A counterexample concerning the -projector onto linear spline spaces

For the -orthogonal projection onto spaces of linear splines over simplicial partitions in polyhedral domains in , , we show that in contrast to the one-dimensional case, where independently of the nature of the partition, in higher dimensions the -norm of cannot be bounded uniformly with respect to the partition. This fact is folklore among specialists in finite element methods and approximation theory but seemingly has never been formally proved.

     

An interpolation error estimate in based on the anisotropic measures of higher order derivatives

In this paper, we introduce the magnitude, orientation, and anisotropic ratio for the higher order derivative (with ) of a function to characterize its anisotropic behavior. The magnitude is equivalent to its usual Euclidean norm. The orientation is the direction along which the absolute value of the -th directional derivative is about the smallest, while along its perpendicular direction it is about the largest. The anisotropic ratio measures the strength of the anisotropic behavior of . These quantities are invariant under translation and rotation of the independent variables. They correspond to the area, orientation, and aspect ratio for triangular elements. Based on these measures, we derive an anisotropic error estimate for the piecewise polynomial interpolation over a family of triangulations that are quasi-uniform under a given Riemannian metric. Among the meshes of a fixed number of elements it is identified that the interpolation error is nearly the minimum on the one in which all the elements are aligned with the orientation of , their aspect ratios are about the anisotropic ratio of , and their areas make the error evenly distributed over every element.

     

On the embedding problem for representations

Let denote the double cover of corresponding to the element in where transpositions lift to elements of order and the product of two disjoint transpositions to elements of order . Given an elliptic curve , let denote its -torsion points. Under some conditions on elements in correspond to Galois extensions of with Galois group (isomorphic to) . In this work we give an interpretation of the addition law on such fields, and prove that the obstruction for having a Galois extension with gives a homomorphism . As a corollary we can prove (if has conductor divisible by few primes and high rank) the existence of -dimensional representations of the absolute Galois group of attached to and use them in some examples to construct modular forms mapping via the Shimura map to (the modular form of weight attached to) .

     

An -local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type

In this paper, an -local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On -quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken norm and norm which are optimal in , suboptimal in are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

     

Rational Gauss-Chebyshev quadrature formulas for complex poles outside

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside to arbitrary complex poles outside . The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside .

     

Two kinds of strong pseudoprimes up to

Let be an odd composite integer. Write with odd. If either mod or mod for some , then we say that is a strong pseudoprime to base , or spsp() for short. Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Conjectured values of were given by us in our previous papers (Math. Comp. 72 (2003), 2085-2097; 74 (2005), 1009-1024).

The main purpose of this paper is to give exact values of for ; to give a lower bound of : ; and to give reasons and numerical evidence of K2- and -spsp's to support the following conjecture: for any , where (resp. ) is the smallest K2- (resp. -) strong pseudoprime to all the first prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: with primes and ; and that a -spsp is an spsp and a Carmichael number of the form: with each prime factor mod .)

     

Comparison theorems of Kolmogorov type and exact values of -widths on Hardy-Sobolev classes

Let be a strip in complex plane. denotes those -periodic, real-valued functions on which are analytic in the strip and satisfy the condition , . Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel'fand, and information -widths of in , , and 2-widths of in , , .

In this paper we continue their work. Firstly, we establish a comparison theorem of Kolmogorov type on , from which we get an inequality of Landau-Kolmogorov type. Secondly, we apply these results to determine the exact values of the Gel'fand -width of in , . Finally, we calculate the exact values of Kolmogorov -width, linear -width, and information -width of in , , .

     

Noether's problem and and its subgroups

We study Noether's problem for various subgroups of the normalizer of a group generated by an -cycle in , the symmetric group of degree , in three aspects according to the way they act on rational function fields, i.e., , and . We prove that it has affirmative answers for those containing properly and derive a -generic polynomial with four parameters for each . On the other hand, it is known in connection to the negative answer to the same problem for that there does not exist a -generic polynomial for . This leads us to the question whether and how one can describe, for a given field of characteristic zero, the set of -extensions . One of the main results of this paper gives an answer to this question.

