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 , .

     

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.

     

Construction of Local Quartic Spline Elements for Optimal-Order Approximation

This paper is concerned with a study of approximation order and construction of locally supported elements for the space of (piecewise polynomial) functions on an arbitrary triangulation of a connected polygonal domain in . It is well known that even when is a three-directional mesh , the order of approximation of is only 4, not 5. The objective of this paper is two-fold: (i) A local Clough-Tocher refinement procedure of an arbitrary triangulation is introduced so as to yield the optimal (fifth) order of approximation, where locality means that only a few isolated triangles need refinement, and (ii) locally supported Hermite elements are constructed to achieve the optimal order of approximation.

     

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.

     

On beta expansions for Pisot numbers

Given a number , the beta-transformation is defined for by (mod 1). The number is said to be a beta-number if the orbit is finite, hence eventually periodic. In this case is the root of a monic polynomial with integer coefficients called the characteristic polynomial of . If is the minimal polynomial of , then for some polynomial . It is the factor which concerns us here in case is a Pisot number. It is known that all Pisot numbers are beta-numbers, and it has often been asked whether must be cyclotomic in this case, particularly if . We answer this question in the negative by an examination of the regular Pisot numbers associated with the smallest 8 limit points of the Pisot numbers, by an exhaustive enumeration of the irregular Pisot numbers in (an infinite set), by a search up to degree in , to degree in , and to degree in . We find the smallest counterexample, the counterexample of smallest degree, examples where is nonreciprocal, and examples where is reciprocal but noncyclotomic. We produce infinite sequences of these two types which converge to from above, and infinite sequences of with nonreciprocal which converge to from below and to the th smallest limit point of the Pisot numbers from both sides. We conjecture that these are the only limit points of such numbers in . The Pisot numbers for which is cyclotomic are related to an interesting closed set of numbers introduced by Flatto, Lagarias and Poonen in connection with the zeta function of . Our examples show that the set of Pisot numbers is not a subset 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.

     

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 Results on Pseudosquares

If is an odd prime, the pseudosquare is defined to be the least positive nonsquare integer such that and the Legendre symbol for all odd primes . In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to . We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that , an inequality that must hold under the extended Riemann Hypothesis.

     

Computing : the Meissel, Lehmer, Lagarias, Miller, Odlyzko method

Let denote the number of primes . Our aim in this paper is to present some refinements of a combinatorial method for computing single values of , initiated by the German astronomer Meissel in 1870, extended and simplified by Lehmer in 1959, and improved in 1985 by Lagarias, Miller and Odlyzko. We show that it is possible to compute in time and space. The algorithm has been implemented and used to compute .

     

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 .

     

Spontaneous Generation of Modular Invariants

It is possible to compute and its modular equations with no perception of its related classical group structure except at . We start by taking, for prime, an unknown ``-Newtonian' polynomial equation with arbitrary coefficients (based only on Newton's polygon requirements at for and ). We then ask which choice of coefficients of leads to some consistent Laurent series solution , (where . It is conjectured that if the same Laurent series works for -Newtonian polynomials of two or more primes , then there is only a bounded number of choices for the Laurent series (to within an additive constant). These choices are essentially from the set of ``replicable functions,' which include more classical modular invariants, particularly . A demonstration for orders and is done by computation. More remarkably, if the same series works for the -Newtonian polygons of 15 special ``Fricke-Monster' values of , then is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise ``spontaneously.'

     

Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation

for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with .

     

Error Bounds for quasi-Monte Carlo integration with nets

We analyze the error introduced by approximately calculating the -dimensional Lebesgue measure of a Jordan-measurable subset of . We give an upper bound for the error of a method using a -net, which is a set with a very regular distribution behavior. When the subset of is defined by some function of bounded variation on , the error is estimated by means of the variation of the function and the discrepancy of the point set which is used. A sharper error bound is established when a -net is used. Finally a lower bound of the error is given, for a method using a -net. The special case of the 2-dimensional Hammersley point set is discussed.

     

Explicit bounds for primes in residue classes

Let be an abelian extension of number fields, with . Let and denote the absolute discriminant and degree of . Let denote an element of the Galois group of . We prove the following theorems, assuming the Extended Riemann Hypothesis:

(1)

There is a degree- prime of such that , satisfying .

(2)

There is a degree- prime of such that generates
the same group as , satisfying .

(3)

For , there is a prime such that , satisfying
.

In (1) and (2) we can in fact take to be unramified in . A special case of this result is the following.

(4)

If , the least prime satisfies
.

It follows from our proof that (1)--(3) also hold for arbitrary Galois extensions, provided we replace by its conjugacy class . Our theorems lead to explicit versions of (1)--(4), including the following: the least prime is less than .

     

On Quadrature Convergence of Extended Lagrange Interpolation

Quadrature convergence of the extended Lagrange interpolant for any continuous function is studied, where the interpolation nodes are the zeros of an orthogonal polynomial of degree and the zeros of the corresponding ``induced' orthogonal polynomial of degree . It is found that, unlike convergence in the mean, quadrature convergence does hold for all four Chebyshev weight functions. This is shown by establishing the positivity of the underlying quadrature rule, whose weights are obtained explicitly. Necessary and sufficient conditions for positivity are also obtained in cases where the nodes and interlace, and the conditions are checked numerically for the Jacobi weight function with parameters and . It is conjectured, in this case, that quadrature convergence holds for .

     

An asymptotic expansion for the incomplete beta function

A new asymptotic expansion is derived for the incomplete beta function , which is suitable for large , small and . This expansion is of the form

where is the incomplete Gamma function ratio and . This form has some advantages over previous asymptotic expansions in this region in which depends on as well as on and .

     

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.

     

Aurifeuillian factorizations and the period of the Bell numbers modulo a prime

We show that the minimum period modulo of the Bell exponential integers is for all primes and several larger . Our proof of this result requires the prime factorization of these periods. For some primes the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.