New Hadamard matrices and conference matrices obtained via Mathon's construction

We give a formulation, via (1, –1) matrices, of Mathon's construction for conference matrices and derive a new family of conference matrices of order 59^2t+1 + 1,t 0. This family produces a new conference matrix of order 3646 and a new Hadamard matrix of order 7292. In addition we construct new families of Hadamard matrices of orders 69^2t+1 + 2, 109^2t+1 + 2, 8499 ^t ,t 0;q ²(q + 3) + 2 whereq 3 (mod 4) is a prime power and 1/2(q + 5) is the order of a skew-Hadamard matrix); (q + 1)q ²9 ^t ,t 0 (whereq 7 (mod 8) is a prime power and 1/2(q + 1) is the order of an Hadamard matrix). We also give new constructions for Hadamard matrices of order 49 ^t 0 and (q + 1)q ² (whereq 3 (mod 4) is a prime power).This work was supported by grants from ARGS and ACRB.Dedicated to the memory of our esteemed friend Ernst Straus.     

Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

A Criterion for Convergence of Weak Greedy Algorithms

We find a class V of sequences such that the condition V is necessary and sufficient for convergence of weak greedy algorithm with weakness sequence for each f and all Hilbert spaces H and dictionaries D. We denote by V the class of sequences x={x _{k k=1} , x _k 0, k=1,2,..., with the following property: there exists a sequence 0=q ₀<q ₁< such that _s=1 2 ^s /q _s )< and _s=1 2^–s _k=1 ^q s ^x _k ²<, where q _s :=q _s –q _s–1.     

Binary hyperidentities of lattices

Summary An equational identity of a given type involves two kinds of symbols: individual variables and the operation symbols. For example, the distributive identity: x (y + z) = x y + x z has three variable symbols {x, y, z} and two operation symbols {+, }. Here the variables range over all the elements of the base set while the two operation symbols are fixed. However, we shall say that an identity ishypersatisfied by a varietyV if, whenever we also allow the operation symbols to range over all polynomials of appropriate arity, the resulting identities are all satisfied byV in the usual sense. For example, the ring of integers Z; +, satisfies the above distributive law, but it does not hypersatisfy the same formal law because, e.g., the identityx + (y z) = (x + y) (x + z) is not valid. By contrast, is hypersatisfied by the variety of all distributive lattices and is thus referred to as a distributive latticehyperidentity. Thus a hyperidentity may be viewed as an equational scheme for writing a class of identities of a given type and the original identities themselves are obtained as special cases by substituting specific polynomials of appropriate arity for the operation symbols in the scheme. In this paper, we provide afinite equational scheme which is a basis for the set of all binary lattice hyperidentities of type 2, 2, .This research was supported by the NSERC operating grant # 8215     

Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Let (0,1) be an irrational, and [a ₁,a ₂,...] be the continued fraction expansion of . Let H be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V>20, then the Hausdorff dimension of the spectrum (H ) is strictly great than zero for any irrational , and is strictly less than 1 if and only if liminf _k(a ₁ a ₂a _k ))^1/k <.     

Existence and Asymptotic Behaviour of Large Solutions of Semilinear Elliptic Equations

In this paper we are interested in the semilinear elliptic equations of the type u=u(,u), on bounded smooth domain of R ⁿ . We also treat existence of positive solution of u=p(x)f(u), which explodes near the boundary of (called large solutions). Our approach is based on potential theory.     

On continuous preservation of norms and areas

Summary We give various properties, examples and equivalent conditions for mapsT of then-dimensional euclidean space into itself (n 2) satisfying the generalised orthogonality equation|Tx Ty| = |x y| for allx, y inR ⁿ , where stands for the usual dot product, and we prove that the only continuous maps verifying this condition are the orthogonal linear transformations.     

Interacting Fock Expansion of Lévy White Noise Functionals

Consider the Lévy white noise space (S ^*,(S ^*),), where S ^* is the Schwartz distributions over R ^d and is a Lévy white noise measure lifted from a 1-dimensional infinitely divisible distribution with finite moments. We give explicit forms and recursion formulas of moment and renormalization kernels for the Lévy white noise measure. By defining inner products (,)_[n] in n-particle spaces, we establish an interacting Fock space _n=0 ⁽ⁿ⁾ and the interacting Fock expansions for Lévy white noise functionals. The usual Fock space (H)= _n=0 can be viewed as a quotient space of the interacting Fock space. As a particular case, we give the interacting Fock expansion for gamma white noise functionals.     

Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients

In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators –(a)+b ₁+b ₂+V with real measurable (singular) coefficients, on an open set R ^N . The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on L ^p () for p only from an interval in [1,), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale L ^p (). We show however that the semigroup is ultracontractive in the scale of weighted spaces L ^p (,²dx) with a suitable weight and derive an upper and lower bound on its integral kernel.     

Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree

Let G=A ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|₌₀=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH in some representation space H . In contrast with the case of PGL(2,), the representation is not irreducible.Let _o (g) be the derivative of the spherical function corresponding to the complementary series of A ut(T). We show that –d(e,ge) and _o (g) come from cohomologous cocycles. Moreover, _o is associated to one of the two (irreducible) special representations of A ut(T).     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

Theorems on Lower Semicontinuity and Relaxation for Integrands with Fast Growth

We prove theorems on the lower semicontinuity and integral representations of the lower semicontinuous envelopes of integral functionals with integrands L of fast growth: c ₁ G(|Du|) + c ₂ L c ₃ G(|Du|) + c ₄ with c ₃ c ₁ > 0 and G : [0, [ [0, [ is an increasing convex function such that vG (v)/G(v) as v and is increasing for large v. Repeating the results for the case of the standard growth (G() = ||^p) the quasiconvexity of integrands characterizes the lower semicontinuity of integral functionals and their quasiconvexifications yield the integral functionals that are lower semicontinuous envelopes.Original Russian Text Copyright © 2005 Sychev M. A.The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00162).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 679–697, May–June, 2005.     

The Long Dimodules Category and Nonlinear Equations

Let H be a bialgebra and _H L^H be the category of Long H-dimodules defined, for a commutative and co-commutative H, by F. W. Long and studied in connection with the Brauer group of a so-called H-dimodule algebra. For a commutative and co-commutative H, _H L^H =_H YD^H (the category of Yetter–Drinfel'd modules), but for an arbitrary H, the categories _H L^H and _H YD^H are basically different. Keeping in mind that the category _H YD^H is deeply involved in solving the quantum Yang–Baxter equation, we study the category _H L^H of H-dimodules in connection with what we have called the D-equation: R¹² R²³ = R²³ R¹², where R End_k(M M) for a vector space M over a field k. The main result is a FRT-type theorem: if M is finite-dimensional, then any solution R of the D-equation has the form R = R_{(M, , )}, where (M, , ) is a Long D(R)-dimodule over a bialgebra D(R) and R_{(M, , )} is the special map R_{(M, , )}(m n) : = n₁ m n₀. In the last section, if C is a coalgebra and I is a coideal of C, we introduce the notion of D-map on C, that is a k-bilinear map : C C / I k satisfying a condition which ensures on the one hand that, for any right C-comodule, the special map R is a solution of the D-equation and, on the other, that, in the finite case, any solution of the D-equation has this form.     

The Neighbourhood of Dihedral 2-Groups

We examine two particular constructions that derive from a 2-group G=G() another 2-group G(*) for the case when G() is one of , , . The constructions (cyclic and dihedral) have the property that x*y=xy for exactly 3/4 of all pairs (x,y)G×G. Work supported by by Grant Agency of Charles University, grant number 269/2001/B-MAT/MFF. The second and the third authors were also supported by the institutional grants MSM 113200007 and MSM 210000010, respectively.Mathematics Subject Classifications (2000) primary: 20D60, secondary: 05B15.     

Positive Solutions of Nonlinear Elliptic Equations in Unbounded Domains in R 2

We study the existence of positive solutions of the nonlinear equation u+f(,u)=0, in D with u=0 on D, where D is an unbounded domain in R ² with a compact nonempty boundary D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.     

Formulating Measure Differential Inclusions in Infinite Dimensions

Measure differential inclusions were introduced by J. J. Moreau to study sweeping processes, and have since been used to study rigid body dynamics and impulsive control problems. The basic formulation of an MDI is d / d (t) K(t) where is a vector measure, an unsigned measure, and K() is a set-valued map with closed, convex values and is hemicontinuous. Note that need not be absolutely continuous with respect to . Stewart extended Moreau's original concept (which applied only to cone-valued K()) to general convex sets, and gave strong and weak formulations of d / d (t) K(t) where K(t) R ⁿ . Here the strong and weak formulations of Stewart are extended to infinite-dimensional problems where K(t) X where X is a separable reflexive Banach space; they are shown to be equivalent under mild assumptions on K().     

Indecomposable Sylow 2-Subgroups of Simple Groups

Let S be a Sylow 2-subgroup of a finite simple group and let S=S₁×S₂××S_k be the direct product and each component S_i, i=1,2,...,k is indecomposable. In this article, we prove that each S_i is also a Sylow 2-subgroup of some simple group. Mathematics Subject Classifications (2000) 20E32, 20D20.     

Cones of highest weight vectors, weight polytopes, and Lusztig's q-analog

We relate the invariant theory of cones of highest weight vectors to weight multiplicities and theirq-analogs. Whenever the action of a maximal torus on the coneC^* has some nice properties, we obtain simple closed formulas for all weight multiplicities and theirq-analogs in the representationsV _n ,n. We find a connection between the character ofV _n and the respective weight polytopes.This research was supported in part by the Alexander von Humboldt Foundation and INTAS Grant No. 93-0893     

Optimal Constant in Approximation by Bernstein Operators

We obtained that for any n N, C = 1 is the smallest constant for which the inequality ||B _n (f) - f|| C ₂(f, 1/n) holds on the class of continuous functions f, as well as on the class of bounded functions f, where B _n is the Bernstein operators of degree n, ₂ is the second order modulus and || || is the sup-norm.