相关论文: Can our number system be improved?

The performance of basis sets made of numerical atomic orbitals is explored in density-functional calculations of solids and molecules. With the aim of optimizing basis quality while maintaining strict localization of the orbitals, as…

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…

We define and study a numerical-range analogue of the notion of spectral set. Among the results obtained are a positivity criterion and a dilation theorem, analogous to those already known for spectral sets. An important difference from the…

You measure the value of a quantity x for a number of systems (cells, molecules, people, chunks of metal, DNA vectors, etc.). You repeat the whole set of measures in different occasions or assays, which you try to design as equal to one…

We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…

An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.

A numeration system originally implies a digitization of real numbers, but in this paper it rather implies a compactification of real numbers as a result of the digitization. By definition, a numeration system with $G$, where $G$ is a…

Like many other voting systems, Majority Judgement suffers from the weaknesses of the underlying mathematical model: Elections as problem of choice or ranking. We show how the model can be enhanced to take into account the complete process…

Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of…

In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found,…

Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…

The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…

The use of the fundamental constants for the definition of the most important measurement units of the International System, was considered a good solution to found it on more solid bases. From further analysis, this solution was found…

Norms with sanctions have been widely employed as a mechanism for controlling and coordinating the behavior of agents without limiting their autonomy. The norms enforced in a multi-agent system can be revised in order to increase the…

We propose a notion of a generalized order, which can be used for the notion of a strict partial order. We introduce a weak order to replace the usual weak order defined from a strict partial order. In a constructive setting, that usual…

As the field of recommender systems has developed, authors have used a myriad of notations for describing the mathematical workings of recommendation algorithms. These notations ap-pear in research papers, books, lecture notes, blog posts,…

Recent studies have shown that the majority of published computational models in systems biology and physiology are not repeatable or reproducible. There are a variety of reasons for this. One of the most likely reasons is that given how…

This paper questions the generally accepted assumption that one can make a random choice that is independent of the rest of the universe. We give a general description of any setup that could be conceived to generate random numbers. Based…

Though Large Language Models (LLMs) have shown remarkable abilities in mathematics reasoning, they are still struggling with performing numeric operations accurately, such as addition and multiplication. Numbers can be tokenized into tokens…

Previous research has found that introductory physics students perform far better on numeric problems than on otherwise equivalent symbolic problems. This paper describes a framework to explain these differences developed by analyzing…