Applied Linear Algebra, Probability and Statistics : A Volume in Honour of C. R. Rao and Arbind K. Lal

This book focuses on research in linear algebra, statistics, matrices, graphs and their applications. Many chapters in the book feature new findings due to applications of matrix and graph methods. The book also discusses rediscoveries of the subject by using new methods. Dedicated to Prof. Calyampudi Radhakrishna Rao (C.R. Rao) who has completed 100 years of legendary life and continues to inspire us all and Prof. Arbind K. Lal who has sadly departed us too early, it has contributions from collaborators, students, colleagues and admirers of Professors Rao and Lal. With many chapters on generalized inverses, matrix analysis, matrices and graphs, applied probability and statistics, and the history of ancient mathematics, this book offers a diverse array of mathematical results, techniques and applications. The book promises to be especially rewarding for readers with an interest in the focus areas of applied linear algebra, probability and statistics. About Rao and Lal Foreword Preface Acknowledgements Contents Editors and Contributors 1 On Some Matrix Versions of Covariance, Harmonic Mean and Other Inequalities: An Overview 1.1 Introduction 1.2 Covariance Inequalities 1.3 Harmonic Mean Inequalities 1.4 General Mean Inequalities References 2 The Impact of Prof. C. R. Rao's Research Used in Solving Problems in Applied Probability 2.1 Professor Rao's Research on Generalized Matrix Inverses 2.2 Professor Hunter Is Exposed to Prof. Rao's Research 2.3 Professor Hunter's Applications of Prof. Rao's Research in the Field of Applied Probability 2.4 Professor Hunter's Links with Prof. Rao 2.5 Final Comments References 3 Upper Bounds for the Euclidean Distances Between the BLUEs Under the Partitioned Linear Fixed Model and the Corresponding Mixed Model 3.1 Introduction 3.2 Upper Bounds for the BLUEs 3.3 Upper Bounds Related to OLSE 3.4 Conclusions References 4 Nucleolus Computation for Some Structured TU Games via Graph Theory and Linear Algebra 4.1 TU Cooperative Game 4.2 Nucleolus for the Talmud Game ch4AM 4.3 The Core and Balanced Collection 4.4 The Lexicographic Center ch4MPS 4.5 Assignment Games ch4NOL 4.6 Domination Power Between Married Couple 4.6.1 The Kernel/Nucleolus of a Standard Tree Game ch4GM1996 4.6.2 Balanced Connected Games ch4SAD1998 4.6.3 Nucleolus for Cyclic Permutation Games ch4SRT2005,ch4TP1984 4.6.4 Brief Remarks on Some References Included References 5 From Linear System of Equations to Artificial Intelligence—The Evolution Journey of Computer Tomographic Image Reconstruction Algorithms 5.1 Introduction 5.2 From Filtered Back Projection to Iterative Reconstruction Technique to Clinical Necessity 5.3 Reconstruction Algorithms as Inverse Problems 5.4 Compressed Sensing-Based CT Image Reconstruction Algorithms 5.5 Nyquist Sampling Versus Non-uniform Sampling 5.6 Current and Future Developments 5.7 Conclusion and Open Question References 6 Shapley Value and Other Axiomatic Extensions to Shapley Value 6.1 Shapley Value 6.2 Multi-choice Shapley Value (ch6HsiaospsRaghavansps1993sps) 6.3 Potential Function and Shapley Value (ch6Hartsps1988sps) References 7 An Accelerated Block Randomized Kaczmarz Method 7.1 Introduction 7.2 Randomized Kaczmarz Method 7.3 Block Accelerated RK 7.4 Convergence and Error 7.5 Applications to Tensor Equations 7.5.1 Tensor BARK 7.5.2 