The GB and GC properties of generalized elliposids

In the paper one investigates the conditions under which the subsets of the form of a Hilbert space, where a0, p>1, =1, ..., possesses the GB or the GC properties.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 104–108, 1983.The author is grateful to V. N. Sudakov for his interest in the paper.     

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.     

On Rounding Error Sums Related to the Circle Problem. ii

For a positive real parameter t, real numbers , , and , we consider sums , where is the rounding error function, i.e.\ . Generalizing and improving the main result of Part I of the paper we show that there exists an absolute constant such that for all , and all . Further, we give applications concerning the circle problem with linear, polynomial, and general weight.     

A Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

Coloured Permutations Containing and Avoiding Certain Patterns

Let be the set of all coloured permutations on the symbols 1, 2, . . . , n with colours 1, 2, . . . , r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let be a subset of d colours; we define to be the set of all coloured permutations . We prove that the number of -avoiding coloured permutations in . We then prove that for any , the number of coloured permutations in which avoid all patterns in except for and contain exactly once equals . Finally, for any , this number equals . These results generalize recent results due to Mansour, Mansour and West, and Simion.AMS Subject Classification: 05A05, 05A15.     

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,

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.     

The Kreîn-Langer problem for Hilbert space operator valued functions on the band

We deal with the Kreîn-Langer problem for -valued functions on the band (–2a, 2a)×, where is the algebra of continuous linear operators on a Hilbert space ,a a finite positive number and a topological Abelian group. We show that every weakly continuous -indefinite function admits a strongly continuous -indefinite continuation to × with the same indefiniteness index . We give a parametrization of the extensions in terms of operator-valued Schur functions.     

Second-Order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients

We consider second-order subelliptic operators with complex coefficients over a connected Lie group G. If the principal coefficients are right uniformly continuous then we prove that the operators generate strongly continuous holomorphic semigroups with kernels K satisfying Gaussian bounds. Moreover, the kernels are Hölder continuous and for each 0, 1 and > 0 one has estimates

A factorization of regular generalized Nevanlinna functions

LetQ be a regular operator valued generalized Nevanlinna function with negative index , i.e. . It is shown that then there exists a rational functionB(z), which collects the generalized poles and zeros ofQ that are not of positive type such that the function

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

On the Densities of Certain Lattice Packings by Parallelepipeds

Given any (non-degenerate) n-dimensional lattice L, let (L) denote the supremum of the numbers such that there exists a lattice packing Q + L of density where Q is some o-symmetric parallelepiped with faces parallel to the coordinate axes. Many efforts have been made to determine or estimate the minimal such density _n taken over all n-dimensional lattices. It is known that 0$$ " align="middle" border="0"> . Here we investigate a sequence of lattices L _n which are known to minimize the function (L) in dimensions n 3 and are likely to provide the minima _n = (L _n ) in certain higher dimensions. We establish the inequality (L _n ) n ^–n/2 which supports the conjecture that lim sup _n ( _n )^{1/(n log n)} is positive.     

K-theory of twisted differential operators on flag varieties

Let be a semisimple Lie algebra overk, an algebraically closed field of characteristic zero, and let be a Cartan subalgebra inside a Borel subalgebra of . LetU be the enveloping algebra of . For letM() denote the corresponding Verma modúle and letU _u=U/AnnM(). LetW be the Weyl group and letW ⁰ be the stabiliser of inW. We prove the following theorem, which affirms a conjecture of T.J. Hodges.Oblatum 16-XII-1994     

Infinite-Time Quenching in a Fast Diffusion Equation with Strong Absorption

This work is concerned with the fast diffusion equation , where 0 < m < 1 and κ < 1. A global positive solution is said to quench regularly in infinite time if for some bounded sequence and some , and if for all compact . It is shown that such regular quenching in infinite time occurs for a large class of initial data if κ > m , whereas it is impossible in one space dimension when κ < −m and the solution is radially symmetric and nondecreasing for x > 0.      

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Analog of the Akhiezer–Krein–Favard Sums for Periodic Splines of Minimal Defect

Let be the space of 2-periodic functions whose (r – 1)th-order derivative is absolutely continuous on any segment and rth-order derivative belongs to L _p, S _2n,m is the space of 2-periodic splines of order m of minimal defect over the uniform partition . In this paper, we construct linear operators such that

An Lp-Lq-Version of Morgan’s Theorem for the Dunkl-Bessel Transform

In this paper, we give an L^p-L^q-version of Morgans theorem for the Dunkl-Bessel transform on More precisely, we prove that for all and then for all measurable function f on the conditions and imply f = 0, if and only if where are the Lebesgue spaces associated with the Dunkl-Bessel transform.Received: November 21, 2003 Revised: April 26, 2004 Accepted: May 28, 2004     

Transversals of Families of Translations of a Planar Convex Compact

Let be a family of translations of a convex compact set such that every two elements of have a common point. Then there exist three points such that each element of contains one of these points. This answers in the affirmative an old question by Grünbaum. Bibliography: 13 titles.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.