Approximating periodic functions in the uniform metric by Jackson type polynomials

Let C be the space of continuous 2π-periodic functions f with the norm . Let , where , be the Jackson polynomials of a function f, E _n (f) be the best approximation of f in the space C by trigonometric polynomials of order n, and let , be the function trigonometrically conjugate to the primitive of f. The paper establishes results of the following types:

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; , ( ), ,

Littlewood-Paley theorem for arbitrary intervals: Weighted estimates

Let 1 < r < 2 and let b is a weight on ℝ such that satisfies the Muckenhoupt condition A_r′/2 (r′ is the exponent conjugate to r). If f_j are functions whose Fourier transforms are supported on mutually disjoint intervals, then

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

Two classes of finite semigroups and monoids involving Lucas numbers

The class of finitely presented groups is an extension of the class of triangle groups studied recently. These groups are finite and their orders depend on the Lucas numbers. In this paper, by considering the three presentations

Approximating periodic functions by interpolation sums of Jackson type

Let

Parabolic twists for the linear algebras <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>−1</Subscript>

New solutions of twist equations for the universal enveloping algebras U (A_n−1) are found. These solutions can be represented as products of full chains of extended Jordanian twists Abelian factors (“rotations”) , and sets of quasi-Jordanian twists . The latter are generalizations of Jordanian twists (with carrier b²) for special deformed extensions of the Hopf algebra U (b²). The carrier subalgebra for the composition is a nonminimal parabolic subalgebra in A _n−1 such that . The parabolic twisting elements are obtained in an explicit form. Details of the construction are illustrated by considering the examples n = 4 and n = 11. Bibliography: 21 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 187–213.     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

On the minimum length of some linear codes

We determine the minimum length n _q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n _q (k, d) = g _q (k, d) + 1 for when k is odd, for when k is even, and for . This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175). This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 17540129.     

Approximations to q-logarithms and q-dilogarithms, with applications to q-zeta values

We construct simultaneous rational approximations to q-series L₁(x₁; q) and L₁(x₂; q) and, if x = x₁ = x₂, to series L₁(x; q) and L₂(x; q), where

On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality holds for α,β > − 1 and n ≥ 1, θ ∈ (0, π), where are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where .      

Behavior of Automorphic L-Functions at the Center of the Critical Strip

Let be the Hecke eigenbasis of the space of -cusp forms of weight 2. Let p be a prime. Let be the Hecke L-series of form . The following statements are proved:

An ultimate extremely accurate formula for approximation of the factorial function

We prove in this paper that for every x ≥ 0,

Surfaces with many solitary points

It is classically known that a real cubic surface in cannot have more than one solitary point (or -singularity, locally given by x ² + y ² + z ² = 0) whereas it can have up to four nodes (or -singularity, locally given by x ² + y ² − z ² = 0). We show that on any surface of degree d ≥ 3 in the maximum possible number of solitary points is strictly smaller than the maximum possible number of nodes. Conversely, we adapt a construction of Chmutov to obtain surfaces with many solitary points by using a refined version of Brusotti’s Theorem. Combining lower and upper bounds, we deduce: , where denotes the maximum possible number of solitary points on a real surface of degree d in . Finally, we adapt this construction to get real algebraic surfaces in with many singular points of type for all k ≥ 1.      

σk-Scalar curvature and eigenvalues of the Dirac operator

On a four-dimensional closed spin manifold (M ⁴, g), the eigenvalues of the Dirac operator can be estimated from below by the total σ₂-scalar curvature of M ⁴ as follows: Equality implies that (M ⁴, g) is a round sphere and the corresponding eigenspinors are Killing spinors.Dedicated to Professor Wang Guangyin on the occasion of his 80th birthday.     

A speciality theorem for curves in P<Superscript>5</Superscript>

Let be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that , where ω _C denotes the dualizing sheaf of . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if , then Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, , and . We give also some partial results in the general case , .      

Hamiltonicity in balancedk-partite graphs

One of the earliest results about hamiltonian graphs was given by Dirac. He showed that if a graphG has orderp and minimum degree at least thenG is hamiltonian. Moon and Moser showed that a balanced bipartite graph (the two partite sets have the same order)G has orderp and minimum degree more than thenG is hamiltonian. In this paper, their idea is generalized tok-partite graphs and the following result is obtained: LetG be a balancedk-partite graph with orderp = kn. If the minimum degree

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain