In this paper we investigate the complexity of matrix multiplication. V. Strassen in 1969 [1] constructed an algorithm to multiply two matrices of order n with the number of arithmetic operations O ( n log2 7 ) , which is asymptotically better than the complexity of the order n 3 of standard matrix multiplication algorithm “line by column”. In subsequent years, active investigations were carried on minimal complexity of various algebraic operations. The results of the researches in this field are well reflected in the book [2]. The situation in the problem of matrix multiplication is quite hard. By the end of the 1980s, with the efforts of many mathematicians complexity of matrix multiplication was reduced to O ( n 2.38) [3], but since then there is no significant progress in this problem. In order to better understand the problems associated with finding fast algorithms for matrix multiplication, this problem is investigated in different directions. One such area is the study minimal complexity of matrix multiplication for small sizes. This study is of interest in itself, but is also linked to the fact that the fast algorithms for matrix multiplication of small size can be recursively used for matrix multiplication of large size. In particular, Strassen’s algorithm uses recursively algorithm for multiplication of two matrices of order 2 with 7 multiplications rather than 8 as in standard algorithm. One can note two special properties of Strassen’s algorithm. Firstly, only the number of multiplications in the algorithm for multiplication of small matrices used for recursion affects the asymptotic complexity of algorithm for multiplication of large matrices. Secondly, matrix elements in recursion are themselves matrices and therefore they do not commute. These two properties have generated studies of bilinear complexity of multiplication of matrices and multiplication in other algebras. The bilinear algorithm must first calculate several products of linear combination of the elements of the first factor by linear combination of the elements of the second factor. Then, all the required expressions must be obtained by linear combinations of these products. The number of products is called bilinear complexity of the algorithm, and the minimum bilinear complexity of all bilinear algorithms that solve this problem is called the bilinear complexity of the problem. It is rather difficult to establish the exact value of the bilinear complexity even for multiplication of two matrices of small size. For example, for the problem of multiplying two 3×3 matrices so far we only know that the bilinear complexity lies between 19 and 23 [4, 5]. It is not difficult to establish the exact value of the bilinear complexity of multiplication of two matrices if at least one of them is only one row or one column. In this paper we investigate the bilinear complexity of multiplication of matrix of size m × 2 by matrix of size 2 × 2 over an arbitrary field. The exact value for the bilinear complexity for the multiplication of such matrices over an arbitrary field is known only when m = 2, 3, 4 [6, 7, 8]. From the result of Strassen it can be easy to get that the bilinear complexity of this problem does not exceed ⌈ 7m 2 ⌉ for an arbitrary field. The same lower bound was obtained in the paper [9], but only for the field with two elements. For arbitrary fields the lower bound 3m + 1 for this problem was obtained in [5]. In this article it is proved that for m ≥ 3 the bilinear complexity of multiplication of m × 2 matrix by 2 × 2 matrix over an arbitrary field can not be less than 3m + 2.

 

Differential geometry methods of have applications in the information files study (families of probability distributions of spaces of quantum states, neural networks, etc.). Research on geometry information back to the S. Rao that based by Fisher information matrix defined the Riemannian metric of probability distributions manifold. Further investigation led to the concept of statistical manifold. Statistical manifold is a smooth finite-dimensional manifold on which a metrically-affine structure, ie, metric and torsion-free linear connection that is compatible with a given metric; while the condition Codazzi. Geometric manifold and the manifold is given statistical structure tensor. In the present study examines the statistical structure of the generated randomized density of the normal distribution and the Cauchy distribution. The study put the allegation that a randomized probability density of the normal distribution can be regarded as the solution of the Cauchy problem for the heat equation, and randomized probability density of the Cauchy distribution can be considered as a solution to the Dirichlet problem for the Laplace equation. Conversely, the solution of the Cauchy problem for the heat equation can be regarded as a randomized probability density of the normal distribution, and the solution of the Dirichlet problem for the Laplace equation as randomized probability density of the Cauchy distribution. The main objective of the study was the fact that for each of these two cases to find the Fisher information matrix components and structural tensor. We found nonlinear differential equations of the first, second and third order for the density of the normal distribution and Cauchy density computational difficulties to overcome. The components of the metric tensor (the Fisher information matrix) and the components of the strain tensor are calculated according to formulas in which there is the log-likelihood function, ie, logarithm of the density distribution. Because of the positive definiteness of the Fisher information matrix obtained inequality, which obviously satisfy the Cauchy problem solution with nonnegative initial conditions in the case of the Laplace equation and the heat equation.

