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.

     

A counterexample concerning the -projector onto linear spline spaces

For the -orthogonal projection onto spaces of linear splines over simplicial partitions in polyhedral domains in , , we show that in contrast to the one-dimensional case, where independently of the nature of the partition, in higher dimensions the -norm of cannot be bounded uniformly with respect to the partition. This fact is folklore among specialists in finite element methods and approximation theory but seemingly has never been formally proved.

     

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 .

     

Divisors in residue classes, constructively

Let be integers satisfying , , , and let . Lenstra showed that the number of integer divisors of equivalent to is upper bounded by . We re-examine this problem, showing how to explicitly construct all such divisors, and incidentally improve this bound to .

     

Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem

Consider the problem with homogeneous Neumann boundary condition in a bounded smooth domain in . The whole range is treated. The Galerkin finite element method is used on a globally quasi-uniform mesh of size ; the mesh is fixed and independent of .

A precise analysis of how the error at each point depends on and is presented. As an application, first order error estimates in , which are uniform with respect to , are given.

     

On Meinardus' examples for the conjugate gradient method

The conjugate gradient (CG) method is widely used to solve a positive definite linear system of order . It is well known that the relative residual of the th approximate solution by CG (with the initial approximation ) is bounded above by

   with

where is 's spectral condition number. In 1963, Meinardus (Numer. Math., 5 (1963), pp. 14-23) gave an example to achieve this bound for but without saying anything about all other . This very example can be used to show that the bound is sharp for any given by constructing examples to attain the bound, but such examples depend on and for them the th residual is exactly zero. Therefore it would be interesting to know if there is any example on which the CG relative residuals are comparable to the bound for all . There are two contributions in this paper:

1. A closed formula for the CG residuals for all on Meinardus' example is obtained, and in particular it implies that the bound is always within a factor of of the actual residuals;
2. A complete characterization of extreme positive linear systems for which the th CG residual achieves the bound is also presented.

     

Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system

In this paper we present some classes of high-order semi-Lagran- gian schemes for solving the periodic one-dimensional Vlasov-Poisson system in phase-space on uniform grids. We prove that the distribution function and the electric field converge in the norm with a rate of

where is the degree of the polynomial reconstruction, and and are respectively the time and the phase-space discretization parameters.

     

On the distinctness of modular reductions of maximal length sequences modulo odd prime powers

We discuss the distinctness problem of the reductions modulo of maximal length sequences modulo powers of an odd prime , where the integer has a prime factor different from . For any two different maximal length sequences generated by the same polynomial, we prove that their reductions modulo are distinct. In other words, the reduction modulo of a maximal length sequence is proved to contain all the information of the original sequence.

     

On the equation

We describe a criterion for showing that the equation has no non-trivial proper integer solutions for specific primes . This equation is a special case of the generalized Fermat equation . The criterion is based on the method of Galois representations and modular forms together with an idea of Kraus for eliminating modular forms for specific in the final stage of the method (1998). The criterion can be computationally verified for primes and .

     

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.

     

An -local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type

In this paper, an -local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On -quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken norm and norm which are optimal in , suboptimal in are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

     

Primitive central idempotents of finite group rings of symmetric groups

Let be a prime. We denote by the symmetric group of degree , by the alternating group of degree and by the field with elements. An important concept of modular representation theory of a finite group is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring , where is a prime power. Here, we describe a new method to compute the primitive central idempotents of for arbitrary prime powers and arbitrary finite groups . For the group rings of the symmetric group, we show how to derive the primitive central idempotents of from the idempotents of . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of which contains the primitive central idempotents. The described results are most efficient for . In an appendix we display all primitive central idempotents of and for which we computed by this method.

     

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.

     

On the polynomial representation for the number of partitions with fixed length

In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

     

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

For any and any non-exceptional modulus , we prove that, for large enough ( ), the interval contains a prime in any of the arithmetic progressions modulo . We apply this result to establish that every integer larger than is a sum of seven cubes.

     

Construction of hyperelliptic function fields of high three-rank

We present several explicit constructions of hyperelliptic function fields whose Jacobian or ideal class group has large -rank. Our focus is on finding examples for which the genus and the base field are as small as possible. Most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high -rank, but one method, applicable to finding large -ranks for odd primes is new and unique to function fields. Algorithms, examples, and numerical data are included.

     

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 .)

     

An interpolation error estimate in based on the anisotropic measures of higher order derivatives

In this paper, we introduce the magnitude, orientation, and anisotropic ratio for the higher order derivative (with ) of a function to characterize its anisotropic behavior. The magnitude is equivalent to its usual Euclidean norm. The orientation is the direction along which the absolute value of the -th directional derivative is about the smallest, while along its perpendicular direction it is about the largest. The anisotropic ratio measures the strength of the anisotropic behavior of . These quantities are invariant under translation and rotation of the independent variables. They correspond to the area, orientation, and aspect ratio for triangular elements. Based on these measures, we derive an anisotropic error estimate for the piecewise polynomial interpolation over a family of triangulations that are quasi-uniform under a given Riemannian metric. Among the meshes of a fixed number of elements it is identified that the interpolation error is nearly the minimum on the one in which all the elements are aligned with the orientation of , their aspect ratios are about the anisotropic ratio of , and their areas make the error evenly distributed over every element.

     

Equal moments division of a set

Let be the minimal positive integer , for which there exists a splitting of the set into  subsets, , , ..., , whose first moments are equal. Similarly, let be the maximal positive integer , such that there exists a splitting of into subsets whose first moments are equal. For , these functions were investigated by several authors, and the values of and have been found for and , respectively. In this paper, we deal with the problem for any prime . We demonstrate our methods by finding for any and for .

     

Modified Taylor reproducing formulas and h-p clouds

We study two different approximations of a multivariate function by operators of the form , where is an -reproducing partition of unity and are modified Taylor polynomials of degree expanded at . The first approximation was introduced by Xuli (2003) in the univariate case and generalized for convex domains by Guessab et al. (2005). The second one was introduced by Duarte (1995) and proved in the univariate case. In this paper, we first relax the Guessab's convexity assumption and we prove Duarte's reproduction formula in the multivariate case. Then, we introduce two related reproducing quasi-interpolation operators in Sobolev spaces. A weighted error estimate and Jackson's type inequalities for h-p cloud function spaces are obtained. Last, numerical examples are analyzed to show the approximative power of the method.