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In the present note a functional calculus $\phi \mapsto \phi(A)$ for self-adjoint definitizable linear relation on Krein spaces is developed. This functional calculus is the proper analogue of $\phi \mapsto \int \phi \, dE$ in the Hilbert…

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts…

In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in several variables. This algorithm is a generalization of a method developed for…

In this work, we will present variants Fixed Point Theorem for the affine and classical contexts, as a consequence of general Brouwer's Fixed Point Theorem. For instance, the affine results will allow working on affine balls, which are…

We study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstelensatz. We study the scheme theoretic properties and…

We show that the coefficients of rational 2-functions are contained in an abelian number field. More precisely, we show that the poles of such functions are poles of order one and given by roots of unity and rational residue.

In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we…

We study the structures of Pfaffian equations and contiguity relations of the hypergeometric function of type $(k+1,k+n+2)$ by using twisted cohomology groups and the intersection form on them. We apply our results to algebraic statistics;…

We give an algorithm to decide whether an algebraic plane foliation F has a rational first integral and to compute it in the affirmative case. The algorithm runs whenever we assume the polyhedrality of the cone of curves of the surface…

The concept of slice regular function over the real algebra $\mathbb{H}$ of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let $\Omega$ be an open subset of $\mathbb{H}$, which intersects…

computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…

We find that all Feynman integrals (FIs), having any number of loops, can be completely determined once linear relations between FIs are provided. Therefore, FIs computation is conceptually changed to a linear algebraic problem. Examples up…

Given an integer $D$ and an ordinary isogeny class of abelian varieties defined over a finite field $\mathbb{F}_q$ with commutative $\mathbb{F}_q$-endomorphism algebra, we provide algorithms for computing all isogenies of degree dividing…

A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives…

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A(\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the space of continuous…

Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary…

Properties of the mappings \begin{align*} C&\mapsto\frac1{(2\pi i)^2}\int_{\Gamma_1}\int_{\Gamma_2}f(\lambda,\mu)\,R_{1,\,\lambda}\,C\, R_{2,\,\mu}\,d\mu\,d\lambda, C&\mapsto\frac1{2\pi i}\int_{\Gamma}g(\lambda)R_{1,\,\lambda}\,C\,…

We provide a new method to calculate the full microlocal description of singularities of Feynman integrals. This is done by associating a unique constructible function to the system of partial differential equations (PDEs) annihilating the…

We introduce a new class of nonlinear equations admitting a representation in terms of Darboux-covariant compatibility conditions. Their special cases are, in particular, (i) the "general" von Neumann equation $i\dot\rho=[H,f(\rho)]$, with…

On a M\"obius surface, as defined by D. Calderbank, we study a variant of the Einstein-Weyl (EW) equation which we call scalar-flat M\"obius EW (sf-MEW). This is a conformally invariant, finite type, overdetermined system of semi-linear…