the asymptotic behavior of the stochastic ginzburg landau equation with additive noise abstract we show that the stochastic ginzburg landau equation with additive noise can be solved pathwise and the unique solution generates a random dynamic system then we prove that the system possesses a compact random attractor in l 2 d when the spatial dimension of d is one and two respectively

the picard contraction mapping method for the parameter inversion of reaction diffusion systems an inverse problem is to determine unknown causes based on observation of their effects such problems often arise in scientific research and engineering practice in this paper we present a parameter inversion method for a class of reaction diffusion systems by the use of the picard contraction mapping we firstly establish an approximate differential equation model which corresponds to the given reaction diffusion system and discuss the contractivity of the picard mapping associated with the approximate model then we provide an algorithm for recovering parameters of the reaction diffusion system based on the collage theorem numerical tests show that the proposed method is efficient

generalized self similarity abstract we prove the existence of l pfunctions satisfying a kind of self similarity condition this is achieved by solving a functional equation by means of the construction of a contractive operator on an appropriate functional space the solution a fixed point of the operator can be obtained by an iterative process making this model very suitable to use in applications such as fractal image and signal compression on the other hand this generalized self similarity equation includes matrix refinement equations of the typef x ckf ax k which are central in the construction of wavelets and multiwavelets the results of this paper will therefore yield conditions for the existence of l p refinable functions in a very general setting

upper semi continuity and regularity of random attractors on p times integrable spaces and applications abstract a sufficient condition for a family of random attractors to be upper semi continuous and regular is obtained when the state space is a p times integrable space with p 2 it is shown that both upper semi continuity and regularity under the topology of p norms do not rely on continuity convergence and absorption of the associated random dynamical systems in l p since the corresponding l 2 properties can offer all as an application of the abstract result it is shown that the family of random attractors for the stochastic reaction diffusion equations on a unbounded domain is upper semi continuous and regular at any point under the topology of p norms

asymptotically autonomous multivalued cauchy problems with spatially variable exponents abstract we study the asymptotic behavior of a non autonomous multivalued cauchy problem of the form u t t div d t u t p x 2 u t u t p x 2 u t f t u t 0 on a bounded smooth domain in r n n 1 with a homogeneous neumann boundary condition where the exponent p c satisfies p min p x 2 we prove the existence of a pullback attractor and study the asymptotic upper semicontinuity of the elements of the pullback attractor a a t t r as t for the non autonomous evolution inclusion in a hilbert space h under the assumptions amongst others that f is a measurable multifunction and d l t is bounded above and below and is monotonically nonincreasing in time the global existence of solutions is obtained through results of papageorgiou and papalini

stochastic perturbation of dynamical systems the weak convergence of measures in this paper we consider the asymptotic behaviour of randomly perturbed discrete dynamical systems we treat this problem by examining the evolution of the corresponding sequences of distributions we show that an average contractive property is sufficient to ensure the weak convergence of the sequence of distributions to a unique stationary measure

fractal image denoising over the past decade there has been significant interest in fractal coding for the purpose of image compression however applications of fractal based coding to other aspects of image processing have received little attention we propose a fractal based method to enhance and restore a noisy image if the noisy image is simply fractally coded a significant amount of the noise is suppressed however one can go a step further and estimate the fractal code of the original noise free image from that of the noisy image based upon a knowledge or estimate of the variance of the noise assumed to be zero mean stationary and gaussian the resulting fractal code yields a significantly enhanced and restored representation of the original noisy image the enhancement is consistent with the human visual system where extra smoothing is performed in flat and low activity regions and a lower degree of smoothing is performed near high frequency components e g edges of the image we find that for significant noise variance spl sigma spl ges 20 the fractal based scheme yields results that are generally better than those obtained by the lee filter which uses a localized first order filtering process similar to fractal schemes we also show that the lee filter and the fractal method are closely related

approximations of fractal sets abstract five approximations of fractal sets a are compared the first three were described by hutchinson barnsley demko and williams respectively two new other approximations are still efficient in all circumstances for any algorithm error estimates in the approximations are found numerical comparisons are also done through classical examples of attractors in the plane

random fixed point equations and inverse problems using collage method for contraction mappings abstract in this paper we are interested in the direct and inverse problems for the following class of random fixed point equations t w x w x w where t x x is a given operator is a probability space and x is a polish metric space the inverse problem is solved by recourse to the collage theorem for contractive maps we then consider two applications i random integral equations and ii random iterated function systems with greyscale maps rifsm for which noise is added to the classical ifsm

