Mathematical Modeling for Computer Applications
Description
Mathematical Modeling for Computer Applications
The mathematical sciences are part of nearly all aspects of everyday life. The discipline has underpinned such beneficial modern capabilities as internet searches, medical imaging, computer animation, numerical weather predictions, and all types of digital communications. This outstanding new volume in the series, "Mathematics and Computer Science," examines the current state of mathematical science and explores the changes needed for the discipline to be in a strong position and able to maximize its contributions.
Besides covering important practical applications for the areas where mathematics and computer science intersect, the editors of this new volume recommend that training for future generations of mathematical scientists should be re-assessed in light of the increasingly cross-disciplinary nature of the mathematical sciences. In addition, because of the valuable interplay between ideas and people from all parts of the mathematical sciences, this group of curated papers emphasizes that universities and governments need to continue to invest in the full spectrum of the mathematical sciences in order for the whole enterprise to continue to flourish long-term. Emphasis on important developments in applied mathematics and modeling, analysis and its applications, applied algebra and its applications, geometry and its applications, algebraic statistics and its applications as well as algebraic topology and its applications are covered. Whether for the veteran engineer, new hire, or student, this is a must-have volume for any library.
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Persons
Biswadip Basu Mallik is a senior assistant professor of mathematics in the Department of Basic Sciences and Humanities at the Institute of Engineering and Management, India. He has been involved in teaching and research for more than 21 years and has published several research papers in various scientific journals along with book chapters with various publishers. He has authored five and has five patents to his credit. He is a managing editor of the Journal of Mathematical Sciences & Computational Mathematics and is an editorial board member and reviewer for several scientific journals.
M. Niranjanamurthy, PhD, is an assistant professor in the Department of Computer Applications, M S Ramaiah Institute of Technology, Bangalore, Karnataka. He earned his PhD in computer science at JJTU, Rajasthan, India. He has over 11 years of teaching experience and two years of industry experience as a software engineer. He has published several books, and he is working on numerous books for Scrivener Publishing. He has published over 60 papers for scholarly journals and conferences, and he is working as a reviewer in 22 scientific journals. He also has numerous awards to his credit.
Sharmistha Ghosh, PhD, is a professor at the Institute of Engineering and Management, India. She received her doctorate in mathematics from the Indian Institute of Technology, India. Her major field of study includes fuzzy and vague databases as well as computational fluid dynamics, and she has published several papers in scientific journals. She is also the editor of several scientific journals and works as reviewer of journals as well as doctoral theses in India as well abroad.
Valentina Emilia Balas, PhD, is a full professor in the Department of Automatics and Applied Software at the Faculty of Engineering, "Aurel Vlaicu" University of Arad, Romania. She holds a PhD cum Laude in applied electronics and telecommunications from the Polytechnic University of Timisoara. She is the author of more than 350 research papers and is the editor-in-chief of two esteemed journals and is an editorial board member for several other national and international publications.
Krishanu Deyasi is an associate professor in the Department of Basic Sciences and Humanities at the Institute of Engineering and Management, India. He earned his PhD from the Indian Institute of Science Education and Research, and he has postdoctoral experience from The Institute of Mathematical Sciences, India. He has written three books and has published papers in scientific journals. He is also an editor for several scientific journals.
Santanu Das is as an assistant professor in the Department of Basic Sciences and Humanities at the Institute of Engineering and Management, India. He completed his undergraduate degree from and is pursuing his doctorate from Jadavpur University.
