Related papers: Inconsistency of Primitive Recursive Arithmetic
This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…
In this note I describe reliability standards for writing and reviewing mathematical papers; these standards are (in my opinion) vital for the progress of mathematics. I give examples of applying the described or other reliability…
We analyze Coquand's game-theoretic interpretation of Peano Arithmetic through the lens of elementary descent recursion. In Coquand's game semantics, winning strategies correspond to infinitary cut-free proofs and cut elimination…
This paper presents a brief survey of the most important and the most remarkable inequalities involving the basic arithmetic functions.
Neither the classical nor intuitionistic logic traditions are perfectly-aligned with the purpose of reasoning about computation, in that neither tradition can permit unconstrained recursive definitions without inconsistency: recursive…
A historical review of the problem of incompleteness in Mathematics since the 20th century is made. The Combinatorial Principle of Paris-Harrington is studied and the way in which it can be codified in the language of Arithmetic.
The aim of this article is to give a rather extensive, and yet nontechnical, account of the birth of the regularity theory for generalized minimal surfaces, of its various ramifications along the decades, of the most recent developments,…
In much discussed work Artemov has recently shown that, for $\mathrm{PA}$, the consistency schema admits a form of uniform verification via selector proofs, despite the unprovability of the corresponding uniform consistency sentence…
The traditional theory of Laplace transformation in its currently prevalent form is unsatisfactory. Its deficiencies can be traced back to a mismatch of the definition intervals of the original function and of the inverse L-transform. A new…
This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
It will be attempted to present a coherent view on the current ideas regarding the origin of the stripe instability. Special emphasis will be put on the problem of how to combine the microscopic pictures, leaning on spin-charge topological…
The notion of slow provability for Peano Arithmetic ($\mathsf{PA}$) was introduced by S.D. Friedman, M. Rathjen, and A. Weiermann. They studied the slow consistency statement $\mathrm{Con}_{\mathsf{s}}$ that asserts that a contradiction is…
Existing performance measures rank delineation algorithms inconsistently, which makes it difficult to decide which one is best in any given situation. We show that these inconsistencies stem from design flaws that make the metrics…
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, root-square means, etc. Some new means recently studied are also presented. Different kinds of refinement of inequalities among these means are…
The aim of this work is to improve Wilker inequalities near the origin and {\pi}/2.
This is the first of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression…
The aim of our paper is to formulate and solve problems concerning multitime multiple recurrence equations. We discuss in detail the generic properties and the existence and uniqueness of solutions. Among the general things, we discuss in…
Measures of irrationality are a numerical way of quantifying how far a given variety is from being rational (or rationally connected, uniruled, etc.). In the last two decades, there has been renewed interest in the study of these…