existence of a global attractor for the parabolic equation with nonlinear laplacian principal part in an unbounded domain abstract in this paper we study the long time behavior of solutions for the parabolic equation with non linear laplacian principal part in r n we prove the existence of a global l 2 r n l r n attractor when n p and the existence of a global l 2 r n l n p n p r n attractor when n p

global attractors for one dimensional p laplacian equation abstract we study the existence of the l 2 r w 1 p r global attractor for an evolution p laplacian equation under additional restrictions on force and source terms we improve the result of a kh khanmamedov existence of a global attractor for the parabolic equation with nonlinear laplacian principal part in an unbounded domain j math anal appl 316 2006 601 615 and correct the result of m yang c sun c zhong existence of a global attractor for a p laplacian equation in r n nonlinear anal 66 2007 1 13 in the one dimensional case

novel techniques in parameter estimation for fractional dynamical models arising from biological systems in recent years both parameter estimation and fractional calculus have attracted a considerable interest parameter estimation of the fractional dynamical models is a new topic in this paper we consider novel techniques for parameter estimation of fractional nonlinear dynamical models in systems biology first a computationally effective fractional predictor corrector method is proposed for simulating fractional complex dynamical models second we convert the parameter estimation of fractional complex dynamical models into a minimization problem of the unknown parameters third a modified hybrid simplex search mhss and a particle swarm optimization pso is proposed finally these techniques are applied to a dynamical model of competence induction in a cell with measurement error and noisy data some numerical results are given that demonstrate the effectiveness of the theoretical analysis

attractors of asymptotically autonomous quasi linear parabolic equation with spatially variable exponents abstract some abstract results on the convergence of nonautonomous pullback attractors in asymptotically autonomous problems are established and then applied to quasi linear parabolic equations with spatially variable exponents in which the parabolic operator is time dependent in particular it is shown that the component subsets of the pullback attractor converge in the hausdorff semi distance to the global autonomous attractor

existence and upper semicontinuity of global attractors for p x laplacian systems abstract this work concerns the study of asymptotic behavior of coupled systems of p x laplacian differential inclusions we obtain that the generalized semiflow generated by the coupled system has a global attractor for each pair of positive diffusion coefficients we also prove that the attractors are upper semicontinuous on positive finite diffusion parameters in particular to get the results we prove that the p x laplacian operator is subdifferential of a convex proper and lower semicontinuous non negative map

existence and upper semicontinuity of pullback attractors for non autonomous p laplacian parabolic problems abstract we study the asymptotic behavior of parabolic p laplacian problems of the form u t t div d t u t p 2 u t u t p 2 u t b t u t in a bounded smooth domain in r n where n 1 p 2 d l t with 0 d t x m a e in t 0 and for each 0 we have d s x d t x c s t for all x s t t for some positive constants and c moreover d d 1 in l t as 1 we prove that for each 0 the evolution process of this problem has a pullback attractor and we show that the family of pullback attractors behaves upper semicontinuously at 1

an automatization of barnsley s algorithm for the inverse problem of iterated function systems we present an automatization of barnsley s manual algorithm for the solution of the inverse problem of iterated function systems ifss the problem is to retrieve the number of mappings and the parameters of an ifs from a digital binary image approximating the attractor induced by the ifs m f barnsley et al described a way to solve manually the inverse problem by identifying the fragments of which the collage is composed and then computing the parameters of the mappings barnsley et al proc nat acad sci usa vol 83 p 1975 7 1986 barnsley fractals everywhere academic 1988 barnsley and hurd l fractal image compression a k peters 1992 the automatic algorithm searches through a finite set of points in the parameter space determining a set of affine mappings the algorithm uses the collage theorem and the hausdorff metric the inverse problem of ifss is related to the image coding of binary images if the number of mappings and the parameters of an ifs with not too many mappings could be obtained from a binary image then this would give an efficient representation of the image it is shown that the inverse problem solved by the automatic algorithm has a solution and some experiments show that the automatic algorithm is able to retrieve an ifs including the number of mappings from a digital binary image approximating the attractor induced by the ifs

