Algebraic properties of separated power series

We study the commutative algebra of rings of separated power series over a ring E and that of their extensions: rings of separated (and more specifically convergent) power series from a field K with a separated E-analytic structure. Both of these collections of rings already play an important role in the model theory of non-Archimedean valued fields and we establish their algebraic properties. This will make a study of the analytic geometry over such fields through the classical methods of algebraic geometry possible.     

An uncertainty inequality for Fourier-Dunkl series

An uncertainty inequality for the Fourier-Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939-2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.     

Circle and divisor problems,and double series of Bessel functions

In approximately 1915, Ramanujan recorded two identities involving doubly infinite series of Bessel functions. The identities were brought to the mathematical public for the first time when his lost notebook was published in 1988, and are connected with the classical, long-standing circle and divisor problems, respectively. We provide a proof of the first identity for the first time by analytically continuing a new kind of Dirichlet series. Delicate estimates of exponential sums are needed, and the new methods we introduce may be of independent interest.     

Baxter-Guttmann-Jensen conjecture for power series in directed percolation problem

A probabilistic model of a flow of fluid through a random medium,percolation model, provides a typical example of statistical mechanical problems which are easy to describe but difficult to solve. While the percolation problem on undirected planar lattices is exactly solved as a limit of the Potts models, there still has been no exact solution for the directed lattices. The most reliable method to provide good approximations is a numerical estimation using finite power-series expansion data of the infinite formal power series for percolation probability. In order to calculate higher-order terms in power series, Baxter and Guttmann [6] and Jensen and Guttmann [33] proposed an extrapolation procedure based on an assumption that thecorrection terms, which show the difference between the exact infinite power series and approximate finite series, are expressed as linear combinations of the Catalan numbers.In this paper, starting from a brief review on the directed percolation problem and the observation by Baxter, Guttmann, and Jensen, we state some theorems in which we explain the reason why the combinatorial numbers appear in the correction terms of power series. In the proof of our theorems, we show several useful combinatorial identities for the ballot numbers, which become the Catalan numbers in a special case. These identities ensure that a summation of products of the ballot numbers with polynomial coefficients can be expanded using the ballot numbers. There is still a gap between our theorems and the Baxter-Guttmann-Jensen observation, and we also give some conjectures.As a generalization of the percolation problem on a directed planar lattice, we present two topics at the end of this paper: The friendly walker problem and the stochastic cellular automata in higher dimensions. We hope that these two topics as well as the directed percolation problem will be of much interest to researchers of combinatorics.     

Quasi-Monte Carlo methods for numerical integration of multivariate Haar series II

The present paper contains a comparison of different classes of multivariate Haar series that have been studied with respect to numerical integration, new properties ofE _s ^α -classes and numerical results. Research supported by the Austrian Science Foundation (FWF), project no. P11143-MAT.     

Quasi-monte carlo methods for numerical integration of multivariate Haar series

In the present paper we study quasi-Monte Carlo methods to integrate functions representable by generalized Haar series in high dimensions. Using (t, m, s)-nets to calculate the quasi-Monte Carlo approximation, we get best possible estimates of the integration error for practically relevant classes of functions. The local structure of the Haar functions yields interesting new aspects in proofs and results. The results are supplemented by concrete computer calculations. Research supported by the Austrian Science Foundation (FWF), project no. P11143-MAT.     

Factorization of Laurent series over commutative rings

We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization.     

On Gevrey solutions of threefold singular nonlinear partial differential equations

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel–Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange–Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series.     

Multi-scale anomaly detection algorithm based on infrequent pattern of time series

In this paper, we propose two anomaly detection algorithms PAV and MPAV on time series. The first basic idea of this paper defines that the anomaly pattern is the most infrequent time series pattern, which is the lowest support pattern. The second basic idea of this paper is that PAV detects directly anomalies in the original time series, and MPAV algorithm extraction anomaly in the wavelet approximation coefficient of the time series. For complexity analyses, as the wavelet transform have the functions to compress data, filter noise, and maintain the basic form of time series, the MPAV algorithm, while maintaining the accuracy of the algorithm improves the efficiency. As PAV and MPAV algorithms are simple and easy to realize without training, this proposed multi-scale anomaly detection algorithm based on infrequent pattern of time series can therefore be proved to be very useful for computer science applications.     

