arXiv subjects

Ahmet A. Husainov

Publications and source records attributed to Ahmet A. Husainov.

18 recordsLinked to original sources

arXiv2026

Non-Abelian homology and homotopy colimit of classifying spaces for a diagram of groups

This paper considers non-Abelian homology groups of a group diagram introduced as homotopy groups of a simplicial change. We prove a theorem stating that the non-Abelian homology groups of a group diagram are isomorphic to the homotopy groups of the homotopy colimit of a classifying space diagram, with the dimension shifted by 1. Bousfield and Kan proved an isomorphism between the homotopy groups of an Abelian simplicial group and the homology groups of this simplicial group. We generalize this to non-Abelian simplicial groups. We also develop a method for finding a non-zero homotopy group of smallest dimension for the homotopy colimit of classifying spaces. For a group diagram over a free category with a zero colimit, we obtain a criterion for the isomorphism of the first non-Abelian and Abelian homology groups.

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arXiv2026

Cech cohomology of partially ordered sets

The article is devoted to a comparison of the Čech cohomology with the coefficients in a presheaf of Abelian groups and the topos cohomology of the sheaf generated by this presheaf for a poset with the Aleksandrov topology. The article consists of three parts. The first part provides information from the theory of cohomology of small categories and cohomology of simplicial sets with systems of coefficients. The second part is devoted to Laudal's Theorem stating that covering cohomology for an arbitrary topological space with coefficients in the presheaf of Abelian groups is isomorphic to the derived limit functors. The third part presents the main results. The criterion for the invariance of cohomology groups of small categories when passing to the inverse image leads to necessary and sufficient isomorphism conditions for the Čech cohomology of an arbitrary presheaf and the topos cohomology of the sheaf generated by this presheaf. In particular, for a finite poset, these conditions reduce to the acyclicity of the upper secrions of Dedekind-MacNeille cuts having a non-empty lower section, and the verification of these conditions is algorithmically computable.

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arXiv2024

Homotopy cofinality for Non-Abelian homology of group diagrams

We prove that a homotopy cofinal functor between small categories induces a weak equivalence between homotopy colimits of pointed simplicial sets. This is used to prove that the non-Abelian homology of a group diagram is isomorphic to the homology of its inverse image under a homotopy cofinal functor. This also made it possible to establish that the non-Abelian homology of group diagrams are invariant under the left Kan extension along virtual discrete cofibrations. With the help of these results, we have constructed a non-Abelian Gabriel-Zisman homology theory for simplicial sets with coefficients in group diagrams. Moreover, we have generalized this theory to presheaves over an arbitrary small category, which play the role of simplicial sets. Sufficient conditions are found for the isomorphism of non-Abelian homology of presheaves over a small category with coefficients in group diagrams. It is proved that the non-Abelian homology of the factorization category for the small category is isomorphic to the Baues-Wirsching homology. A condition is found under which a functor between small categories induces an isomorphism of non-Abelian Baues-Wirsching homology.

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arXiv2023

Homology and cohomology of cubical sets with coefficients in systems of objects

This paper continues the research of the author on the homology of cubical and semi-cubical sets with coefficients in systems of objects. The main result is the theorem that the homology of cubical sets with coefficients in contravariant systems in an Abelian category with exact coproducts is isomorphic to the left satellites of a colimit functor. This made it possible to prove a number of the following new assertions, presented in the paper, about the homology and cohomology of cubical sets with coefficients in systems of objects. These homology are invariant under morphism between cubical sets when passing to the direct image of the system of coefficients. There is a criterion for the invariance of these homologies when passing to the inverse image. These homology generalize the singular cubical homology with local coefficients and the homology of semi-cubical sets with coefficients in contravariant systems. There is a spectral sequence for colimit homologies of cubical sets with coefficients in contravariant systems. The weak equivalence of cubical sets induces an isomorphism of homology with local systems. For a morphism of cubical sets whose inverse fiber morphisms are weak equivalences, there exists a spectral sequence for homology with local systems converging to the homology of the domain of this morphism. The homology of small category with coefficients in a diagram can be calculated as cubical homology. The Baues-Wirsching cohomologies with coefficients in natural systems are isomorphic to cubical cohomologies with coefficients in covariant systems.