Content
Preface xxi
1 Fermatean Fuzzy Entropy Measure with Application in Decision Making Using COPRAS Approach 1
Mansi Bhatia, H. D. Arora, Anjali Naithani and Vijay Kumar
1.1 Introduction 2
1.2 Preliminaries 4
1.3 Novel Fermatean Entropy Measure 5
1.4 Application of Entropy Measure Through COPRAS 7
1.5 Comparative Analysis 12
1.6 Conclusion 13
References 13
2 Some Properties of Cartesian and Lexicographic Products of Soft Graphs 17
Jinta Jose, Bobin George and Rajesh K. Thumbakara
2.1 Introduction 17
2.2 Soft Graphs 18
2.3 Some Properties of Cartesian and Restricted Cartesian Products of Soft Graphs 20
2.4 Some Properties of Lexicographic and Restricted Lexicographic Products of Soft Graphs 23
2.5 Conclusion 26
References 26
3 Advancements in Enhancing Car Object Detection in Complex and Adverse Environmental Conditions Through Deep Learning Techniques 29
Rejuwan Shamim, Biswadip Basu Mallik and Trapty Agarwal
3.1 Introduction 30
3.2 Literature Review 33
3.3 Methodology 35
3.4 Result 44
3.5 Discussion 49
3.6 Conclusion 54
References 57
4 Approximation by Durrmeyer Type Operators Using Polya Distribution 61
Prerna Sharma and Diwaker Sharma
4.1 Introduction 61
4.2 Basic Outcomes 63
4.3 Direct Results 65
4.4 Discussion 73
Disclaimer 73
References 73
5 Solution of Pollutant Dispersion in Porous Medium Under Linear Sorption Using Finite Element Method 75
Rashmi Radha, Tapan Paul, Rakesh Kumar Singh, Nav Kumar Mahato and Mritunjay Kumar Singh
5.1 Introduction 76
5.2 Mathematical Formulation 77
5.3 Numerical Derivation of the Proposed Model Problem by FEM Method 79
5.4 Analytical Derivation of the Proposed Model Equation 81
5.5 Results and Discussion 83
5.6 Conclusion 87
References 88
6 A Comparative Analysis of Fuzzy and Neutrosophic Database Models in Handling Imprecise Queries 91
Doyel Sarkar and Sharmistha Ghosh
6.1 Introduction 92
6.2 Basic Definitions 93
6.3 Processing Imprecise Query using Fuzzy and Neutrosophic Sets 94
6.4 Results and Discussion 98
6.5 Concluding Remarks 98
References 99
7 Tweaked Portfolio Estimation Regarding Indian Securities Exchange: An Empirical Study 101
Abhijit Biswas and Meghdoot Ghosh
7.1 Introduction 102
7.2 Literature Review in Financial Market 102
7.3 Method 103
7.4 Results 107
7.5 Discussion 116
References 116
8 Fixed Point Results Related to Graph Theory 117
Aditya Bhattacharya, Özen Özer, Sonendra Gupta, Ramakant Bhardwaj and Sonam
8.1 Introduction 117
8.2 Preliminaries 118
8.3 Main Results 119
References 128
9 Unleashing GPT-3's Potential in Automatic Text Generation: A Comprehensive Study and Analysis 131
Rejuwan Shamim and Biswadip Basu Mallik
9.1 Introduction 132
9.2 Literature Review 133
9.3 Methodology 135
9.4 Evaluation Metrics 140
9.5 Experimental Results 143
9.6 Discussion 147
9.7 Exploration of the Strengths and Weaknesses of GPT-3 for Automatic Text Generation 148
Conclusion 150
References 151
10 Optimization Techniques and Their Applications in Science and Engineering 153
Ravi Kiran Bagadi, Eali Stephen Neal Joshua, T. Pavankumar, S. NagaMallik Raj and Debnath Bhattacharyya
10.1 Introduction 154
10.2 Related Work 159
10.3 Metaheuristic Optimization Techniques 167
10.4 Multi-Objective Optimization 172
10.5 Stochastic Optimization 176
10.6 Robust Optimization 180
10.7 Applications of Robust Optimization in Science and Engineering 182
10.8 Applications of Optimization Techniques in Science and Engineering 183
10.9 Challenges and Future Directions in Optimization 189
Conclusion 194
References 195
11 On Sum 3-Equitable Labeling of Some Graphs 197
