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D. V. Osipov

Publications and source records attributed to D. V. Osipov.

At least 19 recordsLinked to original sources
arXiv2025

Information Spreading in Random Graphs Evoving by Norros-Reittu Model

The paper is devoted to the spreading of a message within the random graph evolving by the Norros-Reittu preferential attachment model. The latter model forms random Poissonian numbers of edges between newly added nodes and existing ones. For a pre-fixed time $T^*$, the probability mass functions of the number of nodes obtained the message and the total number of nodes in the graph, as well as the distribution function of their ratio are derived. To this end, the success probability to disseminate the message from the node with the message to the node without message is proved. The exposition is illustrated by the simulation study.

math.ST↗
arXiv2024

Local analog of the Deligne-Riemann-Roch isomorphism for line bundles in relative dimension $1$

We prove a local analog of the Deligne-Riemann-Roch isomorphism in the case of line bundles and relative dimension $1$. This local analog consists in computation of the class of $12$th power of the determinant central extension of a group ind-scheme $\mathcal G$ by the multiplicative group scheme over $\mathbb Q$ via the product of $2$-cocyles in the second cohomology group. These $2$-cocycles are the compositions of the Contou-Carrère symbol with the $\cup$-product of $1$-cocycles. The group ind-scheme $\mathcal G$ represents the functor which assigns to every commutative ring $A$ the group that is the semidirect product of the group $A((t))^*$ of invertible elements of $A((t))$ and the group of continuous $A$-automorphisms of $A$-algebra $A((t))$. The determinant central extension naturally acts on the determinant line bundle on the moduli stack of geometric data (proper quintets). A proper quintet is a collection of a proper family of curves over $\mathop{\rm Spec} A$, a line bundle on this family, a section of this family, a relative formal parameter at the section, a formal trivialization of the bundle at the section that satisfy further conditions.

math.AG↗
arXiv2023

Formal Bott-Thurston cocycle and part of a formal Riemann-Roch theorem

The Bott-Thurston cocycle is a $2$-cocycle on the group of orientation-preserving diffeomorphisms of the circle. We introduce and study a formal analog of Bott-Thurston cocycle. The formal Bott-Thurston cocycle is a $2$-cocycle on the group of continuous $A$-automorphisms of the algebra $A((t))$ of Laurent series over a commutative ring $A$ with values in the group $A^*$ of invertible elements of $A$. We prove that the central extension given by the formal Bott-Thurston cocycle is equivalent to the $12$-fold Baer sum of the determinantal central extension when $A$ is a $\mathbb Q$-algebra. As a consequence of this result we prove a part of new formal Riemann-Roch theorem. This Riemann-Roch theorem is applied to a ringed space on a separated scheme $S$ over $\mathbb Q$, where the structure sheaf of the ringed space is locally on $S$ isomorphic to the sheaf ${\mathcal O}_S((t))$ and the transition automorphisms are continuous. Locally on $S$ this ringed space corresponds to the punctured formal neighbourhood of a section of a smooth morphism to $U$ of relative dimension $1$, where an open subset $U \subset S$.

math.AG↗
arXiv2022

Central extensions and Riemann-Roch theorem on algebraic surfaces

We study canonical central extensions of the general linear group of the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. By these central extensions and adelic transition matrices of a rank $n$ locally free sheaf of ${\mathcal O}_X$-modules we obtain the local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf ${\mathcal O}_X^n$. Two various calculations of this difference lead to the Riemann-Roch theorem on $X$ (without the Noether formula).

math.AG↗
arXiv2020

Harmonic analysis on the rank-$2$ value group of a two-dimensional local field

In this work we construct harmonic analysis on free Abelian groups of rank $2$, namely: we construct and investigate spaces of functions and distributions, Fourier transforms, actions of discrete and extended discrete Heisenberg groups. In case of the rank-$2$ value group of a two-dimensional local field with finite last residue field we connect this harmonic analysis with harmonic analysis on the two-dimensional local field, where the latter harmonic analysis was constructed in earlier works by the authors.

