Related papers: Dismal Arithmetic
We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…
Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored. Such objects are ubiquitous in exact computing, and so naturally we would like to have efficient algorithms to handle them. However, with this compact…
This introduction to arithmetic coding is divided in two parts. The first explains how and why arithmetic coding works. We start presenting it in very general terms, so that its simplicity is not lost under layers of implementation details.…
The present work has been designed for students in secondary school and their teachers in mathematics. We will show how with the help of our knowledge of number systems we can solve problems from other fields of mathematics for example in…
Digital System Research has pioneered the mathematics and design for a new class of computing machine using residue numbers. Unlike prior art, the new breakthrough provides methods and apparatus for general purpose computation using several…
Non-linear state estimation and some related topics, like parametric estimation, fault diagnosis, and perturbation attenuation, are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic…
In this paper, we consider a set of similar triangles with parallel sides, along with a set of points in the plane. It turns out that the set $\mathbb{R}_2= \{\pm <x >=\pm (x^2,x,1); x\in\mathbb{R} \}$ describes this set of triangles quite…
Using kicked differential equations of motion with derivatives of noninteger orders, we obtain generalizations of the dissipative standard map. The main property of these generalized maps, which are called fractional maps, is long-term…
We define S(um)anD(ifference) numbers as ordered pairs $(m,\, m+\Delta)$ such that the digital-sum $DS(m(m+\Delta))=\Delta.$ We consider both the decimal and the binary case. If both $m$ and $m+\Delta$ are prime numbers, we refer to SanD…
This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural…
In this paper we deal with a classical problem in elementary number theory, namely repeating decimals. We show how the digits of the period of the decimal representation of any fraction $\frac{k}{m}$, where $k$ and $m$ are positive integers…
Based on a class of associative algebras with zero-divisors which are called real-like algebras by us, we introduce a way of defining automatic differentiation and present different ways of doing automatic differentiation to compute the…
We consider the positive divisors of a natural number that do not exceed its square root, to which we refer as the {\it small divisors\/} of the natural number. We determine the asymptotic behavior of the arithmetic function that adds the…
Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number…
We introduce $p$-derivations and give a few basic ways in which they act like derivatives by numbers.
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…
We discuss several ways of illustrating fundamental concepts in statistical and thermal physics by considering various models and algorithms. We emphasize the importance of replacing students' incomplete mental images by models that are…
We investigate what arithmetic would look like if carry digits into other digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 = 0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45, 47, ... .
In this work we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on…
Our introductory classes in statistics and data science use too much mathematics. The key causal effect which our students want our classes to have is to improve their future performance and opportunities. The more professional their…