Shannon wavelets theory.

Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of theCℓ-functions

Fractals and Hidden Symmetries in DNA

This paper deals with the digital complex representation of a DNA sequence and the analysis of existing correlations by wavelets. The symbolic DNA sequence is mapped into a nonlinear time series. By studying this time series the existence of fractal shapes and symmetries will be shown. At first step, the indicator matrix enables us to recognize some typical patterns of nucleotide distribution. The DNA sequence, of the influenza virus A (H1N1), is investigated by using the complex representation, together with the corresponding walks on DNA; in particular, it is shown that DNA walks are fractals. Finally, by using the wavelet analysis, the existence of symmetries is proven

Signorini Cylindrical Waves and Shannon Wavelets

Hyperelastic materials based on Signorini's strain energy density are studied by using Shannon wavelets. Cylindrical waves propagating in a nonlinear elastic material from the circular cylindrical cavity along the radius are analyzed in the following by focusing both on the main nonlinear effects and on the method of solution for the corresponding nonlinear differential equation. Cylindrical waves' solution of the resulting equations can be easily represented in terms of this family of wavelets. It will be shown that Hankel functions can be linked with Shannon wavelets, so that wavelets can have some physical meaning being a good approximation of cylindrical waves. The nonlinearity is introduced by Signorini elastic energy density and corresponds to the quadratic nonlinearity relative to displacements. The configuration state of elastic medium is defined through cylindrical coordinates but the deformation is considered as functionally depending only on the radial coordinate. The physical and geometrical nonlinearities arising from the wave propagation are discussed from the point of view of wavelet analysis

Shannon Wavelets for the Solution of Integrodifferential Equations

Shannon wavelets are used to define a method for the solution of integrodifferential equations. This method is based on (1) the Galerking method, (2) the Shannon wavelet representation, (3) the decorrelation of the generalized Shannon sampling theorem, and (4) the definition of connection coefficients. The Shannon sampling theorem is considered in a more general approach suitable for analysing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series (connection coefficients)

Second order Shannon wavelet approximation of C2C^2-functions

In this paper, the second order approximation of a C2-function, based on Shannon wavelet functions, is given. The approximation is com- pared with the wavelet reconstruction formula and the error of approxima- tion is explicitly computed

Fractional Calculus and Shannon Wavelet

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for anyL2(ℝ)function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grünwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence

Harmonic wavelet analysis of nonlinear waves

A method based on a multiscale (wavelet) decomposition is proposed for the analysis of nonlinear waves in hyperelastic materials. The wave solution is approximated by a discrete series expansion with respect to the harmonic wavelets, and it is compared with the solution obtained by the method of successive approximations

Harmonic wavelet solution of Poisson's problem with a localized source

A method, based on a multiscale (wavelet) decomposition of the solution is proposed for the analysis of the Poisson problem. The solution is approximated by a finite series expansion of harmonic wavelets and is based on the computation of the connection coefficients. It is shown, how a sourceless Poisson's problem, solved with the Daubechies wavelets, can also be solved in presence of a localized source in the harmonic wavelet basis

Harmonic Wavelet Solution ofPoisson's Problem

The multiscale (wavelet) decomposition of the solution is proposed for the analysis of the Poisson problem. The approximate so- lution is computed with respect to a ¯nite dimensional wavelet space [4, 5, 7, 8, 16, 15] by using the Galerkin method. A fundamental role is played by the connection coe±cients [2, 7, 11, 9, 14, 17, 18], expressed by some hypergeometric series. The solution of the Poisson problem is compared with the approach based on Daubechies wavelets [18]

Complexity on Acute Myeloid Leukemia mRNA Transcript Variant

This paper deals with the sequence analysis of acute myeloid leukemia mRNA. Six transcript variants of mlf1 mRNA, with more than 2000 bps, are analyzed by focusing on the autocorrelation of each distribution. Through the correlation matrix, some patches and similarities are singled out and commented, with respect to similar distributions. The comparison of Kolmogorov fractal dimension will be also given in order to classify the six variants. The existence of a fractal shape, patterns, and symmetries are discussed as well