On non-Archimedean quantum mathematics and non-Archimedean (quantum) computation
We consider selected aspects of (non-Archimedean) quantum mathematics and non-Archimedean (quantum) computation.
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arXiv subjects
Nikolaj Glazunov
Publications and source records attributed to Nikolaj Glazunov.
On the arithmetic and algebraic properties of Minkowski balls and spheres
This paper gives a brief overview of some new work in number theory and algebra, and also studies the arithmetic and algebraic properties of Minkowski balls and spheres. The content of the paper is presented in more detail in the table of Contents and in the Introduction.
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Extremal functions on moduli spaces and applications
Our object of study is extremal functions which are defined by distance functions of convex bodies. These functions take values in the moduli spaces of algebraic and geometric objects associated with these ${\mathbb Z}$-modules (geometric lattices) and with convex bodies. In most cases, convex bodies are $2$-dimensional Minkowski balls whose boundaries are Minkowski curves and we study lattice points on these curves. We define and investigate extremal functions that yield the homogeneous arithmetic minimum of a function in a lattice, the Hermite constant, the critical determinant of a body, optimal packings of bodies, best values of covering constants, and optimal solutions of Diophantine approximation problems. Moreover, for two-dimensional unit Minkowski balls and Minkowski domains we determine the minimal areas of inscribed and circumscribed hexagons.
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On coverings by Minkowski balls in the plane and a duality
Lattice coverings in the real plane by Minkowski balls are studied. We exploit the duality of admissible lattices of Minkowski balls and inscribed convex symmetric hexagons of these balls. An explicit moduli space of the areas of these hexagons is constructed, giving the values of the determinants of corresponding covering lattices. Low and upper bounds for covering constants of Minkowski balls are given. The best known value of covering density of Minkowski ball is obtained. One conjecture and one problem are formulated.
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On direct and inverse limits of Minkowski's balls, domains, and their critical lattices
We construct direct and inverse systems of Minkowski's balls and domains, direct and inverse systems of their critical lattices and calculate their direct and inverse limits.
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On Lattice points and Optimal Packing of Minkowski's Balls and Domains
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.
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On packing of Minkowski balls. II
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.
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Abelian varieties, homogeneous spaces and duality
The article is dedicated to the memory of O.N. Vvedenskii. Vvedenskii's results are presented as well as selected new results of arithmetic algebraic geometry. Elements of ontology of Vvedenskii's research also given.
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Braids, links, cobordisms and formal groups
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.
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On Langlands program, global fields and shtukas
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. 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.
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On Langlands program, related representation and $G$-shtukas
This communication is an introduction to the Langlands Program and to ($G$-) shtukas (over algebraic curves) over function fields. Modular curves and Drinfeld (elliptic) modules and shtukas are used in coding theory. From this point of view the communication is concerned with mathematical models and methods of coding theory. 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 $ {\mathcal Gal} ({\overline K} / K) $ to the dual Langlands group $ \hat G ({\overline {\mathbb 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. 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 $ G $ --shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of $ G $-shtukas.
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($p$-adic) $L$-functions and ($p$-adic) (multiple) zeta values
The article is dedicated to the memory of George Voronoi. It is concerned with ($p$-adic) $L$-functions (in partially ($p$-adic) zeta functions) and cyclotomic ($p$-adic) (multiple) zeta values. The beginning of the article contains a short summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. Ünver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple zeta values is given by H. Furusho. Short history and connections among Galois groups, fundamental groups, motives and arithmetic functions are presented in the talk by Y. Ihara. Results on multiple zeta values, Galois groups and geometry of modular varieties has presented by Goncharov. Interesting unipotent motivic fundamental group is defined and investigated by Deligne and Goncharov. The framework of ($p$-adic) $L$-functions and ($p$-adic) (multiple) zeta values is based on Kubota-Leopoldt $p$-adic $L$-functions and arithmetic $p$-adic $L$-functions by Iwasawa. Motives and ($p$-adic) (multiple) zeta values by Glanois and by Ünver, improper intersections of Kudla-Rapoport divisors and Eisenstein series by Sankaran are reviewed. More fully the content of the article can be found at the following table of contents (plan). Numerical examples are included.
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On A.V. Malyshev's approach to Minkowski's conjecture concerning the critical determinant of the region $|x|^p + |y|^p < 1$ for $p > 1$
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.
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Deformations and elements of deformation theory
This article consisted of an elementary introduction to deformation theory of varieties, schemes and manifolds, with some applications to local and global shtukas and fever to Newton polygons of $p$-divisible groups . Soft problems and results mainly are considered. In the framework we give review of some novel results in the theory of local shtukas, Anderson-modules, global shtukas, Newton polygons of $p$-divisible groups and on deformations of $p$-divisible groups with given Newton polygons.
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On Rigid, Hard and Soft Problems and Results in Arithmetic Geometry
Rigid, hard and soft problems and results in arithmetic geometry are presented. "Soft" and "hard" in our paper are limited to the framework of solutions of quadratic forms over rings of integers of local and global fields, the Hardy-Littlewood-Kloosterman method. Next we consider the notion of rigidity. In the framework we give review of some novel results in the aria.
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Transformations, Dynamics and Complexity
We review and investigate some new problems and results in the field of dynamical systems generated by iteration of maps, β-transformations, partitions, group actions, bundle dynamical systems, Hasse-Kloosterman maps, and some aspects of complexity of the systems.
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Toward a Category Theory Design of Ontological Knowledge Bases
I discuss (ontologies_and_ontological_knowledge_bases / formal_methods_and_theories) duality and its category theory extensions as a step toward a solution to Knowledge-Based Systems Theory. In particular I focus on the example of the design of elements of ontologies and ontological knowledge bases of next three electronic courses: Foundations of Research Activities, Virtual Modeling of Complex Systems and Introduction to String Theory.
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Critical lattices, elliptic curves and their possible dynamics
We present a combinatorial geometry and dynamical systems framework for the investigation and proof of the Minkowski conjecture about critical determinant of the region $ |x|^p + |y|^p < 1, p > 1. $ The application of the framework may drastically reduce the investigation of sufficiently smooth real functions of many variables. Incidentally, we establish connections between critical lattices, dynamical systems and elliptic curves.
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