Related papers: The Convolution Ring of Arithmetic Functions and S…

We prove that star-like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non-autonomous polynomial system or as heteroclinic solutions of a…

It is known that a real function $f$ is convex if and only if the set $$\mathrm{E}(f)=\{(x,y)\in\mathbb{R}\times\mathbb{R};\ f(x)\leq y\},$$ the epigraph of $f$ is a convex set in $\mathbb{R}^2$. We state an extension of this result for…

We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory. Our main result is that the ring of invariants of a splitting algebra under the symmetric group…

Some of the most common mathematical models in biology, chemistry, physics, and engineering, are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand sides. Inspired by notions and results that…

We study a nonlinear analogue of additive commutators, known as \textit{polynomial commutators}, defined by $p(ab) - p(ba)$ for a polynomial $p \in F[x]$ and elements $a, b$ in an algebra $R$ over a field $F$. Originally introduced by…

For a complete lattice $L$ and a relational structure $\mathfrak{X}=(X,(R_i)_I)$, we introduce the convolution algebra $L^{\mathfrak{X}}$. This algebra consists of the lattice $L^X$ equipped with an additional $n_i$-ary operation $f_i$ for…

We study the equivariant cohomology classes of torus-equivariant subvarieties of the space of matrices. For a large class of torus actions, we prove that the polynomials representing these classes (up to suitably changing signs) are…

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect…

In this paper we derive generalizations of different properties of monic polynomial families of binomial type, i.e. families of monic polynomials, for which the binomial theorem holds $$ p_n(\alpha+\beta)=\sum_{k=0}^n…

Let $\mathbb K$ be an algebraically closed field of characteristic zero. Let $V$ be a module over the polynomial ring $\mathbb K[x,y]$. The actions of $x$ and $y$ determine linear operators $P$ and $Q$ on $V$ as a vector space over $\mathbb…

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the…

Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…

Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution.…

Notions of computation can be modelled by monads. Algebraic effects offer a characterization of monads in terms of algebraic operations and equational axioms, where operations are basic programming features, such as reading or updating the…

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study…

We show that the space of logarithmic intertwining operators among logarithmic modules for a vertex operator algebra is isomorphic to the space of 3-point conformal blocks over the projective line. This is considered as a generalization of…

The purpose of this manuscript is to derive two distinct integral representations of the generalized polylogarithm using two different techniques. The first approach involves the Dirichlet series and its Laplace representation, which leads…

We prove the following results: let x,y be (n,n) complex matrices such that x,y,xy have no eigenvalue in ]-infinity,0] and log(xy)=log(x)+log(y). If n=2, or if n>2 and x,y are simultaneously triangularizable, then x,y commute. In both cases…

Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any…

Let $\mathcal C$ be the category of finite graphs. Lov\`{a}sz shows that the semi-ring of isomorphism classes of $\mathcal C$ (with coproduct as sum, and product as multiplication) is embedded into the direct product of the semi-ring of…