On the smallest poles of Igusa's p-adic zeta functions

Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kⁿ. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n=2 and less than −1 for n=3 which occur as the real part of a pole.     

Approximation characteristics for diagonal operators in different computational settings

First we study several extremal problems on minimax, and prove that they are equivalent. Then we connect this result with the exact values of some approximation characteristics of diagonal operators in different settings, such as the best n-term approximation, the linear average and stochastic n-widths, and the Kolmogorov and linear n-widths. Most of these exact values were known before, but in terms of equivalence of these extremal problems, we present a unified approach to give them a direct proof.     

On entire functions whose derivatives are¶integer-valued on geometric progressions

In this paper we study functions which together with all their derivatives take values in a number field ? on a sequence (u _n ) in ? satisfying certain conditions. As one corollary we get a result on the growth rate of entire transcendental functions whose derivatives assume integral values on the geometric progression u ⁿ ,n= 0,1,…. In a second theorem we prove that the lower bound for the growth rate stated in the corollary is best possible. Received: 16 May 2000     

Some New Applications of the Subspace Theorem

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(q ⁿ ) is an algebraic integer for infinitely many n, then h(f(q ⁿ )) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.     

On the Jacobi series

Summary The Jacobi series of a functionf is an expansion in a series of ascending powers of a prescribed polynomialP of degreen in which the coefficients are polynomials of lesser degree. These coefficients are usually expressed as contour integrals or are determined by their interpolatory properties. We show how they may be expressed as generalized derivatives off with respect toP. In so doing we also show how the Jacobi series may be expressed (in yet another way) as a generalized Taylor series. In addition, we obtain a number of interesting relations among the generalized derivatives.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Slice monogenic functions

In this paper we offer a new definition of monogenicity for functions defined on ℝ ⁿ⁺¹ with values in the Clifford algebra ℝ _n following an idea inspired by the recent papers [6], [7]. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra ℝ _n . We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.     

Perturbed nutations,Love numbers and elastic energy of deformation for Earth models 1066A and 1066B

In this paper, the integration method of Takeuchi (1950) is applied to Gilbert and Dziewonski Earth Models1066A and1066B, with the aim of obtaining the functionsF _n G _n andK _n that make up the expression of the displacement vector for the Earth's elastic mantle. The procedure of this integration is valid for any ordern of the perturbing potential expansion in spherical harmonics. The values obtained are applied to recalculate the results of the Hamiltonian theory of Getino and Ferrándiz (1990, 1991a, 1991b) for the rotational motion of an Earth whose elastic mantle follows an earlier Takeuchi's model, and we get new numerical series for nutations in longitude and obliquity. On the other hand, we have calculated the Love numbers, obtaining results which agree perfectly with the experimetal data. Finally, the elastic energy of deformation is computed, obtaining values for any ordern.     

On the best linear approximation methods and the widths of certain classes of analytic functions

We discuss the best linear approximation methods in the Hardy spaceH _q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 2, pp. 186–193, February, 1999.     

On obtaining close estimates in the approximation of functions of many variables by sums of functions of a fewer number of variables

We present a simple method for finding the values of the best approximation of a function of n variables of a given class by means of sums of two functions of a fewer number of variables; we establish close upper and lower bounds for the value of the best approximation to the functionf(x₁, ..., x_n), having the mixed derivativef x₁ ... x_n, by means of sums of a function of n–1 variables.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 105–114, July, 1972.     

On the best approximation of classes of convolutions of periodic functions by trigonometric polynomials

We present sufficient conditions for kernels to belong to the classN _n ^* . In certain cases, this enables us to find exact values of the best approximations of classes of convolutions by trigonometric polynomials in the metrics ofC andL.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1261–1265, September, 1995.     

A New Lower Bound on the Number of Odd Values of the Ordinary Partition Function

The parity of p(n), the ordinary partition function, has been studied for at least a century, yet it still remains something of a mystery. Although much work has been done, the known lower bounds for the number of even and odd values of p(n) for n ≤ N still appear to have a great deal of room for improvement. In this paper, we use classical methods to give a new lower bound for the number of odd values of p(n).     