Strong convergence of two‐dimensional Vilenkin‐Fourier series

We prove that certain means of the quadratical partial sums of the two‐dimensional Vilenkin‐Fourier series are uniformly bounded operators from the Hardy space to the space for We also prove that the sequence in the denominator cannot be improved.     

SOME TOPOLOGICAL PROPERTIES OF BENZENOID Sn,Tn-ISOMERS WITH n-RADICAL CONNECTION

We consider a series of benzenoid isomers obtained by attaching fragments to ann-radical. Some of their topological properties, such as the number of Kekulé patterns and the maximum number of aromatic π-sextets are established. Research of this author is supported in part by the RDG grant of the Pennsylvania State University.     

A remark on very ample linear series

We give a criterium on the existence of (e - 1)-very ample linear series on a general k-gonal curve of genus $g (e \geq 1)$, and we add some general remarks on such series.     

The Bochner-Riesz means for Fourier-Bessel expansions

The Bochner-Riesz means for Fourier-Bessel expansions are analyzed. We prove a uniform two-weight inequality, with potential weights, for these means. The result provides necessary and sufficient conditions for boundedness. Moreover, we obtain some corollaries regarding the convergence of these means and the boundedness of other operators related to the Fourier-Bessel series.     

Comparison of exponential‐logarithmic and logarithmic‐exponential series

We explain how the field of logarithmic‐exponential series constructed in 20 and 21 embeds as an exponential field in any field of exponential‐logarithmic series constructed in 9 , 6 , and 13 . On the other hand, we explain why no field of exponential‐logarithmic series embeds in the field of logarithmic‐exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non‐isomorphic models of Th$(\mathbb {R}_{\mbox{an, exp}})$; the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions.     

Certain summation and transformation formulas for generalized hypergeometric series

We derive summation formulas for generalized hypergeometric series of unit argument, one of which upon specialization reduces to Minton’s summation theorem. As an application we deduce a reduction formula for a certain Kampé de Fériet function that in turn provides a Kummer-type transformation formula for the generalized hypergeometric function _pF_p(x).     

A nonparametric regression cross spectrum for multivariate time series

We consider dependence structures in multivariate time series that are characterized by deterministic trends. Results from spectral analysis for stationary processes are extended to deterministic trend functions. A regression cross covariance and spectrum are defined. Estimation of these quantities is based on wavelet thresholding. The method is illustrated by a simulated example and a three-dimensional time series consisting of ECG, blood pressure and cardiac stroke volume measurements.     

Asymptotic summation of power series

Summary. We give an asymptotic expansion in powers of of the remainder , when the sequence has a similar expansion. Contrary to previous results, explicit formulas for the computation of the coefficients are presented. In the case of numerical series (), rigorous error estimates for the asymptotic approximations are also provided. We apply our results to the evaluation of , which generalizes various summation problems appeared in the recent literature on convergence acceleration of numerical and power series. Received April 22, 1997     

Product formulas and convolution structure for Fourier-Bessel series

One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter ,n=0, 1, 2,, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases and . As a consequence, a positive convolution structure is established for . The method of proof is based on solving a hyperbolic initial boundary value problem.Communicated by Tom H. Koornwinder.     

Integrated mean square properties of density estimation by orthogonal series methods for dependent variables

Summary The rates at which integrated mean square and mean squre errors of nonparametric density estimation by orthogonal series method for sequences of strictly stationary strong mixing random variables are obtained. These rates are better than those known to hold for the independent case and they are shown to hold for Markov processes. In fact our results when specialized to the independent case are improvements over previously known results of Schwartz (1967,Ann. Math. Statist.,38, 1262–1265). An extension of the results to estimation of the bivariate density is also given. Research supported by a faculty summer research grant MS-STAT-42 from the University of Petroleum and Minerals.