math.NT↗
arXiv2019

Lax Pairs for Linear Hamiltonian Systems

In the paper Lax pairs for linear Hamiltonian systems of differential equations are constructed. In particular, Gröbner bases are used for the computations. It is proved that the maps which appear in the construction of Lax pairs are Poisson. Various properties of first integrals of the system which are obtained from the Lax pairs are investigated.

math-ph↗
arXiv2018

Arithmetic surfaces and adelic quotient groups

We explicitly calculate an arithmetic adelic quotient group for a locally free sheaf on an arithmetic surface when the fiber over the infinite point of the base is taken into account. The calculations are presented via a short exact sequence. We relate the last term of this short exact sequence with the projective limit of groups which are finite direct products of copies of one-dimensional real torus and are connected with first cohomology groups of locally free sheaves on the arithmetic surface.

math.AG↗
arXiv2017

On adelic quotient group for algebraic surface

We calculate explicitly an adelic quotient group for an excellent Noetherian normal integral two-dimensional separated scheme. An application to an irreducible normal projective algebraic surface over a field is given.

math.AG↗
arXiv2017

Second Chern numbers of vector bundles and higher adeles

We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic $K$-theory and depends on the canonical $\mathbb{Z}$-torsor of a locally linearly compact $k$-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.

math.AG↗
arXiv2015

Tangent space to Milnor $K$-groups of rings

We prove that the tangent space to the $(n+1)$-th Milnor $K$-group of a ring $R$ is isomorphic to group of $n$-th absolute Kähler differentials of $R$ when the ring $R$ contains $\frac{1}{2}$ and has sufficiently many invertible elements. More precisely, the latter condition is that $R$ is weakly $5$-fold stable in the sense of Morrow.

math.AG↗
arXiv2015

Discrete Heisenberg group and its automorphism group

In this note we give more easy and short proof of a statement previously proved by P. Kahn that the automorphism group of the discrete Heisenberg group ${\rm Heis}(3, \mathbb{Z}) $ is isomorphic to the group $ (\mathbb{Z} \oplus \mathbb{Z}) \rtimes GL(2,\mathbb{Z})$. The method which we suggest to construct this isomorphism gives far more transparent picture of the structure of the automorphism group of the group ${\rm Heis}(3, \mathbb{Z}) $.

math.GR↗
arXiv2013

Noncommutative reciprocity laws on algebraic surfaces: a case of tame ramification

We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a finite last residue field the constructed local central extension is isomorphic to a central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by M. Kapranov.

math.AG↗
arXiv2013

Unramified two-dimensional Langlands correspondence

In this paper we describe the unramified Langlands correspondence for two-dimensional local fields, we construct a categorical analogue of the unramified principal series representations and study its properties. The main tool for this description is the construction of a central extension. For this (and other) central extension we prove noncommutative reciprocity laws (i.e. the splitting of the central extensions over some subgroups) for arithmetic surfaces and projective surfaces over a finite field. These reciprocity laws connect central extensions which are constructed locally and globally.

math.AG↗
arXiv2011

Harmonic analysis and the Riemann-Roch theorem

This paper is a continuation of papers: arXiv:0707.1766 [math.AG] and arXiv:0912.1577 [math.AG]. Using the two-dimensional Poisson formulas from these papers and two-dimensional adelic theory we obtain the Riemann-Roch formula on a projective smooth algebraic surface over a finite field.

math.AG↗
arXiv2010

Harmonic analysis on local fields and adelic spaces II

This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.

math.AG↗
arXiv2008

Harmonic analysis on local fields and adelic spaces I

We develop a harmonic analysis on objects of some category $C_2$ of infinite-dimensional filtered vector spaces over a finite field. It includes two-dimensional local fields and adelic spaces of algebraic surfaces defined over a finite field. The main result is the theory of the Fourier transform on these objects and two-dimensional Poisson formulas.

math.AG↗
arXiv2007

Adeles on n-dimensional schemes and categories C_n

We consider categories C_n, which are very close to the iterated version of Ind Pro limits introduced by A.Bejlinson in his article "How to glue perverse sheaves". We prove that the adelic space on n-dimensional noetherian scheme is an object of C_n.

math.AG↗