Related papers: Periods And Real Numbers
This book introduces the mathematical foundations and techniques that lead to the development and analysis of many of the algorithms that are used in machine learning. It starts with an introductory chapter that describes notation used…
This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck…
Periodic orbits and cycles, respectively, play a significant role in discrete- and continuous-time dynamical systems (i.e. maps and flows). To succinctly describe their shifts when the system is applied perturbation, the notions of…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
Main purposes of the paper are followings: 1) To show examples of the calculations in domain of QFT via ``derivative rules'' of an expert system; 2) To consider advantages and disadvantage that technology of the calculations; 3) To reflect…
A large number of the classical texts dealing with Fourier series more or less state that the hypothesis of periodicity is required for pointwise convergence. In this paper, we highlight the fact that this condition is not necessary.
The goal of this paper is to describe the connections between Perron-Frobenius theory and vertex maps on graphs. In particular, it is shown how Perron-Frobenius theory gives results about the sets of integers that can arise as periods of…
In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…
This article aims to explain essential elements of perturbation theory and their conceptual underpinnings. It is not meant as a summary of popular perturbation methods, though some illustrative examples are given to underline the main…
This primer is intended as an introduction to differential forms, a central object in modern mathematical physics, for scientists and engineers.
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…
A new definition of a real number is that it is a rule which says Yes or No based on whether the real number ought to be in a given rational interval. This is a teaser paper for formalizing, exploring, and generalizing this definition. The…
The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals domains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a…
Preliminary version of a book on univariate real analysis, with 14 chapters and 2 appendices. 1. Real numbers; 2. Limits of real sequences; 3. Series; 4. Limits of real functions. 5. Elementary functions; 6. Continuous functions; 7.…
After giving an overview of the existing theory regarding the periods of sequences defined by linear recurrences over finite fields, we give explicit descriptions of the sets of periods that arise if one considers all sequences over…
This book is devoted to the problem of sequential probability forecasting, that is, predicting the probabilities of the next outcome of a growing sequence of observations given the past. This problem is considered in a very general setting…
This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from…
A modest aim of this pedagogical presentation is to analyze, critically, certain fundamental physical concepts to illustrate the physical principles behind the special theory of relativity and, hence, to also illustrate the limitations of…
This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and…
Lecture notes written for a one-semester course in mathematical relativity aimed at mathematics and physics students. Not meant as an introduction to general relativity, but rather as a complementary, more advanced text.