 2 (0) = 0,

(19)

(det A)g1 (L 1 ) + r g1 (L 1 ) − δ2 g2 (0) = 0,

(20)

sg1 (L 1 ) − δ2 g2 (0) + (det B)g1 (L 1 ) = 0.

(21)

Multiplying (18)–(21) with g1 (L 1 ), −g1 (L 1 ), f 2 (0), − f 2 (0), respectively, and summing it up we find

  δ1 f 1 (L 1 )g1 (L 1 ) − f 1 (L 1 )g1 (L 1 ) + δ2 f 2 (0)g2 (0) − f 2 (0)g2 (0) = 0.

(22)

30

G. Mutlu

This proves L f, g1 =  f, Lg1 , as desired. Moreover, self-adjointness of this quantum graph Hamiltonian in the modified Hilbert space L 2 (Γ ) with inner product (15) can be proved similarly to [1, 3, 9, 10] or [4, 6]. Here the domain of the operator might be taken to be the subspace of the Sobolev space H 2 (Γ ) = H 2 [0, L 1 ] ⊕ H 2 [0, L 2 ] such that L f ∈ L 2 (Γ ) and f satisfies the vertex conditions (10)–(11) and (14). Finally, let us explain the relation with the usual self-adjointness of quantum twostar graph. Obviously, the usual Hilbert space with inner product (1) corresponds to taking δ1 = δ2 = 1 in the modified Hilbert space with inner product (15). We have shown above that in the case δ1 = δ2 = 1 > 0 the operator is self-adjoint. Now, let us show that this actually recovers the well-known self-adjointness criteria i.e., rank(Av Bv ) = |E v | and Av Bv∗ = Bv A∗v for every vertex v. Again for pendant vertices, these two conditions are easy to check. Since δ1 = 0, it easily follows rank(A B) = 2. It is straight forward to check that δ1 = δ2 = 1 implies AB ∗ = B A∗ . Therefore, self-adjointness of quantum two-star graph (in the usual sense) has been proven in a different manner with this approach. Moreover, it can be easily observed that we have defined a self-adjoint quantum two-star graph in a general sense which include the usual self-adjointness as a special case. As a final remark we can say that what have been done for a boundary value transmission problem with one interaction point or equivalently for a quantum two-star graph in this paper can be trivially extended to the transmission problem with finite n > 1 interaction points or a quantum path graph with n + 1 edges, correspondingly.

References 1. Aydemir, K.: Boundary value problems with eigenvalue-dependent boundary and transmission conditions. Bound Value Probl 2014, 131 (2014). https://doi.org/10.1186/1687-2770-2014131. 2. Aydemir, K., Mukhtarov, O.: Asymptotic distribution of eigenvalues and eigenfunctions for a multi-point discontinuous Sturm-Liouville problem. Elec. J. Differ. Equ. 131, 1-14 (2016). 3. Aydemir, K., Mukhtarov, O.S.: Variational principles for spectral analysis of one SturmLiouville problem with transmission conditions. Adv Differ Equ 2016, 76 (2016). https:// doi.org/10.1186/s13662-016-0800-z. 4. Berkolaiko. G., Kuchment P.: Introduction to Quantum Graphs. American Mathematical Society, Rhode Island (2013). 5. Griffith, J.S.: A free-electron theory of conjugated molecules. Part 1.-Polycyclic hydrocarbons. Transactions of the Faraday Society, 49, 345-351 (1953). 6. Kostrykin, V., Schrader, R.: Kirchhoff’s rule for quantum wires. J. Phys. A: Math. Gen. 32, 595-630 (1999). https://doi.org/10.1088/0305-4470/32/4/006. 7. Kuchment, P.: Graph models for waves in thin structures. Waves in Random Media 12(4), R1 (2002). https://doi.org/10.1088/0959-7174/12/4/201. 8. Mugnolo, D.: Semigroup Methods for Evolution Equations on Networks. Springer, Cham (2014). 9. Mukhtarov, O.S, Aydemir, K: New type Sturm-Liouville problems in associated Hilbert spaces. J. Funct. Spaces Appl. 2014, Article ID 606815 (2014). https://doi.org/10.1155/2014/606815.

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10. Mukhtarov, O., Ol˘gar, H., Aydemir, K.: Resolvent operator and spectrum of new type boundary value problems. Filomat 29, 1671-1680 (2015). https://doi.org/10.2298/FIL1507671M. 11. Mukhtarov, O., Ol˘gar, H., Aydemir, K., Jabbarov, I.S.: Operator-pencil realization of one SturmLiouville problem with transmission conditions. Appl. Comput. Math. 17, 284-294 (2018).

A Monge-Kantorovich—Type Norm on a Vector Measures Space Ion Mierlus-Mazilu and Lucian Nita

1 Preliminaries Let X be a Banach space over R and X  its conjugate. Let (T, d) be a compact metric space. We denote by B the Borel subsets of T . If μ : B → X is a countable-additive measure and A ∈ B, we define the variation of μ on A, by the formula:  |μ|(A) = sup



  μ(Ai )(Ai )i is a finite partition of A with Borel subsets .

i

If |μ|(T ) < ∞ we say that μ has bounded variation. We denote by: cabv X = {μ : B → R||μ|(T ) < ∞}. One can prove that  : cabv(X ) → R+ is a norm called the variational norm and (cabv(X ), ) is a Banach space (see [3]). Now, we define the following function spaces: S(X ) = { f : T → X | f is a simple function}; u

T M(X ) = { f : T → X |∃( f n )n ⊂ S(X ) such that f n → f } (the space of totally measurable functions); C(X ) = { f : T → X | f is continuous}. For any A ∈ B, we denote by ϕ A the characteristic function of A.

I. Mierlus-Mazilu (B) · L. Nita Technical University of Civil Engineering, Bucharest, Romania e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_4

33

34

I. Mierlus-Mazilu and L. Nita

2 An Integral for Vector Function with Respect to Vector Measures (see [4])   m Let f = i=1 ϕ Ai xi ∈ S(X ), where xi ∈ X, Ai ∈ B. Let, X . We also, μ ∈ cabv m μ(A define the integral of f with respect to μ by the formula: f dμ = i )(x i ). i=1   f dμ ≤ μ ·  f ∞ , hence, the linear application f → Obviously, we have: f dμ is continuous and can be extended to the closure of S(X ) with respect to u ∞ , that is, to the space T M(X ): if ( f n )n ⊂ S(X ) such that f n → f ∈ T M(X ), we define f dμ = lim f n dμ and the limits do not depend on the sequence of n→∞ simple functions which tends to f . Example 1 We will give now an example of such sequence which will be called the canonical sequence (see [1]), for the case when f ∈ C(X ). Let us denote: X˜ = f (T ); f is continuous and T is compact, hence, X˜ is also compact. That means X˜ is precompact (totally bounded). Consequently, for any 

m∈  m m  m m m N, we will find the elements: x1 = f t1 , x2 = f t2 , . . . , x j (m) = = f t mj(m)     j (m)  de f such that X˜ ⊂ i=1 B xim , m1 . We deduce that tim ∈ Dim = f −1 B xim , m1 and j (m) m i=1 Di = T . We obtain the following partition of T : C1m = D1m , C2m = D2m \D1m , . . . , C mp = D mp \

p−1

Dim

i=1

(we consider only sets Cim which are not empty).  mthose  p m Let yi ∈ f Ci arbitrarily fixed. We define the function f m = i=1 ϕCim yim . If m we take t ∈ T , arbitrarily, then there exists i ∈ {1,. . . , p}  such  that t m∈ C1 i . Then, m m m m both f (t) and yi belong to f Ci . But, f Ci ⊂ f Ai ⊂ B xi , m , hence  f (t) − f m (t) =  f (t) − yim 
1 be a constant. Define a function b : E 2 → [0, ∞) by b(v, Ω) = (v + Ω)m or b(v, Ω) = (max {v, Ω})m . Then (E, b) is a b-m.l.s with parameter s = 2m−1 . Clearly, (E, b) is neither a b-metric, nor metriclike, nor partial b-metric space. Definition 2 ([23]) Let (E, b) be a b-m.l.s with parameter s, and let {vn } be any sequence in E and v ∈ E. Then, the following applies: (a) The sequence {vn } is said to be convergent to v if lim b (vn , v) = b (v, v), n→∞ (b) The sequence {vn } is said to be a Cauchy sequence in (E, b) if lim b (vn , vm ) n,m→∞

exists and is finite, (c) The pair (E, b) is called a complete b-m.l.s if, for every Cauchy sequence {vn } in E, there exists v ∈ E such that lim b (vn , vm ) = lim b (vn , v) = b (v, v)

n,m→∞

n→∞

Definition 3 ([23]) Let (E, b) be a b-m.l.s with parameter s, and a function P : E → E. We say that the function P is continuous if for each sequence {vn } ⊂ E the sequence Pvn → Pv whenever vn → v as n → +∞, that is if lim b (vn , v) = n→+∞

b (v, v) yields lim b (Pvn , Pv) = b (Pv, Pv). n→+∞

Remark 1 In a b-m.l.s with parameter s ≥ 1, if of the sequence {vn } is unique if it exists.

lim

n,m→+∞

b (vn , vm ) = 0 then the limit

Lemma 1 ([8]) Let (E, b) be a b-m.l.s with parameter s ≥ 1. Then, the following applies:

Further Fixed Point Results for Rational Suzuki F-Contractions …

41

(a) If b(v, Ω) = 0, then b(v, v) = b(Ω, Ω) = 0, (b) If {vn } is a sequence such that lim b (vn , vn+1 ) = 0, then we have n→∞

lim b(vn , vn ) = lim b(vn+1 , vn+1 ) = 0

n→∞

n→∞

(c) If v = Ω, then b(v, Ω) > 0. Lemma 2 ([24, 25]) Let (E, b) be a complete b-m.l.s with parameter s ≥ 1, let {vn } be a sequence such that b (vn , vn+1 ) ≤ λb (vn−1 , vn ) for some λ ∈ [0, 1) and each n ∈ N then, {vn } is a b-Cauchy sequence such that lim b (vn , vm ) = 0.

n,m→∞

Lemma 3 ([23, 26]) Let (E, b) be a b-metric-like space with parameter s ≥ 1, and suppose that {vn } is b-convergent to v with b (v, v) = 0. Then for each y ∈ E, we have s −1 b (v, y) ≤ lim inf b (vn , y) ≤ lim sup b (vn , y) ≤ sb (v, y) n→+∞

n→+∞

Definition 4 ([1]) Let (E, b) be a metric space and P : E → E be a mapping. Then is called an F-contraction if there exists a function F : (0, +∞) → R such that (F1 ) F is strictly increasing on (0, +∞), (F2 ) For each sequence {αn }of positive numbers, lim αn = 0 if and only if lim F (αn ) = −∞

n→∞

n→∞

(F3 ) there exists c ∈ (0, 1) such that lim+ αc F (α) = 0,

(F4 ) there exists τ > 0 such that

α→0

τ + F (b (Pv, PΩ)) ≤ F (b (v, Ω)) for all v, Ω ∈ E with Pv = PΩ.

3 Main Results Considering an interesting and useful generalizations of the Banach contaction achieved by Suzuki [11], in the following we give some fixed point theorems related to types of Suzuki and rational contractions. Theorem 1 Let (E, b) be a complete b-m.l.s with parameter s > 1 and let P be a continuous self map on E. If there is an increasing mapping F : (0, +∞) → R and constant τ > 0 such that

42

K. Zoto and I. Vardhami

1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s implies   τ + F s q b (Pu, Pv) ≤ F (αb (v, Pv) Ψ (b (u, Pu) , b (u, v)) + βb (u, v) + γb (v, Pu)) ,

(1) for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 + 2a3 s < 1,q ≥ 2 and Ψ : R+ × R+ → R+ is a continuous function such that Ψ (r, r ) ≤ 1 for r ∈ R+ . Then the function P has a unique fixed point in E. Proof Let be u n+1 = Pu n for n ∈ N ∪ {0}, the Picard sequence induced by function P with initial point u 0 ∈ E. Let us refer to the general case that for each n ∈ N, 0 < b (u n , Pu n ). Then, we have 1 b (u n , Pu n ) < b (u n , Pu n ) = b (u n , u n+1 ) 2s for all n ∈ N ∪ {0} . The condition (1) of the theorem yields     F s q b (u n+1 , u n+2 ) < τ + F s q b (Pu n , Pu n+1 ) ≤ F (a1 b (u n+1 , Pu n+1 ) Ψ (b (u n , Pu n ) , b (u n , u n+1 )) +a2 b (u n , u n+1 ) + a3 b (u n+1 , Pu n )) = F (a1 b (u n+1 , u n+2 ) Ψ (b (u n , u n+1 ) , b (u n , u n+1 )) +a2 b (u n , u n+1 ) + a3 b (u n+1 , u n+1 )) ≤ F (a1 b (u n+1 , u n+2 ) +a2 b (u n , u n+1 ) + 2a3 sb (u n+1 , u n+2 )) .

(2)

If suppose that b (u n , u n+1 ) < b (u n+1 , u n+2 ) , then from inequality (2) we get   F s q b (u n+1 , u n+2 ) < F (a1 + a2 + 2a3 s)b (u n+1 , u n+2 )) < F (b (u n+1 , u n+2 )) . And based in property of F, the inequality above implies b (u n+1 , u n+2 )
0 such that 1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s implies

    b (v, Pu) b (u, Pv) +γ , τ + F s q b (Pu, Pv) ≤ F αb (u, v) + β 2s 2s

(12)

for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 + 2a3 s < 1, q > 2. Then the function P has a unique fixed point in E. Proof It is the same as in Theorem 1. Now, we will propose some corollaries derived from the above theorems that generalize and complement some results for the class of Suzuki rational (s − q)-contractions. Theorem 3 Let (E, b) be a complete b-m.l.s with parameter s > 1 and let P be a continuous self mapping on E. If there is an increasing mapping F : (0, +∞) → R and constant τ > 0 such that

Further Fixed Point Results for Rational Suzuki F-Contractions …

45

1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s implies

    b (u, Pv) + b (v, Pu) , τ + F s q b (Pu, Pv) ≤ F a1 b (u, v) + a2 2s + b (u, v)

(13)

for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 (1 + 2s) < 1, q > 2. Then the function P has a unique fixed point in E. Proof The proof is clear based in Theorem 2, if consider the following inequality a2

b (u, Pv) + b (v, Pu) b (u, Pv) + b (v, Pu) a2 a2 ≤ a2 = b (u, Pv) + b (v, Pu) , 2s + b (u, v) 2s 2s 2s

and take the coefficient a3 = a2 . Corollary 1 Let (E, b) be a complete b-m.l.s with parameter s > 1 and let P be a continuous self mapping on E. If there is an increasing mapping F : (0, +∞) → R and constant τ > 0 such that 1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s which implies     b (u, Pu) b (v, Pv) τ + F s q b (Pu, Pv) ≤ F a1 + a2 b (u, v) + a3 b (v, Pu) , b (u, v) + b (u, Pu) for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 + 2a3 s < 1, q > 2. Then the function P has a unique fixed point in E. Proof Use the function Ψ (k, r ) = k/ (k + r ) for k, r ∈ (0, ∞) in the inequality (1). Corollary 2 Let (E, b) be a complete b- m.l.s with parameter s > 1 and let P be a continuous self mapping on E. If there is an increasing mapping F : (0, +∞) → R and constant τ > 0 such that 1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s implies √     b (v, Pv) b (u, Pu) b (u, v) τ + F s q b (Pu, Pv) ≤ F a1 + a2 b (u, v) + a3 b (v, Pu) , 1 + b (u, v)

for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 + 2a3 s < 1, q > 2. Then the function P has a unique fixed point in E.

46

K. Zoto and I. Vardhami

Proof Use the function Ψ (k, r ) = (1).

√

kt/ (1 + r ) for k, r ∈ (0, ∞) in the inequality

Corollary 3 Let (E, b) be a complete b-m.l.s with parameter s > 1 and let P be a continuous self mapping on E. If there is an increasing mapping F : (0, +∞) → R and constant τ > 0 such that 1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s which implies     b (u, Pu) b (v, Pv) + a2 b (u, v) + a3 b (v, Pu) , τ + F s q b (Pu, Pv) ≤ F a1 1 + b (u, v)

for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 + 2a3 s < 1, q > 2. Then the function P has a unique fixed point in E. Proof Use the function Ψ (k, r ) = k/ (1 + r ) for k, r ∈ (0, ∞) in the inequality (1). Corollary 4 Let (E, b) be a complete b- m.l.s with parameter s > 1 and let P be a continuous self mapping on E. If there is an increasing mapping F : (0, +∞) → R and constant τ > 0 such that 1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s implies   τ + F s q b (Pu, Pv)   b (v, Pv) [b (u, Pu) + b (u, v)] + a2 b (u, v) + a3 b (v, Pu) , ≤ F a1 b (u, Pu) + max {b (u, Pu) b (u, v)} for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 + 2a3 s < 1, q > 2. Then the function P has a unique fixed point in E. Proof Use the function Ψ (k, t) = (k + r ) / (k + max {k, r }) for k, r ∈ (0, ∞) in the inequality (1). Corollary 5 Let (E, b) be a complete b-m.l.s with parameter s > 1 and let P be a continuous self mapping on E. If there is an increasing mapping F : (0, +∞) → R and constant τ > 0 such that 1 b (u, Pu) < b (u, v) and b (Pu, Pv) > 0 2s

Further Fixed Point Results for Rational Suzuki F-Contractions …

47

implies  b (v, Pv) b (u, v) + a2 b (u, v) + a3 b (v, Pu) , max {b (u, Pu) , b (u, v)} (14) for all u, v ∈ E and u = v, where a1 , a2 , a3 ≥ 0 with a1 + a2 + 2a3 s < 1, q > 2. Then the function P has a unique fixed point in E.

 τ + F s b (Pu, Pv) ≤ F a1 

q



Proof Use the function Ψ (k, r ) = r/ max {k, r } for k, r ∈ (0, ∞) in the inequality (1).

4 Conclusion The provided results extend the fixed point results given for rational contractions including classical results of Jaggi, Dass, Gupta and (s − q)-contractions in the setting of b-metric and b-metric-like spaces. They also are a further continuation and fulfillment of the latest results from [7, 21, 22].

References 1. Wardowski, D. (2012) Fixed points of a new type of contractive mappings in complete metric space. Fixed Point Theory Appl., 2012, 94. [Google Scholar] [CrossRef]. 2. Wardowski, D. (2018). Solving existence problems via F-contractions. Proceedings of the American Mathematical Society, 146 (4), 1585–1598. https://doi.org/10.1090/proc/13808. 3. Jaggi D.S. (1977) Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics (8), 223-230. 4. Dass B.K., Gupta S, (1975) An extension of Banach contraction principle through rational expressions. Indian J. Pure Appl. Math., (6), 1455-1458. 5. Hammad, H.A., De la Sen M., (2020) Fixed-point results for a generalized almost (s, q)-Jaggi F- contraction-type on b-metric-like spaces. Mathematics, 8, 63. [Google Scholar] [CrossRef]. 6. De la Sen M., Nikoli´c N., Došenovi´c T., Pavlovi´c M., Radenovi´c S., (2019) Some results on (s − q)-graphic contraction mappings in b-metric-like spaces. Mathematics 2019, 7, 1190. [Google Scholar] [CrossRef]. 7. Mitrovi´c S., Parvaneh V., De La Sen M., Vujakovi´c J., Radenovi´c S., (2021) Some New Results for Jaggi-W-Contraction-Type Mappings on b-Metric-like Spaces. Mathematics 2021, 9, https://doi.org/10.3390/math9161921. 8. Zoto K., Rhoades B.E., Radenovi´c S., (2019) Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations. Math. Slovaca 69, No.1. 9. Zoto K., Radenovi´c S., Ansari A., (2018) On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations. Open Math., 16, 235– 249. 10. Fabiano, N., Dosenovi´c, Raki´c, T. D., Radenovi´c, S. Some new results on (s − q)-Dass-GuptaJaggi type contractive mappings in b-metric-like spaces. Filomat, 2020, (accepted 04/04/2020). 11. Suzuki, T. A new type of fixed point theorems in metric spaces. Nonlinear Anal. 71, 5313-5317 (2009) https://doi.org/10.1016/j.na.2009.04.017.

48

K. Zoto and I. Vardhami

12. Panwar, A., Anita. Metric-like space and Suzuki type common fixed point results with (W.C.C) condition. Asian-Eur. J. Math. 2020, 13, 2050066. [Google Scholar] [CrossRef]. 13. Alsulami, H.H., Karapinar, E., Piri, H. Fixed points of generalized F-Suzuki type contraction in complete b-metric spaces. Discret. Dyn. Nat. Soc. 2015, 2015, 969726. [Google Scholar]. 14. Cosentino, M., Vetro, P. Fixed point result for F-contractive mappings of Hardy-RogersType. Filomat 2014, 28, 715–722. [Google Scholar] [CrossRef]. 15. Kadelburg, Z., Radenovi´c, S. Notes on some recent papers concerning F-contractions in bmetric spaces. Constr. Math. Anal. 2018, 1, 108–112. [Google Scholar] [CrossRef]. 16. Parvaneh, V., Hussain, N., Kadelburg, Z. Generalized Wardowski type fixed point theorems via α-admissible FG-contractions in b-metric spaces. Acta Math. Sci. 2016, 36, 1445–1456. [Google Scholar] [CrossRef]. 17. Piri, H., Kumam, P. Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 2014, 210. [Google Scholar] [CrossRef]. 18. Shukla, S., Radenovi´c, S., Kadelburg, Z. Some fixed point theorems for ordered F-generalized contractions in O − f -orbitally complete partial metric spaces. Theory Appl. Math. Comput. Sci. 2014, 4, 87–98. [Google Scholar]. 19. Wardowski, D., Dung, N.V. Fixed points of F-weak contractions on complete metric spaces. Demonstr. Math. 2014, 47, 146–155. [Google Scholar] [CrossRef]. 20. Popescu, O., Stan, G. Two fixed point theorems concerning F-contraction in complete metric spaces. Symmetry 2020, 12, 58. [Google Scholar] [CrossRef]. 21. Huang, H., Mitrovi´c, Z.D., Zoto, K., Radenovi´c, S. On Convex F-Contraction in b-Metric Spaces. Axioms 2021, 10, 71. https://doi.org/10.3390/axioms10020071. 22. Huang, Huaping, Kastriot Zoto, Zoran D. Mitrovi´c, and Stojan Radenovi´c. 2022. ”Fixed Point Results for Generalized F-Contractions in b-Metric-like Spaces” Fractal and Fractional 6, no. 5: 272. https://doi.org/10.3390/fractalfract6050272. 23. Alghmandi, M.A., Hussain, N., Salimi, P. Fixed point and coupled fixed point theorems on bmetric-like spaces. J. Inequal. Appl. 2013, 2013, 402. [Google Scholar] [CrossRef]. 24. Aleksi´c, S., Mitrovi´c, Z.D., Radenovi´c, S. Picard sequences in b-metric spaces. Fixed Point Theory 2020, 21, 35–46. [Google Scholar] [CrossRef]. 25. Vujakovi´c, J., Aydi, H., Radenovi´c, S., Mukheimer, A. Some remarks and new reults in ordered partial b-metric spaces. Mathematics 2019, 7, 334. [Google Scholar] [CrossRef]. 26. Hussain, N., Roshan, J.R., Parvaneh, V., Kadelburg, Z. Fixed Points of Contractive Mappings in b-Metric-Like Spaces. Sci. World J. 2014, 2014, 471827. [Google Scholar] [CrossRef].

Mutual Generation of the Choice and Majority Functions Elmira Yu Kalimulina

1 Introduction The rapid growth of quantum computers and its application in the field of artificial intelligence has led k-valued (in particular ternary) computing to be relevant again [1]. Also, the research and development of algorithms based on k–valued logic are very relevant in many other fields such as, for example, telecommunications (developing of new protocols [2], choosing optimal network routing scheme [3], data aggregation schemes [4]), symbolic analysis of complex systems, software development and detecting design errors, machine learning, etc. The detailed review of k-valued logic applications was given in [5, 6]. Thus, the problem of a full description of all closed classes of k-valued logic functions is very crucial for progress in many fields of science and engineering. A fundamentally essential problem—the problem of full description of closed classes of three-valued logic functions [7]—must be solved to make the implementation of circuits with the desired functional diagram possible [8]. The famous result by Emil Post relates to the full description of all closed classes of Boolean functions (with respect to the superposition operation) [9]. Later it had been described in detail in [10]. This result let many problems of two-valued logic to be solved. Then the special case of the finite generation of all closed two-valued logic classes with respect to a superposition operation had been proved. But with the transition to a k-valued logic (k > 2) a continuum of closed classes with respect to superposition operation appeared. And in that case a complete description is impossible. There are not finitely generated classes in k-valued logic case (see the example of Yanov and Muchnik [11]). Therefore, the description of all finitely generated classes for k-valued logic is an open problem [12]. There are many results related to the description of family of classes of functions closed with respect to a special operation. The operation of binary superposition determined for the k-valued logic functions on the basis of their representation in the binary number system has been considered in [13]. Criterion of implicit completeness E. Y. Kalimulina (B) V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] URL: https://www.ipu.ru/staff/elmira © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_6

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in three-valued logic in terms of precomplete classes was considered in [14]. Several sufficient conditions of finite generation are known. The most famous of them are: the existence of a majority function in the class, of a choice function, and all unary functions (see [15]). This paper considers the problem of verifying the finite generation of classes containing some subclass of one variable functions. We also give a description of overlattices of classes in Pk containing some precomplete class of unary functions, that has been given earlier by M.A. Posypkin in [16]. The finitely generation of overlattices has been proved. It is also shown that any class consisting of monotone functions and containing all monotone functions of one variable is finitely generated. The finite precompleteness of some closed classes can not be checked under the sufficient assumptions given above. These classes do not contain a choice function, any majority function, all functions of one variable, and the set of any one-variable function that is precompleted on all one-place functions. Some examples of such closed classes have been given in this paper. The proof of finite generation of such classes is based on the constant modelling method proposed in [17]. Let us introduce some standard notation and definitions [18]. Let E k be the set {0, 1, . . . , k − 1}. For every natural number n the set E kn is n-th Cartesian power of a set E k , and the mapping f : E kn → E k is an n-place k-valued logic function. The set of all functions of k-valued logic is denoted by Pk . Let R be an arbitrary set of k-valued logic functions. A superposition of functions over a set R is defined by induction: (1) every function f from R is a superposition over R; (2) if g0 (x1 , . . . , xn ) is superposition over R and if gi (xi,1 , . . . , xi,m i ) is either a superposition over R for any i = 1, . . . , n, or xi,l (1  l  m i ), then a function g0 (g1 (x1,1 , . . . , x1,m 1 ), . . . , gn (xn,1 , . . . , xn,m n )) is superposition over R. The closure (with respect to superposition) of a set R is the set of all superpositions over R. The closure of a set R is denoted by [R]. Obviously, R ⊆ [R]. A set R of k-valued logic functions is called a (functionally) closed class if [R] = R. We call a set of functions Q generates a closed class R (or the class R is generated by a set of functions Q) if [Q] = R. If a closed class R is generated by a finite set of functions, then R is called finitely generated. If the set Q generates a closed class R, then we say the set Q is complete in the class R. A set Q is called a precomplete class in the closed class R if Q ⊆ R, [Q] = R and [Q ∪ { f }] = R holds for every function f that does not belong to Q but belongs to R. Let ein (x1 , . . . , xi , . . . , xn ) denote a k-valued logic function for any natural n and any i, 1  i  n. The values of ein coincide with the values of the variable xi . The functions ein are called selector functions. The function e11 (x) is also denoted by x.

2 Majority Functions and the Choice Function Definition 1 If μ(x, y . . . , y) = μ(y, x . . . , y) = · · · = μ(y, y . . . , x) = y, then a function μ(x1 , . . . , xn ) is called a majority function for any n  3.

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For example, d3 (x1 , x2 , x3 ) = x1 · x2 ∨ x2 · x3 ∨ x1 · x3 is a majority function, where x · y = min(x, y), x ∨ y = max(x, y). The set of all functions of no more than s variables obtained from the function f by identifying variables is denoted by As ( f ) for any function f (x1 , . . . , xn ) and any s, s  1. If s > n, then we set As ( f ) = { f }. The idea of proving the following theorem belongs to K. Baker and A. Pixley [19]. Theorem 1 (see also [20]) If μ(x1 , . . . , xm+1 ) is a majority function of m + 1 variables, where m  2, then we have f ∈ [{μ} ∪ Ak m ( f )] for any function f (x1 , . . . , xn ) ∈ Pk , where k  2. Corollary 1 Let F = [F] ⊆ Pk , k  2, μ(x1 , . . . , xm+1 ) ∈ F, μ be a majority function. Then F is finite generated. Let E ⊂ E k . Let us consider a special case of the majority function. Definition 2 A function g(x1 , . . . , xn ) ∈ Pk , k  3 is called a majority function on the set E if g : E kn → E and g(x, y . . . , y) = g(y, x . . . , y) = · · · = g(y, y . . . , x) = y holds for any n  3 and for all x, y ∈ E. Let us show that the property similar to one considered in Theorem 1 holds for a majority function on the set E. Let PkE for any k  2 denote the set of all functions from Pk taking values from the set E and all selector functions from Pk . Theorem 2 (see also [21]) Let the closed class F ⊆ Pk , k  3 contain a function g(x1 , . . . , xm ), that is a majority on the set E. Then the class F ∩ [PkE ] is finitely generated. Consider the another function called a choice function. Definition 3 (Choice function). If y = i, where i = 0, 1, . . . , k − 1, then the function ϕ(y, x0 , . . . , xk−1 ) = xi is called the choice function in Pk for any k  2. For example, x y ∨ x¯ z is the choice function in P2 . A sufficient condition for the finiteness of class can be obtained via the choice function. And the following theorem gives an answer. Theorem 3 Let ϕ(y, x0 , . . . , xk−1 ) be a choice function, and F is an arbitrary closed class in Pk , k  2 such, that ϕ, 0, . . . , k − 1 ∈ F. Then F is finitely generated. Proof This statement follows from the decomposition, that holds for any arbitrary function f (x1 , . . . , xn ) from the class F. It may be checked by substituting the constants 0, . . . , k − 1 instead of the first variable. Let f (x1 , . . . , xn ) be an arbitrary function from the class F, then f (x1 , . . . , xn ) = ϕ(x1 , f (0, x2 , . . . , xn ), . . . , f (k − 1, x2 , . . . , xn )). Applying a similar decomposition for f (0, x2 , . . . , xn ), …, f (k − 1, x2 , . . . , xn ) and further for all other subfunctions, we can obtain that F ⊆ [{ϕ(y, x0 , . . . , xk−1 ), 0, . . . , k − 1}]. The reverse inclusion is obvious.

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2.1 Mutual Generation of the Choice Function and Majority Functions Theorem 4 Let the choice function ϕ(y, x0 , . . . , xk−1 ) in Pk , k  2 belong to the closed class F. Then F contains some majority function. Proof Let ϕl (z 2 , z 3 ), where l = 0, . . . , k − 1 denote the function obtained from ϕ(y, x0 , . . . , xk−1 ) by the following identification of the variables: y = xl = z 2 , and for all j = l, j ∈ {0, 1, . . . , k − 1} x j = z 3 . Let μ(z 1 , z 2 , z 3 ) = ϕ(z 1 , ϕ0 (z 2 , z 3 ), . . . , ϕk−1 (z 2 , z 3 )). Since ϕ(x, x, . . . , x) = x and ϕ(y, x, . . . , x) = x, then μ(z 2 , z 1 , z 1 ) = ϕ(z 2 , ϕ(z 1 , z 1 , . . . , z 1 ), . . . , ϕ(z 1 , z 1 , . . . , z 1 )) = ϕ(z 2 , z 1 , . . . , z 1 ) = z 1 .

We emphasize that if z 1 = i for any i ∈ {0, 1, . . . , k − 1}, then ϕi (z 2 , z 1 ) = ϕi (z 2 , i) = i, ϕi (z 1 , z 2 ) = ϕ1 (i, z 2 ) = i. Then μ(z 1 , z 2 , z 1 ) = ϕ(z 1 , ϕ0 (z 2 , z 1 ), . . . , ϕk−1 (z 2 , z 1 )) = z 1 , and μ(z 1 , z 1 , z 2 ) = ϕ(z 1 , ϕ0 (z 1 , z 2 ), . . . , ϕk−1 (z 1 , z 2 )) = z 1 . Hence, μ(z 1 , z 2 , z 3 ) is a majority function. The converse statement is not true: the choice function cannot be generated by an arbitrary majority function. However, there are majority functions whose closure the choice function belongs to. Theorem 5 The majority function μ in Pk , k  2, generation a choice function ϕ(y, x0 , . . . , xk−1 ) exists. Proof Let us define the function μ(x1 , . . . , x2k+2 ) as a majority function on sets (x, y . . . , y), (y, x . . . , y),. . . , (y, y . . . , x), x, y ∈ E k . Let i, a ∈ {0, 1, . . . , k − 1}. Then suppose that function μ takes the value a on all sets such that x1 = x2 = i, x2i+3 = x2i+4 = a. It is obvious that a function ϕ(x1 , x3 , . . . , x2k+1 ) = μ(x1 , x1 , x3 , x3 , . . . , x2k+1 , x2k+1 ) is a desired choice function. We denote by M the set of all functions from Pk that are monotone with respect to the linear order (0 < · · · < k − 1). Let for i = 0, 1, . . . , k − 1

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 k − 1, x  i Ji (x) = 0, x < i. Let for i = 0, 1, . . . , k − 1

 ji (x) =

1, x  i 0, x < i.

Definition 4 For any k  3 a monotone function ϕ M ∈ Pk defined as ϕ M (y, x0 , x1 , . . . , xk−1 ) = Jk−1 (y)xk−1 ∨ · · · ∨ J1 (y)x1 ∨ x0 , is called a monotone choice function in Pk . Theorem 6 (property of a monotone choice function) M = [{ϕ M , 0, 1, . . . , k − 1}]. Proof Let f (x1 , . . . , xn ) be an arbitrary function from M. Then f (x1 , . . . , xn ) = Jk−1 (x1 ) f (k − 1, x2 , . . . , xn ) ∨ · · · ∨ J1 (x1 ) f (1, x2 , . . . , xn ) ∨ (0, x2 , . . . , xn ). This equality is verified directly by substituting the values of the variable x1 and using the definition of monotonic function f . Then we apply this expansion to all subfunctions. Hence, it follows that M ⊆ [{ϕ M , 0, 1, . . . , k − 1}]. The reverse inclusion is obvious. Consider a set consisting of functions f for which there exists a number i: 1  i  n such that f (x1 , . . . , xi−1 , k − 1, xi+1 , . . . , xn ) = k − 1 independently of the values of the other variables. This set of functions will be denoted by Fk−1 . It is easy to see that this is a closed class. Note that ϕ M ∈ Fk−1 . It is enough to consider a set in which the value of the variable x0 is equal to k − 1, and the values of the other variables are arbitrary. On any set with this property, the monotone choice function takes a value equal to k − 1. Theorem 7 Let μ to be arbitrary majority function, then μ ∈ / [ϕ M ]. Proof Since [ϕ M ] ⊆ Fk−1 , it is sufficient to show that a majority function doesn’t lie in Fk−1 . Let some majority function μ(x1 , . . . , xn ), n  3 lie in Fk−1 . Then, due to the property that all functions from Fk−1 have, there is a number i : 1  i  n such that if the variable xi = k − 1, then μ(x1 , . . . , xi−1 , k − 1, xi+1 , . . . , xn ) = k − 1 regardless of the values of the other numbers. Consider a set α˜ such that αi = k − 1, and α j = 0 for all j = i, j = 0, 1, . . . , n. Then, on the one hand μ(α) ˜ = 0, since μ is a majority function, and on the other hand μ(α) ˜ = k − 1, since μ is contained in Fk−1 . We’ve got a contradiction. A monotone choice function cannot be generated by an arbitrary monotone majority function. However, there are monotonic majority functions, the closure of which belongs to a monotone choice function.

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Theorem 8 A monotone majority function μ in Pk generating a monotone choice function ϕ M (y, x0 , . . . , xk−1 ) exists. Proof Set k = 3. Define the function μ(x1 , . . . , x8 ) as a majority on the following sets: (x, y . . . , y), (y, x . . . , y), . . . , (y, y . . . , x), x, y ∈ E 3 . Then set function μ = a for any a, b, c ∈ {0, 1, 2} on all sets such that x1 = x2 = 0, x3 = x4 = a; set function μ = max(a, b, c) on all sets such that x1 = x2 = 1, x3 = x4 = a, x5 = x6 = b; and set μ equal to max(a, b, c) on all sets where x1 = x2 = 2, x3 = x4 = a, x5 = x6 = b, x7 = x8 = c. On the other sets, we can redefine the function by monotony. It’s clear that ϕ(x1 , x3 , x5 , x7 ) = μ(x1 , x1 , x3 , x3 , x5 , x5 , x7 , x7 ) is the desired monotone choice function. The majority function of 2n + 2 variables is constructed similarly. By pairwise identification of variables (see k = 3) we obtain a function of n + 1 variable. That function is a desired monotonic choice function.

3 Description of Classes, that Include a Class of Unary Functions, or Pre-complete Class of Unary Functions Firstly, we need the following definitions and notation. We say that a function f (x1 , . . . , xi−1 , xi , xi+1 , . . . , xn ) from Pk essentially depends on the variable xi , if there are such values a1 , a2 , . . . , ai−1 , ai+1 , . . . , an ∈ E k of variables x1 , . . . , xi−1 , xi , xi+1 , . . . , xn such that h(x) = f (a1 , . . . , ai−1 , x, ai+1 , . . . , an ) doesn’t equal to the constant identically. In this case, the variable xi is called essential. A variable is called dummy if the function f (x1 , . . . , xn ) does not depend on it essentially. Let f, g ∈ Pk . We say that f = g if one of them can be obtained from the other by adding or removing dummy variables. Let x˜ to denote the set of numbers x1 , . . . , xn , n  1. Let F be a closed class in Pk , then F(n) is the set of all functions from F that depend on the variables x1 , . . . , xn . F (n) is the set of all functions F taking at most n values. C R(F) is the set of all precomplete classes in the closed class F ⊆ Pk . P Sk is the set of all unary functions taking exactly k values. Definition 5 A function f (x1 , . . . , xn ) that takes no more than two values is called quasilinear if for any number of the variable i, where (1  i  n), and any two elements α, β ∈ E k one of two following relations holds: either for any γ1 , . . . , γi−1 , γi+1 , . . . , γn ∈ E k f (γ1 , . . . , γi−1 , α, γi+1 , . . . , γn ) = f (γ1 , . . . , γi−1 , β, γi+1 , . . . , γn ),

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or for any γ1 , . . . , γi−1 , γi+1 , . . . , γn ∈ E k f (γ1 , . . . , γi−1 , α, γi+1 , . . . , γn ) = f (γ1 , . . . , γi−1 , β, γi+1 , . . . , γn ). The set of all quasilinear functions in Pk will be denoted by L Q k . The problem of enumerating of all closed classes of k-valued logic containing all functions of one variable was solved by G.A. Burle [15]. Theorem 9 (see [15]) Functionally closed classes of functions of k-valued logic that contain all functions of one variable are following classes: Pk (1), L Q k ∪ Pk (1), Pk(2) ∪ Pk (1), . . . , Pk(k−1) ∪ Pk (1), Pk and are only them. The k-valued classes Pk (1), L Q k ∪ Pk (1), Pk(2) ∪ Pk (1), . . . , Pk(k−1) ∪ Pk (1), Pk are called Burle’s classes. The proof of the Theorem 9 is based on the following property of Burle’s classes: Pk (1) is precomplete class in L Q k ∪ Pk (1), L Q k ∪ Pk (1) is precomplete class in Pk(2) ∪ Pk (1), Pk(l) ∪ Pk (1) is precomplete class in Pk(l+1) ∪ Pk (1) for 2  l  k − 2, Pk(k−1) ∪ Pk (1) is precomplete class in Pk . It’s easy to see from this property that all Burle’s classes are finitely generated. In other words, all closed classes of k -valued logic containing all functions of one variable are finitely generated. The problem of finding functionally closed classes containing a given class of functions of one variable was formulated by S.G. Gindikin (as it was point out by G.A. Burle). For classes containing precomplete classes of the set of all one-place functions, this problem was solved by M.A. Posypkin [16]. Theorem 10 ([16]) Let k  3 and C R(P Sk ) = {V1 , . . . , Vrk }. Then the set C R(Pk (1)) consists of classes Pk(k−2) ∪ P Sk , V1 ∪ Pk(k−1) , . . . , Vrk ∪ Pk(k−1) . Corollary 2 Let G ∈ C R(Pk (1)), k  3. Then 1 ∈ G. Proof The Theorem 10 describes the set C R(Pk (1)). Let C R(P Sk ) = {V1 , . . . , Vrk }. Then the set C R(Pk (1)) consists of classes Pk(k−2) ∪ P Sk , V1 ∪ Pk(k−1) , . . . , Vrk ∪ Pk(k−1) . (1) The class Pk(k−2) ∪ P Sk contains one, since 1 ∈ PK(k−2) . (2) Classes V1 ∪ PK(k−1) , . . . , Vrk ∪ Pk(k−1) contain one, since 1 ∈ Pk(k−1) . Remark In the case k = 3, the class Pk(k−2) ∪ P SK coincides with the class of all linear unary functions L 3 (1). An overlattice of a class G is the set of all classes F ⊆ Pk such that G ⊆ F. A complete description of the overlattices of precomplete in Pk (1) classes for k = 3 is described by the following theorem. Theorem 11 ([16]) 1. Let V ∈ C R(P S3 ). Then the closed class V ∪ P3(2) (1) has a finite overlattice consisting of the following classes: V ∪ P3(2) (1), P3 (1);

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V ∪ P3(2) (1) ∪ L Q 3 , P3 (1) ∪ L Q 3 ; V ∪ P3(2) (1) ∪ P3(2) , P3 (1) ∪ P3(2) ; P3 . 2. The class L 3 (1) has a finite overlattice consisting of the following closed classes: L 3 (1), L 3 , P3 (1), L Q 3 ∪ P3 (1), P3(2) ∪ P3 (1), P3 .

4 Conclusion In this paper, the problem of verifying the finite generation of classes containing some subclass of functions of one variable has been considered. We also give a description of the over lattices of classes in Pk containing some precomplete class of unary functions. The finite generation of overlattices has been proved. The completeness problem for this operator has a solution. It is possible to describe the sublattice of closed classes in the general case of closure of functions with respect to the classical superposition operator. In further papers relying on the above results, we plan to show that any class containing any of the precomplete classes of the set of unary functions in P3 is finitely generated. Funding This publication was prepared with the support of the Russian Foundation for Basic Research according to the research project No. 20-01-00575_A.

References 1. Bykovsky, Al. Yu., Heterogeneous Network Architecture For Integration Of Ai And Quantum Optics By Means Of Multiple-valued Logic, Quantum Rep. vol.2., pp. 126–165. (2020) https:// doi.org/10.3390/Quantum2010010. 2. Kalimulina, E.Y.: Application of Multi-Valued Logic Models in Traffic Aggregation Problems in Mobile Networks. IEEE 15th International Conference on Application of Information and Communication Technologies (AICT). pp. 1–6 (2021). https://doi.org/10.1109/AICT52784. 2021.9620244. 3. Kalimulina, E.Y.: Analysis of Unreliable Open Queueing Network with Dynamic Routing. Distributed Computer and Communication Networks. Communications in Computer and Information Science, vol 700. Springer, Cham. (2017). https://doi.org/10.1007/978-3-319-668369_30. 4. Esin, A.; Yavorskiy, R.; Zemtsov, N. Brief Announcement Monitoring Of Linear Distributed Computations. In Distributed Computing; Dolev, S., Ed.; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, Volume 4167. (2006). 5. Kalimulina, E.Y.: Lattice Structure of Some Closed Classes for Three-Valued Logic and Its Applications. MDPI Mathematics. 10, 94. (2022). https://doi.org/10.3390/math10010094. 6. Kalimulina, E.Y.: Lattice Structure of Some Closed Classes for Non-binary Logic and Its Applications. Mathematical Methods for Engineering Applications. ICMASE-2021. Springer Proceedings in Mathematics & Statistics, vol 384. Springer, Cham. (2022). https://doi.org/10. 1007/978-3-030-96401-6_2.

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7. S.V. Yablonskiy, G.P. Gavrilov And V.B. Kudryavtsev “Logical Algebra Functions And Post Classes”. Moscow.: “Nauka”, 1966. 8. B. Parhami and M. McKeown, Arithmetic with binary-encoded balanced ternary numbers, 2013 Asilomar Conference on Signals, Systems and Computers, 2013, pp. 1130–1133, https:// doi.org/10.1109/ACSSC.2013.6810470. 9. Post E.L., Introduction to a general theory of elementary propositions, American Journ. Mathem. V.43, 1921, p. 163–185. 10. Post E.L., Two-valued iterative systems of mathematical logic, Annals of Math. Studies. Princeton Univ. Press, 1942, V.5. 11. Yanov Yu. I., Yanov Yu. I.,On the existence of k-valued closed classes without a finite basis, Dokl. Akad. Nauk SSSR, 1959, 127, 1, p. 44–46. 12. Ugol’nkov, A. B., Some Problems in the Field of Mutivalued Logics, in Proc. X Int. Workshop “Discrete Mathematics and its Applications,” Moscow, MSU, February 1–6, pp. 18–34 (2010). 13. Podol’ko, D.K., A family of classes of functions closed with respect to a strengthened superposition operation. Moscow Univ. Math. Bull. 70, 79–83 (2015). https://doi.org/10.3103/ S0027132215020059. 14. Starostin, M.V., Implicit completeness criterion in three-valued logic in terms of maximal classes, (2021) arXiv:2103.16631pdf. 15. Burle G.A., Classes of k-valued logic containing all functions of one variable, Discrete analysis, iss. 10 (1967). 16. Posypkin M. A., On closed classes containing precomplete classes of the set of all one-place functions, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1997, no. 4, 58–59. 17. Ugolnikov, A.B. On closed Post classes, Proceedings of universities, Mathematics, iss. 7, pp. 79–88 (1988). 18. Enderton Herbert B., First-Order Logic (Ch.2), A Mathematical Introduction to Logic (Second Edition), Academic Press. (2001) https://doi.org/10.1016/B978-0-08-049646-7.50008-4. 19. Baker, K.A., Pixley, A.F., Polynomial interpolation and the Chinese remainder theorem for algebraic systems, Math. Zeitschrift, 143, 165–174 (1975). 20. Marchenkov S. S, Closed classes of Boolean functions. - Fizmatlit, 2000. - S. 130–130. 21. Demetrovics J., Hannak L., and L. Ronyai. On algebraic properties of monotone clones, Order 3, 219–225 (1986).

Dynamical Germ-Grain Models with Ellipsoidal Shape of the Grains for Some Particular Phase Transformations in Materials Science Paulo R. Rios, Harison S. Ventura, and Elena Villa

1 Introduction Formal kinetics is frequently employed to analyze a variety of heterogenous transformations in condensed phases. The early theory developed by Johnson-Mehl [13], Avrami [1–3], and Kolmogorov [15], known as KJMA theory, has constituted the foundation of formal kinetics theories applied today. Heterogeneous transformations may be defined as those transformations in which there is a moving boundary between the transformed and untransformed region. This formalism envisages that the heterogeneous transformations may be decomposed in two stages. The first stage, the nucleation, is that in which the transformed region originates. On the other hand, the second stage, the growth stage, is that in which the transformed region grows consuming the parent matrix. The fundamental way of modelling nucleation and growth transformations is by relying on the physics of the transformations mechanisms. However, this is not always feasible and an alternative treatment is provided by the so-called formal or global kinetics, that is a branch of solid-state transformations theory that deals with nucleation and growth in a phenomenological way, that is, it “prescribes” how nucleation and growth take place. Namely, a birth-and-growth (stochastic) process is a dynamic germ-grain model (e.g., see [7]) used to model situations in which nuclei (germs) are born in time and are located in space randomly, and each nucleus generates a grain evolving in time according with a given growth law. Since, in general, the nucleation is random in time and space, then the transformed region at any time t > 0 will be a random set in Rd , that is a measurable map from a probability space to the space of closed subsets in Rd [7]. Denote by P. R. Rios Escola de Engenharia Industrial Metalúrgica, Universidade Federal Fluminense, Volta Redonda, RJ, Brazil H. S. Ventura Companhia Siderúrgica Nacional, Gerência de Desenvolvimento de Produtos, Volta Redonda, RJ, Brazil E. Villa (B) Department of Mathematics, Università degli Studi di Milano, Milan, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_7

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T j the R+ -valued random variable representing the time of birth of the j-th nucleus, and by X j the Rd -valued random variable representing the spatial location of the nucleus born at time T j ; the sequence N = {(Ti , X i )} is called nucleation process. Let ΘTt j (X j ) be the grain obtained as the evolution up to time t ≥ T j of the nucleus born at time T j in X j ; then, the transformed region Θ t at time t is given by Θt =



ΘTt j (X j ),

t ∈ R+ .

(1)

(T j ,X j )∈N : T j ≤t

The family {Θ t }t is called birth-and-growth process; the materials scientists denote it microstructure of the sample. Since Θ t is a random set, it gives rise to a random measure ν d (Θ t ∩ · ) in Rd for all t > 0, having denoted by ν d the d-dimensional Lebesgue measure in Rd . In particular, it is of interest to consider the expected volume measure E[ν d (Θ t ∩ · )] and its density (i.e., its Radon-Nikodym derivative), called mean volume density of Θ t and denoted by VV , provided it exists:  E[ν d (Θ t ∩ A)] =

VV (t, x)dx

∀A ∈ B Rd ,

(2)

A

where BRd is the Borel σ -algebra of Rd . Whenever VV (t) is independent of x (e.g., under assumptions of homogeneous nucleation and growth), it is called volume fraction at time t. A related quantity is the mean extended volume density at time t, denoted by VE (t, ·) and defined as the density of the mean extended volume measure at time t on Rd :   E[ ν d (ΘTt j (X j ) ∩ A)] = VE (t, x)dx, ∀A ∈ BRd . j:T j ≤t

A

The following further quantities are defined in a similar way: SV (t, ·), the mean surface density at time t, defined to be the density of the mean surface measure at time t: E[Hd−1 (∂Θ ∩ · )]; and S E (t, ·), the mean extended surface density at time  t, defined to be the density of the mean extended surface measure at time t: E[ j:T j ≤t Hd−1 (∂ΘTt j (X j ) ∩ · )]. Here Hd−1 is the n − 1-dimensional Hausdorff measure, while ∂ A is the topological boundary of a set A ⊂ Rd . In other words, the mean extended volume and surface measures represent the mean of the sum of the volume measures and of the surface measures of the grains which are born and grown until time t, supposed free to grow, ignoring overlapping. (See also [21].) Of course, different kinds of nucleation and growth models gives rise to different kinds of processes {Θ t }t . Thus, let {Θ t }t be a birth-and-growth process modelling a particular phase transformation of interest. A problem is to find out explicit formulas for VV and the related quantities above mentioned, associated to Θ t .

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A considerable number of engineering materials are polycrystal: an aggregate of many crystals with size usually below 100μm. Those small crystals are called the grains of the polycrystal, and are often equiaxed; however, because of processing, the grain shape may become anisotropic. For instance, during recrystallization or phase transformations, the new grains may grow in the form of ellipsoids. Indeed, even if the assumption of spherical growth is standard and allows to get more explicit formulas, new regions do not always grow as spheres (e.g., see [11, 23, 24]. Bradley et al. demonstrated in a series of papers [4–6] that a grain boundary nucleated ferrite allotriomorph is best described by an oblate ellipsoid. The reason why these regions grow with ellipsoidal shape is that the growth on the grain boundary plane is faster than the thickening into the austenite. Actually, regarding ellipsoidal grains, one has two related issues. One is the growth of ellipsoidal grains with nuclei located randomly in space, allowed to overlap each other or not. We also mention that when a polycrystal is deformed by cold rolling, the resulting pattern may be modelled as a random union of ellipsoids with a fixed orientation (the major axis is in the direction of the rolling). The other issue is the growth of ellipsoidal grains with nuclei located on random parallel planes. This occurs for example in recrystallization processes after heavy rolling: to a first approximation, one may consider that these anisotropic grains may be approximated by random parallel planes; subsequently a new nucleation takes place on such planes. Finally, we also mention that it is reasonable and it has also been found in experimental works, that the probability of a new nucleus forming very close to another one is likely to be low. Therefore, one might suppose that in some cases there is effectively an “exclusion radius” around each nucleus so that, within that radius, nucleation cannot occur. From a mathematical point of view, such situation may be modelled by assuming hard-core nucleation processes. The three different birth-and-growth processes (ellipsoidal growth with nuclei randomly located in space, on parallel planes, and having an exclusion zone around each nucleus) modelling the real situations above mentioned have been recently faced by the authors in a series of papers (see in particular [17, 25, 27]). Here we propose a survey of them by collecting and specializing the most relevant results concerning the case of ellipsoidal grains.

2 Preliminaries and Notation In this section we fix basic notation and we summarize some preliminary results useful for the sequel. Throughout the paper we work in the Euclidean space Rd , d ≥ 2; Hn is the ndimensional Hausdorff measure, BRd is the Borel σ -algebra of Rd , and ν d denotes the usual d-dimensional Lebesgue volume measure (which coincides with Hd in Rd ). Given a subset A of Rd , Ac denotes the complementary set of A, whereas ∂ A := cl A \ int A its topological boundary. For r ≥ 0 and x ∈ Rd , Br (x) is the closed

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ball with centre x and radius r ; finally, for every integer n > 0, bn denotes the volume of the unit ball in Rn , while S d−1 the unit sphere in Rd .

2.1 Point Process and Germ-Grain Model A nucleation process is said to be site-saturated if the nucleation rate is so fast at the beginning of the transformation that the available nucleation sites are saturated early in the transformation. Therefore, site-saturation essentially signifies that all nuclei are already present at the time origin and no nuclei form later in the transformation. Otherwise we say that the nucleation process is time-dependent. Site-saturated and time-dependent nucleation processes can be modelled by means of (marked) point processes. We give here some basic concepts and definitions useful in what follows, without entering into the details of the mathematical theory of point processes (see [8, 9] for an exhaustive treatment).  is a locally finite collection {X i }i∈N of random A point process in Rd , say Φ,  is a random counting measure, that is a measurable map points; more formally Φ from a probability space (Ω, F, P) into the space of locally finite counting measures    We := E[Φ(A)] on BRd is called intensity measure of Φ. on Rd . The measure Λ(A)  also remind that Φ is called Poisson process whenever has independent increments  n −Λ(A)   e for all A ∈ BRd , n ∈ {0, 1, 2, . . .}. and it is such that P(Φ(A) = n) = Λ(A) n! d A Marked point process in R with marks in a complete and separable metric space K, is a collection Φ = {(X i , K i )}i∈N of random points X i in Rd , each one  : associated with a mark K i ∈ K, with the property that the unmarked process {Φ(B) B ∈ BRd } := {Φ(B × K) : B ∈ BRd } is a point process in Rd . The intensity measure of Φ, say Λ, is a σ -finite measure on BRd ×K defined as Λ(B × L) := E[Φ(B × L)]. It is worth recalling that a marked Poisson point process can be seen as a Poisson point process on the product space Rd × K; the assumption of Poissonian nucleation enables to get more explicit results when dealing with birth-and-growth processes. Notice that a point process N = {X i } in Rd may be taken as model for a sitesaturated nucleation process, whereas a marked point process N = {(Ti , X i )} in R+ with marks in Rd may be taken as model for a time-dependent nucleation process. In this latter case, Λ([0, t] × B) gives the mean number of nuclei which are born in B ∈ BRd during the time interval [0, t]. A random closed set Θ in Rd is a measurable map Θ : (Ω, F, P) −→ (F, σF ), where F denotes the class of the closed subsets in Rd , and σF is the σ -algebra generated by the so called Fell topology, or hit-or-miss topology (e.g., see [7]). Any random closed set in Rd given by a random union of compact random sets (particles) can be represented as germ-grain model. This latter is described by a suitable marked point process Φ in Rd with marks in Kd , the space of compact subsets of Rd . Namely, a germ-grain model is a random set of the type

Dynamical Germ-Grain Models with Ellipsoidal Shape of the Grains for Some …

Θ=



63

X j + Z j,

(3)

(X j ,Z j )∈Φ

where Φ = {(X j , Z j )} j∈N is a marked point process in Rd with marks in K := Kd so that Z j is a compact random set containing the origin, or having the origin as cirucumcentre. It is then clear why the family {Θ t }t∈R+ with Θ t defined as in (1) where the role of X j + Z j and of Φ in (3) are played by ΘTt j (X j ) and by N , respectively, is also called dynamical germ-grain model. Remark 1 We point out, that, in dependence on the growth model, the nucleation point process N has to be chosen in order to well describes the process {Θ t }t accordingly. For instance a point process N = {(Ti , (X i , G i )} in R+ with marks in Rd × R+ may be taken to model the case of time dependent nucleation and spherical growth model with random constant velocity of each nucleus (e.g., see [22] for further insights):  BG i (t−Ti )(X i ) ∀t ≥ 0. Θt = (Ti ,X i ,G i )∈N : Ti ≤t

2.2 The Ellipsoidal Growth Model Dealing with birth-and-growth processes, beside the definition of a suitable nucleation process, one has to define also a growth model. Here we shall consider the following ellipsoidal growth model in R3 : Let us fix a = (a1 , a2 , a3 ) ∈ R3+ ; we denote by   x2 x2 x2 E0 (a) := (x1 , x2 , x3 ) ∈ R3 : 12 + 22 + 32 ≤ 1 a1 a2 a3 the ellipsoid centered in the origin with semi axes a1 , a2 and a3 aligned with the axes x, ˆ yˆ and zˆ , respectively. For any φ ∈ S O(3), where S O(3) is the rotation group in R3 , we denote by E0 (a; φ) the ellipsoid E0 (a) rotated according to φ. We assume that the grains grow with ellipsoidal shape with constant rate G > 0, that is at any time t the grain born at point x at time s and grown up to time t, is given by an ellipsoid centred at x with semiaxes of length a1 G(t − s), a2 G(t − s) and a3 G(t − s) respectively, randomly orientated into the space. By random orientation we mean that the direction φ of the a1 -semiaxis of the ellipsoid is random, accordingly with a given probability distribution Q on S O(3), independent on the spatial location and on the birth time of the corresponding nucleus. Hence, denoted by Θst (x, φ) the grain born in x at time s and grown with orientation φ until time t, we have Θst (x, φ) = x + E0 (G(t − s)a; φ) = x + G(t − s)E0 (a; φ)

∀s ≤ t.

(4)

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Note that for any time t the ratio between the axes of the grain remains constant (that is during the transformation only the size of the grain is varying, not the shape). This models for instance the transformed phase deformed after rolling. Note also that the case of spherical growth with constant velocity follows as particular case by putting a1 = a2 = a3 = 1; of course φ does not make any role, E0 = E0 (φ) = B1 (0) ∀φ ∈ S2 , and Θst (x) = BG(t−s) (x).

