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.     

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.     

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.     

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.     

On the informative power of all possible linear functionals for the discretization of solutions of the Klein-Gordon equation in the metric of <Emphasis Type="Italic">L</Emphasis><Superscript>2,∞</Superscript>

For the Klein-Gordon equation with initial conditions in Nikol’skii classes, we obtain sharp orders of discretization errors with respect to precise information. In particular, these orders imply qualitatively and quantitatively that raising the smoothness of one of the initial conditions relative to the other does not improve the discretization characteristics of the solution of the equation. Bounds for information distortion are given under which the limit possible errors, i.e., discretization errors with respect to precise information, are still maintained. This is important because superfluous computational accuracy used when implementing an algorithm results in an unjustified increase in memory requirements and the number of arithmetic operations without improving the accuracy inherent to the algorithm. The quality of approximation aggregates is also estimated in mean square with respect to probability measures on Nikol’skii classes.     

On the value distribution of <Emphasis Type="Italic">L</Emphasis>-functions of the Selberg class

In this paper, two limit theorems in the sense of the weak convergence of probability measures in the space of meromorphic functions and on the complex plane are proved for L-functions from a subclass of the Selberg class. The explicit form of the limit measures is given.     

Asymptotic behavior of the eigenvalues of the one-dimensional weighted<Emphasis Type="Italic">p</Emphasis>-Laplace operator

In this paper we study the spectral counting function for the weightedp-laplacian in one dimension. First, we prove that all the eigenvalues can be obtained by a minimax characterization and then we show the existence of a Weyl-type leading term. Finally we find estimates for the remainder term. The first author is supported by Universidad de Buenos Aires grant TX48, by ANPCyT PICT No. 03-05009. The second author is supported by CONICET and Universidad de San Andres.     

“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 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.     

Saturation of the linear methods of summation of Fourier series in the spaces <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript><Subscript>φ</Subscript>

We consider the problem of saturation of the linear methods of summation of Fourier series in the spaces S ^p _φ specified by arbitrary sequences of functions defined in a certain subset of the space ℂ. Sufficient conditions for the saturation of the indicated methods in these spaces are established. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 815–828, June, 2008.     

The quantum double of the Yangian of the Lie superalgebra <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">m,n</Emphasis>) and computation of the universal <Emphasis Type="Italic">R</Emphasis>-matrix

The Yangian double DY(A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. We prove the triangular decomposition for Yangian Y(A(m, n)) and its quantum double DY(A(m, n)) as a corollary of the PBW theorem. We introduce normally ordered bases in the Yangian and its dual Hopf superalgebra in the quantum double. We calculate the pairing formulas between the elements of these bases. We obtain the formula for the universal R-matrix of the Yangian double. The formula for the universal R-matrix of the Yangian, which was introduced by V. Drinfel’d, is also obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 185–208, 2005.     

Estimates of the <Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript> norm of the error in the conjugate gradient algorithm

In this paper we derive a formula relating the norm of the l₂ error to the A-norm of the error in the conjugate gradient algorithm. Approximating the different terms in this formula, we obtain an estimate of the l₂ norm during the conjugate gradient iterations. Numerical experiments are given for several matrices. AMS subject classification 65F10, 65G20     

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.     

Estimation of the norm of the derivative of a monotone rational function in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We show that the derivative of an arbitrary rational function R of degree n that increases on the segment [−1, 1] satisfies the following equality for all 0 < ε < 1 and p, q > 1:

Polynomial Form of de Branges Conditions for the Denseness of Algebraic Polynomials in the Space <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>w</Subscript><Superscript>0</Superscript></Stack>

In the criterion for polynomial denseness in the space C _w ⁰ established by de Branges in 1959, we replace the requirement of the existence of an entire function by an equivalent requirement of the existence of a polynomial sequence. We introduce the notion of strict compactness of polynomial sets and establish sufficient conditions for a polynomial family to possess this property. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 305–319, March, 2005.