arXiv subjects

Andronick Arutyunov

Publications and source records attributed to Andronick Arutyunov.

9 recordsLinked to original sources

arXiv2026

Interpolations: a linear algebra approach

This paper presents a unified linear-algebraic framework for interpolation problems, viewing different standart problems (Lagrange polynomials, Hermite splines, and trigonometric interpolation) with the help of dual spaces. By representing interpolation conditions as linear functionals in the dual of a finite-dimensional function space, we establish a general existence and uniqueness theorem. We derive explicit formulas for basis functions dual to given interpolation schemes, enabling efficient construction of interpolants as linear combinations of these bases. The approach is extended to trigonometric interpolation, where we provide closed-form expressions for dual bases and analyze degeneracy conditions. We also address practical scenarios involving heterogeneous sensors, where function values and derivatives are specified at distinct points, and propose a corresponding interpolation scheme. The method is systematic, dimension-aware, and decouples the analysis of interpolation conditions from the choice of basis functions. We cosider examples from kinematics and periodic function interpolation demonstrate the versatility of the approach, suggesting applications in trajectory planning, signal processing, and beyond.

math.NA↗

arXiv2026

Bornological Metrics on Groups

Let $G$ be a countable group. We study left-invariant metrics on $G$ that are not necessarily proper, introducing the notion of a \emph{bornological metric}: a metric $ρ$ such that for every $C>0$ there exists $S_C>0$ with the property that $ρ(x,y)<C$ implies $ρ(gx,gy)<S_C$ for all $g\in G$. We show that each coarse equivalence class of bornological metrics is determined by a bornology on $G$, and that every such class contains a canonical left-invariant representative. The metrizability of a bornology is characterized in terms of countable generation of the associated coarse structure, and a criterion for strong $G$-invariance of a coarse structure is established. As an application, we construct families of improper left-invariant metrics on finitely generated groups that are pairwise non-equivalent and not coarsely equivalent to any proper left-invariant metric.

math.GR↗

arXiv2025

Description of the structure of DG--algebras via characters on a groupoid

We construct a description of graded derivations in group algebras. Using this result for arbitrary graduation of the group algebra, we describe all possible structures of DG algebras. The corresponding examples are given. The description is given in terms of characters on a groupoid analogous to the groupoid of inner action.

math.RA↗

arXiv2023

The derivation problem for one type of bimodules over group algebras

The derivation problem is a familiar one concerning group algebras, particularly $L_1(G)$ and von Neumann algebras. In this paper, we study the Banach bimodule $\ell_p(G)$, which is generated by the $\ell_p$ norm over a specific class of groups with well-organized conjugacy classes. For this case, we will demonstrate that all $\ell_p(G)$ derivations are inner.

math.FA↗

arXiv2023

Grading Structure for Derivations of Group Algebras

In this paper we give a way of equipping the derivation algebra of a group algebra with the structure of a graded algebra. The derived group is used as the grading group. For the proof, the identification of the derivation with the characters of the adjoint action groupoid is used. These results also allow us to obtain the analogous structure of a graded algebra for outer derivations. A non-trivial graduation is obtained for all groups that are not perfect.

math.CO↗

arXiv2022

A combinatorial view on derivations in bimodules

This paper is devoted to derivations in bimodules over group rings using previously proposed methods which are related to character spaces over groupoids. The theorem describing the arising spaces of derivations is proved. We consider some examples, in particular the case of $(σ, τ)$-derivations.

math.RA↗

arXiv2021

Derivations of group rings for finite and FC groups

In this paper we establish decomposition theorems for derivations of group rings. We provide a topological technique for studying derivations of a group ring $A[G]$ in case $G$ has finite conjugacy classes. As a result, we describe all derivations of algebra $A[G]$ for the case when $G$ is a finite group, or $G$ is an FC-group. In addition, we describe an algorithm to explicitly calculate all derivations of a group ring $A[G]$ in case $G$ is finite. As examples, derivations of $\mathbb{Z}_4[S_3]$ and $\mathbb{F}_{2^m}D_{2n}$ are considered.

math.RA↗

arXiv2020

On $(σ,τ)$-derivations of group algebra as category characters

For the space of $(σ,τ)$-derivations of the group algebra $ \mathbb{C} [G] $ of discrete countable group $G$, the decomposition theorem for the space of $(σ,τ)$-derivations, generalising the corresponding theorem on ordinary derivations on group algebras, is established in an algebraic context using groupoids and characters. Several corollaries and examples describing when all $(σ,τ)$-derivations are inner are obtained. Considered in details cases on $(σ,τ)-$nilpotent groups and $(σ,τ)$-$FC$ groups.

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arXiv2018

Cohomology of n-categories and derivations in group algebras

This work represents the concept of an n-groupoid $ Γ^n $ and n-characters $ χ_n $ on n-groupoids as complex-valued maps from spaces of different classes of morphisms satisfying the condition $ χ_n (ψ\circ_k φ) = χ_n (ψ) + χ_n (φ) $ for any possible compositions. A sequence of spaces of n-characters and morphisms between them is constructed and its accuracy is shown. This construction has important application for describing the derivations in a group algebras. In particular, this approach allows us to study the algebra of external derivations from a new point of view, and also to construct some interesting examples. The work was carried out under the guidance of Arutyunov A. A.. And it is based on the Mishchenko A. S. ideas.

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