Odd perfect numbers have a prime factor exceeding

Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding . Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding .

     

The hyperdeterminant and triangulations of the 4-cube

The hyperdeterminant of format is a polynomial of degree in unknowns which has terms. We compute the Newton polytope of this polynomial and the secondary polytope of the -cube. The regular triangulations of the -cube are classified into -equivalence classes, one for each vertex of the Newton polytope. The -cube has coarsest regular subdivisions, one for each facet of the secondary polytope, but only of them come from the hyperdeterminant.

     

New expansions of numerical eigenvalues for by nonconforming elements

The paper explores new expansions of the eigenvalues for in with Dirichlet boundary conditions by the bilinear element (denoted ) and three nonconforming elements, the rotated bilinear element (denoted ), the extension of (denoted ) and Wilson's elements. The expansions indicate that and provide upper bounds of the eigenvalues, and that and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the convergence rate can be obtained, where is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.

     

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 .)

     

Minimality and other properties of the width- nonadjacent form

Let be an integer and let be the set of integers that includes zero and the odd integers with absolute value less than . Every integer can be represented as a finite sum of the form , with , such that of any consecutive 's at most one is nonzero. Such representations are called width- nonadjacent forms (-NAFs). When these representations use the digits and coincide with the well-known nonadjacent forms. Width- nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the -NAF. We show that -NAFs have a minimal number of nonzero digits and we also give a new characterization of the -NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on -NAFs and show that any base 2 representation of an integer, with digits in , that has a minimal number of nonzero digits is at most one digit longer than its binary 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) .

     

Rationality problem of three-dimensional purely monomial group actions: the last case

A -automorphism of the rational function field is called purely monomial if sends every variable to a monic Laurent monomial in the variables . Let be a finite subgroup of purely monomial -automorphisms of . The rationality problem of the -action is the problem of whether the -fixed field is -rational, i.e., purely transcendental over , or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.

     

Finding -strong pseudoprimes

Let be odd primes and . Put

Then we call the kernel, the triple the signature, and the height of , respectively. We call a -number if it is a Carmichael number with each prime factor . If is a -number and a strong pseudoprime to the bases for , we call a -spsp . Since -numbers have probability of error (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of (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.

In this paper, we first describe an algorithm for finding -spsp(2)'s, to a given limit, with heights bounded. There are in total -spsp's with heights . We then give an overview of the 21978 - spsp(2)'s and tabulate of them, which are -spsp's to the first prime bases up to ; three numbers are spsp's to the first 11 prime bases up to 31. No -spsp's to the first prime bases with heights were found. We conjecture that there exist no -spsp's to the first prime bases with heights and so that

which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those -spsp's is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates of -spsp's and their prime factors to Miller's tests, and obtain the desired numbers. At last we speed our algorithm for finding larger -spsp's, say up to , with a given signature to more prime bases. Comparisons of effectiveness with Arnault's and our previous methods for finding -strong pseudoprimes to the first several prime bases are given.

     

Computing weight modular forms of level

For a prime we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space of modular forms of weight and level . For we define a special Hecke stable subspace of which contains the space of modular forms with CM by the ring of integers of and we describe the calculation of the corresponding Brandt matrices.

     

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.

     

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.

     

Balanced multi-wavelets in

The notion of -balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalar-valued discrete polynomial data of order (or degree ), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for is known. In addition, the formulation of the -balancing condition for is so prohibitively difficult to satisfy that only a very few examples for and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the -balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any .

     

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.

     

Weak coupling of solutions of first-order least-squares method

A theoretical analysis of a first-order least-squares finite element method for second-order self-adjoint elliptic problems is presented. We investigate the coupling effect of the approximate solutions for the primary function and for the flux . We prove that the accuracy of the approximate solution for the primary function is weakly affected by the flux . That is, the bound for is dependent on , but only through the best approximation for multiplied by a factor of meshsize . Similarly, we provide that the bound for is dependent on , but only through the best approximation for multiplied by a factor of the meshsize . This weak coupling is not true for the non-selfadjoint case. We provide the numerical experiment supporting the theorems in this paper.

     

Elliptic curves with nonsplit mod representations

We calculate explicitly the -invariants of the elliptic curves corresponding to rational points on the modular curve by giving an expression defined over of the -function in terms of the function field generators and of the elliptic curve . As a result we exhibit infinitely many elliptic curves over with nonsplit mod representations.

     

Real zeros of real odd Dirichlet -functions

Let be a real odd Dirichlet character of modulus , and let be the associated Dirichlet -function. As a consequence of the work of Low and Purdy, it is known that if and , , , then has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if , then has no positive real zeros.

     

Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an -shaped domain

A singularly perturbed reaction-diffusion equation is posed in a two-dimensional -shaped domain subject to a continuous Dirchlet boundary condition. Its solutions are in the Hölder space and typically exhibit boundary layers and corner singularities. The problem is discretized on a tensor-product Shishkin mesh that is further refined in a neighboorhood of the vertex of angle . We establish almost second-order convergence of our numerical method in the discrete maximum norm, uniformly in the small diffusion parameter. Numerical results are presented that support our theoretical error estimate.

     

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 .

     

-estimate for the discrete Plateau Problem

In this paper we prove the convergence rates for a fully discrete finite element procedure for approximating minimal, possibly unstable, surfaces.

Originally this problem was studied by G. Dziuk and J. Hutchinson. First they provided convergence rates in the and norms for the boundary integral method. Subsequently they obtained the convergence estimates using a fully discrete finite element method. We use the latter framework for our investigation.

     

Comparison theorems of Kolmogorov type and exact values of -widths on Hardy-Sobolev classes

Let be a strip in complex plane. denotes those -periodic, real-valued functions on which are analytic in the strip and satisfy the condition , . Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel'fand, and information -widths of in , , and 2-widths of in , , .

In this paper we continue their work. Firstly, we establish a comparison theorem of Kolmogorov type on , from which we get an inequality of Landau-Kolmogorov type. Secondly, we apply these results to determine the exact values of the Gel'fand -width of in , . Finally, we calculate the exact values of Kolmogorov -width, linear -width, and information -width of in , , .