nonlinear extraction of independent components of natural images using radial gaussianization we consider the problem of efficiently encoding a signal by transforming it to a new representation whose components are statistically independent a widely studied linear solution known as independent component analysis ica exists for the case when the signal is generated as a linear transformation of independent nongaussian sources here we examine a complementary case in which the source is nongaussian and elliptically symmetric in this case no invertible linear transform suffices to decompose the signal into independent components but we show that a simple nonlinear transformation which we call radial gaussianization rg is able to remove all dependencies we then examine this methodology in the context of natural image statistics we first show that distributions of spatially proximal bandpass filter responses are better described as elliptical than as linearly transformed independent sources consistent with this we demonstrate that the reduction in dependency achieved by applying rg to either nearby pairs or blocks of bandpass filter responses is significantly greater than that achieved by ica finally we show that the rg transformation may be closely approximated by divisive normalization which has been used to model the nonlinear response properties of visual neurons

existence and roughness of exponential dichotomies of linear dynamic equations on time scales in this paper we define the exponential dichotomy of linear dynamic equations on time scales then we present perturbation theorems on the roughness of exponential dichotomy and develop several explicit sufficient criteria for linear dynamic equations to have an exponential dichotomy as applications of the criteria of exponential dichotomy we derive some new sufficient conditions for the existence of periodic solutions of semi linear dynamic equations and nonlinear dynamic equations on time scales

a combinatorial primal dual approach to semidefinite programs semidefinite programs sdps have been used in many recent approximation algorithms we develop a general primal dual approach to solve sdps using a generalization of the well known multiplicative weights update rule to symmetric matrices for a number of problems such as s parsest c ut and b alanced s eparator in undirected and directed weighted graphs m in u n c ut and m in 2cnf d eletion this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods the design of our primal dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones our ideas have proved useful in quantum computing especially the recent result of jain et al 2011 that qip e pspace

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

a local clustering algorithm for massive graphs and its application to nearly linear time graph partitioning we study the design of local algorithms for massive graphs a local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph we present a local clustering algorithm our algorithm finds a good cluster a subset of vertices whose internal connections are significantly richer than its external connections near a given vertex the running time of our algorithm when it finds a nonempty local cluster is nearly linear in the size of the cluster it outputs the running time of our algorithm also depends polylogarithmically on the size of the graph and polynomially on the conductance of the cluster it produces our clustering algorithm could be a useful primitive for handling massive graphs such as social networks and web graphs as an application of this clustering algorithm we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance our algorithm takes time nearly linear in the number edges of the graph

admissibility and roughness of nonuniform exponential dichotomies for difference equations abstract in this paper we discuss on relationships between nonuniform exponential dichotomies of nonautonomous difference equations and admissibility for classes of weighted bounded solutions and show the roughness of nonuniform exponential dichotomies in the finite dimensional case we give sufficient conditions for existence of nonuniform exponential dichotomies on z 0 or z 0 separately on z 0 and z 0 simultaneously and on the whole z as an application we use this result to discuss the roughness of nonuniform exponential dichotomies for difference equations on z 0 z 0 and the whole z separately and obtain better estimates for both the bound and the nonuniformity degree

the maximum concurrent flow problem the maximum concurrent flow problem mcfp is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently the ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow the mcfp objective is to maximize the throughput subject to the capacity constraints we develop a fully polynomial time approximation scheme for the mcfp for the case of arbitrary demands and uniform capacity computational results are presented it is shown that the problem of associating costs distances to the edges so as to maximize the minimum cost of routing the concurrent flow is the dual of the mcfp a path cut type duality theorem to expose the combinatorial structure of the mcfp is also derived our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm applications include packet switched networks 1 4 8 and cluster analysis 16

iterative gaussianization from ica to random rotations most signal processing problems involve the challenging task of multidimensional probability density function pdf estimation in this paper we propose a solution to this problem by using a family of rotation based iterative gaussianization rbig transforms the general framework consists of the sequential application of a univariate marginal gaussianization transform followed by an orthonormal transform the proposed procedure looks for differentiable transforms to a known pdf so that the unknown pdf can be estimated at any point of the original domain in particular we aim at a zero mean unit covariance gaussian for convenience rbig is formally similar to classical iterative projection pursuit algorithms however we show that unlike in pp methods the particular class of rotations used has no special qualitative relevance in this context since looking for interestingness is not a critical issue for pdf estimation the key difference is that our approach focuses on the univariate part marginal gaussianization of the problem rather than on the multivariate part rotation this difference implies that one may select the most convenient rotation suited to each practical application the differentiability invertibility and convergence of rbig are theoretically and experimentally analyzed relation to other methods such as radial gaussianization one class support vector domain description and deep neural networks is also pointed out the practical performance of rbig is successfully illustrated in a number of multidimensional problems such as image synthesis classification denoising and multi information estimation

an o log k approximate min cut max flow theorem and approximation algorithm it is shown that the minimum cut ratio is within a factor of o log k of the maximum concurrent flow for k commodity flow instances with arbitrary capacities and demands this improves upon the previously best known bound of o log2 k and is existentially tight up to a constant factor an algorithm for finding a cut with ratio within a factor of o log k of the maximum concurrent flow and thus of the optimal min cut ratio is presented

transversality of homoclinic orbits and exponential dichotomies for parabolic equations abstract in this paper we discuss the existence of exponential dichotomies on r of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations and by making use of the results on exponential dichotomies of this paper we investigate the transversality of homoclinic orbits for parabolic equations

nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces abstract in this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew product flows in infinite dimensional spaces we treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families we show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary our results generalize the previous studies on this topic and provide as particular cases many interesting situations among which we mention the detection of the exponential trichotomy of general non autonomous systems

on the asymptotic behavior of discrete dynamical systems an ergodic theory approach abstract the aim of this paper is to present a new and general method for the study of stability expansiveness and dichotomy of dynamical systems based on arguments proceeding from ergodic theory and on admissibility properties we consider a discrete variational system and we associate to it three different families of input output systems corresponding to the investigations for stability expansiveness and dichotomy respectively we introduce some input output conditions relative to ergodic measures and we show that certain nonuniform admissibility properties imply the existence of the uniform asymptotic behaviors first we obtain characterizations for uniform exponential stability as an application we prove that in order to preserve the uniform exponential stability subject to perturbations it is sufficient for the perturbed system to be close enough to the initial system on a measurable set next we deduce criteria for uniform exponential expansiveness based on some exact admissibilities relative to ergodic measures finally we present an interesting application to the study of the uniform exponential dichotomy starting from a nonuniform admissibility property and using stability criteria

strong nonuniform behaviour a datko type characterization abstract we obtain a datko type characterization of several classes of a strong nonuniform exponential behaviour we consider separately the case of exponential contractions expansions and dichotomies in addition we consider both discrete and continuous time as a nontrivial application of our work we study the robustness property of strong nonuniform exponential contractions expansions and dichotomies

regularization operators for natural images based on nonlinear perception models image restoration requires some a priori knowledge of the solution some of the conventional regularization techniques are based on the estimation of the power spectrum density simple statistical models for spectral estimation just take into account second order relations between the pixels of the image however natural images exhibit additional features such as particular relationships between local fourier or wavelet transform coefficients biological visual systems have evolved to capture these relations we propose the use of this biological behavior to build regularization operators as an alternative to simple statistical models the results suggest that if the penalty operator takes these additional features in natural images into account it will be more robust and the choice of the regularization parameter is less critical

psychophysically tuned divisive normalization approximately factorizes the pdf of natural images the conventional approach in computational neuroscience in favor of the efficient coding hypothesis goes from image statistics to perception it has been argued that the behavior of the early stages of biological visual processing e g spatial frequency analyzers and their nonlinearities may be obtained from image samples and the efficient coding hypothesis using no psychophysical or physiological information in this work we address the same issue in the opposite direction from perception to image statistics we show that psychophysically fitted image representation in v1 has appealing statistical properties for example approximate pdf factorization and substantial mutual information reduction even though no statistical information is used to fit the v1 model these results are complementary evidence in favor of the efficient coding hypothesis

discretized characterizations of exponential dichotomy of linear skew product semiflows over semiflows using the method of discretization we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear skew product semiflows we then apply the obtained results to consider the roughness of exponential dichotomy of linear skew product semiflows

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

a zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems the aim of this paper is to present new discrete time characterizations for uniform exponential trichotomy of dynamical systems we obtain necessary and sufficient conditions for the existence of uniform exponential trichotomy of discrete dynamical systems using a set of computational conditions of zabczyk type the main results are applied to deduce new criteria for the detection of the uniform exponential trichotomy of dynamical systems modeled by skew product flows

schaffer spaces and uniform exponential stability of linear skew product semiflows we study the exponential stability of linear skew product semiflows on locally compact metric space with banach fibers our main tool is the admissibility of a pair of the so called schaffer spaces this characterization is a very general one it includes as particular cases many interesting situations among them we can mention some results due to clark datko latushkin van minh montgomery smith randolph rabiger schnaubelt

