Blocks of orthogonal random variables and the strong law of large numbers

(X _k ),k=1,2,... — _k ² >1; (X _k ) , E(X _k X _t )=0 p k<>(p+1) (p,k,l=1, 2, ...) , , ,

Asymptotics of polynomials orthogonal with respect to varying measures

Letd be a finite positive Borel measure on the interval [0, 2] such that >0 almost everywhere; andW _n be a sequence of polynomials, degW _n =n, whose zeros (w _n ,1,,w _n,n lie in [|z|1]. Let d _n <> for each_nN, whered _n =d/|W _n (e ⁱ )|². We consider the table of polynomials _n,m such that for each fixednN the system _n,m,mN, is orthonormal with respect tod _n . If

Spectral synthesis for Euler type operators

A=(a _ij) _i ^j=1 — k-o ,a _ij . :

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

We are concerned with the semilinear polyharmonic model problem (–)K v = v +v|v| ^s–1 inB,D v|B = 0 for ¦|<-K – 1. HereK ,B is the unit ball in ⁿ,n >2K, is the critical Sobolev exponent. Let ₁ denote the first Dirichlet eigenvalue of (-)^K inB. The existence of a positive radial solutionv is shown for

Circularity of Finite Groups without Fixed Points

Let áA, B | A^m=1, Bⁿ=A^t, BAB^-1=A^r?\langle A, B\,\vert\, A^\mu=1,\, B^\nu=A^t,\, BAB^{-1}=A^\rho\rangle where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Realizations of regular polytopes

Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ^{** *–1*} ( ^*), where ^* is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to ^* have a fixed space of dimensionw _G . Then the relations

On the Pólya conjecture concerning the maximum and minimum of the modulus of an entire function of finite order given by a lacunary power series

, (fz) , ,

Interpolation by Lagrange polynomials,B-splines and bounds of errors

m N k=m–2,m–1

Some remarks on Turán's inequality. III: The completion

. 0pq, 1–1/p+1/p0. f(x) — n, [–1,1],

Subadditive functions on modules and subgroups of abelian groups

Thepositive half A ₊ of an ordered abelian groupA is the set {x Ax 0} andM A ₊ is amodule if for allx, y M alsox + y, |x – y| M. If A ₊ \M thenM() is the module generated byM and. S M isunbounded inM if(x M)(y S)(x y) and isdense inM if (x₁, x₂ M)(y S) (x₁ <>₂ x₁ y x₂). IfM is a module, or a subgroup of any abelian group, a real-valuedg: M R issubadditive ifg(x + y) g(x) + g(y) for allx, y M. The following hold:

Christoffel functions,orthogonal polynomials,and Nevai's conjecture for Freud weights

We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials on thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e ^–Q, whereQ: is even and continuous in, Q^" is continuous in (0, ) andQ ^'>0 in (0, ), while, for someA, B,

Maximal polynomial subordination to univalent functions in the unit disk

Let C be a simply connected domain, 0, and let _n,nN, be the set of all polynomials of degree at mostn. By _n() we denote the subset of polynomials p _n withp(0)=0 andp(D), whereD stands for the unit disk {z: |z|<1}, and=" by=">we denote the maximal range of these polynomials. Letf be a conformal mapping fromD onto ,f(0)=0. The main theme of this note is to relate _n (or some important aspects of it) to the imagesf _s (D), wheref _s (z):=f[(1–s)z], 0s<1. for=" instance=" we=" prove=" the=" existence=" of=" a=" universal=">c ₀ such that, forn2c ₀,     

Necessary and sufficient conditions for interpolation with functions having monotoner-th derivatives

,

Relations Between Some Dimensions of Semimodular Lattices

The aim of this paper is to present relations between Goldie, hollow and Kurosh-Ore dimensions of semimodular lattices. Relations between Goldie and Kurosh-Ore dimensions of modular lattices were studied by Grzeszczuk, Okiski and Puczyowski.