High rank elliptic curves with torsion group

We develop an algorithm for bounding the rank of elliptic curves in the family , all of them with torsion group and modular invariant . We use it to look for curves of high rank in this family and present four such curves of rank  and of rank .

     

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.

     

Computational estimation of the order of

The paper describes a search for increasingly large extrema (ILE) of in the range . For , the complete set of ILE (57 of them) was determined. In total, 162 ILE were found, and they suggest that . There are several regular patterns in the location of ILE, and arguments for these regularities are presented. The paper concludes with a discussion of prospects for further computational progress.

     

Korn's inequalities for piecewise vector fields

Korn's inequalities for piecewise vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.

     

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.

     

Convergence of nonconforming -cycle and -cycle multigrid algorithms for second order elliptic boundary value problems

The convergence of -cycle and -cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems.

     

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.

     

A Monte Carlo algorithm for weighted integration over

We present and analyze a new randomized algorithm for numerical computation of weighted integrals over the unbounded domain . The algorithm and its desirable theoretical properties are derived based on certain stochastic assumptions about the integrands. It is easy to implement, enjoys convergence rate, and uses only standard random number generators. Numerical results are also included.

     

Quasi-Monte Carlo integration over

In this paper we show that a wide class of integrals over with a probability weight function can be evaluated using a quasi-Monte Carlo algorithm based on a proper decomposition of the domain and arranging low discrepancy points over a series of hierarchical hypercubes. For certain classes of power/exponential decaying weights the algorithm is of optimal order.

     

-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.

     

Rational double points on supersingular surfaces

We investigate configurations of rational double points with the total Milnor number on supersingular surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular surfaces.

     

-class groups of certain extensions of degree

Let be an odd prime number. In this article we study the distribution of -class groups of cyclic number fields of degree , and of cyclic extensions of degree of an imaginary quadratic field whose class number is coprime to . We formulate a heuristic principle predicting the distribution of the -class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to--part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.

     

On the nonexistence of -cycles for the problem

This article generalizes a proof of Steiner for the nonexistence of -cycles for the problem to a proof for the nonexistence of -cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result of Laurent, Mignotte, and Nesterenko on linear forms in logarithms. Finally numerical calculation of convergents of shows that -cycles cannot exist.

     

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 .

     

Rational Landen transformations on

The Landen transformation preserves the value of an elliptic integral, and its iteration produces the classical arithmetic-geometric mean AGM. We present analogous transformations for rational functions integrated over the whole real line.

     

Normal integral bases for extensions of the rationals

We give an algorithm for constructing normal integral bases of tame Galois extensions of the rationals with group . Using earlier works we can do the same until degree .

     

Solutions of the congruence

To supplement existing data, solutions of are tabulated for primes with and . For , five new solutions 2^{32}$"> are presented. One of these, for , also satisfies the ``reverse' congruence . An effective procedure for searching for such ``double solutions' is described and applied to the range , . Previous to this, congruences are generally considered for any and fixed prime to see where the smallest prime solution occurs.

     

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.

     

Integer points on the curve

We completely solve diophantine equations of the form where is a positive integer, using a reduction to some quartic elliptic equations, which can be solved with well known methods.