2.3 The Causal Cone Notion It is well known and it easily follows by a direct application of Fubini theorem in (2) that VV (t, x) = P(x ∈ Θ t ). The so-called causal cone of a point x at time t, denoted here by C(t, x), plays a fundamental role in evaluating VV (t, x). It is defined as the region (i.e., the subset of the space where the nucleation process N takes values) in which at least one nucleation event has to take place in order to cover the point x at time t. Namely, the following equivalence between events holds: {x ∈ Θ t } ⇐⇒ {N (C(t, x)) > 0}. As a consequence, if N is a Poisson process with intensity measure Λ, then VV (t, x) = 1 − P(N (C(t, x)) = 0) = 1 − e−Λ(C(t,x)) .

(5)

We already pointed out that different birth-and-growth processes are driven by different nucleation processes. Therefore if N takes value in a space E, then the associated causal cone will be a subspace of E; for instance, with reference to the example given in Remark 1, E = R+ × Rd × R+ and C(t, x) := {(s, y, g) ∈ E : s ∈ [0, t], x ∈ Bg(t−s) (y)}. In the simpler case in which the volocity is constant equal to G for each grain, then E = R+ × Rd , and C(t, x) := {(s, y) ∈ E : s ∈ [0, t], x ∈ BG(t−s) (y)}. Note that x ∈ B R (y) if and only if y ∈ B R (x), and that in the subcase of site-saturation E = Rd and C(t, x) = BGt (x).

3 Three Different Models with Ellipsoidal Grains Throughout the paper we shall consider birth-and-growth processes in R3 for applicative reasons, but all our argumentation applies to any dimension d ≥ 2.

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3.1 Ellipsoidal Growth with Space Nucleation Let us consider the birth-and-growth process in R3 driven by a time dependent Poissonian nucleation process in space, and growth model defined as in Sect. 2.2. Hence, let the nucleation process N be a Poisson point process in E = R+ × R3 × S O(3) with intensity measure Λ(d(s, x, φ)) = g(s) f (x)dsdx Q(dφ),

(6)

where g and f are non-negative locally integrable functions on R+ and R3 , respectively, while Q is a probability measure on S O(3). The intensity measure of the type (6) models the fact that the location and the orientation of each grain are independent each other and independent of the corresponding time birth. We remind that Λ([T1 , T2 ] × A × B) is the mean number of nuclei which are born in A during the time interval [T1 , T2 ], and whose associated ellipsoids have orientation in B ⊆ S O(3). The subcase of constant rate I is given by taking g(t) ≡ I > 0. Analogously, the subcase of homogenous spatial location of the nuclei is modelled by choosing f (x) ≡ c > 0. A fixed orientation φ¯ ∈ S O(3) is described by choosing Q(dφ) = δφ¯ (φ)dφ, where δφ¯ (φ) is the classical Dirac-delta ¯ function in φ. Then the transformed region Θ t at time t is the random closed set 

Θ t :=

Θstn (xn , φn ),

(7)

(sn ,xn ,φn )∈N

with Θstn (xn , φn ) as in (4) for any sn ≤ t, and the empty set for any sn > t; therefore C(t, x) := {(s, y, φ) ∈ [0, t] × R3 × S O(3) : x ∈ Θst (y, φ)} = {(s, y, φ) ∈ [0, t] × R3 × S O(3) : y ∈ Θst (x, φ)},

(8)

where the latter equation follows by the central symmetry of the grains. It is easy to show (e.g., see [27]) that VE (t, x) = Λ(C(t, x)),

(9)

and so by (5) and (8):

t

VV (t, x) = 1 − exp −





g(s) 0

 f (y)dy Q(dφ) ds .

S O(3) Θst (x,φ)

The computation of the above integral might be difficult whenever f is not constant. In addition to the situations in which nuclei are located uniformly within the specimen

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(case f constant), it is also of interest to study situations in which nuclei vary along a preferential direction. In [27, Appendix] a generalization of the so-called mean value property for harmonic function has been proved; in the special case of ellipsoids in R3 this implies that  4 h(y)dy = h(x) πa1 a2 a3 3 x+E0 (a)

 for any h ∈ C 2 (R3 , R) such that i ai2 ∂i2 h = 0. Therefore, if we assume f varying along a preferential direction, that is of the type f (x) =

3 

pi xi + q,

(10)

i=1

with p = ( p1 , p2 , p3 ) ∈ R3 , and q ∈ R such that f (x) ≥ 0 for any x in the considered observation window, then the associated volume density VV (t, x) simplifies as   t 4 3 VV (t, x) = 1 − exp − f (x) π G a1 a2 a3 g(s)(t − s)3 ds . (11) 3 0

Figure 1 depicts a particular case of Eq. (11) in which f (x) = 1 and the nucleation is site-saturated. (Site-saturation is modelled by putting g(s) = δ0 (s), the Dirac delta

t function at 0, so that 0 g(s)(t − s)3 ds = t 3 in (11).) Fig. 1 shows the mean volume density as a function of dimensionless time for the growth of regions when those regions are: a ball and oblate ellipsoids of different relations of longer to shorter axis. Before passing to consider the mean surface densities SV and S E , let us recall the notion of support function of a convex body. Definition 1 Let C be a convex body in Rd (that is a compact and convex set containing 0 in its interior). The support function h C of C is the function so defined: h C (v) := sup x · v, v ∈ Rd . x∈C

We point out that, by denoting  a M := max{a1 , a2 , a3 } and  am := min{a1 , a2 , a3 }, the support function of E0 (a; φ) is such that aM  am ≤ h E0 (a;φ) (y) ≤ 

∀y ∈ Rd .

(12)

In [27] the mean surface density has been studied as well; we may summarize the main results in the following Proposition 1 Let Θ t be defined as in (7), with N having intensity measure Λ as in (6). Then (13) SV (t, x) = (1 − VV (t, x))S E (t, x),

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Fig. 1 Comparison of the mean volume density against time (in dimensionless units) between ellipsoidal growth with four different aspect ratios. Spherical growth means a ball growing with: a1 = 1, a2 = 1, a3 = 1. The other ellipsoids are oblate ellipsoids in which the larger axes and smaller axes are: a1 = 2, a2 = 2, a3 = 1, a1 = 4, a2 = 4, a3 = 1, and a1 = 8, a2 = 8, a3 = 1. In all cases, the nucleation took place at points located in space according to a homogeneous Poisson point process

with



t S E (t, x) =

 f (y)H2 (dy)Q(dφ)ds.

g(s) 0

S O(3)

(14)

∂Θst (x,φ))

Moreover, ∂ VV (t, x) = ∂t



t

(1 − VV (t, x))G



g(s) 0

f (y)h E0 (a;φ) (νΘst (x,φ) (y))H2 (dy)Q(dφ)ds,

S O(3) ∂Θst (x,φ)

(15) where h E0 (a;φ) is the support function of the convex body E0 (a; φ), whereas νΘst (x,φ) (y) is the outer normal to Θst (x, φ) at y. Remark 2 In the spherical growth case, that is E0 (a; φ) = B1 (0), it is well known (e.g., see [21]) that : 1 ∂ VV (t, x). SV (t, x) = (16) G ∂t This is in accordance with the above proposition by taking into account that h B1 (0) (y) = 1 for any y ∈ Rd , and that Q is a probability measure in S O(3).

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In the general ellipsoidal growth model, Eq. (16) is not true any more. Actually, by (12), (14) and (15) we have am S E (t, x) ≤ (1 − VV (t, x))G

∂ VV (t, x) ≤ (1 − VV (t, x))G a M S E (t, x), ∂t

and so by (13) 1 ∂ 1 ∂ VV (t, x) ≤ SV (t, x) ≤ VV (t, x). G a M ∂t G am ∂t As a further generalization, one might consider the case of random velocity of the grains, as well. As a matter of fact, in [12] the authors reviewed their experimental measures of growth velocities of individual grains obtained by neutron and 3- dimensional synchrotron X-ray methods, and they concluded that there is compelling evidence to support that “every single grain has its own kinetics different from the other grains”. In accordance to Remark 1, to model the fact that each grain has its own random growth rate, it is sufficient to add a further mark to the nucleation point process N . Namely, let N be now a Poisson point process in E = R+ × R3 × R+ × S O(3), so that any point (si , xi , ξi , φi ) ∈ N represents the nucleus which is born at time si at location xi from which an ellipsoid with orientation φi and velocity ξi develops; that is the associated grain is given by Θsti (xi , ξi , φi ) = xi + ξi (t − s)E0 (a, φi ). Of course the velocity ξi might be dependent on the position, and/or on the orientation φi , and/or on the birth time si ; this would lead to formulas difficult to explicitate. Hence, let us assume independent growth velocity; this means that the random variable {ξi }i have identical probability distribution, say Q 1 on R+ . In particular let us assume that the intensity measure Λ of N is of the type Λ(d(s, x, φ)) = g(s) f (x)dsdx Q 1 (dξ )Q 2 (dφ), with f as in (10), and Q 1 and Q 2 probability measures on R+ and S O(3), respec3  := tively. Moreover let G R+ ξ Q 1 (dξ ) < ∞. By proceeding along the same lines that led to (11), one easily get  VV (x, t) = 1 − exp

4  − f (x) π Ga 1 a2 a3 3

t

 g(s)(t − s) ds . 3

0

Now (11) follows here as particular case. Further generalization may be obtained by arguing similarly to what was done in [26], where the case of random spherical growth has been investigated.

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3.2 Ellipsoidal Growth with Nucleation on Parallel Planes As mentioned in the Introduction, to a first approximation in modelling recrystallization after heavy rolling, one may consider nucleation of ellipsoids on random parallel planes. More precisely, let us consider nucleation on random parallel planes with outer normal vector w = (0, 0, 1) ∈ S 2 , and ellipsoidal growth, as in Sect. 2.2, but with a fixed orientation such that the major axis is parallel to the plane (in order to model major elongation in the direction of the rolling). Let us denote by B(u) := {x ∈ R3 : x3 = u} the plane with outer normal vector w = (0, 0, 1) and distance |u| from the origin, and by δ B(u) (y) the delta function associated to B(u) (which can be seen as a generalization to the well-known deltafunction δx0 associated to a point x0 ∈ Rd ), so that, formally,  δ B(u) (y)dy := H2 (B(u) ∩ A)

∀A ∈ BR3 .

A

We also denote by Nu = {(si , xi )}i the Poisson point process on B(u) with intensity measure Λu (d(t, x)) = g(t) f (x)δ B(u) (x)dtdx with f as in (10). Note that it is zero the probability of having nucleation in B(u)c . Then, the transformed region Θ t,u at time t associated to Nu is given by 

Θ t,u :=

Θsti (xi ),

(17)

(si ,xi )∈Nu

with Θsti (xi ) = xi + G(t − si )E0 (a). To model the nucleation on random parallel planes, let us introduce the point process Ξ = {D j } j on the positive x3 -semiaxis, representing the random distances from the origin of the planes B1 = B(D1 ), B2 = B(D2 ), . . ., respectively. We shall consider the transformed region, say Θ Kt , due to the nucleation on the random planes contained in R2 × [0, K ]. Thus, denoted by Ξ|K := Ξ ∩ [0, K ] the restriction of Ξ to [0, K ], the associated transformed region at time t will be Θ Kt =



Θ t,Di ,

∀t > 0,

(18)

Di ∈Ξ|K

where, for any realization Di = u i , Θ t,u i is defined as in (17). Note that whenever Ξ is a Poisson point process with intensity h = h(u), then

K the mean number of planes in [0, K ] is given by 0 h(u)du. By following the same approach used in [17], the following explicit expression for VV can be proved:

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Theorem 1 ([27, Theorem 10]) Let Ξ = {D j } j be a Poisson point process with intensity h = h(u). Then, for any t > 0, the mean volume density VV (t, x) in x at time t of the transformed region Θ Kt at time t, defined as in (18) is equal to  VV (t.x) = 1 − exp

K −

 VVu (t, x)h(u)du ,

0

where  VVu (t, x) t−

|u−x3 |

 Ga3

− f ((x1 , x2 , u))πa1 a2

  (u − x3 )2 ds1[0,Gta3 ] (|u − x3 |) . g(s) G (t − s) − a32 

2

0

= 1 − exp

2

We point out that the assumption (10) on f is crucial in order to get the above explicit tractable expression for VV . Such formula is further simplified in the particular cases of constant nucleation rate g(s) ≡ I , and of site-saturation (see [27]). Computer simulation results compared with the corresponding analytical formulas are provided in [19].

3.3 Nucleation with Exclusion Zone Around Each Nucleus As mentioned in the Introduction, it has been found in experimental work that the probability of a new nucleus forming very close to another one is likely to be low. For instance, Sudbrack et al. experimentally proved in [20] the existence of exclusion zones in the solid-state: although uniform, the nucleation process was not in agreement with a Poisson point process. In their work, they experimentally determined the so-called pair correlation function of the process (e.g., see [7]) by means of planar sections, and inferred that, actually, it was consistent with each nuclei having an exclusion zone around it in their Ni-Cr-Al superalloy. Hence, as a first approximation, one may suppose that in some cases there is effectively an “exclusion zone” around each nucleus where nucleation cannot occur. In order to model a such situation, one might assume a nucleation process of the hard-core type. A hard-core point process is characterized by the fact that its points have a prescribed minimum distance each other. The Strauss hard-core process and the Matérn point process of type I are the most popular point processes modelling the hard-core property between points, and both of them are defined in terms of an underlying Poisson point process (e.g., see [7] for a classical reference). Roughly speaking, the Matérn I hard-core point process is obtained by deleting every point in the Poisson point process with its nearest neighbor closer than a given hard-core distance. In the classical definition, the underlying

Dynamical Germ-Grain Models with Ellipsoidal Shape of the Grains for Some …

71

Poisson point process is assumed to be homogeneous, but further generalizations to the inhomogeneous case and to different thinning rules are also available in the literature. The Strauss hard-core point process is a particular Gibbs point process, that is a point process whose probability distribution has density with respect to a unit rate Poisson process (e.g., see [10]); the form of such density models the hard-core property. In the recent paper [25], the authors investigated the exclusion zone in solidstate by means of 2D simulation studies employing four different hard-core point processes: Matérn I, Matérn II, Strauss hard-core and Sequential. With reference to the above mentioned work by Sudbrack et al. [20], Fig. 2 below shows the comparison between the pair correlation function of the experimental work in [20] and that one of a simulated Strauss hard-core process. We recall here in simple words the notion of Strauss and Matérn I hard-core point process with minimal interpoint distance R in Rd . We refer to classical literature for more general definitions. Let Z be a Poisson point process in Rd with finite intensity measure Λ(dx) = f (x)dx and probability law distribution PZ on the space (S, S) of locally finite sequences of points in Rd . A finite point process Φ in Rd is said to be a point process with density p with respect to Z if its distribution PX on (S, S) is absolutely continuous with respect to PZ with density p. Without loss of generality, p may be written as p(x) = αq(x), x ∈ S, (19) where q : S → R+ is an integrable function, said interaction function, and α > 0 is a normalizing constant. We remind that in general the normalizing constant α is not explicitly computable. Definition 2 A point process Φ Str with density p with respect to Z as in (19), is called Strauss hard-core process if the interaction function q is of the type

Fig. 2 Comparison of the pair correlation function of the experimental work of Sudbrack et al. [20] with that one of a simulated Strauss hard-core process

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q(x) = 1{xi −x j >R, ∀xi ,x j ∈x} (x),

x ∈ S.

Definition 3 A point process Φ Mat is called Matérn I hard-core point process with underlying Poisson point process Z if Φ Mat := {x ∈ Z : Z \ {x} ∩ B R (x) = ∅}. Remark 3 Note that both the processes Φ Str and Φ Mat defined above may be taken to model centres of non-overlapping balls with radius R/2. It is intuitive that the above definitions may be generalized in order to model random patterns of non-overlapping grains, and so in particular non-overlapping ellipsoids. About generalizations in this direction, we refer to [14, 16] and to [18] for generalizations of the Matérn hard-core and of the Strauss hard-core processes, respectively. Figure 2 suggests the need to investigate more in hard-core and soft-hard-core nucleation processes in formal kinetics theory. A first step in this direction is to consider and compare transformed regions driven by nucleation processes modelled by Matérn I and Strauss hard-core processes. For sake of simplicity, let us consider a site-saturated nucleation process with spherical exclusion zone with radius R, and ellipsoidal growth model as defined in Sect. 2.2. The causal cone C(t, x) of a point x at time t is then given by C(t, x) = {(y, φ) ∈ R3 × S O(3) : x ∈ Θ0t (y, φ)} = {(y, φ) ∈ R3 × S O(3) : y ∈ x + GtE0 (a, φ)} Let N = {(xi , φi )} be a site-saturated nucleation Poisson point process in R3 with marks in S O(3), modelling centres and orientations of the ellipsoidal grains, with intensity measure Λ(d(x, φ)) = f (x)dx Q(dφ). We denote by N Str and by N Mat the marked Strauss and the Matérn hard-core processes with underlying Poisson point process N , respectively. That is the unmarked point process of the locations, Mat , respectively, are the Strauss and the Matérn hard core process Str and N say N with minimal interpoint distance R driven by the unmarked Poisson point process  = {xi } defined as above. Then the intensity measure of N Mat will be of the type N Λ Mat (d(x, φ)) = f Mat (x)dx Q(dφ), whereas the intensity measure of N Str will be of the type Λ Str (d(x, φ)) = f Str (x)dx Q(dφ). It can be shown [18] that • f Mat (x) = f (x)e−Λ(B R (x)) . 4 3 (As a consequence, if f is of the type (10), then f Mat (x) = f (x)e− 3 π R f (x) .) • f Str (x) is not explicitly computable because it turns out to be expressed in terms of the untractable normalizing constant α which appears in Eq. (19). Nevertheless, by specializing a more general result in [18], it holds

Dynamical Germ-Grain Models with Ellipsoidal Shape of the Grains for Some …

f (x)e−Λ(B R (x)) ≤ f Str (x) ≤

1+

f (x) e−Λ(B R (y)∩B R (x)c )

B R (x)

f (y)dy

73

.

Hence, the following relation holds between the intensities of the unmarked involved point processes: ∀x ∈ R3 . f Mat (x) ≤ f Str (x) ≤ f (x) Let us denote by VE,Mat (t, x), VE,Str (t, x) and by VE (t, x) the mean extended volume density at x at time t of the transformed region associated to the nucleation process N Mat , N Str and N , respectively. Hence, by remembering (9), we conclude 



VE,Mat (t, x) = Λ Mat (C(t, x)) =

f Mat (y)dy Q(dφ) S O(3) x+GtE0 (a,φ)





≤ VE,Str (t, x) = Λ Str (C(t, x)) =

f Str (y)dy Q(dφ) S O(3) x+GtE0 (a,φ)





≤ VE (t, x) = Λ(C(t, x)) =

f (y)dy Q(dφ). S O(3) x+GtE0 (a,φ)

We recall that the void probabilities of a point process Φ in Rd are the probabilities of the type P(Φ(A) = 0) for any compact A ⊂ BRd . Unfortunately, for the time being, explicit expressions for the void probabilities of Strauss and Matérn hardcorse processes are not available in the literature. As a consequence we do not have a relation between VV,Str (t, x)(= 1 − P(N Str (C(t, x)) = 0)) and VV,Mat (t, x). Of course both VV,Mat (t, x) and VV,Mat (t, x) are less than VV (t, x) for any (t, x) ∈ R+ × R3 ; by simulation studies in [18] we have VV,Mat (t, x) ≤ VV,Str (t, x), as intuitively one expects by the same relation for the mean extended volume densities. Acknowledgements EV is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). P. R. Rios is grateful to Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPQ, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, CAPES, and Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ, for the financial support.

References 1. Avrami, M.J.: Kinetics of phase change I. General theory. Journal of Chemical Physics, 7(12), 1103-1112 (1939). 2. Avrami, M.J.: Kinetics of phase change. II. Transformation-time relations for random distribution of nuclei. Journal of Chemical Physics, 8(2), 212–224 (1940). 3. Avrami, M.J.: Kinetics of phase change III. Granulation, phase change, and microstructure kinetics of phase change. Journal of Chemical Physics, 9(2), 177–184 (1941) .

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4. Bradley, J.R., Aaronson, H.I.: The stereology of grain boundary allotriomorphs. Metall. Trans., A, Phys. Metall. Mater. Sci., 8A, 317–22 (1977). 5. Bradley, J.R., Rigsbee, J.M., Aaronson, H.I.: Growth kinetics of grain boundary ferrite allotriomorphs in Fe-C alloys. Metall Trans., A, Phys. Metall. Mater. Sci., 8A:323–33 (1977). 6. Bradley, J.R., Aaronson, H.I.: Growth kinetics of grain boundary ferrite allotriomorphs in Fe-C-X alloys. Metall. Trans., A, Phys. Metall. Mater. Sci., 12, 1729–41 (1981). 7. Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications. 3th edition, Wiley (2013). 8. Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Vol. I. 2nd edition. Springer, New York. (2003). 9. Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Vol. II. 2nd edition. Springer, New York (2008). 10. Dereudre, D.: Introduction to the theory of Gibbs point processes. In: Coupier D. (eds), Stochastic Geometry. Lecture Notes in Mathematics, 2237 Springer, Cham (2019). 11. Godiksen, R.B., Rios, P.R., Vandermeer, R.A., Schmidt, S., Juul Jensen, D.: Three-dimensional geometric simulations of random anisotropic growth during transformation phenomena. Scr Mater., 58, 279–82 (2008). 12. Jensen, D.J., Godiksen, R..: Neutron and Synchrotron X-ray Studies of Recrystallization Kinetics. Metall. Mater. Trans. A, 39, 3065–3069(2008). 13. Johnson, W.A., Mehl, R.F. Reaction kinetics in process of nucleation and growth. Transactions AIME 135, 416–442 (1939). 14. Kiderlen, M., Hörig, M.: Matérn’s hard core models of types I and II with arbitrary compact grains. CSGB Research Reports n. 5, Centre for Stochastic Geometry and Advanced Bioimaging, Aarhus University (2013). 15. Kolmogorov, A.N.: On the statistical theory of the crystallization of metals. Isvestiia Academii Nauk SSSR - Seriia Matematicheskaia, 1, 333–359 (1937). 16. Mansson, M., Rudemo, M.: Random Patterns of Nonoverlapping Convex Grains. Advances in Applied Probability, 34, 718–738 (2002). 17. Rios, P.R., Villa, E.: Transformation kinetics for nucleation on random planes and lines. Image Anal. Stereol., 30, 153–165 (2011). 18. Sabatini, S., Villa, E.: On a special class of Gibbs hard-core point processes modelling random patterns of non-overlapping grains. submitted. 19. Silveira de Sá, G.M, da Silva Ventura, H., Assis, W.L.S., Villa, E., Rios, P.R.: Analytical Modeling and Computer Simulation of the Transformation of Ellipsoids Nucleated on Random Parallel Planes. Mat. Res., 23, (2020). 20. Sudbrack, C.,K., Ziebell, T.,D., Noebe, R.,D., Seidman, D.,N.: Effects of a tungsten addition on the morphological evolution, spatial correlations and temporal evolution of a model Ni-Al-Cr superalloy. Acta Mater., 56, 448–463 (2008). 21. Rios, P.R., Villa, E.: Transformation kinetics for inhomogeneous nucleation. Acta Mater., 57, 1199–1208 (2009). 22. Rios, P.R., Villa, E.: On modelling recrystallization processes with random growth velocities of the grains in Materials Science. Image Anal. Stereol., 31, 149–162 (2012). 23. Vandermeer, R.A, Rath, B.B.: Modeling recystallization kinetics in a deformed iron single crystal. Metall Trans, A, Phys Metall Mater Sci., 20, 391–401 (1989). 24. Vandermeer, R.A., Masumura, R.A., Rath, B.B.: Microstructural paths of shape-preserved nucleation and growth transformations. Acta Metall Mater., 39, 383-389 (1991). 25. Ventura, H.,S., Alves, A.L.M., Assis, W.L.S., Villa E., Rios P.R.: Influence of an exclusion radius around each nucleus on the microstructure and transformation kinetics. Materialia, 2, 167–175 (2018). 26. Villa, E., Rios, P.,R.: On modelling recrystallization processes with random growth velocities of the grains in Materials Science. Image Anal. Stereol., 31, 149–162 (2012). 27. Villa, E., Rios, P.R.: On volume and surface densities of dynamical germ-grain models with ellipsoidal growth: a rigorous approach with applications to Materials Science. Stoch. Anal. Appl., 38, 1134–1155 (2020).

Social Interactions and Mathematical Competencies Development Daniela Richtarikova

1 Introduction 1.1 Quotes Visions “Young people have to learn what they need for life” Jan Amos Komensky, 1592–1670, Moravian philosopher, teacher of nations [3]. All living creatures, humans not excluded, live in societies. Social abilities are crucial for good quality of life. “Speech is the garment of thought, and the explanation is its armor.” Antoine Rivarol, 1753–1801, French writer [2]. Speech, verbal and nonverbal communication (gestures, body language) are natural for people. They are means through which we are able to express our ideas, needs, moods, to argue, or convince others about our truth. They are the means which allow to live together with others in one community, the means which push forward development in the fields of practice, science and research, or in common life. “All the great speakers were bad speakers at first” Ralph Waldo Emerson, 1803–1884, American philosopher [2]. Social abilities and skills are cultivated from the very young age in a family, kindergarten, school, or later at work. They stand for a key lifelong competence, which is to be developed in all areas, comprising also education in mathematics. Here it can be found as partial competency C7 Communication in with and about D. Richtarikova (B) Faculty of Mechanical Engineering, Slovak University of Technology, Namestie Slobody 17, Bratislava 812 31, Slovakia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_8

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mathematics of categorization introduced by KOM project [4] and elaborated for tertiary technical education in [1].

1.2 Mathematics at Tertiary Engineering Education At technical universities and other tertiary engineering schools, mathematics represents an unavoidable subject of general natural science preparation, serving as precursive to further specific study. Mathematics I, II comprising selected chapters from Calculus, Linear Algebra, Geometry, etc. are usually taught as the courses of elementary higher mathematics in the very first study semesters. The customary problems, the newcomers usually face to, are the problems connected with transition from secondary to tertiary education including thoroughly new style of teaching and learning, high tempo, heavy and concentrated curricula content demanding higher cognitive abilities, such as ability of abstraction, analysis and synthesis, solving problems of non-specific transfer, etc. In addition, admitted students enter new formed classes of mutually unknown persons coming from different types of high schools with different amount of mathematics teaching hours, and so with different level of skills and knowledge. The just finished academic year 2021/22 revealed really big differences in skills of university freshmen due to not only traditional transition problems mentioned above, but reasoning mainly from the way how secondary schools and predominantly students themselves overcome the distance/on line teaching and learning. Except serious gaps in math knowledge, we observed hard lack of ability and also willingness to communicate as well as in other social interactions, the state which was only very slowly returning to the normal level. We can read it as a consequence of one and half year lack of personal contact, verbal and nonverbal communication emphasized in Slovakia by canceling both traditional written and oral parts of Maturita—the upper secondary school (ISCED 3) leaving exam, due to anti covid measures. A lot of university freshmen were not able to overcome the mathematics shortcomings at all, and during the 2nd semester, which was already taught on site, approximately a half of students did not come to examination period and untimely left the course. With respect to aforementioned, we would like to introduce a selection of collaborative teaching and learning methods, we have a good experience with (see also [5]). They directly develop the communication skills and through last decades they were verified within pedagogy process at the Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava (FME STU).

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2 Development of Social Skills—Collaborative Teaching and Learning 2.1 Discussions The first form we would like to point out is discussion. It is a method which can be carried out in many manners in accordance with particular circumstances and goals. Being well prepared, we execute it in no stress and not violent atmosphere, where students are not afraid to formulate their opinions and attitudes. Warm up or conclusion discussion. Warm up discussions are held at the beginning of the lesson. They are based upon repetition of basic knowledge students have to bring to training. On elementary stage, students are guided by means of Socrates dialogue, questions of type: What’s going on…?, What is your first idea, when…?, What do you think about..?, How does something work…?, Why..?, etc., where students learn to formulate their mathematical statements and opinions, to justify them, to give the chain of arguments, but also to post their own mathematical questions. Just as important is the fact that students build their attitude to mathematics and core the basic concepts here. Considering the current level of students’ knowledge and skills, the questions can be formulated as of the first contact kind, but they are drawn up in the way to help students to discover problems or solutions by themselves. Especially in situations, when students are not accustomed to discuss yet, we strongly recommend to allocate fixed time, not to spend too much time waiting on no answers. Later, the involvement of lecturer can be minimized and the dialogue could be held mainly by students. For preparation, entering homework is effective. Students prepare e.g. three questions with answers which express given mathematical topic content in their best way and the discussion is held upon them. Conclusion discussion. Conclusion discussions are held at the end of the lesson. They summarize the facts, strategies, procedures, conditions and pitfalls of methods. Their main effect goes with systemizing and anchoring of the acquired knowledge and skills. Class discussions on solved tasks. Commonly they appear spontaneously, when a problem is solved or presented centrally on a blackboard. For discussions of this kind we use situations when • • • • • •

the way of solution needs to be explained, brought to reason; or a solution is not unambiguous; presented way of solution is exceptional, students find several ways how to solve it; unexpected or “hidden” mistake appears and students are to reveal it, etc.

This phenomenon could be very nicely used intentionally for development of students critical thinking, their ability to think over and judge whether the result is possible or not, ability to seek more ways of solutions, etc.

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Oral examination by means of discussions. The discussion is moderated by an examiner, who brings up questions and problems with respect to students’ possibilities to present not only their learned knowledge and skills but also their ability to synthesize them, and come to new conclusions or opinions. Students consider this way of examination to be less stressed although they are exposed to direct comparison and in the end they have to evaluate themselves. In mutual discussion students summarize the acquired knowledge on given topic, reveal arising questions and problems, and try to give possible solutions. In master level of study and even for clever students in bachelor study this form is very fruitful, and students like it. For weak students the methodology has to be adapted for instance only to formulations of questions by students and answering them. However, the oral examination by means of discussion helps students to attain to an overview, summary, and sense of the subject. The optimum number of examined group of students is proposed to be up to ten. Students discussions within group work, preparation for exam, etc., and others Since speaking and talking pertain to basic human expressions and needs, discussions serve as main, sometimes unnoticed but natural source for competency development within group work, preparation for exam, etc., and other student activities.

2.2 Group Work The second, also collaborative form we consider to be very effective in improvement of communication is group work. Similarly as discussions, it is very variable. There exist a lot of ways how group work can be conducted. It can be controlled in many pointers, but it can be held also very naturally. The lecturer can decide upon the followed educational goals about the level of control mostly in: 1. Group members selection. Group members are selected by lecturer in random manner or determined with respect to previous results of participants, etc., usually aiming to have balanced groups; or group members are selected by students themselves upon to friendship, previous schoolmates or personal sympathies. 2. Group leader selection. A group leader is usually responsible for outside communication, inner distribution of work and sometimes also for individual evaluation of work within the group. Group leader can be selected by lecturer, or by voting of group members, or it can be a steering position in a group. Anyway, groups can work also intuitively, with not explicitly appointed leader. 3. Tasks solved by groups. The tasks can be the same for all groups, they can be different, or they can be chosen by group members from a given set upon their decision. Solving same tasks allows to compare groups directly and it naturally leads to a form of competition. Having balanced groups, this factor causes very

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high motivation. Having not balanced groups this factor causes high demotivation especially for weak groups or weaker group members. After the comparison of different approaches, we can conclude that “natural” principle of creating groups is the most acceptable by students, and it seems to bring also effective increase of competencies [5]. The essence lies in positive atmosphere, letting students to form their own groups, if possible, differing also in number of students and tolerating even loners. We have recognized that loners are usually students with very serious gaps in their mathematical knowledge and skills or foreign students with language barrier, being ashamed comparing with others. Keeping friendly atmosphere, they often accept it, and after all, they are willing to join some group or form their own group, being able to solve little bit simpler tasks. On the other hand, sometimes the loners are students with lower social affinity, who do not want to cooperate with others, thinking that the others are not as clever as they are, and it is only a waste of time for them.

3 Socializing Activities in Academic Year 2021/2022 At FME STU in Bratislava the winter semester was taught mainly in hybrid manner, when lectures were held on line and practicals were carried out in face to face mode. Keeping the anti covid measures the social activities were realized very carefully. We focused mainly on discussions with all class (warm up/conclusion and discussions on solved problems), where social distances could be kept. The work was very hard, very slowly approaching, since students entered their first year at university being very passive. They weaned off their communication skills, being isolated at home and reliant only on on-line environment during more than whole previous school year. Unfortunately, the schools were again closed in November for the end of semester and all winter examining period. Although, the summer semester we could spend already at school having face to face teaching, the socializing process was in the beginning again hard for the reason of preceding closure. On the other side, after improving the pandemic conditions in the last third of semester, besides the class discussions we were able to practice even the group work, albeit limited. The reactions of students on socializing activities were very nice. For unbelievable 96% of students, discussions were very useful and almost all of them (92%) claimed that discussions helped them to understand mathematical concept. Comparing the methods which were used in Mathematics II course, the most valuable method was solution presented by a teacher (71%). Solution presented by a student, group work and discussions had the same rate 53%. Actually, 71% of students liked to solve problems where mathematical thinking is necessary on contrary to 47% who preferred problems with given procedure. 41% of students liked problems based on argumentation, 35% with different ways of solution, as well as revealing mistakes.

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Considering the final assessment results of the Mathematics II course, 84% of students who completed the course attendance and underwent the socializing activities passed the exam with successful result.

4 Conclusion Presented results can be considered to be very positive. Taking into account also the opinions and attitudes of interested students and teachers on realized collaborative teaching and learning forms and their effects, we can conclude that presented methods substantially participated on the success. Socializing activities play an important role in education. The development of communication in with and about mathematics significantly contributes to overall mathematical maturity. On the contrary, the deficiency of socializing opportunities or even their absence give rice to passivity and loss of acquired skills and knowledge. However, communication activities are time-consuming, but inevitable in mathematics education. They require careful time planning and attention to teaching and learning objectives.

References 1. Alpers, B. et al., A Framework for Mathematics Curricula in Engineering Education. SEFI, Brussels, (2013). 2. Best quotations. https://best-quotations.com/. 3. Komensky, J.A.: Didactica Magna. 2nd edn. SPN, Bratislava (1991). 4. Niss, M., Mathematical competencies and the learning of mathematics: The Danish KOM project. In A. Gagatsis, S. Papastravidis (Eds.), 3rd Mediterranean Conference on Mathematics Education, Athens, Greece: Hellenic Mathematical Society and Cyprus Mathematical Society, 115–124, (2003). 5. Richtarikova, D., Training and Assessment with Respect to Competencies - Forms. In Aplimat 2019, 18th Conference on Applied Mathematics proceedings. pp 986–991. Bratislava, STU in Bratislava, (2019).

Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear Spinor Equations Oktay K. Pashaev

1 Introduction For description of black holes in low dimensional Jackiw-Teitelboim gravity model, two different, but equivalent approaches were proposed. One approach is based on the Resonant NLS equation (RNLS), the Nonlinear Schrödinger equation with quantum potential [1], wich admits solutions in the form of envelope soliton resonances. In another one, we have the system of real valued reaction-diffusion equations [2, 3], gauge equivalent to the Heisenberg model for spin variable s, belonging to the one sheet hyperboloid S L(2, R)/O(1, 1). This system is integrable system, with specific type of solutions called the dissipations [4]. Dissipatons represent non-relativistic particles, with resonant interaction, allowing fusion and fission of single dissipatons and creating a reach, the web type structure of the interaction. In present paper, by using another type of spin model on the same hyperboloid, we construct new model of dissipatons, representing relativistic particles with resonant interaction.

2 The Real Spinor Model 2.1 SL (2, R) Spin Model We start from nonlinear spin model corresponding to time independent LandauLifshitz equation in moving frame [5, 6], v μ ∂μ s = s ∧ ∂ μ ∂μ s,

(1)

O. K. Pashaev (B) Izmir Institute of Technology, Urla, Izmir 35430, Türkiye e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_9

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for spin vector field s(x 0 , x 1 ) ∈ S L(2, R)/O(1, 1), where the velocity vector v = (v 0 , v 1 ) is constant and the space-time psudo-Euclidean metric is diag(1, −1). The model represents non-compact one sheet hyperbolic version of compact Papanicolau spin model [7]. In terms of the light cone variables v + = 21 (v 0 + v 1 ), v − = 21 (v 0 − v 1 ), we have equation v + ∂+ s + v − ∂− s = s ∧ ∂+ ∂− s,

(2)

where ∂± = ∂0 ± ∂1 .

2.2 Integrable Real Spinor Model In present work we treat the case of constant “time-like” velocity two-vector with length μ0 , 1 (3) v + v − = [(v 0 )2 − (v 1 )2 ] ≡ μ20 > 0. 4 Proposition 1 The model (2) with constant velocity two-vector v = (v 0 , v 1 ), satisfying (3), is gauge equivalent to the real valued analog of Thirring model 1 + − + q q p μ0 1 ∂− p − + μ0 q − + q + q − p − μ0 1 + − + p p q −∂+ q + + μ0 p + + μ0 1 + − − p p q ∂+ q − + μ0 p − + μ0

− ∂− p + + μ0 q + +

= 0,

(4)

= 0,

(5)

= 0,

(6)

= 0.

(7)

Proposition 2 The system of equations (4)–(7) is equivalent to the compatibility conditions ∂+ J− − ∂− J+ + [J+ , J− ] = 0, for the linear system of equations ∂± Φ = Φ J± , where the Lax pair in the zero-curvature condition form is

Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear …

    1 1 + − 0 μ10 p − 2 −λ + J+ = p p σ3 + λ , p+ 0 2 μ0     μ2 1 1 1 0 q− , − 20 + q + q − σ3 + J− = 2 λ μ0 λ μ0 q + 0

83

(8) (9)

and λ is the spectral parameter.

3 Hyperbolic Complex Thirring Model By expanding q ± = u 1 ± v1 , p ± = u 2 ± v2 , we combine new functions to the pair of hyperbolic complex valued functions (known also as the double number valued functions) [8], (10) χ1 = u 1 + jv1 , χ2 = u 2 + jv2 , where the hyperbolic imaginary unit satisfies j 2 = 1, j¯ = − j.

(11)

The corresponding conjugate functions are χ¯ 1 = u 1 − jv1 , χ¯ 2 = u 2 − jv2 ,

(12)

χ¯ 1 χ1 = |χ1 |2 = u 21 − v12 , χ¯ 2 χ2 = |χ2 |2 = u 22 − v22 .

(13)

so that

Proposition 3 The system of four equations (4)–(7) can be represented in the form of two hyperbolic complex equations 1 |χ2 |2 χ1 = 0, μ0 1 − j∂− χ2 + μ0 χ1 + |χ1 |2 χ2 = 0. μ0

− j∂+ χ1 + μ0 χ2 +

(14) (15)

Proposition 4 The system (14)–(15) is equivalent to hyperbolic complex Thirring model (16) (− jγ ν ∂ν + m)Ψ + g 2 γ ν (Ψ¯ γν Ψ )Ψ = 0, for hyperbolic complex spinor field Ψ and corresponding hyperbolic Dirac conjugate of it Ψ¯ ,       ψ1 , Ψ¯ = Ψ + γ 0 = ψ¯ 1 ψ¯ 2 γ 0 = ψ¯ 2 ψ¯ 1 , (17) Ψ = ψ2

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√ √ where we denote χ1 ≡ g 2mψ1 , χ2 ≡ g 2 mψ2 , μ0 ≡ m. Here Dirac’s matrices are defined as     01 0 −1 0 1 γ = , γ = , (18) 10 1 0 the Minkowski space-time metric is diag(1, −1) and  γ 0 ∂0 + γ 1 ∂1 =

0 ∂− ∂+ 0

 .

(19)

As would be shown below, this hyperbolic complex counterpart to usual elliptic complex Thirring model, in contrast to the last one admits resonant interaction of dissipative solitons. We notice also that the hyperbolic Thirring model (16) allows the Lax pair in terms of 2 × 2 hyperbolic complex functions, while the equivalent real system (4)–(7) has 2 × 2 Lax pair (8), (9), but for the real valued functions.

4 Hirota Bilinear Form By introducing six real functions g ± , h ± , f ± , we represent our field variables as p± =

g± g± f ± h± h± f ∓ ± = , q = = . f∓ f±f∓ f± f∓f±

Substituting to the system (4)–(7) and using Hirota’s derivatives Dtk Dxl (a(x, t) · b(x, t)) = (∂t − ∂t  )k (∂x − ∂x  )l a(x, t)b(x  , t  )|x=x  ,t=t  , we split equations in form of the bilinear system. Proposition 5 The system of equations (4)–(7) can be represented as bilinear system ± Dt (g ± · f ± ) + μ0 h ± f ∓ = 0, ∓Dx (h ± · f ∓ ) + μ0 g ± f ± = 0,

(20) (21)

μ0 Dt ( f + · f − ) + h + h − = 0, μ0 Dx ( f + · f − ) + g + g − = 0.

(22) (23)

Here x 0 ≡ T , x 1 ≡ X are time and space coordinates in laboratory coordinate systems, x = 21 (X + T ), t = 21 (X − T ) are the light-cone coordinates, so that ∂+ = ∂x , ∂− = −∂t . Corollary 1 For calculating field densities, from Eqs. (22), (23) follow formulas     f+ f+ q + q − (x, t) = −μ0 ln − , p + p − (x, t) = −μ0 ln − . f f t x

Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear …

85

5 The One Dissipaton Solution By Hirota expansion in parameter ε, g ± (x, t) = εg1± (x, t) + ε3 g3± (x, t) + ... ,

3 ± h ± (x, t) = εh ± 1 (x, t) + ε h 3 (x, t) + ... , f ± (x, t) = 1 + ε2 f 2± (x, t) + ε4 f 4± (x, t)... ,

we get the one dissipaton solution. Proposition 6 The one dissipaton solution of system (4)–(7) is ±

p ± (x, t) =

eμ0 η1 g± = μ0 + − , ∓ ∓ f 1 + b2 eμ0 (η1 +η1 )

q ± (x, t) =

a1± eμ0 η1 h± = μ 0 + − , f± 1 + b2± eμ0 (η1 +η1 )

(24)

±

where η1±

=

k1± x

+

ω1± t

+

η1±0

 1 ± =± x − a1 t + η1±0 , a1±

(25)



(26)

ω1± = ∓a1± , k1± = ±1/a1± , b2+ =

(a1+ )2 a1− , (a1+ − a1− )2

b2− =

(a1− )2 a1+ , (a1+ − a1− )2

(27)

and a1± , η1±0 are real constants. Regularity of this solution requires that a1+ > 0, a1− > 0 and as follows k1+ > 0, k1− < 0. Corollary 2 The one dissipaton solution in laboratory coordinate system takes the form √

√ k 2 +1 (T −v X )+ν10 k 2 + 1 ± k ±μ0 √1−v 2 e  p = , X −X 0∓ −vT cosh μ0 k √1−v 2 √

√  14  k 2 +1 (T −v X )+ν10 μ0 k k 2 + 1 ± k ±μ0 √1−v 1+v 2 ± e  q = , X −X 0± −vT 1−v cosh μ0 k √1−v 2 ±



1−v 1+v

 41

μ0 k

(28)

(29)

where ν10 ≡ η1+0 − η1−0 , a1+ − a1− a+a− − 1 k≡  → |v| < 1, , v ≡ 1+ 1− a1 a1 + 1 2 a1+ a1−

(30)

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O. K. Pashaev

and initial positions are √ X 0± =

1 − v2 2k



η1+0

+

η1−0

+ ln

1+v 1−v

 √ k2 + 1 ± k . 4k 2

(31)

Corollary 3 In terms of parameters v, k and X 0 = 21 (X 0+ + X 0− ), dissipaton densities become traveling wave form

+ −

p p =

+ −

q q =

2μ20 k 2 1−v  √  , 1 + v cosh 2μ k X −X √ 0 −vT + k2 + 1 0 1−v 2

(32)

2μ20 k 2 1+v  √  . 1 − v cosh 2μ k X −X 2+1 √ 0 −vT + k 0 2 1−v

(33)

5.1 Integrals of Motion The system (4)–(7) is integrable and admits infinite number of integrals of motion. The physically meaningful are first three integrals, which can be calculated explicitly [9] for one dissipaton solution (28), (29). Proposition 7 The mass, momentum and energy integrals for one dissipaton solution (28), (29) are √

∞ M=

+ −

+ −

( p p + q q ) d x = 2 ln √ −∞ ∞

P=

1

k 2 + 1 − |k|

( p + ∂1 p − + q + ∂1 q − ) d x 1 = √

−∞

E= √

k 2 + 1 + |k|

4k 1 − v2

4kv 1 − v2

,

,

.

(34)

(35) (36)

Corollary 4 The one dissipaton solution represents relativistic particle with corresponding nonlinear mass m 0 , momentum P and energy E, m 0 = 4 sinh

m0v M m0 , P=√ , E=√ , 2 4 1−v 1 − v2

(37)

and relativistic dispersion E 2 = m 20 + P 2 → E =



m 20 + P 2 .

(38)

Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear …

87

5.2 Resonant Interaction The above results allow us to describe collision of two dissipatons with masses m 1 , m 2 and velocities v1 , v2 , producing by fusion one dissipaton with mass m and velocity v. This process is called the resonant interaction of dissipatons. Proposition 8 The conservation laws for fusion of two dissipatons M = M1 + M2 , P = P1 + P2 , E = E 1 + E 2 , are described by algebraic system of equations  

m 21 + 16 + m 1

 

m 22 + 16 + m 2

√ = √

m 2 + 16 + m

m 2 + 16 − m m 21 + 16 − m 1 m 22 + 16 − m 2 m 1 v1 m 2 v2 mv  + = √ , 1 − v2 1 − v12 1 − v22 m1 m2 m  + = √ . 1 − v2 1 − v12 1 − v22

,

(39) (40) (41)

5.3 Y Shaped Resonance Conditions We treat a special case of two equal mass m 1 = m 2 dissipaton collision with equal and opposite velocities v1 = −v2 . Proposition 9 The system (39)–(41) has solution with m 1 = m 2 , v1 = −v2 , v = 0 so that  1 m1 , m = m 1 m 21 + 16. (42) v1 =  2 m 21 + 16 Corollary 5 The solution describes fusion of two dissipatons with opposite momentums P1 = −P2 and the vanishing total momentum P = P1 + P2 = 0. The process creates one dissipaton with mass m in the rest v = 0. Corollary 6 Due to (30), the resonance condition (42) for two, one dissipaton solutions with m 1 , v1 and m 2 , v2 = −v1 restricts parameters of the solutions as a1− = 1, a2− = 1, a2+ =

1 a1+ − 1 a2+ − 1 , v = −v1 . , v = = 1 2 a1+ a1+ + 1 a2+ + 1

(43)

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O. K. Pashaev

6 Two Dissipaton Solution By continuing Hirota’s expansion we get an exact two dissipaton solution. Proposition 10 Two dissipaton solution of bilinear equations (20)–(23) and the corresponding system (4)–(7) is ±

±

+

−

±

+

−

±

± η1 +η1 +η2 ± η2 +η2 +η1 g ± = eη1 + eη2 + c112 e + c221 e , ±

±

+

−

±

+

−

±

± η1 +η1 +η2 ± η2 +η2 +η1 e + d221 e , h ± = a1± eη1 + a2± eη2 + d112 +

−

+

−

+

−

+

−

+

−

+

−

± η1 +η1 ± η1 +η2 ± η2 +η1 ± η2 +η2 ± e + b12 e + b21 e + b22 e + b1122 eη1 +η1 +η2 +η2 f ± = 1 + b11

where in the light cone frame (i = 1,2) ηi± = ki± x + ωi± t + η0±i = ±



 1 ± x − a t + η0±i , i ai±

ωi± = ∓ai± , ki± = ±1/ai± , and in the laboratory frame X − vi j T + η0±i + η0±j , ηi+ + η −j = −2ki j  1 − vi2j and

ai+ − a −j ai+ a −j − 1 ki j =  . , vi j = + − ai a j + 1 2 ai+ a −j

(44)

For regularity of the solution we choose ai+ > 0, ai− > 0, (i = 1, 2) and as follows, velocities in all frames vi j are bounded |vi j | < 1. Parameters of solution are ± = c112

(a2± − a1± )2 (a1∓ )3 (a2± − a1± )2 (a2∓ )3 ± , c = , 221 (a2± − a1∓ )2 (a1+ − a1− )2 (a2∓ − a1± )2 (a2+ − a2− )2

± d112 =

(a2± − a1± )2 (a1∓ )2 a1± a2± ± (a2± − a1± )2 (a2∓ )2 a1± a2± , ± ∓ 2 + − 2 , d221 = (a2 − a1 ) (a1 − a1 ) (a2∓ − a1± )2 (a2+ − a2− )2 bii± = bi+j =

(ai± )2 ai∓ , (i = j), (ai+ − ai− )2

(ai+ )2 a −j (ai+ − a −j )

, bi−j = 2

(a −j )2 ai+ (ai+ − a −j )2

, (i = j),

Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear … ± b1122 =

89

(a2+ − a1+ )2 (a2− − a1− )2 (a1± )2 (a2± )2 a1∓ a2∓ . − a1− )2 (a2+ − a2− )2 (a1+ − a2− )2 (a2+ − a1− )2

(a1+

In above formulas we have chosen μ0 = 1, which corresponds to unit “time-like” vector (3). This can be accomplished by rescaling space-time and field variables: μ0 x 0 ≡ x 0 , μ0 x 1 ≡ x 1 , p ± ≡ μ0 p ± , q ± ≡ μ0 q ± .

(45)

6.1 Y Shaped Resonance Solution By using two dissipaton solution, now we describe fusion of dissipatons as Y shaped resonance. Corollary 7 The two dissipation solution under restriction on parameters (43), such − − − − ± that c112 = c221 = d112 = d221 = b1122 = 0, takes the form  + −

q q =⎡

8v12 (1−v12 )2

⎣cosh

1+

4v1 (X −X 0 ) 1−v12

1−v12 − e 2

+

 + −

p p =⎡

8v12 (1−v12 )2

⎣cosh

1+

4v1 (X −X 0 ) 1−v12

+

1+v12 1−v12

−

+

1−v12 − e 2

1+v12 1−v12

2v12 (T −T0 ) 1−v12

cosh

2v12 (T −T0 ) 1−v12

2 e√ 1−v12

2v12 (T −T0 ) 1−v12

−

+



2v1 (X −X 0 ) 1−v12

cosh

 cosh

2v12 (T −T0 ) 1−v12

2 e√ 1−v12

+ ln

2v1 (X −X 0 ) 1−v12

2v1 (X −X 0 ) 1−v12

cosh

1 2

1−v 1+v

+ 21 e

− ln 1 2

2v1 (X −X 0 ) 1−v12

4v 2 (T −T ) − 1 20 1−v1

1−v 1+v 1 − e 2

+

  ⎤, ⎦

 

4v12 (T −T0 ) 1−v12

⎤. ⎦

This solution is called the Y shaped resonant solution. Corollary 8 The Y shaped resonant solution for q + q − at T → −∞ describes collision of two dissipatons with equal and opposite velocities, the one dissipaton is moving from the left in frame ξ − = X − v1 T , q +q − =

1 2v12  1 − v1 1 − v 2 cosh(2v 1

1

X −X 0L −v1 T 1−v12

)+1

,

and another one is moving from the right in frame ξ + = X + v1 T ,

(46)

90

O. K. Pashaev

q +q − =

1 2v12  1 + v1 1 − v 2 cosh(2v 1

1

X −X 0R +v1 T 1−v12

)+1

.

(47)

After fusion, at time T → ∞ we have only one dissipaton in the rest, q +q − =

1 8v12 . −X 0 2 1 − v12 (1 − v12 ) cosh(4v1 X1−v 2 ) + (1 + v1 )

(48)

1

Initial positions of dissipatons are connected by the mean value X0 =

1 L (X + X 0R ). 2 0

(49)

Corollary 9 The Y shaped resonant solution for p + p − at T → −∞ describes collision of two dissipatons with equal and opposite velocities, the one dissipaton is moving from the left in frame ξ − = X − v1 T , p+ p− =

1 2v12  1 + v1 1 − v 2 cosh(2v 1

1

X −X 0L −v1 T 1−v12

)+1

,

(50)

.

(51)

and another one is moving from the right in frame ξ + = X + v1 T , p+ p− =

1 2v12  1 − v1 1 − v 2 cosh(2v 1

1

X −X 0R +v1 T 1−v12

)+1

After fusion, at time T → ∞ we have only one dissipaton in the rest, p+ p− =

1 8v12 . −X 0 2 1 − v12 (1 − v12 ) cosh(4v1 X1−v 2 ) + (1 + v1 )

(52)

1

Time Reflection Proposition 11 The system of equations (4)–(7) is invariant under time - reflection T → −T, p ± → q ∓ , q ± → p ∓

(53)

T → −T, p + p − → q + q − , q + q − → p + p − .

(54)

and as follows

Hirota Bilinear Method and Relativistic Dissipative Soliton Solutions in Nonlinear …

91

Corollary 10 Due to the time reflection symmetry the Y shaped solution describes also the resonant fission of one dissipaton to two dissipatons, moving in oppposite direction with equal speed.

7 Conclusions In present paper a new relativistic integrable nonlinear model for real, Majorana type spinor fields in 1+1 dimensions was solved by Hirota’s bilinear method. It was shown that exact one dissipaton solution of the model represents a particle-like nonlinear excitation, with relativistic dispersion and highly nonlinear mass. For two dissipative soliton solutions, a resonant character of dissipaton interactions in form of fusion and fission of relativistic particles was demonstrated. In conclusion, we like to note intriguing difference between linear and nonlinear modes for system (4)–(7). The linearized form of the equation (it could be realized in the limit μ0 → ∞) is the Klein-Gordon equation (−∂02 + ∂12 + μ20 )Φ = 0,

(55)

 with tachyonic dispersion E = p 2 − μ20 and the corresponding wave/particle is moving with speed bigger than the characteristic speed (“speed of light”): |v| > 1. In contrast to this, as we have seen the nonlinear dissipaton modes, are moving with speed |v| < 1. This property could shed light on hypothetical particles as tachyons in relativity theory. By taking nonlinear corrections to tachyons it is possible to create bradyon particles moving with speed, below the speed of light. Finally, as we know, by applying Madelung transform, dissipatons of nonrelativistic reaction-diffusion equations can represent resonant envelope solitons of NLS with quantum potential [1]. It implies that should exist another, explicit soliton resonant form for our system (4)–(7) and corresponding soliton resonance solutions. This work was supporting by BAP project 2022IYTE-1-0002.

References 1. Pashaev, O.K., Lee, J.-H.: Resonance Solitons as Black Holes in Madelung Fluid. Mod. Phys. Lett. A, 17, 1601–1619 (2002). https://doi.org/10.1142/S0217732302007995. 2. Martina L., Pashaev O.K., Soliani G.: Integrable dissipative structures in the gauge theory of gravity. Classical and Quantum Gravity, 14, 3179–3186 (1997). https://doi.org/10.1088/02649381/14/12/005. 3. Martina L., Pashaev O.K., Soliani G.: Bright solitons as black holes. Physical Review D: Particles, Fields, Gravitation and Cosmology, 58, 084025(1998). https://doi.org/10.1103/ PhysRevD.58.084025. 4. Pashaev O.K.: Integrable models as constrained topological gauge theory. Nuclear Physics B (Proc. Suppl.), 57, 338–341(1997). https://doi.org/10.1016/S0920-5632(97)00374-5.

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5. Martina L., Pashaev O.K., Soliani G.: Bilinearization of multidimensional topological magnets. J. Phys. A: Math. Gen., 27, 943–954 (1994). https://doi.org/10.1088/0305-4470/27/3/033. 6. Martina L., Pashaev O.K., Soliani G.: Self-dual Chern-Simons solitons in nonlinear σ-models. Mod Physics Lett A, 8, 3241–3250 (1993). https://doi.org/10.1142/S0217732393002166. 7. Papanicolaou N.: Duality rotation for 2-D classical ferromagnets. Phys. Lett. A, 84, 151–154 (1981). https://doi.org/10.1016/0375-9601(81)90742-8. 8. Yaglom I.M.: Complex Numbers in Geometry. Academic Press, New York (1968). 9. Pashaev, O.K., Lee, J.-H.: Relativistic dissipatons in integrable nonlinear Majorana type spinor model. Southeast Asian Bull. Math., 46, 749–770 (2022).

Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s Theorem for Tetrahedron Oktay K. Pashaev

1 Introduction The qubit unit of quantum information in quantum computers is a quantum state from Hilbert space of two-level quantum system, corresponding to binary position system of classical bits. In realistic physical hardware, the information carrying quantum systems typically are multilevel quantum systems. Description of such systems require to introduce more general units of quantum information, like qutrits or qudits, based on three-level and arbitrary d-level quantum systems, correspondingly. These units open access to higher-dimensional Hilbert spaces and provide a wide range of applications in processing of quantum information [1, 5]. The entanglement property of quantum states plays the key role in quantum information processing and applications to dense coding, quantum teleportation etc. Quantification of entanglement for arbitrary pure and mixed multipartite states is a hard and not solved problem for arbitrary qudit states. In special case of two qubit states it is characterized by one real parameter, the concurrence, which is connected with the von Neumann entropy for reduced density matrix [6]. For a real qubit states some simple geometrical arguments relates concurrence with area characteristics [4]. In present paper we apply similar geometrical ideas to a real two-qutrit states and establish a relation of the maximally entangled two-qutrit concurrence with De Gua’s generalization of Pythagoras theorem for triorthogonal tetrahedron areas. First we introduce the qubit, qutrit and generic qudit units of quantum information, determined by position system with bases 2, 3, and arbitrary d, correspondingly. Definition 1 The qubit state |ψ ∈ H = C 2 is a linear combination of |0 and |1 orthonormal states, (1) |ψ = c0 |0 + c1 |1,

O. K. Pashaev (B) Izmir Institute of Technology, Urla, Izmir 35430, Türkiye e-mail: [email protected]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_10

93

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O. K. Pashaev

where |0 =

  1 , 0

|1 =

  0 , 1

(2)

and |c0 |2 + |c1 |2 = 1. Definition 2 The qutrit state |ψ ∈ H = C 3 is a linear combination |ψ = c0 |0 + c1 |1 + c2 |2,

(3)

of three computational states |0, |1 and |2 states, where ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 |0 = ⎝ 0 ⎠ , |1 = ⎝ 1 ⎠ , |2 = ⎝ 0 ⎠ 0 0 1

(4)

and |c0 |2 + |c1 |2 + |c2 |2 = 1. Definition 3 The qudit state |ψ ∈ H = C d is a linear combination |ψ = c0 |0 + c1 |1 + · · · + cd−1 |d − 1,

(5)

of d computational states |0, |1,…, |d − 1, where ⎛ ⎞ ⎛ ⎞ 1 0 ⎜ .. ⎟ ⎜ .. ⎟ |0 = ⎝ . ⎠ , . . . , |d − 1 = ⎝ . ⎠ 0

(6)

1

and |c0 |2 + |c1 |2 + · · · + |cd−1 |2 = 1.

