Search

The paper is devoted to some properites of ramification invariants in infinite Abelian extensions of exponent p for a class of complete discrete valuation fields that includes 2-dimensional local fields of prime characteristic p. In particular, it is proved that the maximal such extension with a prescribed upper bound of ramification breaks has finite depth of ramification, and this depth is computed.

In the present paper, ramification invariants of complete discrete valuation fields of prime characteristic p with imperfect residue field are studied. Using Witt vectors, ramification breaks of cyclic extensions of degree p2 of such a field are calculated. Necessary and sufficient conditions for a pair of natural numbers to be ramification breaks of a cyclic extension of degree p2 are obtained.

An extension of complete dicrete valuation fields with imperfect residue fields is naturally viewed as an epimorphism between algebraic surfaces with a distinguished point. Each regular curve that meets this point with irreducible preimage gives rise to an extension of fields of functions. In this paper, the ramification jump of this extension is considered as a function of a jet of a curve. The Zarisky topology on the set of jets is introduced, and the lower semicontinuity and existence of a common value for the jump are proved.

Let R be a partially ordered commutative ring without zero divisors and with 1/2. Let Gn(R) be the subsemigroup of GLn(R) consisting of matrices with nonnegative elements. In the paper, we describe endomorphisms of this semigroup for n = 2.

In this paper, we prove that every element of the linear group GL14(R) normalizing the Chevalley group of type G 2 over a commutative local ring R without 1/2 belongs to this group up to some multiplier. This allows us to improve our classification of automorphisms of these Chevalley groups showing that an automorphism-conjugation can be replaced by an inner automorphism. Therefore, it is proved that every automorphism of a Chevalley group of type G 2 over a local ring without 1/2 is a composition of a ring and an inner automorphisms.

In this paper, we consider the C∗-algebra Ωmult(ℝn) of multiplicatively weakly oscillating functions and a new wide class of kernels of compact type, which includes the class of SO(n)-invariant kernels. For the Banach algebra generated by operators with kernels of this class and coefficients from Ωmult(ℝn), we construct a symbolic calculus, obtain necessary and sufficient conditions for the presence of the Fredholm property, and propose a method of calculating the index of families. Similar results are obtained for operators with bihomogeneous kernels of compact type and multiplicatively weakly oscillating coefficients, i.e., for operators from the tensor product \( {{\mathfrak{M}}_{p,n}}_{{_1}}\otimes {{\mathfrak{M}}_{p,n}}_{{_2}} \).

In this paper, we define coherent morphisms of chain maps and homology groups of morphisms of this type. We construct strong homology groups of continuous maps of compact metric spaces and prove that for each continuous map f : X → Y , there exists a long exact homological sequence. Moreover, we show that for each inclusion i : A → X of compact metric spaces, there exists an isomorphism $ {{\bar{H}}_n}(i)\approx {{\bar{H}}_n}\left( {X,A} \right) $ .

Let \( \mathbb{F} \) be a finite field of characteristics different from 2. We show that no bijective map transforms the permanent to the determinant when the cardinality of \( \mathbb{F} \) is sufficiently large. Also we determine Gibson barriers (the maximal and minimal numbers of nonzero elements) for convertible (0, 1)-matrices and solve several related problems in different matrix subspaces. Our results are illustrated by examples. This paper is based on the joint work with G. Dolinar, B. Kuzma, and M. Orel.

For simulation of objects of discrete topology, supercomputers are often used. This can be explained by the combinatorial nature of the problems examined, especially in the study of the dynamics of rearrangements of such structures. In this paper, we consider three methods of coding associated with the parallelization of computations: (1) coding of cubic complexes; (2) coding for the calculation of transition probabilities in Markov chains in the reconstructions of triangulations of Euclidean spaces; (3) coding of topological situations in the transformations of complexes, i.e., discrete analogs of homotopic transformation. The base chosen for the geometric and topological objects possesses “minimal” mathematical resources (integer points and primitive vectors in Euclidean spaces).

