New Proof of the Semmes Inequality for the Derivative of the Rational Function

In the open disk of the complex plane, we consider the following spaces of functions: the Bloch space ; the Hardy--Sobolev space ; and the Hardy--Besov space . It is shown that if all the poles of the rational function R of degree n, , lie in the domain , then , where and depends only on . The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions.     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.     

Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series and whose coefficients satisfy the condition for , where the sequence can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If as , then the series is uniformly convergent. If for all , then the sequence of partial sums of this series is uniformly bounded. If the series is convergent for and as , then this series is uniformly convergent. If the sequence of partial sums of the series for is bounded and for all , then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence is lacunary. In the general case, they are not necessary.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

On Exact Constant in the Generalized Poincare Inequality

Some problems concerning an exact constant in the generalized Poincare inequality with are discussed. In particular, the conjecture that the extremal of this problem is symmetric is confirmed. Bibliography: 7 titles.     

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

We study operators (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space are proved with the help of the Lorentz sequence spaces . Sequences of functions belonging to fixed bounded sets in the spaces are characterized for and . The possibility of distinguishing weak type operators (bounded in the space ) from operators factorizable through is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in , is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.     

Sharp Pointwise Interpolation Inequalities for Derivatives

We prove new pointwise inequalities involving the gradient of a function , the modulus of continuity of the gradient , and a certain maximal function and show that these inequalities are sharp. A simple particular case corresponding to and is the Landau type inequality , where the constant 8/3 is best possible and

A Splitting Inequality

We prove that if A is a finite set, and if is an downwards-closed family of subsets of A, and if fx is the proportion of x-element subsets of A in , then fa · fb fa + b – r, if r < . We connect this result with the Weak Threshold Theorem.     

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes with are either bipartite or antipodal, i.e., intersection numbers satisfy either for all for all . In this paper, we show that imprimitive -polynomial association schemes with are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either .     

On Invariance of Some Classes of Holomorphic Functions Under Integrodifferential Operators

The following classes of functions analytic in the unit disk are considered:

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

Hoffman’s Least Error Bounds for Systems of Linear Inequalities

Let E be a normed space, and . Let . We give some exact formulas for 7#x03C4;.     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

A Jackson Type Estimate for Shepard Operators in L p Spaces for p≧ 1

This paper considers the approximation of the Kantorovich–Shepard operators in spaces for . For the Kantorovich–Shepard operators are defined by (1.1). Then

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.