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 , , .

     

Rationality problem of three-dimensional purely monomial group actions: the last case

A -automorphism of the rational function field is called purely monomial if sends every variable to a monic Laurent monomial in the variables . Let be a finite subgroup of purely monomial -automorphisms of . The rationality problem of the -action is the problem of whether the -fixed field is -rational, i.e., purely transcendental over , or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.

     

Notes on some new kinds of pseudoprimes

J. Browkin defined in his recent paper (Math. Comp. 73 (2004), pp. 1031-1037) some new kinds of pseudoprimes, called Sylow -pseudoprimes and elementary Abelian -pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow -pseudoprime to two bases only, where or .

In this paper, in contrast to Browkin's examples, we give facts and examples which are unfavorable for Browkin's observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow -pseudoprimes to the same bases. In Section 3, we give examples of Sylow -pseudoprimes to the first several prime bases for the first several primes . We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow -pseudoprime to the same bases for all . In Section 4, we define to be a -fold Carmichael Sylow pseudoprime, if it is a Sylow -pseudoprime to all bases prime to for all the first smallest odd prime factors of . We find and tabulate all three -fold Carmichael Sylow pseudoprimes . In Section 5, we define a positive odd composite to be a Sylow uniform pseudoprime to bases , or a Syl-upsp for short, if it is a Syl-psp for all the first small prime factors of , where is the number of distinct prime factors of . We find and tabulate all the 17 Syl-upsp's and some Syl-upsp 's . Comparisons of effectiveness of Browkin's observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6.

     

Analysis of PSLQ, an integer relation finding algorithm

Let be either the real, complex, or quaternion number system and let be the corresponding integers. Let be a vector in . The vector has an integer relation if there exists a vector , , such that . In this paper we define the parameterized integer relation construction algorithm PSLQ, where the parameter can be freely chosen in a certain interval. Beginning with an arbitrary vector , iterations of PSLQ will produce lower bounds on the norm of any possible relation for . Thus PSLQ can be used to prove that there are no relations for of norm less than a given size. Let be the smallest norm of any relation for . For the real and complex case and each fixed parameter in a certain interval, we prove that PSLQ constructs a relation in less than iterations.

     

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.

     

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.

     

On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems

The iteratively regularized Gauss-Newton method is applied to compute the stable solutions to nonlinear ill-posed problems when the data is given approximately by with . In this method, the iterative sequence is defined successively by

where is an initial guess of the exact solution and is a given decreasing sequence of positive numbers admitting suitable properties. When is used to approximate , the stopping index should be designated properly. In this paper, an a posteriori stopping rule is suggested to choose the stopping index of iteration, and with the integer determined by this rule it is proved that

with a constant independent of , where denotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of is obtained, and moreover the rate of convergence is derived when satisfies a suitable ``source-wise representation". The results of this paper suggest that the iteratively regularized Gauss-Newton method, combined with our stopping rule, defines a regularization method of optimal order for each . Numerical examples for parameter estimation of a differential equation are given to test the theoretical results.

     

Gaussian elimination is stable for the inverse of a diagonally dominant matrix

Let be a row diagonally dominant matrix, i.e.,

where with We show that no pivoting is necessary when Gaussian elimination is applied to Moreover, the growth factor for does not exceed The same results are true with row diagonal dominance being replaced by column diagonal dominance.

     

Double roots of power series and related matters

For a given collection of distinct arguments , multiplicities and a real interval containing zero, we are interested in determining the smallest for which there is a power series with coefficients in , and roots of order respectively. We denote this by . We describe the usual form of the extremal series (we give a sufficient condition which is also necessary when the extremal series possesses at least non-dependent coefficients strictly inside , where is 1 or 2 as is real or complex). We focus particularly on , the size of the smallest double root of a power series lying on a given ray (of interest in connection with the complex analogue of work of Boris Solomyak on the distribution of the random series ). We computed the value of for the rationals in of denominator less than fifty. The smallest value we encountered was . For the one-sided intervals and the corresponding smallest values were and .

     

Stability analysis for delay differential equations with multidelays and numerical examples

In this paper we are concerned with the asymptotic stability of the delay differential equation

where are constant complex matrices, and 0$"> stand for constant delays . We obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region. We also get similar results for the neutral delay differential equation

where and are constant complex matrices and 0$"> stands for constant delays , . Numerical examples on various circumstances are shown to check our results which are more general than those already reported.

