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.     

Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative polynomials with restricted zeros

Sharp Remez-, Nikolskii-, and Markov-type inequalities are proved for functions of the form

Phenomenology of nonlocal cellular automata

Dynamical systems with nonlocal connections have potential applications to economic and biological systems. This paper studies the dynamics of nonlocal cellular automata. In particular, all two-state, three-input nonlocal cellular automata are classified according to the dynamical behavior starting from random initial configurations and random wirings, although it is observed that sometimes a rule can have different dynamical behaviors with different wirings. The nonlocal cellular automata rule space is studied using a mean-field parametrization which is ideal for the situation of random wiring. Nonlocal cellular automata can be considered as computers carrying out computation at the level of each component. Their computational abilities are studied from the point of view of whether they contain many basic logical gates. In particular, I ask the question of whether a three-input cellular automaton rule contains the three fundamental logical gates: two-input rules AND and OR, and one-input rule NOT. A particularly interesting edge-of-chaos nonlocal cellular automaton, the rule 184, is studied in detail. It is a system of coupled selectors or multiplexers. It is also part of the Fredkin's gate—a proposed fundamental gate for conservative computations. This rule exhibits irregular fluctuations of density, large coherent structures, and long transient times.     

Determination of cocaine contamination on banknotes using tandem mass spectrometry and pattern recognition

Tandem mass spectrometry is used to monitor the contamination of banknotes by cocaine. By introducing a series of banknotes into an instrument a distribution of contamination can be obtained. The distribution of samples arising from defendants where the banknotes have been in close proximity to cocaine should differ from the distribution from the general background population. Peak picking and integration is used to produce a series of intensity readings for a batch of banknotes. By visually inspecting these distribution, and applying a variety of chemometric methods (principal components analysis, cluster analysis and class modelling via Mahalanobis distance) it is possible to discriminate effectively between the two classes of distribution (7157 background notes and 4826 case notes alleged to be from drug dealers). By calculating the Mahalonobis distance over 100 bootstrap iterations, background samples were correctly classified 96.48% of the time, while case samples were correctly classified 89.37% of the time.     

A central limit theorem in nonlinear filtering

It is shown that the probability law of a diffusion process conditioned on weakly corrupted observations is asymptotically Gaussian when properly scaled. The method of proof involves Fisher information matrices and a Cramér-Rao inequality.     

Random frog: An efficient reversible jump Markov Chain Monte Carlo-like approach for variable selection with applications to gene selection and disease classification

The identification of disease-relevant genes represents a challenge in microarray-based disease diagnosis where the sample size is often limited. Among established methods, reversible jump Markov Chain Monte Carlo (RJMCMC) methods have proven to be quite promising for variable selection. However, the design and application of an RJMCMC algorithm requires, for example, special criteria for prior distributions. Also, the simulation from joint posterior distributions of models is computationally extensive, and may even be mathematically intractable. These disadvantages may limit the applications of RJMCMC algorithms. Therefore, the development of algorithms that possess the advantages of RJMCMC methods and are also efficient and easy to follow for selecting disease-associated genes is required. Here we report a RJMCMC-like method, called random frog that possesses the advantages of RJMCMC methods and is much easier to implement. Using the colon and the estrogen gene expression datasets, we show that random frog is effective in identifying discriminating genes. The top 2 ranked genes for colon and estrogen are Z50753, U00968, and Y10871_at, Z22536_at, respectively. (The source codes with GNU General Public License Version 2.0 are freely available to non-commercial users at: http://code.google.com/p/randomfrog/.)     

Automatic hierarchy classification in venation networks using directional morphological filtering for hierarchical structure traits extraction

The extraction of vein traits from venation networks is of great significance to the development of a variety of research fields, such as evolutionary biology. However, traditional studies normally target to the extraction of reticulate structure traits (ReSTs), which is not sufficient enough to distinguish the difference between vein orders. For hierarchical structure traits (HiSTs), only a few tools have made attempts with human assistance, and obviously are not practical for large-scale traits extraction. Thus, there is a necessity to develop the method of automated vein hierarchy classification, raising a new challenge yet to be addressed. We propose a novel vein hierarchy classification method based on directional morphological filtering to automatically classify vein orders. Different from traditional methods, our method classify vein orders from highly dense venation networks for the extraction of traits with ecological significance. To the best of our knowledge, this is the first attempt to automatically classify vein hierarchy. To evaluate the performance of our method, we prepare a soybean transmission image dataset (STID) composed of 1200 soybean leaf images and the vein orders of these leaves are manually coarsely annotated by experts as ground truth. We apply our method to classify vein orders of each leaf in the dataset. Compared with ground truth, the proposed method achieves great performance, while the average deviation on major vein is less than 5 pixels and the average completeness on second-order veins reaches 54.28%.     

Global classical solutions to a non-isothermal dynamical ginzburg-landau model in superconductivity

In this paper we propose and study a generalized time dependent Ginzburg-Landau model for superconductivity in which the temperature appears as a variable rather than a parameter. The model is derived by using the general theory of thermodynamics for non-isothermal phase transitions developed by Penrose-Fife and Alt-Pawlow. The global existence of classical solutions of the model is established by using Leray-Schauder fixed point theorem     

Model selection based on Lorenz and concentration curves,Gini indices and convex order

In order to determine an appropriate amount of premium, statistical goodness-of-fit criteria must be supplemented with actuarial ones when assessing performance of a given candidate pure premium. In this paper, concentration curves and Lorenz curves are shown to provide actuaries with effective tools to evaluate whether a premium is appropriate or to compare two competing alternatives. The idea is to compare the premium income for sub-portfolios gathering low risks (identified as low by means of the premiums under consideration) to the true one, or equivalently, to the actual losses. Numerical illustrations performed on hypothetical data and real ones demonstrate the usefulness of the proposed approach.