The Strong Variant of a Barbashin Theorem on Stability of Solutions for Non-autonomous Differential Equations in Banach Spaces

Among others we shall prove that an exponentially bounded evolution family U = {U(t, s)} _t≥s≥0 of bounded linear operators acting on a Banach space X is uniformly exponentially stable if and only if there exists q [1, ∞) such that

New bounds and conditions for the equation of Nagell-Ljunggren

The general case of the Nagell-Ljunggren equation is

     

Hemin interaction with bare and 4,4'-thio-bis-benzene-thiolate covered n-GaAs (110) electrodes

Cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) investigations on the redox behavior of hemin at bare and 4,4'-thio-bis-benzene-thiolate (TBBT) covered n-GaAs (110) electrodes in dimethylsulfoxide (DMSO) revealed the high irreversibility of the electroreduction process, which appeared to be closely related to the stable adsorbed species strongly interfering with the electronic properties of the semiconducting substrate. The subsequent exploration of the hemin-modified electrodes by second harmonic generation (SHG), X-ray photoelectron spectroscopy (XPS) and atomic force microscopy (AFM) measurements pointed to significant differences between the iron protoporphyrin species adsorbed on the bare- and TBBT-GaAs (110) electrodes. Only Fe(2+) species having a flat configuration with the porphyrin plane oriented parallel to the surface were detected on GaAs, unlike the TBBT-GaAs, where Fe(2+) and Fe(3+) species having both flat and vertical adsorption positions could be observed. These differences originate from the mutual interactions between the solvent, hemin and dithiolate molecules as well as their competition for the surface sites found to play a key role in the electrochemical process under discussion.     

Fractal-like diffractive arrangement with multiple focal points

In this paper we introduce a diffractive structure with a geometry which contains multiple Fresnel zone plates (MFZP) disposed in an arrangement based on a fractal-like rule. The corresponding diffracted intensity in planes perpendicular to the propagation axe presents two or more focal points. Their position and magnitude depend on geometrical parameters of the MFZP and on dimensions in the fractal-like arrangement. In our simulations we also analyze the influence in diffraction pattern of different non-binary phase levels in the MFZP plane. The MFZP structures with different values of geometrical and phase parameters are addressed to an optoelectronic device liquid crystal spatial light modulator (LCSLM), the experimental and simulation results are in a good agreement. The MFZP geometry with better optical parameters in diffraction pattern is then made on glass using electron beam lithography technique.     

Binomial Cycle Bases on Koszul Homology Modules

Let I be a principal p-Borel ideal of the polynomial ring S in variables x over a field. Then the Koszul homology module H ₃(x;S/I) has binomial cycle bases.     

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].     

Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.     

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 .

     

Probability and Fourier Duality for Affine Iterated Function Systems

Let d be a positive integer, and let μ be a finite measure on ? ^d . In this paper we ask when it is possible to find a subset Λ in ? ^d such that the corresponding complex exponential functions e _λ indexed by Λ are orthogonal and total in L ²(μ). If this happens, we say that (μ,Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L ²(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in ? ^d ; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.