Sarang Sadawarte and Sweta Srivastav
11.1 Introduction 197
11.2 Terminology and Notation 198
11.3 Results 199
11.4 Conclusions and Perspectives 205
References 205
12 An Application of Invariant Point Theory in G-Metric Spaces with Special Emphasis on Alpha-Psi Contraction 207
Samriddhi Ghosh, Sonam, Deb Sarkar, Poulami Halder and Ramakant Bhardwaj
12.1 Introduction 207
12.2 Elementaries 210
12.3 Main Result 211
Acknowledgement 216
Collision of Interest 216
References 217
13 Fixed Point Results for Compatible Mapping of Type (a) in Fuzzy Metric Spaces 219
Poulami Halder, Samriddhi Ghosh, Ramakant Bhardwaj, Sonam and Satyendra Narayan
13.1 Introduction 219
13.2 Preliminaries 220
13.3 Compatible Mappings of Type (a) 222
13.4 Main Results 223
Acknowledgement 229
Conflict of Interest 229
References 229
14 Combined Matrices Associated with Soft Digraphs 231
Bobin George, Jinta Jose and Rajesh K. Thumbakara
14.1 Introduction 231
14.2 Soft Digraphs 232
14.3 Combined Adjacency Matrix of a Soft Digraph 234
14.4 Combined Incidence Matrix of a Soft Digraph 236
14.5 Conclusion 238
References 238
15 Refining Medical Text Query Responses: Tailoring Hugging Face's BERT Model for Precise and Swift Medical Question Answering 241
Rejuwan Shamim, Badria Sulaiman Alfurhood and Biswadip Basu Mallik
15.1 Introduction 242
15.2 Related Work 244
15.3 Methodology 248
15.4 Result 253
15.5 Strengths and Weaknesses of the Fine-Tuned BERT Model 255
15.6 Applications 257
15.7 Conclusion 258
References 260
16 Machine Learning Mathematics: A Study on Concepts of Processing Knowledge 263
Prasad Kaviti, Eali Stephen Neal Joshua, N. Venkatram, Dinesh Reddy and Debnath Bhattacharyya
16.1 Introduction 264
16.2 Mathematical Concepts in Machine Learning 264
16.3 Conclusion 299
Disclaimer 299
References 299
17 Nature-Inspired Algorithms to Optimize the Hyper-Parameters of Deep-Learning Networks for Diagnosing Brain Disorders - A Review 301
Manoj Kumar Sharma, M. Shamim Kaiser and Kanad Ray
17.1 Introduction 302
17.2 Literature Review 303
17.3 Applications 307
17.4 Challenges and Future Directions 309
17.5 Conclusions and Perspectives 312
References 312
18 YOLOv8 for Anomaly Detection in Surveillance Videos: Advanced Techniques for Identifying and Mitigating Abnormal Events 317
Rejuwan Shamim, Badria Sulaiman Alfurhood, Trapty Agarwal and Biswadip Basu Mallik
18.1 Introduction 318
18.2 Overview of YOLOv8 and its Advantages in Object Detection 319
18.3 Related Work 321
18.4 YOLOv8 Architecture 324
18.5 Dataset Preparation 327
18.6 Training YOLOv8 for Anomaly Detection 330
18.7 Evaluation Metrics 332
18.8 Results and Discussion 334
18.9 Conclusion 343
References 346
19 Linear Stability and Resonance of Oblate Infinitesimal in the Nearby Region of Triangular Equilibrium Points for Triaxial Primaries in the Elliptic Restricted Three Body Problem 351
Shilpi Dewangan, A. Narayan and Poonam Duggad
19.1 Introduction 352
19.2 Equation of Motion 353
19.3 Position of Triangular Equilibrium Points 355
19.4 Normalization at Hamiltonian for Stability of First Order 356
19.5 Resonance Cases 366
19.6 Conclusion 368
References 370
20 Magnetic Nanofluid Flow with Micro-Organisms and Viscous Dissipation 373
Shweta Mishra, Sharmistha Ghosh and Hiranmoy Mondal
Nomenclature 374
20.1 Introduction 375
20.2 Mathematical Exploration 376
20.3 Similarity Transformation 378
20.4 Heat-Mass Transferal 379
20.5 Discussion of Results 380
20.6 Conclusion 384
References 384
21 The Application of Deep Learning to the Localization of Brain Tumors 387