dependency reduction with divisive normalization justification and effectiveness efficient coding transforms that reduce or remove statistical dependencies in natural sensory signals are important for both biology and engineering in recent years divisive normalization dn has been advocated as a simple and effective nonlinear efficient coding transform in this work we first elaborate on the theoretical justification for dn as an efficient coding transform specifically we use the multivariate t model to represent several important statistical properties of natural sensory signals and show that dn approximates the optimal transforms that eliminate statistical dependencies in the multivariate t model second we show that several forms of dn used in the literature are equivalent in their effects as efficient coding transforms third we provide a quantitative evaluation of the overall dependency reduction performance of dn for both the multivariate t models and natural sensory signals finally we find that statistical dependencies in the multivariate t model and natural sensory signals are increased by the dn transform with low input dimensions this implies that for dn to be an effective efficient coding transform it has to pool over a sufficiently large number of inputs

admissibility and exponential trichotomy of dynamical systems described by skew product flows abstract the aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems the investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input output system we prove that the uniform admissibility of the pair c b r x l 1 r x for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system if p 1 and the pair c b r x l 1 r x is uniformly p admissible then we obtain the uniform exponential trichotomy next we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input output system has unique bounded solutions in certain subspaces finally we obtain that the uniform p admissibility of the pair c b r x l 1 r x is a necessary and sufficient condition for uniform exponential trichotomy

skew product semi flows and non autonomous control systems in this paper we give a necessary and sufficient conditions for exact and approximate controllability of a wide class of linear infinite dimensional non autonomous control systems this is done by employing skew product semi flow technique finally we apply these results to prove the controllability of a broad class of non autonomous reaction diffusion equations in hilbert spaces

o sqrt logn approximation to sparsest cut in tilde o n 2 time this paper shows how to compute o sqrt log n approximations to the sparsest cut and balanced separator problems in tilde o n 2 time thus improving upon the recent algorithm of arora rao and vazirani proceedings of the 36th annual acm symposium on theory of computing 2004 pp 222 231 their algorithm uses semidefinite programming and requires tilde o n 9 5 time our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs the existence of expander flows was also established by arora rao and vazirani proceedings of the 36th annual acm symposium on theory of computing 2004 pp 222 231

faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts this paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities these algorithms are much faster than algorithms discovered previously besides being an important problem in its own right the uniform capacity concurrent flow problem has many interesting applications leighton and rao used uniform capacity concurrent flow to find an approximately sparsest cut in a graph and thereby approximately solve a wide variety of graph problems including minimum feedback arc set minimum cut linear arrangement and minimum area layout however their method appeared to be impractical as it required solving a large linear program this paper shows that their method might be practical by giving an o m 2 log m expected time randomized algorithm for their concurrent flow problem on an m edge graph raghavan and thompson used uniform capacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration an o k 3 2 m n log n expected time randomized algorithm and an o k min n k m n log n log k deterministic algorithm is given for this problem when the channel width is omega log n where k denotes the number of wires to be routed in an n node m edge network

new conditions for non uniform behaviour of linear cocycles over flows abstract we give a characterization of tempered exponential dichotomies for linear cocycles over flows in terms of the spectral properties of certain linear operators we consider noninvertible linear cocycles acting on infinite dimensional spaces and our approach avoids the use of lyapunov norms finally we apply obtained results to give new conditions for uniform exponential stability of linear cocycles

new criteria for exponential expansiveness of variational difference equations abstract the aim of this paper is to obtain general input output conditions for uniform exponential expansiveness of variational difference equations in terms of the complete admissibility of pairs of sequence spaces we introduce a large class q n of banach sequence spaces and we deduce the connections between the complete admissibility of the pair b v n x u n x with u v q n and the uniform exponential expansiveness of a system of variational difference equations we apply our results at the study of the uniform exponential expansiveness of linear skew product flows

robustness of nonuniform μ ν dichotomies in banach spaces abstract this paper focuses on the linear perturbation of dynamical systems in banach spaces which corresponds to a nonautonomous dynamical system admitting a nonuniform dichotomy we establish the robustness or roughness of nonuniform dichotomies in the sense that the linear dynamical system persists under a sufficiently small linear perturbation

spectral theory for general nonautonomous random dispersal evolution operators abstract we investigate the spectral theory of the following general nonautonomous evolution equation t u t x a u t x h t x u t x x d where d is a bounded subset of r n which can be a smooth domain or a discrete set a is a general linear dispersal operator for example a laplacian operator an integral operator with positive kernel or a cooperative discrete operator and h t x is a smooth function on r d we first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal lyapunov exponent of a time dependent dispersal equation is always greater than or equal to that of the time averaged one several results about the principal eigenvalue of time periodic parabolic equations are extended to general time periodic dispersal ones finally the investigation is generalized to random time dependent dispersal equations