In this paper, we study torsion groups containing proper fully invariant subgroups isomorphic to the group itself, the so-called IF-groups. We introduce the concept of an admissible sequence of Ulm–Kaplansky invariants for primary groups by use of which a description of IF-groups is obtained in some important classes of p-groups.

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.

The problem of determination of Abelian groups (up to isomorphism) by their rings of endomorphisms in the class of completely decomposable torsion-free Abelian groups has been solved earlier. For the class of direct sums of rational groups, one can speak of determination of Abelian groups by rational representations of their endomorphism rings up to equality. In this paper, we consider this problem for the class of finite direct sums of rational groups and for some subclasses.

In this paper, we consider the question of determinability of an Abelian group by its automorphism group in the class of completely decomposable block-rigid torsion-free Abelian groups.

The paper is concerned with solarity of intersections of suns with bars (in particular, with closed balls and extreme hyperstrips) in normed linear spaces. A sun in a finite-dimensional (BM)-space (in particular, in l 1(n)) is shown to be monotone path connected. A nonempty intersection of an m-connected set (in particular, a sun in a two-dimensional space or in a finite-dimensional (BM)-space) with a bar is shown to be a monotone path-connected sun. Similar results are obtained for boundedly compact subsets of infinite-dimensional spaces. A nonempty intersection of a monotone path-connected subset of a normed space with a bar is shown to be a monotone path-connected α-sun.

In the paper, we describe automorphisms of stable linear groups E(R) and GL(R) over commutative local rings R with 1/2.

In this paper, we prove that every automorphism of a Chevalley group of type G2 over a commutative local ring without 1/2 is the composition of a ring automorphism and a conjugation by some matrix.

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degree less than d in n + 1 variables in zero characteristic. Consider a linear system on W given by homogeneous polynomials of degree d'. Under the conditions of the first Bertini theorem for W and this linear system, we show how to construct an irreducible divisor in general position from the statement of this theorem. This algorithm is deterministic and polynomial in (dd′)n and the size of the input. This paper is the second part in a tree-part series. Bibliography: 21 titles.

In this paper, we consider a Markov operator (i.e., a contraction preserving the subspace of constants and the nonnegativity of functions) in the L 2 space on the n-dimensional torus that is a special perturbation of the unitary operator corresponding to a hyperbolic toral automorphism. We prove some properties of its spectrum and the spectra of some related operators. Bibliography: 4 titles.

Let Ω be a set and TΩ be the set of all 5-ary functions over Ω. In this paper, by considering five Mann’s compositions on TΩ, we obtain a (2, 5)-semigroup and an abstract characterization of this algebra.

Our aim in this paper is to construct a bounded linear operator that solves the problem of multiple interpolation (interpolation with derivatives). It is proved that such an operator exists for nontangential and sparse interpolation sets if we consider interpolation by analytic functions satisfying the following condition: |f (m)(z 1) − f (m)(z 2)| ≤ ω (|z 1 − z 2|). Bibliography: 8 titles.

The determinant theory for matrices over a pseudo-complemented distributive lattice is presented. Previous results on this topic are special cases of the theorems proved in this paper.

