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.

     

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.

     

On Wendt's determinant

Wendt's determinant of order is the circulant determinant whose -th entry is the binomial coefficient , for . We give a formula for , when is even not divisible by 6, in terms of the discriminant of a polynomial , with rational coefficients, associated to . In particular, when where is a prime , this yields a factorization of involving a Fermat quotient, a power of and the 6-th power of an integer.

     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

Pure product polynomials and the Prouhet-Tarry-Escott problem

An -factor pure product is a polynomial which can be expressed in the form for some natural numbers . We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every -factor pure product has norm at least . We describe three algorithms for determining the least norm an -factor pure product can have. We report results of our computations using one of these algorithms which include the result that every -factor pure product has norm strictly greater than if is , , , or .

     

On the -rank Artin Conjecture

We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which can be generated by given multiplicatively independent numbers. In the case when the given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown-Zassenhaus (J. Number Theory 3 (1971), 306-309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to .

     

Computations of Eisenstein series on Fuchsian groups

We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of as , and also, on non-arithmetic groups, a complex Gaussian limit distribution for when near and , at least if we allow at some rate. Furthermore, on non-arithmetic groups and for fixed with near , our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.

     

New estimates for Ritz vectors

The following estimate for the Rayleigh-Ritz method is proved:

Here is a bounded self-adjoint operator in a real Hilbert/euclidian space, one of its eigenpairs, a trial subspace for the Rayleigh-Ritz method, and a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh-Ritz method, in particular, it shows that if an eigenvector is close to the trial subspace with accuracy and a Ritz vector is an approximation to another eigenvector, with a different eigenvalue. Generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested, and estimates of approximation of eigenspaces and invariant subspaces are proved.

     

Numerical solution of the scalar double-well problem allowing microstructure

The direct numerical solution of a non-convex variational problem () typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem () leading to a (degenerate) convex minimisation problem. The problem () has a minimiser and a related stress field which is known to coincide with the stress field obtained by solving () in a generalised sense involving Young measures. If is a finite element solution, is the related discrete stress field. We prove a priori and a posteriori estimates for in and weaker weighted estimates for . The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments.

     

A continuity property of multivariate Lagrange interpolation

Let be a sequence of interpolation schemes in of degree (i.e. for each one has unique interpolation by a polynomial of total degree and total order . Suppose that the points of tend to as and the Lagrange-Hermite interpolants, , satisfy for all monomials with . Theorem: for all functions of class in a neighborhood of . (Here denotes the Taylor series of at 0 to order .) Specific examples are given to show the optimality of this result.

     

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.

     

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 , .

     

The Diophantine equation

An effective method is derived for solving the equation of the title in positive integers and for given completely, and is carried out for all . If is of the form , then there is the solution , ; in the above range, except for with solution , , there are no other solutions.

     

Modular forms which behave like theta series

In this paper, we determine all modular forms of weights , , for the full modular group which behave like theta series, i.e., which have in their Fourier expansions, the constant term and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in (resp. in ) for the cases (resp. for the cases ). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.

     

On irregular prime power divisors of the Bernoulli numbers

Let ( ) denote the usual th Bernoulli number. Let be a positive even integer where or . It is well known that the numerator of the reduced quotient is a product of powers of irregular primes. Let be an irregular pair with . We show that for every the congruence has a unique solution where and . The sequence defines a -adic integer which is a zero of a certain -adic zeta function originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) -adic expansion of for irregular pairs with below 1000.

     

Canonical construction of finite elements

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, and . Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain , among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for and . At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: , and can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar - and -conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.

     

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.

     

Computing isogeny covariant differential modular forms

We present the computation modulo and explicit formulas for the unique isogeny covariant differential modular form of order one and weight called , an isogeny covariant differential modular form of order two and weight denoted by , and an isogeny covariant differential modular form of order two and weight .

     

Power series expansions for Mathieu functions with small arguments

Power series expansions for the even and odd angular Mathieu functions and , with small argument , are derived for general integer values of . The expansion coefficients that we evaluate are also useful for the calculation of the corresponding radial functions of any kind.

     

Optimal information for approximating periodic analytic functions

Let be a strip in the complex plane. For fixed integer let denote the class of -periodic functions , which are analytic in and satisfy in . Denote by the subset of functions from that are real-valued on the real axis. Given a function , we try to recover at a fixed point by an algorithm on the basis of the information

where , are the Fourier coefficients of . We find the intrinsic error of recovery

Furthermore the -dimensional optimal information error, optimal sampling error and -widths of in , the space of continuous functions on , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class , consisting of all -periodic functions, which are analytic in with -integrable boundary values. In the case sampling fails to yield optimal information as well in odd as in even dimensions.