upgma and the normalized equidistant minimum evolution problem upgma unweighted pair group method with arithmetic mean is a widely used clustering method here we show that upgma is a greedy heuristic for the normalized equidistant minimum evolution neme problem that is finding a rooted tree that minimizes the minimum evolution score relative to the dissimilarity matrix among all rooted trees with the same leaf set in which all leaves have the same distance to the root we prove that the neme problem is np hard in addition we present some heuristic and approximation algorithms for solving the neme problem including a polynomial time algorithm that yields a binary rooted tree whose neme score is within o log2n of the optimum

evolution semigroups translation algebras and exponential dichotomy of cocycles abstract we study the exponential dichotomy of an exponentially bounded strongly continuous cocycle over a continuous flow on a locally compact metric space acting on a banach space x our main tool is the associated evolution semigroup on c 0 x we prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup to show the latter equivalence we establish the spectral mapping annular hull theorem for the evolution semigroup in addition dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 z x here we develop banach algebra techniques and study weighted translation algebras that contain the evolution operators these results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space also exponential dichotomy follows from pointwise discrete dichotomies with uniform constants finally we extend to our situation the classical perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous mild solutions to the inhomogeneous equation

exponential forward splitting for noninvertible difference equations abstract while the basic theory of exponential dichotomies deals with differential and difference equations with uniquely determined forward and backward solutions nowadays applications require to have a corresponding theory for equations whose backward solutions are not guaranteed to exist or to be unique in this paper we study the problem of how to generalize the notion of exponential dichotomy to difference equations whose solutions are only assumed to exist in forward time it turns out that for this kind of equation a straightforward generalization of the notion of exponential dichotomy does not provide satisfactory results but that on the other hand with an additional assumption that desired goal can be reached

uniform and strict persistence in monotone skew product semiflows with applications to non autonomous nicholson systems abstract we determine sufficient conditions for uniform and strict persistence in the case of skew product semiflows generated by solutions of non autonomous families of cooperative systems of odes or delay fdes in terms of the principal spectrums of some associated linear skew product semiflows which admit a continuous separation our conditions are also necessary in the linear case we apply our results to a noncooperative almost periodic nicholson system with a patch structure whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution

roughness of tempered exponential dichotomies for infinite dimensional random difference equations abstract in this paper we study the roughness of tempered exponential dichotomies for linear random dynamical systems in banach spaces such a dichotomy has a tempered bound and describes nonuniform hyperbolicity we prove the roughness without assuming their invertibility and the integrability condition of the multiplicative ergodic theorem we give an explicit bound for the linear perturbation such that the dichotomy is persistent we also obtain explicit forms for the exponent and the bound of tempered exponential dichotomy of the perturbed random system in terms of the original ones and the perturbations

uniformly rd piecewise almost periodic functions with applications to the analysis of impulsive δ dynamic system on time scales we introduce some new concepts of uniformly rd piecewise almost periodic functions on time scales and obtain some properties some new estimation inequalities and completely new theorems are established on time scales for impulsive dynamic systems several applications are given to illustrate our feasible results in the present work we introduce two equivalent concepts of uniformly rd piecewise almost periodic functions on time scales and their equivalence is proved based on this some basic properties of them are obtained then some new criteria of exponential dichotomy are established for homogeneous dynamic system on time scales also some completely new theorems are established on time scales for impulsive almost periodic dynamic systems such as favard s theorem and exponential dichotomy theorem as applications we provide a method to obtain an almost periodic solution for a given nonhomogeneous impulsive dynamic system furthermore we introduce an impulsive non autonomous nicholson s blowflies system model with patch structure and multiple nonlinear harvesting terms for which the existence and exponential stability of almost periodic solutions are studied which shows that our results can be applied feasibly and effectively

existence and robustness of exponential dichotomy of linear skew product semiflows over semiflows in this paper we investigate the exponential dichotomy of linear skew product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on banach bundles we characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy

a version of a theorem of r datko for stability in average in this note we obtain a version of the well known theorem of r datko for the notion of the exponential stability in average we consider both cocycles over flows as well as cocycles over maps

nonuniform behavior and robustness we establish the robustness of linear cocycles in banach spaces admitting a nonuniform exponential dichotomy we first obtain robustness results for positive and negative time by establishing exponential behavior along certain subspaces and showing that the associated sequences of projections have bounded exponential growth we then establish a robustness result in z by constructing explicitly appropriate projections on the stable and unstable subspaces we emphasize that in general these projections may be different from those obtained separately from the robustness for positive and negative time we also consider the case of strong nonuniform exponential dichotomies

criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations abstract we prove that the admissibility of any pair of vector valued schaffer 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 schaffer 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 schaffer 1966 11 n van minh f rabiger and r schnaubelt 1998 26

image compression via joint statistical characterization in the wavelet domain we develop a probability model for natural images based on empirical observation of their statistics in the wavelet transform domain pairs of wavelet coefficients corresponding to basis functions at adjacent spatial locations orientations and scales are found to be non gaussian in both their marginal and joint statistical properties specifically their marginals are heavy tailed and although they are typically decorrelated their magnitudes are highly correlated we propose a markov model that explains these dependencies using a linear predictor for magnitude coupled with both multiplicative and additive uncertainties and show that it accounts for the statistics of a wide variety of images including photographic images graphical images and medical images in order to directly demonstrate the power of the model we construct an image coder called epwic embedded predictive wavelet image coder in which subband coefficients are encoded one bitplane at a time using a nonadaptive arithmetic encoder that utilizes conditional probabilities calculated from the model bitplanes are ordered using a greedy algorithm that considers the mse reduction per encoded bit the decoder uses the statistical model to predict coefficient values based on the bits it has received despite the simplicity of the model the rate distortion performance of the coder is roughly comparable to the best image coders in the literature

minimal sets in monotone and concave skew product semiflows i a general theory abstract the long term dynamics of a general monotone and concave skew product semiflow is analyzed paying special attention to the region delimited from below by the graph of a semicontinuous subequilibrium or by a minimal set admitting a flow extension different possibilities arise depending on the existence and number or absence of minimal sets strongly above the initially fixed one as well as on the coexistence or not of bounded and unbounded semiorbits on the region previous results are unified and extended and significative differences with the sublinear case are established some scenarios which are impossible in the autonomous or periodic cases may occur in this setting

unbounded perturbation of the exponential dichotomy for evolution equations abstract in this paper we prove that the exponential dichotomy for evolution equations in banach spaces is not destroyed if we perturb the equation by small unbounded linear operator this is done by employing a skew product semiflow technique and a perturbation principle from linear operator theory finally we apply these results to partial parabolic equations and functional differential equations

nonlinear image representation for efficient perceptual coding image compression systems commonly operate by transforming the input signal into a new representation whose elements are independently quantized the success of such a system depends on two properties of the representation first the coding rate is minimized only if the elements of the representation are statistically independent second the perceived coding distortion is minimized only if the errors in a reconstructed image arising from quantization of the different elements of the representation are perceptually independent we argue that linear transforms cannot achieve either of these goals and propose instead an adaptive nonlinear image representation in which each coefficient of a linear transform is divided by a weighted sum of coefficient amplitudes in a generalized neighborhood we then show that the divisive operation greatly reduces both the statistical and the perceptual redundancy amongst representation elements we develop an efficient method of inverting this transformation and we demonstrate through simulations that the dual reduction in dependency can greatly improve the visual quality of compressed images

equivalences between nonuniform exponential dichotomy and admissibility abstract relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems it was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy in this paper we find an appropriate admissible pair of classes of lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half lines r separately on both half lines r simultaneously and on the whole line r additionally the maximal admissibility is proved in the case on both half lines r simultaneously

perturbation theory for the approximation of stability spectra by qr methods for sequences of linear operators on a hilbert space abstract in this paper we develop a perturbation analysis for stability spectra lyapunov exponents and sacker sell spectrum for products of operators on a hilbert space both real and complex based upon the discrete qr technique error bounds are obtained in both the integrally separated and non integrally separated cases that correspond to distinct and multiple eigenvalues respectively for a single linear operator we illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation the time varying analogue of gaps between eigenvalues determines the sensitivity of the stability spectra to perturbation

robustness of discrete dynamics via lyapunov sequences for a nonautonomous dynamics with discrete time defined by a sequence of matrices we give a complete characterization of nonuniform exponential contractions and nonuniform exponential dichotomies in terms of lyapunov sequences we note that these include as very special cases uniform exponential contractions and uniform exponential dichotomies due to the central role played by these properties in a substantial part of the theory of dynamical systems in particular in connection with the study of stable and unstable invariant manifolds it is important to have available optimal characterizations that are more amenable to check whether a given dynamics has such a property we also obtain inverse theorems that give explicitly lyapunov sequences for a given contraction or dichotomy as a nontrivial application we establish the robustness under sufficiently small linear perturbations both of nonuniform exponential contractions and nonuniform exponential dichotomies we emphasize that when compared to former work our proof of the robustness property is much simpler

