Related papers: Elementary real analysis without compactness argum…
This paper's central theme is to prove the existence of an n-algebra whose multiplication cannot be expressed employing any binary operation. Furthermore, to prove if two algebras are not isomorphic, this property does not hold for…
The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and…
We demonstrate how a generic automated theorem prover can be applied to establish the non-orderability of groups. Our approach incorporates various tools such as positive cones, torsions, generalised torsions and cofinal elements.
We study criteria for a ring - or more generally, for a small category - to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to…
This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of G\"ollnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at…
Motivated by Andrews' recent work related to Euler's partition theorem, we consider the set of partitions of an integer $n$ where the set of even parts has exactly $j$ elements, versus the set of partitions of $n$ where the set of repeated…
A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…
Bousquet-M\'elou & Eriksson's lecture hall theorem generalizes Euler's celebrated distinct-odd partition theorem. We present an elementary and transparent proof of a refined version of the lecture hall theorem using a simple bijection…
Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinski\u{\i} lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a…
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…
The binary radix expansion of a real number can be used to code the outcome of any series of coin tosses, a fact that provides an intriguing link between number theory, measure theory and statistical physics. Inspired by this fact, a…
We prove a Tverberg type theorem: Given a set $A \subset \mathbb{R}^d$ in general position with $|A|=(r-1)(d+1)+1$ and $k\in \{0,1,\ldots,r-1\}$, there is a partition of $A$ into $r$ sets $A_1,\ldots,A_r$ with the following property. The…
We present short elementary proofs of the well-known Ruffini-Abel-Galois theorems on insolvability of algebraic equations in radicals. These proofs are obtained from existing expositions by stripping away material not required for the…
It was recently shown by Gross, Hacking, and Keel that, in the absence of frozen indices, a cluster A-variety with generic coefficients is the universal torsor of the corresponding cluster X-variety with corresponding coefficients. We…
The G\"ollnitz-Gordon-Andrews identities generalize the partition identities discovered independently by H. G\"ollnitz and B. Gordon. In this article, we present a commutative algebra proof of the G\"ollnitz-Gordon-Andrews identities. More…
The least $r$-gap, $g_r(\lambda)$, of a partition $\lambda$ is the smallest part of $\lambda$ appearing less than $r$ times. In this article we introduce two new partition functions involving least $r$-gaps. We consider a bisection of a…
The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its…
Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly $j$ different multiples of $r$, for a…
In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His…
Differential calculus is not a unique way to observe polynomial equations such as $a+b=c$. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials $a$, $b$ and $c$ satisfying the equation…