相关论文: Wetzel's Problem, Paul Erdos, and the Continuum Hy…

The present work focuses on the study of the renowned Collatz conjecture, also known as the $3x +1$ problem. The distinguished analysis approach lies on the dynamics of an iterative map in binary form. A new estimation of the enlargement of…

The full causal ladder of spacetimes is constructed, and their updated main properties are developed. Old concepts and alternative definitions of each level of the ladder are revisited, with emphasis in minimum hypotheses. The implications…

We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…

This paper intends to survey the vast literature devoted to a problem posed by Wilf in 1978 which, despite the attention it attracted, remains unsolved. As it frequently happens with combinatorial problems, many researchers who got involved…

We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

A brief review of the present-day development of cosmology is presented for mixed physical audience. The universe history is briefly described. Unsolved problems are discussed, in particular, the mystery of the cosmological constant and…

This article discusses epistemological problems in the philosophy of mathematics and issues concerning the reliability of the mathematical literature.

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. This problem has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial…

We present a variation of the water puzzle, which is related to a simple model of marginal utility. The problem has an intriguing solution and can be extended in several directions.

The article presents the detailed analysis of the watch paradox. It is shown that it arose because of unjustified, as it turned out, identification of watch readings at the moment of its return with the time read by it.

We believe the following three ingredients are enough to explain the mystery of the arrow of time: (1). equations of dynamics of gas molecules, (2). chaotic instabilities of the equations of dynamics, (3). unavoidable perturbations to the…

In this small paper we bring together various open problems on geometric multidimensional continued fractions.

We connect a possible solution for the ``cosmological constant problem'' to the existence of a (postulated) conformal fixed point in a fundamental theory. The resulting cosmology leads to quintessence, where the present acceleration of the…

This is a survey of the author's recent results on the Kadison and Halmos similarity problems and the closely connected notion of ``length'' of an operator algebra.

We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of…

We provide a list of (mainly unsolved) problems in ordered and orderable groups. These were originally compiled 10 years ago by the last two authors. New problems have been added to the list. Progress on some of these is noted and…

In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…

In this paper, we consider a conjecture of Erdos and Rosenfeld and a conjecture of Ruzsa when the number is an almost square. By the same method, we consider lattice points of a circle close to the x-axis with special radii.

A detailed and rigorous analysis of G\"odel's proof of his first incompleteness theorem is presented. The purpose of this analysis is two-fold. The first is to reveal what G\"odel actually proved to provide a clear and solid foundation upon…

This note gives results on the existence of semi-continuous solutions of a Fredholm integral equation of the second kind using Tarski's fixed point theorem.