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

classical bounded and almost periodic solutions to quasilinear first order hyperbolic systems in a strip abstract we consider boundary value problems for quasilinear first order one dimensional hyperbolic systems in a strip the boundary conditions are supposed to be of a smoothing type in the sense that the l 2 generalized solutions to the initial boundary value problems become eventually c 2 smooth for any initial l 2 data we investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on r we prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system assuming that the coefficients of the hyperbolic system are almost periodic we prove that the bounded solution is almost periodic also

sparse code shrinkage denoising of nongaussian data by maximum likelihood estimation sparse coding is a method for finding a representation of data in which each of the components of the representation is only rarely significantly active such a representation is closely related to redundancy reduction and independent component analysis and has some neurophysiological plausibility in this article we show how sparse coding can be used for denoising using maximum likelihood estimation of nongaussian variables corrupted by gaussian noise we show how to apply a soft thresholding shrinkage operator on the components of sparse coding so as to reduce noise our method is closely related to the method of wavelet shrinkage but it has the important benefit over wavelet methods that the representation is determined solely by the statistical properties of the data the wavelet representation on the other hand relies heavily on certain mathematical properties like self similarity that may be only weakly related to the properties of natural data

approximating lyapunov exponents and sacker sell spectrum for retarded functional differential equations we consider lyapunov exponents and sacker sell spectrum for linear nonautonomous retarded functional differential equations posed on an appropriate hilbert space a numerical method is proposed to approximate such quantities based on the reduction to finite dimension of the evolution family associated to the system to which a classic discrete qr method is then applied the discretization of the evolution family is accomplished by a combination of collocation and generalized fourier projection a rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra the efficacy of the results is illustrated with some numerical examples

necessary and sufficient criteria for the existence of exponential dichotomy on time scales necessary and sufficient criteria are established for the existence of exponential dichotomies for linear dynamic equations on time scales and are applied to derive some perturbation theorems on the roughness of exponential dichotomy

global attractivity in concave or sublinear monotone infinite delay differential equations abstract we study the dynamical behavior of the trajectories defined by a recurrent family of monotone functional differential equations with infinite delay and concave or sublinear nonlinearities we analyze different sceneries which require the existence of a lower solution and of a bounded trajectory ordered in an appropriate way for which we prove the existence of a globally asymptotically stable minimal set given by a 1 cover of the base flow we apply these results to the description of the long term dynamics of a nonautonomous model representing a stage structured population growth without irreducibility assumptions on the coefficient matrices

existence and robustness of nonuniform h k μ ν dichotomies for nonautonomous impulsive differential equations abstract for the linear nonautonomous impulsive differential equation we propose a new notion called the nonuniform h k dichotomy which extends the notion of dichotomy of uniform and nonuniform types and is related to the theory of nonuniform hyperbolicity we establish a sufficient criterion for the existence of the nonuniform h k dichotomy in terms of appropriate lyapunov exponents for the linear perturbation we develop the robustness or roughness of nonuniform h k dichotomies in the sense that a nonuniform h k dichotomy persists under a sufficiently small linear perturbation

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

smooth robustness of parameterized perturbations of exponential dichotomies we establish the robustness of nonuniform exponential dichotomies in banach spaces under sufficiently small c1 parameterized perturbations moreover we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class c1 on the parameter thus yielding an optimal smoothness

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

randomized approximation schemes for cuts and flows in capacitated graphs we describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs we give a near linear time randomized combinatorial construction that transforms any graph on n vertices into an o n log n edge graph on the same vertices whose cuts have approximately the same value as the original graph s in this new graph for example we can run the tilde o m 3 2 time maximum flow algorithm of goldberg and rao to find an s t minimum cut in tilde o n 3 2 time this corresponds to a 1 epsilon times minimum s t cut in the original graph a related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in tilde o m sqrt n time our algorithm can also be used to improve the running time of sparsest cut approximation algorithms from tilde o mn to tilde o n 2 and to accelerate several other recent cut and flow algorithms our algorithms are based on a general theorem analyzing the concent

an ergodic theorem in the qualitative behavior of non linear semiflows abstract take to be a continuous semiflow on the locally compact metric space and let a be a family of possibly unbounded densely defined closed operators on the banach space x we show that the solutions of the variational equation u a t u t r have a uniform exponential decay if and only if there exist p 1 and a sequence of positive real numbers n n n with n satisfying lim n n n 0 n x p d 0 uniformly w r t for all x x where t t is a strongly continuous cocycle solving in the classical sense the above variational equation thus we generalize some known results obtained by datko and pazy

