Provides a detailed proof of both of Godelfs Incompleteness Theorems
without building on recursion theory
Presents detailed constructions of several standard and non-standard models
of Peano Arithmetic, Presburger Arithmetic, Zermelo Fraenkel set theory and
the real numbers
Contains a self-contained and concise introduction into mathematical logic
and axiomatic set theory which requires almost no prerequisites, whose only
assumption is the notion of finiteness
This book provides a concise and self-contained introduction to the foundations of
mathematics. The first part covers the fundamental notions of mathematical logic, including
logical axioms, formal proofs and the basics of model theory. Building on this, in the second
and third part of the book the authors present detailed proofs of Godelfs classical
completeness and incompleteness theorems. In particular, the book includes a full proof of
Godelfs second incompleteness theorem which states that it is impossible to prove the
consistency of arithmetic within its axioms. The final part is dedicated to an introduction into
modern axiomatic set theory based on the Zermelofs axioms, containing a presentation of Godelf
s constructible universe of sets. A recurring theme in the whole book consists of standard and
non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and
the real numbers. The book addresses undergraduate mathematics students and is suitable for
a one or two semester introductory course into logic and set theory. Each chapter concludes
with a list of exercises.
Mathematics : Mathematical Logic and Foundations
Due 2020-11-24
1st ed. 2020, X, 236 p.
Hardcover
ISBN 978-3-030-52278-0
Product category : Graduate/advanced undergraduate textbook
Defines conditional exceptions differently than in other books
Uses only elementary facts for proof of the Doob?Meyer decomposition
theorem for special cases
Shows how the Euler?Maruyama approximation plays an important role in
proving the uniqueness of martingale problems
This book is intended for university seniors and graduate students majoring in probability
theory or mathematical finance. In the first chapter, results in probability theory are reviewed.
Then, itfollows a discussion of discrete-time martingales, continuous time square integrable
martingales (particularly, continuous martingales of continuous paths), stochastic integrations
with respect to continuous local martingales, and stochastic differential equations driven by
Brownian motions. In the final chapter, applications to mathematical finance are given. The
preliminary knowledge needed by the readerislinear algebra and measure theory. Rigorous
proofs are provided for theorems, propositions, and lemmas. In this book, the definition of
conditional expectations is slightly different than what is usually found inother textbooks. For
the Doob?Meyer decomposition theorem, only square integrable submartingales are considered,
and only elementary facts of the square integrable functions are used in the proof. In
stochastic differential equations, the Euler?Maruyama approximation is used mainly to prove
the uniqueness of martingale problems and the smoothness of solutions of stochastic
differential equations.
Mathematics : Probability Theory and Stochastic Processes
Due 2020-11-29
1st ed. 2020, XII, 218 p.
Hardcover
ISBN 978-981-15-8863-1
Product category : Monograph
Series : Monographs in Mathematical Economics
Constructs a rigorous mathematical approach to linear hereditary problems
of wave propagation theory
Opens unforeseen applications to fractal environments
Presents a classification of near-front asymptotics of solutions to considered equations
The objective of this book is to construct a rigorous mathematical approach to linear hereditary
problems of wave propagation theory and demonstrate the efficiency of mathematical
theorems in hereditary mechanics. By using both real end complex Tauberian techniques for
the Laplace transform, a classification of near-front asymptotics of solutions to considered
equations is given?depending on the singularity character of the memory function. The book
goes on to derive the description of the behavior of these solutions and demonstrates the
importance of nonlinear Laplace transform in linear hereditary elasticity. This book is of
undeniable value to researchers working in areas of mathematical physics and related fields.
Mathematics : Mathematical Physics
Due 2020-12-11
1st ed. 2020, XI, 138 p. 6 illus.
Hardcover
ISBN 978-981-15-8577-7
Product category : Monograph
Presents recent advances in discontinuous and nonlinear dynamical systems,
chaos, and complexity science
Develops the corresponding mathematical theory to apply nonlinear design to
practical engineering
Provides methods for mathematical models with switching, thresholds, and
impulses
Represents a new step into the understanding of complex dynamical systems
and the relation between their structure and functions
This book demonstrates how mathematical methods and techniques can be used in synergy
and create a new way of looking at complex systems.It becomes clear nowadays that the
standard (graph-based) network approach, in which observable events and transportation hubs
are represented by nodes and relations between them are represented by edges, fails to
describe the important properties of complex systems, capture the dependence between their
scales, and anticipate their future developments. Therefore, authors in thisbookdiscuss the new
generalized theories capable to describe a complex nexus of dependences in multi-level
complex systems and to effectively engineer their important functions.The collection of works
devoted to the memory of Professor Valentin Afraimovich introduces new concepts, methods,
and applications in nonlinear dynamical systems covering physical problems and mathematical
modelling relevant to molecular biology, genetics, neurosciences, artificial intelligence as well
as classic problems in physics, machine learning, brain and urban dynamics. The book can be
read by mathematicians, physicists, complex systems scientists, IT specialists, civil engineers,
data scientists, urban planners, and even musicians (with some mathematical background).
Mathematics : Dynamical Systems and Ergodic Theory
Due 2020-12-07
1st ed. 2020, VII, 169 p. 59 illus., 14 illus. in color.
Hardcover
ISBN 978-981-15-9033-7
Product category : Contributed volume
Series : Nonlinear Physical Science
Solutions manual is available to instructors who adopt the textbook for their
course
Second edition revised with new topics, some reworked text, new exercises
Suitable for a one-semester graduate course in integration theory as well as
for independent study
When the first edition of this textbook published in 2011, it constituted a substantial revision of
the best-selling Birkhauser title by the same author, A Concise Introduction to the Theory of
Integration.Appropriate as a primary text for a one-semester graduate course in integration
theory, this GTM is also useful for independent study. A complete solutions manual is available
for instructors who adopt the text for their courses. This second edition has been revised as
follows: 2.2.5 and 8.3 have been substantially reworked. New topics have been added. As an
application of the material about Hermite functions in 7.3.2, the author has added a brief
introduction to Schwartz's theory of tempered distributions in 7.3.4. Section 7.4 is entirely
new and contains applications, including the Central Limit Theorem, of Fourier analysis to
measures. Related to this are subsections 8.2.5 and 8.2.6, where Levy's Continuity Theorem
and Bochner's characterization of the Fourier transforms of Borel probability on N are proven.
Subsection 8.1.2 is new and contains a proof of the Hahn Decomposition Theorem. Finally,
there are several new exercises, some covering material from the original edition and others
based on newly added material.
Mathematics : Measure and Integration
Due 2020-12-14
2nd ed. 2020, XVI, 278 p.
Hardcover
ISBN 978-3-030-58477-1
Product category : Graduate/advanced undergraduate textbook
Series : Graduate Texts in Mathematics
*