This paper is devoted to subexponential estimates in Shirshov’s height theorem. A word W is n-divisible if it can be represented in the form W = W0W1⋯Wn, where W1 ≺ W2 ≺⋯≺ Wn. If an affine algebra A satisfies a polynomial identity of degree n, then A is spanned by non-n-divisible words of generators a1 ≺⋯≺ al. A. I. Shirshov proved that the set of non-n-divisible words over an alphabet of cardinality l has bounded height h over the set Y consisting of all words of degree ≤ n − 1. We show that h < Φ (n, l), where Φ(n, l) = 287l∙n12 log3 n+48. Let l, n, and d ≥ n be positive integers. Then all words over an alphabet of cardinality l whose length is greater than ψ(n, d, l) are either n-divisible or contain the dth power of a subword, where ψ(n, d, l) = 218l(nd)3 log3(nd)+13d2. In 1993, E. I. Zelmanov asked the following question in the Dniester Notebook: Suppose that F2,m is a 2-generated associative ring with the identity xm = 0. Is it true that the nilpotency degree of F2,m has exponential growth? We give the definitive answer to E. I. Zelmanov by this result. We show that the nilpotency degree of the l-generated associative algebra with the identity xd = 0 is smaller than ψ(d, d, l). This implies subexponential estimates on the nilpotency index of nil-algebras of arbitrary characteristic. Shirshov’s original estimate was just recursive; in 1982 a double exponent was obtained, and an exponential estimate was obtained in 1992. Our proof uses Latyshev’s idea of an application of the Dilworth theorem. We think that Shirshov’s height theorem is deeply connected to problems of modern combinatorics. In particular, this theorem is related to the Ramsey theory. We obtain lower and upper estimates of the number of periods of length 2, 3, n − 1 in some non-n-divisible word. These estimates differ only by a constant.

The paper proves inequalities involving the discrete and uniform norms of polynomials on an interval and a circular arc. The proofs are based on Bernstein type inequalities. The inequalities obtained supplement the corresponding results by Rakhmanov, Shekhtman, Sheil-Small, and Dubinin. Bibliography: 8 titles.

In this paper, we give a full description of idempotent elements of semigroups BX (D) defined by semilattices of the class Σ4(X, 7). For the case where X is a finite set, we derive formulas by means of which we can calculate the numbers of idempotent elements of the respective semigroup.

In this paper, we study the properties of prime ideals in semirings of continuous functions with values in the unit interval [0, 1] on topological spaces. We describe maximal and pure ideals of such semirings. We study homomorphisms of semirings of continuous [0, 1]-valued functions. In terms of semirings of functions we characterize some properties of compacta. We show that the theory of ideals in these semirings differs from the case of rings of continuous functions.

In the paper, the scheme of independent particle allocation is considered. A connection is demonstrated between finite-dimensional distributions for the process of the number of particles assigned to the k-th cell and the conditional distribution of independent Poisson distributed random variables. Conditions are specified for the convergence of these distributions to finite-dimensional distributions of the Gaussian process. A certain scheme of dependent particle allocation on a countable set of cells is introduced and investigated.

In this paper, we study the interaction between transferred Chern classes and Chern classes of transferred bundles. We calculate the algebra \( B{P^{*}}\left( {X_{{h\varSigma p}}^p} \right) \) and show that its multiplicative structure is completely determined by the Frobenius reciprocity. We also give some tables of the initial segments of the formal group law in the Morava K-theory which are often useful in calculations.

Let Ω be a bounded open and oriented connected subset of ℝ n which has a compact topological boundary Γ, let C be the Dirac operator in ℝ n , and let ℝ0,n be the Clifford algebra constructed over the quadratic space ℝ n . An ℝ0,n -valued smooth function f : Ω → ℝ0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Holder continuous function f can be decomposed as F + − F − = f on Γ, where the components F ± are extendable to monogenic functions in Ω± with Ω+ := Ω, and Ω− := ℝ n \ (Ω ⋃ Γ), respectively.

We study properties of stable measures with index α > 2, α ∉ ℕ. Such measures are signed ones, and hence, they are not probability measures. We show that in some sense, these signed measures are limit measures for sums of independent random variables. In the last section of the paper, we prove a theorem about large deviations of sums of independent random variables using the positive part of the limit measure. Bibliography: 8 titles.

