Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

Oscillations of certain second order nonlinear differential equations

A second order nonlinear differential equation

Dirichlet norms,capacities and generalized isoperimetric inequalities for Markov operators

We define the notion of p-capacity for a reversible Markov operator on a general measure space and prove that uniform estimates for the ratio of capacity and measure are equivalent to certain imbedding theorems for the Orlicz and Dirichlet norms. As a corollary we get results on connections between embedding theorems and isoperimetric properties for general Markov operators and, particularly, a generalization of the Kesten theorem on the spectral radius of random walks on amenable groups for the case of arbitrary graphs with non-finitely supported transition probabilities.     

Partially finite convex programming,Part I: Quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.     

Influence of air pollution and site conditions on trends of carbon and oxygen isotope ratios in tree ring cellulose

Oxygen and carbon isotopic compositions of tree ring cellulose (δ¹³C_cell and δ¹⁸O_cell) were measured for pines growing at four sites in east Germany. Three sites differed markedly in soil water availability within a short distance and the fourth site served as a reference. The choice of the sites was guided by the desire to detect effects of air pollution on the long-term trend of isotopic compositions and to examine the influence of soil water availability on the relationship between the carbon and oxygen isotope ratios. Locations in east Germany are particularly well suited for the study of pollution effects because there was a steady increase in environmental contamination until the German Reunification in 1990, followed by a sharp decline due to the implementation of stricter environmental standards.

The long-term trend of δ¹³C_cell showed an extraordinary increase in the period 1945–1990 and a rapid decrease after 1990, whereas δ¹⁸O_cell remained nearly constant. The increase of δ¹³C_cell is explained by secondary fractionation caused by phytotoxicity of SO₂. Two effects are mainly responsible for the secondary fractionation under SO₂ exposure: increase of dark respiration, and changes in photosynthate allocation and partitioning. Both effects do not influence δ¹⁸O_cell. Furthermore, a significant positive correlation between the year-to-year variations of carbon and oxygen isotope ratios (δ¹³C_resid and δ¹⁸O_resid) has been found for all sites. The slopes of the relationship between δ¹³C_resid and δ¹⁸O_resid differ insignificantly. It is concluded that this relationship is not influenced by soil water availability but by climatic variables.     

Nonstationary curves in Hilbert spaces and their correlation functions II

Partially supportted by Grant MM-22/1991 of MESC     

Regenerative simulation of TES processes

Regenerative simulation has become a familiar and established tool for simulation-based estimation. However, many applications (e.g., traffic in high-speed communications networks) call for autocorrelated stochastic models to which traditional regenerative theory is not directly applicable. Consequently, extensions of regenerative simulation to dependent time series is increasingly gaining in theoretical and practical interest, with Markov chains constituting an important case. Fortunately, a regenerative structure can be identified in Harris-recurrent Markov chains with minor modification, and this structure can be exploited for standard regenerative estimation. In this paper we focus on a versatile class of Harris-recurrent Markov chains, called TES (Transform-Expand-Sample). TES processes can generate a variety of sample paths with arbitrary marginal distributions, and autocorrelation functions with a variety of functional forms (monotone, oscillating and alternating). A practical advantage of TES processes is that they can simultaneously capture the first and second order statistics of empirical sample paths (raw field measurements). Specifically, the TES modeling methodology can simultaneously match the empirical marginal distribution (histogram), as well as approximate the empirical autocorrelation function. We explicitly identify regenerative structures in TES processes and proceed to address efficiency and accuracy issues of prospective simulations. To show the efficacy of our approach, we report on a TES/M/1 case study. In this study, we used the likelihood ratio method to calculate the mean waiting time performance as a function of the regenerative structure and the intrinsic TES parameter controlling burstiness (degree of autocorrelation) in the arrival process. The score function method was used to estimate the corresponding sensitivity (gradient) with respect to the service rate. Finally, we demonstrated the importance of the particular regenerative structure selected in regard to the estimation efficiency and accuracy induced by the regeneration cycle length.     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Spaces from pieces IV

We are going to investigate simultaneous extensions of various topological structures (i.e. traces on several subsets at the same time are prescribed), also with separation axioms T₀, T₁, symmetry (in the sense of Part I, § 3), Riesz property, Lodato property. The following questions will be considered: (i) Under what conditions is there an extension? (ii) How can the finest extension be described? (iii) Is there a coarsest extension? (iv) Can we say more about extensions of two structures than in the general case? (v) Assume that certain subfamilies (e.g. the finite ones) can be extended; does the whole family have an extension, too? The general categorial results from Part I will be applied whenever possible (even they are not really needed).