A new algorithm for constructing large Carmichael numbers

We describe an algorithm for constructing Carmichael numbers with a large number of prime factors . This algorithm starts with a given number , representing the value of the Carmichael function . We found Carmichael numbers with up to factors.

     

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 .

     

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.

     

On the beta expansion for Salem numbers of degree 6

For a given , the beta transformation is defined for by (mod ). The number is said to be a beta number if the orbit is finite, hence eventually periodic. It is known that all Pisot numbers are beta numbers, and it is conjectured that this is true for Salem numbers, but this is known only for Salem numbers of degree . Here we consider some computational and heuristic evidence for the conjecture in the case of Salem numbers of degree , by considering the set of such numbers of trace at most . Although the orbit is small for the majority of these numbers, there are some examples for which the orbit size is shown to exceed and for which the possibility remains that the orbit is infinite. There are also some very large orbits which have been shown to be finite: an example is given for which the preperiod length is and the period length is . This is in contrast to Salem numbers of degree where the orbit size is bounded by . An heuristic probabilistic model is proposed which explains the difference between the degree- and degree- cases. The model predicts that all Salem numbers of degree and should be beta numbers but that degree- Salem numbers can have orbits which are arbitrarily large relative to the size of . Furthermore, the model predicts that a positive proportion of Salem numbers of any fixed degree will not be beta numbers. This latter prediction is not tested here.

     

Turán's Pure Power Sum Problem

Let be complex numbers, and consider the power sums , . Put , where the minimum is over all possible complex numbers satisfying the above. Turán conjectured that , for some positive absolute constant. Atkinson proved this conjecture by showing . It is now known that , for . Determining whether or approaches some other limiting value as is still an open problem. Our calculations show that an upper bound for decreases for , suggesting that decreases to a limiting value less than as .

     

Convergence rate of approximate solutions to weakly coupled nonlinear systems

We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled systems appear, for example, in the context of resonant waves in gas dynamics equations.

This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or -stability) together with -consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence rate estimates, both global (quantified in terms of the -norms) and local.

We focus our attention on the -stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are -stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of -stability.

     

Computation of -invariants of real quadratic fields

Let be a real quadratic field and an odd prime number which splits in . In a previous work, the author gave a sufficient condition for the Iwasawa invariant of the cyclotomic -extension of to be zero. The purpose of this paper is to study the case of this result and give new examples of with , by using information on the initial layer of the cyclotomic -extension of .

     

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 .

     

Quality Local Refinement of Tetrahedral Meshes Based on 8-Subtetrahedron Subdivision

Let be a tetrahedral mesh. We present a 3-D local refinement algorithm for which is mainly based on an 8-subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, , where , is a positive constant independent of and the number of refinement levels, is any refined tetrahedron of , and is a tetrahedron shape measure. It is also proved that local refinements on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually refined to the number of tetrahedra chosen for refinement is bounded above by a small constant.

     

Constructing nonresidues in finite fields and the extended Riemann hypothesis

We present a new deterministic algorithm for the problem of constructing th power nonresidues in finite fields , where is prime and is a prime divisor of . We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed and , our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if is exponentially large. More generally, assuming the ERH, in time we can construct a set of elements that generates the multiplicative group . An extended abstract of this paper appeared in Proc. 23rd Ann. ACM Symp. on Theory of Computing, 1991.

     

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 .

     

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 perturbations of matrix pencils with real spectra. II

A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let and be two Hermitian matrices, and let and be their eigenvalues arranged in ascending order. Then for any unitarily invariant norm . In this paper, we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra. The much studied case of definite pencils is included in this.

     

Results and estimates on pseudopowers

Let be a positive integer. We say looks like a power of 2 modulo a prime if there exists an integer such that . First, we provide a simple proof of the fact that a positive integer which looks like a power of modulo all but finitely many primes is in fact a power of . Next, we define an -pseudopower of the base to be a positive integer that is not a power of , but looks like a power of modulo all primes . Let denote the least such . We give an unconditional upper bound on , a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that is about for a certain constant . We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than are also given.

     

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 .

     

-complete sequences of integers

An infinite sequence is -complete if every sufficiently large integer is the sum of such that no one divides the other. We investigate -completeness of sets of the form and with nonnegative.

     

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.

     

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 .

     

Expansion and Estimation of the Range of Nonlinear Functions

Many verification algorithms use an expansion , for , where the set of matrices is usually computed as a gradient or by means of slopes. In the following, an expansion scheme is described which frequently yields sharper inclusions for . This allows also to compute sharper inclusions for the range of over a domain. Roughly speaking, has to be given by means of a computer program. The process of expanding can then be fully automatized. The function need not be differentiable. For locally convex or concave functions special improvements are described. Moreover, in contrast to other methods, may be empty without implying large overestimations for . This may be advantageous in practical applications.

     

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.