Pages: 720
ISBN: 978-981-126-168-8 (hardcover)
Hormander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hormander and is intimately connected to the geometry of vector fields.
Motivations for the study of Hormander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hormander operators and vector fields, together with the necessary background and prerequisites.
The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it.
Introduction
Basic Geometry of Vector Fields
Function Spaces
Homogeneous Groups in RN
Hypoellipticity of Sublaplacians
Hypoellipticity of Hormander Operators
Fundamental Solutions
Real Analysis in Locally Doubling Spaces
Sobolev and Holder Estimates on Groups
More Geometry of Vector Fields
Lifting and Approximation
Sobolev and Holder Estimates
Nonvariational Hormander Operators
Appendix: Short Summary of Distribution Theory
Bibliography
Advanced undergraduate and graduate students, researchers in the fields of mathematical analysis and partial differential equations; academic libraries.
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Pages: 560
ISBN: 978-981-125-491-8 (hardcover)
Hyperidentities are important formulae of second-order logic, and research in hyperidentities paves way for the study of second-order logic and second-order model theory.
This book illustrates many important current trends and perspectives for the field of hyperidentities and their applications, of interest to researchers in modern algebra and discrete mathematics. It covers a number of directions, including the characterizations of the Boolean algebra of n-ary Boolean functions and the distributive lattice of n-ary monotone Boolean functions; the classification of hyperidentities of the variety of lattices, the variety of distributive (modular) lattices, the variety of Boolean algebras, and the variety of De Morgan algebras; the characterization of algebras with aforementioned hyperidentities; the functional representations of finitely-generated free algebras of various varieties of lattices and bilattices via generalized Boolean functions (De Morgan functions, quasi-De Morgan functions, super-Boolean functions, super-De Morgan functions, etc); the structural results for De Morgan algebras, Boole-De Morgan algebras, super-Boolean algebras, bilattices, among others.
While problems of Boolean functions theory are well known, the present book offers alternative, more general problems, involving the concepts of De Morgan functions, quasi-De Morgan functions, super-Boolean functions, and super-De Morgan functions, etc. In contrast to other generalized Boolean functions discovered and investigated so far, these functions have clearly normal forms. This quality is of crucial importance for their applications in pure and applied mathematics, especially in discrete mathematics, quantum computation, quantum information theory, quantum logic, and the theory of quantum computers.
Introduction
Some Basic Concepts
Hyperidentities of Lattices
Boole-De Morgan Algebras. Quasi-De Morgan Functions
De Morgan Algebras. De-Morgan Functions
Hyperidentities of Boolean Algebras. Super-Boolean Algebras
Elementary Theories of Super-Boolean Algebras
Hyperidentities of De Morgan Algebras
Functional Completeness Theorem For De Morgan Functions
Bilattices
Super-Boolean Functions and Free Super-Boolean Algebras
Super-De Morgan Functions and Free Super-De Morgan Algebras
Bi-De Morgan Functions and Free Distributive Bilattices
Weakly Idempotent Lattices and Bilattices
A Set-Theoretical Representation for Weakly Idempotent Lattices and Interlaced Weakly Idempotent Bilattices
Bigroups and Gratzer Algebras
Abelian Algebras
Non-Trivial Associative and Distributive Hyperidentities in Q-Algebras
Essentially Non-Trivial Associative and Distributive Hyperidentities in Semigroups
Binary Representations of Semigroups. The Multiplicative Groups of Fields and Hyperidentities
Associative Formulae with Functional Variables
Distributive Formulae with Functional Variables
Other Open Problems
Bibliography
Index
Undergraduate, graduate students, researchers.
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Pages: 338
Series on Knots and Everything
ISBN: 978-981-126-299-9 (hardcover)
One-Cocycles and Knot Invariants is about classical knots, i.e. smooth oriented knots in three-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used in order to construct combinatorial one-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and of the longitude of the knot. The combinatorial 1-cocycles are then lifts of the well-known Conway polynomial of knots and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.
Introduction
The 1-Cocycles from the Conway Polynomial
Proofs
The Examples
Two Forgotten Linear 1-Cocycles
An Eclectic 1-Cocycle
This title will be particularly useful for academic researchers and PhD students.
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Pages: 200
ISBN: 978-981-126-445-0 (hardcover)
The stochastic partial differential equations (SPDEs) arise in many applications of the probability theory. This monograph will focus on two particular (and probably the most known) equations: the stochastic heat equation and the stochastic wave equation.
The focus is on the relationship between the solutions to the SPDEs and the fractional Brownian motion (and related processes). An important point of the analysis is the study of the asymptotic behavior of the p-variations of the solutions to the heat or wave equations driven by space-time Gaussian noise or by a Gaussian noise with a non-trivial correlation in space.
The book is addressed to public with a reasonable background in probability theory. The idea is to keep it self-contained and avoid using of complex techniques. We also chose to insist on the basic properties of the random noise and to detail the construction of the Wiener integration with respect to them. The intention is to present the proofs complete and detailed.
Gaussian Processes and Sheets:
Gaussian Processes
Fractional and Bifractional Brownian Motion
Multiparameter Gaussian Processes
Isonormal Processes and Wiener Integral
Stochastic Heat and Wave Equations with Additive Gaussian Noise:
The Stochastic Heat Equation with Space-Time White Noise
The Stochastic Heat Equation with Correlated Noise In Space
The Stochastic Wave Equation with Space-Time White Noise
Power Variation and Statistical Inference for Solutions to SPDEs:
Variations of the Solution to the Stochastic Heat Equation
Parameter Estimation for the Stochastic Heat Equation via Power Variations
Power Variations and Inference for Stochastic the Wave Equation
Undergraduate, graduate students and researchers working in various areas of probability theory and mathematical statistics.
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Pages: 252
ISBN: 978-981-126-278-4 (hardcover)
The present monograph on stochastic Komatu?Loewner evolutions (SKLEs) provides the first systematic extension of the Schramm?Loewner evolution (SLE) theory from a simply connected planar domain to multiply connected domains by using the Brownian motion with darning (BMD) that has arisen in a recent study of the boundary theory of symmetric Markov processes.
This volume is presented in an accessible manner for the interested researchers and graduate students. It also brings new insights into SLEs as special cases of SKLEs. Mathematically, it can be viewed as a powerful application of stochastic analysis via BMDs to complex analysis.
Multiply Connected Planar Domain and Brownian Motion
Chordal Komatu?Loewner Differential Equation and BMD
Komatu?Loewner Evolution (KLE)
Stochastic Komatu?Loewner Evolution (SKLE)
KLE and its Transformation
Appendix
Researchers and graduate students interested in the Schramm?Loewner evolution (SLE).