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arXiv2023

Homology groups of cubical sets

The paper is devoted to homology groups of cubical sets with coefficients in contravariant systems of Abelian groups. The study is based on the proof of the assertion that the homology groups of the category of cubes with coefficients in the diagram of Abelian groups are isomorphic to the homology groups of normalized complex of the cubical Abelian group corresponding to this diagram. The main result shows that the homology groups of a cubical set with coefficients in a contravariant system of Abelian groups are isomorphic to the values of left derived functors of the colimit functor on this contravariant system. This is used to obtain the isomorphism criterion for homology groups of cubical sets with coefficients in contravariant systems, and also to construct spectral sequences for the covering of a cubical set and for a morphism between cubical sets. This version of the preprint contains a refinement of the proof of Lemma 3.2 corresponding to Lemma 4 from the paper of the same name published in the journal Appl. Category. Struct. 27, 199-216 (2019).

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arXiv2023

Criteria for preserving the category cohomology for the inverse image

The article investigates the question of under what conditions a functor between small categories preserves cohomology groups when passing to the inverse image. For example, it is known that the left adjoint functor preserves the category cohomology with local coefficients or the category cohomology constructed as derived of the limit functor. We give counterexamples showing that the left adjoint functor may not preserve the Baues-Wirsching, Hochschild-Mitchell, and Thomason cohomology. To solve the arising problems, we propose necessary and sufficient conditions for the invariance of these types of cohomology under the transition to the inverse image of a functor between small categories. Moreover, we generalize these cohomology of small categories and find similar criteria for the obtained generalization.

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arXiv2018

Optimum Depth of the Bounded Pipeline

The paper is devoted to studying the performance of a computational pipeline, the number of simultaneously executing stages of which at each time is bounded from above by a fixed number. A look at the restriction as a structural hazard makes it possible to construct an analytical model for calculating the processing time of a given input data amount. Using this model, led to a formula for calculating the optimal depth of a bounded pipeline for a given volume of input data. The formula shows that the optimal depth can get large changes for small changes in the amount of data. To eliminate this disadvantage and to obtain a more convenient formula for optimal depth, a pipeline with a single random hazard is constructed, the mathematical expectation of a random value of the processing time of which approximates the analytical model of the bounded pipeline. In addition, a pipeline with two hazards has been built, the analytical model of which allowed obtaining formulas for calculating the optimal depth of a bounded pipeline with restart for a given amount of data. To check whether the proposed analytical models are consistent with the experiments to calculate the processing time, two methods of computer simulation of bounded pipelines are used, the first of which is constructed as a multi-threaded application, and the second is based on the theory of free partially commutative monoids.

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arXiv2013

Category of asynchronous systems and polygonal morphisms

A weak asynchronous system is a trace monoid with a partial action on a set. A polygonal morphism between weak asynchronous systems commutes with the actions and preserves the independence of events. We prove that the category of weak asynchronous systems and polygonal morphisms has all limits and colimits.

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arXiv2013

Homology and Bisimulation of Asynchronous Transition Systems and Petri Nets

Homology groups of labelled asynchronous transition systems and Petri nets are introduced. Examples of computing the homology groups are given. It is proved that if labelled asynchronous transition systems are bisimulation equivalent, then they have isomorphic homology groups. A method of constructing a Petri net with given homology groups is found.

cs.LO↗

arXiv2013

Time Estimation Model of Concurrent Computing Systems

We consider an asynchronous system with transitions corresponding to the instructions of a computer system. For each instruction, a runtime is given. We propose a mathematical model, allowing us to construct an algorithm for finding the minimum time of the parallel process with a given trace. We consider a problem of constructing a parallel process which transforms the initial state to given and has the minimum execution time. We show that it is reduced to the problem of finding the shortest path in a directed graph with edge lengths equal to 1.

cs.LO↗

arXiv2013

Homology Groups of Pipeline Petri Nets

We study homology groups of elementary Petri nets for the pipeline systems. We show that the integral homology groups of these nets in dimensions 0 and 1 equal the group of integers, and they are zero in other dimensions. We prove that directed homology groups of elementary Petri nets are zero in all dimensions.

