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Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces…
We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in…
We generalize the notion of semi-universality in the classical deformation problems to the context of derived deformation theories. A criterion for a formal moduli problem to be semi-prorepresentable is produced. This can be seen as an…
We present a modular function-based approach to explaining, for primes larger than 3, the exponents that appear in the prime decomposition of the order of the monster finite simple group.
We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal…
Many formal languages include binders as well as operators that satisfy equational axioms, such as commutativity. Here we consider the nominal language, a general formal framework which provides support for the representation of binders,…
The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb Q[x_1, \dots, x_n]$ to a corresponding ideal in $\mathbb F_p[x_1,\dots, x_n]$ where $p$ is a prime number; in other words, the…
By using the elementary symmetric polynomials and some results of number theory, we solve the well known problem of Lehmer on Euler's totient function. As application, we obtain a new characterization of prime numbers.
We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this…
We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to…
The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
One familiar with the Euler zeta function, which established the remarkable relationship between the prime and composite numbers, might naturally ponder the results of the application of this special function in cases where there is no…
Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, Lehmer's generalized Euler numbers are studied to give certain…
We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular…
Based upon properties of ordinal length, we introduce a new class of modules, the binary modules, and study their endomorphism ring. The nilpotent endomorphisms form a two-sided ideal, and after factoring this out, we get a commutative…
The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give…