pullback attractors of non autonomous stochastic degenerate parabolic equations on unbounded domains abstract this paper is concerned with pullback attractors of the stochastic p laplace equation defined on the entire space r n we first establish the asymptotic compactness of the equation in l 2 r n and then prove the existence and uniqueness of non autonomous random attractors this attractor is pathwise periodic if the non autonomous deterministic forcing is time periodic the difficulty of non compactness of sobolev embeddings on r n is overcome by the uniform smallness of solutions outside a bounded domain

longtime robustness of pullback random attractors for stochastic magneto hydrodynamics equations abstract we establish backward compactness of a pullback random attractor pra from a backward limit set compact cocycle we prove that backward compactness of a pra is a necessary and sufficient condition such that its time fibers are upper semi continuous at negative infinity and obtain the minimal compact limit set also we obtain the maximal limit set in the lower semi continuity sense these results greatly reinforce the upper semi continuity result at the finite time given by cui et al 2018 in the application part we show existence backward compactness and longtime robustness of a pra for the non autonomous stochastic magneto hydrodynamics equations perturbed by a backward tempered force moreover we prove that the pra does not change on different attracted universes this fact allows us to establish the finite time robustness of the pra if the force is tempered only

asymptotic behavior of stochastic g navier stokes equations on a sequence of expanding domains the limiting dynamics of stochastic 2d nonautonomous g navier stokes equations defined on a sequence of expanding domains are investigated where the limiting domain is unbounded by generalizing the energy equation method we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact which lead to both existence and upper semicontinuity of the null expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain the limiting dynamics of stochastic 2d nonautonomous g navier stokes equations defined on a sequence of expanding domains are investigated where the limiting domain is unbounded by generalizing the energy equation method we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact which lead to both existence and upper semicontinuity of the null expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain

galerkin method for constrained variational equations and a collage based approach to related inverse problems for a general variational equation with constraints we present both the stability of the corresponding galerkin method and a collage theorem for a related inverse problem in addition we demonstrate the applicability of the results by considering forward and inverse boundary value problems of some linear impulsive differential equations

dynamics of the non autonomous stochastic p laplace equation driven by multiplicative noise we investigate the asymptotic behavior of solutions of the p laplace equation driven simultaneously by non autonomous deterministic forcing and multiplicative white noise on r n we show the tails of solutions of the equation are uniformly small outside a bounded domain which is used to derive asymptotic compactness of solution operators in l 2 r n by overcoming the non compactness of sobolev embeddings on unbounded domains we then prove existence and uniqueness of random attractors and further establish upper semicontinuity of attractors as the intensity of noise approaches zero

regularity and structure of pullback attractors for reaction diffusion type systems without uniqueness abstract in this paper we study the pullback attractor for a general reaction diffusion system for which the uniqueness of solutions is not assumed we first establish some general results for a multi valued dynamical system to have a bi spatial pullback attractor and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories as an application a general reaction diffusion system is proved to have an invariant h v pullback attractor a a r this attractor is composed of all the backwards compact complete trajectories of the system pullback attracts bounded subsets of h in the topology of v and moreover s a s is precompact in v r a non autonomous fitz hugh nagumo equation is studied as a specific example of the reaction diffusion system

fractal block coding a functional approach for image and signal processing in this paper we show how the generalized self similarity model introduced by cabrelli et al in 1 can be used for the coding and reconstruction of digital signals and images we also prove how the block coding techniques introduced by jacquin 2 for measures flt naturally into this functional model allowing us to take advantage of local redundancy of the images as well as to fully automate the encoding and decoding process c

using the collage method to solve odes inverse problems with multiple data sets abstract we consider inverse problems for ordinary differential equations odes where multiple noisy observational data sets are available the inverse problem for a single data set asks us to find an ode or system thereof that admits the interpolated data set as an approximate solution such an inverse problem can be cast within the framework of fixed point equations so that we seek to approximate the data values by a fixed point of an operator related to the odes the collage coding method replaces the minimization of this approximation error by the minimization of the so called collage distance the purpose of this paper is to formulate and explore various methods for dealing with the collection of collage distances corresponding to multiple noisy data sets we present three schemes averaging the parameter estimates obtained by minimizing each of the separate collage distances and two approaches arising from goal programming numerous examples are presented

