On the imbedding of the Waterman class in the class <Emphasis Type="Italic">h</Emphasis><Stack><Subscript>p</Subscript><Superscript>ω</Superscript></Stack>

In this paper, a necessary and sufficient condition for the inclusion of the class ΛBV in the class H _p ^ω is found. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1557–1562, November, 2005.     

On the distribution of the zeros of the cubic <Emphasis Type="Italic">L</Emphasis>-function

The distribution of the real parts of the zeros of the cubic L-function is considered. Some observations based on numerical results are presented. Bibliography: 4 titles.     

On the cubic <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> spline interpolant to the Heaviside function

We prove that the univariate interpolating cubic L ₁ spline to the Heaviside function at three sites to the left of the jump and three sites to the right of the jump entirely agrees with the Heaviside function except in the middle interval where it is the interpolating cubic with zero slopes at the end point. This shows that there is no oscillation near the discontinuous point i.e. no Gibbs’ phenomenon.     

Approximate values of the multiplicities in the arm of the cocharacter sequence of <Emphasis Type="Italic">M</Emphasis><Subscript>2,1</Subscript>

In this paper we study the cocharacter sequence of M _2,1 and obtain estimates for the multiplicities of the irreducible S _n -characters χ^λ, where λ is any partition with at most 5 parts. The author’s work was supported by both the Faculty Research Council of De Paul University and the National Security Agency, under Grant MDA904-500270. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.     

Some remarks concerning the Fourier transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> (ℝ)

Two useful estimates are proved for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus.     

ON THE RATE OF WEIGHTED Lp CONVERGENCE BY THE GRUENWALD INTERPOLATORY OPERATORS

This paper gives the weighted Lp convergence rate estimations of the Gruenwald interpolatory polynomials based on the zeros of Chebyshev polymomials of the first kind,and proves that the order of the estimations is optimal for p≥1.     

Local “superlinearity” and “sublinearity” for the p-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems −Δpu=fλ(x,u), , where Ω is a bounded domain in RN, N>p, and λ>0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)uq+b(x)ur, where 0?q<p−1<r?p^∗−1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in and , a C1,α estimate for equations of the form −Δpu=h(x,u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian.     

Absence of the interpolation property in the calculi <Emphasis Type="Italic">Lα</Emphasis> and <Emphasis Type="Italic">Lf</Emphasis>

We prove the absence of the Craig interpolation property in the polymodal calculi related to some Ershov spaces.     

“Quantizations” of the second Painlevé equation and the problem of the equivalence of its <Emphasis Type="Italic">L</Emphasis>-<Emphasis Type="Italic">A</Emphasis> pairs

We show that the known Flaschka-Newell L-A pair for the second Painlevé equation gives solutions to linear evolutionary equations similar to the quantum Schrödinger equations. Using the Fourier transform for distributions, we derive this pair from the classical Garnier pair.     

On the Loebl–Komlós–Sós conjecture

The Loebl–Komlós–Sós conjecture says that any graph G on n vertices with at least half of vertices of degree at least k contains each tree of size k. We prove that the conjecture is true for paths as well as for large values of k(k ≥ n − 3). © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 269–276, 2000     

Finite groups having the same prime graph as the group <Emphasis Type="Italic">A</Emphasis><Subscript>10</Subscript>

We study solutions to partial differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter transformation group such that all eigenvalues of the infinitesimal matrix are positive. The infinitesimal matrix may contain a nilpotent part. In the asymptotic scale of regularly varying functions, we find conditions under which such differential equations have asymptotically homogeneous solutions in the critical case.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

On the parity of the class number of the 7<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>th cyclotomic field

For an odd prime number p and an integer n ≥ 0, let h _n be the class number of the p ⁿ⁺¹st cyclotomic field Q(). It is known that when p = 3 or 5, h _n is odd for all n ≥ 0. We prove that the same holds also when p = 7. The author was partially supported by Grant-in-Aid for Scientific Research (C), No. 19540005, Japan Society for the Promotion of Science.     

Computation of the zeros of the <Emphasis Type="Italic">L</Emphasis>-function associated with the cubic theta function

The paper considers the L-function associated with the Kubota-Patterson cubic theta function. A method for computing values of the L-function is developed, and some conjectures on the distribution of its zeros are advanced and supported by numerical results. Bibliography: 12 titles. Translated from Zapiski Nauchnykh. Seminarov POMI, Vol. 357, 2008, pp. 180–194.     

Phases of the goldstone multitrace matrix model in the large-<Emphasis Type="Italic">N</Emphasis> limit

We consider the Goldstone Hermitian matrix model with a multitrace term. When defining the solution on two intervals, we introduce a special parameter ξ describing the phase. We discuss the phase existence conditions at ξ = 0 (or 1) and at ξ = 1/2. We calculate the propagator and the vacuum energy in the symmetric case ξ = 1/2. In the general case, we discuss the solution structure and calculate the magnetization and other parameters expressed in terms of the sum of all the intervals. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 457–465, September, 2007.