Constructing representations of higher degrees of finite simple groups and covers

Let be a finite group and an irreducible character of . A simple method for constructing a representation affording can be used whenever has a subgroup such that has a linear constituent with multiplicity 1. In this paper we show that (with a few exceptions) if is a simple group or a covering group of a simple group and is an irreducible character of of degree between 32 and 100, then such a subgroup exists.

     

New integer representations as the sum of three cubes

We describe a new algorithm for finding integer solutions to for specific values of . We use this to find representations for values of for which no solution was previously known, including and .

     

Error bounds on complex floating-point multiplication

Given floating-point arithmetic with -digit base- significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values and can be computed with maximum absolute error . In particular, this provides relative error bounds of and for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur.

We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic.

     

Computing the integer partition function

In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of for for primes up to and small powers of and .

     

Euclidean minima of totally real number fields: Algorithmic determination

This article deals with the determination of the Euclidean minimum of a totally real number field of degree , using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree to and small discriminants, most of which were previously unknown. Tables are given at the end of this paper.

     

Cubature formulas for symmetric measures in higher dimensions with few points

We study cubature formulas for -dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree or and large dimension the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions.

We also show, for any odd degree , that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.

     

Two kinds of strong pseudoprimes up to

Let be an odd composite integer. Write with odd. If either mod or mod for some , then we say that is a strong pseudoprime to base , or spsp() for short. Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Conjectured values of were given by us in our previous papers (Math. Comp. 72 (2003), 2085-2097; 74 (2005), 1009-1024).

The main purpose of this paper is to give exact values of for ; to give a lower bound of : ; and to give reasons and numerical evidence of K2- and -spsp's to support the following conjecture: for any , where (resp. ) is the smallest K2- (resp. -) strong pseudoprime to all the first prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: with primes and ; and that a -spsp is an spsp and a Carmichael number of the form: with each prime factor mod .)

     

Explicit values of multi-dimensional Kloosterman sums for prime powers, II

For any integer fix , and let denote the group of reduced residues modulo . Let , a power of a prime . The hyper-Kloosterman sums of dimension are defined for by

where denotes the multiplicative inverse of modulo .

Salie evaluated in the classical setting for even , and for odd with . Later, Smith provided formulas that simplified the computation of in these cases for . Recently, Cochrane, Liu and Zheng computed upper bounds for in the general case , stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for , relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author's previous determination of these Kloosterman sums using character theory and -adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums.

     

Continuous interior penalty -finite element methods for advection and advection-diffusion equations

A continuous interior penalty -finite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advection-diffusion equations. The analysis relies on three technical results that are of independent interest: an -inverse trace inequality, a local discontinuous to continuous -interpolation result, and -error estimates for continuous -orthogonal projections.

     

Sharp estimates for finite element approximations to elliptic problems with Neumann boundary data of low regularity

Consider a second order homogeneous elliptic problem with smooth coefficients, , on a smooth domain, , but with Neumann boundary data of low regularity. Interior maximum norm error estimates are given for finite element approximations to this problem. When the Neumann data is not in , these local estimates are not of optimal order but are nevertheless shown to be sharp. A method for ameliorating this sub-optimality by preliminary smoothing of the boundary data is given. Numerical examples illustrate the findings.