Nam Le

Free Books on Dynamical Systems

Nam Le
Table of Contents

Arxiv/ Free Books #

1. Lectures on Neural Dynamics - Francesco Bullo #

2. Linear Geometry and Algebra - Taras Banakh #

Abstract: Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So, Linear Geometry studies liners. Imposing some additional axioms on a liner, we obtain some special classes of liners: regular, projective, affine, proaffine, etc. Linear Geometry includes Affine and Projective Geometries and is a part of Incidence Geometry. The aim of this book is to present a self-contained logical development of Linear Geometry, starting with some intuitive acceptable geometric axioms and ending with algebraic structures that necessarily arise from studying the structure of geometric objects that satisfy those simple and intuitive geometric axioms. We shall meet many quite exotic algebraic structures that arise this way: magmas, loops, ternary-ring, quasi-fields, alternative rings, procorps, profields, etc. We strongly prefer (synthetic) geometric proofs and use tools of analytic geometry only when no purely geometric proof is available. Liner Geometry has been developed by many great mathematicians since times of Antiquity (Thales, Euclides, Proclus, Pappus), through Renaissance (Descartes, Desargues), Early Modernity (Playfair, Gauss, Lobachevski, Bolyai, Poncelet, Steiner, Möbius), Late Modernity Times (Steinitz, Klein, Hilbert, Moufang, Hessenberg, Jordan, Beltrami, Fano, Gallucci, Veblen, Wedderburn, Lenz, Barlotti) till our contempories (Hartshorne, Hall, Buekenhout, Gleason, Kantor, Doyen, Hubault, Dembowski, Klingenberg, Grundhöfer).

3. An introduction to graph theory - Darij Grinberg #

Abstract: This is a graduate-level introduction to graph theory, corresponding to a quarter-long course. It covers simple graphs, multigraphs as well as their directed analogues, and more restrictive classes such as tournaments, trees and arborescences. Among the features discussed are Eulerian circuits, Hamiltonian cycles, spanning trees, the matrix-tree and BEST theorems, proper colorings, Turan’s theorem, bipartite matching and the Menger and Gallai–Milgram theorems. The basics of network flows are introduced in order to prove Hall’s marriage theorem.

4. An introduction to reservoir computing - Michael te Vrugt #

Abstract: There is a growing interest in the development of artificial neural networks that are implemented in a physical system. A major challenge in this context is that these networks are difficult to train since training here would require a change of physical parameters rather than simply of coefficients in a computer program. For this reason, reservoir computing, where one employs high-dimensional recurrent networks and trains only the final layer, is widely used in this context. In this chapter, I introduce the basic concepts of reservoir computing. Moreover, I present some important physical implementations coming from electronics, photonics, spintronics, mechanics, and biology. Finally, I provide a brief discussion of quantum reservoir computing.

5. Nonequilibrium and Irreversibility - Giovanni Gallavotti #

Abstract: The work concentrates on relations, which are general and model independent in chaotic system, between time averages of a few (typically {\it very few}) observables. Equilibrium thermodynamics provides a guide and here is attempted to argue that the viewpoint of Sinai-Ruelle-Bowen can be regarded as a generalization to nonequilibrum phenomena of the theory of the ensembles proposing an answer to classical question like which distributions describe the statistics of stationary states (hence extend the analysis selecting canonical, or equivalent distributions, equilibrim between the uncountably many possibilities). The special name “Chaothic Hypothesis” (CH) is given to the above attempt and its mathematical meaning is discussed. General properties are presented and applied (eg. ‘Fluctuation Theorem’, ‘Fluctuation Patterns’, ‘Pairing Symmetry’) and related to the basic Time Reversal symmetry: which presents irreversibility as due to chaotic motion rather than to viscous forces. The case of a simple incompressible fluid is discussed in some detail. The possibility that CH is violated in various cases is considered: and in the end it is suggested that CH is the paradigm of chaotic evolution, as the harmonic oscillators are a paradigm of ordered motions, but of course {\it tertium datur}. The exposition is informal and often restricted to heuristic analysis, with detailed references to the literature and attention to numerical simulations and importance of stressing strongly the discrete models of Physics, trying to imitate the vision of Boltzmann, is widely considered.

