on multiple solutions of a semilinear schrodinger equation with periodic potential abstract this paper is concerned with the semilinear schrodinger equation s u v x u f x u u h 1 r n where v and f are periodic in the x variables f is a superlinear and subcritical nonlinearity and 0 lies in a spectral gap of u v it is shown that if f is odd in u then s has infinitely many large energy solutions the proof relies on a generalized variant fountain theorem for strongly indefinite functionals established in this paper

a variational analysis of the toda system on compact surfaces in this paper we consider the following toda system of equations on a compact surface we will give existence results by using variational methods in a noncoercive case a key tool in our analysis is a new moser trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u1 and u2 2012 wiley periodicals inc

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

linear systems with locally integrable trajectories abstract the locally integrable function space l 1 loc is made a module over the ring of proper rational functions using this module structure the notion of relative dimension is defined naturally for every linear subspace in l 1 loc q it is shown that the property of having finite relative dimension together with the property of having sufficiently many smooth trajectories and the evident property of differentiation invariance characterize the weak solutions sets of linear constant coefficient differential systems among all linear subspaces

differential morphology and image processing image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain we provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems this includes ideas on using partial differential or difference equations pdes to model distance propagation or nonlinear multiscale processes in images we briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics we also review some nonlinear pdes that model the evolution of multiscale morphological operators and use morphological derivatives among the new ideas presented we develop some general 2 d max min sum difference equations that model the space dynamics of 2 d morphological systems including the distance computations and some nonlinear signal transforms called slope transforms that can analyze these systems in a transform domain in ways conceptually similar to the application of fourier transforms to linear systems thus distance transforms are shown to be bandpass slope filters we view the analysis of the multiscale morphological pdes and of the eikonal pde solved via weighted distance transforms as a unified area in nonlinear image processing which we call differential morphology and briefly discuss its potential applications to image processing and computer vision

existence of solution for a class of quasilinear schrodinger equation in rn with zero mass abstract in this paper we use variational methods to establish a berestycki lions type result for two classes of quasilinear schrodinger equations in r n in the zero mass situation

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

on ground states of superlinear p laplacian equations in rn abstract we consider superlinear p laplacian equations in r n with a potential which is periodic or has a bounded potential well without assuming the ambrosetti rabinowitz type condition and the monotonicity of the function t f x t t p 1 we prove the existence of ground states even for the case p 2 our results extend the recent results of li wang and zeng ann inst h poincare anal non lineaire 23 2006 829 837

reconstructions from zero crossings in scale space in computer vision the one parameter family of images obtained from the laplacian of a gaussian filtered version of the image parameterized by the width of the gaussian has proved to be a useful data structure for the extraction of feature data in particular the zero crossings of this so called scale space data are associated with edges and have been proposed by d marr 1982 and others as the basis of a representation of the image data the question arises as to whether the representation is complete and stable the authors survey some of the studies and results related to these questions as well as several studies that attempt reconstructions based on this or related representations they formulate a novel method for reconstruction from zero crossings in scale space that is based on minimizing equation error and they present results showing that the reconstruction is possible but can be unstable they further show that the method applies when gradient data along the zero crossings are included in the representation and they demonstrate empirically that the reconstruction is then stable

on ground states for the kirchhoff type problem with a general critical nonlinearity abstract in this paper we consider the following kirchhoff type problem a b r 3 u 2 u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants and f has a critical growth the aim of this paper is to study the existence of ground state solutions for kirchhoff type equations with a general nonlinearity in the critical growth without the assumption of the monotonicity of the function t f t t 3 moreover we will show that the mountain pass value gives the least energy level and also obtain a mountain pass solution