2 Separability and Entanglement of Two Qubit States The generic two qubit state |ψ = c00 |00 + c01 |01 + c10 |10 + c11 |11,

(7)

where |c00 |2 + |c01 |2 + |c10 |2 + |c11 |2 = 1, rewritten in the form |ψ = |0|φ + |1|ξ ,

(8)

Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s …

95

is characterized by pair of one qubit states |φ ≡ c00 |0 + c01 |1 and |ξ  ≡ c10 |0 + c11 |1. Proposition 1 Two qubit state (7) is separable if and only if states |φ and |ξ  are linearly dependent.

2.1 Concurrence and Area For the rebit states, defined as the qubit states |φ = r00 |0 + r01 |1, |ξ  = r10 |0 + r11 |1, with real coefficients (r00 , r01 ) ≡ r0 and (r10 , r11 ) ≡ r1 , the linear dependence of two dimensional vectors r0 and r1 can be expressed by the area formula in plane. Proposition 2 Two vectors r0 and r1 in plane determine parallelogram with area





r00 r01



.

A = |r0 × r1 | = det

r10 r11

(9)

The normalization condition r20 + r21 = 1

(10)

restricts this area by maximal value Amax = 21 , so that (10) becomes the Pythagoras theorem for the orthogonal vectors, corresponding to rectangle, inscribed to the circle with radius 21 . Corollary 1 The concurrence C defined as





r00 r01





= 2 A C = 2 det

r10 r11

(11)

is bounded 0 ≤ C ≤ 1.

2.2 Concurrence and Reduced Density Matrix The concurrence, as a pure geometrical quantity [4] is connected with physical characteristics of entangled two qubit states, represented by reduced density matrix [6]. Proposition 3 For generic two qubit state, the trace of squared reduced density matrix ρ A and the concurrence C are connected by “Pythagoras theorem” 1 tr ρ 2A + C 2 = 1. 2

(12)

96

O. K. Pashaev

Corollary 2 If C = 0, then tr ρ 2A = 1 and the state is separable. If C = 0, then tr ρ 2A < 1 and the state is entangled (non-separable).

3 Separability and Entanglement of Two Qutrit States Proposition 4 For generic two qutrit state |ψ =



ci j |i j,

(13)

i, j=0,1,2

the density matrix is ρ = |ψψ| =



ci j ci  j  |i ji  j  |.

(14)

i, j,i  , j 

Proposition 5 The reduced density matrix ρ A = tr B ρ for the generic two qutrit state is  (ci0 ci  0 |ii  | + ci1 ci  1 |ii  | + ci2 ci  2 |ii  |) (15) ρA = i,i 

or in matrix form ρ A = ⎛

|c00 |2 + |c01 |2 + |c02 |2 ⎝ c¯00 c10 + c¯01 c11 + c¯02 c12 c¯00 c20 + c¯01 c21 + c¯02 c22

c00 c¯10 + c01 c¯11 + c02 c¯12 |c10 |2 + |c11 |2 + |c12 |2 c¯10 c20 + c¯11 c21 + c¯12 c22

⎞ c00 c¯20 + c01 c¯21 + c02 c¯22 c10 c¯20 + c11 c¯21 + c12 c¯22 ⎠ |c20 |2 + |c21 |2 + |c22 |2

and due to normalization |c00 |2 + |c10 |2 + |c20 |2 + |c01 |2 + |c11 |2 + |c21 |2 + |c02 |2 + |c12 |2 + |c22 |2 = 1, tr ρ A = 1. To decide if the reduced density matrix corresponds to pure or mixed state, we calculate tr ρ 2A . Proposition 6 Trace of the squared reduced density matrix satisfies the “Pythagoras theorem” 1 trρ 2A + C 2 = 1, (16) 2 where the square of the qutrit concurrence is defined by sum

Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s …













1 2

c00 c01

2

c00 c02

2

c01 c02

2 C =

+

+

c10 c11

c10 c12

c11 c12

4

2



2





c c



c c



c c

2 +

00 01

+

00 02

+

01 02



c20 c21 c20 c22 c21 c22

2



2







c10 c11



c10 c12



c11 c12

2















+

+

+

c20 c21

c20 c22

c21 c22

or







c∗∗ c∗∗

2





C =4

c∗∗ c∗∗

, 2

97

(17)

(18)

∗

which includes modulus squares of all 2 × 2 minors of 3 × 3 matrix (C)i j = ci j . Definition 4 The concurrence for two qutrit states is defined as







c∗∗ c∗∗

2



C = 2

c∗∗ c∗∗

.

(19)

∗

As we learn recently, similar expression for concurrence √ was introduced in [2], which is different from our definition by constant factor 3/2. Corollary 3 If C = 0, then tr ρ 2A = 1 and the state is separable. If C = 0, then tr ρ 2A < 1 and the state is entangled (non-separable). Proof For separable two-qutrit states the matrix ci j = ci d j have to be rank one. In this case all 2 × 2 minors are zero and C = 0, so that trρ 2A = 1 and the reduced state is the pure state. If rank of matrix C is bigger than one, then exists at least one non-vanishing 2 × 2 minor, so that C = 0 and as follows trρ 2A < 1. The reduced state in this case becomes mixed.  The value of positive number C shows how far is given state from the pure one and so, how much it is entangled. The maximally entangled state then corresponds to maximal value of C.

4 Geometrical Representation of Two-Retrit State in 3D Space Here we examine generic qutrit states with real coefficients ci = ri , ci j = ri j , i, j = 0, 1, 2, and call them the retrit states. The generic two-retrit state is characterized by triple of one-retrit states |φ0 , |φ1 , |φ2 ,

98

O. K. Pashaev

|ψ = |0|φ0  + |1|φ1  + |2|φ2 ,

(20)

with real coordinates. These coordinates determine the corresponding three vectors in 3D Euclidean space: r0 = (r00 , r01 , r02 ), r1 = (r10 , r11 , r12 ), r2 = (r20 , r21 , r22 ).

(21)

By taking modulus of the cross product of two vectors, |ri × r j | = Ai j we find that it is equal to the area Ai j of the corresponding parallelogram. For our three vectors we have three areas A01 , A02 , A12 and corresponding squares |r0 × r1 | =

A201













r00 r01

2

r00 r02

2

r01 r02

2











,



=

+

+

r10 r11

r10 r12

r11 r12

|r0 × r2 | =

A202













r00 r01

2

r00 r02

2

r01 r02

2











,



=

+

+

r20 r21

r20 r22

r21 r22

2

2













r r

2

r r

2

r r

2 |r1 × r2 |2 = A212 =

10 11

+

10 12

+

11 12

. r20 r21 r20 r22 r21 r22

(22)

(23)

(24)

4.1 Maximally Entangled Retrit States Let us denote A2 = A201 + A202 + A212 ,

(25)

 C = 2 A = 2 A201 + A202 + A212 .

(26)

so that the concurrence is

Substituting this to (16) gives formula tr ρ 2A + 2(A201 + A202 + A212 ) = tr ρ 2A + 2 A2 = 1.

(27)

The formula shows that for pure state, when tr ρ 2A = 1, A = 0 and as follows A01 = A02 = A12 = 0. It means that mutual cross products between three vectors r0 , r1 , r2 vanish, and vectors are directed along the same line. If at least one of the cross products is not zero, then A2 = 0, tr ρ 2A < 1 and the state is mixed. The mixture increases with increasing value of A and reaches maximal value for maximal A.

Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s …

Proposition 7 The maximal value of A2 = Then, for maximally reduced mixed state trρ 2A =

1 3

99

corresponds to r02 = r12 = r22 = 13 .

1 3

(28)

and maximal concurrence for maximally entangled two-retrit state is 2 C=√ . 3

(29)

Proof The areas of parallelograms are maximal for orthogonal vectors, so that A201 = r02 r12 , A202 = r02 r22 , A212 = r12 r22 . Then A2 = A201 + A202 + A212 = r02 r12 + r02 r22 + r12 r22 .

(30)

Applying this formula to normalization condition and its square r02 + r12 + r22 = 1, r04

+ r14

+ r24

+

2(r02 r12

+ r02 r22

+ r12 r22 )

= 1,

(31) (32)

we get relations r04 + r14 + r24 + 2 A2 = 1, A2 = r02 r12 + r02 r22 + r12 r22 . By using r22 = 1 − r02 − r12 we exclude r2 from the last equation so that f (r0 , r1 ) ≡ A2 = r02 + r12 − r04 − r14 − r02 r12 , where |r0 | ≤ 1 and |r1 | ≤ 1. By denoting x ≡ r02 , y ≡ r12 the problem reduces to find extrim values of function F(x, y) ≡ x + y − x 2 − y 2 − x y. By solving the system Fx = 1 − 2x − y = 0, Fy = 1 − 2y − x = 0, we find solution x = y = r02 = r12 = r22 = 13 .

1 3

(33)

and the maximal value A2max = 13 , corresponding to 

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O. K. Pashaev

4.2 Trirectangular Tetrahedron Three vectors r0 , r1 , r2 at origin determine a tetrahedron in 3D Euclidean space. Areas of its faces at origin are 1 1 1 A01 , A 02 = A02 , A 12 = A12 , 2 2 2

A 01 =

and area of the face, opposite to the origin is A =

1 → − 1 → → A = (− r0 − → r 2 ) × (− r1 − − r 2 ) . 2 2

By using formula − → − → − → − → − → − → − → − → − → − → − → − → ( A × B ) · ( C × D ) = ( A · C )( B · D ) − −( A · D )( B · C )

(34)

we find → → → → → → → → A 2 = (− r0 − − r 2 )2 (− r1 − − r2 )2 − ((− r0 − − r 2 ) · (− r1 − − r2 ))2 → → → → → → = (r 2 + r 2 − 2− r ·− r )(r 2 + r 2 − 2− r ·− r ) − (− r ·− r 0

0

2

2

1

→ → → → −− r0 · − r2 − − r2 · − r 1 + r 2 2 )2 .

2

1

2

0

1

The maximally mixed state corresponds to the orthogonal vectors and trirectangular tetrahedron, where all three face angles at one vertex are right angles. In this case all scalar products vanish and the area formula becomes A2 = r02 r12 + r02 r22 + r12 r22 , and A2 =

1 2 2 (r r + r02 r22 + r12 r22 ). 4 0 1

(35)

Then, for areas of faces at origin A201 = |r0 × r1 |2 = r02 r12 , A202 = |r0 × r2 |2 = r02 r22 , A212 = |r1 × r2 |2 = r12 r22 , we get the relation A2 = A201 + A202 + A212 .

(36)

Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s …

101

4.3 De Gua’s Generalization of Pythagoras Theorem for Areas De Gua’s theorem is a three-dimensional analog of the Pythagoras theorem or generalization the Pythagoras theorem to a tetrahedron [3]. For the trirectangular tetrahedron, the square of the area of the face, opposite to the right-angle corner is the sum of the squares of the areas of the other three faces: A2 = A2 01 + A2 02 + A2 12 .

(37)

The last formula corresponds to (35), (36) for maximally mixed reduced state, where A 01 = A01 =

1 |r0 ||r1 |, 2

A 02 = A02 =

1 1 |r0 ||r2 | A 12 = A12 = |r1 ||r2 |. 2 2

This way we have established the relation between maximally entangled two-retrit state and De Gua’s theorem for trirectangular tetrahedron. The relation shows geometrical meaning of concurrence for maximally entangled two qutrit state as the area of opposite site of the tetrahedron.

5 Qutrit Entanglement We extend above results to generic complex two qutrit state, decomposed as |ψ = |0|c0  + |1|c1  + |2|c2 ,

(38)

in terms of one qutrit states |c0  = c00 |0 + c01 |1 + c02 |2, |c1  = c10 |0 + c11 |1 + c12 |2,

(39) (40)

|c2  = c20 |0 + c21 |1 + c22 |2.

(41)

Definition 5 The external (cross) product of two qutrit states |ca  and |cb , (a, b = 0, 1, 2) is a qutrit state, denoted as |ca × cb  and defined by formula |ca × cb  =

2  i, j,k=0

It can be expressed as

i jk ca j cbk |i.

(42)

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O. K. Pashaev







|0 |1 |2





ca0 ca2

ca0 ca1



ca1 ca2











|2. |0 −

|1 +

|ca × cb  = ca0 ca1 ca2 =

cb1 cb2

cb0 cb2

cb0 cb1

cb0 cb1 cb2

(43)

Proposition 8 In terms of cross product of states, the concurrence (19) is  C = 2 c0 × c1 |c0 × c1  + c2 × c0 |c2 × c0  + c1 × c2 |c1 × c2 

(44)

Proof Norms of the cross product states



c c0 × c1 |c0 × c1  =

00 c10



c c2 × c0 |c2 × c0  =

00 c20



c c1 × c2 |c1 × c2  =

10 c20

2



c c01



+

00



c11 c10

2



c c01



+

00 c21

c20

2



c c11



+

10



c21 c20

2



c c02



+

01



c12 c11

2



c c02



+

01 c22

c21

2



c c12



+

11



c22 c21

2 c02



, c12



2 c02



, c22



2 c12



, c22

after substituting to (17) give the concurrence formula (44).



Proposition 9 The inner product of two cross product states satisfies formula A × B|C × D = A|CB|D − A|DB|C

(45)

Corollary 4 Norms of cross product states in (44) are equal c0 × c1 |c0 × c1  = c0 |c0 c1 |c1  − |c0 |c1 |2 , c2 × c0 |c2 × c0  = c0 |c0 c2 |c2  − |c2 |c0 |2 ,

(46) (47)

c1 × c2 |c1 × c2  = c1 |c1 c2 |c2  − |c1 |c2 |2 .

(48)

Proposition 10 For maximally entangled two qutrit states the concurrence is  C = 2 c0 |c0  + c1 |c1  + c2 |c2 .

(49)

Proof The Hermitian inner product determines the Hermitian metrics gi j = ci |c j  = c j |ci  = gi+j .

(50)

By unitary transformation of one qutrit states |c˜i  = Ui j |c j  this metric can be diagonalized, so that all mutual inner products vanish c˜0 |c˜1  = c˜0 |c˜2  = c˜1 |c˜2  = 0, and the metric becomes g˜ i j = diag(c˜0 |c˜0 , c˜1 |c˜1 , c˜2 |c˜2 ). Then, the concurrence (44), due to (46)-(48), reduces to the form (49).  The norms of states for diagonal metric are constrained only by normalization condition

Maximally Entangled Two-Qutrit Quantum Information States and De Gua’s …

tr g = c˜0 |c˜0  + c˜1 |c˜1  + c˜2 |c˜2  = 1.

103

(51)

By denoting these norms as c˜0 |c˜0  = r02 , c˜1 |c˜1  = r12 , c˜2 |c˜2  = r22 , the problem is reduced to the one we have studied above for the two retrit states. Thus, we have the following result. Proposition 11 The concurrence for maximally entangled two-qutrit state is 2 C=√ . 3

(52)

Example As an example we consider two-qutrit state 1 |ψ = √ (|00 + |11 + |22). 3

(53)

√ It is maximally entangled state with concurrence C = 2/ 3. Due to real coefficients, this state is the retrit state and De Gua’s theorem (37) is given by relation 1 1 1 1 = + + , 12 36 36 36

(54)

√ for corresponding trirectangular tetrahedron, with three equal sides 1/ 3.

6 Conclusions The geometrical characterization of entangled states by concurrence and area, allowed us to establish intriguing link between the maximally entangled quantum states in quantum information theory and three-dimensional extension of Pythagoras theorem. Here we note that “Pythagoras theorem” for entanglement (18) can be generalized to an arbitrary two qudit states. The generic two qudit state is determined by d × d complex matrix from coefficients ci j , (i, j = 0, 1, . . . , d − 1). Calculations for the reduced density matrx give following equation 1 − tr

ρ 2A

d−1  d−1



ci j ci j  2



=2

ci  j ci  j  ,  

(55)

i n 0 , the inequality given below is true for any vector norm ..     x n+1 − x ∗  ≤ c x n − x ∗ ρ

(1)

Therefore, the iterative scheme is known to converge to x ∗ with ρth convergence order.

Computational Order of Convergence Definition 2 Provided x ∗ is a root of the equation F(x) = 0 and let x n−1 , x n as well as x n+1 be consecutive iterates closer to the root x ∗ , which is generated by an iterative scheme. Then, the Computational Order of Convergence (C OC)ρ of the numerical algorithm or the iterative scheme can be approximated by x − x ∗  ln xn+1− x ∗  n ρ ≈ x n − x ∗  ln x − x ∗  n−1

(2)

Newton and Secant Methods Newton’s Method Newton iterative method for solving nonlinear equations is given as follows xn+1 = xn −

f (xn ) f  (xn )

(3)

where n = 0, 1, 2, ... Secant Method Secant method for solving nonlinear equations is given as follows xn+1 = xn − where n = 1, 2, ...

f (xn )(xn − xn−1 ) f (xn ) − f (xn−1 )

(4)

Derivative-Free Finite-Difference Homeier Method for Nonlinear Models

107

1.2 Numerical Schemes Homeier Method to Solve Nonlinear Equations in One Variable In order to solve nonlinear equations involving real variables, Homeier developed a third-order convergent as written below xn+1

f (xn ) = xn − 2



1 1 +  f  (xn ) f (yn )

 (5)

where yn = xn − ff (x(xnn)) and n = 0, 1, 2, ... In the Homeier formula, replace the variable x with the complex variable z on both sides; then, the following is the complex form of the Homeier formula. z n+1

f (z n ) = zn − 2



1 1 +  ∗  f  (z n ) f z n+1

 (6)

∗ where z n+1 = z n − ff (z(znn)) and n = 0, 1, 2, ... The Homeier Method to Solve Systems of Nonlinear Equations When F represents a vector-valued function having non-zero derivatives defined on the set D⊂ Rn and x, x0 ∈D, the Homeier method’s extension to resolve nonlinear equation’s systems can be expressed below

x n+1 =x n − where

      1 −1  J F (x n ) +J −1 F x λn+1 F (x n ) 2

  −1   x λn+1 =x n − J F (x n ) F (x n )

(7)

(8)

Here, n = 0, 1, 2, . . . and x n is the nth iterate and J is the Jacobian matrix of F. Secant Method for Systems of Nonlinear Equations When we employ the secant method with respect to nonlinear equations’ systems, we face a special problem with the Jacobian. In the secant method for systems, a vector is present in the denominator. To overcome the problem of taking the inverse of a vector, we follow the most popular secant approximation suggested by C. Broyden. The algorithm is analogous to Newton’s method. However, it replaces the analytic Jacobian with the following approximation. This method is called Broyden’s method Atkinson [7]. ALGORITHM Broyden’s method. Given F Rn → Rn , x 0 ∈ Rn , A0 ∈ Rn×n sk —Initial Step, yk —Yield of current Step, x k+1 —Next iteration FOR k = 0 to a sk = xk+1 − xk yk = F(x k+1 )−F(x k ) for Ak sk = yk

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Ak s k = −F(x k ) x k+1 = x k + s k y k = F(x k+1 )−F(x k ) k sk t Ak+1 = Ak + y k −A sk s tk s k END FOR Here, the final step is used to replace the analytic Jacobian with a matrix.

2 Methodology 2.1 Derivation of DFH from Homeier Method Here we use both backward and forward difference approximations for the derivatives in the same formula appropriately to get an acceptable result. Consider Homeier Method   1 1 f (xn ) +  (9) xn+1 = xn − 2 f  (xn ) f (yn ) where yn = xn − ff (x(xnn)) and n = 0, 1, 2, ... Substituting the forward difference approximation for f  (xn ) ≈

f (xn+1 ) − f (xn ) xn+1 − xn

(10)

and the backward difference approximation for f  (xn+1 ) ≈

f (xn+1 ) − f (xn ) xn+1 − xn

(11)

From equations (9)–(11), we get xn+1

f (xn ) = xn − 2 xn+1



1 f (xn+1 ) − f (xn ) xn+1 − xn

f (xn ) = xn − 2



+

1 f (xn+1 ) − f (xn ) xn+1 − xn

2(xn+1 − xn ) f (xn+1 ) − f (xn )

 (12)

 (13)

Thus, the new iterative formula without derivative terms is xn+1 = xn −

f (xn ) (xn+1 − xn ) f (xn+1 ) − f (xn )

(14)

Derivative-Free Finite-Difference Homeier Method for Nonlinear Models

109

However, it is an implicit method requiring xn+1 term at the (n + 1)th iterative step to determine the (n + 1)th iterate itself. Here, the secant method can be used to replace the term xn+1 on the RHS of the above equation to overcome this difficulty. Thus, the Derivative-Free Homeier method (DFH) is obtained as xn+1

  ∗ − xn f (xn ) xn+1  ∗  = xn − − f (xn ) f xn+1

where ∗ = xn − xn+1

f (xn )(xn − xn−1 ) f (xn ) − f (xn−1 )

(15)

(16)

for n = 1, 2, ...

2.2 Application of DFH for Nonlinear Equations with Complex Roots In the Finite-Difference Homeier formula, we replace the variable x with the complex variable z on both sides to get the following form: z n+1 = z n −

∗ − zn ) f (z n )(z n+1 ∗ f (z n+1 ) − f (z n )

(17)

∗ z n+1 = zn −

f (z n )(z n − z n−1 ) f (z n ) − f (z n−1 )

(18)

where

for n = 1, 2, ...

2.3 Application of DFH for Systems of Nonlinear Equations Select an initial estimate given by x 0 ∈ Rn as well as a non-singular initial Matrix Here, set k := 0 and then repeat the sequence of steps given below A0 ∈ Rn×n  .  until  F x k  < tolerance  ∗ ∗ 1. Solve Ak I k = F X k+1 − F(X k ) f or I k & X k+1 from Broyden’s method 2. X k+1 = X k + I k 3. Y k = F(X k+1 ) − F(X k ) t k Ik 4. Ak+1 = Ak + Y k −A Ik t Ik Ik

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Table 1 Comparison of DFH with secant method & Newton’s for uni-variate nonlinear equations with real roots Function

x0

x1

i

COC

ST 4cosx + e x 0

1.5

x 2 − ex − 3x + 2

2

1

N

DFH ST

NFE N

DFH ST

Root N

DFH

10

7

4

1.904 1.99

2.395 11

9

9

0.904788

8

6

4

1.257 1.56

2.842

9

9

0.257530

9

Table 2 Comparison of DFH with secant method & Newton’s method for nonlinear equations with complex roots Function 2

z0

ze z −sinz + 3cosz + 1

−2+i

(1 − z)3 + 1 1.5+ 0.5i

z1

i

COC

ST

N

DFH

ST

16

13

7

1.5+i 10

9

5

−1+ 1.5i

NFE N

Root

DFH

ST

N

DFH

1.657 1.991

2.42055

17

15

15

1.1199– 0.7028i

1.664 2.002

2.40747

11

11

11

1.500 + 0.8660i

Table 3 Comparison of DFH with Broyden’s method for systems of nonlinear equations Function

x0

A0 

1

2

x − cosy sinx + 0.5y

(0, -0.5)

15x + y 2 − 4z − 13 x 2 + 10y − z − 11 y 3 − 25z + 22

(3,3,2)

i

1 −0.48

BFS



COC DFH

11

7

15 6 −4 ⎢ ⎥ ⎢ 6 10 −1 ⎥ 14 ⎣ ⎦ 0 −18 −25

12

⎡

1

0.5

Root

BFS

DFH

1.78364

2.2331

⎤

1.722963 2.94914

0.53038868– 1.01173734 1.03640452 1.08570343 0.93119446

3 Results and Discussion Results are depicted in Tables 1, 2, and 3. Tables below show the results of the comparison between Newton’s method with Secant, Derivative-Free Homeier and Broyden methods. Where: DFH: Derivative-Free Homeier Method ST: Secant method N: Newton’s method BFS: Broyden’s method for Secant method i: Number of iterations to approximate the root COC: Computational Order of Convergence NFE: Number of Function Evaluations The order of convergence for DFH is near 2.4. It is higher than the secant method and Newton method. Then we applied the method for complex roots. The method was applied to nonlinear equations for complex roots and gave the same satisfactory

Derivative-Free Finite-Difference Homeier Method for Nonlinear Models

111

results. Then the method was extended to nonlinear equations systems. If we implemented the method in nonlinear equations systems, there were some difficulties. In Improved Newton’s method, it was necessary to obtain Jacobian matrices to resolve nonlinear equations systems. However, for this resulting method, the Jacobian matrix becomes a vector. Thus, the challenge was to find the inverse of the vector. So Broyden’s method was used to overcome that difficulty. After following the technique used in Broyden’s method resulting formula for nonlinear equations systems with two and three variables, the results were again encouraging, just like for the one variable case. So, the objective was achieved there as well. However, when we follow Broyden’s method, we cannot do away with the derivative part.

4 Conclusion For all nonlinear equations we have considered, the new algorithm DFH seems more efficient than any other algorithm without the derivatives to numerically solve them. This method gives a computational order of convergence higher compared to any other existing method in derivatives’ absence. Both nonlinear equations having complex roots and systems of nonlinear equations yield the same order of convergence. Evidently, the secant method featured two function evaluations, while the DFH had three. However, according to the computed results (Table 1 as well as Table 2), in the majority of the results, the total number of function evaluations needed is less than or equal to that secant method. Thus, DFH can be considered a superior method in giving faster convergence to find the roots of nonlinear equations with no existence of the derivative for uni-variate nonlinear equations with complex roots as well as for the multivariate systems of nonlinear equations. Since the proposed method is even faster than the universally accepted second order Newton’s method while meeting the requirement of not having the derivative of the function as well, this algorithm will undoubtedly be extremely useful to the scientific and the industrial community.

5 Future Work We will try to apply our DFH method into nonlinear boundary value problem after converting it into system of nonlinear equation.

References 1. Burden R.L. & Faires J.D. Numerical Analysis, 8th Edition, Bob Pirtle, USA, 2005. 2. Frontini M., Sormani E., Modified Newton’s method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156: 345–354, 2003.

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3. Frontini M., Sormani E., Some variant of Newton’s method with third-order convergence, J. Comput. Appl. Math. 140: 419–426, 2003. 4. Homeier H.H.H., On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176: 425–432, 2005. 5. Homeier H.H.H., A modified Newton method for root finding with cubic convergence, J. Comput. Appl. Math. 157: 227–230, 2003. 6. Homeier H.H.H., A modified Newton method with cubic convergence: the multivariate case, J. Comput. Appl. Math. 169: 161–169, 2003. 7. Heenatigala S.L., Weerakoon S., Fernando T.G.I., Finite Difference Weerakoon-Fernando Method to solve nonlinear equations without using derivatives, University of Sri Jayewardenepura, Gangodawila, Nugegoda, Sri Lanka, 2021. 8. Nishani H. P. S., Weerakoon S., Fernando T.G.I. Liyanage M. Third order convergence of Improved Newton’s method for systems of nonlinear equations, 502/E1, Proceedings of the annual sessions of Sri Lanka association for the Advancement of Science, Sri Lanka, 2014. 9. Said Solaiman O., Abdul Karim S.A., Hashim I., Dynamical comparison of several third-order iterative methods for nonlinear equations, Computers, Materials & Continua: 1951–1962, 2021. 10. Said Solaiman O., Hashim I., Optimal eighth-order solver for nonlinear equations with applications in chemical engineering, Intelligent Automation & Soft Computing, 27:379–390, 2021. 11. Said Solaiman O., Hashim I., An iterative scheme of arbitrary odd order and its basins of attraction for nonlinear systems, Computers, Materials & Continua, 66: 1427–1444, 2021. 12. Weerakoon S., Fernando T.G.I., A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett.13: 87–93, 2000.

The Effect (Impact) of Project-Based Learning Through Augmented Reality on Higher Math Classes Cristina M. R. Caridade

1 Introduction How much math does an Engineer have to learn? Mathematics is essential in the training of engineers [1]. In engineering, a student’s poor mastery of mathematics can affect his or her success in other course subjects that require mathematical skills. In this sense, great support is needed for engineering students in their early years, since that is where the mathematical basis of engineers is built. For this reason the mathematics teacher must invest in an innovative teaching adapted to these types of students who like to learn by doing, solving problems and challenges [2]. On the other hand, the lack of motivation that engineering students feel in relation to this discipline influences their poor academic performance. Motivating and keeping engineering students motivated in mathematical disciplines is a challenge for the teacher. According to some authors, the use of a computational tool to model a real problem [3], Gamification and game-based learning [4], early and integrative exposure to technical engineering areas [5–7] and project-based learning [8] are some of the methodologies that improve students’ interest and, consequently, facilitate the teaching-learning process. A recent study [9] brings together important research on innovative teaching and learning practices in engineering mathematics with the aim of a deeper understanding of the existing characteristics in current teaching and learning practices that can help in the implementation of future innovative practices. It was found that practices in mathematics learning lag behind the changes that are taking place in engineering education, and that innovative changes are needed within the current system of teaching mathematics for engineering [9]. It is necessary that the teaching of Mathematics be more creative and stimulating, considering modern society and the interests of students [10]. Project-based learning (PBL) can help increase engagement, improve student interaction, and promote C. M. R. Caridade (B) Coimbra Institute of Engineering, Polytechnic of Coimbra, Coimbra, Portugal Centre For Research in Geo-Space Science (CICGE), Porto, Portugal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_12

113

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C. M. R. Caridade

student success in math. In the process of innovating teaching and learning methods, the use of PBL has become popular in teaching and learning in universities in many countries around the world. PBL helps students have interesting hands-on experiences while developing collaborative skills, problem solving and self-control in studying [11]. In this paper, PBL and its application as a teaching methodology are presented to qualitatively assess its impact on student motivation and performance. The rest of this paper is structured as follows. In the second chapter, the study developed is described. The results achieved are presented and discussed in Chap. 3, with some examples made by the students, and the conclusions and future work are presented in Chap. 4.

2 Methodology In the second semester of 2021/2022, the Calculus 1 course is repeated for all students who did not pass the course in the 1st semester. With the objective of allowing a continuity in the learning of this discipline and that a greater number of students can succeed in the discipline in the academic year. 25 Mechanical Engineering students enrolled in the 2nd semester of the course, where learning planning is mostly practical, as the students have already acquired the theoretical concepts in the 1st semester. In this more practical discipline, with 5 h per week, it is possible to develop a greater number of practical activities using different teaching methodologies. Like all engineering students, Mechanical Engineering students are active people, who learn by doing, who enjoy solving problems and challenges, so why not develop PBL activities in practical math classes. Thus, a 3h practical class with a group PBL activity was planned. The activity was based on the following 4 main tasks: • Define a classroom object that can be considered a solid of revolution. Photograph it and take its real measurements. • Using a planar region defined in the photograph, dynamically model the 3D object in GeoGebra. • Perform calculations of planar areas and volumes of solids of revolution, important contents of the Calculus 1 course. • Insert the 3D object modeled in GeoGebra, in augmented reality. The students performed the PBL activity in groups and submitted their resolutions in the discipline’s Moodle. At the end, they answered an online questionnaire about interest in the developed activity.

The Effect (Impact) of Project-Based Learning Through Augmented Reality …

115

3 Results and Discussions The PBL activity was presented to the students in a Microsoft word file, with blank spaces, where it was possible for the students to introduce the various stages of solving the activity. The images shown are examples of the resolution performed by the students and are in Portuguese. The activity was broken down into several tasks that are described below and accompanied by images. First Task: Have you ever thought that around us we find many objects that can be considered solids of revolution. Do you know what they are? Do you know how to recognize them? In this activity you will identify and create solids of revolution in your everyday life and also calculate the volume of these solids. To start the activity, you must find a real object that can be considered a solid of revolution and photograph it.

Figure 1 shows 2 illustrative examples of this first task. The students chose some objects they had with them in the classroom, such as a small ball (left) and a bottle of water (right). Second Task: You must open GeoGebra on your computer, mobile phone or tablet. Insert the photograph and position it so that the self-reflection axis coincides with the OX or OY axis. Don’t forget to take measurements on the real object to set the scale.

The example shown in Fig. 2 is a pencil case in which the measurements identified in GeoGebra are half of the real measurements, hence the scale is 1:2. Third Task: Then you must mark the symmetry axis and a set of points on the part of the image contour that is reflection invariant with respect to the symmetry axis. Identifies the interpolating polynomial(s) that allow defining this contour and obtaining the reflection of the curve.

Fig. 1 Examples of the first task in PBL activity: choose the object

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Fig. 2 Example of the second task in PBL activity: measurements and scale

Fig. 3 Examples of the third task in PBL activity: definition of the planar region

The photographs are dragged in GeoGebra, until they coincide with the axes of symmetry, and the object is positioned so that its region is symmetrical about one of the axes. In Fig. 3 the objects are symmetrical about the O X axis. To define the curves that limit the region, it is necessary to identify a set of points on the limits of the region and calculate an interpolating polynomial that passes through these points. The polynomial is calculated by the students (upper part of the images in Fig. 3). Sometimes the position of the points needs to be adjusted to get a good interpolating polynomial. Fourth Task: Let’s calculate the area of the planar region.

With the planar region defined by the interpolating polynomial and the axes, the area of this region was calculated through integral calculus, using expressions similar to those presented in Eq. 1. xn Pn (x) − 0 d x

(1)

x0

where, x0 is the x-value of the leftmost point, xn is the x-value of the rightmost point, and Pn (x) is the interpolation polynomial.

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Fig. 4 Example of the fourth task in PBL activity: planar area calculation

Fig. 5 Example of the fifth task in PBL activity: 3D object modeling

In Fig. 4 an example of calculating the planar area is shown. This calculation is done by hand by the students, with the help of the calculator. Fifth Task: Now activate the GeoGebra 3D graphics sheet and create the three-dimensional object.

The 3D object is created in GeoGebra starting from the planar region defined in 2D, and rotating this region dynamically around an axis. The example of the water bottle is shown in Fig. 5 through 5 images corresponding to five rotation angles (45◦ , 90◦ , 180◦ , 270◦ and 360◦ ). In the last image the bottle is fully modeled in a 3D object. Other examples can be seen in Fig. 6.

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Fig. 6 Examples of the fifth task in PBL activity: 3D object modeling

Fig. 7 Example of the sixth task in PBL activity: volume of the solid of revolution

Sixth Task: You just built your revolution solid. Does it look like the real object? Let’s calculate the volume of the solid.

Knowledge of integral calculus can now be applied to determine the volume of the solid of revolution, according to an expression similar to the Eq. 2. xn π×

(Pn (x))2 − (0)2 d x

(2)

x0

where, x0 is the x-value of the leftmost point, xn is the x-value of the rightmost point, and Pn (x) is the interpolation polynomial. Figure 7 represents one of the calculations presented by the students. Seventh Task: You may still be able to visualize your solid of revolution built in GeoGebra in a real context, being able to explore the solid from different angles as you move freely with your device’s camera. We will try? You must use GeoGebra AR for iOS operating system or GeoGebra 3D for Android (free applications). https://www.youtube.com/watch?v=acPMhsiwNKQ. Finally, record a video of your augmented reality, and submit this activity in Moodle.

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Fig. 8 Example of the seventh task in PBL activity: augmented reality video

Fig. 9 Example of the seventh task in PBL activity: augmented reality video

This task, and the most motivating one for the students, was carried out using GeoGebra applications for mobile devices. At this moment the classroom becomes a virtual environment where the 3D objects created by the students were integrated and it was possible to manipulate them. In Figs. 8 and 9 two examples of this task submitted by the students in the activity can be observed. The water bottle (Fig. 8) and the small ball (Fig. 9). To demonstrate that they were videos, 3 prints of each of the videos were collected.

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4 Conclusions and Future Work The teacher must make the students feel comfortable and important. A classroom should be open, positive, and receptive to discussion and disagreement. PBL promotes intrinsic motivation and plays a role in developing critical thinking skills when students are required to explain and teach one another. In addition, students develop a sense of community and commitment when they work together. Through direct observation and analysis of the tasks performed, the teacher verifies that students are engaged, motivated and excited to learn. In reality, students bring to the classroom a variety of motivations and a wide range of demands for attention, commitment, and time. They bring the belief that the teacher is responsible for tapping into this natural desire, providing a classroom environment that fosters a motivation to learn and an excitement that continues from the first day of the semester to the last. From the teacher’s perspective the effect of PBL lessons should be observed in two categories: knowledge and skills. Regarding knowledge, an analysis and correction of the tasks performed by the students was carried out. With a PBL math class in higher education, learning is more concise and rich with more creative solutions and therefore more motivating for students. In relation to the skills acquired by the students, the direct observation of the activities in the classroom, allowed the teacher, and in a qualitative way, to verify the great involvement of students and their willingness to learn in order to carry out the proposed tasks. This reinforces that PBL is a space to arouse students’ interest in Mathematics. From the student’s point of view, and based on the questionnaires carried out after the activity, it appears that students feel involved and would like the remaining classes (not just the experimental classes) to be more interactive, providing greater group work, more fun classes, and thus improving the means of learning. To measure students’ satisfaction, some questions were included in the questionnaire about the satisfaction of the organization of the PBL class, of learning and interaction with their previous knowledge, of the group’s productivity, and of the quality of the tasks requested. In total, the questionnaire consisted of 9 questions. Six “Yes”, “No” or “I don’t know” questions and 3 open-ended questions where students could describe their ideas and personal comments. Of the 25 students surveyed, all of them (100%) indicated that their contribution was important to the group, they were focused on carrying out the tasks and confident in their learning (Fig. 10). They also added, all 25, that they liked to use innovative methods in the classroom such as PBL and that with this type of activities their knowledge is more effective. As for their opinion about the activity, of the 19 students who responded, 4 indicated group work and interaction between colleagues; 12 the 3D object modeling in Geogebra and inclusion in augmented reality and 3 the dynamics of the class and interactivity. Regarding the biggest challenges they encountered in the activity, the 16 students who responded refer to the use of the technological application itself, not indicating any reference to the activity, nor to the PBL methodology. In the ideas and suggestions, only 4 students responded and indicated the realization of more

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Fig. 10 Quiz, questions (1) to (6)

classes of this type. Some quotes from the students that can demonstrate it: “The team spirit of our group.”; “The use of new technologies for modeling objects of our daily life.” Learning new tools like GeoGebra.”; “Continuing to carry out work like this.”; “More activities of this type are good because they help me to improve my performance.” In conclusion PBL proved to be effective and productive to motivate students in learning Mathematics. The higher education system does not seem to be able to prepare students well for the job market. One way to do this is by applying more modern educational approaches such as PBL [12, 13]. PBL allows carrying out activities with students where various areas of knowledge are involved, such as Science, Technology, Engineering and Mathematics, which allows students to prepare for their future insertion in the job market. A connection between mathematics content and PBL projects is possible, so the use of this teaching approach in the classroom will enrich student learning, adding other skills in addition to math skills, such as interpersonal skills related to the process of group. As future work, it is planned to carry out a more exhaustive and detailed study, with students from different engineering areas using the same or another PBL experience.

References 1. André, J.: Ensinar e Estudar Matemática em Engenharia, Imprensa da Universidade de Coimbra (2008). ISBN: 9789898074379. 2. Pinto, C.M.A., Mendonça, J.: DriVE-MATH: Reimagining Education, Open Education Studies, vol. 4, pp. 21–34 (2022). 3. Brandi, A.C., Garcia, R.E.: Motivating engineering students to math classes: Practical experience teaching ordinary differential equations, IEEE Frontiers in Education Conference (FIE), pp. 1–7 (2017) https://doi.org/10.1109/FIE.2017.8190489.

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4. Zabala-Vargas, S.A., García-Mora, L.H., Arciniegas-Hernandez, E., Reina-Medrano, J.I., de Benito-Crosetti, B., Darder-Mésquida, A.:. Strengthening Motivation in the Mathematical Engineering Teaching Processes - A Proposal from Gamification and Game-Based Learning. International Journal of Emerging Technologies in Learning (iJET), 16(06), pp. 4–19 (2021). https://doi.org/10.3991/ijet.v16i06.16163. 5. Foley,J.M., Daly, S., Lenaway, C., Phillips, J.: Investigating Student Motivation and Performance in Electrical Engineering and Its Subdisciplines, IEEE Transaction on Education, pp. 1–7 (2015). 6. González-Martín, A.S., Hernandes-Gomes, G.: Applications in Calculus for enineering. The cases of five teachers with different backgrouns, Inprasitha, M., Changsri, N., Boonsena, N. (Eds.). Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 353–360 (2021). 7. Caridade, C.M.R.: GeoGebra augmented reality: ideas for teaching & learning math. II Internacional Conference on Mathematics and its applications in science and Engineering (ICMASE 2021) 01–12 July 2021, Universidad de Salamanca. 8. Lee, H. , Blanchard, M. R.: Why Teach With PBL? Motivational Factors Underlying Middle and High School Teachers’ Use of Problem-Based Learning. Interdisciplinary Journal of ProblemBased Learning, 13(1) (2019). https://doi.org/10.7771/1541-5015.1719. 9. Pepin, B., Biehler, R., Gueudet, G.: Mathematics in Engineering Education: a Review of the Recent Literature with a View towards Innovative Practices. Int. J. Res. Undergrad. Math. Ed. 7, 163–188 (2021). https://doi.org/10.1007/s40753-021-00139-8. 10. Akhadova, k.S.: Problems of developing Mathemayical competencies of future engineers, Academic research in educational sciences, 3(3), pp. 316–322 (2022). 11. Cuong, T.V., Tuan, N.V.: Project-based Learning Method in Advanced Mathematics for EngineeringStudents in Vietnam: Experimental Research, Universal Journal of Educational Research, Vol. 9, No. 3, pp. 528–539 (2021). https://doi.org/10.13189/ujer.2021.090312. 12. Sattarova, U., Groot, W.; Arsenijevic, J.: Student and Tutor Satisfaction with Problem-Based Learning in Azerbaijan. Educ. Sci., 11, 288 (2021) https://doi.org/10.3390/educsci11060288. 13. Michael Peterson, M: Skills to Enhance Problem-based Learning, Medical Education Online, 2:1, 4289 (1997). https://doi.org/10.3402/meo.v2i.4289.

Performance of Machine Learning Methods Using Tweets ˙ Ilkay Tu˘g and Betül Kan-Kilinç

1 Introduction Data mining, which is used in many different areas, including twitter data, aims to identify patterns and establish relationships to solve problems through data analysis. It includes master data mining, machine learning, classifying parameters, clustering and prediction. Classification refers to the process of finding and predicting a model that identifies and distinguishes classes of data. Hence, data analysis performs the process of extracting information from the data and data structure. The algorithms such as Logistic Regression, Naive Bayes, Support Vector Machines, K-Nearest Neighbor, Decision Trees, Artificial Neural Network, Gradient Boosting Machines and Extreme Gradient Boosting Machines are widely used to solve classification and regression problems. On the other hand, deep learning is a sub section of machine learning algorithms that aims to solve problems such as speech and image recognition. The deep learning model is designed to analyze data with a logic structure similar to the functioning of the human brain, consisting of layers. Deep learning uses Artificial Neural Network algorithm to accomplish such structure [1]. There are many methods in the literature to detect various diseases at an early stage [2, 3]. The COVID-19 virus was first identified in December 2019 in Wuhan, China and continued to spread rapidly around the world by negatively impacting infected individuals, the health systems, and the global economy. During the COVID-19 epidemic, researches on how the disease spreads has been accelerated, and this has paved the way for new research areas [4]. In the literature, there are many studies that use machine learning algorithms especially related to COVID-19 virus [5–8]. In the case of an COVID-19 epidemic, the forecasting modeling is an important element in order to control the epidemic and prevent its spread. Government and public health organizations can prepare themselves according to the forecasts and ˙I. Tu˘g · B. Kan-Kilinç (B) Department of Statistics, Faculty of Science, Eskisehir Technical University, 26470 Eskisehir, Turkey e-mail: [email protected] ˙I. Tu˘g e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_13

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accordingly it becomes easier to take precautions. There are many estimation algorithms for different data types in the literature. In some of them, machine learning models used to detect and characterize user-generated conversations that talk about COVID-19 related symptoms, experiences with accessing tests, and disease recovery using an unsupervised machine learning approach [3, 9, 10]. At the same time, this epidemic, which is closely related to the public, has been frequently discussed on social media. In this period, people also used Twitter, which is one of the social media platforms, to socialize. On Twitter, Hashtag is a tool used to apply a topic filter to the main topic. In addition to their feelings and thoughts, users shared this situation with #COVID-19 in the COVID-19 epidemic, those whose COVID-19 test results were positive or negative. This platform is very convenient for extracting information from data, specifically for large data sets [5]. In this study, the performance of classification algorithms based on different machine learning methods to predict individuals with COVID-19(+) or COVID-19(−) using the emotions among the tweets obtained from Twitter by text mining procedures are compared. LR, NB, SVM, KNN, DT, RF, GBM, XGBM and ANN algorithms were used to extract the accuracy of model performance of each model for the detection and identification of the disease related to the COVID-19 virus, which has been on the agenda recently.

2 Methodology 2.1 Social Media Analysis By using the development of the internet and technology, social networks have expanded and the number of users has increased over time. Social Networks can be defined as the use of social media to connect with known or sometimes unknown acquaintances. Hence people can communicate with their relatives or colleagues by using social network platforms. In addition, it contributes to the big data structure for instant sharing. Twitter, Facebook and Instagram etc. social media applications have become convenient for data analysis. Research in this area ranges from relational statistics to data collected to build models. Instant posts are suitable and convenient for analysis [11]. In this study, Twitter was used to collect the data for analysis.

2.2 Text Mining Text mining is a variation of data mining that tries to find interesting patterns from large databases. Data mining and text mining are similar concepts, but the main difference is that data mining usually deals with structured data whereas text mining works with unstructured or semi-structured data. After social media has a huge data potential, text mining has become the focus of attention. Its main purpose is to obtain

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high-quality information from the processing of large volumes of textual data. In general, text mining and data mining are considered to be similar to each other with the perception that the same techniques can be used in both concepts to do text mining [12–16]. However, both differ in that data mining involves structured data, text deals with specific features and is relatively unstructured and often requires preprocessing. Also, text mining is a field associated with Natural Language Processing [17].

2.3 Machine Learning Machine learning is a subset of techniques which uses statistical methods to improve regression or classification problems. A common task seen in different machine learning applications is to create a nonparametric regression or classification model from the data. Using statistical methods, it is trained to make classifications or predictions and is used in data mining projects. Classification models categorize input data. It predicts different responses, such as whether an email is genuine or spam, or whether a tumor is cancerous. Regression models are used to predict a continuous value. For instance: the size of the house, the price, etc. Learning algorithms are used to discover information and properties of data [18]. The data used consist of two different types, labeled or unlabeled. Labeled data set is used to train an algorithm, and unlabeled data is used to test the trained algorithm (model or system). For this reason, these data is split as training and test sets. The purpose or the problem to be solved lead to machine learning algorithms divided into supervised learning, semi-supervised learning, and unsupervised learning. Supervised machine learning uses classification and regression techniques to develop predictive models. It is the provision of learning by training the system using labeled data. While the system is being trained, inputs and outputs for each sample in the data set are given. In text classification studies, the input represents the content of the text and the output represents its category. The test data set is used to verify the system. In the validation phase of the system, after the model learns with the training set, it assigns any of the outputs in the training data to an unknown test data [19]. Supervised learning methods can be listed as k-Nearest Neighbors, Linear Regression, Logistic Regression, Support Vector Machines, Decision Trees and Random Forest, Artificial Neural Networks, and Naive Bayes. In unsupervised machine learning, the model is trained using unlabeled data while it is being trained. It reveals patterns and relationships in the dataset. Generally, it deals with finding a structure or pattern from unlabeled data and reveals clusters, if any. The purpose of unsupervised learning is not regression or classification because the outputs of the samples in the data set are not known. It is generally used for purposes such as clustering, probability density estimation, finding relationships between features, and dimension reduction. Unsupervised learning methods can be listed as k-Means Clustering, Hierarchical Clustering, Dimensional Reduction, Principal Component Analysis, and Singular Value Decomposition. Semi-Supervised Learning can use both labeled and unlabeled data. It is included in both model families [20].

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Logistic Regression. Logistic regression is a predictive technique used in classification problems where the dependent variable is qualitative (usually binary or multiclassfied as in multiclass logistic regression if the variable has more than 2 classes). The independent variable can be qualitative (in which case coding is required) or quantitative. Its aim is to create a model that allows prediction/explanation [21]. Naive Bayesian. It is a probabilistic classifier used in estimation problems. It is known as a powerful and simple classifier. It consists of two types of probabilities that can be calculated as a set of probabilities by counting the frequency and combinations of values in a training set. It shows the relationship between conditional and marginal probabilities. Naive Bayes classifier is based on Bayes Theorem. Each attribute must be independent. The classifier calculates the probability of each state [1]. Support Vector Machines. Support vector machines (SVM) are mostly used to separate binary classified data. It is a supervised machine learning algorithm that learns to assign tags to objects by example. It can efficiently manage a range of data and high-dimensional problems. It is a machine learning technique also used for time series data for prediction in regression and classification problems. It has excellent generalization ability and is suitable even for small data. SVM uses cores that convert data from low size to high size for easy classification. Planes that can divide classes into a higher dimension are called hyperplanes. In the SVM classifier, specifying parameters is an important step, but there is no specific way to specify it. According to the problem, four kernel functions can be used; linear, radial basis, polynomial and sigmoid. If there is a linearly separable distribution, a linear kernel function is used or a polynomial for a polynomially separable distribution, a radial for a circularly separable distribution and a sigmoid for a special distribution can be other choices. [6]. Artificial Neural Network. Artificial neural networks have been developed by being influenced by the biological nervous system. ANNs are mathematical systems consisting of nodes or neurons that are connected to each other in a weighted manner. Each neuron receives input from “upstream” neurons and outputs to “downstream” neurons. The behavior of ANNs, that is, how they relate input data to output data, is primarily affected by the transfer functions of neurons, how they are connected to each other, and the weights of these connections. It is used in classification, pattern recognition, signal filtering, optimization studies and data compression [22]. Decision Trees. Decision trees are based on the use of a decision tree as a predictive model. It consists of roots, branches and leaves. The decision tree algorithm divides the data set into small or even smaller groups. A decision node can contain one or more branches. A decision tree can consist of both categorical and numerical data. The concepts of decision trees and random forests are similar. A decision tree is effectively created over the entire dataset to produce a tree. A random forest combines many decision trees into a single model [21]. Random Forest. A popular learning Random Forest algorithm is actually an ensemble model of a decision tree. It is mostly used in classification and regression tasks as it helps to solve the over-learning problem and handle missing data values well. The random forest classifier consists of a combination of randomly selected features or attributes at each node to grow a tree. It builds many independent models. Any

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observation is assigned to the most voted class in all tree models in the forest [23]. It aims to improve the performance metric by using multiple trees. Additionally, it recognizes it as an effective technique for predicting missing data and maintaining accuracy [7]. Gradient Boosting. The gradient boosting learning procedure builds new models repeatedly to more accurately predict the response variable [24]. It is used for both regression and classification problems. Unlike bagging, it is a community in which predictions are made sequentially, not independently. There are many versions of the Boosting algorithm, including variants of the classic AdaBoost. The purpose of any created learning algorithm is to define and minimize a loss function [25]. Extreme Gradient Boosting. It is the optimized version of the Gradient Boosting algorithm with various arrangements. It includes model solver and tree learning algorithm. It is a suitable classifier for classification and regression models [8]. In this way, it helps to prevent overfitting by controlling the model complexity by facilitating the learning model. GBM and XGBM use community based weak learners, weak learners are supported by gradient descent method. It aims to correct these errors in the next basic model by giving more weight to the incorrectly predicted data in the previous model [26].

2.4 Performance Metrics Different performance metrics are used to evaluate different machine learning algorithms. The performance metric should be determined depending on the data set and the algorithm. Commonly used metrics for classification problems are accuracy, precision, recall, and ROC curves [27]. One of the most informative ways to evaluate the performance of classifiers is based on confusion matrix analysis as given in Table 1. Accuracy is one of the measures used when it is desired to obtain a single measure by combining sensitivity and specificity. It is the ratio of correct guesses to the total number of guesses. In fact, the ratio of the test’s positives to the total positives is called “accuracy” [28, 29].

Table 1 Confusion matrix Actual Prediction

True Positive(TP) False Negative(FN)

False Positive(FP) True Negative(TN)

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3 Data Collection Twitter can be used to simultaneously study attitudes, trend tracking, and public health. The use of social media data enables the researcher to collect reports on what people experience in social life, about demand and unprompted behavior and views. At the same time, social media offers the opportunity to detect personal expression and human interaction in real time. Because of these aspects, Twitter is widely used for data exploration [30, 31]. In this study, COVID-19 hashtags are detected in order to retrieve tweets using Twitter API. The tweets containing the expressions “#COVID-19”, “#COVID-19 & got & positive” and “#COVID-19 & got & negative” were recorded and a total of 750 tweets in English were analyzed.

3.1 Data Cleaning and Preprocessing During the cleaning of Twitter data, tweets belonging to the same users and tweets that did not contain individual information about COVID-19 belonging to followed publications such as news, TV and magazine were cleaned. Corpus object was created by using the “tm” library in the R program [32]. Unnecessary spaces, url information, numbers, special characters, emojis in sentences were removed by following the text mining steps. The meaningless words and stopwords have been removed.

3.2 Sentiment Analysis and Classification Sentiment analysis or opinion mining plays a role in extracting positive, negative, neutral expressions or opinions from the text. In general terms, the prominent emotional intensity of the text can be analyzed. In this study, the words with positive and negative meanings in Tweets were classified using “tidytext” library in R Studio [33, 34]. During the determination of the model variables, three different variables as X 1 , X 2 and y were created. The variable X 1 represents the frequency of positive words in the text and X 2 represents the frequency of negative words. The variable y consists of the information whether a person is COVID-19(+) or COVID-19(−). All tweets were manually reviewed to record the values of variable y. If there is any information in the tweets that the person or his/her environment is COVID-19(+) or COVID-19(−), it was coded as y = 1 or y = 0, elsewhere. In other words, if it does not contain any information, it is coded as y = 0. Machine learning models were applied to a total of 294 data to predict the case of COVID-19 using emotional expressions. A library of emotional expressions may have some disadvantages. Accordingly, the words in a dictionary constructed by a

Performance of Machine Learning Methods Using Tweets Table 2 Accuracy value for classifiers Tweet Bing Dictionary y “i’ve got 5 d off because 3 of my friends are positive in COVID-19 really i just stay at home. Is it boring? ... but i’m tired wearing a mask tho” “Got COVID-19 positive...” “...Yesterday, she and her family members got tested positive #COVID-19. Today, we desperately find hospital for her...” “Just got my COVID-19 NEGATIVE, so now pure and clean to travel abroad..." “my daughter got the certificate of COVID-19 negative today so is ready to fly back to her school....we were desperately mission ...god bless her” “My child tested negative for COVID-19.”

129

X1

X2

1

0

3

“Positive”

1

0

1

“Positive”

1

0

2

0

2

0

0

3

1

0

1

0

“Positive” “Boring”

“Tired”

“Desperately” “Desperately”

“Negative”

“Pure” “Bless”

“Ready”

“Negative”

“Desperately” “Negative”

researcher might have some positive meanings although it could be used in a negative sense in the texts. In order to avoid such erroneous situations, researchers should use the source dictionary carefully when classifying. Some typical errors are listed in Table 2.

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As seen in Table 2, although the word “positive” is a positive word in the source dictionary, it was classified as negative based on its meaning in the sentence. Likewise, “negative” is classified as positive in the Bing dictionary, even though it is a negative word [33, 34].

4 Results There are different approaches to split the data into test set and train set. In this study, six different ratios were used to examine the classification success of partitioning. Optimization in partitioning can be done manually by selecting candidate values based on the researcher’s intuition and experience, or automatically with an algorithm that recommends candidates. Hyper-parameters giving the best accuracy for the methods can be obtained by cross-validation method [35–37]. In this study, what the best with/without hyper parameters optimization in accuracy values were summarized using default parameters. As seen in Fig. 1, a total of 294 data was divided into train and test sets with different ratios. In the first step, 85% of all observations were used in the train set and15% used for test set. Next, 80% and 75% of all observations and 20% and 25% of all were trained and tested, respectively. The rest of the following ratios of division were: 70:30%, 65:35% and finally 60:40% of all observations for training and testing were used in the analyses, respectively. The purpose of this study is to classify if a person suffers from COVID-19(+) or COVID-19(−) using tweets. The classification if a person suffers from one of the COVID-19 cases are thought to be covered by the given emotional expressions of the same user’s tweets. Hence, the expressions in the tweets containing emotions were divided into two parts; positive and negative emotions. Next, they were assigned to two different variables such as X 1 and X 2 . Here, the variable X 1 represents the number of positive words whereas the variables X 2 represents the number of negative words used in the user’s tweets. The classification of the COVID-19 case were investigated by eight different machine learning methods. For this purpose, several machine learning algorithms such as Logistic Regression, Decision Trees, Support Vector Machines, Random Forest, Gradient Boosting, Extreme Gradient Boosting and Artificial Neural Network were performed successfully for COVID-19 classification. Although there is no specific rule for splitting train and test data, mosf of the researchers use 70:30% ratio for division data sets. In Table 3, the accuracy values obtained from eight different machine learning methods are summarized with respect to common ratios. When the results were examined, it was seen that the XGBM method produced better results in terms of accuracy than the other methods when the data was randomly split into parts as 85:15% and 80:20% for train and test sets. In particular, XGBM method achieved the highest accuracy value as 0.7931 among other methods when

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Fig. 1 Data division

the partition is 80:20%. As per the obtained results from each splits, the accuracy values for almost all models is close to each at split 75:25%. The results of accuracy of XGBM and NB methods were obtained as 0.7534, while the accuracy values of LR, SVM, DT, RF, ANN, GBM were 0.7671. Among all the values for these models, it can be seen that ANN is better when the test set is between 30:40% of all observations. On the other hand, the accuracy values of RF and DT depend on the split ratio. The performance of the models with better accuracy than others emerged

˙I. Tu˘g and B. Kan-Kilinç

132 Table 3 Tweet classification Classifier Train/Test (%) 85/15 80/20 Logistic regression Support vector machine Naive Bayes Decision trees Random forest Artificial neural networks Gradient boost XGradient boost

75/25

70/30

65/35

60/40

0.6977

0.7414

0.7671

0.7356

0.7353

0.7414

0.6977

0.7414

0.7671

0.7126

0.7353

0.7414

0.6977

0.7241

0.7534

0.7126

0.7059

0.7414

0.7209

0.7414

0.7671

0.7356

0.7353

0.6983

0.7209

0.7414

0.7671

0.7356

0.6765

0.6897

0.7209

0.7586

0.7671

0.7356

0.7451

0.6724

0.7209

0.7586

0.7671

0.7586

0.7451

0.7414

0.7442

0.7931

0.7534

0.7471

0.7451

0.7414

as LR, SVM, NB, GBM and XGBM and worked well for text classification and require a few training examples (0.7414) at split 60:40%. However, LR, SVM and NB were far behind at 69.77% when split is 85–15%.