6. Symmetries of Living Systems: Symmetry Fibrations and Synchronization in Biological Networks - Hernan A. Makse, Paolo Boldi, Francesco Sorrentino, Ian Stewart #

Abstract: A symmetry is a `change without change’. As simple as it sounds, this concept is the fundamental cornerstone that unifies all branches of theoretical physics. Virtually all physical laws – ranging from classical mechanics and electrodynamics to relativity, quantum mechanics, and the standard model – can be expressed in terms of symmetry invariances. In this book, we explore whether the same principle can also explain the emergent laws of biological systems. We introduce a new geometry for biological networks and AI architectures, drawing inspiration from the mystic genius of Grothendieck’s fibrations in category theory. We attempt to bridge the gap between physics and biology using symmetries but with a twist. The traditional symmetry groups of physics are global and too rigid to describe biology. Instead, the novel notion of symmetry fibration is local, flexible, and adaptable to evolutionary pressures, providing the right framework for understanding biological complexity. In other words, this more general symmetry invariance is necessary and sufficient to ensure that a given biological network configuration can support a synchronized function. In this book, we review the theoretical progress over the last decades from mathematics, physics, computer science, dynamical systems, and graph theory that has led to the discovery of symmetry fibrations in biological networks. These symmetries act as organizing principles for biological networks. They serve as effective tools for describing the structure of these networks, blending geometry and topology. Fibrations explain how structure dictates function across various biological domains, including the transcriptome, proteome, metabolome, and connectome. Additionally, they facilitate a reduction in the dimensionality of the network, simplifying it into its fundamental building blocks for biological computation.

7. Causal Fermion Systems: An Introduction to Fundamental Structures, Methods and Applications - Felix Finster, Sebastian Kindermann, Jan-Hendrik Treude #

Abstract: This textbook introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and “quantum geometries.” The dynamics is described by a novel variational principle, the causal action principle. The book includes a detailed summary of the mathematical and physical preliminaries. It explains the physical concepts behind the causal fermion system approach from the basics. Moreover, all the mathematical objects and structures are introduced step by step. The mathematical methods used for the analysis of causal fermion systems and the causal action principle are introduced in depth. Many examples and applications are worked out. The textbook is addressed to master and graduate students in mathematics or physics. Furthermore, it serves as a reference work for researchers working in the field.

8. A gentle invitation to the fractional world - Nicola Abatangelo, Serena Dipierro, Enrico Valdinoci #

Abstract: This book is intended as a self-contained introduction to selected topics in the fractional world, focusing particularly on aspects that arise in the study of equations driven by the fractional Laplacian. The scope of this work is not intended to be exhaustive or all-encompassing. We have chosen topics that we believe will appeal to readers embarking on their journey into fractional analysis. It requires only fundamental calculus and a basic understanding of measure theory. In Chapter 1, we introduce the primary object of study, the fractional Laplacian. This operator appears in diverse contexts, prompting multiple definitions and viewpoints, many of which we explore, along with some key identities. A notable distinction between local and nonlocal analysis is that in the latter, explicit calculations are often impractical or impossible. There are anyway some fortunate exceptions which are gathered in Chapter 2, providing useful and instructive examples. Chapter 3 presents an introduction to the important aspect of Liouville-type results. A large portion of this book is devoted to the regularity theory of solutions in Lebesgue spaces. Chapter 4 examines global solutions using Riesz and Bessel potential analysis, capturing the impact of both low and high frequencies on smoothness, decay, and oscillations. These spaces are also flexible enough to provide, as a byproduct, a solid regularity theory in the more commonly used fractional Sobolev spaces. In Chapter 5 we derive the corresponding interior regularity theory for solutions within a bounded domain using appropriate cutoffs and localization techniques. Additionally, technical appendices include auxiliary results used in key proofs.

9. Kinetically constrained models - Ivailo Hartarsky, Cristina Toninelli #

Abstract: The goal of this book is to provide an introduction to the mathematical theory of Kinetically constrained models developed in the last twenty years, intended for both mathematicians and physicists.

10. What is Entropy? - John C. Baez #

Abstract: This short book is an elementary course on entropy, leading up to a calculation of the entropy of hydrogen gas at standard temperature and pressure. Topics covered include information, Shannon entropy and Gibbs entropy, the principle of maximum entropy, the Boltzmann distribution, temperature and coolness, the relation between entropy, expected energy and temperature, the equipartition theorem, the partition function, the relation between expected energy, free energy and entropy, the entropy of a classical harmonic oscillator, the entropy of a classical particle in a box, and the entropy of a classical ideal gas.

