On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms and . Formulas are obtained for the number of representations of a positive integer by the sum of k binary quadratic forms of the kind , of the kind and of the kind .     

Determinants in Triangulated Categories

For a small triangulated category , Bass's K ₁ group is described, and the theorem of the heart is proved. We define the determinant map from to Neeman's , and we compute this map when is the derived category of an Abelian category .     

On Singular Functional Differential Inequalities

Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind and where is a singular Volterra operator, is continuous and positive on ]a, b], is a norm in R ⁿ, and [u]₊ and [u]_– are respectively the positive and the negative part of the vector u R ⁿ.     

Intersections of Convex Bodies

Let be nonempty convex bodies in . Let be vectors in , let , and let . Then is a convex set, and the family of sets is concave. Let . Then for the mean cross-sectional measures W_v (\Phi (\rho )), , the functions are concave on D. (Note that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca% WGxbWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiabfA6agjaacIcacqaH% bpGCcaGGPaGaaiykaiabg2da9iaabAfacaqGVbGaaeiBamaaBaaale% aatCvAUfKttLearyqr1ngBPrgaiuGacqWFRbWAaeqaaOGaeuOPdyKa% aiikaiabeg8aYjaacMcaaaa!4EE7!\[W_0 (\Phi (\rho )) = {\text{Vol}}_k\Phi (\rho )\] is the k-volume.) Bibliography: 2 titles.     

A Constructive Proof of the Generalized Gelfand Isomorphism

Using an analog of the classical Frobenius recursion, we define the notion of a Frobenius -homomorphism. For , this is an ordinary ring homomorphism. We give a constructive proof of the following theorem. Let X be a compact Hausdorff space, the th symmetric power of X, and the algebra of continuous complex-valued functions on X with the sup-norm; then the evaluation map defined by the formula identifies the space with the space of all Frobenius -homomorphisms of the algebra into with the weak topology.     

A Uniqueness Theorem for the Dual Problem Associated to a Variational Problem with Linear Growth

Uniqueness is proved for solutions of the dual problem that is associated with the minimum problem among the mappings with prescribed Dirichlet boundary data and for smooth strictly convex integrands f of linear growth. No further assumptions on f or its conjugate function are imposed, in particular, is not assumed to be strictly convex. A special solution of the dual problem is seen to be a mapping into the image of , which immediately implies uniqueness. Bibliography: 13 titles.     

Majorants and Extreme Points of Unit Balls in Bernstein Spaces

The Bernstein space B ^p () (1 $$ " align="middle" border="0"> 0) is the set of functions from L ^p( ) having Fourier transforms (in the sense of generalized functions) with supports in the compact segment [- , ]. Every function f has an analytic continuation onto the complex plane, which is an entire function of exponential type . The spaces B ^p ()\, are conjugate Banach spaces. Therefore, the closed unit ball in B ^p () has a rich set of extreme (boundary) points: coincides with the weakly ^* closed convex hull of its extreme points. Since, for 1< p< , B ^p () is a uniformly convex space, only the balls and have nontrivially arranged sets of extreme points. In this paper, in terms of zeros of entire functions, we obtain necessary and sufficient conditions of extremeness for functions from .     

Bounds for the Characteristic Functions of the System of Monomials in Random Variables and of Its Trigonometric Analogue

Using a multidimensional analogue of Vinogradov's inequality for a trigonometric integral, the upper bounds are constructed for the moduli of the characteristic functions both of the system of monomials in components of a random vector with an absolutely continuous distribution in and of the system where is uniformly distributed in [0, l]^s.     

Direct Theorem about Approximation on a Family of Two Segments

Introduce the notation: , is the union of two segments [-1,1] and [-1 + ,1+ ], is a noninteger number, is the Hölder class with exponent on The following result announced by the authors in [J. Math. Sci. 117 (2003), No. 3] is proved. There exist numbers a ₁ ( ) , b ₁ ( ) 0 depending only on such that for any there exists a polynomial , such that . Bibliography: 11 titles.     

THE TENSOR CATEGORY OF LINEAR MAPS AND LEIBNIZ ALGEBRAS

We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor-Moore type theorem for Leibniz algebras.     

Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

Conditions of Existence and Uniqueness of Solutions of the Multipoint Boundary Value Problem for a System of Generalized Ordinary Differential Equations

Effective sufficient conditions are established for the solvability and unique solvability of the boundary value problem where is a matrix-function with bounded variation components, is a vector-function belonging to the Carathéodory class corresponding to are continuous functionals (in general nonlinear) defined on the set of all vector-functions of bounded variation.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

Sufficient conditions of solvability and unique solvability of the boundary value problem are established, where are measurable functions and the vector function is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.     

On Some Rings of Arithmetical Functions

In this paper, we consider several constructions which from a given B-product * _B lead to another one We shall be interested in finding what algebraic properties of the ring are shared also by the ring . In particular, for some constructions the rings R _B and will be isomorphic and therefore have the same algebraic properties.     

The Spectrum of Lévy Constants for Quadratic Irrationalities and Class Numbers of Real Quadratic Fields

Let h(d) be the class number of the field and let be the Lévy constant. A connection between these constants is studied. It is proved that if d is large, then the value h(d) increases, roughly speaking, at the rate as grows. A similar result is obtained in the case where the value is close to , i.e., to the least possible value. In addition, it is shown that the interval contains no values of for prime p such that p 3 (mod 4). As a corollary, a new criterion for the equality h(d)=1 is obtained. Bibliography: 14 titles.     

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let be an irreducible crystallographic rootsystem in a Euclidean space V, with ⁺ theset of positive roots. For , , let be the hyperplane . We define a set of hyperplanes . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of is , where n is the rank of and h is the Coxeter number of the finiteCoxeter group corresponding to .     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

Tissus linéaires et théorèmes d'algébrisation de type Abdel-inverse et Reiss-inverse

A d-web in ( ,0) is given by d complex analytic foliations of codimension one in ( ,0) which are in general position. A d-web in ( ,0) is linear if all the leaves are (pieces of) hyperplanes in and is algebraic if it is associated, by duality, to a nondegenerate algebraic curve in of degree d. We characterize linear webs in ( ,0). We give explicit conditions under which a linear d-web in ( ,0) is algebraic and we obtain equations for in this case. Some related problems are discussed and some questions are posed.