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.

     

Sums of heights of algebraic numbers

For , we consider the set . The polynomials are in , with only mild restrictions, and is the Weil height of . We show that this set is dense in for some effectively computable limit point .

     

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 , , .

     

Korovkin tests, approximation, and ergodic theory

We consider sequences of matrices with a block structure spectrally distributed as an -variate matrix-valued function , and, for any , we suppose that is a linear and positive operator. For every fixed we approximate the matrix in a suitable linear space of matrices by minimizing the Frobenius norm of when ranges over . The minimizer is denoted by . We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts:

1.

the sequence is distributed as ,

2.

the sequence is distributed as the constant function (i.e. is spectrally clustered at zero).

The first result is an ergodic one which can be used for solving numerical approximation theory problems. The second has a natural interpretation in the theory of the preconditioning associated to cg-like algorithms.

     

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.

     

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 the convergence of certain Gauss-type quadrature formulas for unbounded intervals

We consider the convergence of Gauss-type quadrature formulas for the integral , where is a weight function on the half line . The -point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials }, where is a sequence of integers satisfying and . It is proved that under certain Carleman-type conditions for the weight and when or goes to , then convergence holds for all functions for which is integrable on . Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.

     

Class group frequencies of real quadratic function fields: The degree 4 case

The distribution of ideal class groups of is examined for degree-four monic polynomials when is a finite field of characteristic greater than 3 with or and is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the -Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.

     

A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

The standard algorithm for testing reducibility of a trinomial of prime degree over requires bits of memory. We describe a new algorithm which requires only bits of memory and significantly fewer memory references and bit-operations than the standard algorithm.

If is a Mersenne prime, then an irreducible trinomial of degree is necessarily primitive. We give primitive trinomials for the Mersenne exponents , , and . The results for extend and correct some computations of Kumada et al. The two results for are primitive trinomials of the highest known degree.

     

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.

     

Nontrivial Galois module structure of cyclotomic fields

We say a tame Galois field extension with Galois group has trivial Galois module structure if the rings of integers have the property that is a free -module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes so that for each there is a tame Galois field extension of degree so that has nontrivial Galois module structure. However, the proof does not directly yield specific primes for a given algebraic number field For any cyclotomic field we find an explicit so that there is a tame degree extension with nontrivial Galois module structure.

     

Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems

In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate in a discrete -weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter . Numerical tests indicate that the rate is sharp for the boundary layer terms. As a by-product, an -uniform convergence of the same order is obtained for the -norm. Furthermore, under the same regularity assumption, an -uniform convergence of order in the norm is proved for some mesh points in the boundary layer region.

     

Involutions and characters of upper triangular matrix groups

We study the realizability over of representations of the group of upper-triangular matrices over . We prove that all the representations of are realizable over if , but that if , has representations not realizable over . This theorem is a variation on a result that can be obtained by combining work of J. Arregi and A. Vera-López and of the authors, but the proof of the theorem in this paper is much more natural.

     

Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime we show a way to construct large families of polynomials , , where is a set of primes of the form mod and is the irreducible polynomial of the Gaussian periods of degree in . Examples of these families when are worked in detail. We also show, given an integer and a prime mod , how to represent by matrices the Gaussian periods of degree in , and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of .

     

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.

     

Computing

Let denote the Von Mangoldt function and . We describe an elementary method for computing isolated values of . The complexity of the algorithm is time and space. A table of values of for up to is included, and some times of computation are given.

     

On the total number of prime factors of an odd perfect number

We say is perfect if , where denotes the sum of the positive divisors of . No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form , where , , ..., are distinct primes and . We prove that if or for all , , then . We also prove as our main result that , where . This improves a result of Sayers given in 1986.

     

Class numbers of real cyclotomic fields of prime conductor

The class numbers of the real cyclotomic fields are notoriously hard to compute. Indeed, the number is not known for a single prime . In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields for the primes . It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue --on the basis of the Cohen-Lenstra heuristics-- that the probability that our table is actually a table of class numbers , is at least .

     

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.