Minus class groups of the fields of the -th roots of unity

We show that for any prime number the minus class group of the field of the -th roots of unity admits a finite free resolution of length 1 as a module over the ring . Here denotes complex conjugation in . Moreover, for the primes we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.

     

An algorithm for constructing a basis for -spline modules over polynomial rings

Let be a polyhedral complex embedded in the euclidean space and , , denote the set of all -splines on . Then is an -module where is the ring of polynomials in several variables. In this paper we state and prove the existence of an algorithm to write down a free basis for the above -module in terms of obvious linear forms defining common faces of members of . This is done for the case when consists of a finite number of parallelopipeds properly joined amongst themselves along the above linear forms.

     

On divisibility of the class number

In this paper, criteria of divisibility of the class number of the real cyclotomic field of a prime conductor and of a prime degree by primes the order modulo of which is , are given. A corollary of these criteria is the possibility to make a computational proof that a given does not divide for any (conductor) such that both are primes. Note that on the basis of Schinzel's hypothesis there are infinitely many such primes .

     

Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

     

Class number bounds and Catalan's equation

We improve a criterion of Inkeri and show that if there is a solution to Catalan's equation

with and prime numbers greater than 3 and both congruent to 3 , then and form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 . Indeed, in light of the results proved here it is reasonable to suppose that if , then and form a double Wieferich pair.

     

Fast evaluation of multiple zeta sums

We show that the multiple zeta sum:

for positive integers with , can always be written as a finite sum of products of rapidly convergent series. Perhaps surprisingly, one may develop fast summation algorithms of such efficiency that the overall complexity can be brought down essentially to that of one-dimensional summation. In particular, for any dimension one may resolve good digits of in arithmetic operations, with the implied big- constant depending only on the set .

     

Subquadratic-time factoring of polynomials over finite fields

New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree over a finite field of constant cardinality in time . Previous algorithms required time . The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree over the finite field with elements, the algorithms use arithmetic operations in .

The new ``baby step/giant step' techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

     

On -amicable pairs

Let denote Euler's totient function, i.e., the number of positive integers and prime to . We study pairs of positive integers with such that for some integer . We call these numbers -amicable pairs with multiplier , analogously to Carmichael's multiply amicable pairs for the -function (which sums all the divisors of ).

We have computed all the -amicable pairs with larger member and found pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other -amicable pairs can be associated. Among these pairs there are so-called primitive -amicable pairs. We present a table of the primitive -amicable pairs for which the larger member does not exceed . Next, -amicable pairs with a given prime structure are studied. It is proved that a relatively prime -amicable pair has at least twelve distinct prime factors and that, with the exception of the pair , if one member of a -amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive -amicable pairs with larger member , the largest pair consisting of two 46-digit numbers.

     

Factoring elementary groups of prime cube order into subsets

Let be a prime and let be the -fold direct product of the cyclic group of order . Rédei conjectured if is the direct product of subsets and , each of which contains the identity element of , then either or does not generate all of . The paper verifies Rédei's conjecture for .

     

Factors of generalized Fermat numbers

A search for prime factors of the generalized Fermat numbers has been carried out for all pairs with and GCD. The search limit on the factors, which all have the form , was for and for . Many larger primes of this form have also been tried as factors of . Several thousand new factors were found, which are given in our tables.-For the smaller of the numbers, i.e. for , or, if , for , the cofactors, after removal of the factors found, were subjected to primality tests, and if composite with , searched for larger factors by using the ECM, and in some cases the MPQS, PPMPQS, or SNFS. As a result all numbers with are now completely factored.

     

Remarks on the Schoof-Elkies-Atkin algorithm

Schoof's algorithm computes the number of points on an elliptic curve defined over a finite field . Schoof determines modulo small primes using the characteristic equation of the Frobenius of and polynomials of degree . With the works of Elkies and Atkin, we have just to compute, when is a ``good" prime, an eigenvalue of the Frobenius using polynomials of degree . In this article, we compute the complexity of Müller's algorithm, which is the best known method for determining one eigenvalue and we improve the final step in some cases. Finally, when is ``bad", we describe how to have polynomials of small degree and how to perform computations, in Schoof's algorithm, on -values only.

