Nikolaj Glazunov - Academia.edu (original) (raw)

Papers by Nikolaj Glazunov

arXiv (Cornell University), Dec 28, 2016

We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning t... more We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning the critical determinant of the region |x| p + |y| p < 1 for p > 1 and Malyshev's method. In the sequel of this article we use these approach and method to present the main result.

We present mathematical models of deformation of elastic shells of mechanical structures of VAP, ... more We present mathematical models of deformation of elastic shells of mechanical structures of VAP, interval methods of calculating tolerances of elastic shells, elements the ontology of problems and methods, algorithms and software implementation. Elements of the ontology of interval analysis for calculating tolerances are also given.

Cybernetics and Systems Analysis, May 1, 2016

The authors consider current problems of the modern theory of dynamic systems on manifolds, which... more The authors consider current problems of the modern theory of dynamic systems on manifolds, which are actively developing. A brief review of such trends in the theory of dynamic systems is given. The results of the algebra of dual numbers, quaternionic algebras, biquaternions (dual quaternions), and their application to the analysis of infinitesimal neighborhoods and infinitesimal deformations of manifolds (schemes) are presented. The theory of differential-algebraic equations over the field of real numbers and their dynamics, as well as elements of trajectory optimization of respective dynamic systems, are outlined. On the basis of connection in bundles, the theory of differential-algebraic equations is extended to algebraic manifolds and schemes over arbitrary fields and schemes, respectively.

Problemi ìnformatizacìï ta upravlìnnâ, Nov 4, 2008

arXiv (Cornell University), May 25, 2023

We investigate lattice packings of Minkowski's balls and domains, as well as the distribution of ... more We investigate lattice packings of Minkowski's balls and domains, as well as the distribution of lattice points on Minkowski's curves which are boundaries of Minkowski's balls. By results of the proof of Minkowski's conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn balls. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. We construct direct systems of these balls, domains and their critical lattices and calculate their direct limits.

Elektronìka ta sistemi upravlìnnâ, Dec 29, 2018

arXiv (Cornell University), Jun 25, 2020

The purpose of this paper is to survey some of the important results on Langlands program, global... more The purpose of this paper is to survey some of the important results on Langlands program, global fields, D-shtukas and finite shtukas which have influenced the development of algebra and number theory. It is intended to be selective rather than exhaustive, as befits the occasion of the 80-th birthday of Yakovlev, 75-th birthday of Vostokov and 75-th birthday of Lurie. Under assumptions on ground fields results on Langlands program have been proved and discussed by Langlands, Jacquet, Shafarevich, Parshin, Drinfeld, Lafforgue and others. This communication is an introduction to the Langlands Program, global fields and to D-shtukas and finite shtukas (over algebraic curves) over function fields. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group G over a global field K, the Langlands correspondence relates automorphic forms on G and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group Gal(K/K) to the dual Langlands groupĜ(Q p). In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. V. Drinfeld and L. Lafforgue have investigated the case of functional global fields of characteristic p > 0 (V. Drinfeld for G = GL 2 and L. Lafforgue for G = GL r , r is an arbitrary positive integer). They have proved in these cases the Langlands correspondence. Under the process of these investigations, V. Drinfeld introduced the concept of a F-bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic p > 0 of the studied cases of the existence of the Langlands correspondence. Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced. G. Anderson has introduced the concept of a t-motive. U. Hartl, his colleagues and students have introduced and have explored the concepts of finite, local and global G-shtukas. In this review article, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of D-shtukas and finite shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of G-shtukas.

arXiv (Cornell University), Dec 28, 2016

We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning t... more We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning the critical determinant of the region |x| p + |y| p < 1 for p > 1 and Malyshev's method. In the sequel of this article we use these approach and method to present the main result.

Cybernetics and Systems Analysis, Jan 30, 2018

The authors consider the problem of recognition of a class of objects by the results of multispec... more The authors consider the problem of recognition of a class of objects by the results of multispectral measurements (spectral brightness of signals) and available spectral and statistical characteristics of the given classes. On the basis of probabilistic and statistical considerations, as well as quantization of continuous distributions, the heuristic recognition criterion is proposed. Based on the criterion, the heuristic method of recognition is presented. Modifications of the method are proposed to improve its reliability and efficiency.