11. Alice’s Adventures in a Differentiable Wonderland – Volume I, A Tour of the Land - Simone Scardapane #

Abstract: Neural networks surround us, in the form of large language models, speech transcription systems, molecular discovery algorithms, robotics, and much more. Stripped of anything else, neural networks are compositions of differentiable primitives, and studying them means learning how to program and how to interact with these models, a particular example of what is called differentiable programming. This primer is an introduction to this fascinating field imagined for someone, like Alice, who has just ventured into this strange differentiable wonderland. I overview the basics of optimizing a function via automatic differentiation, and a selection of the most common designs for handling sequences, graphs, texts, and audios. The focus is on a intuitive, self-contained introduction to the most important design techniques, including convolutional, attentional, and recurrent blocks, hoping to bridge the gap between theory and code (PyTorch and JAX) and leaving the reader capable of understanding some of the most advanced models out there, such as large language models (LLMs) and multimodal architectures.

12. Inverse Problems and Data Assimilation: A Machine Learning Approach - Eviatar Bach, Ricardo Baptista, Daniel Sanz-Alonso, Andrew Stuart #

Abstract: The aim of these notes is to demonstrate the potential for ideas in machine learning to impact on the fields of inverse problems and data assimilation. The perspective is one that is primarily aimed at researchers from inverse problems and/or data assimilation who wish to see a mathematical presentation of machine learning as it pertains to their fields. As a by-product, we include a succinct mathematical treatment of various topics in machine learning.

13. The Lanczos algorithm for matrix functions: a handbook for scientists - Tyler Chen #

Abstract: Lanczos-based methods have become standard tools for tasks involving matrix functions. Progress on these algorithms has been driven by several largely disjoint communities, resulting many innovative and important advancements which would not have been possible otherwise. However, this also has resulted in a somewhat fragmented state of knowledge and the propagation of a number of incorrect beliefs about the behavior of Lanczos-based methods in finite precision arithmetic. This monograph aims to provide an accessible introduction to Lanczos-based methods for matrix functions. The intended audience is scientists outside of numerical analysis, graduate students, and researchers wishing to begin work in this area. Our emphasis is on conceptual understanding, with the goal of providing a starting point to learn more about the remarkable behavior of the Lanczos algorithm. Hopefully readers will come away from this text with a better understanding of how to think about Lanczos for modern problems involving matrix functions, particularly in the context of finite precision arithmetic.

14. New Book: Tensor Decompositions for Data Science #

Abstract: This book is intended for a graduate-level course in a data-science domain such as mathematics, computer science, engineering, statistics, physics, neuroscience, etc. It is written so that it can be used flexibly. It can be adapted for a subunit in a longer class or can stand on its own in a full semester course. We include substantial background material in linear algebra, optimization, and probability and statistics in the hopes of making the contents widely accessible. The book includes links to several real-world datasets to be used as examples for experiments in the book, grounding the material and providing a playground for student experimentation.

15. Calculus and applications - Teo Banica #

Abstract: This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f’(x)$ and of the integral $\int_a^bf(x)dx$. Then we investigate the case of the complex functions $f:\mathbb C\to\mathbb C$, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, $f:\mathbb R^N\to\mathbb R^M$ or $f:\mathbb R^N\to\mathbb C^M$ or $f:\mathbb C^N\to\mathbb C^M$, with general theory, integration results, maximization questions, and basic applications to physics.

16. Stochastic Partial Differential Equations, Space-time White Noise and Random Fields - Robert C. Dalang, Marta Sanz-Solé #

Abstract: This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). The volume consists of two blocks: the core matter (Chapters 1 to 5) and the appendices (A, B and C). Chapter 1 introduces the subject, with a discussion of isonormal Gaussian processes, space-time white noise, and motivating examples of SPDEs. Chapter 2 presents a theory of stochastic integration with respect to space-time white noise. Chapter 3 deals with SPDEs with additive noise. In Chapter 4, we study a general class of SPDEs, in which additive and multiplicative nonlinearities appear. In Chapter 5, we present a selection of important topics in the theory of SPDEs, that have been the subject of much research over the last twenty years. Appendix A summarises the main results from the theory of stochastic processes and stochastic analysis that are used throughout the book. Appendix B is devoted to a systematic presentation of properties of fundamental solutions and Green’s functions associated to the classical linear differential operators (heat, fractional heat and wave operators). Appendix C is a toolbox section. Each chapter is followed by a “Notes” section, which gives historically important references, original sources and points towards other related important contributions.

