Related papers: Decomposition of P{\l}onka sums into direct system…
In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new…
The theory of support varieties gives a rich supply of examples of thick subcategories of the stable module category of a finite group algebra. We study direct sum decompositions of such categories. We give examples where there are finer…
In this article we propose an original classification method for unobscured imaging systems unfolded in two dimensions. This classification is based on a study of off-axis properties, and relies on topology and algorithm of real algebraic…
For any field $K$ and for a completely arbitrary graph $E$, we characterize the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it…
A module is said to be \textsf{distributive} if the lattice of its submodules is distributive. A direct sum of distributive modules is called a \textsf{semidistributive} module. In this paper we consider rings $A$ such that all right…
Two methods can be used to calculate explicitly the Killing form on the Lie algebras. The first one is a direct calculation of the traces of the generators in a matrix representation of the algebra, and the second one is the usage of the…
By giving the definition of the sum of a series indexed by a set on which a group acts, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function, and a few other series indexed by $\Z^k$ has an…
A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular…
We count the number of all Rota-Baxter operators on a finite direct sum $A = F\oplus F\oplus \ldots \oplus F$ of fields and count all of them up to conjugation with an automorphism. We also study Rota-Baxter operators on $A$ corresponding…
Scheme-theoretic methods are used to classify ternary quadratic forms with values in line bundles over arbitrary schemes and to canonically determine the isomorphisms between them. The association of a quadratic bundle to its even Clifford…
We classify general systems of polynomial equations with a single solution, or, equivalently, collections of lattice polytopes of minimal positive mixed volume. As a byproduct, this classification provides an algorithm to evaluate the…
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
We give a homological interpretation of the coefficients of the Hilbert series for an algebra associated with a directed graph and its dual algebra. This allows us to obtain necessary conditions for Koszulity of such algebras in terms of…
We introduce an efficient algorithmic procedure for implementing the direct formula that represents the product of splines in the B-spline basis. We first demonstrate the relevance of this direct approach through numerical evidence showing…
Homological algebra is often understood as the translator between the world of topology and algebra. However, this branch of mathematics is worth studying by itself, given that it provides fascinating perspectives about other disciplines,…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an $\omega$-categorical algebra $\mathfrak{A}$. There are $\omega$-categorical groups where this problem is…
Proper relation algebras can be constructed using $\mathbb{Z}/p\mathbb{Z}$ as a base set using a method due to Comer. The cycle structure of such an algebra must, in general, be determined \emph{a posteriori}, normally with the aid of a…
We investigate the arithmetic nature of P-recursive sequences through the lens of their D-finite generating functions. Building on classical tools from differential algebra, we revisit the integrality criterion for Motzkin-type sequences…
Absolute algebras are a new type of algebraic structures, endowed with a meaningful notion of infinite sums of operations without supposing any underlying topology. Opposite to the usual definition of operadic calculus, they are defined as…