UDC 510.67 We present a complete description of ℵ0-categorical quite o-minimal theories introduced earlier as an extension of the class of o-minimal theories but “quite” inherited various their properties. Bibliography :6 titles. Let L be a countable first order language. Throughout the paper, we consider L-structures and assume that L contains a binary relation symbol < interpreted as a linear ordering in these structures. For arbitrary subsets A and B of a structure M we write A

In this paper, we consider provably secure cryptographic constructions in the context of circuit complexity. Based on the ideas of provably secure trapdoor functions developed in (Hirsch, Nikolenko, 2009; Melanich, 2009), we present a new linear construction of a provably secure trapdoor function with order of security 5/4. Besides, we present an in-depth general study of the gate elimination method for the case of linear functions. We also give a nonconstructive proof of lower bounds on the circuit complexity of linear Boolean functions and upper bounds on circuit implementations of linear Boolean functions, obtaining specific constants. Bibliography: 53 titles.

This paper studies the problem on the frequency of the event where a graph with a fixed number of vertices contains induced subgraphs that are isomorphic to distance graphs in spaces of definite dimensions. In particular, the case of plane and three-dimensional spaces is considered.

Relatively recent and new results on the enumeration of classes of projective congruent quadrics and classes of projective equivalent quadrics of projective spaces over a local ring R = 2R with nilpotent principal maximal ideal are reflected in the paper. For the case of the basic ring R with nonprincipal maximal ideal, the enumerations of quadrics of projective plane are given up to projective equivalence.

In this paper, we describe automorphisms of the lattice \( {\mathbb A} \) of all subalgebras of the semiring ℝ+[x] of polynomials in one variable over the semifield ℝ+ of nonnegative real numbers. It is proved that any automorphism of the lattice A is generated by an automorphism of the semiring ℝ+[x] that is induced by a substitution x ⟼ px for some positive real number p. It follows that the automorphism group of the lattice \( {\mathbb A} \) is isomorphic to the group of all positive real numbers with multiplication. A technique of unigenerated subalgebras is applied.

In this paper, we study G-spaces, i.e., topological spaces with actions of a compact group G, from the point of view of the theory of retracts. We recall definitions and properties of absolute retracts, absolute neighborhood retracts, absolute extensors, and absolute neighborhood extensors with respect to the equivariant weakly hereditary class \( \mathfrak{G} \) of G-spaces. Also, we give basic notions of the theory of topological transformation groups.

This paper discusses Abelian groups (modules) isomorphic to their endomorphism groups (modules). A necessary and sufficient condition is given according to which the commutativity of the endomorphism ring of a group G follows from the isomorphism G ≅ End G.

The aim of this paper is to generalize results on dimension polynomials of difference modules over difference rings for a wider class of rings of difference operators. We introduce the notion of quasi-commutativity, which generalizes the notion of commutativity and enables one to consider wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established.

We define the length of a finite system of generators of a given algebra \( \mathcal{A} \) as the smallest number k such that words of length not greater than k generate \( \mathcal{A} \) as a vector space, and the length of the algebra is the maximum of the lengths of its systems of generators. In this paper, we obtain a classification of matrix subalgebras of length 1 up to conjugation. In particular, we describe arbitrary commutative matrix subalgebras of length 1, as well as those that are maximal with respect to inclusion.

In this paper, we study the problem on the existence of nonmeasurable automorphisms of finite-dimensional and infinite-dimensional Lie groups over the field of real numbers and also over non-Archimedean local fields. The nonmeasurability of automorphisms is considered relative to real-valued measures and also measures with values in non-Archimedean local fields. Their existence is proved and a procedure for their construction is given. Their application to the construction of nonmeasurable irreducible unitary representations is demonstrated.

In this paper, we study k-cutsets in k-connected graphs. We introduce generalized flowers and prove some fundamental statements describing their structure. After that, we consider generalized flowers in the case where k = 4. For k = 4, we give a complete description of 4-cutsets belonging to a generalized flower. Bibliography: 10 titles.

