The Global Attractor and Its Dimension Estimation of Generalized Kolmogorov-Petrovlkii-Piskunov Equation

In this paper, the initial boundary value problem of a class of nonlinear generalized Kolmogorov-Petrovlkii-Piskunov equations is studied. The existence and uniqueness of the solution and the bounded absorption set are proved by the prior estimation and the Galerkin finite element method, thus the existence of the global attractor is proved and the upper bound estimate of the global attractor is obtained.

THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS

Letτbe a generalized Thue-Morse substitution on a two-letter alphabet{a,b}:τ(a)=ambm,τ(b)=bmam for the integer m≥2.Letξbe a sequence in{a,b}Z that is generated byτ.We study the one-dimensional Schr?dinger operator Hm,λon l2(Z)with a potential given by v(n)=λVξ(n),whereλ>0 is the coupling and Vξ(n)=1(Vξ(n)=-1)ifξ(n)=a(ξ(n)=b).LetΛ2=2,and for m>2,letΛm=m if m≡0 mod 4;letΛm=m-3 if m≡1 mod 4;letΛm=m-2if m≡2 mod 4;letΛm=m-1 if m≡3 mod 4.We show that the Hausdorff dimension of the spectrumσ(Hm,λ)satisfies that dimHσ(Hm,λ)>logΛm/(log 64m+4).It is interesting to see that dimHσσ(Hm,λ)tends to 1 as m tends to infinity.

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

THE HAUSDORFF DIMENSION AND MEASURE IN GENERALIZED GEOMETRIC GRAPH DIRECTED CONSTRUCTION

Abthors introduce the notation of generalized geometric constructions in Rm generated by a directed graph G and by a sequence of similarity ratios which are labelled with the edges of this graph. In this paper, it is obtained the Hausdorff dimension and measure of this construction object for some cases.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

Gelfand-Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules for Classical Lie Algebras

Let g be a classical complex simple Lie algebra and q be a parabolic subalgebra.Let M be a generalized Verma module induced from a one dimensional representation of q.Such M is called a scalar generalized Verma module.In this paper,we will determine the reducibility of scalar generalized Verma modules associated to maximal parabolic subalgebras by computing explicitly the Gelfand-Kirillov dimension of the corresponding highest weight modules.

DIMENSIONS OF MEASURE ON GENERAL SIERPINSKI CARPET

Let S = Pi(i=1)(infinity){0, 1, ..., r - 1} and (R) over bar the general Sierpinski carpet, Let mu be the induced probability measure on (R) over bar of <(mu)over tilde> on S by phi, where phi is the natural surjection from S onto (R) over bar and <(mu)over tilde> is the infinite product probability measure corresponding to probability vector (b(0), ..., b(r-1)) with b(i) = a(i)(logn) (m-1)/m(alpha). Authors show that dim(H) mu = (C) under bar(L)(mu) = (C) over bar(L)(mu) = (C) under bar(mu) = (C) over bar C(mu) = alpha.

Application of General Fractal Dimension to Coupling Fault Diagnosis

This paper presents the coucept of general and sensitive dimension, and also proposes the calculation formula of the general dimension least squares method. By calculating and analyzing the power spectrum and general dimension from the fault sample, the relationship is achieved between sample status and the general dimension from vibration signals of the equipment so as to provide reference to fault diagnosis. Furthermore, a correlation function of general dimension is proposed, and calculations are carried out for a monitor signal and samples signal. The diagnosis method based on fractal theory is effective through the concrete examples of the steam electric generating set fault diagnosis, and the correlation coefficient of general dimension between a monitor signal and samples signal can improve the accuracy for fault diagnosis.

Correlation Dimension in Fault Diagnosis of 600 MW Steam Turbine Generator

GP algorithm of correlation dimension computation is ameliorated which overcomes the shortage of traditional one. Improved process of GP algorithm takes the influence of temporal correlative pairs of points on correlation dimension into account and promotes the computational efficiency prominently. Iterative SVD method is applied to remove the influence of noise on the result of correlation dimension. The faults of steam flow turbulence and oil film disturbance which occur in 600 MW Steam Turbine Generator are analyzed and whose correlation dimensions are computed. More distinct quantitative index than FFT is gained to distinguish two faults and it’s of little importance to apply correlation dimension to study the influence of various factors on steam flow turbulence fault for nonexistence of convergent floor in correlation integral curve, which presents a new way to learn the operational function of large capacity steam turbine generator and carry out comprehensive condition monitoring.

THE BOUNDS OF THE GENERAL M AND J SETS AND THE ESTIMATIONS FOR THE HAUSDORFF'S DIMENSION OF THE GENERAL J SET

The bounds of the general M and J sets were analytically offered. Some of, the bounds were optimal in certain meaning. It not only solved the primary problem of the construction of fractal sets by escape time algorithm, and followed from the conclusion, but also offered two estimations of some special Julia set's Hausdorff's dimension by approximate linearization method.

CONNECTION BETWEEN THE ORDER OF FRACTIONAL CALCULUS AND FRACTIONAL DIMENSIONS OF A TYPE OF FRACTAL FUNCTIONS

The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.

THE P-COTORSION DIMENSIONS OF MODULES AND RINGS

Let R be a ring. We define a dimension, called P-cotorsion dimension, for modules and rings. The aim of this article is to investigate P-cotorsion dimensions of modules and rings and the relations between P-cotorsion dimension and other homological dimensions. This dimension has nice properties when the ring in consideration is generalized morphic.

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy pro jection technique

This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method （FEM） by recursive application of the one-dimensional （1D） element energy projection （EEP） technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element （FE） model is reached. This conceptual dimension-by- dimension （D-by-D） discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional （2D） and three-dimensional （3D） problems can be obtained over the domain. Numerical examples of 3D Poisson＇s equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.

The Integrable in Liouville Sense of a Finite-dimensional Hamilton System

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RESEARCH FROM SINGLE FRACTAL DIMENSION TO GEN-ERALIZED FRACTAL DIMENSION OF SPEECH

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THE PACKING MEASURE OF A CLASS OF GENERALIZED SIERPINSKI CARPET

For 1/4< a <(?)/4, let S1(x) =ax, S2(x)=1-a+ax, x∈[0,1]. Ca is the attractor of the iteratedfunction system {S1,S2}, then the packing measure of Ca×Ca isPs(a)(Ca×Ca) = 4·2s(a)(1-a)s(a),where s(a) = -loga4.

A NOTE ON GENERALIZED MORAN SET

Let E be a generallied Moran set. The authors determine the dim E and get the sufficient and necessary condition for E being a s-set.

Generalized ExcellentExtension and Hom ologicalDim ension

We introduce generalized excellent extension of ring R in [4]. If S is a generalized excellent extension of R, we prove that (1) wgldim S≤ wgldim R and glodim S≤ gldim R; (2) the relations between rings R and rings S.

The Use of Yanai’s Generalized Coefficient of Determination to Reduce the Number of Variables in DEA Models

This paper proposes a new method to reduce the dimensionality of input and output spaces in DEA models. The method is based on Yanai’s Generalized Coefficient of Determination and on the concept of pseudo-rank of a matrix. In addition, the paper suggests a rule to determine the cardinality of the subset of selected variables in a way to gain the maximal discretionary power and to suffer a minimal informational loss.

On Rings with Finite Global Gorenstein Dimensions

As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou's equality of global Gorenstein dimension is given.