Spectral parameter power series for arbitrary order linear differential equations

Let L be the n‐th order linear differential operator Ly=?₀y⁽ⁿ⁾+?₁y^(n?1)+?+?_ny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of n‐th order initial value problems and spectral problems.     

Embedding directed and undirected partial cycle systems of index λ > 1

Recent results have found small embeddings for partial m-cycle systems of order n with λ= 1. However, if λ> 1 then the best known techniques produce embeddings that are often quadratic functions of both m and n and linear functions of λ. In this article we obtain embeddings for partial m-cycle systems of order n, and of partial directed m-cycle systems, for all values of m. These embeddings are independent of λ and linear in both n and m. © 1993 John Wiley & Sons, Inc.     

The mean field traveling salesman and related problems

The edges of a complete graph on n vertices are assigned i.i.d. random costs from a distribution for which the interval [0, t] has probability asymptotic to t as t→0 through positive values. In this so called pseudo-dimension 1 mean field model, we study several optimization problems, of which the traveling salesman is the best known. We prove that, as n→∞, the cost of the minimum traveling salesman tour converges in probability to a certain number, approximately 2.0415, which is characterized analytically.     

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

Let k be a positive number and t _k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t _2k (n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t ₁₂(n), t ₁₆(n), t ₂₀(n), t ₂₄(n), and t ₂₈(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t ₃₂(n). In addition, we derive some identities involving the Ramanujan function (n), the divisor function ₁₁(n), and t ₂₄(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.     

Tight bounds for the vertices of degree k in minimally k‐connected graphs

For minimally k‐connected graphs on n vertices, Mader proved a tight lower bound for the number of vertices of degree k in dependence on n and k. Oxley observed 1981 that in many cases a considerably better bound can be given if is used as additional parameter, i.e. in dependence on m, n, and k. It was left open to determine whether Oxley's more general bound is best possible. We show that this is not the case, but give a closely related bound that deviates from a variant of Oxley's long‐standing one only for small values of m. We prove that this new bound is best possible. The bound contains Mader's bound as special case.     

Approximation of classes of periodic functions with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.     

Special moments

In this article, we show that a linear combination of n independent, unbiased Bernoulli random variables {X_k} can match the first 2n moments of a random variable Y which is uniform on an interval. More generally, for each p2, each X_k can be uniform on an arithmetic progression of length p. All values of lie in the range of Y, and their ordering as real numbers coincides with dictionary order on the vector (X₁,…,X_n). The construction involves the roots of truncated q-exponential series. It applies to a construction in numerical cubature using error-correcting codes [G. Kuperberg, Numerical cubature using error-correcting codes, arXiv:math.NA/0402047]. For example, when n=2 and p=2, the values of are the 4-point Chebyshev quadrature formula.     

AnO(m + n log n) Algorithm for the Maximum-Clique Problem in Circular-Arc Graphs

We present an algorithm to compute, inO(m + n log n) time, a maximum clique in circular-arc graphs (withnvertices andmedges) provided a circular-arc model of the graph is given. If the circular-arc endpoints are given in sorted order, the time complexity isO(m). The algorithm operates on the geometric structure of the circular arcs, radially sweeping their endpoints; it uses a very simple data structure consisting of doubly linked lists. Previously, the best time bound for this problem wasO(m log log n + n log n), using an algorithm that solved an independent subproblem for each of thencircular arcs. By using the radial-sweep technique, we need not solve each of these subproblems independently; thus we eliminate the log log nfactor from the running time of earlier algorithms. For vertex-weighted circular-arc graphs, it is possible to use our approach to obtain anO(m log log n + n log n) algorithm for finding a maximum-weight clique—which matches the best known algorithm.     

Chebyshev’s alternance in the approximation of constants by simple partial fractions

Uniform approximation of real constants by simple partial fractions on a closed interval of the real axis is studied. It is proved that a simple partial fraction of best approximation of degree n for a constant is unique and coincides with this constant at n nodes lying on the interval; moreover, there is a Chebyshev alternance consisting of n + 1 points.