Journal of Mathematical Sciences, Dec 1, 1988

The equalities (7.1) and (7.2) enable us to evaluate the right-hand side of the formulas of Theor... more The equalities (7.1) and (7.2) enable us to evaluate the right-hand side of the formulas of Theorem i for m ~ k m i (mod8). LITERATURE CITED i.

arXiv (Cornell University), Dec 10, 2012

Abelian varieties and p-divisible groups of Minkowski’s conjecture concerning critical lattices o... more Abelian varieties and p-divisible groups of Minkowski’s conjecture concerning critical lattices of the region, its characteristic p > 0 analogues and expansions.

2015 IEEE International Conference Actual Problems of Unmanned Aerial Vehicles Developments (APUAVD), 2015

This paper deals with a short overview of the problem of shape optimization of elastic bodies of ... more This paper deals with a short overview of the problem of shape optimization of elastic bodies of mechanical structures of VAP, the ontology of problems and methods of optimizing the shape of elastic body in terms of buckling load with an emphasis on algebraic methods. Elements of the ontology of interval analysis for calculating tolerances are also given.

Cybernetics and Systems Analysis, 2019

New results are presented and a brief review is given for new methods of the theory of dynamic sy... more New results are presented and a brief review is given for new methods of the theory of dynamic systems on manifolds over local fields and formal groups over local rings. For the analysis of n-dimensional manifolds and dynamic systems on such manifolds, formal structures are used, in particular, n-dimensional formal groups. Infinitesimal deformations are presented in terms of formal groups. The well-known one-dimensional case is extended and n-dimensional (n ³ 1) analytic mappings of an open p-adic polydisc (n-disk) D p n are considered. The n-dimensional analogs of modules arising in formal and non-Archimedean dynamic systems are introduced and investigated and their formal-algebraic structure is presented. Rigid structures, objects, and methods are outlined. From the point of view of systems analysis, new, namely formal and non-Archimedean, faces and structures of systems, mappings and iterations of mappings between these faces and structures are introduced and investigated.

Proceedings of the Bulgarian Academy of Sciences

We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conj... more We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conjecture about the critical determinant we divide Minkowski balls into 3 classes: Minkowski balls, Davis balls and Chebyshev–Cohn balls. We investigate lattice packings of these balls on planes with varying Minkowski metric and search among these packings the optimal packings. In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev–Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding balls.

arXiv (Cornell University), Feb 3, 2023

This is the continuation of the author's ArXiv presentation ''On packing of Minkowski balls. I". ... more This is the continuation of the author's ArXiv presentation ''On packing of Minkowski balls. I". In section 2 we investigate lattice packings of Minkowski balls and domains. By results of the proof of Minkowski conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. Direct limits of direct systems of Minkowski balls and domains and their critical lattices are calculated.

Cornell University - arXiv, Jul 13, 2022

V.V. Sharko in his papers and books has investigated functions on manifolds and cobordism. Braids... more V.V. Sharko in his papers and books has investigated functions on manifolds and cobordism. Braids intimately connect with functions on manifolds. These connections are represented by mapping class groups of corresponding discs, by fundamental groups of corresponding punctured discs, and by some other topological or algebraic structures. This paper presents selected algebraic methods and results of braids, links, cobordism connect with investigations by V.V. Sharko. These includes group theoretic results on braids and links, infinitesimal braid group relations and connections as well as connections on coherent sheaves on smooth schemes, a sketch of our algorithm for constructing of Lazard's one dimensional universal commutative formal group and selected results on applications of commutative formal groups to cobordism theory. as with the theory of cobordism, took place in 1970-1971. during his internship with academician A.A. Markov. (A.A. Markov was the head of the laboratory at the Computing Center of the USSR Academy of Sciences, and, at the same time, the head of the department of mathematical logic at Moscow State University). A.A. Markov gave the description of the set of isotopy classes of oriented links in R 3 in terms of braids. For manifolds of the dimension grater than 3 A.A. Markov has proved the undecidability of the problem of homeomorphy.

arXiv (Cornell University), Dec 28, 2016

We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning t... more We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning the critical determinant of the region |x| p + |y| p < 1 for p > 1 and Malyshev's method. In the sequel of this article we use these approach and method to present the main result.

