Semigroups, Algebras and Operator Theory: ICSAOT 2022, CUSAT, India, March 28–31 (Springer Proceedings in Mathematics & Statistics, 436)

This book contains chapters on a range of topics in mathematics and mathematical physics, including semigroups, algebras, operator theory and quantum mechanics, most of them have been presented at the International Conference on Semigroup, Algebras, and Operator Theory (ICSAOT-22), held at Cochin, Kerala, India, from 28–31 March 2022. It highlights the significance of semigroup theory in different areas of mathematics and delves into various themes that demonstrate the subject’s diverse nature and practical applications. One of the key features of the book is its focus on the relationship between geometric algebra and quantum mechanics. The book offers both theoretical and numerical approximation results, presenting a comprehensive overview of the subject. It covers a variety of topics, ranging from C∗-algebraic models to numerical solutions for partial differential equations. The content of the book is suitable for active researchers and graduate students who are just beginning their studies in the field. It offers insights and practical applications that would be valuable to anyone interested in the mathematical foundations of physics and related fields. Overall, this book provides an excellent resource for anyone seeking to deepen their understanding of the intersections between mathematics and physics. Preface Acknowledgements Contents Contributors Semigroups Cross-Connections in Clifford Semigroups 1 Introduction 2 Normal Categories and Cross-Connections in Clifford Semigroups 3 Cross-Connections of a Semilattice References Non-commutative Stone Duality 1 Introduction 2 Classical Stone Duality 2.1 Boolean Algebras 2.2 Finite Boolean Algebras 2.3 Arbitrary Boolean Algebras 2.4 Generalized Boolean Algebras 3 Boolean Inverse Semigroups 4 Boolean Groupoids 5 From Boolean Groupoids to Boolean Inverse Semigroups 6 From Boolean Inverse Semigroups to Boolean Groupoids 7 Non-commutative Stone Duality 7.1 Properties of Prime Filters 7.2 Properties of Compact-Open Local Bisections 7.3 Proof of the First Part of the Main Theorem 7.4 Proof of the Main Theorem 8 Special Cases 9 Unitization 10 Final Remarks References Group Lattices Over Division Rings 1 Introduction 2 Group Lattices 3 G-Lattices Over a Division Ring 4 Schreier Extension and Associated Group Lattices References Identities in Twisted Brauer Monoids 1 Introduction 2 Twisted Brauer Monoids 2.1 Definition 2.2 Background 2.3 Presentation for script upper B Subscript n Superscript taumathcalBτn 2.4 The Monoid script upper B Subscript n Superscript plus or minus taumathcalBpmτn and its Identities 2.5 Structure Properties of the Monoid script upper B Subscript n Superscript plus or minus taumathcalBpmτn 3 Reduction Theorem for Identity Checking 4 Co-NP-Completeness of Identity Checking in script upper B Subscript n Superscript taumathcalBτn with n greater than or equals 5nge5 5 Related Results and Further Work 5.1 Checking Identities in Twisted Partition Monoids 5.2 Open Questions References Structure of the Semi-group of Regular Probability Measures on Locally Compact Hausdorff Topological Semiroups 1 Introduction 2 Preliminaries 3 Regularity Question 4 Identification of Elements in upper P left parenthesis upper G right parenthesisP(G) Which Are Algebraically Regular/Not Regular 5 Probability Measures on Stochastic/Doubly Stochastic Matrices 6 Concluding Remarks References Compatible and Discrete Normal Categories 1 Introduction 2 Preliminaries 2.1 Categories 2.2 Normal Categories 2.3 Normal Duals and Cross Connections 2.4 Category of Principal Left Ideals of a Regular Semigroup 3 Compatible and Discrete Normal Categories 3.1 Compatible Duals References A Range of the Multidimensional Permanent on (0, 1)-Matrices 1 Introduction and Main Definitions 2 Preliminaries 3 Resembling Matrices 4 Increasing the Dimension 5 Increasing of Dimension and Consecutive Values of Permanent References Generalized Essential Submodule Graph of an upper RR-module 1 Introduction and Preliminaries 2 Generalized Essential Submodule Graph 3 Conclusion References Algebras On Category of Lie Algebras 1 Introduction 2 Preliminaries 3 Group Lie Algebras and Plesken Lie Algebras 4 Category of Lie Algebras References On Lattice Vector Spaces over a Distributive Lattice 1 Introduction 2 Lattice Vector Spaces 3 Congruence Relation and Linear Transformations on l.v.s. 4 Conclusion References Operator Theory Application of Geometric Algebra to Koga's Work on Quantum Mechanics 1 Introduction 2 Fundamentals of Geometric Algebra 2.1 Tensor Product 3 The Four-Dimensional Gradient 4 The Dirac-Hestenes Equation 5 A Solution to the Dirac-Hestenes Equation References Resolvent Algebra in Fock-Bargmann Representation 1 Introduction 2 CCR and Resolvent Algebra 3 Fock-Bargmann Space and Toeplitz Operators 4 Resolvent Algebra in Fock-Bargmann Representation 4.1 Correspondence Theory in the Fock-Bargmann Space 4.2 Computing script upper D 0 Superscript bold tmathcalD0t 5 Infinite Dimensional Symplectic Space References left parenthesis n comma epsilon right parenthesis(n,ε)-Condition Spectrum of Operator Pencils 1 Introduction 2 On Some Properties of left parenthesis n comma epsilon right parenthesis(n,ε)-Condition Spectrum of Operator Pencils 3 left parenthesis n comma backslash e right parenthesis(n,ε)-Condition Spectral Mapping Theorem 4 Stability Analysis of Operator Equations Using left parenthesis n comma backslash e right parenthesis(n,ε)-Condition Spectrum References Numerical Solution of One-Dimensional Hyperbolic Telegraph Equation Using Collocation of Cubic B-Splines 1 Introduction 2 Cubic Splines and Boundary Modification 3 B-Spline Collocation 4 Convergence Analysis of the Numerical Scheme 5 Stability Analysis 5.1 Stability of the Proposed Method 6 Numerical Experiments 7 Discussion References