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.

     

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 .

     

Prime decomposition in the anti-cyclotomic extension

For an imaginary quadratic number field and an odd prime number , the anti-cyclotomic -extension of is defined. For primes of , decomposition laws for in the anti-cyclotomic extension are given. We show how these laws can be applied to determine if the Hilbert class field (or part of it) of is -embeddable. For some and , we find explicit polynomials whose roots generate the first step of the anti-cyclotomic extension and show how the prime decomposition laws give nice results on the splitting of these polyniomials modulo . The article contains many numerical examples.

     

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 .

     

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.

     

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 .)

     

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.

     

A search for Fibonacci-Wieferich and Wolstenholme primes

A prime is called a Fibonacci-Wieferich prime if , where is the th Fibonacci number. We report that there exist no such primes . A prime is called a Wolstenholme prime if . To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes . Wolstenholme, in 1862, proved that for all primes . It is estimated by a heuristic argument that the ``probability' that is Fibonacci-Wieferich (independently: that is Wolstenholme) is about . We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient modulo .

     

Structured data-sparse approximation to high order tensors arising from the deterministic Boltzmann equation

We develop efficient data-sparse representations to a class of high order tensors via a block many-fold Kronecker product decomposition. Such a decomposition is based on low separation-rank approximations of the corresponding multivariate generating function. We combine the interpolation and a quadrature-based approximation with hierarchically organised block tensor-product formats. Different matrix and tensor operations in the generalised Kronecker tensor-product format including the Hadamard-type product can be implemented with the low cost. An application to the collision integral from the deterministic Boltzmann equation leads to an asymptotical cost - in the one-dimensional problem size (depending on the model kernel function), which noticeably improves the complexity of the full matrix representation.

     

On the embedding problem for representations

Let denote the double cover of corresponding to the element in where transpositions lift to elements of order and the product of two disjoint transpositions to elements of order . Given an elliptic curve , let denote its -torsion points. Under some conditions on elements in correspond to Galois extensions of with Galois group (isomorphic to) . In this work we give an interpretation of the addition law on such fields, and prove that the obstruction for having a Galois extension with gives a homomorphism . As a corollary we can prove (if has conductor divisible by few primes and high rank) the existence of -dimensional representations of the absolute Galois group of attached to and use them in some examples to construct modular forms mapping via the Shimura map to (the modular form of weight attached to) .

     

On univoque Pisot numbers

We study Pisot numbers which are univoque, i.e., such that there exists only one representation of as , with . We prove in particular that there exists a smallest univoque Pisot number, which has degree . Furthermore we give the smallest limit point of the set of univoque Pisot numbers.

     

On geometric interpolation by planar parametric polynomial curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree can interpolate given points in is confirmed for under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order can be achieved as soon as the interpolating curve exists.

     

Asymptotically fast group operations on Jacobians of general curves

Let be a curve of genus over a field . We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of . After a precomputation, which is done only once for the curve , the algorithms use only linear algebra in vector spaces of dimension at most , and so take field operations in , using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to . This represents a significant improvement over the previous record of field operations (also after a precomputation) for general curves of genus .

     

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.

     

Efficient CM-constructions of elliptic curves over finite fields

We present an algorithm that, on input of an integer together with its prime factorization, constructs a finite field and an elliptic curve over for which has order . Although it is unproved that this can be done for all , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in , where is the number of distinct prime factors of . In the cryptographically relevant case where is prime, an expected run time can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order and nextprime.

     

Tractability of quasilinear problems II: Second-order elliptic problems

In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the Helmholtz equation in the -dimensional unit cube, in which depends linearly on , but nonlinearly on . Here, both and  are -variate functions from a reproducing kernel Hilbert space with finite-order weights of order . This means that, although  can be arbitrarily large, and  can be decomposed as sums of functions of at most  variables, with independent of .

In this paper, we apply our previous general results to the Helmholtz equation, subject to either Dirichlet or Neumann homogeneous boundary conditions. We study both the absolute and normalized error criteria. For all four possible combinations of boundary conditions and error criteria, we show that the problem is tractable. That is, the number of evaluations of and  needed to obtain an -approximation is polynomial in  and , with the degree of the polynomial depending linearly on . In addition, we want to know when the problem is strongly tractable, meaning that the dependence is polynomial only in  , independently of . We show that if the sum of the weights defining the weighted reproducing kernel Hilbert space is uniformly bounded in  and the integral of the univariate kernel is positive, then the Helmholtz equation is strongly tractable for three of the four possible combinations of boundary conditions and error criteria, the only exception being the Dirichlet boundary condition under the normalized error criterion.

     

K3 surfaces with Picard number three and canonical vector heights

In this paper we construct the first known explicit family of K3 surfaces defined over the rationals that are proved to have geometric Picard number . This family is dense in one of the components of the moduli space of all polarized K3 surfaces with Picard number at least . We also use an example from this family to fill a gap in an earlier paper by the first author. In that paper, an argument for the nonexistence of canonical vector heights on K3 surfaces of Picard number was given, based on an explicit surface that was not proved to have Picard number . We redo the computations for one of our surfaces and come to the same conclusion.

     

Robust norm equivalencies for diffusion problems

Additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem. These solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition. To obtain a robust iterative scheme, it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem.

In this paper we present such a robust norm equivalence for the model problem with a scalar diffusion coefficient in . Our estimates involve only very weak information about , and the results are applicable for a large class of diffusion coefficients. Namely, we require to be in the Muckenhoupt class , a function class well-studied in harmonic analysis.

The presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems.

     

Numerically satisfactory solutions of hypergeometric recursions

Each family of Gauss hypergeometric functions

for fixed (not all equal to zero) satisfies a second order linear difference equation of the form

Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different values) can be transformed into each other. In this way, only with four basic difference equations can all other cases be obtained. For each of these recurrences, we give pairs of numerically satisfactory solutions in the regions in the complex plane where , and being the roots of the characteristic equation.

     

On the largest prime divisor of an odd harmonic number

A positive integer is called a (Ore's) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than is even. If Ore's conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than must be divisible by a prime greater than .