Tensor BARK in the Fourier Domain 7.6 Numerical Examples 7.6.1 Comparison Between ARK and BARK 7.6.2 T-BARK Performance References 8 Nullity of Graphs—A Survey and Some New Results 8.1 Basic Results on Nullity 8.2 Bounds for the Nullity of a Graph 8.3 Energy and Nullity 8.4 Graphs with Nullity 0 and 1 8.5 Graph Operations Preserving Nullity 8.6 The Nullity Set of a Family of Graphs 8.7 Graphs with Maximum Nullity 8.8 Conclusion References 9 Some Observations on Algebraic Connectivity of Graphs 9.1 Introduction 9.2 Describing the Bottleneck Matrix 9.3 The Sliding Principle and its Application References 10 Orthogonality for Biadjoints of Operators 10.1 Introduction 10.2 Main Results References 11 Permissible Covariance Structures for Simultaneous Retention of BLUEs in Small and Big Linear Models 11.1 Introduction 11.2 The Case when V0 is the Identity Matrix 11.3 The General Case of V0 11.4 A Related Sidetrack 11.5 OLSE Equals BLUE Under the Small Model, What About the Big Model? 11.6 Conclusions References 12 On Some Special Matrices and Their Applications in Linear Complementarity Problem 12.1 Introduction 12.2 Preliminaries 12.3 The Class of Hidden Z-Matrices 12.3.1 Various Game Theory-Based Characterizations of Hidden Z-Matrices 12.4 The Class of Positive Subdefinite Matrices 12.5 The Class of Generalized Positive Subdefinite Matrices 12.6 The Class of WGPSBD Matrices 12.7 The Class of GPSBD Matrices of Level k 12.8 The Class of Fully Copositive Matrices 12.9 The Class of Singular N0-Matrices 12.10 Almost Type Classes of Matrices 12.10.1 A Generalization of Almost Type Class of Matrices 12.11 Principal Pivot Transforms of some Classes of Matrices 12.12 A List of Open Problems References 13 On Nearest Matrix with Partially Specified Eigen-Structure 13.1 Introduction 13.2 SVD of Sylvester Operator 13.3 Minimal Perturbation 13.4 Clarke Subdifferential of Singular Value Functions 13.5 Singular Value Characterization of Eigenvalue Distance Problem References 14 Equality of BLUEs for Full, Small, and Intermediate Linear Models Under Covariance Change, with Links to Data Confidentiality and Encryption 14.1 Introduction 14.2 Full Model mathcalM12 and Small Model mathcalM1 14.3 Reparametrised Full Model mathcalM12t and Small Model mathcalM1 14.4 Models Intermediate Between mathcalM12t and mathcalM1 14.5 BLUEs and Their Covariance 14.6 BLUPs for Mixed Models 14.7 Data Cloning 14.8 Conclusions References 15 Statistical Inference for Middle Censored Data with Applications 15.1 Introduction 15.2 Non-parametric Estimation for Middle Censored Data 15.3 Middle Censoring Under Parametric Models 15.3.1 Middle Censoring Under an Exponential Setup 15.3.2 Middle Censored Competing Risks Data Under Exponential Distribution 15.3.3 Middle Censoring Under a Few Other Parametric Models 15.4 Inference for Discrete Middle Censored Data 15.4.1 Maximum Likelihood Estimator 15.4.2 Bayesian Inference 15.4.3 Discrete Lifetime Data Under Middle Censoring and Presence of Covariates 15.4.4 Middle Censoring in the Multinomial Distribution with Applications 15.5 A General Censoring Scheme for Circular Data 15.5.1 Non-parametric Estimation of Censored Data 15.6 Additive Risks Regression Model for Middle-Censored Data 15.6.1 Martingale Method of Estimation 15.7 Conclusions References 16 Matrix Partial Orders Based on the Secondary-Transpose 16.1 Introduction 16.2 Preliminaries 16.3 Projectors and Generalized