Iterated function systems, Ruelle operators, and invariant projective measures

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space comes with a finite-to-one endomorphism which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets in of the same cardinality which generate complex Hadamard matrices.

Our harmonic analysis for these iterated function systems (IFS) is based on a Markov process on certain paths. The probabilities are determined by a weight function on . From we define a transition operator acting on functions on , and a corresponding class of continuous -harmonic functions. The properties of the functions in are analyzed, and they determine the spectral theory of . For affine IFSs we establish orthogonal bases in . These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in .

     

Continuous-time Kreiss resolvent condition on infinite-dimensional spaces

Given the infinitesimal generator of a -semigroup on the Banach space which satisfies the Kreiss resolvent condition, i.e., there exists an such that for all complex with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated -semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like . Furthermore, we show that for every there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like . As a consequence, we find that for with the standard Euclidian norm the estimate cannot be replaced by a lower power of or .

     

The monic integer transfinite diameter

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval . The monic integer transfinite diameter is defined as the infimum of all such supremums. We show that if has length , then .

We make three general conjectures relating to the value of for intervals of length less than . We also conjecture a value for where . We give some partial results, as well as computational evidence, to support these conjectures.

We define functions and , which measure properties of the lengths of intervals with on either side of . Upper and lower bounds are given for these functions.

We also consider the problem of determining when is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.

     

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.

     

Limiting set of second order spectra

Let be a self-adjoint operator acting on a Hilbert space . A complex number is in the second order spectrum of relative to a finite-dimensional subspace iff the truncation to of is not invertible. This definition was first introduced in Davies, 1998, and according to the results of Levin and Shargorodsky in 2004, these sets provide a method for estimating eigenvalues free from the problems of spectral pollution. In this paper we investigate various aspects related to the issue of approximation using second order spectra. Our main result shows that under fairly mild hypothesis on the uniform limit of these sets, as increases towards , contain the isolated eigenvalues of of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine nonpollution and approximation at a very high level of generality.

     

Lower bounds for the condition number of a real confluent Vandermonde matrix

Lower bounds on the condition number of a real confluent Vandermonde matrix are established in terms of the dimension , or and the largest absolute value among all nodes that define the confluent Vandermonde matrix and the interval that contains the nodes. In particular, it is proved that for any modest (the largest multiplicity of distinct nodes), behaves no smaller than , or than if all nodes are nonnegative. It is not clear whether those bounds are asymptotically sharp for modest .

     

Integer transfinite diameter and polynomials with small Mahler measure

In this work, we show how suitable generalizations of the integer transfinite diameter of some compact sets in give very good bounds for coefficients of polynomials with small Mahler measure. By this way, we give the list of all monic irreducible primitive polynomials of of degree at most with Mahler measure less than and of degree and with Mahler measure less than .

     

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions

Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

     

Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

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 , , .

     

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.

     

The 192 solutions of the Heun equation

A machine-generated list of local solutions of the Heun equation is given. They are analogous to Kummer's  solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with  singular points as the Coxeter group  . Each of the expressions is labeled by an element of  . Of the ,  are equivalent expressions for the local Heun function  , and it is shown that the resulting order- group of transformations of  is isomorphic to the symmetric group . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

     

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.

     

Largest known twin primes and Sophie Germain primes

The numbers are twin primes. The number is a Sophie Germain prime, i.e. and are both primes. For , the numbers , and are all primes.

     

Optimal logarithmic energy points on the unit sphere

We study minimum energy point charges on the unit sphere in , , that interact according to the logarithmic potential , where is the Euclidean distance between points. Such optimal -point configurations are uniformly distributed as . We quantify this result by estimating the spherical cap discrepancy of optimal energy configurations. The estimate is of order . Essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term in the asymptotical expansion of the optimal energy. Previously, this was known for the unit sphere in only. Furthermore, we present an upper bound for the error of integration for an equally-weighted numerical integration rule with the nodes forming an optimal logarithmic energy configuration. For polynomials of degree at most this bound is as . For continuous functions of satisfying a Lipschitz condition with constant the bound is as .

     

Even moments of generalized Rudin-Shapiro polynomials

We know from Littlewood (1968) that the moments of order of the classical Rudin-Shapiro polynomials satisfy a linear recurrence of degree . In a previous article, we developed a new approach, which enables us to compute exactly all the moments of even order for . We were also able to check a conjecture on the asymptotic behavior of , namely , where , for even and . Now for every integer there exists a sequence of generalized Rudin-Shapiro polynomials, denoted by . In this paper, we extend our earlier method to these polynomials. In particular, the moments have been completely determined for and , for and and for and . For higher values of and , we formulate a natural conjecture, which implies that , where is an explicit constant.

     

Jacobi sums and new families of irreducible polynomials of Gaussian periods

Let 2$">, an -th primitive root of 1, mod a prime number, a primitive root modulo and . We study the Jacobi sums , , where is the least nonnegative integer such that mod . We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families , , of irreducible polynomials of Gaussian periods, , of degree , where is a suitable set of primes mod . We exhibit examples of such families for several small values of , and give a MAPLE program to construct more of them.

     

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.

     

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.

     

Computing the arithmetic genus of Hilbert modular fourfolds

The Hilbert modular fourfold determined by the totally real quartic number field is a desingularization of a natural compactification of the quotient space , where PSL acts on by fractional linear transformations via the four embeddings of into . The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight , is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results.