 

The article considers two initial-boundary value problems for a singularly perturbed parabolic equation ε 2 (ut − ∆u) = f(u, x, y, t, ε), (x, y, t) ∈ g × (0 < t ≤ T), where ε is a small positive parameter, ∆ is the Laplas operator, for cases where degenerate equation f(u, x, y, t, 0) = 0 has root u = φ(x, y, t) of multiplicity 2 or 3. Conditions are determined in each case for the problem to have a solution of boundary-layer type. Asymptotics are found and justified for epsilon ε −→ 0, which consist of a regular part and a few boundary-layer series. Unlike a well-known case of a single root of degenerate equation, boundarylayer solution asymptotic in case of a multiple root is constructed as a series in fractional powers of the small parameter, and these powers as well as scales of boundary layer variables depend on a multiplicity of the degenerate equation root. Another substantial difference is that three-zones of the boundary layer in a neighborhood of the initial time exist. These zones differ in decay behavior of boundary-level functions and scales of boundary-level value. The existing algorithm for constructing boundary-layer functions in case of a single root gives unacceptable results and needs significant modifications. This holds for boundary-layer functions describing boundary-layer behavior of the solution in a neighborhood of initial time as well as for angular boundarylayer functions, which play role in a neighborhood of the curve ∂g × (t = 0). Proposed here modified algorithm allows for constructing unified boundarylevel functions for all three boundary-layer zones. This is an advantage of this method in comparison to the method of coordination of asymptotic decomposition, where a separate asymptotic is found for each zone and then they are coordinated with each other. Asymptotic justification (i.e. existence of the solution with this asymptotic) is made using the asymptotic method of differential inequalities, that is lower and higher solutions of the problem are found using formal asymptotics.

 

The aim of the article is an overview of some important results in the theory of quadratic forms, and algebraic groups, and which had an impact on the development of the theory of numbers. The article focuses on selected tasks and is not exhaustive. A mathematical structures, methods and results, including the new ones, related in some extent with research of V.P. Platonov. The content of the article is following. In the introduction drawn attention to the classic researches of Korkin, Zolotarev and Voronoi on the theory of extreme forms and recall the relevant definitions. In section 2 "Quadratic forms and lattices"presented the necessary definitions, the results of the lattices and quadratic forms over the field of real numbers and over the ring of rational integers. Section 3 "Algebraic groups"contains a representation of the class of lattices in a real space as factors of algebraic groups, as well as the version of Mahler’s compactness criterion of such factors. Bringing the results of the compactness of factors of orthogonal groups of quadratic forms which do not represent zero rationally, and the definitions and concepts related to the quaternion algebras over rational numbers. These results explicitly or implicitly are used in the works of V. P. Platonov and in sections 4 and 5. Section 4 "Heegner points and their generalizations" provides an overview of new research in the direction of finding Heegner points and their generalizations. Section 5 summarizes some new research and results on the Hasse principle for algebraic groups. For the reading of the article may be a useful another article which has published by the author in the Chebyshevsky sbornik, vol. 16, no. 3, in 2015. I am deeply grateful to N. M. Dobrovolskii for help and support under the preparation of the article for publication.