box counting dimensions and upper semicontinuities of bi spatial attractors for stochastic degenerate parabolic equations on an unbounded domain abstract the paper contributes box counting dimensions and upper semicontinuities of random bi spatial attractors for stochastic degenerate parabolic equations on the whole euclid space under some weak assumptions for the force and the nonlinearity we first prove the existence of a unique l 2 d 0 1 2 l q random attractor for any q 2 p 2 i 2 where p 1 is the order of the nonlinearity and i is a given integer such that the force is i 1 times integrable by using truncation and splitting techniques and also induction methods we then prove that a priori estimate is uniform with respect to the density of noise which leads to the upper semicontinuity result of the obtained attractors as the density tends to a constant including zero under the topology of the terminative space furthermore we give a new framework to discuss the bound of l q box counting dimensions of random attractors for spde on an unbounded domain

fractal feature analysis and classification in medical imaging following b b mandelbrot s fractal theory 1982 it was found that the fractal dimension could be obtained in medical images by the concept of fractional brownian motion an estimation concept for determination of the fractal dimension based upon the concept of fractional brownian motion is discussed two applications are found 1 classification 2 edge enhancement and detection for the purpose of classification a normalized fractional brownian motion feature vector is defined from this estimation concept it represented the normalized average absolute intensity difference of pixel pairs on a surface of different scales the feature vector uses relatively few data items to represent the statistical characteristics of the medial image surface and is invariant to linear intensity transformation for edge enhancement and detection application a transformed image is obtained by calculating the fractal dimension of each pixel over the whole medical image the fractal dimension value of each pixel is obtained by calculating the fractal dimension of 7 7 pixel block centered on this pixel

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

fourier frequencies in affine iterated function systems abstract we examine two questions regarding fourier frequencies for a class of iterated function systems ifs these are iteration limits arising from a fixed finite families of affine and contractive mappings in r d and the ifs refers to such a finite system of transformations or functions the iteration limits are pairs x where x is a compact subset of r d the support of and the measure is a probability measure determined uniquely by the initial ifs mappings and a certain strong invariance axiom the two questions we study are 1 existence of an orthogonal fourier basis in the hilbert space l 2 x and 2 explicit constructions of fourier bases from the given data defining the ifs

random attractors of reaction diffusion equations with multiplicative noise in lp in this paper we study the random dynamical system rds generated by the reaction diffusion equation with multiplicative noise and prove the existence of a random attractor for such rds in l p d for any p 2

the collage coding method and its application to an inverse problem for the lorenz system abstract a lorenz system inverse problem featuring noisy partial data sets is considered via the method of collage coding collage coding is a rigorously established approach to such problems with its roots in fractal imaging and its theoretical basis being banach s fixed point theorem and some recent related results for comparison purposes results for this inverse problem obtained by using other methods are mentioned a framework for treating general inverse problems involving partial data sets in a similar way is also outlined

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

random dynamics of stochastic p laplacian equations on rn with an unbounded additive noise abstract this article is concerned with the random dynamics of solutions of the non autonomous parabolic p laplacian equations on r n driven by an unbounded additive noise where the nonlinearity has a p q growth exponents including the general polynomial type functions firstly by using an inductive method we establish the higher order integrability of difference of solutions near the initial time for arbitrary space dimension secondly based on this very estimate the asymptotical compactness of solutions in l p r n l q r n for any p q 2 is proved by a new technique instead of the usual truncation estimate method and then we obtain the existence of pullback attractor in l p r n l q r n for p laplacian equations with an unbounded additive noise thirdly we prove that the obtained l 2 pullback attractor is attracting and translation bounded in l r n topology for any 2 and arbitrary space dimension n n 1

a generalized fractal transform for measure valued images abstract fractal image coding generally seeks to express an image as a union of spatially contracted and greyscale modified copies of subsets of itself generally images are represented as functions u x and the fractal coding method is conducted in the framework of 2 or in this paper we formulate a method of fractal image coding on measure valued images at each point x x is a probability measure over the range of allowed greyscale values we construct a complete metric space y d y of measure valued images x m r g where x is the base or pixel space and m r g is the set of probability measures supported on the greyscale range r g a generalized fractal transform m is formulated over the metric space y d y under suitable conditions m y y is contractive implying the existence of a unique fixed point measure valued function m