     

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.

     

Finding finite -sequences faster

A -sequence is a sequence of positive integers such that the sums , , are different. When is a power of a prime and is a primitive element in then there are -sequences of size with , which were discovered by R. C. Bose and S. Chowla.

In Theorem 2.1 I will give a faster alternative to the definition. In Theorem 2.2 I will prove that multiplying a sequence by integers relatively prime to the modulus is equivalent to varying . Theorem 3.1 is my main result. It contains a fast method to find primitive quadratic polynomials over when is an odd prime. For fields of characteristic 2 there is a similar, but different, criterion, which I will consider in ``Primitive quadratics reflected in -sequences', to appear in Portugaliae Mathematica (1999).

     

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

     

Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix , where the polynomials satisfy a three-term recurrence relation. If are the Chebyshev polynomials , then coincides with . This paper presents a new fast algorithm for the computation of the matrix-vector product in arithmetical operations. The algorithm divides into a fast transform which replaces with and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes with almost the same precision as the Clenshaw algorithm, but is much faster for .

     

Steiner systems

It is proved that there are precisely 4204 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

It is further proved that there are precisely 38717 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

     

Power series with restricted coefficients and a root on a given ray

We consider bounds on the smallest possible root with a specified argument of a power series with coefficients in the interval . We describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when is rational. When we show that the smallest disc containing two roots has radius coinciding with the smallest double real root possible for such a series. It is clear from our computations that the behaviour is more complicated for smaller . We give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument. We conclude by briefly discussing variants of the beta-numbers (where the defining integer sequence is generated by taking the nearest integer rather than the integer part). We show that the conjugates, , of these pseudo-beta-numbers either lie inside the unit circle or their reciprocals must be roots of power series; in particular we obtain the sharp inequality .

     

Detecting perfect powers in essentially linear time

This paper (1) gives complete details of an algorithm to compute approximate th roots; (2) uses this in an algorithm that, given an integer , either writes as a perfect power or proves that is not a perfect power; (3) proves, using Loxton's theorem on multiple linear forms in logarithms, that this perfect-power decomposition algorithm runs in time .

     

A comparison of regularizations for an ill-posed problem

We consider numerical methods for a ``quasi-boundary value' regularization of the backward parabolic problem given by

where is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value by adding , where is a small parameter. We show how this leads very naturally to a reformulation of the problem as a second-kind Fredholm integral equation, which can be very easily approximated using methods previously developed by Ames and Epperson. Error estimates and examples are provided. We also compare the regularization used here with that from Ames and Epperson.

We consider numerical methods for a ``quasi-boundary value' regularization of the backward parabolic problem given by

where is positive self-adjoint and unbounded. The regularization, due to Clark and Oppenheimer, perturbs the final value by adding , where is a small parameter. We show how this leads very naturally to a reformulation of the problem as a second-kind Fredholm integral equation, which can be very easily approximated using methods previously developed by Ames and Epperson. Error estimates and examples are provided. We also compare the regularization used here with that from Ames and Epperson.

     

Multigrid and multilevel methods for nonconforming elements

In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming rotated finite elements in two space dimensions. In particular, for the case of square partitions and the Laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the -cycle with any number of smoothing steps and close-to-optimal condition number estimates for -cycle preconditioners. This is in contrast to most of the other nonconforming finite element discretizations where only results for -cycles with a sufficiently large number of smoothing steps and variable -cycle multigrid preconditioners are available. Some numerical tests, including also a comparison with a preconditioner obtained by switching from the nonconforming rotated discretization to a discretization by conforming bilinear elements on the same partition, illustrate the theory.