Sur l'homotopie rationnelle des espaces fonctionnels

Let X be a nilpotent space such that it exists k1 with H^p (X,) = 0 p > k and H^k (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of Y^X (resp. is isomorphic as Lie algebra, to H^* (X,) (_* (Y) ) (resp.⁺ (X,) (_* (Y) )). If X is formal and Y -formal, then the spaces Y^X and are -formal. Furthermore, if dim _* (Y) is infinite and dim H^* (Y,Q) is finite, then the sequence of Betti numbers of grows exponentially.     

On Interpolation Sequences of One Class of Functions Analytic in the Unit Disk

We establish a criterion for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and satisfying the relation

Nondiagonal Hermite–Sobolev Orthogonal Polynomials

In this work, we study algebraic and analytic properties for the polynomials { Q _n } _{n 0}, which are orthogonal with respect to the inner product where , R such that – ² > 0.     

From Ginzburg-Landau to Gross-Pitaevskii

In this note we consider the Gross-Pitaevskii equation i _t ++(1–²)=0, where is a complex-valued function defined on ^N×, and study the following 2-parameters family of solitary waves: (x, t)=e ^it v(x ₁–ct, x), where and x denotes the vector of the last N–1 variables in ^N . We prove that every distribution solution , of the considered form, satisfies the following universal (and sharp) L -bound:

Boundary Asymptotics for Orthogonal Rational Functions on the Unit Circle

Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points { _k } _{k = 1} included in a compact subset of the unit disk, we consider the orthogonal rational functions _n that are obtained by orthogonalization of the sequence { 1, z / ₁, z ² / ₂, ... } where , with respect to the inner product In this paper we discuss the behaviour of _n (t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all _k = 0.     

Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _i . We refer to E ₀ and E _D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We show

On Some Proximinal Subspaces of Modular Spaces

Let (, , ) be a complete measure space, L⁰ the vector lattice of -measurable real functions on , : L⁰ [0, )] a lattice semimodular, the corresponding modular space, S₀ the ideal generated by and 0,{\text{ }}\exists {\text{ }}s \in {\text{ }}S_{\text{0}} {\text{ such that }}\rho \left( {\frac{{x - s}}{\user1{\lambda }}} \right) < \infty } \right\}$$ " align="middle" border="0"> . In X consider the distance 0:\rho \left( {\frac{{x - y}}{\user1{\lambda }}} \right) \leqq \user1{\lambda }} \right\}$$ " align="middle" border="0"> and, if is convex, the distances d_L, d_o subordinated to the Luxemburg and Amemiya-Orlicz norms, respectively. We give necessary and sufficient conditions for H(S_o) in order to be proximinal in X with the distances d, d_L and d_o.     

On the Boundedness of a Recurrence Sequence in a Banach Space

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: where |y _n} and | _n } are sequences bounded in B, and A _k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition is satisfied, then the sequence |x _n} is bounded for arbitrary bounded sequences |y _n} and | _n } if and only if the operator has the continuous inverse for every z C, |z| 1.     

Piecewise-Convex Maximization Problems

A function F:Rⁿ R is called a piecewise convex function if it can be decomposed into , where f _j:Rⁿ R is convex for all jM={1,2...,m}. We consider subject to xD. It generalizes the well-known convex maximization problem. We briefly review global optimality conditions for convex maximization problems and carry one of them to the piecewise-convex case. Our conditions are all written in primal space so that we are able to proposea preliminary algorithm to check them.     

On Rounding Error Sums Related to the Circle Problem

For a large real parameter t and 0 a b we consider sums where is the rounding error function, i.e. (z) = z - [z] - 1/2. We generalize Huxley's well known estimate by showing that holds uniformly in 0 a b . Fruther, we investigate an analogous question related to the divisor problem and show that the inequality , which (due to Huxley) holds uniformly in 0 a b , and which is in general not true for 1 a b t, is true uniformly in 0 a b .     

Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

Interpolation Sequences for the Class of Functions of Finite η-Type Analytic in the Unit Disk

We establish conditions for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and such that

Aperiodische Wahrscheinlichkeitsmaße auf topologischen Gruppen

Several conditions are shown to be equivalent to the aperiodicity of a regular probability measure on a locally compact, separated topological GroupG. In particular, is aperiodic if and only if the sequence ( ( ⁽ⁿ⁾ denoting then-th convolution power of ) is convergent for any nonvoid open subsetU ofG with compact closure. It is always assumed that the support of generatesG as a closed semigroup.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

über Wurzeln und Logarithmen beschrÄnkter Ma\e

Ohne ZusammenfassungBezeichnungen und Symbole G lokalkompakte topologische Gruppe - M(G)/R(G)/P(G)/ regulÄre komplexe/reelle/positive Ma\e/ - Q(G)/W(G) Ma\e mit · l/Wahrscheinlichkeitsma\e - _x Punktma\: _x(f)=f(x) - v Faltung, — Bekanntlich bildet M(G) bezüglich der Faltung eine Banachalgebra; - Involution in M(G), , wobei — die Komplexkonjugierte bezeichnet - × diskreter Anteil eines Ma\es, - T _gm Faltungsoperator auf L ² (G) (bezüglich des linken Haarschen Ma\es), f.ü. - p(·)/q(·)/u(·) - exp(·.) Exponentialfunktion, exp - normal/unitÄr/symmetrisch/positiv definit bezeichnet man ein Ma\ , wenn der Faltungsoperator T diese Eigenschaft besitzt - invertierbar hei\t M(G), wenn ein vM(G) existiert, so da\ v = v= _e - _1/n n-te Wurzel von ₁ hei\t wenn( _1/n)ⁿ= ₁ - ₁ hei\t unendlich teilbar wenn zu jedem natürlichen n eine n-te Wurzel _1/n von existiert - N Menge der natürlichen Zahlen     

A proof of Shelah's partition theorem

A self contained proof of Shelah's theorem is presented: If is a strong limit singular cardinal of uncountable cofinality and 2 > ⁺ then .     

On the relation between Fourier and Leont’ev coefficients with respect to smirnov spaces

Yu. Melnik showed that the Leontev coefficients _f () in the Dirichlet series 2}}$$ " align="middle" border="0"> of a function f E ^p (D), 1 < p < , are the Fourier coefficients of some function F L ^p , ([0, 2]) and that the first modulus of continuity of F can be estimated by the first moduli and majorants in f. In the present paper, we extend his results to moduli of arbitrary order.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 517–526, April, 2004.     

Über die Riemannsche Einbettungsgeometrie der Veronesemannigfaltigkeiten

Let E be a n-dimensional euclidean vector space. The subset V _k ⁿ ={x ... x | x E} of ^kE is called a Veronesemanifold. The scalar product of E induces a euclidean structure on ^kE. Passing to the corresponding projective space , one may consider as a riemannian submanifold of the space form . In this paper we study properties of the pair of riemannian manifolds.