17. Dynamic Programming: Finite States - Thomas J. Sargent, John Stachurski #

Abstract: This book is about dynamic programming and its applications in economics, finance, and adjacent fields. It brings together recent innovations in the theory of dynamic programming and provides applications and code that can help readers approach the research frontier. The book is aimed at graduate students and researchers, although most chapters are accessible to undergraduate students with solid quantitative backgrounds.

18. Resources of the Quantum World - Gilad Gour #

Abstract: This book delves into the burgeoning field of quantum resource theories, a novel and vibrant area of research within quantum information science that seeks to unify diverse quantum phenomena under a single framework. By recognizing various attributes of physical systems as “resources,” this approach offers a fresh perspective on quantum phenomena, transforming our understanding and application of concepts such as quantum entanglement, coherence, and more. With a focus on the pedagogical, the book aims to equip readers with the advanced mathematical tools and physical principles needed to navigate and contribute to this rapidly evolving field. It covers a wide range of topics, from the foundational aspects of quantum mechanics and quantum information to detailed explorations of specific resource theories, including entanglement, asymmetry, and thermodynamics. Through rigorous mathematical exposition and a unique axiomatic approach, the book provides deep insights into the operational and conceptual frameworks that underpin quantum resource theories, making it an invaluable resource for graduate students, early-career researchers, and anyone interested in the cutting-edge developments in quantum information science.

19. Funktionalanalysis Teil I - Christoph Bock #

Abstract: Roughly spoken, Functionalanalysis means the study of the category of infinite-dimensional vectorspaces over the field of real or complex numbers, together with their linear maps. In most cases, one further needs a topological structure on such a vectorspace, because then, you can consider the continuous linear maps between such spaces. The name Functionalanalysis is due to the fact, that in the beginning of the theory, the authors wanted to expand Calculus onto functionals of spaces of functions. Functionalanalytical results give the possibility to solve problems in the Theory of (Partial) Differential Equations, in Complex Analysis or in Quantum Mechanics. But the aim of this lines is not to explain the applications. We will discuss the mathematical theory of almost metric spaces, normed vector spaces and algebras, spaces of continuous resp. $p$-integrable functions as well as reflexive and uniformly convex spaces.

We added that, in the case $p \in {]}0,1{[}$, $L^p$ is the completion of the compactly supported continuous functions (with the obvious metric), too. Actually, the proof is the same as in the case $p \in {[}1, \infty{[}$.

20. Algebraic Topology for Data Scientists - Michael S. Postol #

Abstract: This book gives a thorough introduction to topological data analysis (TDA), the application of algebraic topology to data science. Algebraic topology is traditionally a very specialized field of math, and most mathematicians have never been exposed to it, let alone data scientists, computer scientists, and analysts. I have three goals in writing this book. The first is to bring people up to speed who are missing a lot of the necessary background. I will describe the topics in point-set topology, abstract algebra, and homology theory needed for a good understanding of TDA. The second is to explain TDA and some current applications and techniques. Finally, I would like to answer some questions about more advanced topics such as cohomology, homotopy, obstruction theory, and Steenrod squares, and what they can tell us about data. It is hoped that readers will acquire the tools to start to think about these topics and where they might fit in.

21. Discrete and Continuous Weak KAM Theory: an introduction through examples and its applications to twist maps - Maxime Zavidovique #

Abstract: The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear, but where central ideas and results persist. It can therefore serve as a good introduction to (continuous) weak KAM theory. After a general exposition of the general abstract theory, several examples are studied. The last section is devoted to the historical problem of conservative twist maps of the annulus. At the end of the first three Chapters, the relations between the results proved in the discrete setting and the analogous theorems of classical weak KAM theory are discussed. Some key differences are also highlighted between the discrete and classical theory. Those results are new. The text also contains other results never published before, such as the convergence of solutions of discounted equations for degenerate perturbations.

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