On a Problem of Byrnes concerning Polynomials with Restricted Coefficients

We consider a question of Byrnes concerning the minimal degree of a polynomial with all coefficients in which has a zero of a given order at . For , we prove his conjecture that the monic polynomial of this type of minimal degree is given by , but we disprove this for . We prove that a polynomial of this type must have , which is in sharp contrast with the situation when one allows coefficients in . The proofs use simple number theoretic ideas and depend ultimately on the fact that .

     

Optimal information for approximating periodic analytic functions

Let be a strip in the complex plane. For fixed integer let denote the class of -periodic functions , which are analytic in and satisfy in . Denote by the subset of functions from that are real-valued on the real axis. Given a function , we try to recover at a fixed point by an algorithm on the basis of the information

where , are the Fourier coefficients of . We find the intrinsic error of recovery

Furthermore the -dimensional optimal information error, optimal sampling error and -widths of in , the space of continuous functions on , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class , consisting of all -periodic functions, which are analytic in with -integrable boundary values. In the case sampling fails to yield optimal information as well in odd as in even dimensions.

     

On the rapid computation of various polylogarithmic constants

We give algorithms for the computation of the -th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of or on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of , the billionth hexadecimal digits of and , and the ten billionth decimal digit of . These calculations rest on the observation that very special types of identities exist for certain numbers like , , and . These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for :

     

On Wendt's determinant

Wendt's determinant of order is the circulant determinant whose -th entry is the binomial coefficient , for . We give a formula for , when is even not divisible by 6, in terms of the discriminant of a polynomial , with rational coefficients, associated to . In particular, when where is a prime , this yields a factorization of involving a Fermat quotient, a power of and the 6-th power of an integer.

     

Further tabulation of the Erdös-Selfridge function

For a positive integer , the Erdös-Selfridge function is the least integer such that all prime factors of exceed . This paper describes a rapid method of tabulating using VLSI based sieving hardware. We investigate the number of admissible residues for each modulus in the underlying sieving problem and relate this number to the size of . A table of values of for is provided.

     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

Stieltjes polynomials and Lagrange interpolation

Bounds are proved for the Stieltjes polynomial , and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials . This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials . Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of , and for the extended Lagrange interpolation process with respect to the zeros of in the uniform and weighted norms. The corresponding Lebesgue constants are of optimal order.

     

The exponent of discrepancy is at most

We study discrepancy with arbitrary weights in the norm over the -dimensional unit cube. The exponent of discrepancy is defined as the smallest for which there exists a positive number such that for all and all there exist points with discrepancy at most . It is well known that . We improve the upper bound by showing that

This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent is .

     

Power integral bases in a parametric family of totally real cyclic quintics

We consider the totally real cyclic quintic fields , generated by a root of the polynomial

Assuming that is square free, we compute explicitly an integral basis and a set of fundamental units of and prove that has a power integral basis only for . For (both values presenting the same field) all generators of power integral bases are computed.

     

Equivalent formulae for the supremum and stability of weighted pseudoinverses

During recent decades, there have been a great number of research articles studying interior-point methods for solving problems in mathematical programming and constrained optimization. Stewart and O'Leary obtained an upper bound for scaled pseudoinverses of a matrix where is a set of diagonal positive definite matrices. We improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses. Forsgren further generalized these results to derive the supremum of weighted pseudoinverses where is a set of diagonally dominant positive semidefinite matrices, by using a signature decomposition of weighting matrices and by applying the Binet-Cauchy formula and Cramer's rule for determinants. The results are also extended to equality constrained linear least squares problems. In this paper we extend Forsgren's results to a general complex matrix to establish several equivalent formulae for , where is a set of diagonally dominant positive semidefinite matrices, or a set of weighting matrices arising from solving equality constrained least squares problems. We also discuss the stability property of these weighted pseudoinverses.

     

Pure product polynomials and the Prouhet-Tarry-Escott problem

An -factor pure product is a polynomial which can be expressed in the form for some natural numbers . We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every -factor pure product has norm at least . We describe three algorithms for determining the least norm an -factor pure product can have. We report results of our computations using one of these algorithms which include the result that every -factor pure product has norm strictly greater than if is , , , or .

     

Bounds for Multiplicative Cosets over Fields of Prime Order

Let be a positive integer and suppose that is an odd prime with . Suppose that and consider the polynomial . If this polynomial has any roots in , where the coset representatives for are taken to be all integers with , then these roots will form a coset of the multiplicative subgroup of consisting of the th roots of unity mod . Let be a coset of in , and define . In the paper ``Numbers Having Small th Roots mod ' (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for of the form , where is the Euler phi-function. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson are proven to be optimal when is a power of or when

     

On the -divisibility of Fermat quotients

The authors carried out a numerical search for Fermat quotients vanishing mod , for , up to . This article reports on the results and surveys the associated theoretical properties of . The approach of fixing the prime rather than the base leads to some aspects of the theory apparently not published before.

     

Computing Canonical Heights With Little (or No) Factorization

Let be an elliptic curve with discriminant , and let . The standard method for computing the canonical height is as a sum of local heights . There are well-known series for computing the archimedean height , and the non-archimedean heights are easily computed as soon as all prime factors of have been determined. However, for curves with large coefficients it may be difficult or impossible to factor . In this note we give a method for computing the non-archimedean contribution to which is quite practical and requires little or no factorization. We also give some numerical examples illustrating the algorithm.

     

A stochastic particle method for the McKean-Vlasov and the Burgers equation

In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is three-fold. First, we consider a McKean-Vlasov equation in with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure , the solution to the McKean-Vlasov equation. The simulation of the stochastic system with particles provides a discrete measure which approximates for each time (where is a discretization step of the time interval ). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of . We show that the convergence rate is for the approximation in of the cumulative distribution function at time , and of order for the approximation in of the density at time ( is the underlying probability space, is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rate . This part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.

     

Computational experiences on the distances of polynomials to irreducible polynomials

In this paper we deal with a problem of Turán concerning the `distance' of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial of degree there exists a monic polynomial with deg() = deg() such that is irreducible over and the `distance' of and is .

     

The Diophantine equation

An effective method is derived for solving the equation of the title in positive integers and for given completely, and is carried out for all . If is of the form , then there is the solution , ; in the above range, except for with solution , , there are no other solutions.

     

Numerical solution of isospectral flows

In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation

where is a symmetric matrix, is a skew-symmetric matrix function of and is the Lie bracket operator. We show that standard Runge-Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order.

     

New estimates for Ritz vectors

The following estimate for the Rayleigh-Ritz method is proved:

Here is a bounded self-adjoint operator in a real Hilbert/euclidian space, one of its eigenpairs, a trial subspace for the Rayleigh-Ritz method, and a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh-Ritz method, in particular, it shows that if an eigenvector is close to the trial subspace with accuracy and a Ritz vector is an approximation to another eigenvector, with a different eigenvalue. Generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested, and estimates of approximation of eigenspaces and invariant subspaces are proved.

     

Accelerated polynomial approximation of finite order entire functions by growth reduction

Let be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of by incorporating information about the growth of for . We consider ``near polynomial approximation' on a compact plane set , which should be thought of as a circle or a real interval. Our aim is to find sequences of functions which are the product of a polynomial of degree and an ``easy computable' second factor and such that converges essentially faster to on than the sequence of best approximating polynomials of degree . The resulting method, which we call Reduced Growth method (-method) is introduced in Section 2. In Section 5, numerical examples of the -method applied to the complex error function and to Bessel functions are given.