An algorithm for matrix extension and wavelet construction

This paper gives a practical method of extending an matrix , , with Laurent polynomial entries in one complex variable , to a square matrix also with Laurent polynomial entries. If has orthonormal columns when is restricted to the torus , it can be extended to a paraunitary matrix. If has rank for each , it can be extended to a matrix with nonvanishing determinant on . The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.

     

Computing the canonical height on K3 surfaces

Let be a surface in given by the intersection of a (1,1)-form and a (2,2)-form. Then is a K3 surface with two noncommuting involutions and . In 1991 the second author constructed two height functions and which behave canonically with respect to and , and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights . We discuss how the geometry of the surface is related to formulas for the local heights, and we give practical algorithms for computing the involutions , , the local heights , , and the canonical heights , .

     

On integral bases in relative quadratic extensions

Let be an algebraic number field and a quadratic extension with . We describe a minimal set of elements for generating the integral elements of as an module. A consequence of this theoretical result is an algorithm for constructing such a set. The construction yields a simple procedure for computing an integral basis of as well. In the last section, we present examples of relative integral bases which were computed with the new algorithm and also give some running times.

     

Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers

For totally positive algebraic integers of degree , we consider the set of values of , where is the Mahler measure of . C. J. Smyth has found the four smallest values of and conjectured that the fifth point is . We prove that this is so and, moreover, we give the sixth point of .

     

Simultaneous Pell Equations

Let and be positive integers with . We shall call the simultaneous Diophantine equations

simultaneous Pell equations in and . Each such pair has the trivial solution but some pairs have nontrivial solutions too. For example, if and , then is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible to show that the number of solutions is always finite, and it is possible to give a complete list of them. In this paper we report on the solutions when .

     

Integral Cohomology and Detection of -basic 2-groups

In this paper we compute the cohomology of all -basic 2-groups with integral coefficients twisted by the orientation character . We also calculate appropriate restiction maps and thus prove that the cohomology of any -basic group is detected by subgroups isomorphic to one of five types, and we provide a sample application of this main theorem.

     

Which Circulant Preconditioner is Better?

The eigenvalue clustering of matrices and is experimentally studied, where , and respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function . Some illustrations are given to show how the clustering depends on the smoothness of and which preconditioner is preferable. An original technique for experimental exploration of the clustering rate is presented. This technique is based on the bisection idea and on the Toeplitz decomposition of a three-matrix product , where is a Toeplitz matrix and is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of is not greater than 4, provided that and are symmetric.

     

Further investigations with the strong probable prime test

Recently, Damgård, Landrock and Pomerance described a procedure in which a -bit odd number is chosen at random and subjected to random strong probable prime tests. If the chosen number passes all tests, then the procedure will return that number; otherwise, another -bit odd integer is selected and then tested. The procedure ends when a number that passes all tests is found. Let denote the probability that such a number is composite. The authors above have shown that when and . In this paper we will show that this is in fact valid for all and .

     

Some New Error Estimates for Ritz--Galerkin Methods with Minimal Regularity Assumptions

New uniform error estimates are established for finite element approximations of solutions of second-order elliptic equations using only the regularity assumption . Using an Aubin--Nitsche type duality argument we show for example that, for arbitrary (fixed) sufficiently small, there exists an such that for

Here, denotes the norm on the Sobolev space . Other related results are established.