5 Discussion and Conclusion In this study, a classification problem related to the #COVID-19 sentiments as expressed in tweets by users was examined. The data were obtained by text mining methods and preprocessed due to unstructured forms for the analysis. After examining the tweets expressing the emotions of the users, the number of emotion specifying words were used as potential predictors for #COVID-19 case. Additionally the COVID-19 case was used as a dependent variable taking on binary values such as (±), whereas the predictor variables (X 1 , X 2 ) were created to represent the number of positive emotions and negative emotions, respectively. To predict the COVID-19 case, eight different machine learning models were investigated. Accuracy values of these models were calculated with different split ratios. In this study, different splits of training data used to train each classifier and the performance of the classifier on the test set were investigated. According to the results, the XGBM is the best model that classifies the sentiment analysis of tweets with an accuracy of 79.31% at 80:20 split (often called Pareto principle). However

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depending on the training and testing ratios it was employed, the methodology may change. Although it all depends on the data, a simulation study will have better information in order to generalize the results. Acknowledgement This work was supported by Eski¸sehir Technical University Scientific Projects Commissions under the grade no 22ADP026.

References 1. Alsaffar D., Alfahhad A., Alqhtani B., Alamri, L., Alansari, S., ¸ Alqahtani, N., & Alboaneen, D.A. (2019). Machine and deep learning algorithms for twitter spam detection, Proceedings of the International Conference on Advanced Intelligent Systems and Informatics, pp. 483–491. 2. Chandra, S.K., & Bajpai, M.K. (2019). Mesh free alternate directional implicit method based three dimensional super-diffusive model for benign brain tumor segmentation, Computers & Mathematics with Applications, Vol. 77, No. 12, pp. 3212–322. 3. Singh, K. K., Kumar, S., Dixit, P., & Bajpai, M.K. (2020). Kalman filter based short term prediction model for COVID-19 spread, Applied Intelligence, Vol. 51, pp. 2714–2726. 4. Gill, S.E., dos Santos, C.C., O’Gorman, DB., Carter, D.E., Patterson, E.K., Slessarev, M., Martin, C., Daley, M., Miller, M.R., Cepinskas, G., & Fraser, D.D. (2020). Transcriptional profiling of leukocytes in critically ill COVID-19 patients: implications for interferon response and coagulation, Intensive Care Medicine Experimental, Vol. 8, No. 75. 5. Delizo, J.P.D., Abisado. M.B., & De Los Trinos, M.I.P. (2020). Philippine twitter sentiments during COVID-19 pandemic using multinomial Naïve-Bayes, International Journal of Advanced Trends in Computer Science and Engineering, Vol. 9, No. 1. 6. Saba, T., Abunadi, I., Shahzad, M.N., & Khan, A.R. (2020). Machine learning techniques to detect and forecast the daily total COVID-19 infected and deaths cases under different lockdown types, Microscopy Research and Technique. Vol. 84, No. 17, pp. 1462–1474. 7. Madani, Y., Eritali, M., Boukhalene, & B. (2021). Using artificial intelligence techniques for detecting COVID-19 epidemic fake news in Moroccan tweets, Results in Physics, Vol. 25, 104266. 8. Chen, W., Fu, K., Zuo, J., et al. (2017). Radar emitter classification for large data set based on weighted-xgboost, IET Radar, Sonar & Navigation, Vol. 11, No. 8, pp. 1203–1207. 9. Zhong L., Mu L., Li, J., Wang J., Yin, Z.,& Liu, D. (2020). Early prediction of the 2019 novel coronavirus out break in the mainland China based on simple mathematical model, IEEE Access, Vol. 8, 51761–51769. 10. Lu, H.M., Zeng D., & Chen H. (2010). Prospective infectious disease out break detection using markov switching models, IEEE Transactions on Knowledge and Data Engineering, Vol. 22, No. 4, pp. 565–577. 11. Elmar, G. (2015). Going Public on Social Media, Social Media + Society, pp. 1–2. 12. Gupta, V., & Lehal, G.S. (2009). A survey of text mining techniques and applications, Journal of Emerging Technologies in Web Intelligence, Vol. 1, No. 1, pp. 60–76. 13. Akilan, A. (2015). Text mining: challenges and future directions, 2nd International Conference on Electronics and Communication Systems (ICECS), pp. 1679–1684. 14. Sukanya, M., & Biruntha, S. (2012). Techniques on text mining, International Conference on Advanced Communication Control and Computing Technologies (ICACCCT), pp. 269–271. 15. Navathe, S.B., & Ramez, E. (2000). Data warehousing and data mining, Fundam, Database Syst., pp. 841–872. 16. Ergul-Aydin Z., Kami¸sli-Öztürk Z., Erzurum-Çiçek Z.˙I. (2021). Turkish Sentiment Analysis For Open and Distance Education Systems, TOJDE, Vol. 22(3), 124–138.

134

˙I. Tu˘g and B. Kan-Kilinç

17. Salloum, S.A., Al-Emran, M., Monem, A.A., & Shaalan, K. (2018). Using text mining techniques for extracting information from research articles, Intelligent Natural Language Processing: Trends and Applications, pp. 373–397. 18. Sun. S., Cao, Z., Zhu, H., & Zhao, J. (2020). A Survey of optimization methods from a machine learning perspective, IEEE Transactions on Cybernetics, Vol. 50, No. 8, pp. 3668–3681. 19. Kotsiantis, S. B. (2007). Supervised machine learning: A review of classification techniques, Informatica, pp. 249–268. 20. Bilgin, M. (2017). Gerçek veri setlerinde klasik makine ö˘grenmesi yöntemlerinin performans analizi, Akademik Bili¸sim. 21. Wills, S., Underwood, C.J., & Barrett, P.M. (2020). Learning to see the wood for the trees: machine learning, decision trees and the classification of isolated theropod teeth, Palaeontology, Vol. 64, No. 1, pp. 75–99. 22. Haykin, S., (2008). Neural networks and learning machines, Pearson 3rd edn. 23. Breiman, L., (2001). Random Forests, Machine Learning, Vol. 45, pp. 5–32. 24. Freund, Y., Schapire, R.E. (1997). Decision-theoretic generalization of on-line learning and an application to boosting, Journal of Computer and System Sciences, Vol. 55, No. 1, pp. 119–139. 25. Natekin, A., & Knoll A. (2013). Gradient boosting machines, a tutorial, Front in Neurorobotics, Vol. 7, No. 21. 26. Zhao, W., Li, J., Zhao, J., Zhao, D., Lu, J., & Wang, X. (2020). XGB model: Research on evaporation duct height prediction based on XGBoost algorithm, RADIOENGINEERING, Vol. 29, No. 1, pp. 81–93. 27. Powers, D.M. (2011). Evaluation: From Precision, Recall and F-measure to ROC, Informedness, Markedness and Correlation;Technical Report SIE-07-001; School of Informatics and Engineering. 28. Flach, P.A. (2003). The geometry of ROC space: Understanding machine learning metrics through ROC isometrics, In Proceedings of the 20th International Conference on Machine Learning (ICML-03). 29. Hossin, M., and Sulaiman, M.N. (2015). A review on evaluation metrics for data classification evaluations, International Journal of Data Mining & Knowledge Management Process (IJDKP), Vol. 5, No. 2. 30. McCormick, T.H., Lee, H., Cesare, N., Shojaie, A., & Spiro, E.S. (2017). Using Twitter for demographic and social science research: Tools for data collection and processing, Sociological Methods & Research, Vol. 46, No. 3, pp. 390–421. 31. Misra, P., & Gupta, J. (2021). Impact of COVID-19 on Indian migrant workers: decoding twitter data by text mining, The Indian Journal of Labour Economics, pp. 1–17. 32. Feinerer, I., & Hornik, K. (2020). tm: Text Mining Package, R package version 0.7-8, 2020. 33. Silge, J., Robinson, D., & Hester. J. (2016). Tidytext: Text Mining Using Dplyr, Ggplot2, and Other Tidy Tools, The Journal of Open Source Software, Vol. 1, No. 3. 34. Hu, M., & Liu, B. (2004). Mining and summarizing customer reviews, Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (KDD-2004). 35. Goodfellow, I., Bengio, & Y., Courville, A. (2016). Deep Learning, The MIT Press. 36. Feurer, M., & Hutter, F. (2019). Hyperparameter optimization, The NeurIPS ’18 Competition, pp. 3–33. 37. Hertel, L., Baldi, P., & Gillen, D.L. (2021). Reproducible hyperparameter optimization, Journal of Computational and Graphical Statistics, Vol. 00, No. 0, pp. 1–16.

A Note on Special Matrices Involving k-Bronze Fibonacci Numbers Paula Catarino and Sandra Ricardo

1 Introduction Numerical sequences are a source of very interesting and challenging mathematical problems and have attracted the attention of many researchers. Many developments have been done not only concerning the well-known Fibonacci sequence or the Lucas sequence, but also many works have emerged on other sequences of numbers, polynomials, quaternions, octonions, sedenions, etc. We refer, for example, to the works on hybrid numbers [1, 2], applications of Fibonacci and Lucas numbers [3], Leonardo’s numbers [4], k-Pell generalized numbers of order m, where m is a non-negative integer [5], Gaussian Fibonacci sequences [6], Gaussian Lucas sequences [7], Gaussian Pell sequences and Gaussian Pell-Lucas sequences [8], Gaussian Jacobsthal sequences [9], and Gaussian Bronze Fibonacci sequences [10]. Newer developments appeared recently concerning third-order Bronze Fibonacci numbers [11], on Vietoris’numbers sequence [12], on generalized sequences of numbers known as k-Fibonacci numbers, k-Jacobsthal numbers, k-Pell numbers, balancing numbers, k-telephone numbers, hyper k-pell numbers and incomplete numbers [13]. In 1985, Levesque [14] deduced an important formula for linear recurrences, known as the Binet formula, which is obtained in terms of the roots of the characteristic equation associated with the recurrence relation. Binet’s formula allows one to find the general term of a sequence, without having to resort to other terms of the sequence, thus being an important tool to study some properties of the considered sequence. P. Catarino · S. Ricardo (B) Department of Mathematics, University of Trás-os-Montes e Alto Douro, 5001-801 Vila Real, Portugal e-mail: [email protected] P. Catarino e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_14

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In the present work, we take as our starting point the Bronze Fibonacci sequence, {B Fn }n≥0 , listed in the Online Encyclopedia of integer sequences [15] as the sequence A006190, and defined by the following recurrence relation B Fn+2 = 3B Fn+1 + B Fn , with the initial conditions B F0 = 0 and B F1 = 1. We consider a generalization of the Bronze Fibonacci sequence, in which the recurrence formula depends on one parameter k: (1) B Fk,n+2 = 3B Fk,n+1 + k B Fk,n , k ∈ N, with the initial conditions B Fk,0 = 0 and B Fk,1 = 1. We call this new sequence the k-Bronze Fibonacci sequence and denote it by {B Fk,n }n≥0 . The particular case in which k = 4 is also listed in the Online Encyclopedia of integer sequences [15] as the sequence A015521. There are many known connections between determinants or permanents of tridiagonal matrices and the Fibonacci numbers, Lucas numbers and Pell numbers (see for instance, [16–20]). In this paper, our goal is to give alternative ways to determine the general term of the k-Bronze Fibonacci sequence involving some special tridiagonal matrices, their determinants and also their permanents.

2 The k-Bronze Fibonacci Sequence Consider the k-Bronze Fibonacci sequence, defined as in (1). Table 1 presents the k-Bronze Fibonacci numbers B Fk,n , for 0 ≤ n ≤ 6. Next we find the Binet formula for the k-Bronze Fibonacci sequence, which will be important for our considerations in Sect. 3.4. Theorem 1 (Binet’s formula). The nth term of the sequence {B Fk,n }n≥0 is given by B Fk,n =

r1n − r2n , r1 − r2

Table 1 The k-Bronze Fibonacci numbers B Fk,n , for 0 ≤ n ≤ 6 B Fk,0 = 0 B Fk,1 = 1 B Fk,2 = 3 B Fk,3 = 32 + k B Fk,4 = 33 + 6k B Fk,5 = 34 + 33 k + k 2 B Fk,6 = 35 + (34 + 33 )k + 32 k 2

(2)

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where r1 and r2 are the roots of the characteristic equation associated with the recurrence relation (1). Proof The characteristic equation associated with the recurrence relation (1) is x 2 − 3x − k = 0, which has two roots r1 =

3+

√

9 + 4k 2

and

r2 =

3−

√ 9 + 4k . 2

(3)

Therefore, the general solution of equation (1) is given by B Fk,n = c1 r1n + c2 r2n , for some coefficients c1 and c2 . Giving to n the values n = 0 and n = 1 we obtain, 1 1 and c2 = − r1 −r . respectively, c1 + c2 = 0 and c1 r1 + c2 r2 = 1. It follows c1 = r1 −r 2 2 So, using these values in the above expression of B Fk,n , we get the required result.  

3 Tridiagonal Matrices and the k-Bronze Fibonacci Numbers In this section, we consider some special tridiagonal matrices whose determinant or permanent allows us to find the general term of the k-Bronze Fibonacci sequence. We start by introducing a general family An , n = 1, 2, 3, . . . , of tridiagonal matrices and find their successive determinants and permanents. Then, in Sects. 3.2, 3.3 and 3.4, we propose special tridiagonal matrices for which we investigate connections between their determinants or permanents and the k-Bronze Fibonacci sequence.

3.1 Determinants and Permanents of a Tridiagonal Matrix In general, given A = (ai j ) a square n × n matrix, we define, in a standard manner, the determinant of A as n  sgn(σ )aiσ (i) , |A| = σ ∈Sn i=1

and the permanent of A as per(A) =

n  σ ∈Sn i=1

aiσ (i) ,

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where Sn is the symmetric group consisting of all permutations σ of {1, 2, . . . , n}, with σ = (σ1 , σ2 , . . . , σn ). These two definitions look very similar, differing only in the presence (in the definition of |A|) and absence (in the definition of per(A)) of the factor sgn(σ ), which gives the sign of permutations. That is, the permanent of a matrix is analogous to the determinant of that matrix, but with all the signs used in the Laplace expansion of minors positive. Consider the following family of n × n tridiagonal matrices An , n = 1, 2, 3, . . ., given by: ⎛ ⎞ a1,1 a1,2 ⎜ a2,1 a2,2 a2,3 ⎟ ⎜ ⎟ ⎜ ⎟ . . ⎜ ⎟ . a3,2 a3,3 ⎟ . An = ⎜ (4) ⎜ ⎟ .. .. ⎜ ⎟ . . ⎜ ⎟ ⎝ an−1,n−1 an−1,n ⎠ an,n−1 an,n n×n The sucessive determinants |An | and permanents per(An ), n ∈ N, can be described, respectively, by the following recurrence relations: |A1 | = a1,1 |A2 | = a2,2 a1,1 − a2,1 a1,2 |An | = an,n |An−1 | − an,n−1 an−1,n |An−2 |, n ≥ 3,

(5)

and per(A1 ) = a1,1 per(A2 ) = a2,2 a1,1 + a2,1 a1,2 per(An ) = an,n per(An−1 ) + an,n−1 an−1,n per(An−2 ), n ≥ 3,

(6)

where equalities (5) and (6) follow by cofactor expansion on the last column and then the last row (see, respectively, Theorem 1 in [16] and Lemma 3 in [21]). A square n × n matrix A is called convertible if there exists an n × n matrix H , whose elements consist only of the numbers 1 and −1, such that per(A) = |A ◦ H |, where ◦ denotes the Hadamard product of A and H . The matrix H is called a converter of A.

3.2 Special Tridiagonal Matrices—Type Number One We consider the family of n × n tridiagonal matrices Mk,n , n = 1, 2, 3, . . ., given by:

A Note on Special Matrices Involving k-Bronze Fibonacci Numbers

⎛

Mk,n

3 ⎜ −k ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

1 3 −k

0 1 3 .. .

139

⎞ ⎟ ⎟ ⎟ 1 ⎟ ⎟ .. .. ⎟ . . . ⎟ ⎟ ⎟ −k 3 1 ⎠ 0 − k 3 n×n

(7)

We obtain the successive determinants |Mk,1 | = 3 = B Fk,2 |Mk,2 | = 3|Mk,1 | + k = 32 + k = B Fk,3 |Mk,3 | = 3|Mk,2 | + k|Mk,1 | = 33 + 6k = B Fk,4 |Mk,4 | = 3|Mk,3 | + k|Mk,2 | = 34 + 33 k + k 2 = B Fk,5 .. . |Mk,n+1 | = 3|Mk,n | + k|Mk,n−1 |, from which we conclude the next result. Proposition 1 If Mk,n is the n × n tridiagonal matrix considered in (7), then the nth k-Bronze Fibonacci number is given by B Fk,n = |Mk,n−1 |, n ≥ 2.

(8)

Proof Using an inductive reasoning, the base step is obvious. If we assume that the result is true for 2 ≤ n < N , then |Mk,N | = 3|Mk,N −1 | + k|Mk,N −2 | = 3B Fk,N + k B Fk,N −1 = B Fk,N +1 , where the last equality follows from the recurrence definition of the k-Bronze Fibonacci numbers.   We also consider a family of given by: ⎛ 3 ⎜k ⎜ ⎜ ⎜ ⎜ Pk,n = ⎜ ⎜ ⎜ ⎜ ⎝

n × n tridiagonal matrices Pk,n , n = 1, 2, 3, . . ., 1 3 k

0 1 3 .. .

⎞ ⎟ ⎟ ⎟ 1 ⎟ ⎟ .. .. ⎟ . . . ⎟ ⎟ ⎟ k 3 1⎠ 0 k 3 n×n

(9)

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We obtain the successive permanents per(Pk,1 ) = 3 = B Fk,2 per(Pk,2 ) = 3 per(Pk,1 ) + k = 32 + k = B Fk,3 per(Pk,3 ) = 3 per(Pk,2 ) + k per(Pk,1 ) = 33 + 6k = B Fk,4 per(Pk,4 ) = 3 per(Pk,3 ) + k per(Pk,2 ) = 34 + 33 k + k 2 = B Fk,5 .. . per(Pk,n+1 ) = 3 per(Pk,n ) + k per(Pk,n−1 ), from which we conclude the next result. Proposition 2 If Pk,n is the n × n tridiagonal matrix considered in (9), then the nth k-Bronze Fibonacci number is given by B Fk,n = per(Pk,n−1 ), n ≥ 2.

(10)

Proof Using an inductive reasoning, similar to the proof of Proposition 1, the result is easily obtained.   Let Hn , n ∈ N, be the square n × n matrix given by ⎛

1 ⎜ −1 ⎜ ⎜ Hn = ⎜ 1 ⎜ .. ⎝ . 1

1 1 −1 .. . 1

1 1 1 .. .

... ... ... .. .

1 1 1 .. .

⎞ 1 1⎟ ⎟ 1⎟ ⎟ .. ⎟ .⎠

1 ... − 1 1

.

(11)

n×n

Clearly, Hn is a converter of Pk,n , that is, per(Pk,n ) = |Pk,n ◦ Hn |. Thus, we get the following result: Proposition 3 If Pk,n−1 is the (n − 1) × (n − 1) tridiagonal matrix considered in (9) and Hn−1 is given accordingly with (11), then the nth k-Bronze Fibonacci number is given by (12) B Fk,n = per(Pk,n−1 ) = |Pk,n−1 ◦ Hn−1 |, n ≥ 2.

3.3 Special Tridiagonal Matrices—Type Number Two In this subsection, we give an alternative way, using determinants, to compute B Fk,n . Given a general sequence {xn }n≥0 satisfying the second order linear recurrence xn+2 = Axn+1 + Bxn , n ≥ 0,

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with x0 = C, x1 = D, and A, B, C, D real numbers, consider the associated (n + 1) × (n + 1) tridiagonal matrix defined as ⎛

Rn+1

C ⎜ −1 ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

D 0 −1

0 B A .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ . ⎟ ⎟ ⎟ ⎟ −1 A B ⎠ − 1 A (n+1)×(n+1)

B .. .. . .

(13)

Accordingly with [20] (see also [18]), the nth term of the sequence {xn }n≥0 is obtained as the determinant of the matrix Rn+1 , that is, xn = |Rn+1 | for n ≥ 0. If we consider the second order linear recurrence (1) defining the k-Bronze Fibonacci sequence, we get A = 3, B = k, C = 0 and D = 1 and, so, the associated (n + 1) × (n + 1) tridiagonal matrix Rk,n+1 = (rk;i, j )(n+1)×(n+1) , becomes ⎛

Rk,n+1

0 ⎜ −1 ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

1 0 −1

0 k 3 .. .

⎞ ⎟ ⎟ ⎟ k ⎟ ⎟ .. .. . ⎟ . . ⎟ ⎟ ⎟ −1 3 k ⎠ − 1 3 (n+1)×(n+1)

(14)

The next result, relating the determinant of this matrix with the nth term of the k-Bronze Fibonacci sequence, follows. Proposition 4 If Rk,n is the (n + 1) × (n + 1) tridiagonal matrix considered in (14), then the nth k-Bronze Fibonacci number is given by B Fk,n = |Rk,n+1 |, n ≥ 0. Proof Clearly, |Rk,1 | = 0 = B Fk,0 |Rk,2 | = 1 = B Fk,1 |Rk,3 | = 3 = B Fk,2 |Rk,4 | = 9 + k = B Fk,3

(15)

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By an inductive reasoning assume that B Fk,n−1 = |Rk,n |, n ≥ 0. Therefore, |Rk,n+1 | = rk;n+1,n+1 |Rk,n | − rk;n+1,n rk;n,n+1 |Rk,n−1 | = 3|Rk,n | − (−1)k|Rk,n−1 | = 3B Fk,n−1 + k B Fk,n−2 = B Fk,n ,  

thus yielding the result.

We consider also the tridiagonal matrix Sk,n+1 = (sk;i, j )(n+1)×(n+1) given by ⎞ ⎛ 0 1 0 ⎟ ⎜1 0 k ⎟ ⎜ ⎟ ⎜ 1 3 k ⎟ ⎜ ⎟ ⎜ . . . .. .. .. Sk,n+1 = ⎜ . (16) ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ 1 3 k⎠ 1 3 (n+1)×(n+1) The next result, relating the permanent of this matrix with the nth term of the k-Bronze Fibonacci sequence, follows. Proposition 5 If Sk,n is the (n + 1) × (n + 1) tridiagonal matrix considered in (16), then the nth k-Bronze Fibonacci number is given by B Fk,n = per(Sk,n+1 ), n ≥ 0.

(17)

Proof Clearly, per(Sk,1 ) = 0 = B Fk,0 per(Sk,2 ) = 1 = B Fk,1 per(Sk,3 ) = 3 = B Fk,2 per(Sk,4 ) = 3 per(Sk,3 ) + k per(Sk,2 ) = 3 + k = B Fk,3 .. . per(Sk,n+1 ) = 3 per(Sk,n ) + k per(Sk,n−1 ), Assume that the result holds for 0 ≤ n ≤ l. Then per(Sk,l+1 = sk;l+1,l+1 per(Sk,l ) + sk;l+1,l rk;l,l+1 per(Sk,l−1 ) = 3 per(Sk,l ) + k per(Sk,l−1 ) = 3 B Fk,l−1 + k B Fk,l−2 = B Fk,l , thus proving the result.

 

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We observe that the square (n + 1) × (n + 1) matrix Hn+1 , defined accordingly with (11), is a converter of Sk,n+1 , that is per(Sk,n+1 ) = |Sk,n+1 ◦ Hn+1 |. Therefore, we obtain the following result: Proposition 6 If Sk,n is the (n + 1) × (n + 1) tridiagonal matrix considered in (16) and H1 is given by (11), then the nth k-Bronze Fibonacci number is given by B Fk,n = per(Sk,n+1 ) = |Sk,n+1 ◦ Hn+1 |, n ≥ 0.

(18)

3.4 Special Tridiagonal Matrices—Type Number Three: Toeplitz Matrices In [18], the authors considered the n × n tridiagonal Toeplitz matrices Tn and Un defined as ⎞ ⎛ α+β β ⎟ ⎜ α α+β β ⎟ ⎜ ⎟ ⎜ . . ⎟ , ⎜ . α α+β (19) Tn = ⎜ ⎟ ⎟ ⎜ . . . . ⎝ . . β ⎠ α α + β n×n and

⎞

⎛

α+β β ⎜ −α α + β β ⎜ ⎜ . − α α + β .. Un = ⎜ ⎜ ⎜ .. .. ⎝ . .

β −α α+β

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

(20)

n×n

where α and β are real or complex numbers such that αβ = 0 and (α + β)2 = 4αβ. It was proven that (see [18], Theorem 1, pp. 12–13) |Tn | =

α n+1 − β n+1 , n ≥ 1, α−β

(21)

and (see [18], Theorem 3, p. 5) per(Un ) =

α n+1 − β n+1 , n ≥ 1. α−β

(22)

Focusing attention on the k-Bronze Fibonacci sequence we are considering in this paper, we construct the following n × n tridiagonal Toeplitz matrices, for k, n ∈ N:

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⎞

⎛

Tk,n

r1 + r2 r2 ⎜ r1 r + r2 r2 1 ⎜ ⎜ . r1 r1 + r2 . . =⎜ ⎜ ⎜ .. .. ⎝ . .

r2 r1 r1 + r2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Uk,n

r1 + r2 r2 ⎜ −r1 r1 + r2 r2 ⎜ ⎜ . − r1 r1 + r2 . . =⎜ ⎜ ⎜ .. .. ⎝ . .

(23)

n×n

⎞

⎛

and

,

r2 − r1 r1 + r2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

(24)

n×n

where r1 and r2 are the roots of the characteristic equation obtained in (3). We observe that r1 + r2 = 3 and r1r2 = −k = 0, thus implying that (r1 + r2 )2 = 4r1r2 . The Binet Formula (2) and equality (21) allow to establish the (n + 1)th k-Bronze Fibonacci number as the determinant of the matrix Tk,n . Proposition 7 The (n + 1)th k-Bronze Fibonacci number is given by B Fk,n+1 = |Tk,n |, n ≥ 1.

(25)

Similarly, using the Binet Formula (2) and equality (22), we get the (n + 1)th k-Bronze Fibonacci number as the permanent of the matrix Uk,n . Proposition 8 The (n + 1)th k-Bronze Fibonacci number is given by B Fk,n+1 = per(Uk,n ), n ≥ 1.

(26)

Again, we observe that matrix Hn is a converter of Uk,n . Therefore, we obtain the result: Proposition 9 If Uk,n is the n × n tridiagonal matrix considered in (16) and Hn is given by (11), then the nth k-Bronze Fibonacci number is given by B Fk,n+1 = per(Uk,n ) = |Uk,n ◦ Hn |, n ≥ 1.

(27)

Acknowledgments The first author is member of the Research Centre CMAT-UTAD (Polo of Research Centre CMAT—Centre of Mathematics of University of Minho) and also a collaborating member of the Research Centre CIDTFF—Research Centre on Didactics and Technology in the Education of Trainers of University of Aveiro. The second author is member of the Research Centre ISR—Institute of Systems and Robotics of University of Coimbra. This research was partially financed by Portuguese Funds through FCT—Fundação para a Ciência e a Tecnologia, within the Projects UIDB/00013/2020, UIDP/00013/2020 and UIDB/00194/2020.

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References 1. Catarino, P.: On k-Pell hybrid numbers. J. Discrete Math. Sci. Cryptogr. 22(1), 83–89 (2019). 2. Kizilates, C.: A new generalization of Fibonacci hybrid and Lucas hybrid numbers. Chaos Solitons Fractals. 130, 5pp. (2020). https://doi.org/10.1016/j.chaos.2019.109449 3. Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley-Interscience, New York (2001). 4. Catarino, P., Borges, A.: On Leonardo numbers. Acta Math. Univ. Comenian. 89(1), 75–86 (2019). 5. Catarino, P., Vasco, P.: The Generalized order−m(k-Pell) numbers. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. 20(1), 55–65 (2020). 6. Berzsenyi G.: Gaussian Fibonacci Numbers. Fibonacci Quart. 15(3), 233–236 (1977). 7. Jordan J.H.: Gaussian Fibonacci and Lucas Numbers. Fibonacci Quart. 3, 315–318 (1965). 8. Halici, S., Öz, S.: On some Gaussian Pell and Pell-Lucas numbers. Ordu Univ. J. Sci. Tech. 6(1), 8–18 (2016). 9. Asc M., Gurel E.: Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Numbers. Ars Combin. 111, 53–63 (2013). 10. Kartal, M. Y.: Gaussian Bronze Fibonacci Numbers. Ejons int. j. math. eng. natural sci. 13, 19–25 (2020). 11. Akbiyik, M., Alo, J.: On Third-Order Bronze Fibonacci Numbers. Mathematics. 9(20): 2606 (2021). 12. Catarino, P., Almeida, R.: A Note on Vietoris’ Number Sequence. Mediterr. J. Math. 19(41) (2022). 13. Catarino, P., Campos, H.: From Fibonacci Sequence to More Recent Generalisations. In: Yilmaz, F., Queiruga-Dios, A., Santos Sánchez, M.J., Rasteiro, D., Gayoso Martínez, V., Martín Vaquero, J. (eds.) Mathematical Methods for Engineering Applications, ICMASE 2021. Springer Proceedings in Mathematics & Statistics, vol. 384, pp. 259–269 (2022). https://doi. org/10.1007/978-3-030-96401-6_24 14. Levesque, C.: On m-th order linear recurrences. Fibonacci Quart. 23(4), 290–295 (1985). 15. Sloane N. J. A.: The on-line encyclopedia of integer sequences. http://oeis.org/ 16. Cahill, N.D., Narayan, D.A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants. Fibonacci Quart. 42(3), 216–221 (2004). 17. Falcon, S.: On the generating matrices of the k-Fibonacci numbers. Proyecciones. 32(4), 347– 357 (2013). 18. Kılıç, E., Tasci, D.: On the Second Order Linear Recurrences by Tridiagonal Matrices. Ars Combin. 91, 11–18 (2009). 19. Kılıç, E., Tasci, D.: On the generalized Fibonacci and Pell sequences by Hessenberg matrices. Ars Combin. 94, 161–174 (2010). 20. Kılıç, E., Tasci, D., Haukkanen, P.: On the generalized Lucas sequences by Hessenberg matrices. Ars Combin. 95 (2010). 21. Kılıç, E., Tasci, D.: On the Permanents of Some Tridiagonal Matrices with Applications to the Fibonacci and Lucas Numbers. Rocky Mountain J. Math. 37(6), 1953–1969 (2007). https:// doi.org/10.1216/rmjm/1199649832

Influence of the Collaboration Among Predators and the Weak Allee Effect on Prey in a Modified Leslie-Gower Predation Model Alejandro Rojas-Palma and Eduardo González-Olivares

1 Introduction One of the important features of the dynamics of the food chain or food webs is the interactions between predators and their prey. Nevertheless, social interactions among members of the same species are also a significant part of the past and present of many populations. These factors could significantly change population dynamics and change their typical behavior, to quote [1]. Examples of these social phenomena are the Allee effect, which influences the number of prey, or the hunting cooperation of predators. However, there isn’t enough research to accurately assess the effects of these two phenomena in that dynamic relationship when they both work concurrently in the interaction. Additionally, several mathematical formulations have been provided for each of these events, as seen in previous works [2], diverse mathematical forms for the same phenomenon can produce changes in the properties of the system describing the interaction. Then, in the future, a comparative study employing a different model for these phenomena must also be carried out. An interesting goal for the modelers is to establish how these mathematical expressions affect the dynamics of the systems [3– 5]. So, in this work, a modified Leslie-Gower type model is analyzed, in which the prey growth rate is affected by a weak Allee effect, and there exists hunting cooperation or collaboration among predators. A. Rojas-Palma (B) Departamento de Matemáticas, Física y Estadística, Faculdad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile e-mail: [email protected] E. González-Olivares Pontificia Universidad Católica de Valparaíso, ValparaÃso, Chile e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_15

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1.1 Hunting Cooperation The collaboration or hunting cooperation between predators to capture their prey is a social behavior that is receiving increasing attention from modelers [3, 6]. This and other collective phenomena can have strong consequences on the relationship between species, in addition, modify the dynamic properties of the models that describe such interactions [7]. It is well known that competition between predators produces a stabilizing effect on the system described by the model since there is a large set in the parameter space where the system has a single equilibrium point in the phase plane, which is globally asymptotically stable (gas) [8]. Meanwhile, cooperation can give rise to more complex and unusual dynamics [9]. As shown in [8], it is possible to prove that for a certain subset of parameter values, the size of the predator population goes to infinity when the prey population becomes extinct. This situation apparently contradicts the idea of a realistic model, especially if it is assumed that the predators are specialists, that is, the prey is their only source of food. However, this could be a desirable effect when the prey is a pest, for instance. Cooperation or collaboration between predators is quite frequent in nature, representing a mechanism developed through evolution to improve hunting skills and chances of survival [10]. For example, wolves that follow bison [11], hyenas that chase buffalo, African wild dogs (lincaons) in search of zebras, corals consumed by a type of starfish [9, 12]. Collaboration or cooperation between predators to capture their prey is a behavior that has been mathematically analyzed in few articles [3, 8, 9, 13], but that will be incorporated in the model studied here. It will be assumed that in cooperative hunting, the functional response depends on both the density of prey and predators. We assume that cooperative predators benefit from their behavior, so the success of prey attacks increases with predator density. We will represent this assumption by replacing, in Leslie’s original model, the constant attack rate q by a density-dependent term [3], given by the function:   a h (x, y) = (q + ay) x = q 1 + y x q where x = x (t) indicates the prey population size for t ≥ 0 and a > 0 is a parameter that describes the cooperation of predators in hunting [3, 5]. We will refer to the qa y term as the cooperation term among predators. If a = 0, we obtain a predator-prey model without hunting cooperation. If a < 0, this would correspond to the interference or competition among the predators; in this case the term q + ay > 0 must be fulfilled, to represent a predation model [14, 15]. However, in this case, another interpretation is possible for the function h (x, y), assuming that a fraction xr of the prey population makes use of a physical refuge [16]. In this case the size of the population available to be consumed is x − xr . Assuming

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that the population in coverage is proportional to the encounter between both species, we have xr = σ x y, with σ > 0 [17]. So the functional response is a h (x, y) = q (x − xr ) y = q (x − σ x y) y = q (1 − σ y) x y, with σ = − . q

1.2 Allee Effect The Allee effect is a phenomenon in ecology characterized by a correlation between population size or density and the mean individual fitness (often measured as per capita population growth rate) of a population or species [18–20]. This phenomenon occurs in some species at low population densities, when the per capita growth rate is an increasing function of population abundance. At large population sizes, this rate is negative, as is the case in the logistic equation for all population sizes [14, 21]. The Allee effect can occur due to a wide range of biological phenomena, such as reduced vigilance against predation, social thermoregulation, mating difficulty, and poor feeding at low densities. However, various other causes can lead to this phenomenon (see Table 1 in [18] or Table 2.1 in [22]). The most common mathematical way to describe this effect is the nonlinear differential equation [21, 23]. dx dt

 =r 1−

x K



(x − m) x,

(1)

where x = x (t) indicates the size of a population for t ≥ 0. The parameter r is the intrinsic growth rate of and K indicates the environmental carrying capacity (with the same meanings as in the logistic equation); m is the parameter associated with the Allee effect. Equation (1) describes an Allee effect, when the per capita growth rate x1 ddtx is negative for values of the variable x close to zero, which occurs, if and only if, −K < m 0, the parameter is called minimum viable population or extinction threshold and there is a strong Allee effect [1, 15, 24]. Clearly, if 0 < x < m in equation (1), it has ddtx < 0, which implies that the population is going extinct. If m ≤ 0, it has a weak Allee effect [1, 2, 24]. The weak Allee effect does not induce an extinction threshold, so small populations can be maintained over time. Different mathematical expressions have been proposed for this social phenomenon, although many of them are topologically equivalent [2]. Some of these equations are: ax = r (1 − Kx )x − x+b , with a, b > 0, which is deduced in [23, 25] and also proposed in [20]. )x, with m, b > 0, proposed in [18] and used in pre2. ddtx = r (1 − Kx )(1 − m+b x+b dation models [24].

1.

dx dt

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= r (1 − Kx )x − bx exp (−cx) , with b, c > 0, derived by J. R. Philip in 1957 [26], giving probabilistic arguments. dx dt

On the other hand, it has been shown that two or more Allee effects produced by different causes can generate mechanisms that act simultaneously in the same population (see Table 2 in [27] or Table 2.2 in [22]). The combined influence of some of these phenomena is called the multiple Allee effect [27, 28].

2 Model Proposition The model to be studied is a modification of the Leslie-Gower model [29, 30], being described by the following bidimensional system of autonomous differential equations of the Kolmogorov type [31, 32]: X μ (x, y) :

 dx dt dy dt

  = r 1 − Kx (x − m) x − (q + ay) x y y y = s 1 − nx+c

(2)

where x = x (t) and y = y (t) represent the sizes of the prey and predator populations, respectively, for t ≥ 0, with μ = (r, K , q, a, s, n, c, m) ∈ R7+ × ]−K , K [. They have the following ecological meanings: – r and s indicate the intrinsic growth rate of the population of prey and predators, respectively, – K is the environmental carrying capacity of the prey, – q is the predator consumption rate, – m, is the Allee parameter, named extinction threshold of the Allee effect or the minimum viable population [15, 21, 22, 33] when m > 0. – n represents the energy quality provided by prey as food for predators, – c indicates the maximum available size of the alternative feed, – α is the cooperation of the predator in the constant hunting. System (2) is defined in the first quadrant, that is,   Ω = (x, y) ∈ R2 : x ≥ 0, y ≥ 0 . The equilibrium points of the system (2) or singularities of the vector field X μ (x, y) are: (0, 0), (m, 0), when m > 0, (K , 0), (0, c), and those that are at the intersection of the isoclines x

y = nx + c and − ay 2 − qy + r 1 − (x − m) = 0. K that is, if (xe , ye ) is a positive equilibrium point, the absisa xe satisfies the polynomial equation

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p (x) = r + K an 2 x 2 − (K r + mr − K nq − 2K acn) + K ac2 + qc + mr = 0.

Thus, different dynamical cases arise when considering a strong or weak Allee effect, that is, if m > 0, m < 0, or m = 0. The system (2) has some important properties, for example, there is a positively invariant region Γ ⊂ Ω where the dynamics of interest are exhibited, it can also be shown that the solutions are uniformly bounded so that none of the populations can grow indefinitely, condition that points to the realism of the model. It is also possible to prove that, in the case m < 0 (weak Allee effect) the solutions are permanent. The proof of these results can be found in [34]. In this work, we will focus on the asymptotic behavior of solutions, as well as on local bifurcations.

2.1 Topological Equivalence To simplify the calculations we will make a change of variables and a rescaling of the time, described below: Lemma 1 Topologically equivalent system System (2) is topologically equivalent to the Kolmogrov-type system [31, 32] Yν (u, v) :

 du dτ dv dτ

  = (1 − u) (u − M) − Q (1 + Av) v u (u + C) = S (u + C − v ) v

where ν = (Q, A, S, C, M) ∈ R4+ × ]−1, 1[ with C = an K and S = r sK . q

c nK

,M=

m , K

Q=

(3) qn r

Proof Let x = K u and y = n K v. Replacing in (3)  Uν (u, v) :

  = r (1 − u) (Ku − m) − (q + an K v) n K v K u K du dt v n K dv n K v, = s 1 − n Kn Ku+c dt

Factoring and simplifying, we have ⎧  ⎨ du = r K (1 − u) u − dt

Uν (u, v) : dv ⎩ = s 1 − v c v, dt u+

m K



− 1+

nK

Carrying out the time rescaling change given by τ= is finally obtained,

du dτ du rK = , c t, u + nK dt dτ dt

an K q

v



qn v r

u

,A=

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 Vν (u, v) : defining M =

m , K

du dτ dv dτ

S=

  = (1 − u) u − mK − 1 +   = r sK u + ncK − v v, s rK

,C =

c nK

,A=

an K q

an K q

and Q =

v



qn v r

qn , r

 u+

c nK



u

system (3) is obtained.

Remark 1 The system (3) or the vector field Yν (u, v) is defined in   Ω¯ = (u, v) ∈ R2 : u ≥ 0, v ≥ 0 . Thus, we have constructed a diffeomorphism ϕ : Ω¯ × R → Ω × R, such that  ϕ (u, v, τ ) =

u + ncK τ K u, n K v, rK

 = (x, y, t) .

The Jacobian matrix of the function ϕ is ⎛

K 0 Dϕ (u, v, τ ) = ⎝ 0 n K 1 0 rK and det Dϕ (u, v, τ ) = time orientation.

nK r

 u+

c nK



⎞ 0 0 ⎠

u+ ncK rK

> 0, then, the diffeomorphism preserves the

2.2 Equilibria Existence In the weak Allee effect case (M ≤ 0), the equilibrium points of the system (3) or the vector field Yν (u, v) are (0, 0), (1, 0), (0, C), and those that are at the intersection of the nullclines v = u + C and (1 − u) (u − M) − Q (1 + Av) v = 0. This implies that the abscissa u e of the positive equilibrium points (u e , ve ), satisfies the polynomial equation P (u) = (AQ + 1) u 2 − a1 u + a0 = 0.

(4)

with a1 = M + 1 − Q (2 AC + 1) and a0 = M + C Q ( AC + 1) . According to Descartes’ rule of sign, the polynomial P (u) has two, one, or no positive real roots, depending on the sign of parameters M and C. Let

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Δ = (a1 )2 − 4 (AQ + 1) (a0 ) = M 2 − 2b1 M + b2 with b1 = Q + 2 AQ + 2 AC Q + 1 and b2 = (Q − 1)2 − 4C Q ( A + AC + 1). In first place, we consider different cases for M < 0 1. Assuming a0 > 0 and a1 > 0, then the polynomial P (u) can have 1.1 two real positive roots, if and only if, Δ > 0, which are given by: u1 =

√

√

1 1 a1 − Δ and u 2 = a1 + Δ , 2 (AQ + 1) 2 (AQ + 1)

with 0 < u 1 < u 2 < 1. 1.2 A real positive root, if and only if, Δ = 0, which is u∗ =

1 a1 . 2 (AQ + 1)

1.3 No real positive root, if and only if, Δ < 0. 2. Assuming a0 > 0 and a1 ≤ 0, then, the polynomial P (u) has no positive real roots. 3. Assuming a0 < 0, for any sign of a1 , the polynomial P (u) has a positive real root, which is given by u2 =

√

1 a1 + Δ . 2 (AQ + 1)

Suppose now that M = 0, then 1. Assuming a1 > 0, then the polynomial P (u) can have 1.1 two real positive roots, if and only if, Δ > 0, which are given by: u1 =

√

√

1 1 a1 − Δ and u 2 = a1 + Δ , 2 (AQ + 1) 2 (AQ + 1)

with 0 < u 1 < u 2 < 1. 1.2 A real positive root, if and only if, Δ = 0, which is u∗ =

1 − Q (2 AC + 1) . 2 (AQ + 1)

1.3 No real positive root, if and only if, Δ < 0. 2. Assuming a1 ≤ 0, then, the polynomial P (u) has no positive real roots. To study the local stability of hyperbolic equilibrium points, the Jacobian matrix is required, which is:

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 DYν (u, v) =

 DYν (u, v)11 −Qu (2 Av + 1) (C + u) , Sv S (C + u − 2v)

(5)

with, DYν (u, v)11 = (M + 1 − 2u)u(u + C) + ((1 − u)(u − M) − Q(1 + Av)v) (2u + C).

3 Main Results System (3) was studied in [34] for the case of the Strong Allee effect (M > 0), for which local stability of the equilibria as well as the existence of Hopf, BogdanovTakens, heteroclinic and homoclinic bifurcations were determined. Now we will study the case of weak Allee effect (M ≤ 0) in which the number of singularities in the first quadrant decreases. We will determine if the conditions for the existence of local bifurcations obtained in the Strong case hold, that is, if the conditions depend on the sign of the parameter M and its influence on the dynamics.

3.1 Saddle-Node Bifurcation at Origin A saddle-node bifurcation is a collision of two equilibria in dynamical systems. In systems generated by autonomous differential equations, this occurs when the jacobian matrix of the collide equilibrium has one zero eigenvalues. Lemma 2 The point (0, 0) is an hyperbolic repeller when M < 0. If M = 0 the system undergoes a saddle-node bifurcation in the origin. Proof The result for M < 0 is given by linearization (Hartman-Grobman Theorem) evaluating the Jacobian matrix (5). Furthermore, where the system is twodimensional, its local stability is determined by the trace and the determinant of this matrix. As   MC 0 DYν (0, 0) = , 0 SC with detDYν (0, 0) = −M SC 2 and trDYν (0, 0) = (−M + S) C. If M < 0, detDYν (0, 0) > 0, trDYν (0, 0) > 0, and the point (0, 0) is an hyperbolic repeller. Suppose now that M = 0. In this case, the equilibrium point at the origin (0, 0) and the axial equilibrium point (M, 0) collapse. The jacobian matrix becomes  DYν (0, 0) =

0 0 0 SC

 ,

which has a null eigenvalue. The eigenvector corresponding to zero eigenvalue is θ = (1, 0)T . We note that DYν (0, 0) has a unique simple null eigenvalue and none

Influence of the Collaboration Among Predators and the Weak Allee Effect on Prey … Fig. 1 For M = 0, the origin presents a saddle node bifurcation. In the figure, the dotted lines represent the nulclines of the system. The green lines represent the stable and unstable manifolds of the equilibrium point (0, 0)

155

0.2 (0,C)

0.1

0

(0,0)

-0.1

0

0.1

u of the other eigenvalues have real part equal to zero. Also, we have 

2C 0 θ D Yν (0, 0) (θ, θ ) = (1, 0) S −2S 2

 (1, 0)T = 2C = 0,

(6)

where right and left eigenvalues coincide in this case, these two conditions ensure the existence of a saddle-node bifurcation (see Fig. 1).

3.2 Equilibria Stability Lemma 3 Local stability of axial equilibrium points 1. The equilibrium point (1, 0) is a hyperbolic saddle point for all parameter values. 2. The equilibrium point (0, C) is a 2.1. 2.2. 2.3. 2.4.

hyperbolic attractor point if M = 0. hyperbolic saddle point if M < 0 and M + Q(1 + AC)C < 0. hyperbolic attractor node if M < 0 and M + Q(1 + AC)C > 0 non-hyperbolic attractor if M < 0 and M + Q(1 + AC)C = 0

Proof By linearization (Hartman–Grobman Theorem) evaluating the Jacobian matrix (5) and considering Determinant-trace Theorem. 

1. We have DYν (1, 0) =

− (1 − M) (1 + C) −Q (1 + C) 0 S (1 + C)



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where detDYν (1, 0) = − (1 − M) (1 + C)2 < 0, for all M ≤ 0, then (1, 0) is a hyperbolic saddle point. 2. Where   −M − Q (1 + AC) C 0 DYν (0, C) = , SC −SC with det DYν (0, C) = (M + Q (1 + AC) C) SC, and tr DYν (0, C) = − (M + Q (1 + AC) C + S) C. 2.1. If M = 0, then det DYν (0, C) = (Q (1 + AC) C) SC 2 > 0 and tr DYν (0, C) = − (M + Q (1 + AC) C + S) C < 0, the equilibrium (0, C) is a hyperbolic attractor point. 2.2. If M < 0 and M + Q (1 + AC) C < 0, then det DYν (0, C) = (M + Q (1 + AC) C) SC < 0, and the equilibrium is a hyperbolic saddle point. 2.3. If M < 0 and M + Q (1 + AC) C > 0, then det DYν (0, C) = (M + Q (1 + AC) C) SC > 0, tr DYν (0, C) = − (M + Q (1 + AC) C + S) < 0 and the equilibrium (0, C) is a hyperbolic attractor node (see Fig. 2). 2.4. If M < 0 and M + Q (1 + AC) C = 0 then det DYν (0, C) = 0 and the linearization method does not provide a solution. We note there is a zero

1

v

Fig. 2 For M = −0.1, C = 0.3, Q = 0.5, A = 0.2, S = 1 the equilibrium point (0, C) is a hyperbolic attractor node. Where no positive equilibria exist, the point is globally asymptotically stable in this case. The equilibrium point (1, 0) is a hyperbolic saddle point

(0,C)

0

(0,0)

(1,0)

0

1

u

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eigenvalue and other real and negative, so the equilibrium point is a nonhyperbolic attractor. Now, from Sect. 2.2 we know that two positive equilibria can exist simultaneously, which are determined by the roots of the polynomial P (u). For simplicity, we consider in the next results the case M = 0 with Δ ≥ 0. Lemma 4 The equilibrium (u 1 , u 1 + C) is a hyperbolic saddle point. Proof Evaluating the equilibrium (u 1 , u 1 + C) in the jacobian matrix (5) we obtain det DYν (u 1 , u 1 + C) = −Su 1 (C + u 1 )2

√

Δ < 0,

and according to the trace and determinant theorem, the equilibrium (u 1 , u 1 + C) is a hyperbolic saddle point. Before determining the nature of the equilibrium point (u 2 , u 2 + C), we must show some properties of the system, related to the stable and unstable manifolds of equilibrium (u 1 , u 1 + C). We denote by W+s (u 1 , u 1 + C) the upper stable manifold and W+u (u 1 , u 1 + C) the right unstable manifold of the point (u 1 , u 1 + C), respectively and let W+u (1, 0) be the upper unstable manifold of the saddle point (1, 0). The next technical result was presented in [34] for the strong Allee case. However, they are still valid in the weak Allee effect case since the demonstration does not depend on the sign of the M parameter, by this reason, the proof will be omitted. Proposition 1 Homoclinic and Heteroclinic curves 1. There is a homoclinic curve [35, 36] determined by the curves W+s (u 1 , u 1 + C) and W+u (u 1 , u 1 + C), that is, there exists a subset of parameter values for which the two manifolds coincide. 2. There are conditions in the parameter space for which a heteroclinic curve γ [35, 36] is generated determined by the upper stable manifold W+s (u 1 , u 1 + C) and the upper unstable manifold W+u (1, 0). The nature of the equilibrium point (u 2 , u 2 + C) depends on the relative positions of the upper stable manifold W+s (u 1 , u 1 + C) and the upper unstable manifold W+u (1, 0) from the saddle point (1, 0) . Theorem 1 Nature of the equilibrium point (u 2 , u 2 + C) Let (u ∗∗ , v s ) ∈ W+s (u 1 , u 1 + C) and (u ∗∗ , v u ) ∈ W+u (1, 0), with 0 < u 1 < u ∗∗ < 1. 1. Suppose that v s > v u . The point (u 2 , u 2 + C) is 1.1 an attractor, if and only if, S > u 2 (−2u 2 + 1) , 1.2 a repellor, if and only if, S < u 2 (−2u 2 + 1) , 1.3 a weak focus, if and only if, S = u 2 (−2u 2 + 1) . 2. Suppose that v s < v u , then the equilibrium point (u 2 , u 2 + C) is

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2.1 an attractor surrounded by an unstable limit cycle. 2.2 a hyperbolic repeller node or focus and the trajectories of system (3) have the point (0, C) as their ω − limit, which is an almost globally stable equilibrium [37, 38]. Proof 1. Suppose v s > v u . Then, W+s (u 1 , u 1 + C) is above W+u (1, 0). The nature depends on the sign of the trace. It has: trDYν (u 2 , u 2 + C) = (u 2 (−2u 2 + 1) − S) (u 2 + C) , whose sign depends on the factor: T1 = u 2 (−2u 2 + 1) − S. The thesis is fulfilled, according to the sign of T1 . 2. Suppose v s < v u . Then, W+s (u 1 , u 1 + C) is below of W+u (1, 0). 2.1 if T1 < 0, then the equilibrium point (u 2 , u 2 + C) is hyperbolic attractor (focus or node). Then, the ω − limit of W+u (u 1 , u 1 + C) is the point (u 2 , u 2 + C), or an attractor limit cycle, surrounding an unstable limit cycle and the equilibrium point (u 2 , u 2 + C). 2.2 if T1 > 0, then the equilibrium point (u 2 , u 2 + C) is hyperbolic repellor (focus or node). So the solutions with α − limit in the neighborhood of point (u 2 , u 2 + C) have as ω − limit an attractor (or stable) limit cycle (see Fig. 3), when (u 2 , u 2 + C) is focus, or else, the point (0, C), when (u 2 , u 2 + C) is a node.

1

P2

v

Fig. 3 For M = 0, Q = 0.4875, A = 0.0005, C = 0.025, S = 0.019 the equilibrium point (u 2 , u 2 + C) is a hyperbolic repellor sorrounded by a stable limit cycle. The equilibrium point (u 1 , u 1 + C) is a hyperbolic saddle point

Stable limit cycle

(0,C)

0

P1 (1,0)

(0,0)

0

1

u

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3.3 Hopf and Bogdanov–Takens Bifurcation We have seen that the stability of (u 2 , u 2 + C) depends on the sign of tr (DYν (u 2 , u 2 + C)). Here, we consider S as a bifurcation parameter. Now, the equilibrium point looses its stability when the sign of tr (DYν (u 2 , u 2 + C)) changes from negative to positive due to variation of S. Solving tr (DYν (u 2 , u 2 + C)) = 0, we get Proposition 2 There is a Hopf bifurcation [35, 36] at the equilibrium point (u 2 , u 2 + C) for the value of the bifurcation S = u 2 (−2u 2 + 1). Proof The co-existing equilibrium (u 2 , u 2 + C) of the system (3) undergoes a Hopf bifurcation at S if DYν (u 2 , u 2 + C; S) > 0 and the transversality condition [35, 36] ∂ (trDYν (u 2 , u 2 + C)) | S = 0, ∂S is verified. In this case, we have tr (DYν (u 2 , u 2 + C)) = 0 implies (u 2 (−2u 2 + 1) − S)(u 2 + C) = 0, then S = u 2 (−2u 2 + 1). From Theorem above det DYν (u 2 , u 2 + C; S)) > 0, also ∂ (trDYν (u 2 , u 2 + C)) = − (u 2 + C) = 0, ∂S which proves the statement. We prove that when the equilibrium point (u 2 , u 2 + C) is a repeller, a stable limit cycle can be generated by Hopf bifurcation. This limit cycle can be broken and the solutions will tend over time to the attractor point (0, C) as was shown in the Theorem 1 (Fig. 4). The stability of the limit cycle, formed due to Hopf bifurcation, is determined by the first Lyapunov coefficient l1 . The Hopf bifurcation is supercritical and results in a stable limit cycle for l1 < 0. On the other hand, the Hopf bifurcation is subcritical and results in an unstable limit cycle for l1 > 0. The analysis of the weakness of the focus and the obtaining of the Liapunov coefficients will be considered in subsequent works. Bogdanov–Takens (BT) bifurcation [35, 36], occurs at the intersection of the Hopf and saddle-node bifurcation curves if we examine the two-dimensional parametric plane. According to our observations, the system (3) experiences saddle-node bifurcation in response to changes in the parameter M, particularly when M ≤ 0. In Lemma 4 above, we prove that the interior equilibrium point (u 1 , u 1 + C) is a saddle point. Furthermore, in Proposition 2, the existence of Hopf bifurcation at the equilibrium point (u 2 , u 2 + C) was shown. Finally, in Sect. 2.2 we establish that there is parameter conditions for which the interior equilibrium points collapse 1 a1 , (u 1 = u 2 = u ∗ ). In particular, when a0 , a1 > 0 and Δ = 0 we have u ∗ = 2(AQ+1) and we obtain the following result

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Fig. 4 Figure represents a critical transition through one-parameter bifurcation diagram. ‘H’ means Hopf bifurcation and ‘LP’ means limit point. For the parameter set Q = 0.4875, A = 0.0005, C = 0.025, S = 0.019 there is a Hopf bifuration for M = 0 in the interior positive equilibrium point (u 2 , u 2 + C). In (0, 0) there is a a neutral saddle labeled H. This is not a bifurcation point for the equilibrium, since it is a saddle point

Theorem 2 The equilibrium point (u ∗ , u ∗ + C) is 1. a repellor saddle-node, if and only if, u ∗ (−2u ∗ + 1) > S, 2. an attractor saddle-node, if and only if, u ∗ (−2u ∗ + 1) < S, 3. a cusp point, if and only if, u ∗ (−2u ∗ + 1) = S, Proof Suppose that a BT bifurcation is observed around (u ∗ , u ∗ + C) for parameter values S and C. Then the following conditions must hold detDYν (u ∗ , u ∗ + C) = 0 and trDYν (u ∗ , u ∗ + C) = 0. When detDYν (u ∗ , u ∗ + C) = 0, it is clear that the points (u 1 , u 1 + C) and (u 2 , u 2 + C) coincide. Where   tr DYν u ∗ , u ∗ + C = (u ∗ (−2u ∗ + 1) − S)(u ∗ + C) = 0, then the statement is fulfilled, according to the sign of the factor u ∗ (−2u ∗ + 1) − S. As a remark, the determinant and the trace of the Jacobian matrix evaluated in the corresponding equilibrium must vanish for the Bogdanov–Takens Bifurcation to exist, and both requirements must be met simultaneously. This suggests that both eigenvalues cancel and the saddle point becomes a cusp point. At the cusp bifurcation point two branches of saddle-node bifurcation curve meet tangentially, forming a semicubic parabola. For nearby parameter values, the system can have two equilibria that collide and disappear pairwise via the saddle-node bifurcations (Fig. 5). The codimension of a bifurcation is the number of parameters that must be varied for the bifurcation to occur. Saddle-node and Hopf bifurcations are codimensionone generic local bifurcations. Nevertheless, because the normal forms can be written with only one parameter, transcritical and pitchfork bifurcations are frequently

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0.2

0.15

u

0.1

0.05

BT

0

CP -1

-0.8

-0.6

-0.4

-0.2

0

0.2

M

Fig. 5 Figure represents a critical transition through one-parameter bifurcation diagram. ‘BT’ means Bogdanov-Takens bifurcation and ‘CP’ means cusp point. For the parameter set Q = 0.4875, A = 0.0005, C = 0.97, S = 0.01 there is a BT bifuration for M = −0.47 in the interior positive equilibrium point (0.02, 0.98)

thought of as codimension-one. The Bogdanov–Takens bifurcation presented in this work is a codimension-two bifurcation. However, the proof of this statement requires normal form results and the Center manifold theorem and will be given in future research.

4 Conclusions In this work, we have studied a modification of the Leslie-Gower model [29, 30], considering collaboration between predators and weak Allee effect, extending the results obtained in a previous article [5, 34]. A special emphasis was given to the study of some local bifurcations in the system (2), complementing the results previously published by the authors for the strong Allee effect case [34]. For any parameter conditions, there is an attractor equilibrium point on the vertical axis, but there can also be a locally stable positive equilibrium point. These findings imply that both populations could coexist or else the prey population disappears, while the predator population persists because of having an alternative food. However, both populations will not become extinct simultaneously, where the origin is always a saddle point (hyperbolic or no hyperbolic). In addition, when there is a limit cycle obtained by Hopf bifurcation, then the populations subsist in the environment, continuously varying the population sizes. Since the analyzed model is based on the linear functional response, which does not produce oscillations if the collaboration between predators to capture their prey is not taken into account, these oscillations can only be induced due to cooperation. We note that there are no significant differences in the dynamics of the studied system, comparing the model studied in this work, with the one that considers the strong Allee effect [34]. In general, there is no qualitative difference, except that it has one less equilibrium point. This implies that the behavior of the system depends

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mainly on the collaboration between generalist predators, rather than on the Allee effect acting on the prey population. Some differences to be highlighted in the case of a weak Allee effect in the prey population are: 1. Permanence: It is possible to prove that the solutions, in this case, remain above a minimum value (positive lower bound) unlike the strong case, where there is a threshold of extinction of the population. 2. Saddle-node bifurcation: It was proved that conditions exist for the collapse of the singularities (0, 0) and (M, 0), indicating that although the singularity (M, 0) does not belong to the Ω region is still influential (indirectly) in the dynamical behavior.