We present mathematical models of deformation of elastic shells of mechanical structures of VAP, ... more We present mathematical models of deformation of elastic shells of mechanical structures of VAP, interval methods of calculating tolerances of elastic shells, elements the ontology of problems and methods, algorithms and software implementation. Elements of the ontology of interval analysis for calculating tolerances are also given.

Cybernetics and Systems Analysis, May 1, 2016

The authors consider current problems of the modern theory of dynamic systems on manifolds, which... more The authors consider current problems of the modern theory of dynamic systems on manifolds, which are actively developing. A brief review of such trends in the theory of dynamic systems is given. The results of the algebra of dual numbers, quaternionic algebras, biquaternions (dual quaternions), and their application to the analysis of infinitesimal neighborhoods and infinitesimal deformations of manifolds (schemes) are presented. The theory of differential-algebraic equations over the field of real numbers and their dynamics, as well as elements of trajectory optimization of respective dynamic systems, are outlined. On the basis of connection in bundles, the theory of differential-algebraic equations is extended to algebraic manifolds and schemes over arbitrary fields and schemes, respectively.

Problemi ìnformatizacìï ta upravlìnnâ, Nov 4, 2008

arXiv (Cornell University), May 25, 2023

We investigate lattice packings of Minkowski's balls and domains, as well as the distribution of ... more We investigate lattice packings of Minkowski's balls and domains, as well as the distribution of lattice points on Minkowski's curves which are boundaries of Minkowski's balls. By results of the proof of Minkowski's conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn balls. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. We construct direct systems of these balls, domains and their critical lattices and calculate their direct limits.

Elektronìka ta sistemi upravlìnnâ, Dec 29, 2018

arXiv (Cornell University), Jun 25, 2020

The purpose of this paper is to survey some of the important results on Langlands program, global... more The purpose of this paper is to survey some of the important results on Langlands program, global fields, D-shtukas and finite shtukas which have influenced the development of algebra and number theory. It is intended to be selective rather than exhaustive, as befits the occasion of the 80-th birthday of Yakovlev, 75-th birthday of Vostokov and 75-th birthday of Lurie. Under assumptions on ground fields results on Langlands program have been proved and discussed by Langlands, Jacquet, Shafarevich, Parshin, Drinfeld, Lafforgue and others. This communication is an introduction to the Langlands Program, global fields and to D-shtukas and finite shtukas (over algebraic curves) over function fields. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group G over a global field K, the Langlands correspondence relates automorphic forms on G and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group Gal(K/K) to the dual Langlands groupĜ(Q p). In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. V. Drinfeld and L. Lafforgue have investigated the case of functional global fields of characteristic p > 0 (V. Drinfeld for G = GL 2 and L. Lafforgue for G = GL r , r is an arbitrary positive integer). They have proved in these cases the Langlands correspondence. Under the process of these investigations, V. Drinfeld introduced the concept of a F-bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic p > 0 of the studied cases of the existence of the Langlands correspondence. Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced. G. Anderson has introduced the concept of a t-motive. U. Hartl, his colleagues and students have introduced and have explored the concepts of finite, local and global G-shtukas. In this review article, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of D-shtukas and finite shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of G-shtukas.

arXiv (Cornell University), Dec 28, 2016

We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning t... more We present A.V. Malyshev's approach to Minkowski's conjecture (in Davis's amendment) concerning the critical determinant of the region |x| p + |y| p < 1 for p > 1 and Malyshev's method. In the sequel of this article we use these approach and method to present the main result.

Cybernetics and Systems Analysis, Jan 30, 2018

The authors consider the problem of recognition of a class of objects by the results of multispec... more The authors consider the problem of recognition of a class of objects by the results of multispectral measurements (spectral brightness of signals) and available spectral and statistical characteristics of the given classes. On the basis of probabilistic and statistical considerations, as well as quantization of continuous distributions, the heuristic recognition criterion is proposed. Based on the criterion, the heuristic method of recognition is presented. Modifications of the method are proposed to improve its reliability and efficiency.

