Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

The smoothing Newton method for solving a system of nonsmooth equations , which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the th step, the nonsmooth function is approximated by a smooth function , and the derivative of at is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if is semismooth at the solution and satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is -uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

     

Multilevel Additive Schwarz Method for the Version of the Galerkin Boundary Element Method

We study a multilevel additive Schwarz method for the - version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the - version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns . We prove that the condition number of the multilevel additive Schwarz operator behaves like . As a direct consequence of this we also give the results for the -level preconditioner and also for the - version with quasi-uniform meshes. Numerical results supporting our theory are presented.

     

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 .

     

Orbits of algebraic numbers with low heights

We prove that the two smallest values of are and , for any algebraic integer.

     

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 .

     

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.

     

Classification of integral lattices with large class number

A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of -elementary lattices, for small prime level are obtained. For instance, the class number of -dimensional -elementary even lattices of determinant is ; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.

     

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.

     

On the Diophantine equation

If and are positive integers with and , then the equation of the title possesses at most one solution in positive integers and , with the possible exceptions of satisfying , and . The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.

     

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 .