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arXiv2012

Generalized Asynchronous Systems

The paper is devoted to a mathematical model of concurrency the special case of which is asynchronous system. Distributed asynchronous automata are introduced here. It is proved that the Petri nets and transition systems with independence can be considered like the distributed asynchronous automata. Time distributed asynchronous automata are defined in standard way by the map which assigns time intervals to events. It is proved that the time distributed asynchronous automata are generalized the time Petri nets and asynchronous systems.

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arXiv2011

The Homology Groups of a Partial Trace Monoid Action

The aim of this paper is to investigate the homology groups of mathematical models of concurrency. We study the Baues-Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with an action on a set, we will build a cubical complex of free Abelian groups with homology groups isomorphic to the integral homology groups of the action category. It allows us to solve the problem posed by the author in 2004 of the constructing an algorithm for computing homology groups of the CE nets. We describe the algorithm and give examples of calculating the homology groups.

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arXiv2011

The Cubical Homology of Trace Monoids

This article contains an overview of the results of the author in a field of algebraic topology used in computer science. The relationship between the cubical homology groups of generalized tori and homology groups of partial trace monoid actions is described. Algorithms for computing the homology groups of asynchronous systems, Petri nets, and Mazurkiewicz trace languages are shown.

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arXiv2011

The study of distributed computing algorithms by multithread applications

The material in this note is used as an introduction to distributed algorithms in a four year course on software and automatic control system in the computer technology department of the Komsomolsk-on-Amur state technical university. All our the program examples are written in Borland C/C++ 5.02 for Windows 95/98/2000/NT/XP, and hence suit to compile and execute by Visual C/C++. We consider the following approaches of the distributed computing: the conversion of recursive algorithms to multithread applications, a realization of the pairing algorithm, the building of wave systems by Petri nets and object oriented programming.

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arXiv2009

Cubical Homology of Asynchronous Transition Systems

We show that a set with an action of a locally finite-dimensional free partially commutative monoid and the corresponding semicubical set have isomorpic homology groups. We build a complex of finite length for the computing homology groups of any asynchronous transition system with finite maximal number of mutually independent events. We give examples of computing the homology groups.

math.KT↗

arXiv2008

Global Dimension of Polynomial Rings in Partially Commuting Variables

For any free partially commutative monoid $M(E,I)$, we compute the global dimension of the category of $M(E,I)$-objects in an Abelian category with exact coproducts. As a corollary, we generalize Hilbert's Syzygy Theorem to polynomial rings in partially commuting variables.

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arXiv2008

On the Cubical Homology Groups of Free Partially Commutative Monoids

We study a Leech homology of a locally bounded free partially commutative monoid $M(E,I)$. Given a contravariant natural system of abelian groups $F$ on $M(E,I)$ we build a precubical set $T(E,I)$ with a homological system of abelian groups $\bar{F}$ and prove that the Leech homology groups $H_n(M(E,I),F)$ are isomorphic to the cubical homology groups $H_n(T(E,I),\bar{F})$, $n\geq 0$. As a consequence we have confirmed a conjecture that if the free partially commutative monoid does not contain $>n$ mutually commuting generators, then its the homological dimension $\leq n$. We have built the complexes of finite length for a computation of the Leech homology of such monoids and the Hochschild homology of their monoid rings. The results are applied to the homology of asynchronous transition systems. We give the positive answer to a question that the homological dimension of the asynchronous system does not greater than the maximal number of its mutually independent events. We have built the complex for computing the integral homology groups of an asynchronous transition system by the Smith normal form of integer matrices.

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