References 1. González-Olivares E, Mena-Lorca J, Rojas-Palma A, Flores JD (2011) Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey. Applied Mathematical Modelling 35(1):366–381, https://doi.org/10.1016/j.apm.2010.07.001, https:// www.sciencedirect.com/science/article/pii/S0307904X10002507 2. González-Olivares E, nez BGY, Mena-Lorca J, Ramos-Jiliberto R (2007) Modelling the Allee effect: Are the different mathematical forms proposed equivalents? In: Mondaini R (ed) Proc. of the 2006 International Symposium on Mathematical and Computational Biology: BIOMAT 2006, Editora E-papers, pp 53–71 3. Teixeira Alves M, Hilker FM (2017) Hunting cooperation and Allee effects in predators. Journal of Theoretical Biology 419:13–22, https://doi.org/10.1016/j.jtbi.2017.02.002, https://www. sciencedirect.com/science/article/pii/S0022519317300577 4. Martínez-Jeraldo N, Rozas-Torres E, González-Olivares E (2021) A Leslie-Gower-type predation model considering generalist predators and the Allee effect on prey. Selecciones Matemáticas 8(01):147–160, https://doi.org/10.17268/sel.mat.2021.01.14, https://revistas.unitru.edu. pe/index.php/SSMM/article/view/3712 5. Ye P, Wu D (2020) Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system. Chinese Journal of Physics 68:49–64, https://doi.org/10.1016/j.cjph. 2020.07.021, https://www.sciencedirect.com/science/article/pii/S0577907320302392 6. Jang SRJ, Zhang W, Larriva V (2018) Cooperative hunting in a predator-prey system with Allee effects in the prey. Natural Resource Modeling 31(4):e12,194, https://doi.org/10.1111/ nrm.12194, https://onlinelibrary.wiley.com/doi/abs/10.1111/nrm.12194 7. González-Olivares E, Cabrera-Villegas J, Lepe FC, Rojas-Palma A (2019) Competition among predators and Allee effect on prey, their influence on a Gause-type predation model. Mathematical Problems in Engineering 2019:1–19, https://doi.org/10.1155/2019/3967408, https:// www.hindawi.com/journals/mpe/2019/3967408/ 8. González-Olivares E, Valenzuela-Figueroa S, Rojas-Palma A (2019) A simple Gause-type predator-prey model considering social predation. Mathematical Methods in the Applied Sciences 42(17):5668–5686, https://doi.org/10.1002/mma.5372, https://onlinelibrary.wiley.com/ doi/abs/10.1002/mma.5372 9. Antonelli P, Kazarinoff N (1984) Starfish predation of a growing coral reef community. Journal of Theoretical Biology 107(4):667–684, https://doi.org/10.1016/S0022-5193(84)80138-1, https://www.sciencedirect.com/science/article/pii/S0022519384801381 10. Francomano E, Hilker FM, Paliaga M, Venturino E (2018) Separatrix reconstruction to identify tipping points in an eco-epidemiological model. Applied Mathematics and Computation 318:80–91, https://doi.org/10.1016/j.amc.2017.07.022, https://www.sciencedirect.com/

Influence of the Collaboration Among Predators and the Weak Allee Effect on Prey …

11.

12.

13.

14.

15.

16. 17.

18.

19.

20. 21. 22. 23.

24.

25. 26. 27.

28.

163

science/article/pii/S0096300317304733, recent Trends in Numerical Computations: Theory and Algorithms MacNulty DR, Tallian A, Stahler DR, Smith DW (2014) Influence of group size on the success of wolves hunting bison. Plos One 9(11):1–8, https://doi.org/10.1371/journal.pone.0112884, https://doi.org/10.1371/journal.pone.0112884 Antonelli P, Lin X (1990) Bifurcation analysis on a coral-starfish model. Mathematical and Computer Modelling 13(6):35–44, https://doi.org/10.1016/0895-7177(90)90007-A, https:// www.sciencedirect.com/science/article/pii/089571779090007A Antonelli P, Kazarinoff N (1988) Modelling density-dependent aggregation and reproduction in certain terrestrial and marine ecosystems: A comparative study. Ecological Modelling 41(3):219–227, https://doi.org/10.1016/0304-3800(88)90029-4, https://www.sciencedirect. com/science/article/pii/0304380088900294 Bazykin AD, Khibnik AI, Krauskopf B (1998) Nonlinear Dynamics of Interacting Populations. World Scientific, https://doi.org/10.1142/2284, https://www.worldscientific.com/doi/abs/10. 1142/2284 González-Olivares E, Rojas-Palma A (2020) Influence of the strong Allee effect on prey and the competition among predators in Leslie-Gower type predation models. Selecciones Matemáticas 7(02):302–313, https://doi.org/10.17268/sel.mat.2020.02.12, https://revistas.unitru.edu. pe/index.php/SSMM/article/view/3206 Maynard-Smith J (1974) Models in Ecology. Cambridge University Press González-Olivares E, González-Yañez B, Becerra-Klix R, Ramos-Jiliberto R (2017) Multiple stable states in a model based on predator-induced defenses. Ecological Complexity 32:111–120, https://doi.org/10.1016/j.ecocom.2017.10.004, https://www.sciencedirect.com/ science/article/pii/S1476945X17300867 Boukal DS, Berec L (2002) Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters. Journal of Theoretical Biology 218(3):375– 394, https://doi.org/10.1006/jtbi.2002.3084, https://www.sciencedirect.com/science/article/ pii/S0022519302930845 Stephens PA, Sutherland WJ (1999) Consequences of the Allee effect for behaviour, ecology and conservation. Trends in Ecology Evolution 14(10):401–405, https:// doi.org/10.1016/S0169-5347(99)01684-5, https://www.sciencedirect.com/science/article/pii/ S0169534799016845 Stephens PA, Sutherland WJ, Freckleton RP (1999) What is the Allee effect? Oikos 87(1):185– 190, http://www.jstor.org/stable/3547011 Kot M (2001) Elements of Mathematical Ecology. Cambridge University Press, https://doi. org/10.1017/CBO9780511608520 Courchamp F, Berec L, Gascoigne J (2008) Allee Effects in Ecology and Conservation. Oxford biology readers, OUP Oxford Dennis B (1989) Allee effects: Population growth, critical density, and the chance of extinction. Natural Resource Modeling 3(4):481–538, https://doi.org/10.1111/j.1939-7445.1989. tb00119.x, https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1939-7445.1989.tb00119.x Boukal DS, Sabelis MW, Berec L (2007) How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theoretical Population Biology 72(1):136–147, https://doi.org/10.1016/j.tpb.2006.12.003, https://www.sciencedirect. com/science/article/pii/S0040580906001687 Thieme HR (2003) Mathematics in Population Biology. Princeton University Press, http:// www.jstor.org/stable/j.ctv301f9v Philip JR (1957) Sociality and sparse populations. Ecology 38(1):107–111, http://www.jstor. org/stable/1932132 Berec L, Angulo E, Courchamp F (2007) Multiple Allee effects and population management. Trends in Ecology Evolution 22(4):185–191, https://doi.org/10.1016/j.tree.2006.12.002, https://www.sciencedirect.com/science/article/pii/S0169534706003818 Boukal DS, Berec L (2009) Modelling mate-finding Allee effects and populations dynamics, with applications in pest control. Population Ecology 51(3):445–458, https://

164

29. 30.

31. 32.

33.

34.

35. 36.

37.

38.

A. Rojas-Palma and E. González-Olivares doi.org/10.1007/s10144-009-0154-4, https://esj-journals.onlinelibrary.wiley.com/doi/abs/10. 1007/s10144-009-0154-4 Leslie PH (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35(3/4):213–245, http://www.jstor.org/stable/2332342 Leslie PH, Gower JC (1960) The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika 47(3/4):219–234, http://www.jstor.org/stable/ 2333294 Freedman H (1980) Deterministic Mathematical Models in Population Ecology. Monographs and textbooks in pure and applied mathematics, M. Dekker Goh BS (1980) Management and Analysis of Biological Populations, Developments in Agricultural and Managed Forest Ecology, vol 8. Elsevier, https://doi.org/ 10.1016/B978-0-444-41793-0.50006-X, https://www.sciencedirect.com/science/article/pii/ B978044441793050006X Courchamp F, Clutton-Brock T, Grenfell B (1999) Inverse density dependence and the Allee effect. Trends in Ecology Evolution 14(10):405–410, https://doi.org/10.1016/S01695347(99)01683-3, https://www.sciencedirect.com/science/article/pii/S0169534799016833 González-Olivares E, Rojas-Palma A (2021) Influencia del efecto Allee en las presas y de la colaboración entre los depredadores en un modelo de depredación del tipo Leslie-Gower. Modelamiento Matemático de Sistemas Biológicos (MMSB) 1(2):21–29, https://revistammsb. utem.cl/articulos/ Chicone C (2008) Ordinary Differential Equations with Applications. Texts in Applied Mathematics, Springer New York Perko L, Perko L (2001) Differential Equations and Dynamical Systems. Texts in Applied Mathematics, Springer New York, https://link.springer.com/book/10.1007/978-1-4613-00038 Monzón P (2005) Almost global attraction in planar systems. Systems Control Letters 54(8):753–758, https://doi.org/10.1016/j.sysconle.2004.11.014, https://www.sciencedirect. com/science/article/pii/S0167691104002099 Rantzer A (2001) A dual to lyapunov’s stability theorem. Systems Control Letters 42(3):161–168, https://doi.org/10.1016/S0167-6911(00)00087-6, https://www.sciencedirect. com/science/article/pii/S0167691100000876

Experience in Teaching Mathematics to Engineers: Students Versus Teacher Vision Cristina M. R. Caridade

1 Introduction To capture students’ attention and interest, teachers must adopt alternatives to traditional teaching. Some researchers indicate the need for teachers to use new, more student-centered teaching approaches as a way to face these challenges [1–3]. Collaborative Learning is an active learning strategy, which allows for meaningful learning in which the student takes an active role in their learning process [4], developing a wide range of competence in the group, such as sharing ideas, interaction, collaboration and discussion between elements of the same group (Fig. 1). In collaborative learning students have the opportunity to work independently of each other and build their own meaning so that they can contribute to the group discussion promoting critical thinking [5, 6]. Despite the advantages of collaborative learning, there is little evidence of its implementation or effectiveness in higher education institutions [6], since, at this academic level, collaborative work is limited almost exclusively to participation in research projects on technical topics. It is pertinent that a greater reflection on didactic knowledge is also initiated in higher education institutions. Collaborative learning emerges as an opportunity for higher education teachers in the areas of Science, Technology, Engineering and Mathematics (STEM), to investigate their own practice, in addition to sharing, developing and improving their teaching experiences with other [7]. On the other hand, GeoGebra is a technological tool, free and open, that can be used in Mathematics classes, as it allows the visualization of geometric representations, contributing to the understanding of concepts seen in theoretical classes [8–10]. Revolution solids are one of the syllabus of first-year engineering students, and are C. M. R. Caridade (B) Coimbra Institute of Engineering, Polytechnic of Coimbra, Portugal, Centre For Research in Geo-Space Science (CICGE), Porto, Portugal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_16

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Fig. 1 Collaborative learning. https://acsieu.org/wp/collaboration/the-benefits-of-collaborativelearning/

very important in areas such as Mechanical, Civil, Electrical, etc. However, students have a lot of difficulties visualizing them in three dimensions, so with the use of GeoGebra, students’ visual representations are more effective, which shows that GeoGebra can improve students’ visual representation skills [11, 12]. Thus, this paper intends to present the reflections of a higher education mathematics teacher about her participation in a collaborative mathematics class for engineers where GeoGebra is integrated. Students’ perspectives will also be presented through the activities developed during the collaborative class and through two quizzes.

2 Methodology During the second semester of the 2021/2022 academic year, the Calculus 1 course is repeated for all students who did not succeed in the 1st semester. It is an opportunity offered by the school as a continuation of the 1st semester and which has the objective of having a greater number of students pass the subject during the academic year. In this second semester, classes are more focused on practices and laboratories, since the theoretical contents have already been developed in the 1st semester, so that students can train with a greater amount of exercises and problems. This year, only 25 Mechanical Engineering students signed up to take the course. The experience described here was carried out in a 3-hour class using collaborative learning where students, following the activity guide proposed by the teacher, explored one of the course contents. Students worked in groups (6 groups) using computers (GeoGebra, Excel, MicrosoftWord, etc.), calculators and mobile phones. The topic chosen to explore was the construction of solids of revolution, the calculation of planar areas and volumes of solids of revolution following the learning plan:

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Fig. 2 Examples of parts of the task developed by students

• Choosing an object that can be considered a solid of revolution. • Photograph the object, measure it and insert the photograph into the GeoGebra application. • Position the object on the axes and set its scale. • Build the interpolating polynomial that limits the planar region. • Model the 3D object in GeoGebra starting from the planar region and rotating around one of the coordinate axes. • Insert the 3D object into augmented reality using the GeoGebra AR application on the mobile phone. Students complete all group activity following a guide. In Fig. 2 two examples of the guide from two different groups are represented. On the left, the task that consists of a photo of an object that can be considered a solid of revolution and on the right, a sequence of prints of the task in which it is necessary to insert the 3D object created in GeoGebra into augmented reality. Students are also asked to apply the knowledge acquired in the Calculus 1 course on the application of integrals to calculate planar areas and volumes of solids of revolution. In this sense, the students performed these calculations by hand with the help of a calculator or GeoGebra and presented them in the activity guide. In the Fig. 3 on the left the student calculates the volume of the solid of revolution by hand (top) and confirms its value using GeoGebra (bottom). In the example shown on the right, the student performs his calculations on the calculator by defining the interpolating polynomial (top) and the calculation of the planar area and volume of the solid of revolution (bottom). During the activity, students share ideas, opinions and ways to solve the tasks. They use various tools and applications on computers and mobile phones. Their discussions and exchange of ideas begin to be more structured throughout the activity. They help each other in their learning. The teacher circulates around the classroom, accompanies the students, observes and notes the behavior and performance of the

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Fig. 3 Examples of parts of the task developed by students

students. For each student, the teacher will have to indicate an individual and group score, the final result being an average of these two values. The teacher has the task of guiding students by letting them navigate their own learning. Before and after the activity, students answered two quizzes anonymously, the first to find out what knowledge students already bring about GeoGebra, revolution solids and augmented reality, the second quiz to get students’ opinions about the activity that just accomplished. Due to the purpose and nature of this study, the investigation fits into a qualitative and interpretive approach, based on a participant observation design [13, 14].

3 Results and Discussions The two quizzes, which took place before and after the task, were answered by 25 students. In the first quiz, with the answer options “Yes”, “No” and “I don’t know”, the students commented on their experience with the use of GeoGebra and GeoGebra AR or GeoGebra 3D (depending on the operating system of their mobile phones or tablets), the use of the mobile phone in a school environment and the use of GeoGebra to create a solid of revolution. They answered the following questions: (1) Have I done GeoGebra activities in other contexts?; (2) Have I done GeoGebra AR or GeoGebra 3D activities in other contexts?; (3) Have I used my mobile phone in an activity of another discipline?; (4) Have I used GeoGebra to create solids of revolution?

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Fig. 4 First quiz, questions (1) on left and (2) on right

Fig. 5 First quiz, questions (3) and (4)

To the first question 64% (16 students) answered “No”, and to the second question 96% (24 students) answered “No” (Fig. 4). It was observed during the class where the activity was carried out that most students had no experience with GeoGebra, much less with GeoGebra AR or Geogebra 3D. That is, even the students who answered “Yes” to both questions had clear difficulties during the tasks. In the answer to question (3), 40% answer “No” and 60% “Yes” (Fig. 5 on left). Which leads to the conclusion that the use of mobile phones in the school environment is already starting to be a reality. Finally, in question (4), represented in Fig. 5 on right, 88% of the students answer “No” before using Geogebra to create solids of revolution. The second quiz was carried out at the end of the activity and consists of ten questions. The first seven questions with “Yes”, “No” or “I don’t know” answers were as follows: (1) Was my contribution valuable to the group during the task?; (2) Was I committed and focused on completing the task?; (3) Am I confident in what I learned about the solid of revolution?; (4) Did I enjoy doing the task in GeoGebra?; (5) Did I enjoy doing the task in GeoGebra AR or Geogebra 3D?; (6) Do I like innovative working methods implemented in the classroom?; (7) Do I think active learning makes my learning more effective? These questions were answered by 25 students. In questions (1), (2), (3), (6), (7), all 25 students answered “Yes” (Fig. 6 on left). In questions (4), (5), 96% (25 students)

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Fig. 6 Final quiz, questions (1) to (7)

of the students answered “Yes” and only 1 student (4%) answered “No” (Fig. 6 on right). The students’ answers indicate that the activity developed, different from the usual, was to their liking and that they were committed to carrying out the proposed tasks and learning effectively. Only one student in question (4) and (5) stated that he did not like using GeoGebra and GeoGebra AR or GeoGebra 3D. Perhaps because it was the first time he had contact with these applications. With the last three development questions, it is intended to appreciate the impact of the activity developed in the students’ teaching and learning process. The open-ended questions were as follows: (7) What did you like most about the activity?; (8) What was your biggest challenge during the activity?; (9) Do you have any suggestions for the next classes? In question (7), 24% (6 students) did not answer, 16% (4 students) indicated that what they liked most about the activity was the group work and the interaction between colleagues, 48% (12 students) said that the modeling of 3D objects in Geogebra and the inclusion of the 3D object in augmented reality, and, finally, 12% (3 students) mentioned that the class dynamics and interactivity was what they liked the most. The following quotes are examples of these responses given by students: “The team spirit of our group.” “The form we create the object in GeoGebra and its insertion in the augmented reality of our classroom.” “The use of new technologies for modeling objects of our daily life.”

Regarding the challenges felt by students during the activity, addressed in question (8), 7 students (28%) did not respond, and the remaining students indicate the difficulty felt in carrying out the proposed activity (2 students) and the use of GeoGebra (16 students). In the case mentioned about the use of GeoGebra, the students refer to the difficulty of using it in general (3 students) and using it on the mobile phone (13 students) as a challenge. The following quotes demonstrate these students’ views: “Learning new tools like GeoGebra.” “Using GeoGebra AR on my mobile phone.” “Model my team object with GeoGebra.”

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Fig. 7 Word cloud

Regarding question (9) about suggestions for future classes, of the 25 students, 76% (19 students) did not answer the question, 8% (2 students) identified not having suggestions and 16% (4 students) said they would like to have more classes of this type. From the answers obtained, it is evident that the students liked this activity carried out in the classroom, so different from the usual and reinforce that they would like to have more classes of this type. This idea is evident in the following quotes: “Continuing to carry out work like this.” “More activities of this type are good because they help me to improve my performance.”

At the end of the activity, and in order to be able to make a general assessment of the students’ interest and motivation during the activity, they were asked to identify 3 words that could describe their view of the activity they had just carried out. The word cloud in Fig. 7 shows the result obtained. Words such as “GeoGebra”, “interesting”, “innovative”, “interactivity”, “different” and “fun” are the most mentioned by students and therefore appear more clearly in the image in Fig. 7. This suggests that the students’ experience during the activity was quite enriching, both in terms of involvement and in terms of learning mathematics. Which confirms the answers obtained in the final quiz. At the beginning of the activity, the teacher observed that the students started having some difficulties in identifying objects in their backpacks that were considered solids of revolution. Hence a little help from the teacher was needed. Discussions and exchange of opinions were more frequent in this first task, since it was necessary to choose only one object for the whole group. However, this aspect was not mentioned by the students as like a challenge. Later the teacher observed that the students showed many difficulties in using GeoGebra. This difficulty would be expected, as it was the first time that the students used this application, an aspect mentioned by the students as a challenge encountered. The calculations that the students performed by hand and

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where it would be important to use the knowledge acquired in the 1st semester of the course, were very well achieved, although some help was needed from the teacher. The modeling of the 3D object was one of the tasks that most pleased the students (second quiz, question (8)), because at this stage they are able to visualize a solid that corresponds to the object they have on their desks. The most exciting and fun task was the insertion of the 3D object created in GeoGebra into augmented reality. Here the teams worked as a whole and everyone wanted to try it out on their mobile phone. However, some difficulties arose from the download of the application on the mobile phone to the creation of the video in augmented reality. Which again was not identified by the students as a challenge (second quiz, question (9)). Throughout the activity, it was possible to observe that all students worked as a team in an active and participatory way, showing that the experience they were carrying out was pleasant and motivating, even for those in which mathematics is the subject they least enjoy. Involvement, dedication and humor were always present in all groups, allowing them to overcome difficulties in a positive way. The analysis of the students’ answers to the quizzes confirmed the behavior observed in the classroom.

4 Final Consideration Collaborative classes are excellent for developing skills in both teachers and students. During the collaborative class described in this paper, the student was motivated to develop reasoning, critical thinking, knowledge and experience, gradually feeling high levels of motivation and participation. The use of collaborative learning in higher education is an opportunity for the teacher to analyze in greater depth the teaching methodologies of mathematics in a practical context, where the dynamics of the classroom assumes particular importance. In this way, it was intended in this paper to identify the students’ vision in confrontation with the teacher’s vision during the realization of a collaborative learning experience. The results presented demonstrate that students benefit a lot from this type of learning, even those who are less good at mathematics. Aspects such as motivation, commitment and dedication are found in each group along with reasoning, critical thinking and knowledge. In a class based on collaborative learning in mathematics, students are led to perform mathematics problems without realizing it. Classroom learning is transformed. To understand the degree of knowledge acquired by the students in the activity described here, on the calculation of planar areas and volumes of solids of revolution, the grades obtained by each of the students in this syllabus during the activity and in the exam are represented in the Fig. 8. The 25 students who attended the activity had positive grades (>70%). 3 of these students did not take the exam and the rest (22 students) also had positive grades (≥70%) and 8 students obtained 100% in the exam. This demonstrates the impact of this strategy on the success of students in the Calculus 1 program.

Experience in Teaching Mathematics to Engineers: Students Versus Teacher Vision

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Fig. 8 Activity and exam grades

As final considerations, the experience in teaching mathematics to engineers seen by the students and the teacher is very similar. The use of active learning, such as collaborative learning, is extremely important in higher education, and especially in mathematics teaching. Because with it, the students are more motivated and involved in their learning, which was possible to observe by the teacher and confirmed by the quizzes made to the students. The development of skills such as speaking and writing mathematics, using tools and technologies, defending opinions and managing a group are also explored in these active learning. What was easily observed during the classes by the teacher and in the activities of the groups. Mathematical skills, such as problem solving, critical thinking, among others, are intrinsically obtained by the students, as it was possible to verify in the resolution of the activities by the groups. This experience allowed a mathematics teacher to confront his vision of an collaborative learning class with the vision of his engineering students. The result obtained was quite satisfactory for the students, as it pleased and allowed them to achieve success, for the teacher, as it was an enriching experience that allowed him to observe the classroom from another perspective. Experiences of this type in mathematics classes in engineering courses are often difficult to implement, due to the teaching load of the subject being reduced for the program to be taught and the high number of students per class. On the other hand, in this type of classes, students need more time to achieve the proposed objectives, and sometimes they do not have the necessary mathematical foundations to be able to overcome the proposed tasks. Therefore, this decision becomes a challenge for the teacher. When, how and in what way to implement collaborative learning in classes.

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In the next academic year, the author intends to apply the same experience to another engineering course with a larger number of students and to develop another activity with collaborative learning, for another theme of the syllabus of the Calculus 1 discipline.

References 1. Keiler, L.S.; Teachers’s roles and identities in student-centered classrooms. IJ STEM Ed 5, 34 (2018). https://doi.org/10.1186/s40594-018-0131-6 2. Hernández-de-Menéndez, M., Vallejo Guevara, A., Tudónez, J.C. et al.: Active learning in engineering education. A review of fundamentals, best practices and experiences. Int J Interact Des Manuf 13, 909–922 (2019). https://doi.org/10.1007/s12008-019-00557-8 3. Lesik, S.A.: Do developmental mathematics programs have a causal impact on student retention? An application of discrete-time survival and regression-discontinuity analysis, Researchin Higher Education, 48(5), 583–608, (2007). https://doi.org/10.1007/s11162-006-9036-1 4. Herrera-Pavo, M.A., Collaborative learning for virtual higher education, Learning, Culture and Social Interaction, Elsevier, 28 (2021). 5. Smith, B.L., MacGregor, J.T.: What Is Collaborative Learning? Washington Center for Improving the Quality of Undergraduate Education. pp. 1–11 (1992). https://www.researchgate.net/ profile/Jean-Macgregor/publication/242282475_What_is_Collaborative_Learning/links/ 53f279060cf2bc0c40eaa8be/What-is-Collaborative-Learning.pdf 6. Gokhale, A.: Collaborative Learning Enhances Critical Thinking’. Journal of Technology Education, 7(1. (1995) http://scholar.lib.vt.edu/ejournals/JTE/jte-v7n1/gokhale.jte-v7n1.html 7. Le, H., Jeroen Janssen, J., Wubbels, T.: Collaborative learning practices: teacher and student perceived obstacles to effective student collaboration, Cambridge Journal of Education, 48:1, 103–122, (2018) https://doi.org/10.1080/0305764X.2016.1259389 8. Adhikari, G.P.: Effect of using GeoGebra software on students’ achievement at university level, Scholars’ Journal, 3, pp. 47–60. 2020. https://doi.org/10.3126/SCHOLARS.V3I0.37129 9. Houssam, K., JurdakMurad E.J.: Effect of GeoGebra collaborative and iterative professional development on in-service secondary mathematics teachers’ practices. Cerme10 TWG 15 Teaching mathematics with resources and technologyAt, (2021). 10. Caridade, C.M.R.: GeoGebra augmented reality: ideas for teaching & learning math. II Internacional Conference on Mathematics and its applications in science and Engineering (ICMASE 2021) 01–12 July 2021, Universidad de Salamanca, (2021). 11. Azizah, A.N., Kusmayadi,T.A., Fitriana, L.: The Effectiveness of Software GeoGebra to Improve Visual Representation Ability, Journal of Physics: Conference Series, 1808 (2021) 012059, (2021). https://doi.org/10.1088/1742-6596/1808/1/012059 12. Wen, Y.: Augmented reality enhanced cognitive engagement: designing classroom-based collaborative learning activities for young language learners. Education Tech Research Dev 69, 843–860 (2021). https://doi.org/10.1007/s11423-020-09893-z 13. Bogdan, R., Bikle, S.: Qualitative Research in Education: an introduction to theory and methods, (M. J. Alvarez, S. B. Santos & T. M. Baptista, Transl.). Porto: Porto Editora, 1994. 14. Jorgensen, D.L.: Participant observation: A methodology for human studies, Newbury Park: Sage, 1989.

On Some Q-Dual Bicomplex Jacobsthal Numbers Serpil Halıcı and Sule Curuk

1 Introduction and Preliminaries The set of complex numbers with coefficients from complex numbers is denoted by BC or C2 and this set is defined as BC = {z 1 + jz 2 : j2 = −1}.

(1)

i and j are two different imaginary units and are commutative units between them (see [1–3]). The bicomplex number system (BC, +, .) is a commutative ring with zero as additive identity and unity as multiplicative identity but BC is not a field due to presence of zero divisors. The zero divisors in BC are in the form {a(1 ± ij) : a ∈ C}. Since any bicomplex number Z can be written as Z = z 1 + jz 2 = x1 + ix2 + jx3 + jix4 , the sets BC and R4 are isomorphic and so the set of units {1, i, j, ij} is a basis in 4-dimensional real space. The set of bicomplex numbers with coefficients from Jacobsthal sequence is defined by (2) BJn = {G Jn + jG Jn+2 : j2 = −1} where G Jn = Jn + iJn+1 and Jn are nth Gaussian Jacobsthal number and nth Jacobsthal numbers, respectively[4]. Similarly, the set of bicomplex Jacobsthal-Lucas numbers is (3) Bjn = {G jn + jG jn+2 : j2 = −1}

S. Halıcı · S. Curuk (B) Pamukkale University, Kinikli Campus, 20100 Denizli, Turkey e-mail: [email protected] S. Halıcı e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_17

175

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where G jn = jn + i jn+1 and jn is nth Jacobsthal-Lucas number. In [5], bicomplex Jacobsthal-Lucas numbers and important identities related to these numbers are given. In [6], the author studied bicomplex third-order Jacobsthal numbers and gave the Binet formula, generating function and some properties of this sequence. In [2], the authors studied dual bicomplex numbers and examined their some important properties. In [7], the author examined the Fibonacci and Lucas numbers in detail. In [1], the author defined dual bicomplex Fibonacci and Lucas numbers and discussed their properties. In [8], the authors described q-Fibonacci and Lucas dual bicomplex numbers, and they gave some useful properties and identities related to these numbers. In [9], the author defined the q-Fibonacci bicomplex numbers and the q-Lucas bicomplex numbers. Then, he gave some algebraic properties of the q-Fibonacci bicomplex numbers and the q-Lucas bicomplex numbers. In [10], Pashaev and Nalci studied quantum harmonic oscillator for Golden calculus and Binet-Fibonacci calculus. In [11], Kızılates and Cekim derived families of multilinear and multilateral generating functions for Fibonacci and Lucas polynomials based on q-integers. In [12], Akkus and Kizilaslan examined the quantum approach to Fibonacci quaternions in a study they conducted in 2019. In [13], Kizilates and Polatli studied q-Fibonacci octonions and q-Lucas octonions and gave Binet formula to find the nth term. Also, they gave exponential generating function and some known identities involving these numbers. In this study, we introduce the dual bicomplex numbers with coefficients from q-Jacobsthal sequence. By examining the q-versions of these numbers, we give some useful identities related with them. As it is known, q numbers play an important role in many fields of mathematics, physics and especially number theory.

2 Dual Bicomplex Numbers with Coefficients from q-Jacobsthal Numbers In this section, using all the studies on q-forms, we give the bicomplex numbers with coefficients from q-Jacobsthal and Jacobsthal-Lucas numbers. Definition 1 The nth terms of the bicomplex number sequence with coefficients from q-Jacobsthal and Jacobsthal-Lucas numbers can be defined as follows, respectively. BJn (α, q) =

1 n α ([n]q + iα[n + 1]q + jα 2 [n + 2]q + ijα 3 [n + 3]q ) 2

(4)

and Bjn (α, q) = α n [(1 + q n ) + iα(1 + q n+1 ) + jα 2 (1 + q n+2 ) + ijα 3 (1 + q n+3 )]. (5)

On Some Q-Dual Bicomplex Jacobsthal Numbers

177

Also, for q- Jacobsthal-Lucas bicomplex numbers we write [2n]q [2n + 2]q [2n + 4]q [2n + 6]q + iα n+1 + jα n+2 + ijα n+3 . [n]q [n + 1]q [n + 2]q [n + 3]q (6) The q-analogue of a complex number n is defined Bjn (α, q) = α n

[n]q =

1 − qn = 1 + q + q 2 + . . . + q n+1 , q = 1. 1−q

(7)

For detailed information about q integers, refer to the reference [14]. In the definition below, we examine dual bicomplex numbers with coefficients from q-Jacobsthal and Jacobsthal-Lucas numbers. Definition 2 The nth terms of the q-Jacobsthal and Jacobsthal-Lucas dual bicomplex number sequences can be defined as follows, respectively. DBJn (α, q) = BJn (α, q) + BJn+1 (α, q)

(8)

DBjn (α, q) = Bjn (α, q) + Bjn+1 (α, q).

(9)

and

We define dual bicomplex numbers with q-Jacobsthal number coefficients DBJn (α, q) =

1 n α (A + α B), 2

(10)

where A and B are A = [n]q + iα[n + 1]q + jα 2 [n + 2]q + ijα 3 [n + 3]q ,

(11)

B = [n + 1]q + iα[n + 2]q + jα 2 [n + 3]q + ijα 3 [n + 4]q .

(12)

Also, we can write DBJn (α, q) =

1 n α {(α − q n γ ) + α(α − q n+1 γ )}. 2

(13)

Similar to this definition, it is possible to define dual bicomplex numbers with qJacobsthal Lucas coefficients. That is, for DBjn (α, q) DBjn (α, q) = α n (C + α D).

(14)

Where C and D are follows. C = (1 + q n ) + iα(1 + q n+1 ) + jα 2 (1 + q n+2 ) + ijα 3 (1 + q n+3 ), D = (1 + q n+1 ) + iα(1 + q n+2 ) + jα 2 (1 + q n+3 ) + ijα 3 (1 + q n+4 ).

(15) (16)

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Thus, we get DBjn (α, q) = α n {(α + q n γ ) + α(α + q n+1 γ )}.

(17)

We can also write the above formulas C and D as follows. [2n + 4]q [2n + 6]q [2n]q [2n + 2]q + iα + jα 2 + ijα 3 , [n]q [n + 1]q [n + 2]q [n + 3]q

(18)

[2n + 2]q [2n + 6]q [2n + 8]q [2n + 4]q + iα + jα 2 + ijα 3 . [n + 1]q [n + 2]q [n + 3]q [n + 4]q

(19)

C= D=

The algebraic operations in the sets DBJn and DBjn are given in the following lemma. Lemma 1 For m, n ∈ N, (i) DBJn (α, q) ± DBJm (α, q) is 1 {α(α n−1 ± α m−1 ) − γ (α n−1 q n ± α m−1 q m ) 1−q +[α(α n ± α m ) − γ (α n q n+1 ± α m q m+1 )]}. (ii) DBjn (α, q) ± DBjm (α, q) is α(α n ± α m ) + γ ((αq)n ± (αq)m ) + [α(α n+1 ± α m+1 ) + γ ((αq)n+1 ± (αq)m+1 )].

(iii) DBJn (α, q)DBJm (α, q) is α m+n−2 {(α 2 − (q m + q n )αγ + q m+n γ 2 ) (1 − q)2 +α(2α 2 − (1 + q)(q m + q n )αγ + 2q m+n+1 γ 2 )}. (iv) DBjn (α, q)DBjm (α, q) is α m+n {(α 2 + (q m + q n )αγ + q m+n γ 2 ) + α(2α 2 + (1 + q)(q m + q n )αγ + 2q m+n+1 γ 2 )}.

It is necessary to talk about the conjugates and therefore the norms of these numbers that we have just defined according to the imaginary units in the set BC. For the DBJn (α, q), the conjugate equations are DBJn (α, q) + DBJin (α, q) = 2{(Jn (α, q) +  Jn+1 (α, q) + j(Jn+2 (α, q) +  Jn+3 (α, q))}, DBJn (α, q) + DBJjn (α, q)

(20)

On Some Q-Dual Bicomplex Jacobsthal Numbers

179

= 2{(Jn (α, q) +  Jn+1 (α, q) + i(Jn+1 (α, q) +  Jn+2 (α, q))},

(21)

DBJn (α, q) + DBJijn (α, q) = 2{(Jn (α, q) +  Jn+1 (α, q) + ij(Jn+3 (α, q) +  Jn+4 (α, q))}.

(22)

Similarly, for the DBjn (α, q), the conjugate equations are DBjn (α, q) + DBjin (α, q) = 2{( jn (α, q) +  jn+1 (α, q) + j( jn+2 (α, q) +  jn+3 (α, q))},

(23)

DBjn (α, q) + DBjjn (α, q) = 2{( jn (α, q) +  jn+1 (α, q) + i( jn+1 (α, q) +  jn+2 (α, q))},

(24)

DBjn (α, q) + DBjijn (α, q) = 2{( jn (α, q) +  jn+1 (α, q) + ij( jn+3 (α, q) +  jn+4 (α, q))}.

(25)

Binet’s formula is a formula used to find the nth term of sequence. In the following theorem, we give the Binet formulas for the dual bicomplex q-Jacobsthal and qJacobsthal-Lucas numbers. Theorem 1 For the DBJn (α, q) and DBjn (α, q), we have i) DBJn (α, q) = α n−1 (

α − qnγ 1−q

) + α n (

α − q n+1 γ 1−q

),

ii) DBjn (α, q) = (α n α + (αq)n γ ) + (α n+1 α + (αq)n+1 γ ), where α = 1 + i α + j α 2 + ij α 3 , γ = 1 + i αq + j (αq)2 + ij (αq)3 . Proof DBJn (α, q) =

1 n α {([n]q + i α[n + 1]q + j α 2 [n + 2]q + ij α 3 [n + 3]q ) 2

+α([n + 1]q + i α[n + 2]q + j α 2 [n + 3]q + ij α 3 [n + 4]q )}, DBJn (α, q) =

1 n 1 α {(1 − q n + i α(1 − q n+1 ) + j α 2 (1 − q n+2 ) + ij α 3 (1 − q n+3 )) 2 1−q

+α(1 − q n+1 + i α(1 − q n+2 ) + j α 2 (1 − q n+3 ) + ij α 3 (1 − q n+4 ))},

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DBJn (α, q) =

αn {(1 + iα + jα 2 + ijα 3 ) − q n (1 + iαq + j(αq)2 + ij(αq)3 ) 2(1 − q)

+α[(1 + iα + jα 2 + ijα 3 ) − q n+1 (1 + iαq + j(αq)2 + ij(αq)3 )]}, DBJn (α, q) = α n−1 (

α − qnγ 1−q

) + α n (

α − q n+1 γ 1−q

).

Similarly, for DBjn (α, q) we get DBjn (α, q) = α n {[(1 + q n ) + i α(1 + q n+1 ) + j α 2 (1 + q n+2 ) + ij α 3 (1 + q n+3 )], +α[(1 + q n+1 ) + i α(1 + q n+2 ) + j α 2 (1 + q n+3 ) + ij α 3 (1 + q n+4 )]} DBjn (α, q) = α n (1 + iα + jα 2 + ijα 3 ) + (αq)n (q + iαq 2 + jα 2 q 3 + ijα 3 q 4 ), +[α n+1 (1 + iα + jα 2 + ijα 3 ) + (αq)n+1 (q + iαq 2 + jα 2 q 3 + ijα 3 q 4 )]. If necessary operations are made on the last equation, we obtain that DBjn (α, q) = (α n α + (αq)n γ ) + (α n+1 α + (αq)n+1 γ )  

so that the theorem is proved.

In the following corollary, we give some relations between the elements of these two sequences. Corollary 1 For DBJn (α, q) and DBjn (α, q), we have DBJn+1 (α, q) + 2DBJn−1 (α, q) = DBjn (α, q),

(26)

DBjn+1 (α, q) + 2DBjn−1 (α, q) = 9DBJn (α, q),

(27)

DBJn+1 (α, q) + DBJn (α, q) = α n α(1 + α),

(28)

DBjn+1 (α, q) + DBjn (α, q) = 3α n α(1 + α).

(29)

Proof It’s easy to see that these equations are true, but let’s give a proof of one. DBJn+1 (α, q) + 2DBJn−1 (α, q) is equal to this: αn ( =

α − q n+1 γ 1−q

) + α n+1 (

α − q n+2 γ 1−q

) + 2{α n−2 (

α − q n−1 γ 1−q

) + α n−1 (

α − qnγ 1−q

)}

1 {(α n−1 α(α + 1) − (αq)n−1 γ (1 + αq 2 )) + (α n α(α + 1) − (αq)n γ (1 + αq 2 ))}, 1−q

On Some Q-Dual Bicomplex Jacobsthal Numbers

181

DBJn+1 (α, q) + 2DBJn−1 (α, q) = 2(α n−1 α − α −1 (αq)n−1 γ ) + 2(α n α − α −1 (αq)n γ ),

DBJn+1 (α, q) + 2DBJn−1 (α, q) = (α n α + (αq)n γ ) + (α n+1 α + (αq)n+1 γ ), DBJn+1 (α, q) + 2DBJn−1 (α, q) = DBjn (α, q).  

Thus, the proof is completed. 2

For a0 , a1 , a2 , . . . ∈ R, the function f (x) = a0 + a1 x + a2 x2! + · · · is called the exponential generating function. The exponential generating functions for the DBJn (α, q) and DBjn (α, q) numbers are given in the following theorem. Theorem 2 The exponential generating function for the sequences DBJn (α, q) and DBjn (α, q) are follows. i)

∞ 

DBJn (α, q)

n=0

ii)

∞ 

1 xn = {(αeαx − γ eαq x ) + (αeαx − γ qeαq x )}, n! α(1 − q)

DBjn (α, q)

n=0

xn = (αeαx + γ eαq x ) + (αeαx + γ qeαq x ). n!

Proof If the Binet’s formula for DBJn (α, q) is used in the generating function definition, the following equalities are obtained:     n ∞   α − qnγ α − q n+1 γ x xn n−1 n α = + α , DBJn (α, q) n! 1 − q 1 − q n! n=0 n=0

∞ 

=

∞ ∞ γ  α  n−1 x n xn − α α n−1 q n 1 − q n=0 n! 1 − q n=0 n!

∞ ∞ n γ  α  n xn n n+1 x − , α α q + 1 − q n=0 n! 1 − q n=0 n!



=

  γ γ α α eαx − eαq x +  eαx − qeαq x . α(1 − q) α(1 − q) α(1 − q) α(1 − q)

Thus, we obtain that ∞  n=0

DBJn (α, q)

1 xn = {(αeαx − γ eαq x ) + (αeαx − γ qeαq x )}. n! α(1 − q)

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Similarly, the exponential generating function of the dual bicomplex q-JacobsthalLucas sequence is ∞ 

∞

DBjn (α, q)

n=0 ∞ 

 xn xn = (α n α + (αq)n γ + (α n+1 α + (αq)n+1 γ )) , n! n! n=0

DBjn (α, q)

n=0

+(α

∞ 

∞ ∞   xn xn xn αn (αq)n =α +γ n! n! n! n=0 n=0

α n+1

n=0 ∞ 

DBjn (α, q)

n=0

∞  xn xn +γ (αq)n+1 ), n! n! n=0

xn = (αeαx + γ eαq x ) + (αeαx + γ qeαq x ). n!  

Thus, the proof is completed.

In the following theorem, we give the Catalan’s identities for the sequences DBJn (α, q), DBjn (α, q). Theorem 3 For n, r ∈ Z+ and n ≥ r , we have (i) DBJn+r (α, q)DBJn−r (α, q) − DBJ2n (α, q) =

α 2(n−1) αγ q n (1 − q)2

(2 − q r + q −r )

1 

(α(1 + q))k ,

k=0

(ii) DBjn+r (α, q)DBjn−r (α, q) − DBj2n (α, q) = α 2n αγ q n (q r + q −r − 2)

1 

(α(1 + q))k .

k=0

Proof DBJn+r (α, q)DBJn−r (α, q) − DBJ2n (α, q) is equal to = +

α 2n−2 (α 2 − αγ q n−r − αγ q n+r + q 2n γ 2 − α 2 + 2αγ q n − q 2n γ 2 ) (1 − q)2

α 2n− (α 2 − αγ q n−r +1 − αγ q n+r + q 2n+1 γ 2 + α 2 − αγ q n−r − αγ q n+r +1 + q 2n+1 γ 2 (1 − q)2

−2α 2 + 2αγ q n+1 + 2αγ q n − 2q 2n+1 γ 2 )

On Some Q-Dual Bicomplex Jacobsthal Numbers

183

Here, after the necessary operations and simplifications, we can write the following equations. DBJn+r (α, q)DBJn−r (α, q) − DBJ2n (α, q) =

+

α 2n−1 αγ (1 − q)2

α 2n−2 αγ (1 − q)2

q n (2 − q r + q −r )

q n (2 + 2q − q r +1 − q −r + q r − q −r +1 ),

DBJn+r (α, q)DBJn−r (α, q) − DBJ2n (α, q) =

α 2(n−1) αγ q n (1 − q)2

(2 − q r + q −r )

1 

(α(1 + q))k .

k=0

Thus, the claim is proved. Similarly, DBjn+r (α, q)DBjn−r (α, q) − DBj2n (α, q) is α 2n q n−r αγ + α 2n q n+r αγ − 2α 2n q n αγ +α 2n+1 (q n−r +1 αγ + q n+r αγ + q n−r αγ + q n+r +1 αγ − 2q n+1 αγ − 2q n αγ ) = α 2n q n αγ (q r + q −r − 2) + α 2n+1 αγ q n (q r + q −r + q −r +1 + q r +1 − 2q − 2) DBjn+r (α, q)DBjn−r (α, q) − DBj2n (α, q) = α 2n αγ q n (q r + q −r − 2)

1 

(α(1 + q))k

k=0

 

Thus, the proof is completed.

With the help of this theorem, we can also write the Cassini identity, which is a special case of the Catalan equation. Corollary 2 (i) DBJn+1 (α, q)DBJn−1 (α, q) − DBJ2n (α, q) =

α 2(n−1) αγ q n (1 − q)2

(2 − q −1 − q)

1  (α(1 + q))k , k=0

(ii) DBjn+1 (α, q)DBjn−1 (α, q) − DBj2n (α, q) = α 2n αγ q n (q −1 + q − 2)

1  (α(1 + q))k . k=0

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In the following theorem, we give the d’Ocagne’s identities for the sequences DBJn (α, q), DBjn . Theorem 4 For m, n ∈ Z+ , we get (i) DBJm (α, q)DBJn+1 (α, q) − DBJn (α, q)DBJm+1 (α, q) =

1 α m+n−1 αγ (q − 1)(q m − q n )  (α(1 + q))k , (1 − q)2 k=0

(ii) DBjm (α, q)DBjn+1 (α, q) − DBjn (α, q)DBjm+1 (α, q) = α m+n+1 αγ (1 − q)(q m − q n )

1  (α(1 + q))k . k=0

Proof (i) If we use the Binet formula, then      α − qmγ α − q m+1 γ m−1 m + α DBJm (α, q)DBJn+1 (α, q) = α 1−q 1−q      α − q n+1 γ α − q n+2 γ + α n+1 αn 1−q 1−q      α − qnγ α − q n+1 γ + α n DBJn (α, q)DBJm+1 (α, q) = α n−1 1−q 1−q      α − q m+1 γ α − q m+2 γ m m+1 + α . α 1−q 1−q Let us subtract the second from the first equality. Then, we get α m+n−1 αγ (1 − q)2

{(q m+1 + q n − q m − q n+1 )

+α(q m+2 + q m+1 + q n+1 + q n − q n+2 − q m − q n+1 − q m+1 )}. After necessary simplifications, we have DBJm (α, q)DBJn+1 (α, q) − DBJn (α, q)DBJm+1 (α, q) =

1 α m+n−1 αγ (q − 1)(q m − q n )  (α(1 + q))k . (1 − q)2 k=0

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(ii) Now, we have the value DBjm (α, q)DBjn+1 (α, q) − DBjn (α, q)DBjm+1 (α, q) is follows. DBjm (α, q)DBjn+1 (α, q) is {(α m α + (αq)m γ ) + (α m+1 α + (αq)m+1 γ )} {(α n+1 α + (αq)n+1 γ ) + (α n+2 α + (αq)n+2 γ )} and DBjn (α, q)DBjm+1 (α, q) is {(α n α + (αq)n γ ) + (α n+1 α + (αq)n+1 γ )} {(α m+1 α + (αq)m+1 γ ) + (α m+2 α + (αq)m+2 γ )}. If we subtract these terms side by side, then we have α m+n+1 αγ (q n+1 + q m − q m+1 − q n ) + α(q n+2 + q m − q m+2 − q n ). DBjm (α, q)DBjn+1 (α, q) − DBjn (α, q)DBjm+1 (α, q) = α m+n+1 αγ (1 − q)(q m − q n )

1  (α(1 + q))k k=0

 

which is the desired result.

In text theorem, we give the following identity which known as the Honsberger’s identity or the Convolution theorem in the literature. Theorem 5 For m, n ∈ Z+ , we have (i) DBJn−1 (α, q)DBJm (α, q) + DBJn (α, q)DBJm+1 (α, q) =

α m+n−3 (A + α B) (1 − q)2

(ii) DBjn−1 (α, q)DBjm (α, q) + DBjn (α, q)DBjm+1 (α, q) = α m+n+1 (C + α D). Where A = α 2 − q m+k αγ − q n+k−1 αγ + q m+n+2k γ 2 , B = α 2 − q m+k+1 αγ − q n+k αγ + q m+n+2k+1 γ 2 , C = α 2 + q m+k αγ + q n+k−1 αγ + q m+n+2k−1 γ 2 , D = α 2 + q m+k+1 αγ + q n+k αγ + q m+n+2k+1 γ 2 .

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Proof Let us first look at the first claim of the theorem.      α − q n−1 γ α − qnγ + α n−1 DBJn−1 (α, q)DBJm (α, q) = α n−2 1−q 1−q      α − qmγ α − q m+1 γ m−1 m + α , α 1−q 1−q      α − qnγ α − q n+1 γ n−1 n + α DBJn (α, q)DBJm+1 (α, q) = α 1−q 1−q      α − q m+1 γ α − q m+2 γ + α m+1 . αm 1−q 1−q If we add these equations side by side, then DBJn−1 (α, q)DBJm (α, q) + DBJn (α, q)DBJm+1 (α, q) =

α m+n−3 {(α 2 − q m αγ − q n−1 αγ + q m+n−1 γ 2 ) (1 − q)2 +α 2 (α 2 − q m+1 αγ − q n αγ + q m+n+1 γ 2 )

+α((2α 2 − q m+1 αγ − q n−1 αγ − q m αγ − q n αγ + 2q m+n γ 2 +α 2 (2α 2 − q m+2 αγ − q n αγ − q m+1 αγ − q n+1 αγ + 2q m+n+2 γ 2 ))}. DBJn−1 (α, q)DBJm (α, q) + DBJn (α, q)DBJm+1 (α, q) =

α m+n−3 (A + α B) (1 − q)2

which is desired result. (ii) DBjn−1 (α, q)DBjm (α, q) = {(α n−1 α + (αq)n−1 γ ) + (α n α + (αq)n γ )} {(α m α + (αq)m γ ) + (α m+1 α + (αq)m+1 γ )}. DBjn (α, q)DBjm+1 (α, q) is {(α n α + (αq)n γ ) + (α n+1 α + (αq)n+1 γ )} {(α m+1 α + (αq)m+1 γ ) + (α m+2 α + (αq)m+2 γ )}.

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Let’s add up these two equations. We get α m+n−1 {(α 2 + q m αγ + q n−1 αγ + q m+n−1 γ 2 ) + α 2 (α 2 + q m+1 αγ + q n αγ + q m+n+1 γ 2 )

+α(2α 2 + q m+1 αγ + q ( n − 1)αγ + q m αγ + q n αγ + 2q m+n γ 2 +α 2 (2α 2 + q m+2 αγ + q n αγ + q m+1 αγ + q n+1 αγ + 2q m+n+2 γ 2 ))}. So, we obtain that DBjn−1 (α, q)DBjm (α, q) + DBjn (α, q)DBjm+1 (α, q) = α m+n+1 (C + α D).  

Thus, the proof is completed.

In the following theorem, we give the Vajda’s identity, which is one of the generalizations of the Catalan’s identity. Theorem 6 For n ∈ Z+ , we have (i) DBJn+i (α, q)DBJn+ j (α, q) − DBJn (α, q)DBJn+i+ j (α, q) is α 2(n−1)+i+ j q n αγ (1 − q)2

(q j − 1)(q i − 1)

1  (α(1 + q))k . k=0

(ii) DBjn+i (α, q)DBjn+ j (α, q) − DBjn (α, q)DBjn+i+ j (α, q) is α 2n+i+ j q n αγ (q j − 1)(1 − q i )

1 

(α(1 + q))k .

k=0

Proof Using Binet formula, we can give the second side of the first equality with the following equality: α 2(n−1)+i+ j q n αγ (1 − q)2

{(q i+ j + 1 − q j − q i ) + α(q i+ j + 1 + q + q i+ j+1 − q j+1 − q i − q j − q i+1 )}.

DBJn+i (α, q)DBJn+ j (α, q) − DBJn (α, q)DBJn+i+ j (α, q) =

α 2(n−1)+i+ j q n αγ (1 − q)2

(q j − 1)(q i − 1)(1 + α(1 + q)).

DBJn+i (α, q)DBJn+ j (α, q) − DBJn (α, q)DBJn+i+ j (α, q) =

α 2(n−1)+i+ j q n αγ (1 − q)2

(q j − 1)(q i − 1)

1  k=0

(α(1 + q))k .

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For the other equality, if the Binet formula is applied to the left side of the equality, the right side of the equality is as follows: α 2n+i+ j q n αγ (q j + q i − q i+ j − 1) +α 2n+i+ j+1 q n αγ (q j+1 + q i + q j + q i+1 − q i+ j+1 − 1 − q i+ j − q). DBjn+i (α, q)DBjn+ j (α, q) − DBjn (α, q)DBjn+i+ j (α, q) = α 2n+i+ j q n αγ (1 − q i )(q j − 1)(1 + α(1 + q)), DBjn+i (α, q)DBjn+ j (α, q) − DBjn (α, q)DBjn (α, q) = α 2n+i+ j q n αγ (q j − 1)(1 − q i )

1  (α(1 + q))k . k=0

Thus, the proof is completed.

 

3 Conclusions In this study, we define bicomplex sequences whose coefficients are dual q-Jacobsthal and Jacobsthal-Lucas numbers. Then we obtain some important properties involving elements of newly defined numbers. We give some commonly used equations for sequences of numbers. We also derive the Binet formulas to find the nth terms for these sequences. In addition to this, we give some basic identities for these numbers, such as Cassini and Catalan identities, which have an important place in the literature.

References 1. Babadag, F., Fibonacci, Lucas Numbers with Daul Bicomplex Numbers. Journal of Informatics and Mathematical Sciences, 10(1-2), 161–172, (2018). 2. Halici, S., Curuk, S., On dual bicomplex numbers and their some algebraic properties. Journal of Science and Arts, 19(2), 387–398, (2019). 3. Luna-Elizarraras, M. E., Shapiro, M., Struppa, D. C., Vajiac, A., Bicomplex numbers and their elementary functions. Cubo (Temuco), 14(2), 61–80, (2012). 4. Halici, S., Oz, S., On some Gaussian Pell and Pell-Lucas numbers. Ordu Üni. Science and Tech. Journal, 6(1), (2016). 5. Halici, S., On Bicomplex Jacobsthal-Lucas Numbers. Journal of Mathematical Sciences and Modelling, 3(3), 139–143, (2020). 6. Cerda-Morales, G., On bicomplex third-order Jacobsthal numbers. Complex Variables and Elliptic Equations, 1–13, (2021). 7. Koshy, T., Fibonacci and Lucas numbers with applications., John Wiley Sons, (2001).

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8. Kome, C., S. Kome, Catarino, P., Quantum Calculus Approach to the Dual Bicomplex Fibonacci and Lucas Numbers. Journal of Mathematıcal Extensıon, 16, (2021). 9. Aydın, F. T. q-Fibonacci bicomplex and q-Lucas bicomplex numbers. Notes on Number Theory and Discrete Mathematics 28(2), (261–275). 10. Pashaev, O.K.,Nalci, S., Golden quantum oscillator and Binet-Fibonacci calculus. Journal of Physics A: Mathematical and Theoretical, 45(1), 015303(23pp), (2012). 11. Kızılate¸s, C., Cekim, B., New families of generating functions for q-Fibonacci and the related polynomials. Ars Combinatoria, 136, (2018). 12. Akkus, I., Kizilaslan, G., Quaternions: Quantum calculus approach with applications. Kuwait Journal of Science, 46(4), 1–13, (2019). 13. Kızılate¸s, C., Polatlı, E., New families of Fibonacci and Lucas octonions with q-integer components. Indian Journal of Pure and Applied Mathematics, 52(1), 231–240, (2021). 14. Kac, V. G., Cheung, P., Quantum calculus, Vol. 113. New York: Springer, (2002).

The Moore-Penrose Inverse in Rickart ∗-Rings Mehsin Jabel Atteya

1 Introduction The original concept of Moore-Penrose inverse began with the work of E. H. Moore between 1910 and 1920. Basically, Moore studied the general mutual of any matrix and employs it to solve systems of linear equations [1]. In 1955, R. Penrose rediscover it later [2]. Nowadays, it is called the Moore-Penrose inverse. Indeed, in the last decades, the Moore-Penrose inverse has found a wide range of applications in many areas of science and became a useful tool for dealing with optimization. problems, data analysis, the solution of linear integral equations, etc. It is well known that the inverse of a square complex matrix with nonzero determinant may be expressed in terms of the adjoint of the matrix. In 1920, E. H. Moore [1] extended this classical notion to provide a formula for what is now termed the Moore-Penrose inverse of an arbitrary complex matrix. Several others have also provided representations of the elements of this generalized inverse in terms of rational functions of certain determinants of submatrices of the given matrix. In [3], more new characterizations of EP elements in rings are given by Mosi´c and Djordjevi´c, which involve powers of their group and Moore-Penrose inverse. In [4], Tian and Wang presented some necessary and sufficient conditions such that A ∈ C to be an EP matrix, which also involve powers of their group and Moore-Penrose inverse, where C stands for the set of all n × n matrices over the field of complex numbers. Motivated by the above statements, in this paper, we will show that the existence of the Moore-Penrose inverse of an element in a ring R is closely related with powers of some Hermite elements, idempotents and projections. Recently, Zhu, The author is grateful to Al-Mustansiriyah University, the Republic of Iraq and beholden to the reviewer(s) for his/their accuracy with professionally reading the article. M. J. Atteya (B) Department of Mathematics, College of Education, Mustansiriyah University, Baghdad, Iraq e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_18

191

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Chen and Pedro Patrício [5] introduced the concepts of left ∗-regular and right ∗regular. We call an element a ∈ R is left (right) ∗-regular if there exists x ∈ R such that a = aa ∗ ax(a = xaa ∗ a). They proved that a ∈ R † if and only if a is left ∗regular if and only if a is right ∗-regular. Motivated by these results, we will give more equivalent conditions of the Moore-Penrose of an element in a ring. If a, b are invertible in a semigroup with the unit, then the rule (ab)−1 = b−1 a −1 is known as the reverse order rule for the ordinary inverse. In the case of the Moore-Penrose inverse in a ring with involution, the rule (ab)† = b† a † is not always satisfied. Greville [6] proved that (ab)† = b† a † holds for complex matrices if and only if a † commutes with bb∗ and bb† commutes with aa ∗ (see also Boullion and Odell [7]). Prasad and Bapat [8] found necessary and sufficient conditions for the existence of the weighted MP-inverse for matrices. As matter of fact, the basic existence theorem for the Moore-Penrose inverse in the setting of rings with involution was given in [9]. The aims of this paper are to study certain identities related to the Moore-Penrose inverse of an element in a ∗-ring and Rickart ∗-ring R. Further, as an application, we characterize some properties concerning an element in a ∗-ring and Rickart ∗-ring R.

