Ergodic Space Research Articles

Weighted pseudo almost periodic functions, convolutions and abstract integral equations

This paper deals with a systematic study of the convolution operator Kf=f⁎k defined on weighted pseudo almost periodic functions space PAP(X,ρ) and with k∈L1(R). Upon making several different assumptions on k,f and ρ, we get five main results. The first two main results establish sufficient conditions on k and ρ such that the weighted ergodic space PAP0(X,ρ) is invariant under the operator K. The third result specifies a sufficient condition on all functions (k,f and ρ) such that the Kf∈PAP0(X,ρ). The fourth result is a sufficient condition on the weight function ρ such that PAP0(X,ρ) is invariant under K. The hypothesis of the convolution invariance results allows to establish a fifth result related to the translation invariance of PAP0(X,ρ). As a consequence of the fifth result, we obtain a new sufficient condition such that the unique decomposition of a weighted pseudo almost periodic function on its periodic and ergodic components is valid and also for the completeness of PAP(X,ρ) with the supremum norm. In addition, the results on convolution are applied to general abstract integral and differential equations.

Weighted Stepanov-Like Pseudoperiodicity and Applications

By the weighted ergodic space, we propose a new class of functions called weighted Stepanov-like pseudoperiodic function and explore its properties. Furthermore, the existence and uniqueness of the weighted pseudoperiodic solution to fractional integro-differential equations and nonautonomous differential equations are investigated. Some interesting examples are presented to illustrate the main findings.

An atomically quantized hierarchy of shear transformation zones in a metallic glass

Quasistatic measurements of room-temperature anelastic relaxation were used to characterize the properties of shear transformation zones (STZs) in amorphous Al86.8Ni3.7Y9.5 in the dilute limit. Using a combination of nanoindenter cantilever bending and mandrel bend relaxation techniques, anelastic relaxation was measured over times ranging from 1 s to 3 × 107 s. Direct spectrum analysis yields relaxation-time spectra, which display seven distinct peaks. The results were analyzed using a linear dashpot-and-spring model, combined with transition-state theory, to yield several STZ properties. These reveal a quantized hierarchy of STZs that differ from each other by one atomic volume. Potential STZs occupy a large volume fraction of the solid. They access their ergodic space, with the ratio of forward-to backward jump rates ranging from 1.03 to 4.3 for the range of stress values used.

Non-equilibrium Thermodynamic Separation Theory of Nonlinear Chromatography I. Local Lagrangian Approach

The matrix forms of local Lagrangian approach (LLA) are developed based on Lagrangian description for single-component in nonlinear, non-ideal chromatography. A local thermodynamic path (LTP) is designed based on essential physical principles, such as the Lagrangian description, the local equilibrium assumption and the thermodynamic state functions. With the LTP, the iteration equations of fully thermodynamic states on time sequence in the matrix forms are obtained with the Markov character. And the convergence, compatibility and stability of the LLA based on the LTP are discussed with some theoretical analysis and numerical experiments, and the stability condition of the LLA is given. The algorithm of the LLA in the vector form is shown as the computer program to simulate the elution profiles affected by a few of factors, space-distribution, axial diffusions, injection samples, etc. According to the LLA, the corresponding relationships are established between the trajectories of discrete time state and discrete time control vectors in the ergodic space. And a compendium algorithm of multistage decision problems concerning the optimal control of nonlinear, non-idea chromatography is given with Bellman's dynamic programming to find the optimal trajectories of state vector and control vector. The matrix forms of the LLA remove the gap between preparative chromatography theories and Markov decision processes or optimal control approaches based on discrete time states.

A chaos search immune algorithm with its application to neuro-fuzzy controller design

In this paper, a chaos search immune algorithm (CSIA) is proposed by integrating the chaos optimization algorithm and the clonal selection algorithm. First, optimization variables are expressed by chaotic variables through solution space transformation. Then, taking advantages of the ergodic and stochastic properties of chaotic variables, a chaos search is performed in the neighbourhoods of high affinity antibodies to exploit local solution space, and the motion of the chaotic variables in their ergodic space is used to explore the whole solution space. Furthermore, a generalized radial basis function neuro-fuzzy controller (GRBFNFC) is constructed and designed automatically by the proposed CSIA. Application of the CSIA-designed GRBFNFC to real-time control of an inverted pendulum system is discussed. Experimental results demonstrate that the designed GRBFNFC has very satisfactory performance. � 2005 Elsevier Ltd. All rights reserved.

Dielectric nonlinearity of relaxor ferroelectric ceramicsat low alternating-current-drive amplitudes

The dielectric nonlinear response of(PbMg1/3Nb2/3O3)0.9(PbTiO3)0.1(0.9PMN-0.1PT) relaxor ceramics was investigated under different ac-drive voltages. It was observed that: (i) the dielectricpermittivity is independent of the ac-field amplitude at hightemperatures; (ii) with increasing ac-drive amplitude, thepermittivity maximum increases, and the temperature of the maximumshifts to lower temperature; (iii) the nonlinear effect is weakenedwhen the measurement frequency increases. The influence ofincreasing ac-drive amplitude was found to be similar to that ofdecreasing frequency. It is believed that the dielectricnonlinearities of relaxors at low drive amplitudes can be explainedby the phase transition theory of the ergodic space undergoing a succession of shrinkages. A Monte Carlo simulation was performed on the flips ofmicro-polarizations at low ac-drive amplitudes to verify the theory.

Phase transition of ergodic space shrinking in succession and relaxor ferroelectrics

A special phase transition, the phase transition of ergodic space shrinking in succession (abbreviated as ES 3 phase transition), is proposed according to the freezing process of the ordered microregion dipoles in relaxors. Its features are discussed. The special characteristics of relaxors are explained from the viewpoint of the ES 3 phase transition.

Dielectric mechanism in complex perovskite

After analyzing the characteristics of interactions between the polar microregions in the complex perovskite relaxor ferroelectrics, a physical model is built up in which the effective interaction energy parameters have a Gaussian distribution. The dielectric mechanism of the materials is studied by using the Monte Carlo computer simulation. The concepts of "frozen dipole" and "slow dipole" are proposed according to the distribution of relaxation time of dipole flipping, the diffuse phase transition (DPT), frequency dispersion and polarization behavior, and other relaxor properties are explained systematically. It is proposed that the relaxor characteristics are caused by the ergodic space shrinking in succession, which arises from the inhomogeneous crystal structure of this kind of material.