 

This work consists of two main parts. In the first part, which presents the introduction, given a fairly comprehensive overview of the theory of the hyperbolic Zeta-function of lattices. Unlike earlier reviews is that, firstly, most of the results of the General theory particularized to two-dimensional case. This is done because the main goal of this lattice is quadratic fields. And these lattices are two-dimensional. Secondly, the first explicit form of the functional equation for hyperbolic Zeta-function of one and two diagonal lattices. In the second part we investigate the behavior of the hyperbolic Zetafunction of the lattice Λ(t) of the quadratic field when the growth parameter t. For applications of the theory of hyperbolic Zeta-function lattices to estimate the error of the approximate integration on the class of Eα s by using generalized parallelepipedal nets with weights it is important to have assessment through growing the determinant of the lattice. In this work, we derived a new asymptotic formula for the hyperbolic Zeta function lattices of quadratic fields. The peculiarity of this formula is that it has a main two-term member and remaining a member with the assessment of incoming constants. In this formula more specific correlation between the hyperbolic Zeta function of lattices of quadratic fields and quadratic field characteristics as: the Zeta function of the Dedekind principal ideals of a quadratic field, the derivative of the Zeta-function of Dedekind principal ideals of a quadratic field, quadratic field by the regulator and the fundamental unit of the quadratic field.

 

In this paper the problem of the width for verbal subgroups in different classes of groups is considered. We give a review the results obtained in this direction. The width of the verbal subgroups V (G) is equal to а least value of m ∈ N ∪ {+∞} such that every element of the subgroup V (G) is represented as the product of at most m values of words V ±1 . The results about the width of verbal subgroups for free products and other free group constructions, such as free products with amalgamation and HNN-extensions are indicated. A. H. Rhemtulla solved the question of conditions for infinity of the width of any proper verbal subgroups in free products. V. G. Bardakov and I. V. Dobrynina received similar results for the free products with amalgamation and HNN-extensions, for which associated subgroups are different from the base group. Also, V. G. Bardakov completely solved the problem of the width of verbal subgroups in the group of braid. Many mathematicians studied the width of verbal subgroups generated by words from commutator subgroup for some classes of groups. R. I. Grigorchuk found conditions for infinity such verbal subgroups of free products with amalgamation and HNN-extensions, for which associated subgroups are different from the base group. D. Z. Kagan obtained the corresponding results on width of verbal subgroups generated by words from commutator subgroup for groups with one defining relation and two generators, having a non-trivial center. Authors obtained the results about infinity of the width of verbal subgroups for groups with certain presentations, as well as for anomalous products of various types of groups. Also many results about verbal subgroups of Artin and Coxeter groups and graph groups are considered in the article.

 

Let G be a connected reductive group acting on an irreducible normal algebraic variety X. We study equivariant geometry of the cotangent vector bundle X, and we apply these results to study of a little Weyl group. The aim of this paper is to extend various results of E. B. Vinberg, who constructed a rational Galois cover of T ∗X of quasiaffine X by means of cotangent bundle to the so-called variety of generic horosheres. It is well-known that the example of a flag variety shows that these results could not be generalized directly. We develop the results of D. A. Timashev [18], who obtained the generalizations of the results of Vinberg to the class of varieties wider than quasiaffine but smaller than quasiprojective. We construct a family of horospheres of a smaller dimension in X which are called degenerate, and a variety Hor parameterizing this family, which has the same dimension as the variety parametrizing generic horosheres. Moreover in the quasiaffine case our construction shows that the familly of degenerate horosheres coincides with the familly of generic ones. We show that for constructed family of horosheres there exists a rational Gequivariant symplectic Galois covering of cotangent vector bundles T ∗Hor 99K T ∗X. It is proved that the extension of the fields of rational functions corresponding to this cover is a finite Galois extension with the Galois group isomorphic to the little Weyl group. As an application we get the description of the image of the moment map of T ∗X and the image of the normalized moment map by means of purely geometric methods. The first description of the image of the normalized moment map was obtained by F. Knop, nevertheless his proof is non-elementary since it involves the methods of differential operators.