2 Preliminaries Let R be a unital ring with involution, that is a ring with an involution a → a ∗ is an anti-isomorphism of degree 2 satisfying (a ∗ )∗ = a, (ab)∗ = b∗ a ∗ and (a + b)∗ = a ∗ + b∗ . We say that b ∈ R is the Moore-Penrose inverse of a ∈ R, if the following hold: aba = a, bab = b, (ab)∗ = ab and (ba)∗ = ba. There is at most one b such that above four equations hold. If such an element b exists, it is denoted by a † . The set of all Moore-Penrose invertible elements will be denoted by R † . An element a ∈ R is called an idempotent if a 2 = a, also, a is called a projection if a 2 = a = a ∗ . A Rickart ∗-ring is a ∗-ring in which the right annihilator of every element is generated by a projection, as a right ideal in S. Every Rickart ∗-ring contains unity. For each element a in a Rickart ∗-ring, there is a unique projection e such that ae = a and ax = 0 if and only if ex = 0, called a right projection of a. Moreover, a is called normal if aa ∗ = a ∗ a, also, R sa denoted to the set of all self-adjoint elements of R i.e. a ∗ = a. Furthermore, a is called a Hermite element if a ∗ = a, a is said to be an EP element if a ∈ R † ∩ R  and a † = a  . The set of all EP elements will be denoted by ´ = a, (a a) ´ ∗ = a a. ´ The set of R E P . a´ is called a {1, 3}-inverse of a if we have a aa {1,3} . Similarly, an element aˆ ∈ R all {1, 3}-invertible elements will be denoted by R ˆ The set of all {1, 4}-invertible is called a {1, 4}-inverse of a if a aa ˆ = a, (aa) ˆ ∗ = aa. {1,4} . An element a ∈ R is ∗-cancellable if a ∗ ax = 0 elements will be denoted by R implies ax = 0 and yaa ∗ = 0 implies ya = 0. Additionally, let a ∈ R, we call a is well-supported if there exist a projection p ∈ R such that ap = 0 and a ∗ a + p is invertible. we call a is co-supported if there exist a projection q ∈ R such that qa = 0 and aa ∗ + q is invertible. Let a ∈ R, we call a is weak-supported if there exists b ∈ R

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such that ab = 0 and a ∗ a + b is invertible. we call a is coweak-supported if there exists c ∈ R such that ac = 0 and aa ∗ + c is invertible. Further, R sa the set of all self-adjoint elements of R i.e. a ∗ = a. An idempotent element p is called right (resp. left) semi-central if pa = pap (resp. ap = pap), for all a ∈ A [10]. p ∈ A is called central if pa = ap each a ∈ A [10]. Moreover, R is a reduced Rickart ∗-ring. That means, Rickart ∗-ring having no nonzero nilpotent element. We write the notation [x, y] for the commutator x y − yx and x ◦ y for anticommutator x y + yx.

3 The Main Results We start with the following necessary and sufficient conditions for the elements a and b of a ∗-ring R. Proposition 1 Let a, b ∈ R, then the following elements satisfy, (i) (ii) (iii) (iv)

[a, b]∗ = [b∗ , a ∗ ], (a ◦ b)∗ = (b∗ ◦ a ∗ ), [a, b]† = [b† , a † ], (a ◦ b)† = (b† ◦ a † ).

Proof (i) In this branch, we calculate the action of ∗ on a Lie structure into the following identity, [a, b]∗ = (ab − ba)∗ = (ab)∗ − (ba)∗ , applying the property of ∗ on a ring R, we conclude that [a, b]∗ = b∗ a ∗ − a ∗ b∗ = [b∗ , a ∗ ] for all a, b ∈ R. Likewise, for ∗ on Jordan structure in the Branch (ii). (iii) This branch has calculation the action of † on a Lie structure, when we have the following identity, [a, b]† = (ab − ba)† = (ab)† − (ba)† , employing the property of † on a ring R, we deduce that [a, b]† = b† a † − a † b† = [b† , a † ] for all a, b ∈ R. Applying the same manner, for † on Jordan structure in the Branch (iv).   Definition 1 Let R be a ∗-ring. An element a ∈ R is called a semiprojection element if a is both (a Hermition and idempotent) i.e. projection element and nilpotent element with index n > 2 in a ring R, that is a = a ∗ = a 2 and a n = 0, n > 2, a ∈ N. To be closer to the above definition, we demonstrate the following example. Example 1 Let R = M2 (R), where  R denotes   the setof real numbers. Now, we x y w −y define the involution mapping: ∗ : → . z w −z x Now, we determine whether R is ∗-ring or not.    ∗ ∗    ∗ x y x y w −y , then = = (i) (a ∗ )∗ = a. Let a = z w z w −z x   x y = a. z w

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    x y h g (ii) (a + b)∗ = a ∗ + b∗ . Assume a = and b = , zw q p     ∗  ∗ x y h g x +h y+g then (a + b)∗ = + = = zw q p z+q w+ p   w + p −(y + g) −(z + q) x + h     w −y p −g = + = a ∗ + b∗ . −z x −q h To show the last relation, we consider        ∗ ∗ x y h g xh + yq xg + yp = (iii) (ab)∗ = b∗ a ∗ . Then, from the left side z w q p zh + wq wzg + wp 

 zg + wp −(xg + yp) . −(zh + wq) xh + yq

From therightside, weconclude  that  ∗ ∗ x y h g zg + wp −(xg + yp) ∗ ∗ b a = = . z w q p −(zh + wq) xh + yq Obviously, R is ∗-ring. Based on the above information, we emphasize   on x 0 Definition 1. Hence, let y = z = 0 and x = w = x yields a = , such 0x that x 2 = x and x n = 0, n > 2. Without a doubt, we deduce that a = a ∗ = a 2 , where x 2 = x and a n = 0, where x n = 0, n > 2. We observe that, a is semiprojection element of R. Remark 1 Let R be a reduced Rickart ∗-ring and S be an ideal of R. Then (i) For a ∈ S, b ∈ R, b the Moore-Penrose inverse (or MP-inverse) of a such that a ∗ b∗ = 0 then b∗ a ∗ = 0. Suppose a ∗ b∗ = 0. Then (b∗ a ∗ )2 = (b∗ a ∗ )(b∗ a ∗ ) = 0. Since R is ∗-ring, we conclude that ∗ ∗ ∗ ∗ b (a b a ) = 0. Right-multiplying by b∗ with applying b is the Moore-Penrose inverse (or MP-inverse) of a, we arrive to b∗ (a ∗ b∗ a ∗ )b∗ = b∗ = 0. (ii) Every semiprojection in R is central. Let t be a semiprojection, then t (1 − t) = (1 − t)t = 0 and (t x(1 − t))n = t x(1 − t)t x(1 − t) = 0 for any x ∈ R. Using the fact that R is reduced, this can be written as (t x(1 − t))2 = t x(1 − t)t x(1 − t) = 0 which yields t x(1 − t) = 0. It follows that t x = t xt and t x ∗ = t x ∗ t. This implies that, xt = t xt = t x, for all x ∈ R. Consequently, t is a central projection. Theorem 1 In a Rickart ∗-ring R sa , every central idempotent is a semiprojection. Proof It is known in the literature, and we give a simple proof here. Assume that R is a Rickart ∗-ring R sa . Henceforth, using the same arguments as we have used in the proof of Remark 1 (ii). This completes the proof of theorem.   Proposition 2 Let a ∈ R sa Then the following conditions are equivalent for any n ∈ N +:

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(i) a ∗ ∈ R † , (ii) There exists a semiprojection q ∈ R such that qa = 0 and (a ∗ a)n + q is invertible; (iii) There exists a semiprojection q ∈ R such that qa = 0 and (a ∗ a)n + q is left invertible; (iv) There exists an idempotent f ∈ R such that f a = 0 and (a ∗ a)n + f is invertible; (v) There exists an idempotent f ∈ R such that f a = 0 and (a ∗ a)n + f is left invertible; (vi) There exists c ∈ R such that ca = 0 and (a ∗ a)n + c is invertible; (vii) There exists c ∈ R such that ca = 0 and (a ∗ a)n + c is left invertible. In this case, a † = a ∗ yi (aa ∗ )2n−1 yi∗ , i ∈ {1, 2, 3}, where 1 = y1 ((aa ∗ )n + q) = y2 ((aa ∗ )n + f ) = y3 ((aa ∗ )n + c), for some y1 , y2 , y3 ∈ R. Proof Branch (i) implies to Branch (ii). Based on the hypothesis that a ∈ R sa then a = a ∗ . At beginning, we assume a ∈ R † and q = 1 − aa † . Also, from our hypothesis a = a ∗ = a 2 and a n = 0, n > 2, a ∈ N and qa = (1 − aa † )a = 0. Applying Lemma 2.3 in [11], we deduce that aa ∗ (a † )∗ a † = (a † )∗ a † aa ∗ , Furthermore, (aa ∗ + q)((a † )∗ a † + 1 − aa † ) = (aa ∗ + 1 − aa † )((a † )∗ a † + 1 − aa † ) = aa ∗ (a † )∗ a † + aa ∗ (1 − aa † ) + (1 − aa † )(a † )∗ a † + (1 − aa † )2 = aa ∗ (a † )∗ a † + 1 − aa † = aa † + 1 − aa † = 1: ((aa ∗ )2 + q)[((a † )∗ a † )2 + 1 − aa † ] = ((aa ∗ )2 + 1 − aa † )[((a † )∗ a † )2 + 1 − aa † ] = (aa ∗ (a † )∗ a † )2 + (aa ∗ )2 (1 − aa † ) + (1 − aa † )((a † )∗ a † )2 + (1 − aa † )2 = (aa † )2 + 1 − aa † = aa † + 1 − aa † = 1.

Step by step, we arrive to ((aa ∗ )n + q)[((ay)∗ ay)n+1 − aa † ] = 1. Consequently, (aa ∗ )n + p is invertible. The prove of Branch (ii) which yields Branch (iii) straightforward. Hence, it omitted. Now, we should prove Branch (iii) leads to Branch (i). By reason of q is semiprojection, we deduce that q 2 = q = q ∗ . Based on the hypothesis, we conclude that pa = 0 and (aa ∗ )n + q is left invertible. Consequently, for some y1 ∈ R we see that 1 = y1 ((aa ∗ )n + q). Employing the fact, pa = 0, we observe that a = y1 ((aa ∗ )n + q)a = y1 (aa ∗ )n a ∈ R(aa ∗ )n a. Now, in view of Theorem 3.1(4) in [11] is achieved and a † = a ∗ y1 (aa ∗ )n−1 a[y1 (aa ∗ )n−1 a]∗ = a ∗ y1 (aa ∗ )n−1 aa ∗ (aa ∗ )n−1 y1∗ = a ∗ y1 (aa ∗ )2n−1 y1∗ . We employ the result of Branch (i) implies to Branch (ii), to determine the relation between Branch (iv) and Branch (i). Therefore, we suppose f = q = 1 − aa † , then by the result of Branch (i) implies to Branch (ii), we find that f 2 = f ∈ R, f a = 0. Hence, (aa ∗ )n + f is invertible. Obviously, Branch (vi) yields Branch (vii). Finally, we determine the relation between Branch (vii) and Branch (i). From the hypothesis, we conclude that ca = 0 and (aa † )n + c is left invertible. Hence,

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for some y3 ∈ R we see that 1 = y3 ((aa ∗ )n + c). Based on the fact ca = 0, we deduce that a = y3 ((aa ∗ )n + c)a = y3 (aa ∗ )n a ∈ R(aa ∗ )n a. Again, in view of Theorem 3.1(4) in [11] is satisfied and a † = a ∗ y3 (aa ∗ )n−1 a[y3 (aa ∗ )n−1 a]∗ = a ∗ y3 (aa ∗ )n−1 aa ∗ (aa ∗ )n−1 y3∗ = a ∗ y3 (aa ∗ )2n−1 y3∗ . This completes the proof of theorem.   Applying the same strategies which used in the proof of Proposition 2, one can proof the following theorem. Theorem 2 Let a ∈ R sa and n ∈ N ∗ . Then the following conditions are equivalent: (i) a ∈ R † , (ii) There exists a semiprojection p ∈ R sa such that ap = 0 and (a ∗ a)n + p is invertible, (iii) There exists a semiprojection p ∈ R sa such that ap = 0 and (a ∗ a)n + p is right invertible, (iv) There exists an idempotent e ∈ R sa such that ae = 0 and (a ∗ a)n + e is invertible, (v) There exists an idempotent e ∈ R sa such that ae = 0 and (a ∗ a)n + e is right invertible, (vi) There exists b ∈ R sa such that ab = 0 and (a ∗ a)n + b is invertible (vii) There exists b ∈ R sa such that ab = 0 and (a ∗ a)n + b is right invertible. In this case, a † = xi (a ∗ a)2n−1 xi a ∗ , i ∈ {1, 2, 3}, where 1 = ((aa ∗ )n + p)x1 = ((aa ∗ )n + e)x2 = ((aa ∗ )n + b)x3 , for some x1 , x2 , x3 ∈ R. In view of Definition 3.1 in [12], an element a ∈ R is named a left supported (resp. a right supported) by projection if a = f a (resp. a = ah) for some projection f ∈ R (resp. h ∈ R). It is supported by projection if it is both left and right supported by a projection. Applying the same manner of the proof of Proposition 2 and Theorem 2, we get the following result. Proposition 3 Suppose a ∈ R sa . Then the following conditions are equivalent: (i) (ii) (iii) (iv) (v)

a ∈ R†, a is weak-supported, a is right weak-supported, a is coweak-supported a is left coweak-supported.

Theorem 3 Let a ∈ R sa . Then we have the following results: (i) a is {1, 3}-invertible with {1, 3}-inverse x if and only if x ∗ aa ∗ = a, (ii) a is {1, 4}-invertible with {1, 4}-inverse y if and only if a ∗ ay ∗ = a. To proof this result. It is sufficient using the same arguments as which used in the page (201) at [13], with employing the concept of R sa .

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Theorem 4 Let R be a ∗-ring, a ∈ R sa and n ∈ N+ . Then the following conditions are equivalent: (i) a ∗ is MP-invertible, (ii) a ∗ is a left ∗-cancellable, right supported by semiprojection and aa ∗ is group invertible, (iii) a ∗ is a right ∗-cancellable, left supported by semiprojection and a ∗ a is group invertible, (iv) a ∗ is a ∗-cancellable, supported by semiprojection and both a ∗ a and aa ∗ are group invertible. Proof To prove the Branch (i) implies to other branches. We assume that a ∗ is MPinvertible. Based on the hypothesis that a ∈ R sa then a = a ∗ . Hence, we deduce a is MP-invertible. For any arbitrary elements x and y in ∗-ring R such that, a ∗ ax = a ∗ ay. Consequently, ax = aa † ax = (aa † )∗ ax = (a † )∗ a ∗ ax. Obviously, applying the relation a ∗ ax = a ∗ ay which is in the last term give us ax = a †∗ a ∗ ay. Then ax = (aa † )∗ ay = aa † ay = ay. By reason of a ∈ R sa , we conclude that a ∗ is a left ∗-cancellabl. By the same manner for a ∗ is a right ∗-cancellabl. Suppose h = a † a and f = aa † , the h and f are semiprojection with a = aa † a. Then a = a(a † a) = ah, also, a = (aa † )a = f a. Thus, we find that ah = f a. In other words, a ∗ right and left supported by a semiprojection. Now, based on the fact that a ∗ a and aa ∗ are Hermitian, also, a ∗ a and aa ∗ are EP elements. Both reason make the term a ∗ a (resp. aa ∗ ) is a group invertible. In fact, this result indicate to the Branches (ii) and (iii). Since the Branches (ii) and (iii) yield (iv). That means the Branch (iv) satisfied. To prove Branch (ii) implies Branch (i), we assume a ∗ is a left ∗-cancellable, right supported by semiprojection and aa ∗ is group invertible. Therefore, a = ah for some  semiprojection h. Suppose x = (a ∗ a) a ∗ , Thus, a ∗ ahxa = a ∗ a(a ∗ a) a ∗ a = a ∗ a = a ∗ ah. Based on the fact that a ∗ is a left ∗-cancellable and a ∈ R sa , we deduce (i) (ii) (iii) (iv)

axa = ahxa = ah = a.   xax = (a ∗ a) a ∗ a(a ∗ a) a ∗ = (a ∗ a) a ∗ = x. ∗ ∗  ∗ ∗ ∗  ∗ (ax) = (a(a a )a ) = a(a a )a = ax. (xa)∗ = ((a ∗ a  )a ∗ a)∗ = a ∗ a(a ∗ a  ) = (a ∗ a  )a ∗ a = xa.

Consequently, a (resp. a ∗ ) is MP-invertible and x = a † . Branch (iii) yields Branch (i). Let a ∗ be a right ∗-cancellable, left supported by semiprojection and a ∗ a be group invertible. Hence, a ∗ = f a ∗ . Basically, a ∈ R sa then a = a ∗ . Rewrite this relation as a = f a for some semiprojection f . Assume x = a ∗ (a ∗ a) . Therefore, ax f aa ∗ = aa ∗ (aa ∗ ) a ∗ a = a ∗ a = f a ∗ a

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By reason of a ∗ (resp. a) is a right ∗-cancellable, we find that (i) (ii) (iii) (iv)

axa = ax f a = f a = a. xax = a ∗ (aa ∗ ) aa ∗ (aa ∗ ) = a ∗ (aa ∗ ) = x. (ax)∗ = (aa ∗ (aa ∗ ) )∗ = (aa ∗ ) aa ∗ = aa ∗ (aa ∗ ) = ax. (xa)∗ = (a ∗ (aa ∗ ) a)∗ = (a ∗ (aa ∗ ) a = xa.

We arrive to that, a (resp. a ∗ ) is MP-invertible and x = a † . Finally, Branch (iv) to Branch (i) is straightforward. This completes the proof of theorem.  

4 Conclusions In this paper we considered Moore-Penrose invertible elements in Rickart rings with involution. Precisely, we characterized MP-invertible normal, EP elements, R sa the set of all self-adjoint elements of a ring with involution R and semiprojection elements in terms of equations involving their Moore-Penorse and group inverse. However, we demonstrated the new technique in proving the results. In fact, we applied a purely algebraic technique, involving different characterizations of the Moore-Penrose inverse.

References 1. E. H. Moore: On the reciprocal of the general algebraic matrix, Abstract, Bull. Amer. Math. Sot., 26, 394–395(1920). 2. R. Penrose: A generalized inverse of matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 51(3), 401–413(1955). 3. D. Mosi´c, D.S.Djordjevi´c: New characterizations of EP, generalized normal and generalized Hermitian elements in rings, Appl. Math. Comput., 218, no. 12, 6702–6710 (2012). 4. Y.G. Tian, H.X. Wang: Characterizations of EP matrices and weighted-EP matrices, Linear Algebra Appl., 434, 1295–1318 (2011). 5. Huihui Zhu, Jianlong Chen, Pedro Patrício: Further results on the inverse along an element in semigroups and rings, Linear Multilinear Algebra, 64:3, 393–403 (2016), https://doi.org/10. 1080/03081087.2015.1043716. 6. T. N. E. Greville: Note on the generalized inverse of a matrix product, SIAM Rev., 8, 518– 521(1966). 7. T. L. Boullion and P. L. Odell: Generalized inverse matrices, wiley-interscience, New York,(1971). 8. K. M. Prasad, R. B. Bapat: The generalized Moore-Penrose inverse, Linear Algebra Appl., 165, 59–69 (1992). 9. K. P. S. Bhaskara Rao: The theory of generalized inverses over commutative rings, Taylor and Francis, London, (2002). 10. G.F. Birkenmeier, J. Y. Kim and J.K. Park: Principally quasi-Baer rings, Comm. in Algebra, 29, 639–660(2001). https://doi.org/10.1081/agb-100001530. 11. Sanzhang Xua, Jianlong Chenb: The Moore-Penrose inverse in rings with involution, Filomat 33, 18 , 5791–5802(2019). https://doi.org/10.2298/FIL1918791X.

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12. Wannisa Apairat: Moore-Penrose inverse and normal elements in rings, MSc. thesis, Prince of Songkla University, (2017). 13. R.E. Hartwig: Block generalized inverses, Arch. Retional Mech. Anal., 61, no.3, 197–251 (1976).

k-Oresme Polynomials and Their Derivatives Serpil Halıcı, Zehra Betül Gür, and Elifcan Sayın

1 Introduction In [1], Horadam defined Wn = Wn (W0 , W1 ; p, q) numbers by the second order linear homogeneous recurrence relation Wn+2 = pWn+1 − qWn

(1)

for integers p, q and n ≥ 0. By choosing W0 , W1 , p and q properly, some well-known number sequences can be obtained such as Fibonacci numbers Fn = Wn (0, 1; 1, −1) and Pell numbers Pn = Wn (0, 1; 2, −1). Nicole Oresme extended the values of p and q to be rational numbers and defined a new number sequence called Oresme numbers, by taking W0 = 0, W1 = 21 , p = 1 and q = 14 in the Horadam sequence [2]. We write the sequence   1 1 {On } = Wn 0, ; 1, 2 4 

as 

 1 2 3 4 3 5 6 7 0, , , , , , , , , ... . 2 4 8 16 8 32 64 128

S. Halıcı (B) · Z. B. Gür · E. Sayın Pamukkale University, 20160 Denizli, Turkey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_19

201

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From (1), the recurrence relation of Oresme numbers is written as On = On−1 −

1 On−2 4

with O0 = 0, O1 = 21 . The following properties are derived by the authors in [1, 2]. n−1 

 Oi = 4

i=0

n−1 

(−1)i Oi =

i=0 n−1  i=0

 1 − On+1 , 2

  1 4 − + (−1)n (On+1 − 2On ) , 9 2

O2i+1 =

1 (10 + 5O2n−1 − 16O2n ) . 9

In [3], Cook generalized the Oresme numbers with respect to k ≥ 2 and defined k-Oresme numbers by (k) − On(k) = On−1

1 (k) O , n≥2 k 2 n−2

where O0(k) = 0, O1(k) = k12 . Notice that On(k) = Wn (0, k1 ; 1, k12 ) and On(2) = On . Using standard techniques to solve recurrence relation, Binet formula can be deduced and written as n  n

 √ √ k − k2 − 4 k + k2 − 4 1 (k) − On = √ 2k 2k k2 − 4 with k 2 − 4 > 0. Some important properties and identities of k-Oresme numbers are given by authors (see [3–6]). For example, (k) (k) On−1 − (On(k) )2 = − On+1

1 , n ≥ 1. k 2n

(2)

In [7], By using (1), Soykan defined the generalized Oresme numbers Wn = Wn (W0 , W1 ) by the second-order recurrence relation 1 Wn = Wn−1 − Wn 4 wtih the initial conditions W0 = c0 , W1 = c1 not all being zero.

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For a real number x = 0, Oresme polynomials are defined by the following relation [5]. ⎧ 1 ⎪ ⎨ n = 0, 1, x (3) On+1 (x) = 1 ⎪ ⎩ On (x) − On−1 (x) n ≥ 2. 2 x Some terms are   1 1 x 2 − 1 x 2 − 2 x 4 − 3x 2 + 1 x 4 − 4x 2 + 3 0, , , , , , , ... x x x3 x3 x5 x5 Solving Binet formula of the auxiliary equation (3), we get the Binet formula On (x) = √ √

1 x2

−4

(α n − β n ),

where x 2 − 4 > 0 and α, β = x± 2xx −4 . The relation between Oresme polynomials and Fibonacci numbers can be given by On (3) = 3Fn3 in [5]. In [5], Morales defined a new matrix corresponding to k-Oresme polynomials by 2



1 − x12 M= 1 0



and derived some special identities and properties by considering powers of this matrix which play important roles in our present paper. In this study, Oresme polynomials and k-Oresme polynomials are extended to a new sequence of rational functions that called k-Oresme polynomials and various properties are given.

2 k-Oresme Polynomials nth k-Oresme polynomial denoted by On(k) (x) has the following recurrence relation by initial conditions O0(k) (x) = 0, O1(k) (x) = kx1 . (k) (k) On+2 (x) = On+1 (x) −

1 k2 x 2

On(k) (x),

where x ∈ R and n ∈ N. O1(k) (x) = O2(k) (x) =

1 k2 x 2 − 1 k2 x 2 − 2 (k) , O3(k) (x) = , O (x) = , ... 4 kx k3x 3 k3x 3

are first few terms of the k-Oresme polynomials.

(4)

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Taking k = 1 and x = 1 in (4) respectively, one can get Oresme polynomials and k-Oresme numbers. Theorem 1 The Binet formula for k-Oresme polynomials is of the form On(k) (x)

 n−1  2 2 i 2   n k x −4 1 α(x)n − β(x)n = n−1 = √ . 2 kx i=0 2i + 1 k2 x 2 k2 x 2 − 4

Proof Since the roots of the characteristic equation of (4) are α(x) = β(x) =

√ kx− k 2 x 2 −4 2kx

On(k) (x)

=√

(5)

√ kx+ k 2 x 2 −4 , 2kx

, we get 

1

kx +

k2 x 2 − 4

√

k2 x 2 − 4 2kx



n −

kx −

n

√ k2 x 2 − 4 . 2kx

Then by using the Binomial theorem and some elementary operations, we obtain ⎛ (k) On (x) =

1

n    n



⎝  2n k 2 x 2 − 4 i=0 i

=

i −

n  i=0

i ⎞   2 2 n k x − 4 ⎠ (−1)i kx i

2i+1  √ 2 2 n k x −4 . 2i + 1 kx

 n−1 2 

2



√ 2n k 2 x 2 − 4

i=0

Hence On(k) (x)

k2 x 2 − 4 kx

=

 n−1 2 



1 2n−1 kx

i=0

 2 2 i n k x −4 2i + 1 k2 x 2 

which proves the theorem. For instance, one can easily see that  2 2 i    2 2 1   1 3 k x −4 k x −4 1 = O3(k) (x) = 3 + = 2x2 2x2 2i + 1 4kx k 4kx k i=0 by taking n = 3 in (5). Proposition 1 For kx > 2, then lim

n→∞

(k) On+1 (x)

On(k) (x)

= α(x) =

kx +

√

k2 x 2 − 4 . 2kx

(6)

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205

Proof By using (5), we get lim

(k) On+1 (x)

n→∞ O (k) (x) n

and lim

n→∞

(k) On+1 (x)

On(k) (x)

α(x)n+1 − β(x)n+1 n→∞ α(x)n − β(x)n

= lim

α(x)n+1 − β(x)n+1 n→∞ α(x)n − β(x)n

= lim

1 α(x)n 1 α(x)n

.

n  α(x) − β(x) α(x) lim = lim n .  n→∞ O (k) (x) n→∞ β(x) n 1 − α(x)

Then we have

(k) On+1 (x)

(7)

For every kx > 2, one can observe that β(x) < α(x) which implies  lim

n→∞

β(x) α(x)

n = 0.

By (7) and (8), we have the result.   1 − k 21x 2 and n ∈ Z+ , we have Theorem 2 For O = 1 0 

(k) (x) − kx1 On(k) (x) kx On+1 O = . (k) kx On(k) (x) − kx1 On−1 (x) n

(8) 

(9)

Proof It is clearly seen that (9) holds for n = 1. Now assume for induction that (9) is true for all integers such that m ≤ n. Then we can write and calculate

(k) (k) (x) − k 21x 2 On(k) (x)) − kx1 On+1 (x) kx(On+1 . =O O= (k) (x)) − kx1 On(k) (x) kx(On(k) (x) − k 21x 2 On−1 

O

n+1

n

By using the recurrence relation of the k-Oresme polynomials, we obtain O

n+1

which completes the proof.

 (k) (k) (x) − kx1 On+1 (x) kx On+2 . = (k) (x) − kx1 On(k) (x) kx On+1 

Theorem 2 helps us to derive some well-known identities by using some determinant properties. Next two theorems give us this important identities.

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Theorem 3 (Cassini’s Identity) For n ∈ Z+ , we have (k) (k) (x)On−1 (x) − (On(k) (x))2 = − On+1

1 . (kx)2n

(10)

Proof Observe that det (O n ) = (det (O))n and det (O n ) = − for O =

1 , (kx)2n

  1 − k 21x 2 . Also we can see from (9) that 1 0

(k) (x) − kx1 On(k) (x) kx On+1 (k) (k) det (O ) = = −On+1 (x))On−1 (x)) + On(k) (x)2 . (k) (x) kx On(k) (x)) kx1 On−1 

n

Combining these equalities we get the desired result.



Obviously, we get (2) from (10) by taking x = 1. Theorem 4 (Honsberger’s Formula) For n, m ∈ Z+ (k) (k) On+m (x) = kx On(k) (x)Om+1 (x) −

1 (k) O (x)Om(k) (x), kx n−1

(11)

where n, m ≥ 1. Proof By using O n+m = O n O m and (9) we obtain 

(k) (x) − kx1 On(k) (x) kx On+1 O O = (k) (k) (x) kx On (x) − kx1 On−1 n

m

and O

n+m



(k) kx Om+1 (x) − kx1 Om(k) (x) (k) (x) kx Om(k) (x) − kx1 Om−1



(k) (k) kx On+m+1 (x) − kx1 On+m (x) = . (k) (k) (x) − kx1 On+m−1 (x) kx On+m

Equating the entries of both matrices, we have (k) (k) (k) (x) = k 2 x 2 On(k) (x)Om+1 (x) − On−1 (x)Om(k) (x) kx On+m

which proves the Theorem.



In particular, we obtained the following expressions, for m = n and m = n − 1 respectively:

k-Oresme Polynomials and Their Derivatives

207

  (k) (k) O2n (x) = kx On(k) (x) 2On+1 (x) − On(k) (x) , (k) (x) = kx On(k) (x)2 − O2n−1

1 (k) O (x)2 . kx n−1

(12) (13)

Proposition 2 (General Bilinear Formula) For the integers a, b, c, d and t such that a + b = c + d, we have  1  (k) (k) (k) (k) (k) (k) (k) (k) O Oa (x)Ob (x) − Oc (x)Od (x) = (x)O (x) − O (x)O (x) . a−t c−t b−t d−t (kx)2t

(14)

Proof Using (5) and α(x)β(x) =

1 k2 x 2

we write

(k) (k) (k) On+r (x)On+s (x) − On(k) (x)On+r +s (x) =

(α(x)r − β(x)r ) (α(x)s − β(x)s ) (15) (k 2 x 2 − 4)(kx)2n

which implies (k) (k) (k) On+r (x)On+s (x) − On(k) (x)On+r +s (x) =

1 O (k) (x)Os(k) (x). (kx)2n r

(16)

Taking n = c, r = a − c, s = b − c and n = c − t, r = a − c, s = b − c in (15) and (16) respectively, we calculate 1 (k) O (k) (x)Ob−c (x) (kx)2c a−c

(17)

1 (k) O (k) (x)Ob−c (x). (kx)2(c−t) a−c

(18)

(k) (x) = Oa(k) (x)Ob(k) (x) − Oc(k) (x)Oa+b−c

and (k) (k) (k) (k) Oa−t (x)Ob−t (x) − Oc−t (x)Oa+b−c−t (x) =

Finally taking d = a + b − c we complete the proof.



As a result of this proposition, we have the following Corollary. Corollary 1 (d’Ocagne’s Identity) For integers m and n with m ≥ n ≥ 0 , we have (k) (k) (x)Om(k) (x) − On(k) (x)Om+1 (x) = On+1

1 O (k) (x). (kx)2n+1 m−n

(19)

Proof Using the Proposition 2 with n + 1 + m = n + m + 1, we can write (k) (k) (x)Om(k) (x) − On(k) (x)Om+1 (x) = On+1

  1 (k) (k) O (x) − O (x) . m−n+1 m−n+2 (kx)2n−1

Finally, the proof is finished by using the relation (4).



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In the end of this section, we give the generating function of k-Oresme polynomials in the next theorem. Theorem 5 Generating function of the k-Oresme polynomials is

f (z) =



z kx

Oi(k) (x)z i =

1−z+

i≥0

z2 2 k x2

,

(20)

where z ∈ R. Proof The following equations can be seen by some elementary operations. f (z) = O0(k) (x) + z O1(k) (x) + z 2 O2(k) (x) + · · · z f (z) = z O0(k) (x) + z 2 O1(k) (x) + z 3 O2(k) (x) + · · · z2 z2 z3 z4 (k) (k) f (z) = O (x) + O (x) + O (k) (x) + · · · 0 1 k2 x 2 k2 x 2 k2 x 2 k2 x 2 2 Then f (z) − z f (z) + O0(k) (x)

+z



z2 k2 x 2

O1(k) (x)

−

f (z) equals to O0(k) (x)



 +z

2

O2(k) (x)

 +z 3 O3(k) (x) − O2(k) (x) +

−

O1(k) (x)

1 + 2 2 O0(k) (x) k x



 1 (k) O (x) + ... k2 x 2 1

Using the relation (4), we get f (z) − z f (z) +

z2 k2 x 2

f (z) =

z . kx



3 Derivatives In this section, we examined derivative of the k-Oresme polynomials and deduced the generating function of this derivative sequence. The followings are some terms of the sequence of the derivatives. O0(k) (x) = 0, O1(k) (x) = O2(k) (x) = By differentiating (4), we get

1 3 − k2 x 2 (k)

, O (x) = , ... 3 kx 2 k3x 4

k-Oresme Polynomials and Their Derivatives

209

d (k) 2 1 (k) (k) (k) On+1 (x) = On+1 (x) = On(k) (x) + 2 3 On−1 (x) − 2 2 On−1 (x) , dx k x k x

(21)

which is the recurrence relation of derivatives. In the next theorem, the generating function of this derivative sequence is given. One can notice the relation between k-Oresme polynomials and their derivatives. Theorem 6 Generating function of On(k) (x) is of the form f (z)2 g(z) = −k z



z2 1−z− 2 2 k x

 ,

(22)

where z ∈ R. Proof Observing that  i≥0

Oi(k) (x) z i =

d  (k) O (x)z i d x i≥0 i

and using (20), we have   z2 1 − z −  (k) kz k2 x 2 Oi (x) z i = − 2 2  2 . k x z2 i≥0 1−z− 2 2 k x In the last equation, using f (z) completes the proof.



4 Conclusion In this paper, we defined a new polynomial family of Oresme numbers and studied some properties and identities of this poliynomials and their derivative by using some combinatorial operations. It would be interesting to study k-Oresme polynomials with negative indices and research their properties and higher order derivatives of this polynomial.

References 1. Horadam, A.F.: Basic Properties of a Certain Generalized Sequence of Numbers. Fibonacci Quart. 3(3), 161–176 (1965). 2. Horadam, A.F.: Oresme Numbers. Fibonacci Quart. 12(3), 267–271 (1974).

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3. Cook, C.K.: Some sums related to sums of Oresme numbers. In: Applications of Fibonacci Numbers, Proceedings of the Tenth International Research Conference on Fibonacci Numbers and their Applications, vol. 9, pp. 87–99. Kluwer Academic Publishers, Dordrecht (2004). https://doi.org/10.1007/978-0-306-48517-6_10 4. Liana, A.S., Wloch, I.: Oresme Hybrid numbers and Hybrationals. Kragujev. J. Math. 48(5), 747–753 (Accepted in 2021). 5. Morales, G.C.: Oresme Polynomials and Their Derivatives. arXiv:1904.01165 [math.CO]. (2019). 6. Sentürk, ¸ G.Y., Gürses, N., Yüce, S.: A New Look on Oresme Numbers: Dual-Generalized Complex Component Extension. In: Conference Proceeding of 18th International Geometry Symposium, vol. 4(2), pp. 206–214. C-POST, Malatya (2021). 7. Soykan, Y.: Generalized Oresme Numbers. Earthline J. Math. Sci. 7(2), 333–367 (2021). https:// doi.org/10.34198/ejms.7221.333367

k-Oresme Numbers and k-Oresme Numbers with Negative Indices Serpil Halıcı, Elifcan Sayın, and Zehra Betül Gür

1 Introduction A. F. Horadam defined the sequence of numbers Wn = Wn (W0 , W1 ; p, q) with the recurrence relation (1) Wn+2 = pWn+1 − qWn , n ≥ 0 for integers p, q different from zero [1]. Nicole Oresme defined a new sequence of numbers denoted by {On } by expanding the integer coefficients p, q in the Horadam sequence to rational numbers [2]. The reccurence relation of this number sequence is On = On−1 −

1 1 On−2 ; O0 = 0 , O1 = . 4 2

(2)

The negative and positive indexed terms of the Oresme sequence can be given in the following Table 1. In his study, Horadam gave both linear and nonlinear correlations related to Oresme numbers and also gave the generating function of these numbers. Cook examined the k-Oresme numbers by using the Oresme numbers. This sequence is defined by the following reccurence relation for k ≥ 2 , with the initial conditions O1,k = k1 , O0,k = 0. On,k = On−1,k −

1 On−2,k . k2

(3)

S. Halıcı · E. Sayın (B) · Z. B. Gür Deparment of Mathematics, Pamukkale University, Denizli, Turkey e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_20

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Table 1 The elements of sequence {On } n ... −3 −2 −1 On

···

−24

−8

−2

0

1

2

3

...

0

1 2

1 2

3 8

···

It can be seen that by taking k = 2 the sequence {On,k } gives the classical Oresme numbers. In [3], Binet’s formula for k-Oresme numbers is given as On,k = √



1

k+

√

k2 − 4

k2 − 4 2k



n −

k−

n  √ k2 − 4 2k

(4)

where k 2 − 4 > 0. The following properties are derived by the authors in [3, 4].  On+2,k =

k2 − 1 k2 

On+3,k =

 On+3,k =

(−1) Oi,k

i=1

n  i=1

O2i+1,k

k2 − 1 k2

k2 − 1 k2

(5)

On+1,k −

1 On,k . k2

(6)

On+2,k −

1 On,k . k4

(7)







(8)

 

1 k2 n+1 − + (−1) On+2,k − 2On+1,k . = 2 2k + 1 k

(9)

Oi,k = k 2

i=1

i

1 On−1,k , n ≥ 1. k4

On+1,k −

 1 − On+2,k . k

n 

n 



k2 = 2 2k + 1





k2 + 1 k2 + 1 2 + O2n+1,k − k O2(n+1),k . k k2

(10)

Oresme numbers have recently been the subject of re-examination of some studies and continue to be examined. One of these works belongs to Gürses et al. (see [5]). Cerda Morales, studied a new sequence of polynomials called Oresme polynomials which is a generalization of Oresme numbers. He obtained some identities including

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the general bilinear index-reduction formula of these numbers by using the matrix methods for Oresme polynomials [6]. Mc Laughlin gave some important combinatorial identities using the trace and determinant of any second-order matrix [7]. Inspired by all the studies mentioned above, in this study we also discussed new identities containing k-Oresme numbers and negative indexed k-Oresme numbers.

2 k-Oresme Numbers The generating matrix defined by using the initial values of the k- Oresme numbers is  −1  1 k2 O= . (11) 1 0 In the following theorem, we give a matrix identity which plays an important role to obtain well-known identities in next results. Theorem 1 For n ∈ Z+ , we have  k On+1,k O = k On,k n

−On,k  k −On−1,k . k

Proof The proof is easily seen by using the induction method.

(12)  

The results in the next two theorems can be obtained by using the previous theorem. Corollary 1 For n ∈ Z+ , we have the followings.     On+1,k n On,1 = , 1. O On,0 On,k

2 2. On+1,k On−1,k − On,k = − (k)12n . Proof The proof is easily seen by using Theorem 1 and some determinant properties.   Theorem 2 (Honsberger’s Formula) For n, m ∈ Z+ , we have On+m,k = k On,k Om+1,k −

1 On−1,k Om,k . k

(13)

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Proof By Theorem 1, it can be seen that   k Om+1,k − k1 Om,k k On+1,k − k1 On,k O O = k On,k − k1 On−1,k k Om,k − k1 Om−1,k n

m

equals  2 k On+1,k Om+1,k − On,k Om,k k 2 Om+1,k On,k − Om,k On−1,k n+m

Since O obtain

1 k2 1 k2

On,k Om−1,k − On+1,k Om,k . On−1,k Om−1,k − On,k Om,k

 k On+m+1,k − k1 On+m,k = , equating the entries of both matrices, we k On+m,k − k1 On+m−1,k On+m,k = k On,k Om+1,k −

1 On−1,k Om,k k  

which completes the proof.

Proposition 1 (General Bilinear Formula) For tha integers a,b,c,d and t such that a + b = c + d, we have Oa,k Ob,k − Oc,k Od,k =

1 (Oa−t,k Ob−t,k − Oc−t,k Od−t,k ). k 2t

Proof By the Binet Formula, On+r,k On+s,k − On,k On+r +s,k equals 1 [(α n+r − β n+r )(α n+s − β n+s ) − (α n − β n )(α n+r +s − β n+r +s )]. k2 − 4 Since αβ =

1 , k2 x 2

we write the last equation as 1 [(αr − β r )(α s − β s )]. (k 2 − 4)(k)2n

Thus, we have On+r,k On+s,k − On,k On+r +s,k =

1 Or,k Os,k . (k)2n

(14)

By taking n = c, r = a − c, s = b − c and n = c − t, r = a − c, s = b − c in (14) respectively, we get Oa,k Ob,k − Oc,k Oa+b−c,k = and

1 Oa−c,k Ob−c,k . (k)2c

(15)

k-Oresme Numbers and k-Oresme Numbers with Negative Indices

Oa−t,k Ob−t,k − Oc,k Oa+b−c,k =

1 (k)2(c−t)

215

Oa−c,k Ob−c,k .

(16)

Finally, taking d = a + b − c in Eqs. (15) and (16) we complete the proof.

 

Theorem 3 For x ∈ R, the generating function of the k-Oresme number is f (x) =

∞ 

Oi,k x i =

i=0

x k

1−x +

x2 k2

.

Proof The following equations can be observed clearly. f (x) = O0,k + x O1,k + x 2 O2,k + x 3 O3,k · · ·

−x f (x) = −x O0,k − x 2 O1,k + x 3 O2,k + x 4 O3,k · · ·

x2 x2 x3 x4 x5 f (x) = O + O + O + O3,k · · · 0,k 1,k 2,k k2 k2 k2 k2 k2 If we make necessary arrangements and some operations, the expression f (x) − 2 x f (x) + xk 2 f (x) is written as     1 1 O0,k + x(01,k − 00,k ) + x 2 O2,k − O1,k + 2 O0,k + x 3 O3,k − O2,k + 2 O1,k + · · · k k

By using (3) we get f (x) =

x k

1−x +

x2 k2

.  

In the following theorem we give some summation formulas of Oresme numbers. Theorem 4 For n ≥ 0, we have the followings.  1 − On+2,k , k

(17)

  1 k2 n+1 − + (−1) (On+2,k − 2On+1,k ) , 2k 2 + 1 k

(18)

n  i=0

n  (−1)i Oi,k = i=0

 Oi,k = k

2

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n 

O2i+1,k

i=0

n 

k2 = 2 2k + 1

O2i,k =

i=0



 k2 + 1 k2 + 1 2 O2n+1,k − k O2n+2,k ) , + k k2

k2 k − (k 2 + 1)O2n+2,k + O2n+1,k ) . 2k 2 + 1

(19)

(20)

Proof We will show the Eq. (17) by using induction. Obviously for n = 0, the equation holds. Assume that the equation is true for ∀m ≤ n. Then by (3), we obtain n+1 

Oi,k =

n 

i=0

 Oi,k + On+1,k = k 2

i=0

 1 − On+2,k + On+1,k k

Hence n+1 

 Oi,k = k 2

i=0

1 1 − On+2,k + 2 On+1,k k k



 = k2

 1 − On+3,k . k

The Eq. (18) can be shown similarly by using induction. To obtain the Eqs. (19) and (20), observe that n 

O2i,k

i=0

and

n  i=0

O2i+1

1 = 2

1 = 2

2n+1 

Oi,k +

2n+1 

 (−1)i Oi,k ,

i=0

i=0

2n+1 

2n+1 

i=0

Oi,k −

 (−1)i Oi,k .

i=0

  Now for Oresme numbers, we give an important result given by Laughlin in [7]. Let α − β = 0 and n−1  [ 2 ] n αn − β n T n−2m−1 (T 2 − 4D)m /2n−1 = zn = 2m + 1 α−β m=0

for An = z n A − z n−1 D I2

k-Oresme Numbers and k-Oresme Numbers with Negative Indices

where T is the trace and D is the determinant of the matrix A =

217

  ab cd

Then, for yn =

 [ n2 ]   n−i i

i=0

T n−2i (−D)i ,

we have  An =

 yn − dyn−1 byn−1 . cyn yn − ayn−1

Halici and Akyüz considered the Horadam sequence and some summation formulas involving the terms of the Horadam sequence. The authors derived combinatorial identities by using the trace, the determinant and the nth power of a special matrix. Inspired by these studies, in the following theorem, we give some combinatorial identities involving k-Oresme numbers. Theorem 5 For n ≥ 1, we have the following.   [ n2 ]  [ n2 ]    n−i 1 i n n −i 2 1  − 2 k ( k − 4)i . = n−1 i 2i k n − i 2 i=0 i=0

(21)

Proof The generating matrix of k- Oresme numbers can be calculated in terms of yn as  O = n

where,

 yn − k12 yn−1 . yn yn − yn−1

 [ n2 ]   n−i yn = . i i=0

For k > 2, from the equation |O − λI | = 0 λ1 =

k+

√ √ k2 − 4 k − k2 − 4 and λ2 = 2k 2k

are obtained. To get the result, we will use the following equation. λn1 + λn2 = 2yn − yn−1 .

(22)

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The left-hand side of the Eq. (22) can be written as n    n−i  n 1 i=0

i

2

√  √ i i n    n−i  n 1 k2 − 4 k2 − 4 + − i 2k 2 2k i=0

which equals  [ n2 ]   n    i  i i  n 1 1  2 1  n i 2 2−4 . = (k) k − 4 − − k − 4 k i 2n (k)i 2n−1 i=0 2i i=0 Similarly, we can write the right-hand side of the Eq. (22) as n−1     [ [ n2 ]  2 ]  n−i n−i −1 1 i 1 i − 2 − − 2 2 i i k k i=0 i=0

which equals n−1      [ [ n2 ]  2 ]  1 i n−i 1 i n − i n − 2i − 2 . − 2 − 2 k k i n − 2i n − i i=0 i=0

Since

n−2i n−i

=

n−2[ n2 ] n−[ n2 ]

= 0, we get

2yn − yn−1

    [ n2 ]   n−i 1 i n − 2 . = i k n−i i=0

By using these equations in (22), we obtain     [ n2 ]   [ n2 ]   i  n−i 1 i n n 1  − 2 (k)−i = n−1 k2 − 4 i 2i k n−i 2 i=0 i=0  

which completes the proof. Theorem 6 For k ≥ 2, we have  On+3,k =

k2 − 1 k2

 On+2,k −

1 On,k . k4

(23)

Proof The proof is clear by using the recurrence relation of k-Oresme numbers.  

k-Oresme Numbers and k-Oresme Numbers with Negative Indices

219

3 k-Oresme Numbers with Negative Indices In this section, we study the negative-indexed terms of the k-Oresme numbers denoted by O−n,k . Definition 1 For n ∈ Z+ , the recurrence relation of the negative-indexed k-Oresme numbers is O−n,k = k 2 (O−n+1,k − O−n+2,k ),

(24)

where O−1,k = −k O0,k = 0. The first three terms of this sequence obtained by (24) are O−1,k = −k, O−2,k = −k 3 , O−3,k = k 3 − k 5 . Note that, taking k = 2 in (24), one can obtain the Oresme numbers with negative indices. Since the characteristic √ equation of the recurrence relation (24) is k 2 x 2 − k 2 x + √ k+ k 2 −4 k− k 2 −4 and β = . 1 = 0, its roots are α = 2k 2k Recall that Binet’s formula of k-Oresme numbers is On,k = √k12 −4 (α n − β n ) and

αβ = k12 . If we substitute n by −n in the Binet’s formula for the k-Oresme numbers, we can get the Binet’s formula for negative-indexed Oresme numbers. That is, 1 O−n,k = √ 2 k −4



1 1 − n αn β 

1

= −k √ k2 − 4 2n



k+

1 =√ 2 k −4



αn − β n (αβ)n



n  n  √ √ k − k2 − 4 k2 − 4 − . 2k 2k

From the last equation, we get 1

O−n,k = −k √ k2 − 4 2n



k+

n  n  √ √ k − k2 − 4 k2 − 4 − 2k 2k

(25)

which is the Binet’s formula for negative-indexed k-Oresme numbers. We can define the generating matrix of negative-indexed k-Oresme numbers as 0 1 . O= −k 2 k 2 

(26)

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Theorem 7 For the matrix O, we have  k O−n+1,k − k1 O−n,k . O = k O−n,k − k1 O−n−1,k n

(27)

Proof Obviously the Eq. (27) holds for n = 1. Suppose fot the induction that the equation is true for ∀m ≤ n. Then we have  On+1 =

k O−n,k k O−n−1,k

 k O−(n+1)+1,k −k O−n+1,k − k O−n,k = −k O−n,k − k O−n−1,k kO −(n+1),k

 2 − kk (O−n,k − O−n+1,k ) 2

− kk (O−n−1,k − O−n,k )

.

By using (24), we get On+1 =

 k O−(n+1)+1,k − k1 O−(n+1),k k O−(n+1),k − k1 O−(n+1)−1,k  

which completes the proof.

Furthermore, the Cassini’s identity can also be obtained by using this theorem. Observing the fact that (det (O))n = det (On ) and (det (O))n = (k)2n ,

(28)

we get − O−n+1,k O−n−1,k + (O−n,k )2 = (k)2n

(29)

Theorem 8 For n, m ∈ Z+ , we have the following. (k) (k) (k) O−(n+m) = k O−n O−m+1 −

1 (k) O O (k) k −n−1 −m

Proof Using the fact On+m = On Om and (27) we can write On Om =

  k O−m+1,k − k1 O−m,k k O−n+1,k − 1k O−n,k k O−n,k − 1k O−n−1,k k O−m,k − k1 O−m−1,k

and On+m =

 k O−(n+m)+1,k − k1 O−(n+m),k . k O−(n+m),k − k1 O−(n+m)−1,k

(30)

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221

By equating of the matrices On+m and On Om we get k O−(n+m),k = k 2 O−n,k O−m+1,k − O−n−1,k O−m,k Thus, O−(n+m),k = k O−n,k O−m+1,k −

1 O−n−1,k O−m,k . k  

Theorem 9 For x ∈ R, the generating function for negative-indexed k-Oresme number is g(x) =

∞ 

O−i,k x i = −

i=0

kx . 1 − xk 2 + x 2 k 2

Proof By observing g(x) = O0,k + x O−1,k + x 2 O−2,k + x 3 O−3,k · · · ,

−xk 2 g(x) = −xk 2 O0,k − x 2 k 2 O−1,k − x 3 k 2 O−2,k − x 4 k 2 O−3,k · · · and x 2 k 2 g(x) = x 2 k 2 O0,k + x 3 k 2 O−1,k + x 4 k 2 O−2,k + x 5 k 2 O−3,k · · · we can write the expression g(x) − xk 2 g(x) + x 2 k 2 g(x) as O0,k + x(O−1,k − k 2 O0,k ) + x 2 (O−2,k − k 2 O−1,k + k 2 O0,k ) + · · · Then using (24), we have g(x) − xk 2 g(x) + x 2 k 2 g(x) = −kx  

which completes the proof. Theorem 10 For n ≥ 1, we have the followings. n 

O−i,k = −k(1 − k O−n+1,k ),

(31)

i=0

n  (−1)i O−i,k = i=0



1 k + (−1)n (k 2 O−n,k + O−n−1,k ) , +1

2k 2

(32)

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S. Halıcı et al. n 

O−(2i+1),k

i=0

n 

  −k + 2k 2 O−2n−1,k + (k 2 + k 4 − 1)O−2n,k 1 −k + , (33) = 2 2k 2 + 1

O−2i,k =

i=0

k2 k − (k 2 + 1)O−2n−2,k + O−2n−1,k ) . 2 2k + 1

(34)

Proof Proof of the Equations of (32) and (33) is similar to the proof of the (17) and (18) by induction. To show the Eqs. (34) and (35), observe that n 

O−2i+1,k

i=0

1 = 2

2n+1  i=0

O−i,k −

2n+1 

 (−1) O−i,k ) i

i=0

and n  i=0

O−2i,k

  2n 2n  1  i = O−i,k + (−1) O−i,k ) . 2 i=0 i=0  

Theorem 11 For n ≥ 1, we have  [ n2 ]  [ n2 ]    n−i n n 1  2 n−2i 2 i (k 2 )n−i (k 2 − 4)i . (k ) (−k ) = n−1 i 2i n − i 2 i=0 i=0 Proof The proof is the similar to proof of the Theorem 5.

(35)  

Moreover, it can be noticed that the results given for k-Oresme numbers in this study satisfy for the Oresme numbers by taking k = 2.

4 Conclusion In this paper, we have defined a matrix for k-Oresme numbers and studied some properties and identities of these matrices using some combinatorial operations. In addition, we have defined k-Oresme numbers with negative indices. We have obtained some new formulas including these numbers by defining a matrix.

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References 1. A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly, 1965, 3(3), 161–176. 2. Oresme, N., Quaestionessuper Geometriam Euclides, H. Busard, Ed., Brill, Leiden, 1961. 3. Cook, C. K., Some sums related to sums of Oresme numbers, Applications of Fibonacci Numbers, Springer„ 2004 ,9, 87–89. 4. Horadam, A. F., Oresme Numbers, The Fibonacci Quarterly, 1974, 12(3), 267–271. 5. Sentürk, ¸ G.Y., Gürses, N., Yüce, S., A New Look on Oresme Numbers: Dual-Generalized Complex Component Extension, Conference Proceeding Science and Technology, 2018, 1(1), 254–265. 6. Morales, G.C., Oresme Polynomials and Their Derivatives, arXiv:1904.01165 [math.CO], 2019. 7. Laughlin, J. Mc., Combinatorial identities deriving from the nth power of a 2 × 2 matrix, Integers: Electronic Journal of Combinatorial Number Theory, 2004, 4, 1–15. 8. Liana, A. S., Wloch, I., Oresme Hybrid numbers and Hybrationals, Kragujevac Journal of Mathematics, 2024, 48(5), 747–753. 9. Melham, R.S., Shannon, A. G., Some summation identities using generalized Q-matrices, The Fibonaci Quaterly, 1995, 33(1), 64–73. 10. Akyuz, Z., Halici, S., On some combinatorial identities involving the terms of generalized Fibonacci and Lucas sequences, Hacettepe Journal of Mathematics and Statistics, 2013, 42(4), 431–435 11. Akyuz, Z., Halici, S. (2012). Some identities deriving from the nth power of a special matrix. Advances in Difference Equations, 2012, 1, 1–6.

A Note on k-Telephone and Incomplete k-Telephone Numbers Paula Catarino, Eva Morais, and Helena Campos

Mathematics Subject Classification 11B37 · 11B83 · 05A10

1 Introduction The Fibonacci numbers are one of the most well-known sequences of positive numbers, in part due to all the applications (e.g. [1–4]). The Fibonacci sequence {Fn }n is defined by the following second order recurrence relation Fn = Fn−1 + Fn−2 , n ≥ 2 with initial terms F0 = 0 and F1 = 1, which can be expressed as the sum (n−1)/2 

Fn =

 i=0

 n−i −1 . i

Several authors have investigated this sequence and generalizations of this sequence. The incomplete Fibonacci sequence {Fn (s)}n was introduced by Filipponi [5] and is given by P. Catarino · E. Morais (B) · H. Campos Department of Mathematics, University of Trás-os-Montes e Alto Douro, 5001-801 Vila Real, Portugal e-mail: [email protected] P. Catarino e-mail: [email protected] H. Campos e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_21

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     s   n−i −1 n−1 , 0≤s≤ ; n ∈ IN . Fn (s) = i 2 i=0 The k−Fibonacci sequence {Fk,n }n , for k ∈ IR+ , is defined by [6] Fk,n = k Fk,n−1 + Fk,n−2 , n ≥ 2 with Fk,0 = 0 and Fk,1 = 1, that verifies the explicit formula [7] (n−1)/2 

Fk,n =

 i=0

 n − i − 1 n−2i−1 k . i

Similarly, the incomplete k−Fibonacci sequence is defined by [8] Fnk,n

     l   n − i − 1 n−2i−1 n−1 k ; n ∈ IN . = , 0≤l ≤ i 2 i=0

The same investigation methodology has been applied to other sequences, as the Pell {Pn }n sequence, a generalization of the Fibonacci sequence defined by Pn = 2Pn−1 + Pn−2 , n ≥ 2 with P0 = 0 and P1 = 1, for which Catarino and Campos [9], Ramírez [8, 10], and Ramírez and Sirvent [11] investigated the incomplete k−Pell, k−Pell-Lucas and modified k−Pell numbers. A recent research about the incomplete version of the Leonardo integer sequence was done by Catarino and Borges [12] with the establishment of several properties and identities. This work focuses on other number sequences of which the telephone and ktelephone numbers are two of their generalizations. We begin with the explicit formulas of the telephone numbers and the k-telephone numbers sequences, and then we introduce and study the incomplete versions of both sequences.

2 The Telephone and k-Telephone Numbers This section focuses on telephone and k-telephone numbers. The telephone numbers, also known as involution numbers, are given by Tn = Tn−1 + (n − 1)Tn−2 , n ≥ 2

(1)

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227

with initial terms T0 = T1 = 1. The recurrence relation (1) of the sequence {Tn }n was found by Heinrich August Rothe in 1800 [13] when counting the involutions (that is, permutations that are their own inverse) in a set of n elements. This sequence can also be seen as the number of possible patterns of connections between the n subscribers of a telephone service, where each subscriber can only be connected with another one, therefore the designation telephone numbers. Another application of the telephone numbers is to graph theory, with Tn given by the number of matchings in a complete graph with n vertices. In recreational mathematics, the nth telephone number Tn is the number of ways to place n rooks on an n × n chessboard such that no two rooks attack each other and such that the configuration of the rooks is symmetric under a diagonal reflection of the board. In recreational mathematics, the nth telephone number Tn is the number of possible positionings of n rooks on an n × n chessboard such that no two rooks attack each other (they are not in the same column or row) and they are symmetrically disposed about a diagonal of the board. The telephone numbers may be expressed as [14] n/2 

Tn =

 n (2i − 1)!! . 2i

 i=0

(2)

From Eq. (2) it is easily obtained the following explicit formula of the telephone numbers, which will be more useful throughout this text n/2



Tn =

2i (n

i=0

n! . − 2i)!i!

(3)

The nth telephone number Tn is given by the value at zero of the nth derivative of the exponential generating function (e.g., [14–16]) given by e0.5x

2

+x

=

∞ 

Tn

n=0

xn . n!

(4)

Thus, the Poisson generating function of the telephone numbers is 2

e0.5x =

∞  n=0

Tn e−x

xn . n!

