New convergence results on the generalized Richardson extrapolation process GREP for logarithmic sequences

Let as , where and are known for for some 0$">, but and the are not known. The generalized Richardson extrapolation process GREP is used in obtaining good approximations to , the limit or antilimit of as . The approximations to obtained via GREPare defined by the linear systems , , where is a decreasing positive sequence with limit zero. The study of GREP for slowly varying functions was begun in two recent papers by the author. For such we have as with possibly complex and . In the present work we continue to study the convergence and stability of GREPas it is applied to such with different sets of collocation points that have been used in practical situations. In particular, we consider the cases in which (i) are arbitrary, (ii) , (iii) as for some 0$">, (iv) for all , (v) , and (vi) for all .

     

Solving norm equations in relative number fields using -units

In this paper, we are interested in solving the so-called norm equation , where is a given arbitrary extension of number fields and a given algebraic number of . By considering -units and relative class groups, we show that if there exists at least one solution (in , but not necessarily in ), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the -units that are norms are norms of -units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.

     

Upper bounds for the prime divisors of Wendt's determinant

Let be an even integer, . The resultant of the polynomials and is known as Wendt's determinant of order . We prove that among the prime divisors of only those which divide or can be larger than , where and is the th Lucas number, except when and . Using this estimate we derive criteria for the nonsolvability of Fermat's congruence.

     

On a problem of Byrnes concerning polynomials with restricted coefficients, II

As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length of a polynomial with all coefficients in which has a zero of a given order at . In that paper we showed that for all and showed that the extremal polynomials for were those conjectured by Byrnes, but for that rather than . A polynomial with was exhibited for , but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that is one of the 7 values or . Here we prove that without determining all extremal polynomials. We also make some progress toward determining . As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if is a primitive th root of unity where is a prime, then the condition that all coefficients of be in , together with the requirement that be divisible by puts severe restrictions on the possible values for the cyclotomic integer .

     

Some numerical computations concerning spinor zeta functions in genus at the central point

We numerically compute the central critical values of odd quadratic character twists with respect to some small discriminants of spinor zeta functions attached to Seigel-Hecke eigenforms of genus 2 in the first few cases where does not belong to the Maass space. As a result, in the cases considered we can numerically confirm a conjecture of Böcherer, according to which these central critical values should be proportional to the squares of certain finite sums of Fourier coefficients of .

     

On the least prime primitive root modulo a prime

We derive a conditional formula for the natural density of prime numbers having its least prime primitive root equal to , and compare theoretical results with the numerical evidence.

     

Numerical calculation of the density of prime numbers with a given least primitive root

In this paper the densities of prime numbers having the least primitive root , where is equal to one of the initial positive integers less than 32, have been numerically calculated. The computations were carried out under the assumption of the Generalised Riemann Hypothesis. The results of these computations were compared with the results of numerical frequency estimations.

     

The irreducibility of some level 1 Hecke polynomials

Let be the characteristic polynomial of the Hecke operator acting on the space of level 1 cusp forms . We show that is irreducible and has full Galois group over  for and ,  prime.

     

Approximation by quadrilateral finite elements

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order in and order in is that the given space of functions on the reference element contain all polynomial functions of total degree at most . In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree . We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

     

Positivity preserving finite element approximation

We consider finite element operators defined on ``rough' functions in a bounded polyhedron in . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of and approximating general functions which do not. We give impossibility results for approximation of general functions to more than first order accuracy at extreme points of . We also give impossibility results about invariance of positive operators on finite element functions. This is in striking contrast to the well-studied case without positivity.

     

On the primality of

For each prime , let be the product of the primes less than or equal to . We have greatly extended the range for which the primality of and are known and have found two new primes of the first form ( ) and one of the second (). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.

     

Finite SAGBI bases for polynomial invariants of conjugates of alternating groups

It is well-known, that the ring of polynomial invariants of the alternating group has no finite SAGBI basis with respect to the lexicographical order for any number of variables . This note proves the existence of a nonsingular matrix such that the ring of polynomial invariants , where denotes the conjugate of with respect to , has a finite SAGBI basis for any .     

Computation of several cyclotomic Swan subgroups

Let denote the locally free class group, that is the group of stable isomorphism classes of locally free -modules, where is the ring of algebraic integers in the number field and is a finite group. We show how to compute the Swan subgroup, , of when , a primitive -th root of unity, , where is an odd (rational) prime so that and 2 is inert in We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial ring; we do the computations obtaining for several primes a nontrivial divisor of These calculations give an alternative proof that the fields for =11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

     

A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case

A new upper bound is provided for the L-norm of the difference between the viscosity solution of a model steady state Hamilton-Jacobi equation, , and any given approximation, . This upper bound is independent of the method used to compute the approximation ; it depends solely on the values that the residual takes on a subset of the domain which can be easily computed in terms of . Numerical experiments investigating the sharpness of the a posteriori error estimate are given.

     

Special prime numbers and discrete logs in finite prime fields

A set of primes involving numbers such as , where and , is defined. An algorithm for computing discrete logs in the finite field of order with is suggested. Its heuristic expected running time is for , where as , , and . At present, the most efficient algorithm for computing discrete logs in the finite field of order for general is Schirokauer's adaptation of the Number Field Sieve. Its heuristic expected running time is for . Using rather than general does not enhance the performance of Schirokauer's algorithm. The definition of the set and the algorithm suggested in this paper are based on a more general congruence than that of the Number Field Sieve. The congruence is related to the resultant of integer polynomials. We also give a number of useful identities for resultants that allow us to specify this congruence for some .

     

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.

     

Linear quintuple-product identities

In the first part of this paper, series and product representations of four single-variable triple products , , , and four single-variable quintuple products , , , are defined. Reduced forms and reduction formulas for these eight functions are given, along with formulas which connect them. The second part of the paper contains a systematic computer search for linear trinomial identities. The complete set of such families is found to consist of two 2-parameter families, which are proved using the formulas in the first part of the paper.

     

Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of only. For example, when polynomials of degree are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order in the -norm, whereas the post-processed approximation is of order ; if the exact solution is in only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order in , where is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

     

Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos, Joó and Komornik in 1990, is the determination of for Pisot numbers , where

Although the quantity is known for some Pisot numbers , there has been no general method for computing . This paper gives such an algorithm. With this algorithm, some properties of and its generalizations are investigated.

A related question concerns the analogy of , denoted , where the coefficients are restricted to ; in particular, for which non-Pisot numbers is nonzero? This paper finds an infinite class of Salem numbers where .

     

Polynomial factorization over

We describe algorithms for polynomial factorization over the binary field , and their implementation. They allow polynomials of degree up to to be factored in about one day of CPU time, distributing the work on two processors.