 

An abelian group is called semisimple if it is the additive group of a semisimple ring. R. A. Beaumont and D. A. Lawver have formulated the description problem for semisimple groups. We consider vector semisimple groups in the present paper. Vector groups are direct products ∏ i∈I Ri of torsion free abelian groups Ri (i ∈ I) of rank 1. The semisimple vector groups ∏ i∈I Ri are described in the present paper in the case where I is a not greater than countable set. A multiplication on an abelian group G is a homomorphism µ: G⊗G → G, we denote it as µ(g1 ⊗ g2) = g1 × g2 for g1, g2 ∈ G. The group G with a multiplication × is called the ring on the group G and it is denoted as (G, ×). It is shown that every multiplication on a direct product of torsion free rank-1 groups is determined by its restriction on the direct sum of these groups. In particular, the following statement takes place. Lemma 3. Let I be a not greater than countable set, G = ∏ i∈I Ri and S = ⊕ i∈I Ri . Let × be a multiplication on the group G. If the restriction of this multiplication on S is zero, then the multiplication itself is zero. Let ∏ i∈I Ri be a vector group. We use the following notations: t(Ri) is the type of the group Ri , I0 is the set of indices i ∈ I such that t(Ri) is an idempotent type with an infinite number of zero components. If k ∈ I, then I0(k) is the set of indices i ∈ I0 such that t(Ri) ≥ t(Rk). Theorem 1. Let I be a not greater than countable set. A reduced vector group ∏ i∈I Ri is semisimple if and only if 1) there are no groups Ri (i ∈ I) of an idempotent type, where the number of zero components is finite; 2) the set I0(k) is infinite for every group Rk of the not idempotent type. Note that the set of types of groups Ri (i ∈ I) is an invariant of the group G = ∏ i∈I Ri , if I is a not greater than countable set. Therefore, this description doesn’t depend on the decomposition of the group G into a direct product of rank-1 groups.

A ring is said to be a ring on an abelian group G, if its additive group coincides with the group G. A subgroup of the group G is called the absolute ideal of G, if it is an ideal of every ring on the group G. If every ideal of a ring is an absolute ideal of its additive group, then the ring is called the AI-ring. If there exists at least one AI-ring on a group G, then the group G is called the RAI-group. We consider rings on almost completely decomposable abealian groups (acd-groups) in the present paper. A torsion free abelian group is an acd-group, if it contains a completely decomposable subgroup of finite rank and of finite index. Every acd-group G contains the regulator A, which is completely decomposable and fully invariant. The finite quotient group G/A is called the regulator quotient of the group G, the order of the group G/A is called the regulator index. If the regulator quotient of an acd-group is cyclic, then the group is called the crq-group. If the types of the direct rank-1 summands of the regulator A are pairwise incomparable, then the groups A and G are called rigid. If all these types are idempotent, then the group G is of the ring type. The main result of the present paper is that every rigid crq-group of the ring type is an RAI-group. Moreover, the principal absolute ideals are completely described for such groups. Let G be a rigid crq-group of the ring type. A subgroup A is the regulator of the group G, the quotient G/A = ⟨d + A⟩ is the regulator quotient and n is the regulator index. A decomposition A = ⊕ τ∈T(G) Aτ of the regulator A into a direct sum of rank-1groups Aτ determines the set T(G) = T(A) of critical types of the groups A and G. Then for every τ ∈ T(G), there exists an element eτ ∈ Aτ such that A = ⊕ τ∈T(G) Rτ eτ , where Rτ (τ ∈ T(G)) is a subring of the field of rational numbers containing the unit. Moreover, the definition of natural near-isomorphism invariants mτ (τ ∈ ∈ T(G)) of the group G naturally implies that every element g ∈ G can be written in the divisible hull of the group G in the following way g = ∑ τ∈T(G) rτ mτ eτ , where rτ are elements of the ring Rτ which are uniquely determined by a fixed decomposition of the regulator A. Every description of RAI-groups is based on a description of principal absolute ideals of the groups. The least absolute ideal ⟨g⟩AI containing an element g is called the principal absolute ideal generating by g. The following theorem describes principal absolute ideals. Theorem 1. Let G be a rigid crq-group of the ring type with a fixed decomposition of the regulator, g = ∑ τ∈T(G) rτ mτ eτ ∈ G. Then ⟨g⟩AI = ⟨g⟩ + ⊕ τ∈T(G) rτAτ . Note that the elements rτ (τ ∈ T(G)) in the representation of the element g ∈ G are determined uniquely up to an invertible factor of Rτ . Therefore, the representation of the principal absolute ideal doesn’t depend on the decomposition of the regulator. Theorem 2. Every rigid crq-group G of the ring type is an RAI-group. In this case, for every integer α соprime to n there exists an AI-ring (G, ×) such that the equality d × d = αd takes place in the quotient ring (G/A, ×), where d = d + A, G/A = ⟨d⟩.

