相关论文: Diophantine and Non-Diophantine Arithmetics: Opera…

Computing is a high-level process of a physical system. Recent interest in non-standard computing systems, including quantum and biological computers, has brought this physical basis of computing to the forefront. There has been, however,…

Deep learning has generated diverse perspectives in astronomy, with ongoing discussions between proponents and skeptics motivating this review. We examine how neural networks complement classical statistics, extending our data analytical…

What science does, what science could do, and how to make science work? If we want to know the answers to these questions, we need to be able to uncover the mechanisms of science, going beyond metrics that are easily collectible and…

Mathematics is an essential element of physics problem solving, but experts often fail to appreciate exactly how they use it. Math may be the language of science, but math-in-physics is a distinct dialect of that language. Physicists tend…

Many real-world dynamic systems, both natural and artificial, are understood to be performing computations. For artificial dynamic systems, explicitly designed to perform computation - such as digital computers - by construction, we can…

Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's `machine model' which formalises computing with real numbers. A fundamental…

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from…

Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are…

The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be…

The goal of the work is to take on and study one of the fundamental tasks studying Bidiophantine polygons (let us call a polygon Diophantine, if the distance between each two vertex of those is expressed by a natural number and we say that…

In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…

Curiously overlooked in physics is its dependence on the transmission of numbers. For example the transmission of numerical clock readings is implicit in the concept of a coordinate system. The transmission of numbers and other logical…

This book is not meant to be another compendium of select inequalities, nor does it claim to contain the latest or the slickest ways of proving them. This project is rather an attempt at describing how most functional inequalities are not…

Pre-college mathematics modeling instruction often frames mathematics as being separated from reasoning about the real world -- and commonly treats reasoning mathematically and reasoning about the real-world context as separate stages of a…

We describe a theory of finite sets, and investigate the analogue of Dedekind's theory of natural number systems (simply infinite systems) in this theory. Unlike the infinitary case, in our theory, natural number systems come in differing…

Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with…

Standard economic theory uses mathematics as its main means of understanding, and this brings clarity of reasoning and logical power. But there is a drawback: algebraic mathematics restricts economic modeling to what can be expressed only…

This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions.…

We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…

In any other circumstance, it might make sense to define the extent of the terrain (Data Science) first, and then locate and describe the landmarks (Principles). But this data revolution we are experiencing defies a cadastral survey. Areas…