     

The Abel Lemma and the -Gosper Algorithm

Chu has recently shown that the Abel lemma on summation by parts reveals the telescoping nature of Bailey's bilateral summation formula. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the -Gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas.

     

Real zeros of real odd Dirichlet -functions

Let be a real odd Dirichlet character of modulus , and let be the associated Dirichlet -function. As a consequence of the work of Low and Purdy, it is known that if and , , , then has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if , then has no positive real zeros.

     

New expansions of numerical eigenvalues for by nonconforming elements

The paper explores new expansions of the eigenvalues for in with Dirichlet boundary conditions by the bilinear element (denoted ) and three nonconforming elements, the rotated bilinear element (denoted ), the extension of (denoted ) and Wilson's elements. The expansions indicate that and provide upper bounds of the eigenvalues, and that and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the convergence rate can be obtained, where is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.

     

Continuous interior penalty -finite element methods for advection and advection-diffusion equations

A continuous interior penalty -finite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advection-diffusion equations. The analysis relies on three technical results that are of independent interest: an -inverse trace inequality, a local discontinuous to continuous -interpolation result, and -error estimates for continuous -orthogonal projections.

     

On the polynomial representation for the number of partitions with fixed length

In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

     

Computation of the -part of the ideal class group of certain real abelian fields

Under Greenberg's conjecture, we give an efficient method to compute the -part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the -part of the ideal class group of the maximal real subfield of in the range and . In order to explain our method, we show an example whose ideal class group is not cyclic.

     

spline wavelets on triangulations

In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.

     

Minimality and other properties of the width- nonadjacent form

Let be an integer and let be the set of integers that includes zero and the odd integers with absolute value less than . Every integer can be represented as a finite sum of the form , with , such that of any consecutive 's at most one is nonzero. Such representations are called width- nonadjacent forms (-NAFs). When these representations use the digits and coincide with the well-known nonadjacent forms. Width- nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the -NAF. We show that -NAFs have a minimal number of nonzero digits and we also give a new characterization of the -NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on -NAFs and show that any base 2 representation of an integer, with digits in , that has a minimal number of nonzero digits is at most one digit longer than its binary representation.

     

Finding -strong pseudoprimes

Let be odd primes and . Put

Then we call the kernel, the triple the signature, and the height of , respectively. We call a -number if it is a Carmichael number with each prime factor . If is a -number and a strong pseudoprime to the bases for , we call a -spsp . Since -numbers have probability of error (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of (the smallest strong pseudoprime to all the first prime bases). If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement.

In this paper, we first describe an algorithm for finding -spsp(2)'s, to a given limit, with heights bounded. There are in total -spsp's with heights . We then give an overview of the 21978 - spsp(2)'s and tabulate of them, which are -spsp's to the first prime bases up to ; three numbers are spsp's to the first 11 prime bases up to 31. No -spsp's to the first prime bases with heights were found. We conjecture that there exist no -spsp's to the first prime bases with heights and so that

which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those -spsp's is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates of -spsp's and their prime factors to Miller's tests, and obtain the desired numbers. At last we speed our algorithm for finding larger -spsp's, say up to , with a given signature to more prime bases. Comparisons of effectiveness with Arnault's and our previous methods for finding -strong pseudoprimes to the first several prime bases are given.

     

The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

Computing weight modular forms of level

For a prime we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space of modular forms of weight and level . For we define a special Hecke stable subspace of which contains the space of modular forms with CM by the ring of integers of and we describe the calculation of the corresponding Brandt matrices.

     

Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems

In many physical situations, a few specific eigenvalues of a large sparse generalized eigenvalue problem are needed. If exact linear solves with are available, implicitly restarted Arnoldi with purification is a common approach for problems where is positive semidefinite. In this paper, a new approach based on implicitly restarted Arnoldi will be presented that avoids most of the problems due to the singularity of . Secondly, if exact solves are not available, Jacobi-Davidson QZ will be presented as a robust method to compute a few specific eigenvalues. Results are illustrated by numerical experiments.