Inverses 16.4 The s-Order and †s-Order 16.5 Factors with Reference to the Order Relations 16.6 A New Decomposition References 17 On Products of Graph Matrices 17.1 Introduction 17.1.1 Product of Graphs (Graph Product) 17.2 Matrix Product of Graphs 17.3 Realization of A(G)A(H) (Modulo 2) 17.4 Some Matrix Equations of Graphs 17.5 Realization of A(G)A(GkP) 17.6 Companion Search Algorithm 17.7 Conclusion and Open Problems References 18 On Rao's Weighted Distributions for Modeling the Dynamics of Wildfires and Air Pollution 18.1 Historical Background 18.2 Weighted Distributions 18.3 Weighted Systems for Data Fusion 18.4 Application: Wildfires and Air Pollution Data 18.5 Discussion References 19 Characterization of Q-Matrices Using Bordered Matrix Algorithm 19.1 Introduction 19.2 Preliminaries 19.2.1 Lemke's Algorithm and Continuity of Solution z 19.2.2 Some Q and Q0-Matrices 19.3 Main Results 19.3.1 Bordered Matrix Algorithm (BMA) 19.3.2 Characterization of Q-Matrices Using BMA 19.4 Conclusion References 20 Descending Endomorphisms of Some Families of Groups 20.1 Introduction 20.2 Descending Endomorphisms of Symmetric Groups 20.3 Descending Endomorphisms of Dihedral Groups 20.4 Descending Endomorphisms of Hamiltonian Groups 20.5 Descending Endomorphisms of Dicyclic Groups References 21 On Circulant Partial Hadamard Matrices 21.1 Introduction 21.2 Preliminaries 21.2.1 Hadamard Matrix 21.2.2 Partial Hadamard Matrices 21.2.3 Row Regular Matrices 21.2.4 Circulant Partial Hadamard Matrices 21.3 Main Work 21.3.1 New Bounds of r for Circulant Partial Hadamard Matrix r-H(ktimesn) in Terms of n Which Implies k=4 21.3.2 Some Results on Circulant Partial Hadamard Matrix r-H(ktimesn) Obtained with the Help of Column Sum Restrictions and k=3 21.3.3 Results on Circulant Partial Hadamard Matrices Which Reflect Types of Column Sums 21.3.4 Some Results on Augmentation of Circulant Partial Hadamard Matrices 21.3.5 Non-existence of Circulant Skew Hadamard Matrix of Order n>1 21.3.6 One Part of Conjecture Given by Craigen et al. ch212craigen for the Existence of 2-H(ktimes2k) 21.4 Thoughts and Conclusion References 22 On Weak Hypervector Spaces Over a Hyperfield 22.1 Introduction 22.2 Preliminaries 22.3 Some Examples of Weak Hypervector Spaces 22.4 Properties of Subhyperspaces 22.5 Properties of Linear Transformations 22.6 Annihilators in Hypervector Spaces 22.7 Conclusion References 23 Generalized Lie Triple Derivations of Trivial Extension Algebras 23.1 Introduction 23.2 Preliminaries 23.3 Generalized Lie Triple Derivations 23.4 Application to Triangular Algebras References 24 On Ideals of Compatible ΘΓ N-Group 24.1 Introduction 24.2 Preliminaries 24.3 Compatible ΘΓ N-Group References 25 The Range Column Sufficiency and the Pseudo-SSM Property of Linear Transformation on Euclidean Jordan Algebra 25.1 Introduction 25.2 Preliminaries 25.2.1 Euclidean Jordan Algebra 25.2.2 The Linear Complementarity Concepts 25.2.3 Z Property 25.2.4 Group Inverse 25.3 The Range Column Sufficiency Property and the pSSM Property 25.4 Equivalence of rCCS and pSSM Properties for Some Special Transformations 25.5 The Relaxation Transformation 25.6 Conclusion References 26 On the Proofs of Formulae by Mahoverlineavoverlineira and Brahmagupta 26.1 On Mahoverlineavoverlineira Formulae for Sum of the Squares and Sum of the Cubes of n Terms of an Arithmetic Sequence 26.2 A Study of Cyclic Quadrilaterals References