 

One important problem is studying of lattices that naturally associated with universal algebra. In this article is considered algebras ⟨A, p, f⟩ with one Mal’tsev operation p and one unary operation f acting as endomorphism with respect to operation p. We study properties of congruence lattices of algebras ⟨A, p, f⟩ with Mal’tsev operation p that introduced by V. K. Kartashov. This algebra is defined as follows. Let ⟨A, f⟩ be an arbitrary unar and x, y ∈ A. For any element x of the unar ⟨A, f⟩ by f n (x) we denote the result of f applied n times to an element x. Also f 0 (x) = x. Assume that Mx,y = {n ∈ N ∪ {0} | f n (x) = f n (y)} and also k(x, y) = min Mx,y, if Mx,y ̸= ∅ and k(x, y) = ∞, if Mx,y = ∅. Assume further p(x, y, z) def = { z, if k(x, y) 6 k(y, z) x, if k(x, y) > k(y, z). It is described a structure of coatoms in congruence lattices of algebras ⟨A, p, f⟩ from this class. It is proved congruence lattices of algebras ⟨A, p, f⟩ has no coatoms if and only if the unar ⟨A, f⟩ is connected, contains one-element subunar and has infinite depth. In other cases congruence lattices of algebras ⟨A, p, f⟩ has uniquely coatom. It is showed for any congruences θ ̸= A × A and φ ̸= A × A of algebra ⟨A, p, f⟩ fulfills θ ∨ φ < A × A. Necessary and sufficient conditions when a congruence lattice of algebras from given class is complemented, uniquely complemented, relatively complemented, Boolean, generalized Boolean, geometric are obtained. It is showed any non-trivial congruence of algebra ⟨A, p, f⟩ from this class has no complement. It is proved that congruence lattices of any algebra ⟨A, p, f⟩ from given class is dual pseudocomplemented lattice.

 

The problem of classification of irreducible representations is a very complicated, "wild" problem for some groups like maximal unipotent, Borel and parabolic subgroups of the finite simple groups of Lie type. In 1962, A. A. Kirillov discovered the orbit method that establishes a one to one correspondence between the irreducible representations of a nilpotent Lie group and the coadjoit orbits. In 1977, D. Kazhdan modified the orbit method to be true for finite unipotent groups. However, the orbit method does not solve the problem, since the problem of classidication of the coadjpit orbits is a "wild" problem again. In 1995–2003, C. Andre constructer the theory of basic characters for the unitriangular group UT(n, Fq). These characters are not irreducible, but they have many common features with the irreducible characters. The Andre theory was simplified be Ning Yan in 2003. In 2008, P. Diaconis and I. Isaacs formulated the general notion of a supercharacter theory and constructed the supercharacter theory for algebra groups, its precial case is the Andre theory of basic characters. The general problem is to construct for a given group a supercharacter theory that as close to the theory of irreducible characters as possible. Many papers were devoted to the supercharacter theory. Up today the case of abelian groups is studied in details; the connection with Gauss, Kloosterman and Ramanujan sums is investigated. The supercharacter theories for maximal unipotent subgroups in orthogonal and symplectic groups are constructed. The problems of restriction and superinduction is solved for the basic characters. The problem of construction of a supercharacter theory for the parabolic subgroups is still open. In § 1–2 of the present paper, we present the authors proof of the main statements of the supercharacter theory for algebra groups, following the context of the paper of P. Diaconis and I. Isaacs. In §3, we announce the authors results on the supercharacter theory for the finite groups of triangular type, for which the theory of P. Diaconis and I.Isaacsas is a special case. We obtain the analog of A. A. Kirillov formula for irreducible characters. We show that the restriction of the supercaracter on a subgroup of triangular type is a sum supercharacters of these subgroup. As in the case of algebra group, the induction does not work for supercharacters. We defined a superinduction, obeying the main properties of induction including the Frobenius formula.

