Computing the multiplicative group of residue class rings

Let be a global field with maximal order and let be an ideal of . We present algorithms for the computation of the multiplicative group of the residue class ring and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group modulo , where denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.

     

On the absolute Mahler measure of polynomials having all zeros in a sector. II

Let be an algebraic integer of degree , not or a root of unity, all of whose conjugates are confined to a sector . In the paper On the absolute Mahler measure of polynomials having all zeros in a sector, G. Rhin and C. Smyth compute the greatest lower bound of the absolute Mahler measure ( of , for belonging to nine subintervals of . In this paper, we improve the result to thirteen subintervals of and extend some existing subintervals.

     

Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants

In the Laurent expansion

of the Riemann-Hurwitz zeta function, the coefficients are known as Stieltjes, or generalized Euler, constants. [When , (the Riemann zeta function), and .] We present a new approach to high-precision approximation of . Plots of our results reveal much structure in the growth of the generalized Euler constants. Our results when for , and when for (for such as 53/100, 1/2, etc.) suggest that published bounds on the growth of the Stieltjes constants can be much improved, and lead to several conjectures. Defining , we conjecture that is attained: for any given , for some (and similarly that, given and , is within of for infinitely many ). In addition we conjecture that satisfies for 1$">. We also conjecture that , a special case of a more general conjecture relating the values of and for . Finally, it is known that for . Using this to define for all real 0$">, we conjecture that for nonintegral , is precisely times the -th (Weyl) fractional derivative at of the entire function . We also conjecture that , now defined for all real arguments 0$">, is smooth. Our numerical method uses Newton-Cotes integration formulae for very high-degree interpolating polynomials; it differs in implementation from, but compares in error bounding to, Euler-Maclaurin summation based methods.

     

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.

     

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 , , .

     

On strong tractability of weighted multivariate integration

We prove that for every dimension and every number of points, there exists a point-set whose -weighted unanchored discrepancy is bounded from above by independently of provided that the sequence has for some (even arbitrarily large) . Here is a positive number that could be chosen arbitrarily close to zero and depends on but not on or . This result yields strong tractability of the corresponding integration problems including approximation of weighted integrals over unbounded domains such as . It also supplements the results that provide an upper bound of the form when .

     

A new approximation technique for div-curl systems

In this paper, we describe an approximation technique for div-curl systems based in where is a domain in . We formulate this problem as a general variational problem with different test and trial spaces. The analysis requires the verification of an appropriate inf-sup condition. This results in a very weak formulation where the solution space is and the data reside in various negative norm spaces. Subsequently, we consider finite element approximations based on this weak formulation. The main approach of this paper involves the development of ``stable pairs" of discrete test and trial spaces. With this approach, we enlarge the test space so that the discrete inf-sup condition holds and we use a negative-norm least-squares formulation to reduce to a uniquely solvable linear system. This leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low Sobolev regularity (e.g., with ). The resulting algebraic equations are symmetric, positive definite and well conditioned. A second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned. Some numerical results are also presented.

     

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) .

     

A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions

Let , , be a dimensional slab. Denote points by , where and . Denoting the boundary of the slab by , let

where is an ordered sequence of intervals on the right half line (that is, b_{n}$">). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let . Let and denote respectively the cone of bounded, positive harmonic functions in and the cone of positive harmonic functions in which satisfy the Dirichlet boundary condition on and the Neumann boundary condition on .

Letting , the main result of this paper, under a modest assumption on the sequence , may be summarized as follows when :

1. If , then and are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if with 2$">.

2. If and , then and is one-dimensional. In particular, this occurs if .

3. If , then and the set of minimal elements generating is isomorphic to (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if with .

When , as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for is as above.

     

The approximation of the Maxwell eigenvalue problem using a least-squares method

In this paper we consider an approximation to the Maxwell's eigenvalue problem based on a very weak formulation of two div-curl systems with complementary boundary conditions. We formulate each of these div-curl systems as a general variational problem with different test and trial spaces, i.e., the solution space is and components in the test spaces are in subspaces of , the Sobolev space of order one on the computational domain . A finite-element least-squares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the div-curl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given.

     

Values of the Legendre chi and Hurwitz zeta functions at rational arguments

We show that the Hurwitz zeta function, , and the Legendre chi function, , defined by

and

respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained as a corollary to this result. Among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments. Also, these transform relations make it likely that other results can be easily recovered and unified in a more general context.

     

Wavelet bases in

Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of . These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces . Moreover, -free vector wavelets are constructed and analysed. The relationship between and are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.

Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains . As an application, we obtain wavelet multilevel preconditioners in and .

     

Local behaviour of polynomials

In this paper we study the local behaviour of a trigonometric polynomial around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in . The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form are also proved.

     

A priori error estimates for Galerkin approximations to porous medium and fast diffusion equations

Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation

on bounded convex domains are considered. The range of the parameter includes the fast diffusion case . Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in norm with an error controlled by for and for . For the fully discrete problem, a global convergence rate of in norm is shown for the range . For , a rate of is shown in norm.

     

An old conjecture of Erdos-Turán on additive bases

There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:

Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and

Suppose that

If 0$"> for all , then is unbounded.

Our main purpose is to show that the sequence cannot be bounded by . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.

     

Games and general distributive laws in Boolean algebras

The games and are played by two players in -complete and max -complete Boolean algebras, respectively. For cardinals such that or , the -distributive law holds in a Boolean algebra iff Player 1 does not have a winning strategy in . Furthermore, for all cardinals , the -distributive law holds in iff Player 1 does not have a winning strategy in . More generally, for cardinals such that , the -distributive law holds in iff Player 1 does not have a winning strategy in . For regular and , implies the existence of a Suslin algebra in which is undetermined.

     

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.

     

All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

All first-order averaging or gradient-recovery operators for lowest-order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain in . Given a piecewise constant discrete flux (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux (that is the gradient of the exact displacement), recent results verify efficiency and reliability of

in the sense that is a lower and upper bound of the flux error up to multiplicative constants and higher-order terms. The averaging space consists of piecewise polynomial and globally continuous finite element functions in components with carefully designed boundary conditions. The minimal value is frequently replaced by some averaging operator applied within a simple post-processing to . The result provides a reliable error bound with .

This paper establishes and so equivalence of and . This implies efficiency of for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound established for tetrahedral finite elements appears striking in that the shape of the elements does not enter: The equivalence is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli's lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices.

     

Jacobi sums and new families of irreducible polynomials of Gaussian periods

Let 2$">, an -th primitive root of 1, mod a prime number, a primitive root modulo and . We study the Jacobi sums , , where is the least nonnegative integer such that mod . We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families , , of irreducible polynomials of Gaussian periods, , of degree , where is a suitable set of primes mod . We exhibit examples of such families for several small values of , and give a MAPLE program to construct more of them.

     

The -approximation order of surface spline interpolation

We show that if the open, bounded domain has a sufficiently smooth boundary and if the data function is sufficiently smooth, then the -norm of the error between and its surface spline interpolant is ( ), where and is an integer parameter specifying the surface spline. In case , this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the -approximation order of surface spline interpolation is .