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2024 | Book

Prologue Read chapter Read first chapter

Exercises in Applied Mathematics

With a View toward Information Theory, Machine Learning, Wavelets, and Statistical Physics

Author: Daniel Alpay

Publisher: Springer International Publishing

Book Series : Chapman Mathematical Notes

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This text presents a collection of mathematical exercises with the aim of guiding readers to study topics in statistical physics, equilibrium thermodynamics, information theory, and their various connections. It explores essential tools from linear algebra, elementary functional analysis, and probability theory in detail and demonstrates their applications in topics such as entropy, machine learning, error-correcting codes, and quantum channels. The theory of communication and signal theory are also in the background, and many exercises have been chosen from the theory of wavelets and machine learning. Exercises are selected from a number of different domains, both theoretical and more applied. Notes and other remarks provide motivation for the exercises, and hints and full solutions are given for many. For senior undergraduate and beginning graduate students majoring in mathematics, physics, or engineering, this text will serve as a valuable guide as theymove on to more advanced work.

Table of Contents

Frontmatter
Chapter 1. Prologue
Abstract
Abstract
Daniel Alpay

Algebra

Frontmatter
Chapter 2. Complements in Linear Algebra
Abstract
We first present a number of elementary (and less elementary) exercises on complex numbers. We also discuss briefly the skew field of quaternions and other systems of numbers. We tried to limit analytic methods to a minimum in this chapter. Still, we use the existence of eigenvalues for a matrix, i.e., the fundamental theorem of algebra.
Daniel Alpay
Chapter 3. Positive Semi-Definite Matrices
Abstract
The present chapter deals essentially with positive semi-definite matrices. It is the longest chapter of the book, but the reader should be aware that barely the surface of the topic has been touched. Positive semi-definite matrices play a key role in numerous domains, of which we mention in particular machine learning in its various facets. See Remark 3.5.2 on the terminology positive versus positive semi-definite.
Daniel Alpay
Chapter 4. Algebra and Error-Correcting Codes
Abstract
Algebraic structures, such as groups, rings, fields, and ideals, play an important role in different parts of the present book and are discussed in this chapter. We begin with a related section on sets and functions between sets.
Daniel Alpay

Analysis

Frontmatter
Chapter 5. Complements in Real and Complex Analysis
Abstract
Among the tools from real analysis to be mastered by a student of machine learning, statistical physics, and thermodynamics, we mention in particular convexity, the main properties of partial derivatives and differentials, and the theory of Lagrange multipliers. Liouville’s theorem on systems of ordinary differential equations plays also an important role. In this chapter, we recall some elementary facts from the theory of functions. Other results include, for instance, the existence of a continuous logarithm for a complex-valued function of a complex variable.
Daniel Alpay
Chapter 6. Complements in Functional Analysis
Abstract
Functional analysis deals with properties of spaces of functions (or of sequences), rather than of individual objects. To study a specific function is doing real (or complex, or more generally hypercomplex) analysis. To study a vector space of all functions with a given common property pertains to functional analysis and uses the fact that this space may be a Hilbert, Banach, or Fréchet space, or have another structure. This structure will determine notions such as convergence of sequences and characterization of continuous functionals on the given space.
Daniel Alpay

Probability and Applications

Frontmatter
Chapter 7. Probability Theory
Abstract
In probability theory, events are subsets of an underlying set \(\Omega \). Denote by \(\mathcal {P}(\Omega )\) the set of all subsets of \(\Omega \), together with the empty set \(\emptyset \). When \(\Omega \) is not finite, one will not in general take all subsets of \(\Omega \) as possible events but restrict to a family of sets \(\mathcal {C}\subset \mathcal {P}(\Omega )\), called a sigma-algebra; see Definition 7.5.1.
Daniel Alpay
Chapter 8. Entropy: Discrete Case
Abstract
The main character of this chapter is the entropy function (1.​2.​9),
$$\displaystyle H=-\sum _{k=1}^Kp_k{\log _2}p_k, $$
introduced via the source partition theorem. Most of the chapter is devoted to the case of random variables defined on a finite probability space (but see Remark 8.1.10 for an interpretation of the result in an infinite probability space).
Daniel Alpay
Chapter 9. Thermodynamics
Abstract
Abstract
Daniel Alpay
Backmatter
Metadata
Title
Exercises in Applied Mathematics
Author
Daniel Alpay
Copyright Year
2024
Electronic ISBN
978-3-031-51822-5
Print ISBN
978-3-031-51821-8
DOI
https://doi.org/10.1007/978-3-031-51822-5

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