Let T be an infinite set of prime numbers, \( \mathcal{M} \) be a set of groups \( \left\{ {\left. {\mathbb{Z}(p)} \right|p \in T} \right\} \). An Abelian group A is said to be \( \mathcal{M} \)-large if $$ {\text{Hom}}\left( {A,\;\mathop { \bigoplus }\limits_{p \in T} \mathbb{Z}(p)} \right) = {\text{Hom}}\left( {A,\;\prod\limits_{p \in T} {\mathbb{Z}(p)} } \right). $$ This paper presents a characterization of \( \mathcal{M} \)-large torsion-free and mixed groups.

The paper suggests new two-sided bounds for the Perron root of a weakly irreducible nonnegative matrix, which depend on the circuits of length no less than two in the associated directed graph and, in some cases, improve Al’pin’s bounds, suggested in 1995. Two approaches to deriving circuit bounds are considered. Bibliography: 5 titles.

The paper considers the eigenvalve problem for a polynomial m × n matrix F(μ) of rank ρ. Algorithms. allowing one to reduce this problem to the generalized matrix eigenvalve problem are suggested. The algorithms are based on combining rank factorization methods with the method of hereditary pencils. Methods for exhausting the subspaces of polynomial solutions of zero index from the matrix nullspaces and for isolating the regular kernel from F(μ), with its subsequent linearization, are proposed. Bibliography: 6 titles.

The paper is concerned with Shimura sums related to modular forms with multiplicative coefficients which are products of Dedekind η-functions of various arguments. Several identities involving Shimura sums are established. The type of identity obtained depends on the splitting of primes in certain imaginary quadratic number fields.

This paper considers several cycle detection algorithms. Proofs of their correctness are given, bounds for complexity are obtained, some number theory applications like the factorization of integers and the discrete log problem are examined.

In this paper, we obtain an upper estimate for the cardinality of a subset of the discrete torus over a field of three elements of which any four points do not form a nonsingular parallelogram.

An important property of a scale-free network which allows one to communicate between its nodes is the connectivity. From the simulation results it is known that such networks are resistant to random damage. It is also known from experiments that usually disconnecting such a graph is obtained by isolating a vertex. In this paper, we deal with a problem of isolating a vertex both experimentally and theoretically. For the theoretical results, we use the LCD model defined by Bollobas and Riordan. Among others, we prove that even after random deleting of nα vertices (mα edges) for α < (d − 1)/d, where d is the out-degree, with high probability, there are no isolated vertices in the remaining graph.

In this paper, we discuss the algebraic independence and algebraic relations, first, for reciprocal sums of even terms in Fibonacci numbers \( \sum\limits_{n = 1}^\infty {F_{2n}^{ - 2s}} \), and second, for sums of evenly even and unevenly even types \( \sum\limits_{n = 1}^\infty {F_{4n}^{ - 2s}}, \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}} \). The numbers \( \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}}, \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 4}} \), and are shown to be algebraically independent, and each sum \( \sum\limits_{n = 1}^\infty {F_{4n - 2}^{ - 2s}} \left( {s \geqslant 4} \right) \) is written as an explicit rational function of these three numbers over \( \mathbb{Q} \). Similar results are obtained for various series of even type, including the reciprocal sums of Lucas numbers \( \sum\limits_{n = 1}^\infty {L_{2n}^{ - p}}, \sum\limits_{n = 1}^\infty {L_{4n}^{ - p}} \), and \( \sum\limits_{n = 1}^\infty {L_{4n - 2}^{ - p}} \).

This present paper is a survey of recent value-distribution results for periodic Hurwitz zeta-functionsis. These results include probabilistic limit theorems, universality, and the functional independence of periodic Hurwitz zeta-functions. Both one-dimensional and joint value-distribution theorems are concerned.

In this paper, two limit theorems in the sense of the weak convergence of probability measures in the space of meromorphic functions and on the complex plane are proved for L-functions from a subclass of the Selberg class. The explicit form of the limit measures is given.