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

on superlinear p x laplacian equations in rn abstract we consider the p x laplacian equations in r n the nonlinearity is superlinear but does not satisfy the ambrosetti rabinowitz type condition we obtain ground states of the equations improving a recent result of fan x l fan p x laplacian equations in r n with periodic data and nonperiodic perturbations j math anal appl 341 2008 103 119 we also establish a bartsch wang type compact embedding theorem for variable exponent spaces then a multiplicity result for the equations is proved for odd nonlinearity

evolution equations for continuous scale morphological filtering multiscale signal analysis has emerged as a useful framework for many computer vision and signal processing tasks morphological filters can be used to develop nonlinear multiscale operations that have certain advantages over linear multiscale approaches in that they preserve important signal features such as edges the authors discuss several nonlinear partial differential equations that model the scale evolution associated with continuous space multiscale morphological erosions dilations openings and closings these equations relate the rate of change of the multiscale signal ensemble as scale increases to a nonlinear operator acting on the space of signals the nonlinear operator is characterized by the shape and dimensionality of the structuring element used by the morphological operators generally taking the form of a nonlinear function of certain partial differential operators

an asymptotically periodic and asymptotically linear schrodinger equation with indefinite linear part abstract we consider the semilinear schrodinger equation u v x u f x u x r n u h 1 r n where v x is asymptotically periodic and sign changing f x u is asymptotically periodic in x and asymptotically linear as u under some weak assumptions on v and f we prove that the above problem has a nontrivial solution

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

asymptotically optimal algorithms for running max and min filters on random inputs given a d dimensional array of size n d and an integer p the running max or min filter is the set of maximum or minimum elements within a d dimensional sliding window of edge length p inside the array this problem is useful in many signal processing applications such as pattern analysis adaptive signal processing and morphological analysis the current best algorithm for computing the one dimensional 1 d max or min filter due to the work of h yuan and m j atallah running max min filters using 1 o 1 comparisons per sample ieee trans pattern anal mach intell vol 33 no 12 pp 2544 2548 dec 2011 uses 1 o 1 comparisons per sample in the worst case as a direct consequence the d dimensional max or min filter max and min filters respectively can be computed in d o 1 2d o 1 respectively comparisons per sample in this paper we first present an algorithm for computing d dimensional max and min filters simultaneously on i i d inputs that uses 1 5 o 1 expected comparisons per sample this is the first algorithm on i i d inputs that gets rid of the dependence on d in the dominating term with respect to n and p of the expected number of comparisons needed it is also asymptotically optimal when d is a fixed constant as n rightarrow infty and p rightarrow infty we also consider the dynamic version of the problem of d dimensional max and min filters simultaneously on i i d inputs where we want to maintain the filters after changes in the input array we design a linear sized data structure that stores precomputed information for efficient update using o p d 1 log 2 p expected comparisons per update

spatially variant morphological restoration and skeleton representation the theory of spatially variant sv mathematical morphology is used to extend and analyze two important image processing applications morphological image restoration and skeleton representation of binary images for morphological image restoration we propose the sv alternating sequential filters and sv median filters we establish the relation of sv median filters to the basic sv morphological operators i e sv erosions and sv dilations for skeleton representation we present a general framework for the sv morphological skeleton representation of binary images we study the properties of the sv morphological skeleton representation and derive conditions for its invertibility we also develop an algorithm for the implementation of the sv morphological skeleton representation of binary images the latter algorithm is based on the optimal construction of the sv structuring element mapping designed to minimize the cardinality of the sv morphological skeleton representation experimental results show the dramatic improvement in the performance of the sv morphological restoration and sv morphological skeleton representation algorithms in comparison to their translation invariant counterparts

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

superlinear schrodinger operators abstract we find nontrivial and ground state solutions for the nonlinear schrodinger equation under conditions weaker than those previously assumed

existence and multiplicity of nontrivial solutions for a class of schrodinger kirchhoff type equations abstract in the present paper the following schrodinger kirchhoff type problem p a b r n u 2 d x u v x u f x u in r n u x 0 as x is studied when n 2 3 4 and v x 1 the existence theorem of nontrivial weak solutions for problem p is obtained when n 1 2 3 and v x is more general two existence theorems of nontrivial weak solutions and a sequence of high energy weak solutions for problem p are obtained

on the state of behaviors the theme of the present paper is the study of the concept of state and the corresponding state maps in the context of willems behavioral theory we concentrate on markovian system and their representation in terms of first order difference or differential systems we follow by a full analysis of the special case of state systems the embedding of a linear system in a state system via the use of state maps arriving at state representations or equivalently to a realization theory for behaviors minimality is defined and characterized and a state space isomorphism theorem is established realization procedures based on the shift realization are developed as well as a rigorous analysis of the construction of state maps the paper owes much to rapisarda and willems p rapisarda j c willems state maps for linear systems siam j contr optim 35 1997 1053 1091

solutions of superlinear elliptic equations and their morse indices we consider here solutions of semilinear second order elliptic equations with superlinear nonlinearities and we present some relationships between their morse indices and some qualitative properties in particular we show that for uo subcritical nonlinearities bounds on solutions are equivalent to bounds on their morse indices