(5)

Chowla, Herstein, and Moore [14] also obtained the asymptotic formulas in n √ Tn ∼ n Tn−1

and

(n/e)n/2 en Tn ∼ 21/2 e1/4

1/2

.

(6)

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For any positive real number k, the k-telephone sequence {Tk,n }n∈IN is recurrently defined by Tk,0 = 1, Tk,1 = k, Tk,n = kTk,n−1 + (n − 1)Tk,n−2 .

(7)

The first terms of this sequence are 1, k, k 2 + 1, k 3 + 3k, k 4 + 6k 2 + 3, k 5 + 10k 3 + 15k, . . . Notice that for k = 1, we have the telephone numbers T1,n = Tn . The explicit formula for this sequence is given in the next result. Proposition 1 The {Tk,n }n∈IN sequence satisfies the following explicit formula n/2

Tk,n =

 i=0

2i (n

n! k n−2i . − 2i)!i!

(8)

Proof This result will be proved by induction on n. It is clear that Eq. (8) is verified for n = 0 and n = 1 Tk,0 =

0  i=0

Tk,1 =

0  i=0

0! 0! 0 k 0−2i = 0 k =1 2i (0 − 2i)!i! 2 0!0! 1! 0! 1 k 1−2i = 0 k =k. 2i (1 − 2i)!i! 2 0!0!

Now suppose that Eq. (8) holds for all non-negative integer up to n ≥ 1; we prove that it still holds for n + 1, that is (n+1)/2

Tk,n+1 =

 i=0

From Eq. (7), we have

2i (n

(n + 1)! k n+1−2i . + 1 − 2i)!i!

(9)

A Note on k-Telephone and Incomplete k-Telephone Numbers

229

Tk,n+1 = kTk,n + nTk,n−1 n/2

=k

 i=0

n! k n−2i + n i 2 (n − 2i)!i!

n/2



=

2i (n

i=0

For n odd,

n 2

(n−1)/2

Tk,n+1 =

 i=0

(n−1)/2

=

 i=0

(n+1)/2

=

 i=0

=

 i=0

(n+1)/2

=

 i=0

=

2

=

2i (n

n+1 , 2

n! k n−1−2i − 1 − 2i)!i!

so



n + 1 − 2i 2i + n+1 n+1



(n + 1)! k n+1−2i 2i (n + 1 − 2i)!i!

n/2 

n/2  i=0

=

i=0

 n+1 

(n + 1)! k n+1−2i 2i (n + 1 − 2i)!i!

i=0

=

and



n! n! k n+1−2i − (−1) k n+1 2(i−1) (n + 1 − 2i)!(i − 1)! 2 (n + 1)!(−1)!

thus Eq. (9) is true    if nis odd. = For n even, n2 = n+1 2 Tk,n+1 =

n−1 2

i=0

(n−1)/2

(n − 1)! k n−1−2i − 1 − 2i)!i!

n! n!

k n+1−(n+1) k n+1−2i − (n+1)/2 2i (n − 2i)!i! ! 2 (n − (n + 1))! n+1 2

i=0

=

2

2i (n

(n+1)/2  n! n! n+1−2i k k n+1−2i + i (i−1) (n + 1 − 2i)!(i − 1)! 2 (n − 2i)!i! 2 i=1



(n+1)/2

 n−1 



(n−1)/2  n! n! n+1−2i k k n−1−2i + i i (n − 1 − 2i)!i! 2 (n − 2i)!i! 2 i=0

(n+1)/2

+

n! k n+1−2i + − 2i)!i!

(n−1)/2

n/2  i=0

−

n 2

and

 n−1  2

=

n 2

− 1, so

n/2−1  n! n! n+1−2i k k n−1−2i + i i (n − 1 − 2i)!i! 2 (n − 2i)!i! 2 i=0

 n! n! k n+1−2i + k n+1−2i i i−1 (n + 1 − 2i)!i! 2 (n − 2i)!i! 2 i=1 n/2

 n! n! k n+1−2i + k n+1−2i i i−1 (n + 1 − 2i)!i! 2 (n − 2i)!i! 2 i=0

2−1 (n

n/2

n! k n+1 + 1)!(−1)!

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Table 1 The incomplete Tnl for 1 ≤ n ≤ 6 n 0 1 2 3 4 5 6

=

l 0

1

2

3

1 1 1 1 1 1 1

2 4 7 11 16

10 26 61

76

n/2  i=0

=

n/2  i=0

(n + 1)! k n+1−2i 2i (n + 1 − 2i)!i!



n + 1 − 2i 2i − n+1 n+1



(n + 1)! k n+1−2i 2i (n + 1 − 2i)!i! 

and also Eq. (9) is true if n is even.

3 The Incomplete Telephone Numbers In this section we present a new integer sequence related with the explicit formula (3) of {Tn }n∈IN . Definition 1 The incomplete telephone numbers are defined by Tnl =

l  i=0

n! , i 2 (n − 2i)!i!

0≤l ≤

n 2

.

In Table 1, some values of incomplete telephone numbers are provided. Some special cases of Eq. (10) are Tn0 = 1, Tn1 =

n≥0

n2 − n + 2

, n≥2 2 2 (n − 1)(n − 5n + 10) Tn2 = 1 + , 8 n/2 = Tn , Tn 

Tn

n−2 2

=

n≥4

n≥0

n! , n even Tn − n!! Tn − n!!, n odd

In the next result we state the recurrence relation satisfied by this sequence.

(10)

A Note on k-Telephone and Incomplete k-Telephone Numbers

231

Proposition 2 The recurrence relation of the incomplete telephone numbers Tnl is given by  =

l+1 Tn+2

l+1 Tn+1

+ (n +

1)Tnl ,

0≤l ≤

 n−1 , n≥2. 2

(11)

The relation (11) can be transformed into the nonhomogeneous recurrence relation (n + 1)! l l . (12) = Tn+1 + (n + 1)Tnl − l Tn+2 2 (n − 2l)!l! Proof Using Definition 1, we obtain for n ≥ 2 and 0 ≤ l ≤ l+1 + (n + 1)Tnl = Tn+1

l+1  i=0

 n−1  2

l  (n + 1)! n! + (n + 1) i i 2 (n + 1 − 2i)!i! 2 (n − 2i)!i! i=0

l+1  (n + 1)! n! + (n + 1) i (n + 1 − 2i)!i! i−1 (n − 2(i − 1))!(i − 1)! 2 2 i=0 i=1  l+1   (n + 1)! (n + 1)! (n + 1)! = + i−1 − −1 i (n + 1 − 2i)!i! 2 2 (n − 2i + 2)!(i − 1)! 2 (n + 2)!(−1)! i=0   l+1  n − 2i + 2 2i (n + 2)! × + = −0 2i (n − 2i + 2)!i! n+2 n+2 i=0

=

=

l+1 

l+1  i=0

(n + 2)! l+1 = Tn+2 2i (n − 2i + 2)!i!

thus the recurrence relation (11) is true. The relation (12) follows easily l l = Tn+1 + (n + 1)Tnl−1 Tn+2 l = Tn+1 + (n + 1)Tnl − (n + 1) l + (n + 1)Tnl − = Tn+1

2l (n

n! − 2l)!l!

(n + 1)! 2l (n − 2l)!l!



Two properties involving the incomplete telephone numbers are stated in the results below. Proposition 3 The incomplete telephone numbers sequence verifies s    s i=0

i

l+i l+s Tn+i (n + 1)s−i = Tn+2s , 0≤l ≤

n−s−1 . 2

(13)

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P. Catarino et al.

Proof This result will be proved by induction on s. For s = 0 0    0

i

i=0

l+i Tn+i (n + 1)0−i =

  0 l+0 l+0 T (n + 1)0 = Tnl = Tn+2·0 0 n+0

For s = 1 1    1 i=0

i

l+i Tn+i (n + 1)1−i =

    1 l 1 l+1 Tn (n + 1)1−0 + T (n + 1)1−1 0 1 n+1

l+1 l+1 l+1 = (n + 1)Tnl + Tn+1 = Tn+2 = Tn+2·1

Considering the result is true for all j < s + 1, we prove it for s + 1. We have  s+1   s + 1 l+i Tn+i (n + 1)s+1−i i i=0   s+1    s s l+i + Tn+i = (n + 1)s+1−i i i − 1 i=0  s+1   s+1    s l+i s l+i s+1−i Tn+i (n + 1) Tn+i = + (n + 1)s+1−i i i − 1 i=0 i=0   s    s l+i s l+s+1 Tn+i (n + 1)s+1−i + = Tn+s+1 (n + 1)0 i s + 1 i=0 s    s l+i+1 Tn+i+1 (n + 1)s+1−i−1 + i i=−1   s    s l+i+1 s l+s Tn+i+1 (n + 1)s−i =(n + 1)Tn+2s + Tnl (n + 1)s+1 + i −1 i=0 l+s l+1+s =(n + 1)Tn+2s + Tn+1+2s l+s+1 l+s+1 =Tn+2s+2 = Tn+2(s+1)



The next result is concerning about the sum of s consecutive elements of the lth column of the array shown in Table 1. Proposition 4 The incomplete telephone numbers sequence verifies s−1  i=0

l+1 l+1 l Tn+i (n + 1) = Tn+s+1 − Tn+1 , 0≤l ≤

n−s−1 . 2

(14)

A Note on k-Telephone and Incomplete k-Telephone Numbers

233

Proof (by induction on s) s = 1 0 

l+1 l+1 l Tn+i (n + 1) = Tnl (n + 1) = Tn+2 − Tn+1

i=0

Considering the result is true for all j < s + 1, we prove it for s + 1. We have: s 

l Tn+i (n + 1) =

i=0

= =

s−1 

l l Tn+i (n + 1) + Tn+s (n + 1)

i=0 l+1 Tn+s+1 l+1 Tn+s+2

l+1 l − Tn+1 + Tn+s (n + 1) l+1 − Tn+1



4 The Incomplete k-Telephone Numbers In this section we present a generalization of the incomplete telephone numbers sequence. We start with the following definition. Definition 2 The incomplete k-telephone numbers are defined by l Tk,n =

l  i=0

n/2

Note that Tk,n 0 = kn , Tk,n 1 = Tk,n

1  i=0

n! k n−2i , i 2 (n − 2i)!i!

0≤l ≤

n 2

.

(15)

= Tk,n and some special cases of (15) are:

n≥0 n! k n−2i 2i (n − 2i)!i!

n! k n−2 = kn + 2(n − 2)!   1 , = kn 1 + 2 2k (n − 2)(n − 1) 2 = kn + Tk,n

n>2

n! k n−2 k n−4 + 2 2(n − 2)(n − 1) 2 (n − 4)!2!

k n−4 k n−2 + 3 = kn + 2(n − 2)(n − 1) 2 (n − 4)(n − 3)(n − 2)(n − 1)   1 1 n + 3 4 , =k 1+ 2 2k (n − 2)(n − 1) 2 k (n − 4)(n − 3)(n − 2)(n − 1)

n>4

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l for 1 ≤ n ≤ 6 Table 2 The incomplete Tk,n

n n n n n n n n

=0 =1 =2 =3 =4 =5 =6



l 0

1

1 k k2 k3 k4 k5 k6

k2 + 1 k 3 + 3k k 4 + 6k 2 k 5 + 10k 3 k 6 + 15k 4

n−2 2





Tk,n

=

n−2

2

3

k 4 + 6k 2 + 3 k 5 + 10k 3 + 15k k 6 + 15k 4 + 45k 2

k 6 + 15k 4 + 45k 2 + 15



2 

i=0

n! k n−2i 2i (n − 2i)!i!

  n! n−2 n2 = Tk,n −  n  n n k 2 2 (n − 2 2 )! 2 ! ⎧ n−2 n2 n! ⎪ , n even ⎪Tk,n − 2n/2 (n−2 n )! n ! k ⎨ 2 2   n−1 = n−2 2 n! ⎪ , n odd ⎪ ⎩Tk,n − 2(n−1)/2 n−2 n−1 ! n−1 ! k 2

2

⎧ n! ⎪ , n even ⎨Tk,n − 2n/2 n ! 2 = n!   k , n odd ⎪ ⎩Tk,n − n−1

=

2(n−1)/2

n! Tk,n − n!!

2

!

, n even

Tk,n − n!!k , n odd

(n ≥ 2)

By Definition (15), we have on Table 2 the first incomplete k-Telephone Numbers. The recurrence relation which is satisfied by this sequence is shown as follows. Proposition 5 The k−incomplete telephone numbers sequence verifies  l+1 Tk,n+2

=

l+1 kTk,n+1

+ (n +

l 1)Tk,n ,

0≤l ≤

 n−1 , n ≥ 2. 2

Proof The incomplete k-telephone numbers are defined by l Tk,n

=

l  i=0

n! k n−2i , 2i (n − 2i)!i!

0≤l ≤

n 2

.

(16)

A Note on k-Telephone and Incomplete k-Telephone Numbers

235

Using this definition, we have l+1 l =k kTk,n+1 + (n + 1)Tk,n

l+1  i=0

= =

l+1 

(n + 2)!

i=0

2i (n − 2i + 2)!i!

l+1 

 ×

 (n + 1)! n! k n+1−2i + (n + 1) k n−2i i 2 (n + 1 − 2i)!i! 2i (n − 2i)!i! l

i=0



n − 2i + 2 2i + k n−2i+2 n+2 n+2

(n + 2)! k n−2i+2 2i (n − 2i + 2)!i!

i=0 l+1 = Tk,n+2



A nonhomogeneous version of the recurrence relation established in the previous proposition is contained in the following result. Proposition 6 The relation (16) can be transformed into the nonhomogeneous recurrence relation l l l Tk,n+2 = kTk,n+1 + (n + 1)Tk,n −

2l+1 (n

(n + 1)! k n−2l − 1 − 2l)!(l + 1)!

(17)

Proof By (16) and Definition 2, we get l l l − kTk,n+1 − (n + 1)Tk,n Tk,n+2 l+1 l l l = kTk,n+1 + (n + 1)Tk,n − kTk,n+1 − (n + 1)Tk,n

l+1 l = k Tk,n+1 − Tk,n+1  l+1  l   (n + 1)! (n + 1)! n+1−2i n+1−2i k k =k − 2i (n + 1 − 2i)!i! 2i (n + 1 − 2i)!i! i=0 i=0

(n + 1)! k n+2−2(l+1) 2l+1 (n + 1 − 2(l + 1))!(l + 1)! (n + 1)! k n−2l = l+1 2 (n − 1 − 2l)!(l + 1)! =



Two combinatorial properties about this sequence are presented in the next two results. Proposition 7 The incomplete k-telephone numbers sequence verifies l+s Tk,n+2s

=

s    s i=0

i

l+i Tk,n+i (n + 1)s−i k i

(18)

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P. Catarino et al.

Proof We will prove the result by induction on s. For s = 0, we have 0    0

i

i=0

l+i Tk,n+i (n + 1)−i k i =

  0 l l T (n + 1)0 k 0 = Tk,n 0 k,n

For s = 1, we have 1    1 i=0

i

l+i Tk,n+i (n

+ 1)

    1 l 1 l+1 1 0 k = T (n + 1) k + T (n + 1)0 k 1 0 k,n 1 k,n+1

1−i i

l+1 l = (n + 1)Tk,n + kTk,n+1 l+1 = Tk,n+2

Now suppose that the proposition is true for all j < s + 1 and we shall prove this identity for s + 1. We have  s+1   s+1

l+i (n + 1)s+1−i k i Tk,n+i i i=0   s+1    s s l+i = (n + 1)s+1−i k i + Tk,n+i i i − 1 i=0  s+1   s+1    s l+i s l+i Tk,n+i (n + 1)s+1−i k i + Tk,n+i = (n + 1)s+1−i k i i i − 1 i=0 i=0 l+s l+s+1 = (n + 1)Tk,n+2s + kTk,n+2s+1 l+s+1 = Tk,n+2s+2



Proposition 8 For n ≥ 2l + 2 s−1 

l+1 l+1 l Tk,n+i (n + 1)k s−i−1 = Tk,n+s+1 − k s Tk,n+1 .

(19)

i=0

Proof We will prove this result by induction on s. For s = 1 l+1 l+1 l l (n + 1)k 1−0−1 = (n + 1)Tk,n = Tk,n+2 − kTk,n+1 Tk,n

Now suppose that it is true for all j < s + 1 and we shall prove it for s + 1. Using Proposition 6, we obtain

A Note on k-Telephone and Incomplete k-Telephone Numbers

237

l+1 l+1 l+1 l+1 l Tk,n+s+2 − k s+1 Tk,n+1 = (n + 1)Tk,n+s + kTk,n+s+1 − k s+1 Tk,n+1

l+1 l+1 l + (n + 1)Tk,n+s = k Tk,n+s+1 − k s Tk,n+1

=k

s−1 

l l Tk,n+i (n + 1)k s−i−1 + (n + 1)Tk,n+s

i=0

=

s 

l Tk,n+i (n + 1)k s−i



i=0

5 Conclusions In this work we studied generalizations of the telephone numbers sequence similar to the methodology that was used for other sequences, like the Fibonacci, Pell, and Leonardo sequences (e.g., [6, 9, 12]). We also obtained meaningful properties for the new sequences, such as recurrence relations. As a future work, we are planning to explore possible applications of the introduced sequences to other areas of Mathematics and Engineering. Acknowledgements The research of the first two authors was partially financed by Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020. Also the first and the third authors thanks the Portuguese Funds through FCT (Fundação para a Ciência e a Tecnologia) within the Project UIDB/ CED/00194/2020.

References 1. Koshy, T.: Fibonacci and Lucas Numbers with Applications Vol 1. John Wiley & Sons, New Jersey (2010) 2. Mason, J.F., Hudson, R.H.: A generalization of Euler’s Formula and its connection to Fibonacci Numbers. In: F.T. Howard (ed.) Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applications, vol. 9, pp. 177–185. Springer Netherlands, Dordrecht (2004) 3. Somasundaram, K., SUMITRA, P.: Compression of Image using Fibonacci Code(FC) in JPEG2000. Int. J. Eng. Sci. Technol. 2(12), 7311–7319 (2010) 4. Srinivasan, T.P.: Fibonacci sequence, golden ratio, and a network of resistors. Am. J. Phys. 60(5) 461 (1992) 5. Filipponi, P.: Incomplete Fibonacci and Lucas numbers. Rend. Circ. Mat. Palermo 45(2), 37–56 (1996) 6. Falcón, S., Plaza, A.: On the Fibonacci k−numbers. Chaos Solitons Fractals 32(5), 1615–1624 (2007) 7. Falcón, S., Plaza, A.: On k−Fibonacci sequences and polynomials and their derivatives. Chaos Solitons Fractals 39(3), 1005–1019 (2009) 8. Ramírez, J.L.: Incomplete k-Fibonacci and k-Lucas numbers. Chinese J. of Math. 2013, article ID 107145 (2013) 9. Catarino, P., Campos, H.: Incomplete k−Pell, k−Pell-Lucas and Modified k−Pell Numbers. Hacet. J. Math. Stat. 46(3), 361–372 (2017)

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10. Ramírez, J.L.: Incomplete generalized Fibonacci and Lucas polynomials. Hacet. J. Math. Stat. 44(2) 363–373 (2015) 11. Ramírez, J.L., Sirvent, V.: Incomplete tribonacci numbers and polynomials. J. Integer Seq. 17 article 14.4.2 (2014) 12. Catarino, P., Borges, A.: A note on incomplete Leonardo Numbers. Integers 20, article A43 (2020) 13. Knuth, D.E.: The Art of Computer Programming Vol 3. Sorting and Searching, AddisonWesley, Reading (1973) 14. Chowla, S., Herstein, I.N., Moore, W.K.: On recursions connected with symmetric groups I. Can. J. Maths. 3, 328–334 (1951) 15. Banderier, C., Bousquet-Melou, M., Denise, A., Flajolet, P., Gardy , D., Goutou-Beauchamps, D.: Generating functions for generating trees. Discrete Math. 246(1–3), 29–55 (2002). 16. Kutler, M.B., Vinroot, C.R.: On q-analogs of recursions for the number of involutions and prime order elements in symmetric groups. J. Integer Seq., [electronic only] 13(3) (2010) http://eudml.org/doc/230628

Extended Exponential-Weibull Mixture Cure Model for the Analysis of Cancer Clinical Trials Adam Braima Mastor, Oscar Ngesa, Joseph Mung’atu, Ahmed Z. Afify, and Abdisalam Hassan Muse

1 Introduction Life-time data analysis has applications in many different fields, including business, engineering, finance, and health, among others [1, 2]. For modeling lifetime data, there are a number of probability models that may be used, including log-logistic, beta, gamma, Weibull, exponential, and others. Additionally, modeling lifetime data using these typical approaches is usually insufficient [3, 4]. The Weibull distribution has been used to cope with several challenges in a wide range of survival data and to model lifespan data. The Weibull distribution, with its negatively and positively skewed density forms, is the primary option when modeling monotone hazard rates [5]. The parameters of this distribution’s tremendous flexibility allow for a range of techniques, all of which have the same key property: A. B. Mastor (B) · A. H. Muse Department of Mathematics (Statistics Option) Programme, Pan African University, Institute for Basic Sciences, Technology and Innovation (PAUSTI), Nairobi 62000-00200, Kenya e-mail: [email protected] A. H. Muse e-mail: [email protected] O. Ngesa Mathematics Statistics and Physical Sciences Department, Taita Taveta University, Voi, Kenya e-mail: [email protected] J. Mung’atu Department of Mathematics, Jomo Kenyatta University of Agriculture and Technology, Juja, Kenya A. Z. Afify Department of Statistics, Mathematics and Insurance, Benha University, Banha, Egypt e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_22

239

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The hazard rate is a monotone function that can be decreasing, constant, or growing [6]. The Weibull distribution is inappropriate for survival data with a non-monotone failure rate function. As a result, scientists explore for extensions and modifications of the Weibull distribution. Mudholkar et al. [7] developed a three parameter model by exponentiating the Weibull distribution known as the exponentiated Weibull distribution. Silva et al. [8] proposed the beta modified Weibull distribution. Bourguignon et al. [9] introduced the Weibull-G family. Pinho et al. [10] developed the Gamma-exponentiated Weibull distributions (GEW). Xie et al. [11] developed a three-parameter modified Weibull extension. Lee et al. [12] introduced the beta-Weibull model, which can be incorporated to data sets with non-monotone and monotone hazard rate functions (hrf) and has the exponential, exponentiated Weibull, and exponentiated exponential models as sub-models, among others. Cordeiro et al. [13] defined and investigated a novel four-parameter modification of the Weibull distribution that can simulate a bathtubshaped failure rate forms. Cordeiro et al. [14] developed the exponential-Weibull distribution. Additionally, a common scenario in the study of survival data, particularly in cancer research, is when a portion of the population is not exposed to the problem event. Researchers often choose cure fraction models over parametric models in this scenario if the survival time distribution for vulnerable individuals is known. The study of survival data including long-term survivors makes use of cure fraction models, which are regarded as an enlarged form of conventional survival models. Since the 1940s, these models have been a subject of study. The mixture and nonmixture cure are the two primary groups of cure fraction models. The population is separated into two categories, susceptible (cured) and unsusceptible (uncured), according to the mixed cure model’s underlying assumption. Boag [15] first proposed the mixed cure model, then Berkson [16] and Gag further improved it after three years. Based on the above discussion, this study, present probability distribution, named the extended exponential-Weibull distribution, as a extension of the exponentialWeibull distribution using Lehmann-Type II generator approach. Then we will develop the extended exponential-Weibull (ExEW) mixture cure model. The motivation of this paper is to propose a maximum likelihood estimates method for analyzing the four-parameter extended exponential-Weibull (ExEw) distribution where cured subjects. The remainder of the article is structured as follows: Sect. 2 discusses the basic lifetime functions of the proposed distribution and its sub-models. Section 3 presents cure model. The estimation of model parameters is investigated in Sect. 4. Section 5 presents an application of the proposed model to real-life data and model comparison. In Sect. 6 concluding remarks are presented.

Extended Exponential-Weibull Mixture Cure Model …

241

2 The Extended Exponential-Weibull (ExEW) Distribution The ExEW distribution is an extension of the exponential-Weibull distribution which is as presented in Mastor et al. [17]. It is a continuous probability distribution with positive support R on a subset of (0, ∞), and it has a four parameters. The distribution can handle the five fundamental hazard rate function forms (increasing, constant, decreasing, bathtub shapes, and unimodal) and it is formulated by using Lehmanntype II approach. Let X ∼ E x E W (a, b, c, α) then the cumulative distribution function (CDF) of the ExEW distribution can be defined as follows F(x) = 1 − exp [−α(ax + bx c )], x > 0.

(1)

Where a > 0, c > 0, are the shape parameters, b > 0, is scale parameter, and the additional shape parameter is α > 0.

2.1 Sub-models The distribution is made up of many fundamental sub-models that are often utilized in parametric survival modeling. These include exponential (E) distribution, Weibull (W) distribution, exponentiated exponential (EE) distribution [18], and the exponential-Weibull (EW) distribution [14] (Fig. 1 and Table 1). 1. The probability density function (pdf) corresponding to Eq. 1 takes the form f (x) = α(a + bcx c−1 ) exp [−α(ax + bx c )], x > 0.

(2)

The ExEW distribution’s pdf forms are shown in Fig. 1 for various parameter selections. The pdf of the ExEW distribution can be symmetrical, asymmetrical, unimodal, J, and reversed-J shapes. 2. The survival function (sf) corresponding to Eq. 1 is as follows: S(x) = exp [−α(ax + bx c )]. Table 1 Sub-models of ExEW(a, b, c, α) distribution Model a b E W EE EW

a 0 a a

0 b 0 b

(3)

c

α

0 c 0 c

1 1 α 1

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Fig. 1 Pdf of the ExEW distribution with various choices of the parameters

The sf is a non-increasing function of time t with the following properties. 1. S(x) = 1 for x = 0 2. S(x) = 0 for x = ∞ 3. The hazard rate function (hrf) of the proposed distribution is expressed as: h(x) = α(a + bx c−1 ).

(4)

Figure 2 shows the hrf which is clearly decreasing, increasing, remaining constant, or indicating a more complicated process, such as a bath-tubshaped hazard rate and unimodal. 4. The cumulative hazard rate function is obtained as: H (x) = α(ax + bx c ), x > 0.

(5)

The hrf is clearly decreasing, increasing, constant, J-shaped, and reverse Jshaped as shown in Fig. 2

3 Cure Model Two basic categories of cure models-mixture cure and non-mixture cure modelshave been proposed in the literature to suit lifetime data from medical investigations. These models may be used to analyze real-world data in areas apart from medicine,

Extended Exponential-Weibull Mixture Cure Model …

243

Fig. 2 Hrf of the ExEW distribution with different values of the parameters

including economics, reliability, criminology, education, marketing, and sociology. The modeling strategy varies depending on the event that interests the researcher; the general aim is to observe the passage of time until the occurrence, although for some subjects, the event will never happen. In this part, we present the mixture cure.

3.1 Mixture Cure Model The technique that is commonly used to model data with long-term survivors is the mixture cure model (MCM). The advantages of the MCM is that it allow covariates to affect patients who are cured differently from how long vulnerable people survive. On the other hand, this model is unable to confirm the proportional hazard function characteristic. Additionally, the MCM does not seem to have a biological explanation, particularly in investigations of cancer recurrence. Let T denote the event’s time occurrence and let ρ ∈ (0, 1) be a probability of being cured.

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Additionally, suppose that S ∗ (t) and f ∗ (t) are survival and density functions, respectively, Thus, the entire population survival function is S(mcm) (t) = ρ + (1 − ρ)S ∗ (t),

(6)

and the corresponding probability density function is f (mcm) (t) = (1 − ρ) f ∗ (t),

(7)

then from Eqs. 3 and 6 the S(mcm) (t) for ExEw distribution can be written as follows S(mcm) (t) = ρ + (1 − ρ).exp [−α(ax + bx c )],

(8)

and from Eqs. 2 and 7 the f (mcm) (t) or ExEw distribution can be written as follows f (mcm) (t) = (1 − ρ)α(a + bcx c−1 ) exp [−α(ax + bx c )].x > 0.

(9)

4 Parameter Estimation Assume we observe the pair t j , δ j in a random sample of size m, where δ is the censoring indicator variable, which has values of zero and one for censored and uncensored observations, respectively, and j = 1, . . . , m. The likelihood function for the MCM model is given in Eq. 6 is L MC M (θ ) =

m   δ  1−δ j (1 − ρ) f ∗ (t j ) j ρ + (1 − ρ)S ∗ (t j ) .

(10)

j=1

The maximum likelihood estimates of θ can be obtained by maximising Eqs. (10) directly, with respect to the parameter vector θ . By directly maximizing the total loglikelihood function with the help of the tools MATHEMATICA, R, and MATLAB, the parameter estimations can be derived. R software is the program that is used in this paper.

5 Application To illustrate the applicability of the proposed model, we analyze the rebuilt IPASS clinical trial data that was published by Argyropoulos and Unruh [19]. The data set is available in the R package AHSurv see https://cran.r-project.org/web/packages/ AHSurv/index.html. The data base covers the time frame from March 2006 to April 2008, which is consisting 1217 observations. The TTT transform plot in Fig. 3

Extended Exponential-Weibull Mixture Cure Model …

245

Fig. 3 The TTT plot of the rebuilt IPASS clinical trial data

shows a concavity pattern, illustrating the data’s increasing hazard rate shape. This demonstrates that the hrf in Fig. 2 is suitable for analyzing this data. The proposed model is compared to the another-models such as: Burr XII (BXII), WEibull (W), log-logistic(LL) and the generalized log-logistic [4] models. The pdf of the Burr XII (BXII) distribution is f (x)abx a−1 (1 + x a )−(b)+1 , x > 0.

(11)

The pdf of the WEibull (W) distribution is f (x) = acx a−1 exp(−cx a )x > 0.

(12)

The pdf of the log-logistic(LL) distribution is f (x) =

ac(cx)a−1 x > 0. [1 + (cx)a ]2

(13)

The pdf of the generalized log-logistic (GLL) distribution is ac(cx)a−1 ca

[1 + (bx)a ] ba +1

,x > 0

(14)

The analytical techniques are used to determine which mixture cure mode (MCM) matches the data the best. The Akaike information criterion (AIC). The AIC is define as follows AI C = 2(k − ), (15)

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Table 2 Parameters of competing models estimated bˆ model aˆ cˆ ExEW BXII W LL GLL

0.019 (0.025) 2.716 (0.060) 7.458 (0.252) 1.573 (0.058) 0.140 (0.007)

1.829 (0.000) 0.005 (0.325) 1.359 (0.042) 6.574 (0.323) 1.422 (0.072)

αˆ

ρ

1.368 (0.000) –

0.035 (0.000) –

–

–

–

–

0.044 (0.035)

–

0.001 (0.094) –24.382 (0.026) 0.010 (0.013) 0.144 (0.024) –0.013 (0.030)

Table 3 The analytical performance measures for comparing distributions Model AIC ExEW BXII W LL GLL

5704.828 5803.840 5708.006 5712.876 5708.614

Table 2 displays the ML estimates of the fitted models’ parameters with standard error. ExEW MCM model more closely matches the data from the rebuilt IPASS clinical trial than any others comparable MCM models. Table 3 shows that the ExEW MCM model has the lowest AIC values when compared to the other MCM models. means that the ExEW MCM model provides the greatest fit to the data.

6 Conclusion For right-censored data, we proposed the extended exponential-Weibull mixture cure model in this paper. Using real data set, we compared the various susceptible distributions with mixed models. According to the rebuilt IPASS clinical trial data, we showed that the proposed mixture cure model is the best one when compared to other models for modeling real data.

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References 1. Ahmad, Z., Mahmoudi, E., Hamedani, G., Kharazmi, O.: New methods to define heavy-tailed distributions with applications to insurance data. Journal of Taibah University for Science 14(1), 359–382 (2020) 2. Nasiru, S., Mwita, P.N., Ngesa, O.: Exponentiated generalized power series family of distributions. Annals of Data Science 6(3), 463–489 (2019) 3. Nasiru, S., Mwita, P.N., Ngesa, O.: Exponentiated generalized exponential dagum distribution. Journal of King Saud University-Science 31(3), 362–371 (2019) 4. Muse, A.H., Mwalili, S., Ngesa, O., Almalki, S.J., Abd-Elmougod, G.A.: Bayesian and classical inference for the generalized log-logistic distribution with applications to survival data. Computational intelligence and neuroscience 2021 (2021) 5. He, W., Ahmad, Z., Afify, A.Z., Goual, H.: The arcsine exponentiated-x family: validation and insurance application. Complexity 2020 (2020) 6. Santana, T.V., Ortega, E.M., Cordeiro, G.M.: Generalized beta Weibull linear model: Estimation, diagnostic tools and residual analysis. Journal of Statistical Theory and Practice 13(1), 1–23 (2019) 7. Mudholkar, G.S., Srivastava, D.K.: Exponentiated weibull family for analysing bathtub failurerate data. IEEE transactions on reliability 42(2), 299–302 (1993) 8. Silva, G.O., Ortega, E.M., Cordeiro, G.M.: The beta modified Weibull distribution. Lifetime data analysis 16(3), 409–430 (2010) 9. Bourguignon, M., Silva, R.B., Cordeiro, G.M.: The weibull-g family of probability distributions. Journal of data science 12(1), 53–68 (2014) 10. Pinho, L., Cordeiro, G., Nobre, J.: The gamma-exponentiated Weibull distribution. Journal of Statistical Theory and Applications 11(4), 379–395 (2012) 11. Xie, M., Tang, Y., Goh, T.N.: A modified weibull extension with bathtubshaped failure rate function. Reliability Engineering & System Safety 76(3), 279–285 (2002) 12. Lee, C., Famoye, F., Olumolade, O.: Beta-weibull distribution: some properties and applications to censored data. Journal of modern applied statistical methods 6(1), 17 (2007) 13. Carrasco, J.M., Ortega, E.M., Cordeiro, G.M.: A generalized modified weibull distribution for lifetime modeling. Computational Statistics & Data Analysis 53(2), 450–462 (2008) 14. Cordeiro, G.M., Ortega, E.M., Lemonte, A.J.: The exponential-Weibull lifetime distribution. Journal of Statistical Computation and simulation 84(12), 2592–2606 (2014) 15. Boag, J.W.: Maximum likelihood estimates of the proportion of patients cured by cancer therapy. Journal of the Royal Statistical Society. Series B (Methodological) 11(1), 15–53 (1949) 16. Berkson, J., Gage, R.P.: Survival curve for cancer patients following treatment. Journal of the American Statistical Association 47(259), 501–515 (1952) 17. Mastor, A.B.S., Ngesa, O., Mung’atu, J., Alfaer, N.M., Afify, A.Z., et al.: The extended exponential Weibull distribution: properties, inference, and applications to real-life data. Complexity 2022 (2022) 18. Nadarajah, S.: The exponentiated exponential distribution: a survey. Springer (2011) 19. Argyropoulos, C., Unruh, M.L.: Analysis of time to event outcomes in randomized controlled trials by generalized additive models. PLoS One 10(4), 0123784 (2015)

On the Statistical Properties of the Deformed Algebras on the Jackson q-Derivative Mehmet Niyazi Çankaya

1 Introduction The deformation is an overwhelmingly evolving topic. The deformation for summation and subtraction is highly used in the mathematical analysis. Especially, the role of subtraction is very important if the base of the analysis is taken into account. The different deformations based on group theory and its application in the time scale calculus are proposed in Refs. [1–3]. One of the techniques to construct the deformation can use the entropy function. The advantage of entropy is that it can produce the generalized logarithm which has different properties. One of the generalized logarithm which is called deformed logarithm because of the nature of entropy function is the logq derived by Tsallis entropy [4, 5]. Entropy is widely used for many areas in the science. A generalized entropy function is defined by the following form:  f (x)Λ ( f (x)) dx,

(1)

where Λ represents the generalized logarithm and f is a probability density function. If the random variable X is discrete, then the entropy function is defined in the form: n 

p(xi )Λ ( p(xi )) ,

(2)

i=1

where p is a probability function. M. N. Çankaya (B) Department of International Trading and Finance, Faculty of Applied Sciences, U¸sak University, Bir Eylül Campus, 64200 Usak, Turkey e-mail: [email protected] URL: https://akbis.usak.edu.tr/Home/Index/mehmet.cankaya © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_23

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Note that since the deformed summation is based on the estimation of the correlation coefficient, the expression performing the summation should be called as the deformed summation based on the correlation coefficient. The reason why we need such deformation and stochasticity for total values of two variables is that the outcomes of experiment cannot be measured accurately due to many reasons such as gravity of universe, mistaken results while performing measurement. Further, the correct measurement is impossible because it depends on the number π [3, 4, 6] (see Ref. [7]). The organization of the paper is the following. Initially, preliminaries about entropy and estimation method are given. Section 3 provides the theoretical results. Then the numerical method for finding the values of summation and subtraction is examined by Sect. 4. Afterwards, numerical results of the summation and the subtraction are given by figures and tables. The paper finalizes with the concluding remarks and brief discussion of the results.

2 Preliminaries 2.1 The Correlation Coefficient for Random Variables X and Y The dependence between two variables X and Y can be measured by the correlation coefficient if there is a linear relationship between X and Y [8]. Cov(X, Y ) V ar (X )V ar (Y ) E[(X − μ X )(Y − μY )] = σ X σY E(X Y ) − E(X )E(Y ) = . σ X σY

ρ(X, Y ) = √

(3)

2.2 Time Scale Calculus: Jackson q-Derivative Time scale calculus covers the union of discrete and continuous cases of the support set of the function at the same time. In the time scale calculus, there are two operators which are forward and backward. These operators are the generalizations of the classical derivative. It is possible to define Jackson q-derivative by using the forward or backward operators. Let us use the forward operator σ (x) to propose the graininess function τ (x) = σ (x) − x = q x − x. Thus, we define the following Jackson q-derivative given by the following expression [1, 2]:

On the Statistical Properties of the Deformed Algebras on the Jackson q-Derivative

g(q x) − g(x) . x(q − 1)

251

(4)

2.3 Tsallis Entropy Abe provided the procedure which generates an entropy functional from the function given by [9–11]: Dq g(z) = Dq f −z =

f −qz − f −z . −z(q − 1)

(5)

If z = −1, then the functional form of Tsallis entropy is obtained as follows: T ( f, q) =

1 − f 1−q fq − f = −fq = f q logq ( f ). q −1 1−q

(6)

2.4 Tsallis Non-extensive Statistic for the Deformed Summation A deformed summation based on Tsallis non-extensive statistics is presented in the following form [4]: x + y + (1 − q)x y. (7)

2.5 M-Estimations of Location and Scale Parameters The parameters for location (μ) and scale (σ ) are central tendency and central dispersion measures to summarize the data set. These parameters can be estimated by means of different estimation methods. The most common estimation method which can be free from the assumption about the distribution of the data set is the robust estimation method known as M-estimation. In order to define M-estimation method which is a generalization of maximum likelihood estimation (MLE) method, we can use the definition of MLE method. The definition of MLE includes a probability density function (PDF). Note that it is not essential to use a PDF in the framework of M-estimation. Let us introduce a PDF which includes the location and scale parameters. Thus, we have a family for the joint estimation of location and scale parameters. The PDF for the parameters μ and σ is given by the following form:

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f (x; μ, σ ) =



1 h σ

 x −μ . σ

(8)

The likelihood function for function f (x; μ, σ ) is given by L(μ, σ, X) =

n 

f (xi ; μ, σ ),

(9)

i=1

where the vector X represents the observations x1 , x2 , . . . , xn . n is the number of sample size drawn from the population. It is assumed that the observations are assumed to be member of the function f (x; μ, σ ) in the corresponding population. The log-likelihood function for function f (x; μ, σ ) is given by log(L(μ, σ ; X)) = =

n  i=1 n 

log( f (xi ; μ, σ )) −ρ(xi ; μ, σ ).

(10)

i=1

When the log-likelihood function, i.e., log(L(μ, σ, X)), is maximized with respect to parameters μ and σ , we can have the M-estimators of these parameters after the tools in robust estimation method are used. Thus, the M-estimators μˆ and σˆ of parameters μ and σ are given by the following expression, respectively, n wi xi , μˆ = i=1 n i=1 wi and

 σˆ =

n 1 wi (xi − μ)2 n i=1

(11) 1/2 ,

(12)

where wi ∈ [0, 1] is a weighting function produced by wi = ψ(xi )/xi . The function  ψ is the derivative of objective function ρ, that is, ψ = ρ . The derivative is taken according to parameters μ and σ and so only one score function ψ is obtained [12–14].

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3 Theoretical Results 3.1 The Deformed Summation Based on Tsallis q-Entropy If the deformed summation is replicated at k-times, then we have the following expression: x + y + k(1 − q)x y, (13) where k is an integer number. Let k be chosen as non-integer number. Then, we can have a reparametrized form for the expression (1 − q). That is, a new expression is k(1 − q), which corresponds another value of q, that is, q ∗ = k(1 − q).

3.2 The Deformed Summation and Subtraction Based on Tsallis q-Entropy and Correlation Coefficient with Applications on Derivative and Integral Since the correlation coefficient includes the product of random variables X and Y , the following expression can be given by using Eq. (3). E(X Y ) = ρσ X σY + E(X )E(Y ).

(14)

Let us rewrite Eq. (14) as follows: E(X Y ) = ρσ X σY + μ X μY ,

(15)

where μ X = E(X ) and μY = E(Y ) represent the location parameter for the variables X and Y . Since μ X and μY are parameters, it is necessary to use estimators for these parameters. Many different estimators can be suggested for the parameters μ X and μY . For example, the robust forms can be suggested. Secondly, let us write the expectation form of the deformed summation which is given by the following form: E(X ) + E(Y ) + (1 − q)E(X Y ).

(16)

Equation (16) can be suggested, because the variable can be regarded as the random variable. Note that if the random variables X and Y represent only one observation, the expectation (E) can be dropped out. Thus, we have X + Y + (1 − q)X Y . The summation based on the correlation coefficient is given by E(X ) + E(Y ) + (1 − q)(ρσ X σY + μ X μY ),

(17)

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which is the theoretical representation for the deformed summation. If E(X ) and ˆ ) and E(Y ˆ ) and the estimators for the parameters ρ, E(Y ) are replaced by E(X σ X , σY , μ X and μY are suggested, then we have the following expression: ˆ ) + E(Y ˆ ) + (1 − q)(ρˆ σˆ X σˆ Y + μˆ X μˆ Y ). E(X

(18)

ˆ ) and E(Y ˆ ) show the first order moment, they can be moment estimators Since E(X ˆ ) and E(Y ˆ ), the for the expected values of random variables X and Y . For E(X alternative estimators such as robust estimators, harmonic and geometric mean, etc., can be used. Alternative form of Eq. (18) can be suggested if the replicated form is applied for the deformed summation based on the correlation coefficient: ˆ ) + E(Y ˆ ) + (1 − q)k(ρˆ σˆ X σˆ Y + μˆ X μˆ Y ). E(X

(19)

If k = 1, Eq. (19) drops to Eq. (18). The first order moment without location parameter μ, that is, E(X ) or E(Y ) and also the moment for the product of two variables X and Y must exist and finite for the proposed probability (density) function; because,

b

b as it is well known, E(X ) = a x f (x)dx < ∞ and E(Y ) = a y f (y)dy < ∞ or n n ˆ ) = i=1 ˆ ) = i=1 wi xi < ∞ and E(Y wi yi < ∞. empirically E(X The deformed subtraction based on the correlation coefficient can be proposed by using the Eq. (19). It is defined as: ˆ ) − E(Y ˆ ) E(X 1 + (1 − q)k μˆ −1 ˆ X σˆ Y + μˆ X μˆ Y ) Y (ρˆ σ

.

(20)

If it is replicated at p-times, then we have the following expression: ˆ ) − E(Y ˆ ) E(X p[1 + (1 − q)k μˆ −1 ˆ X σˆ Y + μˆ X μˆ Y )] Y (ρˆ σ

.

(21)

Note that p can also be chosen as non-integer number. Since we have subtraction from the deformed form of summation, it is possible to apply for the definition of derivative based on the linear sense. A derivative based on the linear sense can be given by the following form: d dρˆ f (x) = p[1 + (1 − q)k μˆ −1 f (x). ˆ X σˆ Y + μˆ X μˆ Y )] Y (ρˆ σ dρˆ x dx

(22)

The inverse of derivative is the definition of the integral [15, 16]. Thus, it is possible to propose a new integral based on the expression given by Eq. (22):  ρˆ

 f (x)dρˆ x =

f (x) p −1 (1 + (1 − q)k μˆ −1 ˆ X σˆ Y + μˆ X μˆ Y ))−1 dx. Y (ρˆ σ

(23)

On the Statistical Properties of the Deformed Algebras on the Jackson q-Derivative

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Since Eq. (23) is an example for variable transformation having Jacobian, i.e., dρˆ = Adx, A is expression in the Eq. (23), it is possible to apply it on the engineering applications such as image analysis, etc. For example, the shape of an image can be resized to have different forms of shape of the image and other topics in the regionalized variables [17]. Equation (22) which is based on the linear sense can be transferred to time scale calculus. When the definition of time scale calculus is applied, we have the following expression: Δ

dρˆ f (τ (x)) − f (x) f (x) = p[1 + (1 − q)k μˆ −1 . ˆ X σˆ Y + μˆ X μˆ Y )] Y (ρˆ σ dρˆ x τ (x) − x

(24)

Equation (24) is an example for the forward operator based on the deformed subtraction. An alternative definition can be backward form from time scale calculus. The backward form of Eq. (24) can be given by the following form: ∇

dρˆ f (x) − f (ξ(x)) ˆ X σˆ Y + μˆ X μˆ Y )] f (x) = p[1 + (1 − q)k μˆ −1 . Y (ρˆ σ dρˆ x x − ξ(x)

(25)

It is possible to define the expected form of the forward and backward forms of derivatives given by Eqs. (24)–(25). Thus, we can have the following expressions for Eqs. (24)–(25), respectively: ˆ f (τ (x))) − E( ˆ f (x)) dρˆ E( f (x) = p[1 + (1 − q)k μˆ −1 ˆ X σˆ Y + μˆ X μˆ Y )] , Y (ρˆ σ ˆ (x)) − E(x) ˆ dρˆ x E(τ (26) and E(Δ)

ˆ f (x)) − E( ˆ f (ξ(x))) dρˆ E( ˆ X σˆ Y + μˆ X μˆ Y )] f (x) = p[1 + (1 − q)k μˆ −1 , Y (ρˆ σ ˆ ˆ dρˆ x E(x) − E(ξ(x)) (27) where the variables x and y can be given by using the regression equation between these two variables. An arbitrary function f which is convex or concave can be used. In other words, the non-decreasing or non-increasing functions should be discarded, because the difference at where the function f gives the same values for the different points. That is, the tangent must be different from zero to have a value for difference of the function f at the distinct two points. Let us propose a derivative when q-deformation of subtraction is applied to the Eq. (26). Thus, it is possible to express the following form: E(∇)

E(Δq )

dρˆ f (x) = p[1 + (1 − q)k μˆ −1 ˆ X σˆ Y + μˆ X μˆ Y )] Y (ρˆ σ dρˆ x ˆ f (τ (x))) − E( ˆ f (x)) 1 + (1 − q) E(x) ˆ E( ˆ (x)) − E(x) ˆ E(τ

ˆ f (x)) 1 + (1 − q) E(

.

(28)

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The backward form of Eq. (28) is as follows: E(∇q )

dρˆ ˆ X σˆ Y + μˆ X μˆ Y )] f (x) = p[1 + (1 − q)k μˆ −1 Y (ρˆ σ dρˆ x ˆ f (x)) − E( ˆ f (ξ(x))) 1 + (1 − q) E(ξ(x)) ˆ E( ˆ ˆ E(x) − E(ξ(x))

ˆ f (ξ(x))) 1 + (1 − q) E(

.

(29)

Equations (26)–(27) and (28)–(29) are applied to the non-linear system in a stochastic process such as queuing theory, estimation of location and scale parameters. The qdeformed difference form of the expressions in the Eqs. (24)–(25) can be proposed, as given by Equations (28)–(29). It should be noted that the entropy functions based on the derivatives are examples of different weight functions from maximum likelihood estimation method [7, 9, 13].

4 Numerical Results The theoretical results are applied for the numerical evaluation. The codes written with Matlab software are provided to use them for the different aims in the researches. The codes for the deformed summation and subtraction based on the correlation coefficient and the goodness of fit test are given as well. Since the values of x are distributed uniformly from the intervals [x − ε, x + ε], the weighting function should be chosen as the uniform distribution in the interval [0, 1]. On the other hand, uniform distribution can be replaced by alternative distributions such as cumulative distribution function of normal, Student t distributions, etc. [8]. Even if the regularity conditions of MLE method are not satisfied when the support set of the chosen probability density function depends on the parameters such as uniform distribution on the interval [a, b], M-estimations for the locationscale family can be proposed for any function [13]. Consequently, it is possible to use the Eqs. (11) and (12) for the estimations of location μ and scale σ parameters, respectively. The empirical distributions of the deformed summation and subtraction based on the statistical property are tested according to which theoretical distribution shows the best accommodation. Note that the Eqs. (13) and (19) are basically the variable transformation. In addition, Eq. (19) is summation of random variables due to the number of observations. If we have a summation for the n-observations, it cannot be easy to generate a new probability function for this new summation which also includes the n observations from the variables x and y (see details in Ref. [8]). Since the variable transformation for the analytical expression of the deformed summation based on the correlation coefficient is not tractable, the computational evaluation for determining which theoretical distribution can accommodate with the empirical distribution is necessary. Additionally, in order to examine the moment property of the empirical distribution, we test the skewness and kurtosis of the empirical dis-

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257

tribution. If the skewness and kurtosis of the corresponding distribution are around zero and three, respectively, which are the theoretical values of the normal distribution, then the empirical distribution is assumed to come from the normal distribution. Note that the number of replication for determining the empirical distribution of the deformed summation is important to get an accurate decision about where the empirical distribution comes from. For this reason, the number of replication is chosen as 105 . The simulation study is performed in the following order: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

The number of the first replication is 1000. The number of the second replication is 105 . q = 0.99; x=2.99:0.001:3.01; nx=length(x);ny=length(x);y=5+x+normrnd(0,0.01,1,nx); wx=rand(1,nx);wy=rand(1,ny); numx=wx.*x;numy=wy.*y; mx=sum(numx)/sum(wx);my=sum(numy)/sum(wy); dx=x-mx;dy=y-my;exy=sum(dx.*dy)/nx; sx=sqrt(sum(wx.*(x-mx).ˆ2)/nx);sy=sqrt(sum(wy.*(y-my).ˆ2)/ny); r = exy/(sx*sy); deformed-summation = mx + my + (1-q) * (r * sx * sy + mx * my); [summation form]: deformed-subtraction =(mx - my) + (1 + (1-q) * (1/my) * (r * sx * sy + mx * my))

14. [division form]: deformed-subtraction = (mx - my) / (1 + (1-q) * (1/my) * (r * sx * sy + mx * my))

The codes given above can be changed for the subtraction given by Eq. (20), as given by Line 14 in the codes above. Line 5 has a regression equation y = β0 + β1 x + u, where β0 = 5, β1 = 1 and u ∼Normal(μ = 0, σ = 0.01). Note that since the regression form is used, the location parameter μ in the normal distribution has to be chosen zero. The scale parameter σ is chosen as 0.01, because the variability of y in the regression equation given at fixed values of x is tried to be small in order to decrease the degree of numerical magnitude of the variable y in the deformed summation. In addition, the main deformation parameter q is chosen as 0.99, because the role of the added term in Eq. (19) is tried to be decreased. However, its role should be observed as well as small degree. One can prefer to choose the different values of parameters to have different values of the deformed summation or subtraction in Lines 12–14. The first replication is used to recompute the second replication for performing to get the mean of tests used for the goodness of fit. In the computation of the second replication, deformed-summation is computed 105 times. The computational distribution of deformed-summation can be determined by using the goodness of fit test replicated 1000 times. For each test values and their corresponding pvalues of Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) test statistics [8], Tables 1, 2 and 3 provide the results to detect whether or not the empirical distribution of the deformed summation and subtraction are normally distributed.

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Table 1 Goodness of fit test statistics for the deformed summation KS test p-value of KS AD test 0.0021

0.7281

0.4814

p-value of AD 0.7714

Table 2 Goodness of fit test statistics for the deformed subtraction if line 13 is used KS test p-value of KS AD test p-value of AD 0.0022

0.7229

0.4796

0.7716

Table 3 Goodness of fit test statistics for the deformed subtraction if line 14 is used KS test p-value of KS AD test p-value of AD 0.0022

0.7235

0.4818

0.7755

According to the results of p-values of KS and AD test statistics in Tables 1, 2 and 3, the computational distribution of deformed-summation can be assumed to be normal. Further, if we examine skewness and kurtosis to represent the normality of deformed-summation, these values are 0.0165 − 3.0072, 0.0120 − 3.0006, and 0.0008 − 3.0022, respectively for the performed simulation which are represented by Figs. 1, 2 and 3 and Tables 1, 2 and 3. The skewness and kurtosis computed by using the moment with integer order show that the computational distribution of deformed-summation can be implied to be normal. 1. NCDF=normcdf(defsum,mean(defsum),std(defsum)); 2. test_NCDF=[defsum,NCDF]; [p-KS,test-KS]=kstest(defsum,’CDF’,test_NCDF);

3. dist = makedist(’normal’,’mu’,mean(defsum),’sigma’,std(defsum)); 4. [p-AD,test-AD]=adtest(defsum,’Distribution’,dist); defsum represents deformed-summation. Note that since q = 0.99 and the value of x is around 3, the chosen step size is 0.001. Additionally, the value of y as a second number in the deformed summation is computed by using the linear regression and its error term with normal distribution from μ = 0 and σ = 0.01. It is reasonable to expect that the computational distribution of the deformed-summation is normal with the maximum likelihood estimators for location μ and scale σ . However, the chosen weight functions wx=Uniform(0,1) and wy=Uniform(0,1) for the variables x and y, respectively, in the deformed summation are uniform distribution with the values of lower and upper parameters which are 0 and 1, respectively, and the values are represented by uniform distribution as well; because the values of x are as follows: x=2.99:0.001:3.01. Even if the values of the parameter q are changed, the empirical distribution of the deformed summation follows the normal distribution; because, the added term in Eq. (19) is a kind of shifting form or location. Note that when the normal distribution is compared with Student t-distribution, the normal distribution is not heavy-tailed

On the Statistical Properties of the Deformed Algebras on the Jackson q-Derivative

259

Values of the deformed summation based on correlation coefficient

11.255

11.25

11.245

11.24

11.235

11.23

11.225

0

1

2

3

4

5

6

7

8

9

10 4

x 10

Replication number

Fig. 1 The values of deformed summation based on correlation coefficient

Values of the deformed subtraction based on correlation coefficient

−4.74

−4.745

−4.75

−4.755

−4.76

−4.765

−4.77

−4.775

0

1

2

3

4

5

6

7

8

9

10 4

x 10

Replication number

Fig. 2 The values of deformed subtraction based on correlation coefficient if the summation form in line 13 is used

function, which shows that the results of the deformed summation tend to go to location. If we have a data set around highly intensive around location, the deformed summation or subtraction is capable of tending to go to one value, which shows that the statistic of the deformed summation or subtraction is consistent. Consequently, ones can use these statistics for their researches in economical, biological, social, etc. experiments safely.

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Values of the deformed subtraction based on correlation coefficient

−4.835

−4.84

−4.845

−4.85

−4.855

−4.86

−4.865

−4.87

0

1

2

3

4

5

6

7

8

9

10 4

x 10

Replication number

Fig. 3 The values of deformed subtraction based on correlation coefficient if the division form in line 14 is used

5 Conclusions and Discussions The deformed summation based on the Tsallis entropy has been used to propose a new deformed summation and subtraction. If we have product of two random variables, it is possible to imply the correlation coefficient. We have proposed the deformed summation and subtraction based on the correlation coefficient because of the product of two random variables. Since we can have the deformed subtraction, it is possible to define new derivative and its inverse form which is integral. The numerical simulations have been performed for the summation and subtraction cases and it has been shown that the computational distributions of summation and subtraction are normal after the goodness of fit tests, skewness and kurtosis are applied. The conditional probability and its group property with Markovian statistic will be studied. The application of the deformed summation and subtraction based on the correlation coefficient will be conducted to construct the differential equations. The integrals based on the deformed sense can be applied to define new moment estimators to manage the different degrees of moments which represent the frequency at the empirical distributions which can also be member of a theoretical distribution. Thus, the generalized moment estimators will be studied for the estimation of parameters and manage the precise fitting performance when the non-extensive statistic is used. A package in open access R Software will be prepared for the proposed theoretical results. Especially, the definition of Tsallis q-entropy is based on the Jackson q-derivative. If we change the definition of calculus, we can have the generalized entropy functions proposed by Ref. [7]. In future work, we provide a collection or a family for the

On the Statistical Properties of the Deformed Algebras on the Jackson q-Derivative

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deformed summation and subtraction based on the existing entropy functions based on q-sense in the time scale calculus.