 

A new approach to the torsion problem in the Jacobians of hyperelliptic curves over the field of rational numbers was offered by Platonov. This new approach is based on the calculation of fundamental units in hyperelliptic fields. The existence of torsion points of new orders was proved with the help of this approach. The full details of the new method and related results are contained in [2]. Platonov conjectured that if we consider the S consisting of finite and infinite valuation and change accordingly definition of the degree of S-unit, the orders of torsion Q-points tend to be determined by the degree of fundamental S-units. The main result of this article is the proof of existence of the fundamental S-units of large degrees. The proof is based on the methods of continued fractions and matrix linearization based on Platonov’s approach. Efficient algorithms for computing S-units using method of continued fractions have been developed. Improved algorithms have allowed to construct the above-mentioned fundamental S-units of large degrees. As a corollary, alternative proof of the existence of torsion Q-points of some large orders in corresponding Jacobians of hyperelliptic curves was obtained.

 

In this article we study Hamiltonian simple algebras and lattices of Hamiltonian closed subalgebras in class of algebras with one operator. Obtained for algebras with arbitrary basic signature results are used for the description of Hamiltonian simple algebras and lattices of Hamiltonian closed subalgebras from class of unars with Mal’tsev operation that by V. K. Kartashov were defined. Unar with Mal’tsev operation is an algebra with one Mal’tsev operation p(x, y, z) and one unary operation acting as endomorphism with respect to operation p(x, y, z). Universal algebra A is called Hamiltonian if every subuniverse of A is a block of some congruence of the algebra A. A. G. Pinus defined a Hamiltonian closure on an arbitrary universal algebra. Precisely, the Hamiltonian closure B of a subalgebra B of a universal algebra A is the smallest subalgebra of algebra A containing B that coincides with some block of some congruence on algebra A. Subalgebra B of universal algebra A is called Hamiltonian closed if B = B. Set of all Hamiltonian closed subalgebras of algebra A with added empty set is lattice with respect to inclusion. A universal algebra A is called a Hamiltonian simple algebra if B = A for each non-empty and non-one-element subalgebra B of A. We found necessary conditions of Hamiltonian simplicity for arbitrary algebras with one operator and idempotent basic operations of positive arity. For these algebras families of their subalgebras forming chains with respect to inclusion in their lattices of Hamiltonian closed subalgebras are constructed. We also found necessary conditions of Hamiltonian simplicity for arbitrary algebras with one operator and with connected unary reduct. It is showed these conditions are not sufficient. For arbitrary algebras with one operator and idempotent basic operations necessary conditions of their lattice of Hamiltonian closed subalgebras is chain are obtained. We found necessary and sufficient conditions of Hamiltonian simplicity for unars with Mal’tsev operation that by V. K. Kartashov were defined. The structure of lattices of Hamiltonian closed subalgebras for algebras from this class is described. For these lattices necessary and sufficient conditions of their distributivity and modularity are obtained. We also found necessary and sufficient conditions when a lattice of Hamiltonian closed subalgebras of algebras from given class is a chain. The structure of atoms and coatoms of such lattices is described.

 

In this paper the exact value of a index of convergence for the mean-value of the complete rational arithmetical for the arithmetical function, satisfying the functional equation of Gaussian type, is found. In particular, the Bernoulli’s polynomials satisfy for this functional equation. A similar result holds for the complete rational trigonometric sums (Hua Loo-keng, 1952). The deduction of the main result of the paper leads of the elementary method. We owe to I. M. Vinogradov for the demonstration of fruitful results and profit of it. The complete rational arithmetic sums are the analogue the oscillatory integral of a periodic function, for example, trigonometric functions. In 1978 similar results for the exact value of the index of convergence of the trigonometric integral were obtained (G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov). In nowadays for a multivariate problem there are successful to get only upper and lower estimates for the index of convergence of appropriate sums and integrals.

 

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The article is dedicated to the scientific work and to the teaching activities of well-known Soviet mathematician, one of the founders of the Soviet school of the history of mathematics M. Ya. Vygodskii (1898–1965) who worked during many years in the institutes of Tula.