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

object based image analysis using multiscale connectivity this paper introduces a novel approach for image analysis based on the notion of multiscale connectivity we use the proposed approach to design several novel tools for object based image representation and analysis which exploit the connectivity structure of images in a multiscale fashion more specifically we propose a nonlinear pyramidal image representation scheme which decomposes an image at different scales by means of multiscale grain filters these filters gradually remove connected components from an image that fail to satisfy a given criterion we also use the concept of multiscale connectivity to design a hierarchical data partitioning tool we employ this tool to construct another image representation scheme based on the concept of component trees which organizes partitions of an image in a hierarchical multiscale fashion in addition we propose a geometrically oriented hierarchical clustering algorithm which generalizes the classical single linkage algorithm finally we propose two object based multiscale image summaries reminiscent of the well known morphological pattern spectrum which can be useful in image analysis and image understanding applications

nonlinear degenerate elliptic equation with hardy potential and critical parameter abstract some embedding inequalities in hardy sobolev spaces with general weight functions were proved and a positive answer to an open problem raised by brezis vazquez was given in the weighted hardy sobolev spaces the existence of nontrivial many solutions to the corresponding nonlinear degenerated elliptic equations with hardy potential and critical parameter under conditions weaker than ambrosetti rabinowitz condition was obtained

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

asymptotically linear schrodinger equation with potential vanishing at infinity we are concerned with the existence of bound states and ground states of the following nonlinear schrodinger equation

ground state solutions for generalized quasilinear schrodinger equations without ar condition abstract in this paper we study the following generalized quasilinear schrodinger equations div g 2 u u g u g u u 2 v x u h x u x r n where n 3 g r r is an even differentiable function satisfying lim t g t t 1 0 for some 1 h x t is a given function behaving like t 2 1 at infinity which does not satisfy the a r condition combining the change of variables and the monotone method developed by jeanjean in 16 we obtain the existence of positive ground state solutions for the given problem

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

existence and nonexistence of positive solutions for a class of generalized quasilinear schrodinger equations involving a kirchhoff type perturbation with critical sobolev exponent in this paper by variational methods we study the existence and nonexistence of positive solutions for a class of generalized quasilinear schrodinger equations involving a kirchhoff type perturbation with critical sobolev exponent in particular the first result is based on the ambrosetti rabinowitz condition moreover the second results are established via the poho aev identity finally the last results are not based on the ambrosetti rabinowitz condition and we use generalized conditions to prove the results

a multiscanning approach based on morphological filtering it is argued that the mathematical morphology method seems to be more reasonable and powerful in studying certain multiscaling vision problems than the approach that uses derivatives of gaussian shaped filters of different sizes to show the validity of this method the authors concentrated on an application that involves forming scale space image of a 2 d shape using morphological opening filtering a proof is given to show that morphological opening filtering has a property of not introducing additional zero crossings as one moves to a coarser scale this is a different result from the conclusion by a l yuille and t a poggio ibid vol pami 8 jan 1986 that the gaussian filter is the only filter with this property in addition opening filtering is computationaly simpler than the gaussian filter

on ground state solutions for the schrodinger poisson equations with critical growth abstract in this paper we study the existence of ground state solutions for the following schrodinger poisson equation u v x u u u q 1 u u 4 u in r 3 u 2 in r 3 where is a positive parameter under some certain assumptions on v we prove that for every 0 and q 2 5 such a class of schrodinger poisson equation with critical growth has at least a positive ground state solution via variational methods some recent results from the literature are extended