References 1. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser Basel (2001) 2. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser Basel (2003) 3. Bogopol’skij, O. V.: Introduction to group theory (Vol. 6). European Mathematical Society (2008) 4. Tsallis, C.: Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World. Springer, New York (2009) 5. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988). https://doi.org/10.1007/BF01016429 6. Korbel, J.: Rescaling the nonadditivity parameter in Tsallis thermostatistics. Physics Letters A, 381(32), 2588–2592 (2017). https://doi.org/10.1016/j.physleta.2017.06.033 7. Çankaya, M. N.: Derivatives by ratio principle for q-sets on the time scale calculus. Fractals, 2140040 (2021). https://doi.org/10.1142/S0218348X21400405 8. Gut, A.: Probability: a graduate course. (Vol. 200, No. 5). New York: Springer (2005) 9. Wada, T., Suyari, H.: A two-parameter generalization of Shannon-Khinchin axioms and the uniqueness theorem. Phys. Lett. A 368, 199–205 (2007). https://doi.org/10.1016/j.physleta. 2007.04.009 10. Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A, 373, 2516–2519 (2009). https://doi.org/10.1016/j.physleta.2009.05.026 11. Abe, S.: A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics. Phys. Lett. A, 224, 326–330 (1997). https://doi.org/10.1016/S03759601(96)00832-8 12. Çankaya, M.N., Korbel, J.: Least informative distributions in maximum q-log-likelihood estimation. Physica A, 509, 140–150 (2018). https://doi.org/10.1016/j.physa.2018.06.004 13. Godambe, V.P.: An optimum property of regular maximum likelihood estimation. Ann. Math. Statist. 31, 1208–1211 (1960). https://doi.org/10.1214/aoms/1177705693 14. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., Stahel, W. A.: Robust statistics: the approach based on influence functions. John Wiley & Sons. Vol. 196 (2011). 15. Thomas, G.B., Weir, M.D., Hass J., Giordano, F.R.: Thomas’ Calculus. Addison-Wesley (2005) 16. Borges, E.P.: A possible deformed algebra and calculus inspired in nonextensive thermostatistics. Physica A, 340, 95–101 (2004). https://doi.org/10.1016/j.physa.2004.03.082 17. Cruz-Orive, L. M.: On the precision of systematic sampling: a review of Matheron’s transitive methods. Journal of Microscopy, 153(3), 315–333 (1989). https://doi.org/10.1111/j.13652818.1989.tb01480.x

Quaternion Algebras and the Role of Quadratic Forms in Their Study Nechifor Ana-Gabriela

1 Bilinear Forms Since the theory of quadratic forms and that of symmetric bilinear forms remain essentially the same, we will set out some of their properties and highlight how they lead us to characterize quaternion algebras. Let V be a vector space over the field K. Definition 1 [1] We say that the function g : V × V → K is a bilinear functional or bilinear form in V , if it is linear in both arguments, that means it satisfies the following conditions: 1. g(αx + βy, z) = αg(x, z) + βg(y, z)

(1)

g(x, αy + βz) = αg(x, y) + βg(x, z)

(2)

2. for any x, y, z ∈ V and α, β ∈ K. A bilinear form g : V × V → K can be classify as been: – symmetric, if g(x, y) = g(y, x), ∀x, y ∈ V

(3)

g(x, y) = −g(y, x), ∀x, y ∈ V

(4)

– antisymmetric, if

N. Ana-Gabriela (B) Ovidius University, Constanta, Romania e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_24

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Proposition 1 [1] A bilinear form g : V × V → K is antisymmetric if and only if g(x, x) = 0, for any x ∈ V . Further, we define the symmetric bilinear space over the field K as the pair (V, g), where V is a vector space with an associated symmetric bilinear form. We consider (V1 , g1 ) and (V2 , g2 ) two symmetric bilinear spaces. It is called isometry [2], an injective linear trasnformation σ : V1 → V2 , such that g2 (σ (x), σ (y)) = g1 (x, y),

(5)

for any x, y ∈ V1 . The composition of isometries is obviously, an isometry. At the same time, we say about two spaces (V1 , g1 ) and (V2 , g2 ), they are isometric or isomorphic (denoted by (V1 , g1 ) ∼ = (V2 , g2 )), if there is a bijective isometry σ : V1 → V2 . As a remark, we have that isometry is an equivalence relation. Definition 2 Let x, y ∈ V be two vectors. We say that they are orthogonal (or perpendicular to each other) if g(x, y) = g(y, x) = 0.

(6)

Also, the subsets V1 , V2 of V are called orthogonal if g(x, y) = 0 for all x ∈ V1 , y ∈ V2 and we denoted by V1 ⊥ V2 . For each V1 of V , is associated its orthogonal space: V1 ⊥ = {x ∈ V | x ⊥ V1 }.

(7)

Lemma 1 [2] V1 ⊥ is a subspace of V . In case of V ⊥ = 0, i.e. for each vector x = 0 there exists a vector y such that g(x, y) = 0, we talk about a regular symmetric bilinear space (V, g). It can also be called nondegenerate or nonsingular and otherwise, it said to be singular. We notice that in a regular space, zero is the only vector perpendicular to all other vectors. We call the radical of (V, g), the subspace V ⊥ . For two bilinear spaces (V1 , g1 ), (V2 , g2 ), we define the orthogonal sum of them, denoted by V1 ⊥V2 , as been the direct sum V = V1 ⊕ V2 (i.e. V = V1 × V2 and V1 ∩ V2 = {0}, V1 , V2 considered subspaces of V ), with associated bilinear form:

Quaternion Algebras and the Role of Quadratic Forms in Their Study

g : V1 ⊕ V2 → K, g((x1 , x2 ), (y1 , y2 )) = g1 (x1 , y1 ) + g2 (x2 , y2 ).

265

(8)

It can be remarked that, if g1 s¸i g2 are symmetric, then g is also symmetric. Definition 3 [2] A bilinear form g : V × V → K is called anisotropic, if g(x, x) = 0 implies x = 0, for any x ∈ V , and otherwise, it is called isotropic. For an isotropic bilinear form, we also talk about an isotropic bilinear space (V, g) (or in other words a space that represents zero). A subspace V of V is considered to be totally isotropic, if g(x, y) = 0, for any x, y ∈ V , i.e. V ⊂ V ⊥ and in case of g is an isotropic bilinear form, then (V, g) will always contains a nonzero totally isotropic subspace. Definition 4 We say that a bilinear space (V, g) represents the scalar α ∈ K, if there exists a vector x = 0 such that g(x, x) = α and we will call it universal space, when (V, g) represents all α ∈ K, α = 0 Lemma 2 If a bilinear space is isotropic, then it is universal.

1.1 The Associated Matrix of a Bilinear Form We consider now V be a n−dimensional fix B = {e1 , ..., en } a basis n vector space, we xi ei and y = nj=1 y j e j . in V and two arbitrary vectors x = i=1 The bilinear form expression g, for the vectors x, y is given by [1]: g(x, y) = g

 n

xi ei ,

i=1

n  j=1

=

n 

 yjej

=

n  n 

xi y j g(ei , e j ) =

i=1 j=1

ai j xi y j = X T AY

(9)

i, j=1

where A = (ai j )i, j=1,n is the matrix of bilinear form g relative to the basis B. Remark 1 The matrix of a symmetric bilinear form is a symmetric matrix: ai j = g(ei , e j ) = g(e j , ei ) = a ji , i, j = 1, n.

(10)

Otherwise, if the bilinear form is antisymmetric, then its matrix is also antisymmetric:

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ai j = g(ei , e j ) = −g(e j , ei ) = −a ji , i, j = 1, n.

(11)

Transformation of the matrix of a bilinear form when the basis is changed Theorem 1 Let be g : V × V → K a bilinear form and we consider the bases B1 and B2 in the vector space V , with the transition matrix from B1 to B2 denoted by C. If A1 ∈ Mn (K ) and A2 ∈ Mn (K) are the matrices of g relative to the bases B1 and B2 , then the connection between them is: A2 = C × A1 × C T .

(12)

Corollary 1 The bilinear space (V, g) is regular, if the associated matrix A of g is invertible. Definition 5 The rank of a bilinear form is the rank of its associated matrix. In case of rank g =dim V = n we talk about a nondegenerate bilinear form g : V × V → K. Instead, if rank g and as a remark, we have that every bilinear space is isometric to the space < α1 , ..., αn >.

Quaternion Algebras and the Role of Quadratic Forms in Their Study

267

2 Quadratic Forms We have now come to one of the main subjects, and namely, quadratic forms for whose definition we consider V be a vector space over the field K with characteristic other than 2. Definition 6 [1] It is called quadratic form on V , a map h : V → K with the property there exists a symmetric bilinear form g : V × V → K such that h(x) = g(x, x), ∀x ∈ V . The symmetric bilinear form g that defines the quadratic form h is called bilinear form associated with h and it has the following form: g(x, y) =

1 [h(x + y) − h(x) − h(y)] 2

(13)

Further, we define the quadratic space over the field K as the pair (V, h), where V is a vector space with an associated quadratic form h : V → K. An isometry between two quadratic spaces (V1 , h 1 ) and (V2 , h 2 ) is defined as an injective linear transformation σ : V1 → V2 that satisfies h 2 (σ (x)) = h 1 (x), ∀x ∈ V . The following proposition reffers to different ways of renotation the orthogonal sum defined on bilinear spaces, by using quadratic forms. Proposition 3 Let (V1 , g1 ),(V2 , g2 ) be two bilinear spaces and V = V1 ⊥V2 the orthogonal sum of V1 and V2 . h 1 , h 2 , h. If h 1 , h 2 , h are their associated quadratic forms, we can write h = h 1 ⊥h 2 , instead of V = V1 ⊥V2 . We denote m × h = h⊥...⊥h

 , where m ∈ N. by m times

Another definition is for quadratic forms h 1 : V1 → K and h 2 : V2 → K which are isometric, [3], if there exists a vector space isomorphism σ : V1 → V2 such that h 2 (σ (x)) = h 1 (x), ∀x ∈ V1 and we denote h 1 h 2 . The map σ is called an isometry. Definition 7 [4] We say that a quadratic form represents the scalar α ∈ K, if there exists x ∈ V, x = 0 such that h(x) = α. If h represents all non-zero elements of K, then the space (V, h) is called universal. Definition 8 [2] A quadratic form h : V → K is anisotropic if h(x) = 0 implies x = 0, for all x ∈ V . Otherwise, it is called isotropic. Definition 9 [2] We say that a quadratic space (V, h) is positive definite if h(x) > 0, ∀x ∈ V and x = 0. Conversly, it is called negative definite.

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2.1 The Associated Matrix of a Quadratic Form We consider a n−dimensional vector space V , with a basis B = {e1 , ..., en } and n xi ei a vector from V . x = i=1 The analytical expression of a quadratic form h is given by: h(x) = g(x, x) = g

 n

xi ei ,

n 

i=1

 xjej

=

j=1

n  n 

ai j xi x j = X T AX,

(14)

i=1 j=1

where A = (ai j )i, j=1,n is the associated matrix of symmetric bilinear form g and also, it is the asssociated matrix of quadratic form h. It is called the canonical expression of a quadratic form, its expression in a basis in which its matrix is a diagonal matrix. Definition 10 [1] The rank of a quadratic form is the rank of its associated matrix. A quadratic form is called nondegenerate, if its rank is equal to the dimension of vector space on which it is defined. If the rank of form is smaller than the dimension of vector space, then it is called degenerate. Proposition 4 Let h : V → K be a quadratic form with rank h = r ≤ n =dim V . Then the matrix of quadratic form in the corresponding canonical basis B1 ⎛

α1 ⎜0 ⎜ ⎜ .. ⎜ . ⎜ A1 = ⎜ ⎜0 ⎜0 ⎜ ⎜ .. ⎝ .

0 α2 .. . 0 0 .. .

... ... .. .

... ... .. .

0 0 .. .

αr 0 .. .

... ... .. .

... ... .. .

⎞ 0 0⎟ ⎟ .. ⎟ .⎟ ⎟ 0⎟ ⎟, 0⎟ ⎟ .. ⎟ .⎠

0 0 ... 0 ... 0

that is, the canonical expression of h is: h(x) = α1 · x1 2 + α2 · x2 2 + · · · + αr · xr 2 =

r  i=1

for x = (x1 , x2 , ..., xn ) B1 ∈ V .

αi · xi 2 ,

(15)

Quaternion Algebras and the Role of Quadratic Forms in Their Study

269

Definition 11 The number of positive coefficients in a canonical expression of a quadratic form denoted by p, is called the positive index of inertia and the number q of negative coefficients is called the negative index of inertia. Now, if we denote by p = positive index of inertia, q = negative index of inertia and comparing these indices with the dim V = n and rang h = r , we will classify the quadratic forms as follows: 1. 2. 3. 4. 5.

positive defined, iff p = r = n; semi-positive defined, iff p = r < n; negative defined, iff q = r = n; semi-negative defined, iff q = r < n; undefined, iff p = 0 and q = 0.

3 Quaternions One method to define and classify quaternion algebras is by using quadratic forms. Above, we brought to the fore a brief theory of quadratic forms and now, we will try to explain their connection to quaternion algebras. Forwards, we had to consider the quaternion algebra over an arbitrary field K with char K = 2 to ensure we get a non-commutative algebra. In the characteristic 2, this construction gives a non-central algebra (remark i j = ji) and therefore, not a quaternion algebra.   as the generalized Definition 12 [5] Let α, β ∈ K. We define H(α, β) = α,β K quaternion algebra over the field K, if there exists two generators i, j with the defining relations: (16) i 2 = α, j 2 = β, i j = − ji If we denote k := i j, we obtain k 2 = (i j)(i j) = −i 2 j 2 = −αβ ∈ K,

(17)

ik = −ki = α j, k j = − jk = βi

(18)

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and the multiplication table for a quaternion algebra which is determined by the multiplication rules, linearity and associativity as follows: · 1 i j k

1 1 i j k

i i α −k -α j

j j k β βi

k k αj −βi −αβ

In the  case where K = R and α = β = −1, we denote by H := H(−1, −1) =  −1,−1 the real quaternion algebra. R From the multiplication rules among {i, j, k} presented above, it is clear that H(α, β) is spanned by {1, i, j, k} over K. Proposition 5 [5] {1, i, j, k} forms a K−basis for H(α, β) and so, it is a 4dimensional algebra over the field K. In the following [3], we present two possibilities for generalized quaternion algebra, which depend on the choice of K, α and β:   1. α,β is a division algebra (an algebra in which every non-zero element is invertK ible);   2. α,β is isomorphic with M2 (K), the algebra of all 2 × 2 matrix with elements K from K. In this case, we say that the quaternion algebra is split. We consider now an arbitrary quaternion a ∈ H(α, β), of the form a = a0 + a1 i + a2 j + a3 k. Definition 13 We define the conjugate of element a: a = a0 − a1 i − a2 j − a3 k

(19)

and we consider the following elements: t (a) = a + a = 2a0 ∈ K

(20)

n (a) = aa = a02 − αa12 − βa22 + αβa32 ∈ K,

(21)

and called the trace, respectively the norm of a ∈ H(α, β). As a remark, we have that the map a → a is also called the bar involution on H(α, β).

Quaternion Algebras and the Role of Quadratic Forms in Their Study

271

Since t(a) = a + a = t(a) and n (a) = aa = aa = n (a), it is noted that the norm and the trace of a are both scalars. We further define g(a, b) =

t(ab) ab + ba = ∈ K, 2 2

(22)

that is a symmetric bilinear form on H(α, β). So, (H(α, β), g) becomes a quadratic space over K and the quadratic form associated with this bilinear form is given by: h(a) = g(a, a) =

t(aa) 2aa = = aa = n(a). 2 2

(23)

Therefore, the quadratic form n is known as the norm form of the quaternion algebra. Using the standard notations of quadratic forms theory, given above, the form n can be written as < 1, −α, −β, αβ >. In the case of Hamilton’s quaternions H, the norm form is a0 2 + a1 2 + a2 2 + a3 2 which is a sum of four squares, so < 1, 1, 1, 1 >. Proposition 6 If n(a) = 0, then the element a is invertible and its inverse is



1 n(a)

 a.

Indeed a is invertible if and only if n(a) = 0, because otherwise n(a) = 0 implies that a is zero divisor. Therefore, we have:   Theorem 3 [3] The quaternion algebra α,β is a division algebra if and only if K for any a ∈ H(α, β), the relation n(a) = 0 implies a = 0.   is a division algebra if and Using the notions of quadratic forms theory, α,β K only if its norm form is anisotropic.     Theorem 4 [3] The quaternion algebras α,β and αK,β are isomorphic as algeK bras if and only if their norm forms are isometric as quadratic forms. In what follows, we present some examples relating to isomorphism:     β,α is isomorphic to , because their norm forms, < Example 1 1. α,β K K isometric.  −β, αβ > and < 1, −β, −α, αβ >, are 1, −α, ∗ is isomorphic to M (K) for any α ∈ K (where K∗ is the set of non-zero 2. α,1 2 K elements −1, α > is isotropic.  of K), because the norm form < 1, −α,  ∗ is isomorphic to M (K) for any α ∈ K , α = 1, because the norm form 3. α,1−α 2 K < 1, −α, α − 1, α(α − 1) > is isotropic.

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Some concrete examples are:   Example 2 1. The complex numbers C: α,β is isomorphic to M2 (C) for any nonC zero α, β ∈ C, because the norm form < 1, −α, −β, αβ > is always isotropic.   is always split. So α,β C   2. The real numbers R: α,β is isomoprhic to H for any α < 0 and β < 0. OtherR   wise, α,β is split. R   3. The finite fields Fn with n elements: α,β is always split, because due to WedF derburn’s theorem, any finite division ring is commutative.

4 Conclusions Since the study of quaternion algebras is closely related to the algebraic theory of quadratic forms, it is extremely necessary to know the basic notions related to bilinear and quadratic forms in order to facilitate the understanding of particular results. The main idea of this paper is to study and discover as many properties of bilinear and quadratic forms as possible. We brought to the fore elementary concepts of them and implicitly, we analyzed both symmetrical bilinear and quadratic spaces. As applications, we turned our attention to different connections and properties of matrices associated with bilinear and quadratic forms. During the last section, we highlighted how these quadratic forms can lead to the characterization of quaternion algebras and we focused on their construction and properties.

References 1. D. Fetcu: Elemente de algebr˘a liniar˘a, geometrie analitic˘a s¸i geometrie diferen¸tial˘a, Ia¸si, Casa Editorial˘a Demiurg, pp. 65–71 (2009) 2. Winfried Scharlau: Quadratic and Hermitian Forms, Berlin, Heidelberg, New York, SpringVerlag, pp. 1–8 (1985) 3. David W. Lewis: Quaternion Algebras and the Algebraic Legacy of Hamilton’s Quaternions, Irish Math, Soc. Bulletin 57, pp. 43–46 (2006) 4. John Voight: Quaternion algebras, v.0.0.26, March 27, p. 50 (2021) 5. T.Y. Lam: Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society Providence, Rhode Island, p. 51 (2004)

Multicovariance and Multicorrelation for p-variables Mehmet Niyazi Çankaya

1 Introduction The correlation is observed if there is a jointly distributed random variables, that is, f (x1 , x2 ). The merit of correlation is constructed by using the covariance and variance formulae. The covariance and variance depend on the first and second moments of random variables, respectively. If the existence of moments with integer and r non-integer cases is satisfied by the chosen function g(X 1r1 , X 2r2 , . . . , X pp ) and the joint distribution of the p-random variables X 1 , X 2 , . . . , X p , then the existence of covariance function can be satisfied. Thus, the correlation coefficient for the pvariables can be guaranteed as well [1, 2]. As it is well-known in the general setting, y = x corresponds to an exact and linear relationship between the dependent variable y and the independent variable x. If we use a general expression, that is y = f (x), then the expression f (x) will be responsible to show the non-linear relationship according to the chosen function f . Throughout the paper, the chosen function f is linear type expression [2]. The main of this study is to get a coefficient which is responsible to show what the degree of relationship among p-variables is as. If the value of this coefficient notated by ρ is high, then it means that the degree of relationship among p-variables is very high. Otherwise, there can exist a relationship, however it cannot be evaluated by using the expression which is responsible to show a dependence based on the linear sense. The sketch of the paper is as follows. The first Sect. 2 represents the correlation coefficient formula when it is used to show the relationship between two variables. Section 3 introduces the basic regression model which is based on the linear type. M. N. Çankaya (B) Department of International Trading and Finance, Faculty of Applied Sciences, U¸sak University, Bir Eylül Campus, 64200 Usak, Turkey e-mail: [email protected] URL: https://akbis.usak.edu.tr/Home/Index/mehmet.cankaya © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_25

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Section 3.4 provides the results of simulation study. The algorithm and its corresponding codes compatible with MATLAB 2013a platform can be used by researchers to perform their simulation studies and the real data application. The last section is divided for the conclusion and discussions with further studies.

2 Correlation and Multicorrelation The correlation in the vector calculus is an inner product of the two variables [3]. That is, the following expression can be given to represent the correlation between two variables: > . (1) ρ(X 1 , X 2 ) = √ < X 1 >< X 2 > If the moment expression is used to express the correlation coefficient in a calculation form, the following expression with definition of covariance and variance is a way to propose an analytical expression to get the numerical expression for the linear relationship between two variables. The correlation coefficient is defined by ρ(X 1 , X 2 ) = √ = 

Cov(X 1 , X 2 ) V ar (X 1 )V ar (X 2 ) E[(

X 1 −μ X 1 sX1

)(

X 2 −μ X 2 sX2

)]

X −μ X −μ E[( 1s X X 1 )2 ]E[( 2s X X 2 )2 ] 1 2

:= √

E[Z 1 Z 2 ] E[Z 1 ]E[Z 2 ]

E[(X 1 − μ X 1 )(X 2 − μ X 2 )] σX1 σX2 E(X 1 X 2 ) − E(X 1 )E(X 2 ) = , σX1 σX2 =

(2)

1/2  1/2  1 n j = n j i=1 |x ji − μˆ X ji |2 , j = 1, 2, · · · , p. If we where σ X j = V ar (X j ) reflect the inner product for the p-variables case represented by the following form: ρ(X 1 , X 2 , . . . , X p ) =

> , {< X 1 > · < X 2 > · · · · · < X p >}1/ p

(3)

the correlation coefficient for the used multicovariance which is product of the pvariables and variance based on the p-normed distance instead of using Euclidean norm, i.e., (x − μ)2 , for p-variables is defined by

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Cov(Z 1 , Z 2 , · · · , Z p ) r p {V ar (|Z 1 | )V ar (|Z 2 |r p ) · · · V ar (|Z p |r p )}1/ p p E[ j=1 Z rj ] sign(β1 )  1/ p p rp) V ar (|Z | j j=1

ρ(Z 1 , Z 2 , · · · , Z p ) = sign(β1 ) =

= sign(β1 )

E[Z 1r Z 2r · · · Z rp ] (V ar (|Z 1 |r p )V ar (|Z 2 |r p ) · · · V ar (|Z p |r p ))1/ p

,(4)

n n X j −μ X |x ji − μˆ X ji | < ∞ and μˆ X ji = n1 i=1 x ji where Z j = s X j < ∞, s X j = n1 i=1 j < ∞. sign(β1 ) represents the sign of β1 . Since p-variables are used, norm should be changed as from 2 in Eq. (2) to p in Eq. (4). Then, we have the variance formula based on the p-normed space given by following form: n 1 |z ji | p }1/ p , j = 1, 2, . . . , p. V ar (|Z j |) = { n i=1

(5)

If Eq. (5) is applied to the denominator of ρ(Z 1 , Z 2 , . . . , Z p ), then we have the following expression:

1/ p 1/ p 1/ p

n n n 1 1 1 p p p |z 1i | · |z 2i | · ··· · |z pi | , j = 1, 2, . . . , p. n i=1 n i=1 n i=1 (6) If the number of sample size for each variable cannot be same with number n for sample size, then we have the following expression: 1/ p 1/ p 1/ p

np n1 n 1 2 1 1 |z 1i | p · |z 2i | p · ··· · |z pi | p , j = 1, 2, . . . , p. n 1 i=1 n 2 i=1 n p i=1 (7) Note that Eqs. (6) and (7) can be updated for the case where the empirical expressions of moments in these Eqs. (6) and (7) can be rewritten for the |z rpi | and r ∈ R. The power r is a kind of order of random variable where we can update the position of random variable in which it is shifted or located with the transformed form of the power r as a kind of variable transformation and generally it can be implied that the vector movement in the inner product of > cannot change the linear dependence among p-variables in the p-dimensional vector calculus if the power r is chosen as near to 1 which can keep the property of linear dependence among p-dimensional vector calculus [3–5]. Note that while calculating s X j , the absolute value of distance x ji − μˆ X ji is taken; because, numerical values of the distance must be small values to avoid the big values of numerator and denominator of ρ(Z 1 , Z 2 , . . . , Z p ). Note that r can be chosen to be close to left to 1 in order to decrease the numerical values of huge numbers. In other words, the moment expression can also be updated; thus a new type random variable

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has been derived and the (linear) relationship among the new ones will be examined. Further, the value of r will determine the degree of relationship and the structure of dependence of the p-variables, as is seen from the new expression z r which is a kind of variable transformation. However, one can prefer to choose r to be the left side of 1. Alternatively, the power r can be chosen from a probability (density) function. The reason why we choose an extra situation for the power r is that the degree of dependence can be managed according to the chosen values of power r . In other words, the power r makes a new variable regarded as the variable transformation. Alternatively, the power r can play a same role with error term including from the regression equation. However, the main role is based on variable transformation. Note that two expressions which are numerator and denominator in Eq. (4) can take big values if the number of p-variables are increased. When these expressions have huge numbers such as e + 202 in the engineering representation, etc., then the denominator of Eq. (4) can go to infinity. When it occurs, then the correlation coefficient ρ(Z 1 , Z 2 , . . . , Z p ) can give value which is zero. To overcome this problem, x j −μ X rescaling or resizing, that is, z j = s X j , on the numerical values of observations j should be applied to get small new values of observations. Thus, the linear dependence among p-variables is kept.

2.1 Conditions for the Existence of Moments Since the correlation coefficient depends on the definition of moments with the integer or non-integer orders, the following conditions are necessary to guarantee the existence of ρ(Z 1 , Z 2 , . . . , Z p ) theoretically. – E[Z 1r Z 2r · · · Z rp ] < ∞. – V ar (|Z 1 |r p ) < ∞, V ar (|Z 2 |r p ) < ∞, · · · , V ar (|Z p |r p ) < ∞. The extra conditions are necessary to have finite values for the numerator and denominator of ρ(Z 1 , Z 2 , · · · , Z p ). – (V ar (|Z 1 |r p )V ar (|Z 2 |r p ) · · · V ar (|Z p |r p ))1/ p < ∞. Note that if the theoretical existence of ρ(Z 1 , Z 2 , . . . , Z p ) is satisfied, then the estiˆ 1 , Z 2 , . . . , Z p ), can be guaranteed. mated form of ρ(Z 1 , Z 2 , . . . , Z p ), i.e., ρ(Z Even if there does not exist the expectation of r th moment of random variables X 1 and X 2 in the definition of correlation coefficient for two variables x1 and x2 , it is reasonable to propose the expectation of r th moment of random variables X 1 , X 2 , . . . , X p to produce a new formula for the joint form of the variables x1 , x2 , . . . , x p . That is, it is a variable transformation from X j to X rj , j = 1, 2, . . . , p for joint distribution of p-variables.

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3 Regression Equation The dependence can be created by using the regression equation between two variables. A regression equation in the linear sense is defined by the following form: x j+1 = β0 + β1 x j + ε, j = 1, 2, . . . , p,

(8)

where β0 and β1 are regression coefficients. The subscript j is used to represent the p-variables. Note that each variable x j+1 depends on the x j which is the sequence of linear dependence produced by the Eq. (8). The left-hand side of Eq. (8), i.e., xi+1 , is dependent variable. The right-hand side of Eq. (8), i.e., xi , is independent variable. Due to recursive replication, each part can be regarded as the dependent variable [5, 6]. When we have a data set generated by using the Eq. (8), the representation of Eq. (8) is given by the following form: x j+1,i = β0 + β1 x j,i + εi , j = 1, 2, . . . , p, i = 1, 2, · · · , n,

(9)

where the intercept β0 and the tangent β1 are parameters of regression equation based on linear sense. p and n are numbers of variables and sample size, respectively [6].

3.1 Generating Linearly Dependent Variables The following algorithm is responsible to generate the linearly dependent variables given by Eq. (9): Line Line Line Line

1: 2: 3: 4:

for j = 1:number of variables e=normrnd(0,1,number of sample size,1); x(:,j+1) = b0 + b1 * x(:,j) + e; end

Note that the variable x in the algorithm generates the next variable x which can be regarded as the dependent variable. If the number of variables is increased, then the quantitative values of variable x turn out to be big. To overcome this problem, alternative situation given below can be proposed for the line 3 in the algorithm given above. Line 3: mx(j) = mean(x(:,j)); Line 4: x(:,j+1) = b0 + b1 * (x(:,j) - mx(j)) + e;

The expression (x(:,j) - mx(j)) provides to have small values for the next x values in the ( p + 1)-order case. Additionally, an alternative version of the above algorithm can be proposed as the following form:

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Line 3: mx(j) = mean(x(:,j)); Line 4: stdx(j)=sum(abs(x(:,j)-mx(j)))/number of sample size; Line 5: x(:,j+1) = b0 + b1 * (x(:,j) - mx(j)) / sx(j) + e;

Note that this algorithm is proposed for the theoretical sense. According to the structure of the data set, it is possible to consult this algorithm whether or not the relationship among p-variables exists in reality. This is a way to have a standardized form, , of the natural data set. The different estimators for the location μ and i.e., z = x−μ σ scale σ parameters can be used to have the standardized form.

3.2 The Transformation on the Variables: z and z r The variable transformation can be necessary to have small values of variables. Since the linear dependence structure is applied to have next variable at index j, the forthcoming values of the next variable x j start to be evolved hugely according to the chosen values of β1 especially. After applying the variable transformation z r , the codes are given by the following lines: Line Line Line Line Line Line

1: 2: 3: 4: 5: 6:

for j = 1:number of variables + 1 mx(j)=mean(x(:,j)); stdx(j)=sum(abs(x(:,j)-mx(j))) / number of sample size; z(:,j)=abs(x(:,j)-mx(j))/stdx(j); rr(j)=normrnd(0.99,0.01,1,1); z(:,j)=z(:,j).ˆrr(j);end

3.3 The Codes for Computational Evaluation of Correlation Coefficient The following additional codes are provided to observe how the computational process is conducted in the section of simulation study. The following lines are used to compute the numerator of Eq. (4). Line Line Line Line

1: 2: 3: 4:

for k=1:number of sample size diffproduct(k)=prod((z(k,:)))/number of sample size; end exy=sum(diffproduct);

The following lines are used to compute the denominator of Eq. (4) and the correlation coefficient for p-variables. Line Line Line Line Line

1: 2: 3: 4: 5:

for i = 1:number of variables + 1 rotpvarx(i)=(sum(abs(z(:,i)).ˆp)/number of sample size)ˆ(1/p); end prodrotpvarx=prod(rotpvarx); corvalues=sign(b1)*exy/prodrotpvarx;

Multicovariance and Multicorrelation for p-variables

279

3.4 Simulation Study The powers of moment, i.e., r1 , r2 , . . . , r p , make the different degree of momentum. If these powers are changed, the magnitude of quantitative values of random variables are changed, as expected. In other words, it can correspond to the error term in the regression equation if it is generally commented. We prefer this situation and it can correspond the error term in the regression in the reassignment values, however it is very useful to have the revalued and smaller values when they are compared with the first order moment which is z r =1 . In the simulation, we have different tryings. One of tryings we did is that the value of β1 are changed. It should be noted that even if the value of β1 has been changed, the linear type dependence can be kept, but the numerical values for the numerator and denominator of ρ(Z ˆ 1 , Z 2 , . . . , Z p ) can go the big values. If it is so, the value of ˆ 1, Z2, . . . , Z p ) ρ(Z ˆ 1 , Z 2 , . . . , Z p ) can go to zero, because the denominator of ρ(Z goes to infinity. Another situation we have is that if the numerator and denominator ˆ 1 , Z 2 , . . . , Z p ), for of ρ(Z ˆ 1 , Z 2 , . . . , Z p ) are infinity, the correlation coefficient, ρ(Z p-variables becomes NaN, i.e., Not a Number. Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 represent the values of the computed values of correlation coefficient repeated 105 times. Since these values are computed for p-variables at recursive replication, the computed correlation values are based on the multicorrelation in Eq. (4). If Fig. 1a, b are compared, it is observed that since the value of β1 is increased from 3 in (a) case of Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 1.1

Correlation values for the p−variables

Correlation values for the p−variables

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

1

2

4

3

5

6

7

8

9

1 0.9 0.8 0.7 0.6 0.5 0.4 0

10 4 x 10

1

2

3

4

5

6

7

8

9

10 4 x 10

7

8

9

10 4 x 10

Replication number

Replication number

1.2

Correlation values for the p−variables

Correlation values for the p−variables

Fig. 1 Correlation values for the p = 300 and n = 30 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0

1

2

3

4

5

6

7

8

9

10 4 x 10

1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0

Replication number

Fig. 2 Correlation values for the p = 300 and n = 50

1

2

3

4

5

6

Replication number

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M. N. Çankaya 1.1

1.1 1 Correlation values for the p−variables

Correlation values for the p−variables

1 0.9 0.8 0.7 0.6 0.5 0.4

0.9 0.8 0.7 0.6

0.3 0.5 0.2 0.1 0

4

3

2

1

6

5

Replication number

7

8

9

0.4 0

10 4 x 10

5

4

3

2

1

Replication number

10 4 x 10

9

8

7

6

Fig. 3 Correlation values for the p = 300 and n = 60 1.05

1.1

1 Correlation values for the p−variables

Correlation values for the p−variables

1 0.9 0.8 0.7 0.6 0.5

0.95 0.9 0.85 0.8 0.75 0.7 0.65

0.4 0

0.6

1

2

3

4

5

Replication number

6

7

8

9

0.55 0

10 4 x 10

2

1

3

4

5

Replication number

6

7

8

9

10 4 x 10

7

8

9

10 4 x 10

8

9

10 4 x 10

Fig. 4 Correlation values for the p = 300 and n = 90 1.1

1.1

1 Correlation values for the p−variables

Correlation values for the p−variables

1 0.9 0.8 0.7 0.6 0.5

0.8 0.7 0.6 0.5

0.4 0

0.9

1

2

3

4

5

Replication number

6

7

8

9

10 4 x 10

0.4 0

1

2

3

4

5

Replication number

6

Fig. 5 Correlation values for the p = 300 and n = 100 1.05

1.1

1 Correlation values for the p−variables

Correlation values for the p−variables

1 0.9 0.8 0.7 0.6 0.5

0.9 0.85 0.8 0.75 0.7 0.65

0.4 0

0.95

0.6

1

2

3

4

5

Replication number

6

7

8

9

10 4 x 10

0.55 0

1

2

3

4

5

Replication number

6

7

Fig. 6 Correlation values for the p = 300 and n = 150

and 12 to 5 in (b) case of Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, the degree of linear dependence can be increased. As it is expected, the quantitative magnitude of the multicorrelation coefficient can have high linear dependence.

Multicovariance and Multicorrelation for p-variables

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1.1

1.1 1 Correlation values for the p−variables

Correlation values for the p−variables

1 0.9 0.8 0.7 0.6 0.5 0.4

0.9 0.8 0.7 0.6

0.3 0.5 0.2 0.1 0

1

2

3

4

5

Replication number

6

7

8

9

0.4 0

10 4 x 10

1

2

3

4

1

2

3

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1

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10 4 x 10

5

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7

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10 4 x 10

5

6

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8

9

10 4 x 10

5

6

7

8

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10 4 x 10

Replication number

Fig. 7 Correlation values for the p = 350 and n = 30 1.1

1.2 1.1 Correlation values for the p−variables

Correlation values for the p−variables

1 1 0.9 0.8 0.7 0.6 0.5

0.9 0.8 0.7 0.6

0.4 0.5 0.3 0.2 0

1

2

3

4

5

Replication number

6

7

8

9

0.4 0

10 4 x 10

Replication number

Fig. 8 Correlation values for the p = 350 and n = 50 1.1

1.2 1.1 Correlation values for the p−variables

Correlation values for the p−variables

1 1 0.9 0.8 0.7 0.6 0.5

0.9 0.8 0.7 0.6

0.4 0.5 0.3 0.2 0

1

2

3

4

5

Replication number

6

7

8

9

0.4 0

10 4 x 10

Replication number

Fig. 9 Correlation values for the p = 350 and n = 60 1.2

1.2 1.1 1

Correlation values for the p−variables

Correlation values for the p−variables

1.1

0.9 0.8 0.7 0.6 0.5

1 0.9 0.8 0.7

0.4 0.6 0.3 0.2 0

1

2

3

4

5

Replication number

6

7

8

9

10 4 x 10

0.5 0

Fig. 10 Correlation values for the p = 350 and n = 90

Replication number

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M. N. Çankaya 1.05

1.1

1 Correlation values for the p−variables

Correlation values for the p−variables

1 0.9 0.8 0.7 0.6 0.5

0.95 0.9 0.85 0.8 0.75 0.7 0.65

0.4 0

0.6

1

2

3

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Replication number

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Replication number

Fig. 11 Correlation values for the p = 350 and n = 100 1.05

1.1

1 Correlation values for the p−variables

Correlation values for the p−variables

1 0.9 0.8 0.7 0.6 0.5 0.4 0

0.95 0.9 0.85 0.8 0.75 0.7 0.65

1

2

3

4

5

Replication number

6

7

8

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10 4 x 10

0

Replication number

Fig. 12 Correlation values for the p = 350 and n = 150 Table 1 The statistics for the replicated correlation coefficients at 105 times when p = 300 β1 = 3 n 30 50 60 90 100 150 M SD

0.9311 0.1136

0.9316 0.1069

0.9329 0.1034

0.9329 0.0998

0.9333 0.0988

0.9339 0.0953

β1 = 5 n M SD

30

50

60

90

100

150

0.9762 0.0545

0.9766 0.0508

0.9769 0.0495

0.9774 0.0471

0.9775 0.0466

0.9774 0.0450

n: Sample size, M: Mean, SD: Standard deviation

Tables 1 and 2 show the summarize statistics for the replicated correlation coefficients at 105 times in the Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 when the numbers of p-variables are 300 and 350. Tables 1 and 2 also provide the value of parameter β1 in Equation (9) to produce the correlated artificial data set. Thus, the mean and standard deviation of the computed correlation values for p-variables are observed to see the role of values β1 which are 3 and 5 in the analytical expression of regression equation in Eq. (9). As it is expected, when the sample size n is increased, the standard deviation decreases. In our case, β0 has been chosen as zero, because the value of next variable, i.e., the variable x j , should not go to huge numbers. Note that the chosen value for

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Table 2 The statistics for the replicated correlation coefficients at 105 times when p = 350 β1 = 3 n 30 50 60 90 100 150 M SD

0.9304 0.1141

0.9327 0.1062

0.9325 0.1046

0.9333 0.1000

0.9328 0.0996

0.9336 0.0955

β1 = 5 n M SD

30

50

60

90

100

150

0.9765 0.0541

0.9769 0.0506

0.9770 0.0495

0.9773 0.0474

0.9774 0.0470

0.9776 0.0450

parameter β1 does already determine to have the huge numbers. In the meanwhile, it should be noted that a rescaling on the overall of variables from x1 , x2 , . . . , x p can be applied. For example, x = x/(9 · 1011 ) can be chosen to go to small values of the variables in order to avoid the problem about the infinity for the numerator and denominator of ρ(Z ˆ 1 , Z 2 , . . . , Z p ). It is surprisingly noted that even if a rescaling on all of p-variables are applied, the structure of linear dependence can be kept and the simulation results show that the correlation coefficient ρ(Z ˆ 1, Z2, . . . , Z p ) is performable to give values which are near from the left hand right of 1, which is mainly showing the existence of linear dependence among the p-variables; because, since we conduct a simulation study which includes the linear dependence structure in Eq. (9) and the formulation in Eq. (4) for the correlation coefficient written as codes is performed, we can observe that the results of correlation coefficient will become to be near from the left hand right of 1. Further, the number of p-variables can be increased by the codes given in the paper.

4 Conclusions and Discussions The correlation coefficient for the p-variables has been proposed to evaluate the relationship among p-variables at the same time. In the case of huge values for the artificial data sets, the shifted form, i.e., x − μ, ˆ and the standardized form, i.e., x j −μˆ X z j = s X j , have been proposed. Additionally, when we have huge numbers, the j powers of standardized forms, i.e., z rj , can also be used by researchers even if it becomes a new variable. The covariance and correlation formulae are based on the inner product of two vectors represented by X 1 and X 2 . If we change the rule of product and the rules in vector calculus are applied, then we have different type of covariance and correlation formulae which will be proposed.

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The rescaling can correspond to the squeezing on the manifolds which are dimensions x1 , x2 , . . . , x p in the vector calculus. Even if a rescaling such as y = x/a, where a is a huge number, is applied, the linear dependence among variables represented by x can be followed or kept by new variable y which is obtained by applying the rescaling as a variable transformation or squeezing on the manifolds [7]. A package in open access R Software will be prepared to analyse and perform a controlling among the similarity of slices of images, the detection of health process, climate change throughout time, etc. If there exists an accommodation or linear movement/dependence among the slices or process, then it can be detected and controlled by using the correlation coefficient for p-variables as outputs throughout the time. If ρˆ is close to left to 1, the components of system can accommodate each other well and so it shows that the system is consistent.

References 1. Prakasa Rao, B. L. S.: Hoeffding identity, multivariance and multicorrelation. A Journal of theoretical and applied statistics, 32(1), 13–29 (1998). https://doi.org/10.1080/ 02331889808802650 2. Gut, A.: Probability: a graduate course. (Vol. 200, No. 5). New York: Springer (2005) 3. Matthews, P. C.: Vector calculus. Springer Science & Business Media (2000) 4. Díaz, W., Cuadras, C. M.: On a multivariate generalization of the covariance. Communications in Statistics-Theory and Methods, 46(9), 4660–4669 (2017). https://doi.org/10.1080/ 03610926.2015.1056368 5. Rao, B. P., Dewan, I.: Associated sequences and related inference problems. Handbook of statistics, 19, 693–731 (2001). https://doi.org/10.1016/S0169-7161(01)19022-X 6. Chatterjee, S., Hadi, A. S.: Regression analysis by example. John Wiley & Sons (2006) 7. Cruz-Orive, L. M.: On the precision of systematic sampling: a review of Matheron’s transitive methods. Journal of Microscopy, 153(3), 315–333 (1989). https://doi.org/10.1111/j.13652818.1989.tb01480.x

An Individual Work Plan to Influence Educational Learning Paths in Engineering Undergraduate Students M. E. Bigotte de Almeida, J. R. Branco, L. Margalho, M. J. Cáceres, and A. Queiruga-Dios

1 Introduction Differential and Integral Calculus curricular units (CDI-UC) are responsible for the theoretical basis necessary for professionals in areas of Engineering, and for this reason, they are presented in most of the degrees, offered in Higher Education Institutions. However, it has been observed that this basic science is the cause of high failure rates in these degrees, resulting in different problems, such as absenteeism and, consequently, dropout, both from classes and examinations. The failure and dropout rates in CDI-UC have highlighted the need to question which methodologies and teaching procedures are being applied, which learning environments are being developed, and which assessment practices are being used that best allow students to be co-responsible in their educational processes, which may affect their academic success, and which lead to significant learning [1–3].

M. E. B. de Almeida (B) · J. R. Branco · L. Margalho Polytechnic Institute of Coimbra, Coimbra Institute of Engineering, 3030-199 Coimbra, Portugal e-mail: [email protected] J. R. Branco e-mail: [email protected] L. Margalho e-mail: [email protected] M. J. Cáceres Department of Didactics of Mathematics and Experimental Sciences, University of Salamanca, 37008 Salamanca, Spain e-mail: [email protected] A. Queiruga-Dios Department of Applied Mathematics, University of Salamanca, 37700 Salamanca, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications, Springer Proceedings in Mathematics & Statistics 414, https://doi.org/10.1007/978-3-031-21700-5_26

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These issues related to the failure of mathematics in engineering education and the negative impact of these difficulties on the success of the CDI-UC are a problem to which we have devoted our attention and investigation. Most students entering engineering degrees present insufficient mathematics knowledge and heterogeneity of mathematical training. Those limitations and asymmetries hinder their integration into higher education, which motivates the definition of alternative paths, for these students, in their learning process that could help them to succeed [4–9]. Coimbra Engineering Institute (ISEC) is an organic unit of the Polytechnic Institute of Coimbra that offers degrees in engineering, such as Biomedical, Bioengineering, Civil, Electrical (normal and post-work regime), Electromechanical, Engineering and Industrial Management, Informatics (normal, post-work regime and European course) and Mechanical. A study carried out in the CDI-UC, integrated into the curricular plan of engineering degrees, taught in the 1st year, evidenced a low average pass rate and a higher dropout rate [10]. The data presented come from the analysis over a period of 7 academic years, from 2011/2012 to 2017/2018, and evidence that Biomedical presents the best results with a mean of 73.10% (between 62.5% and 88.2%), with Electrical showing lower results than the other undergraduates (between 27.9% and 52.1%), presenting an average of 42.24%. It is also verified that the overall average pass rate in the first semester is 58.40%, with a standard deviation of 14.34% and an average deviation of 12.4%. The successful integration of students in CDI-UC subjects is verified by the need to reconcile the mathematical basis of the knowledge acquired during secondary education with the knowledge considered essential for attending the 1st year of engineering degree [11, 12]. These results highlight the need for early detection of students’ difficulties, through the application of a diagnostic test that allows timely intervention to avoid demotivation and consequent dropout [13–15].

2 The Diagnostic Test Given the real situation, it became evident for maths teachers from ISEC to build a diagnostic tool, applied at the entry of higher education, which helps to characterize the student’s level of mathematics syllabus and identify flaws in the set of essential knowledge in mathematics, considered essential for the full integration of students in CDI-UC. After several editions, that included revision and changes, the final version of the Diagnostic Test (DT) is from 2013/2014 and it is a result of a cooperative work with the Dublin Institute of Technology, so that comparative studies can be conducted in both countries [13]. According to Core Zero outcomes from SEFI guidelines, Mathematics for the European Engineer—A Curriculum for the TwentyFirst Century [16] and Basic and Secondary Education programs in Portugal, the 20 questions of the final version of DT, were regrouped considering the different topics from Algebra, Analysis and Calculus and Geometry and Trigonometry.

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Table 1 DT results, by year, from 2013/2014 to 2021/2022 academic years Academic year Dimension Mean value St. deviation 433 399 494 397 225 93 209 238 325 2813

9.50 9.91 10.35 10.57 10.21 9.55 12.23 12.31 11.18 10.55

4.17 3.82 4.47 3.79 4.25 4.42 4.46 4.33 4.57 4.31

15 10 5 0

Diagnostic Test Results

20

2013/2014 2014/2015 2015/2016 2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 All

Biological

Biomedical

Civil

Electrical

Electromec.

Indust.

Informatics

Mechanics

Degree

Fig. 1 Boxplot from DT results from different degrees

DT was carried out in the first week of CDI-UC classes of the different ISEC engineering degrees, except in the first semester of academic year 2016/2017, when it was carried out at the time of student’s registration at ISEC. Due to the pandemic situation experienced, in the academic year 2020/2021 DT was carried out remotely. In Table 1 we present the number of DT answers, the mean value and standard deviation of DT, from 2013/2014 to the present. The last line presents also the mean and standard deviation of all the DT results. According to the Portuguese system, results are up to 20 values. A mean value of 10.55 values shows that mathematical knowledge is far below what we would like to get. In Fig. 1 we compare the distribution of the DT results from the different degrees. We observe that Biomedical is the degree with better results so it will be considered as a benchmark. 84.3% of the students from Biomedical degree come from science and technology courses in secondary education, meaning that the majority of them contain in their syllabus the elementary knowledge considered essential for a good integration in CDI-UC. Informatics degree has the larger number of students in our sample (1812) and also corresponds to the degree that has the largest number of vacancies for access to higher education in ISEC. Taking these assumptions into account, it was decided, in what follows, to focus the study on these two degrees in order to avoid bias.

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Fig. 2 Informatics and Biomedical DT results vs. global DT results (mean values), from 2013/2014 to 2021/2022 academic years

In Fig. 2 we compare the results from Informatics and Biomedical DT with the global DT results (mean values). Since only 52.8% of Informatics students provide from science and technology courses, in secondary education (the remaining ones provide from technological and professional courses), the diversity of background training in mathematics on those courses explains the gaps and the consequent low results on DT. Concerning to Biomedical, we remark that since 2018/2019 the majority of the students entering this degree came from technological and professional courses. We also remark that, concerning to Informatics, 2018/2019 results were obtained on different conditions, because DT was not done on 1st week of classes and only students that came to CeAMatE, to overcome their mathematical difficulties, respond to DT. It is thus evident that a good mathematical support is essential for a good integration in the CDI-UC, which will imply a higher approval rate. Also, some strategies in the classroom or outside can be implemented in parallel with the development of the course in overcoming previously identified difficulties [16, 17].

3 CeAMatE DT results provide information about the mathematical contents that should be worked with the student, making possible to build an individualized plan to assess the evolution of mathematical knowledge. This plan will describe the evolution of the student’s learning, through its monitoring in the Support Center for Mathematics in Engineering (CeAMatE), located in ISEC. This Centre is a space dedicated to accompanying students to overcome difficulties in basic and elementary knowledge essential for full integration into engineering courses. The aim of this Centre is to pro-

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vide support and learning resources that allow students to overcome their difficulties, through independent study [18, 19]. MathCentre (https://www.mathcentre.ac.uk/) is considered by the scientific community to be a reference for all Higher Education Institutions that intend to develop support strategies for students. Since all materials are available it was decided that the pedagogical resources to be applied in CeAMatE would be obtained from the MathCentre repository. Also, a set of reference texts in Portuguese are produced by CeAMatE teachers. In this way, CeAMatE allows the construction of an academic course that promotes the development of students’ independent study skills, with the joint responsibility of building their own educational paths. It also facilitates the construction of learning and acquisition of new knowledge through the availability of various activities and resources aimed at overcoming students’ difficulties. This project started in the academic year 2015/2016 and is directed to all the ISEC students that don’t have the basic mathematical knowledge, necessary for a good attendance in the mathematics curricular units, especially those of CDI-UC and students that want specialized help to overcome their lack of fundamentals on Mathematics. The basic instrument in the monitoring methodology applied in CeAMatE is the DT. The information of the DT referring to basic knowledge that the student has and needs to overcome, serves as a baseline to conclude the evolution of the student about his specific learning. As DT evaluates basic knowledge on mathematics, we considered that students with results below 12 values on DT (60%) need complementary support to proceed to the standardization of knowledge in higher education. So, all students who obtained a result under 60% were advised to enroll CeAMatE, to obtain complementary mathematical knowledge. Furthermore, DT provides specific information about the mathematical content that should be worked on with the student during the follow-up period at CeAMatE and which will involve a greater level of effort. For this purpose, an Individual Work Plan is built [20, 21]. This document includes a training plan, consisting of a selected set of study sheets, texts, and exercises. This Individual Work Plan is reformulated as the student progresses, with greater or lesser success, while attending support at the defined study time. The evaluation of this work is carried out, in each student’s presence at CeAMatE, through self-proposed tasks, promoting continuous monitoring through the definition of solid and structured training. Additionally, periodic realizations of adapted versions of the DT imply assessment and reformulation of the Individual Work Plan. This process runs until the student reaches the minimum required (90% = 18 values) to be considered able to integrate the syllabus contents of the mathematics curricular units. A maths teacher accompanies the students and helps to clarify doubts and to guide the moments of autonomous study, in order to make the learning process more profitable. On each visit of the student, data is collected on the date and time of stay in the physical space, as well as information on the topic worked on, difficulties encountered, and support needed to overcome them.

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In this work, we present qualitative results and reflections, on Biomedical and Informatics students’ performance, from the first 7 working years of CeAMatE.

4 Results The data collection of the DT was made at the beginning of each semester, since CeAMatE’s beginning, on October 14th, 2015. We have data referring to 7 years, from 2015/2016 to the present, concerning 124 Biomedical Engineering students and 1425 Informatics Engineering students. All students who do not match at least 12 (60%) of the DT questions are advised to enroll in CeAMatE. The definition of 60% level of correct answers is not equivalent to the student possessing the basic and complementary knowledge, essential to the full integration on CDI-UC, but only to define a limit of alert for standardization of knowledge at the entrance to higher education. From the data analysis, we conclude that Biomedical students have better results (Fig. 3), which corroborates what has been said before. In our opinion, students that don’t achieve more than 60% on DT need, at least, 20 h of continuous work at CeAMatE (2 h per week for the 10 weeks from the semester) to overcome their mathematical difficulties. Figure 4 shows that, on average, students don’t achieve that needed time. Again, Biomedical students do not participate in CeAMatE as they mostly come from secondary school courses that have mathematics with the minimum requirements. We remark that students from Biomedical only attended the Centre at 2021/2022 academic year, and those students accessed ISEC through a professional path. As mentioned in [10], CDI-UC approval rates on Biomedical Engineering are the highest among ISEC’s degrees, with a mean value in the order of 73.10%. This corroborates the results obtained in the DT and the low demand of these students for complementary support in mathematics. Figure 5 shows the percentage of Informatics students, depending on the results of the DT and the CDI-UC evaluations. It appears that these rates are higher for students who attend the Center. This result may indicate greater confidence in students who

Fig. 3 Informatics and Biomedical DT results, from 2015/2016 to 2021/2022 academic years

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Fig. 4 Attendance to CeAMatE from Biomedical vs Informatics students, since 2015/2016 to 2021/2022 academic years

propose to do extra work, to present themselves for evaluation, feeling more secure in their basic knowledge of mathematics necessary for CDI-UC. Observing the data referring to students who had less than 12 on DT, those who were advised to attend CeAMatE since they did not have the necessary requirements to integrate CDI-UC, it appears that the rate of students who access the evaluation (67.76%) and who approve (47.62%) is greater for students who comply with the recommended guidelines for a better recovery of learning in mathematics (more than 20 h of attendance at CeAMatE). These results are presented in Fig. 6.

5 Conclusions Studies carried out at ISEC show that students had low and irregular attendance rates at CeAMatE. However, our results show that the more hours the student attended the Center, the greater the probability that he will attend the CDI assessment and pass. These results seem to reveal students’ awareness of their limitations, but also reveal their inertia to overcome these difficulties.

Fig. 5 CDI-UC results from Informatics students, depending on the results

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Fig. 6 Informatics students (%) on CDI-UC evaluations, depending on the number of hours at CeAMatE

The gaps in basic knowledge of mathematics essential to students’ full integration into these courses and the student’s behaviours and attitudes are the main concerns expressed by the studies carried out in ISEC. In our opinion, this is not a problem from ISEC or even Portuguese students, but a social behaviour. For most of them, the focus is to approve and not the learning process, so they believe that they can achieve it even if they don’t have enough skills.

References 1. Barbosa, M.A.: O Insucesso no Ensino e Aprendizagem na Disciplina de Cálculo Diferencial e Integral. Mestrado em Educação, Pontifícia Universidade Católica do Paraná, Curitiba, Brazil (2004). 2. Rezende, W.M.: O Ensino de Cálculo: Dificuldades de Natureza Epistemológica. Tese Doutorado em Educação, Faculdade de Educação, Universidade de São Paulo, USP, São Paulo, Brazil (2003). 3. Rosa, O.S.: Aspectos Motivacionais do Cálculo Diferencial e Integral. Programa de PósGraduação Stricto Sensu, Mestrado Profissional em Educação Matemática, Universidade de Severino Sombra, Vassouras, Brazil (2011). 4. Koch, D., Herrin, G.D.: Intervention strategy for improving success rates in calculus. Proceedings of the 2006 ASEE Annual Conference and Exposition, Chicago, Illinois - USA, 775 (2006). 5. Gill, O., O’Donoghue, J.: The mathematical deficiencies of students entering third level: An item by item analysis of student diagnostic tests. Proceedings of the Second National Conference on Research in Mathematics Education, St. Patrick’s College, Dublin - Ireland, pp 228–239 (2007). 6. Hieb, J., Lyle, K., Ralston, P., Chariker, J.: Predicting performance in a first engineering calculus course: implications for interventions. International Journal of Mathematical Education in Science and Technology, 46(1): 40–55 (2015). 7. Nite, S.B., Capraro, R.M., Capraro, M.M., Allen, G.D., Pilant, M., Morgan, J.: A bridge to engineering: a personalized precalculus (bridge) program. Proceedings of the 2015 IEEE Fron-

An Individual Work Plan to Influence Educational Learning Paths …

8.

9. 10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

293

tiers in Education Conference (FIE), El Paso - USA, pp 1–6. (2015). https://doi.org/10.1109/ FIE.2015.7344358 Cardella, M.: Which mathematics should we teach engineering students? An empirically grounded case for a broad notion of mathematical thinking. Teaching Mathematics and its Applications, 27(3):150–159 (2008) Tackling the Mathematics Problem. London Mathematical Society, Institute of Mathematics and its Applications, Royal Statistical Society (1995). Bigotte de Almeida M.E., Queiruga-Dios, A., Cáceres, M.J.: Differential and Integral Calculus in First-Year Engineering Students:A Diagnosis to Understand the Failure. Mathematics, 9(1):61 (2021). Fidalgo, C., Bigotte de Almeida, M.E., Rasteiro, D.M.L.: Understanding the difficulties in mathematics of engineering students in the transition from second to third level education. Proceeedings of the 4th ATEE Winter Conference, Coimbra - Portugal, pp 477–487 (2012). Bigotte de Almeida, M.E., Coelho, A.: Mathematics in engineering: the perspectives of students. In: K. Kumpulainen & A. Tom (eds.), Proceedings of the 22nd Annual Conference of European Teacher Education Network (ETEN), Helsinki - Finland, pp 21–26 (2012). Carr, M., Fidalgo, C., Bigotte de Almeida, M.E., Branco, J., Santos, V., Murphy, E., Ni Fhloinn, E.: Mathematics diagnostic test in engineering: An international comparison between Ireland and Portugal. European Journal of Engineering Education, 40(5): 546–556 (2014). https://doi. org/10.1080/03043797.2014.967182 Bigotte de Almeida, M.E., Branco, J.R., Margalho, L., Queiruga-Dios, A., Cáceres, M.J.: Understanding the Level of Mathematics Knowledge of Students Who Joined ISEC. In: Yilmaz, F., Queiruga-Dios, A., Santos Sánchez, M.J., Rasteiro, D., Gayoso Martínez, V., Martín Vaquero, J. (eds) Mathematical Methods for Engineering Applications. ICMASE 2021. Springer. Proceedings in Mathematics & Statistics, 384. Springer, Cham (2022). https://doi. org/10.1007/978-3-030-96401-6_23 Bigotte de Almeida, M.E., Santos, V., Campos, H., Costa, C.: Comparative studies between four higher intitutions: undergraduate students at informatic engineering. Proceedings of the 13th annual International Conference of Education, Research and Innovation (ICERI 2020), online conference, Seville - Spain, pp 7214–7219 (2020). https://doi.org/10.21125/iceri.2020. 1543 Mustoe, L., Lawson, D. (Eds.) (2002). Mathematics for the European Engineer. A Curriculum for the Twenty-First Century. A Report by the SEFI Mathematics Working Group. Brussels: SEFI (2002). Available online: http://www.kmf.fgg.uni-lj.si/Matematika/Core Bigotte de Almeida, M.E., Santos, V.: Centro de apoio à matemática nos cursos de engenharia, Proceedings of the XV Congreso Internacional Gallego-Português de Psicopedagogía / II Congreso de la Asociación Científica Internacional de Psicopedagogía, Coruña - Spain, pp 224-233 (2019). Bigotte, M.E., Branco, J.R.: A program to promote mathematical knowledge for students’ integration in Engineering degrees: CeAMatE. Proceedings of the 3rd International Conference of the Portuguese Society for Engineering Education (CISPEE 2018), Aveiro - Portugal, pp 35–40 (2018). Bigotte de Almeida, M.E., Santos, V.: Strategies for supporting mathematics in engineering, Proceedings of the 12th annual International Conference of Education, Research and Innovation, Seville - Spain, pp 11181–11185 (2019). https://doi.org/10.21125/iceri.2019.2764 Bigotte de Almeida, M.E., Gomes, A., Branco, J.R., Pessoa, T.: The influence of educational learning paths in academic success of mathematics in engineering undergraduate. Proceedings of the 2016 IEEE Frontiers in Education Conference Proceedings (FIE), Eire - USA, 1, pp 1–6 (2016). https://doi.org/10.1109/FIE.2016.7757453 Bigotte de Almeida, M.E., Branco, J.R., Fidalgo, C., Margalho, L.: Mathematics academic success in engineering undergraduate: an individual work plan to influence educational learning paths. Proceedings of the 14th International Conference of Education, Research and Innovation (ICERI 2021), Seville - Spain, pp 8950–8956 (2021).