Charanarur Panem, Srinivasa Rao Gundu, J. Vijaylaxmi and Biswadip Basu Mallik
21.1 Introduction 388
21.2 Proposed Method 389
21.3 Conclusion 395
Acknowledgement 395
References 395
22 Contextual Information Retrieval using Root Word Stemming in Indian Languages 397
Chandamita Nath and Bhairab Sarma
22.1 Introduction 397
22.2 Literature Review 399
22.3 Problem Statement 399
22.4 Experimental Work 400
22.5 Conclusion and Future Work 402
Disclaimer 403
References 404
23 Recent Advances in Object Detection Based on YOLO-V 4 and Faster RCNN: A Review 405
Anwesa Das, Atanu Nandi and Ishani Deb
23.1 Introduction 406
23.2 Literature Review 407
23.3 Proposed Methodology 409
23.4 Result and Discussion 413
23.5 Conclusion 416
References 417
24 Detection of Leukemia Using Transfer Learning 419
Biplab Kanti Das, Chanchal Ghosh, Joydeb Sheet and Himadri Sekhar Dutta
24.1 Introduction 420
24.2 Literature Survey 421
24.3 Dataset 423
24.4 Proposed Methodology 424
24.5 Pre-Processing of Image 425
24.6 Deep Learning Model Generation 426
24.7 Classification 430
24.8 Results and Analysis 430
24.9 Conclusion and Future Direction 434
References 434
25 IoMT - An ML-Based Patient Monitoring System with Prediction of Health Condition 437
Kakali Das, Sagnik Ghosh and Himadri Sekhar Dutta
25.1 Introduction 438
25.2 System Architecture 440
25.3 Machine Learning Algorithm 443
25.4 Results and Discussion 444
25.5 Conclusion 449
References 449
26 Eulerian Soft Graphs 453
Jinta Jose, Bobin George and Rajesh K. Thumbakara
26.1 Introduction 453
26.2 Soft Graphs 454
26.3 Eulerian Soft Graphs 454
26.4 Conclusion 460
References 461
27 An Algorithm to Solve an Exponential Diophantine Equation 463
Subramani K. and Srinivasa Prasanna
27.1 Introduction 464
27.2 Example 466
27.3 Conclusion 475
Acknowledgements 475
References 475
28 Optimal Number of Emergency Facility and Its Positioning Using Nature-Based Algorithm: A Case of Mumbai City 477
K.V. Ajaygopal and Rakesh Verma
28.1 Introduction 478
28.2 Literature Review 479
28.3 Real-Life Application 484
28.4 Results and Discussion 487
28.5 Conclusion 489
References 490
Appendix 492
29 Chicken Swarm Optimization Algorithm-Based Propagation Delay Estimation in Transmission Path of UWASN 501
A. Kannappan and R.M. Bommi
29.1 Introduction 502
29.2 General Biology Behavior of Chicken Swarm 503
29.3 Chicken Swarm Pseudocode 506
29.4 Results and Discussion 507
29.5 Conclusion 508
References 509
30 Frequency Analysis of Coreference Resolution 513
Mridusmita Das and Apurbalal Senapati
30.1 Introduction 514
30.2 Existing Literature 515
30.3 Resource Building for the English and Assamese Languages 517
30.4 Description of the Data Sets 517
30.5 Frequency Analysis of Coreference Relations 518
30.6 Experiment and Result 520
30.7 Conclusion 520
References 520
31 Hate Neologism in Election Context in India 523
Sujit Das and Apurbalal Senapati
31.1 Introduction 524
31.2 Related Work 525
31.3 Corpus Creation 526
31.4 Methodology 528
31.5 Result 529
31.6 Conclusion 530
References 530
Index 533
1
Fermatean Fuzzy Entropy Measure with Application in Decision Making Using COPRAS Approach
Mansi Bhatia1*, H. D. Arora1, Anjali Naithani1 and Vijay Kumar2
1Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Uttar Pradesh, Noida, India
2Department of Mathematics, Manav Rachna International Institute of Research and Studies, Faridabad, India
Abstract