exponential dichotomy and lp lq admissibility on the half line abstract we give necessary and sufficient conditions for uniform exponential dichotomy of discrete evolution families in terms of the admissibility of the pair l p n x l 0 q n x we prove that the admissibility of the pair l p n x l 0 q n x is a sufficient condition for uniform exponential dichotomy of a discrete evolution family this condition becomes necessary for discrete evolution families with uniform exponential growth if and only if p q as consequences we obtain necessary and sufficient conditions for uniform exponential dichotomy of evolution families

lyapunov functions for trichotomies with growth rates abstract we consider linear equations x a t x that may exhibit stable unstable and central behaviors in different directions with respect to arbitrary asymptotic rates e c t determined by a function t for example the usual exponential behavior with t t is included as a very special case and when t log t we obtain a polynomial behavior we emphasize that we also consider the general case of nonuniform exponential behavior which corresponds to the existence of what we call a nonuniform exponential trichotomy this is known to occur in a large class of nonautonomous linear equations our main objective is to give a complete characterization in terms of strict lyapunov functions of the linear equations admitting a nonuniform exponential trichotomy this includes criteria for the existence of a nonuniform exponential trichotomy as well as inverse theorems providing explicit strict lyapunov functions for each given exponential trichotomy in the particular case of quadratic lyapunov functions we show that the existence of strict lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the lyapunov functions as an application of the characterization of nonuniform exponential trichotomies in terms of strict lyapunov functions we establish the robustness of nonuniform exponential trichotomies under sufficiently small linear perturbations we emphasize that in comparison with former works our proof of the robustness is much simpler even when t t

robust nonuniform dichotomies and parameter dependence abstract we establish the robustness of linear cocycles with an exponential dichotomy under sufficiently small lipschitz perturbations in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations we consider cocycles in banach spaces as well as the general case of nonuniform exponential dichotomies and also the general case of an exponential behavior e c n given by an arbitrary sequence n including the usual exponential behavior n n as a very special case moreover we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter and in fact that they are locally lipschitz on finite dimensional parameters

robustness via lyapunov functions for nonautonomous linear equations x a t x we give a complete characterization of nonuniform exponential dichotomies in terms of strict quadratic lyapunov functions nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies in particular we construct explicitly strict lyapunov functions for each exponential dichotomy as a nontrivial application we establish in a simple and direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations this represents a considerable simplification of former work

on the scaling limits at zeros of solutions of parabolic equations abstract we are concerned with the local scaling limits near zero points of solutions of general parabolic equations we show that at each zero point of finite vanishing order the solution is asymptotic to a homogeneous polynomial solution of the osculating equation with constant coefficients this provides useful information about the local structure of zero sets of solutions in particular we give upper bounds for hausdorff dimensions of zero sets of weak solutions our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems

lp r x lq r x admissibility and exponential dichotomies of cocycles we analyze the existence of eventually no past exponential and ordinary dichotomies of an exponentially bounded strongly continuous cocycle t t r over a continuous semiflow our main tool is the lp r x lq r x admissibility condition i e there exist p q 1 p q 1 such that for each input f lp r x and there exists x x such that the output u x f r x u t x f t x 0t s t s f s ds belongs to lq r x we prove that the above admissibility condition implies that the output is bounded above by the input but nonuniformly with respect to requiring that the boundedness to be uniform with respect to we prove that the above admissibility condition assures the existence of a no past exponential dichotomy for t t r variants for ordinary dichotomy and also complete characterizations for the exponential dichotomy of cocycles are obtained it is worth to note that we involve a concept of a no past exponential dichotomy for cocycles weaker than the well known concept defined by sacker and sell 1994 33 our definition of exponential dichotomy follows partially the definition given in chow and leiva 1996 2 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 perron 1930 23 daleckij and krein 1974 6 massera and schaffer 1966 14 nguyen van minh rabiger and r schnaubelt 1998 18