     

Finite SAGBI bases for polynomial invariants of conjugates of alternating groups

It is well-known, that the ring of polynomial invariants of the alternating group has no finite SAGBI basis with respect to the lexicographical order for any number of variables . This note proves the existence of a nonsingular matrix such that the ring of polynomial invariants , where denotes the conjugate of with respect to , has a finite SAGBI basis for any .     

Parallel integer relation detection: Techniques and applications

Let be a vector of real numbers. An integer relation algorithm is a computational scheme to find the integers , if they exist, such that . In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing.

This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Single- and multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.

     

The Postage Stamp Problem: An algorithm to determine the

Given an integral ``stamp" basis with and a positive integer , we define the -range as

. For given and , the extremal basis has the largest possible extremal -range

We give an algorithm to determine the -range. We prove some properties of the -range formula, and we conjecture its form for the extremal -range. We consider parameter bases , where the basis elements are given functions of . For we conjecture the extremal parameter bases for .

     

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 .)

     

New convergence results on the generalized Richardson extrapolation process GREP for logarithmic sequences

Let as , where and are known for for some 0$">, but and the are not known. The generalized Richardson extrapolation process GREP is used in obtaining good approximations to , the limit or antilimit of as . The approximations to obtained via GREPare defined by the linear systems , , where is a decreasing positive sequence with limit zero. The study of GREP for slowly varying functions was begun in two recent papers by the author. For such we have as with possibly complex and . In the present work we continue to study the convergence and stability of GREPas it is applied to such with different sets of collocation points that have been used in practical situations. In particular, we consider the cases in which (i) are arbitrary, (ii) , (iii) as for some 0$">, (iv) for all , (v) , and (vi) for all .

     

On the optimal convergence rate of universal and nonuniversal algorithms for multivariate integration and approximation

We study the maximal rate of convergence (mrc) of algorithms for (multivariate) integration and approximation of -variate functions from reproducing kernel Hilbert spaces . Here is an arbitrary kernel all of whose partial derivatives up to order satisfy a Hölder-type condition with exponent . Algorithms use function values and we analyze their rate of convergence as tends to infinity. We focus on universal algorithms which depend on , , and but not on the specific kernel , and nonuniversal algorithms which may depend additionally on .

For universal algorithms the mrc is for both integration and approximation, and for nonuniversal algorithms it is for integration and with for approximation. Hence, the mrc for universal algorithms suffers from the curse of dimensionality if is large relative to , whereas the mrc for nonuniversal algorithms does not since it is always at least for integration, and for approximation. This is the price we have to pay for using universal algorithms. On the other hand, if is large relative to , then the mrc for universal and nonuniversal algorithms is approximately the same.

We also consider the case when we have the additional knowledge that the kernel has product structure, . Here are some univariate kernels whose all derivatives up to order satisfy a Hölder-type condition with exponent . Then the mrc for universal algorithms is for both integration and approximation, and for nonuniversal algorithms it is for integration and with for approximation. If or for all , then the mrc is at least , and the curse of dimensionality is not present. Hence, the product form of reproducing kernels breaks the curse of dimensionality even for universal algorithms.

     

Real zeros of Dedekind zeta functions of real quadratic fields

Let be a primitive, real and even Dirichlet character with conductor , and let be a positive real number. An old result of H. Davenport is that the cycle sums are all positive at and this has the immediate important consequence of the positivity of . We extend Davenport's idea to show that in fact for , 0$"> for all with so that one can deduce the positivity of by the nonnegativity of a finite sum for any . A simple algorithm then allows us to prove numerically that has no positive real zero for a conductor up to 200,000, extending the previous record of 986 due to Rosser more than 50 years ago. We also derive various estimates explicit in of the as well as the shifted cycle sums considered previously by Leu and Li for . These explicit estimates are all rather tight and may have independent interests.

     

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.

     

Modular exponentiation via the explicit Chinese remainder theorem

Fix pairwise coprime positive integers . We propose representing integers modulo , where is any positive integer up to roughly , as vectors . We use this representation to obtain a new result on the parallel complexity of modular exponentiation: there is an algorithm for the Common CRCW PRAM that, given positive integers , , and in binary, of total bit length , computes in time using processors. For comparison, a parallelization of the standard binary algorithm takes superlinear time; Adleman and Kompella gave an expected time algorithm using processors; von zur Gathen gave an NC algorithm for the highly special case that is polynomially smooth.

     

Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form

where the vector of functions is in and is a finitely supported sequence of matrices called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the subdivision scheme associated with , i.e., the convergence of the sequence in the -norm.

Our main result characterizes the convergence of a subdivision scheme associated with the mask in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the -convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations.

Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.