Due to the influence of media and easy access to other mediums uncertainty has significantly impacted decision making process in every field and aspect of human life. Over the last few decades improved fuzzy decision making models have proved themselves as more efficient tool in handling decision making as compared to classical set theory. One such concept is the notion of Fermatean fuzzy sets (FFS) which states that the sum of cubes of membership and non-membership is restricted to unity. FFS is an extension of Pythagorean fuzzy sets (PFS) and intuitionistic fuzzy sets (IFS). FFS can also handle situations where IFS and PFS fails and hence can be easily combined with different decision making techniques to process uncertain information and hence simplify complex decisions. The purpose of this article is to propose a novel entropy measure that can be used to solve real life application using COmplex PRoportional ASsessment (COPRAS). Weights are calculated using fuzzy entropy and example is taken to numerically evaluate the rationality of the measure. Finally, practical examples are given to validate the reliability and effectiveness of the proposed measure.
Keywords: COPRAS, decision-making, entropy measure, fermatean fuzzy sets
1.1 Introduction
Multi Criteria Decision Making (MCDM) is a branch of Operation Research (OR) which deals with decision making situations involving conflicting criteria. It is used in wide varieties of fields but not limited to management, education, medical and environment. Humans have limitations when it comes to conflicting decision making situations hence MCDM techniques makes it convenient to evaluate quantitatively and makes sound decisions. There are various types of MCDM techniques like AHP, SAW, TOPSIS, COPRAS, ELECTRE, VIKOR, etc. which has been combined time to time with different fuzzy sets to answer and explore the uncertainty in the path of decision making.
In real life data is not in crisp form hence a need was felt by Zadeh [1] to extend the classical set theory of George Cantor to the fuzzy set theory that can address the uncertainty involved in practical situations. Unlike crisp set where an element either belongs to a set or it does not, fuzzy set theory also takes into account all the values that falls between them. FS has proved to be an ideal tool to handle the vagueness but a situation is possible where belongingness of an object may be accompanied with hesitancy. This problem was solved with the use of Intuitionistic Fuzzy Set (IFS) that incorporated the hesitancy by Atanassov [2, 3].
Later on, it was observed that for some situations Atanassov's theory was not satisfied and hence Yager [4-7] formulated the theory of Pythagorean Fuzzy Sets (PFS) which states that the aggregate of square of membership and non-membership lies between 0 and 1. In 2017, Yager [8] extended the concept of IFS and PFS with the introduction of q-rung orthopair fuzzy sets. If is the measure of belongingness, is the measure of non-belongingness function and t(?) is the hesitancy then and . For q = 1 the inequality reduces to which is the condition of IFS, for q = 2 it becomes inequality satisfied by PFS and q=3 denotes the Fermatean Fuzzy sets i.e., . The theory of FFS given in 2020 by Senapati and Yager [9, 10] is the latest addition to FS theory. The whole theoretical concept can be explained with the help of example as follows: Let <0.6, 0.9> be a fuzzy set then 0.6+0.9 ? 1 and 0.62 + 0.92 = 0.36 + 0.81 > 1 but 0.63 + 0.93 = 0.216 + 0.729 <1. The example showed a situation where IFS, PFS fails but FFS satisfies the case. The evolution of Fuzzy theory can also be understood with the help of flow Figure 1.1 given below.