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

almost optimal local graph clustering using evolving sets spectral partitioning is a simple nearly linear time algorithm to find sparse cuts and the cheeger inequalities provide a worst case guarantee for the quality of the approximation found by the algorithm a local graph partitioning algorithm finds a set of vertices with small conductance i e a sparse cut by adaptively exploring part of a large graph g starting from a specified vertex for the algorithm to be local its complexity must be bounded in terms of the size of the set that it outputs with at most a weak dependence on the number n of vertices in g previous local partitioning algorithms find sparse cuts using random walks and personalized pagerank spielman and teng 2013 andersen et al 2006 in this article we introduce a simple randomized local partitioning algorithm that finds a sparse cut by simulating the volume biased evolving set process which is a markov chain on sets of vertices we prove that for any 0 and any set of vertices a that has conductance at most for at least half of the starting vertices in a our algorithm will output with constant probability a set of conductance o we prove that for a given run of the algorithm the expected ratio between its computational complexity and the volume of the set that it outputs is vol a 1 2polylog n where vol a v ad v is the volume of the set a this gives an algorithm with the same guarantee up to a constant factor as the cheeger s inequality that runs in time slightly superlinear in the size of the output this is the first sublinear in the size of the input time algorithm with almost the same guarantee as the cheeger s inequality in comparison the best previous local partitioning algorithm by andersen et al 2006 has a worse approximation guarantee of o log n and a larger ratio of 1 polylog n between the complexity and output volume as a by product of our results we prove a bicriteria approximation algorithm for the expansion profile of any graph for 0 0 finds a set s of volume vol s o k1 and expansion s o k as a new technical tool we show that for any set s of vertices of a graph a lazy t step random walk started from a randomly chosen vertex of s will remain entirely inside s with probability at least 1 s 2 t this itself provides a new lower bound to the uniform mixing time of any finite state reversible markov chain

globally positive solutions of linear parabolic pdes of second order with robin boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with robin boundary conditions are unique up to multiplication by a positive constant

exponential dichotomy of difference equations and applications to evolution equations on the half line abstract for a sequence of bounded linear operator a n n 0 on a banach space x we investigate the characterization of exponential dichotomy of the difference equations v n 1 a n v n we characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations v n 1 a n v n f n in l space then we apply the results to study the exponential dichotomy of evolution families generated by evolution equations

novel perceptually uniform chromatic space chromatically perceptive observers are endowed with a sense of similarity between colors for example two shades of green that are only slightly discriminable are perceived as similar whereas oth

exponential trichotomy and p admissibility for evolution families on the real line the aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line we prove that if p 1 and the pair c b r x c c r x is uniformly p admissible for an evolution family open image in new window u t s t s then open image in new window is uniformly exponentially trichotomic after that we analyze when the uniform p admissibility of the pair c b r x c c r x becomes a necessary condition for uniform exponential trichotomy as applications of these results we study the uniform exponential dichotomy of evolution families we obtain that in certain conditions the admissibility of the pair c b r x l p r x for an evolution family open image in new window u t s t s is equivalent with its uniform exponential dichotomy

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

input output admissibility and exponential trichotomy of difference equations abstract we propose a new method for the study of the asymptotic behavior of difference equations in infinite dimensional spaces providing characterizations for the property of uniform exponential trichotomy we deduce the structure of the stable unstable and bounded subspace and prove the uniqueness of the projection families we introduce a new admissibility concept with respect to a discrete input output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy throughout the paper there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations

robustness of general dichotomies abstract for a linear equation v a t v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates e c t for some function t this includes as a special case the usual exponential behavior when t t we also consider the general case of nonuniform exponential dichotomies we establish the robustness of the exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy

exponential dichotomy and dichotomy radius for difference equations abstract the aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces we classify the classes of input and output spaces respectively and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems by an example we show that the obtained results are the most general in this topic as an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input output operators acting on sequence spaces which are invariant under translations

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

nonuniform mean square exponential dichotomies and mean square exponential stability abstract in this paper the existence conditions of nonuniform mean square exponential dichotomy nms ed for a linear stochastic differential equation sde are established the difference of the conditions for the existence of a nonuniform dichotomy between an sde and an ordinary differential equation ode is that the first one needs an additional assumption nonuniform lyapunov matrix to guarantee that the linear sde can be transformed into a decoupled one while the second does not therefore the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity this paper is also concerned with the mean square exponential stability of nonlinear perturbation of a linear sde under the condition of nonuniform mean square exponential contraction nms ec for this purpose the concept of second moment regularity coefficient is introduced this concept is essential in determining the stability of the perturbed equation and hence we deduce the lower and upper bounds of this coefficient our results imply that the lower and upper bounds of the second moment regularity coefficient can be expressed solely by the drift term of the linear sde

admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line abstract we give new criteria for nonuniform dichotomy of nonautonomous systems on the whole line in terms of admissibility relative to an integral equation in our approach the input space i r x is an intersection of spaces that can be successively minimized and the output space c r x can be one of some well known spaces of continuous functions using computational arguments we show that the admissibility of c r x i r x leads to a nonuniform exponential dichotomy we expose a complete analysis of the connections between admissibility and nonuniform dichotomy on the whole line and we also discuss several interesting consequences moreover we obtain the explicit expression of the growth rates for dichotomy in terms of the initial exponential growth and the norm of the input output operator finally we present a direct application of the main result in the case of evolution families which admit uniform exponential growth

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time