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 , , .

     

Normal integral bases for extensions of the rationals

We give an algorithm for constructing normal integral bases of tame Galois extensions of the rationals with group . Using earlier works we can do the same until degree .

     

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.

     

On the greatest prime factor of with effective constants

Let denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of exceeding in which the constants are effectively computable. As a result we prove that it is possible to calculate a value such that for every x_0$"> there is a with the greatest prime factor of exceeding . The novelty of our approach is the avoidance of any appeal to Siegel's Theorem on primes in arithmetic progression.

     

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.

     

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.

     

Elliptic curves with nonsplit mod representations

We calculate explicitly the -invariants of the elliptic curves corresponding to rational points on the modular curve by giving an expression defined over of the -function in terms of the function field generators and of the elliptic curve . As a result we exhibit infinitely many elliptic curves over with nonsplit mod representations.

     

Practical solution of the Diophantine equation

Let be an odd prime and , positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation can be easily reduced to the resolution of the unit equation over . The solutions of the latter equation are given by Wildanger's algorithm.

     

Polyharmonic splines on grids and their limits

Radial Basis Functions (RBF) have found a wide area of applications. We consider the case of polyharmonic RBF (called sometimes polyharmonic splines) where the data are on special grids of the form having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines on such grids for the limiting process 0.$"> For a large class of data functions defined on it turns out that there exists a limit function This limit function is shown to be a polyspline of order on strips. By the theory of polysplines we know that the function is smooth up to order everywhere (in particular, they are smooth on the hyperplanes , which includes existence of the normal derivatives up to order while the RBF interpolants are smooth only up to the order The last fact has important consequences for the data smoothing practice.

     

Solutions of the congruence

To supplement existing data, solutions of are tabulated for primes with and . For , five new solutions 2^{32}$"> are presented. One of these, for , also satisfies the ``reverse' congruence . An effective procedure for searching for such ``double solutions' is described and applied to the range , . Previous to this, congruences are generally considered for any and fixed prime to see where the smallest prime solution occurs.

     

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.

     

Balanced multi-wavelets in

The notion of -balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalar-valued discrete polynomial data of order (or degree ), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for is known. In addition, the formulation of the -balancing condition for is so prohibitively difficult to satisfy that only a very few examples for and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the -balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any .

     

New irrationality measures for -logarithms

The three main methods used in diophantine analysis of -series are combined to obtain new upper bounds for irrationality measures of the values of the -logarithm function

when and .

     

Some heuristics and results for small cycles of the discrete logarithm

Brizolis asked the question: does every prime have a pair such that is a fixed point for the discrete logarithm with base ? The first author previously extended this question to ask about not only fixed points but also two-cycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on and . In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results.

     

Extension of a class of periodizing variable transformations for numerical Integration

Class variable transformations with integer , for numerical computation of finite-range integrals, were introduced and studied by the author in the paper [A. Sidi, A new variable transformation for numerical integration, Numerical Integration IV, 1993 (H. Brass and G. Hämmerlin, eds.), pp. 359-373.] A representative of this class is the -transformation that has been used with lattice rules for multidimensional integration. These transformations ``periodize' the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even . In the present work, we extend these transformations to arbitrary values of , and give a detailed analysis of the resulting transformed trapezoidal rule approximations. We show that, with suitable , they can be very useful in different situations. We prove, for example, that if the integrand function is smooth on the interval of integration and vanishes at the endpoints, then results of especially high accuracy are obtained by taking to be an odd integer. Such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration. We also illustrate some of the results with numerical examples via the extended -transformation.

     

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 .)

     

Fast algorithms for component-by-component construction of rank- lattice rules in shift-invariant reproducing kernel Hilbert spaces

We reformulate the original component-by-component algorithm for rank- lattices in a matrix-vector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost , in contrast with the original algorithm which has construction cost . Herein is the number of dimensions and the number of points (taken prime). In contrast to other approaches to speed up construction, our fast algorithm computes exactly the same quantity as the original algorithm. The presented algorithm can also be used to construct randomly shifted lattice rules in weighted Sobolev spaces.

     

Approximating the number of integers without large prime factors

denotes the number of positive integers and free of prime factors . Hildebrand and Tenenbaum gave a smooth approximation formula for in the range , where is a fixed positive number . In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate . The computational complexity of this algorithm is . We give numerical results which show that this algorithm provides accurate estimates for and is faster than conventional methods such as algorithms exploiting Dickman's function.

     

High rank elliptic curves with torsion group

We develop an algorithm for bounding the rank of elliptic curves in the family , all of them with torsion group and modular invariant . We use it to look for curves of high rank in this family and present four such curves of rank  and of rank .

     

Littlewood polynomials with high order zeros

Let be the minimal length of a polynomial with coefficients divisible by . Byrnes noted that for each , and asked whether in fact . Boyd showed that for all , but . He further showed that , and that is one of the 5 numbers , or . Here we prove that . Similarly, let be the maximal power of dividing some polynomial of degree with coefficients. Boyd was able to find for . In this paper we determine for .