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

using the collage method to solve one dimensional two point boundary value problems at steady state abstract we consider an inverse problem for two point boundary value problems bvps d u f x q x u x u a u a u b u b where the goal is to recover an approximation of the physically meaningful functions of x and parameters in the differential operator d we rewrite the bvp inverse problem as an initial value problem ivp inverse problem and recast this inverse problem in terms of approximating a target function by the fixed point of a contractive operator on a complete metric space we formulate different collage coding frameworks to solve such problems although the approach has broad applicability we focus on two model problems approximating the variable diffusivity in a steady state heat flow equation and approximating the flexural rigidity in a steady state vibrational beam we present two examples each treated with three different frameworks

asymptotically autonomous dynamics for parabolic equations abstract given an asymptotically autonomous system it is shown that the components of the pullback attractor converge to the global attractor if and only if the pullback attractor is forward compact this result reduces two uniformness conditions in the theoretical result given by kloeden and simsen j math anal appl 2015 2017 some constructions from the forward limit set of a pullback attractor are established some applications are given for two classes of non autonomous quasi linear parabolic equations one has a parabolic operator with spatially variable exponents and another has a weakly dissipative nonlinearity for two equations it is shown that the pullback attractor is forward compact and upper semi continuous to the global attractor moreover there is an inclusion between the global attractor and the forward limit set of pullback attractors

l2 lp random attractors for stochastic reaction diffusion equation on unbounded domains abstract in this paper the existence of l 2 l p random attractor is established for a stochastic reaction diffusion equation on the whole space r n this random attractor is a compact and invariant tempered set which attracts every tempered random subset of l 2 in the topology of l p the nonlinearity f is supposed to satisfy some growth of arbitrary order p 1 where p 2 the l 2 l p asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions

pathwise upper semi continuity of random pullback attractors along the time axis abstract the pullback attractor of a non autonomous random dynamical system is a time indexed family of random sets typically having the form a t t r with each a t a random set this paper is concerned with the nature of such time dependence it is shown that the upper semi continuity of the mapping t a t for each fixed has an equivalence relationship with the uniform compactness of the local union s i a s where i r is compact applied to a semi linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that in order to prove the above locally uniform compactness and upper semi continuity no additional conditions are required in which sense the two properties appear to be general properties satisfied by a large number of real models

power domains and iterated function systems we introduce the notion of weakly hyperbolic iterated function system ifs on a compact metric space which generalises that of hyperbolic ifs based on a domain theoretic model which uses the plotkin power domain and the probabilistic power domain respectively we prove the existence and uniqueness of the attractor of a weakly hyperbolic ifs and the invariant measure of a weakly hyperbolic ifs with probabilities extending the classic results of hutchinson for hyperbolic ifss in this more general setting we also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure extending the corresponding algorithms for hyperbolic ifss we then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent ifs and obtain an algorithm to generate the invariant distribution on the digitised screen the generalised riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution for hyperbolic recurrent ifss and lipschitz maps one can estimate the integral up to any threshold of accuracy

solution of an inverse problem for fractals and other sets abstract the problem of providing succinct approximate descriptions of given bounded subsets of rn can be solved by application of the contraction mapping principle

contractive multifunctions fixed point inclusions and iterated multifunction systems abstract we study the properties of multifunction operators that are contractive in the covitz nadler sense in this situation such operators t possess fixed points satisfying the relation x t x we introduce an iterative method involving projections that guarantees convergence from any starting point x 0 x to a point x x t the set of all fixed points of a multifunction operator t we also prove a continuity result for fixed point sets x t as well as a generalized collage theorem for contractive multifunctions these results can then be used to solve inverse problems involving contractive multifunctions two applications of contractive multifunctions are introduced i integral inclusions and ii iterated multifunction systems

inverse problems for random differential equations using the collage method for random contraction mappings in this paper we are concerned with differential equations with random coefficients which will be considered as random fixed point equations of the form t w x w x w w w here t wxx x is a random integral operator w f p is a probability space and x is a complete metric space we consider the following inverse problem for such equations given a set of realizations of the fixed point of t possibly the interpolations of different observational data sets determine the operator t or the mean value of its random components as appropriate we solve the inverse problem for this class of equations by using the collage theorem for contraction mappings

random attractors for stochastic semi linear degenerate parabolic equations with additive noise in lq this paper studies the random dynamical system generated by the stochastic semi linear degenerate parabolic equations with additive noise in a bounded domain email protected r n and proves the existence of random attractors for such random dynamical system in l q d for any q 2 the obtained results improve some recent ones for this problem