Entropy is defined as the degree of uncertainty or randomness. It is used to measure the fuzziness within the fuzzy sets. Entropy is directly proportional to the amount of information conveyed by an object that is high value of entropy implies more information and low value indicates less information. Entropy is used to produce the parameter weights based on the decision matrix. Entropy always gets confused with probability but in case of probability we deal with events that might or might not occur but entropy deals with truth values only. Entropy has been combined with many fuzzy sets as per the requirement to get desired information. Hung and Yang [11] proposed two entropies for IFS. Verma and Sharma in 2014 [12] explored entropy of order - a for IFS. Rahimi et al. [13] suggested entropy approach for IFS based on supplier selection. Yuan and Zeng [14] proposed an entropy function based on special function for IFS with its applications in regional collaborative innovation. Pandey et al. [15] introduced novel entropy based on IFS for feature selection that tends to improve data management by enhancing machine learning. Further, entropy has been applied with PFS to enhance the decision making by many researchers. Yang and Hussain [16] in 2018 applied fuzzy entropy on PFS with its applications in MCDM. A new entropy measure proposed by Thao et al. [17] in 2019 based on PFS.IN 2021 Gandotra [18] gave us two entropy measures first with R. Kumar to assess best automative company and then with Kizielewicz [19] and others to solve various applications in MCDM. Chaurasiya and Jain [20] proposed the entropy measure on PFS and solved health care waste management problem. Mohagheghi and Mousavi [21] suggested entropy approach on Interval valued PFS for sustainable project decision. FFS are the most recent type of fuzzy sets and since they have an advantage over existing fuzzy sets it is expanding its roots quickly in decision making problems to solve the real life problems faced by humans. Use of entropy in FFS relatively new and only few researchers have explored this area like Murat Kirisci [22] has proposed new entropy and distance measure based on FFS with its applications in medical diagnosis. Chang et al. [23] have suggested a new entropy method based on FFS using risk assessment.
Figure 1.1 Fuzzy sets.
Entropy has been applied with different fuzzy sets for giving information on various decision making problems using Lagrange entropy [24] or MCDM techniques like TOPSIS [25], VIKOR [26], COPRAS [27] etc. The COmplex PRoportional ASsessment (COPRAS) method was introduced by Zavadskas [28] in 1994. Many entropy measures have been proposed in the literature like by Wei [30], Wang [31] using IFS, Xue et al. [29] on the basis of PFS etc. Many fuzzy techniques have been combined with COPRAS to solve decision making problems. In this article we have formulated entropy measure using COPAS method to provide an easy way for decision making problems.
The manuscript has been organized as follows: Section 1.2 helps you to understand the basic fundamentals used in the article. Section 1.3 is dedicated to the proposed entropy measure and its properties. Section 1.4 describes the COPRAS approach and the application of the measure. Section 1.5 shows comparative analysis with other authors. At last Section 1.6 concludes the article followed by references.
1.2 Preliminaries
1.2.1 Intuitionistic Fuzzy Sets
In a universe of discourse X, an intuitionistic fuzzy set can be stated as
(1.1)Where is the degree of affiliation and is the degree of non-affiliation and the both the function satisfies the relation
(1.2)Degree of hesitancy is the part of membership or non - membership or both and is specified as and it expresses the insufficient information or uncertainty.
1.2.2 Pythagorean Fuzzy Sets
Let be a PFS in X then it can be stated as
(1.3)Where and is the degree of inclusion and non-inclusion and the membership.
Given ? be the element the set satisfies the relation .
(1.4)The measure of uncertainty in this case is outlined as .
1.2.3 Fermatean Fuzzy Sets
(1.5)
is called Fermatean fuzzy set if there exists an element such that and where is called the measure of inclusion, is the measure of non-inclusion and is the degree of indeterminacy or hesitancy.
1.3 Novel Fermatean Entropy Measure
In this segment we have projected a new entropy measure based on FFS and shown its legitimacy by proving the properties for the proposed measure.
1.3.1 Entropy
Let be Fermatean fuzzy set then the entropy on is defined as
(1.6)The proposed measure satisfies the following properties:
The entropy is said to be Fermatean entropy if
- is a crisp set.
- where and are FFS such that that...