Journal of Mathematical Sciences, Dec 1, 1988

The equalities (7.1) and (7.2) enable us to evaluate the right-hand side of the formulas of Theor... more The equalities (7.1) and (7.2) enable us to evaluate the right-hand side of the formulas of Theorem i for m ~ k m i (mod8). LITERATURE CITED i.

arXiv (Cornell University), Dec 10, 2012

Abelian varieties and p-divisible groups of Minkowski’s conjecture concerning critical lattices o... more Abelian varieties and p-divisible groups of Minkowski’s conjecture concerning critical lattices of the region, its characteristic p > 0 analogues and expansions.

2015 IEEE International Conference Actual Problems of Unmanned Aerial Vehicles Developments (APUAVD), 2015

This paper deals with a short overview of the problem of shape optimization of elastic bodies of ... more This paper deals with a short overview of the problem of shape optimization of elastic bodies of mechanical structures of VAP, the ontology of problems and methods of optimizing the shape of elastic body in terms of buckling load with an emphasis on algebraic methods. Elements of the ontology of interval analysis for calculating tolerances are also given.

Cybernetics and Systems Analysis, 2019

New results are presented and a brief review is given for new methods of the theory of dynamic sy... more New results are presented and a brief review is given for new methods of the theory of dynamic systems on manifolds over local fields and formal groups over local rings. For the analysis of n-dimensional manifolds and dynamic systems on such manifolds, formal structures are used, in particular, n-dimensional formal groups. Infinitesimal deformations are presented in terms of formal groups. The well-known one-dimensional case is extended and n-dimensional (n ³ 1) analytic mappings of an open p-adic polydisc (n-disk) D p n are considered. The n-dimensional analogs of modules arising in formal and non-Archimedean dynamic systems are introduced and investigated and their formal-algebraic structure is presented. Rigid structures, objects, and methods are outlined. From the point of view of systems analysis, new, namely formal and non-Archimedean, faces and structures of systems, mappings and iterations of mappings between these faces and structures are introduced and investigated.

Proceedings of the Bulgarian Academy of Sciences

We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conj... more We investigate lattice packings of Minkowski balls. By the results of the proof of Minkowski conjecture about the critical determinant we divide Minkowski balls into 3 classes: Minkowski balls, Davis balls and Chebyshev–Cohn balls. We investigate lattice packings of these balls on planes with varying Minkowski metric and search among these packings the optimal packings. In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev–Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding balls.

arXiv (Cornell University), Feb 3, 2023

This is the continuation of the author's ArXiv presentation ''On packing of Minkowski balls. I". ... more This is the continuation of the author's ArXiv presentation ''On packing of Minkowski balls. I". In section 2 we investigate lattice packings of Minkowski balls and domains. By results of the proof of Minkowski conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. Direct limits of direct systems of Minkowski balls and domains and their critical lattices are calculated.

Cornell University - arXiv, Jul 13, 2022

V.V. Sharko in his papers and books has investigated functions on manifolds and cobordism. Braids... more V.V. Sharko in his papers and books has investigated functions on manifolds and cobordism. Braids intimately connect with functions on manifolds. These connections are represented by mapping class groups of corresponding discs, by fundamental groups of corresponding punctured discs, and by some other topological or algebraic structures. This paper presents selected algebraic methods and results of braids, links, cobordism connect with investigations by V.V. Sharko. These includes group theoretic results on braids and links, infinitesimal braid group relations and connections as well as connections on coherent sheaves on smooth schemes, a sketch of our algorithm for constructing of Lazard's one dimensional universal commutative formal group and selected results on applications of commutative formal groups to cobordism theory. as with the theory of cobordism, took place in 1970-1971. during his internship with academician A.A. Markov. (A.A. Markov was the head of the laboratory at the Computing Center of the USSR Academy of Sciences, and, at the same time, the head of the department of mathematical logic at Moscow State University). A.A. Markov gave the description of the set of isotopy classes of oriented links in R 3 in terms of braids. For manifolds of the dimension grater than 3 A.A. Markov has proved the undecidability of the problem of homeomorphy.