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

existence of multiple solutions of kirchhoff type equation with sign changing potential abstract in this paper we study the existence of infinitely many nontrivial solutions for a class of nonlinear schrodinger kirchhoff type equation a b r n u 2 dx u v x u f x u in r n u x 0 as x where constants a 0 b 0 and the potential v x is allowed to be sign changing under general superlinear assumption on nonlinearity f x u we establish the existence of infinitely many solutions via variational methods which unifies and improves the recent results of wu 2011 9

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

measuring the fractal dimension of signals morphological covers and iterative optimization fractals can model many classes of time series data the fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales the covering methods are a class of efficient approaches to measure the fractal dimension of an arbitrary fractal signal by creating multiscale covers around the signal s graph in this paper we develop a general method that uses multiscale morphological operations with varying structuring elements to unify and extend the theory and digital implementations of covering methods it is theoretically estab lished that for the fractal dimension computation covering one dimensional signals with planar sets is equivalent to mor phologically transforming the signal by one dimensional func tions which reduces the computational complexity from quad ratic in the signal s length to linear then a morphological covering algorithm is developed and applied to discrete time signals synthesized from weierstrass functions fractal inter polation functions and fractional brownian motion further for deterministic parametric fractals depending on a single pa rameter related to their dimension we develop an optimization method that starts from an initial estimate and iteratively con verges to the true fractal dimension by searching in the param eter space and minimizing a distance between the original sig nal and all such signals from the same class experimental results are also provided to demonstrate the good performance of the developed methods

berestycki lions conditions on ground state solutions for kirchhoff type problems with variable potentials based on the strategy introduced by chen and tang adv nonlinear anal 9 496 515 2020 and some new tricks we prove that the nonlinear problem of kirchhoff type a b r3 u 2dx u v x u f u x r3 in h1 r3 admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a simple minimax characterization of the ground state energy our results improve and extend recent theorems in several directions

nonlinear multiresolution signal decomposition schemes ii morphological wavelets interest in multiresolution techniques for signal processing and analysis is increasing steadily an important instance of such a technique is the so called pyramid decomposition scheme this paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis the proposed theory is based on the following ingredients 1 the pyramid consists of a finite or infinite number of levels such that the information content decreases toward higher levels and 2 each step toward a higher level is implemented by an information reducing analysis operator whereas each step toward a lower level is implemented by an information preserving synthesis operator one basic assumption is necessary synthesis followed by analysis yields the identity operator meaning that no information is lost by these two consecutive steps several examples of pyramid decomposition schemes are shown to be instances of the proposed theory a particular class of linear pyramids morphological skeleton decompositions the morphological haar pyramid median pyramids etc furthermore the paper makes a distinction between single scale and multiscale decomposition schemes i e schemes without or with sample reduction finally the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks

steep potential well may help kirchhoff type equations to generate multiple solutions abstract in this paper we consider a nonlinear kirchhoff type problem with steep potential well a r 3 u 2 d x b u v x u f x u p 2 u in r 3 u h 1 r 3 where a b 0 2 p 4 v c r 3 r and f l r 3 r such problem cannot be studied by applying variational methods in a standard way even by restricting its corresponding energy functional on the nehari manifold because palais smale sequences may not be bounded in this paper we introduce a novel constraint method to prove the existence of one and two positive solutions under the different assumptions on v respectively we conclude that steep potential well v may help kirchhoff type equations to generate multiple solutions which has never been involved before

the existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p laplacian type without the ambrosetti rabinowitz condition abstract in this paper we study the existence of a nontrivial solution to the following nonlinear elliptic boundary value problem of p laplacian type p p u f x u x u 0 x where p 1 r 1 r n is a bounded domain and p u d i v u p 2 u is the p laplacian of u f c 0 r 1 r 1 is p superlinear at t 0 and subcritical at t we prove that under suitable conditions for all 0 the problem p has at least one nontrivial solution without assuming the ambrosetti rabinowitz condition our main result extends a result for p for when p 2 given by miyagaki and souto 2008 in 8 to the general problem p where p 1 meanwhile our result is stronger than a similar result of li and zhou 2003 given in 15

schrodinger kirchhoff equation involving double critical nonlinearities abstract in this paper we are interested in weak solutions of the following schrodinger kirchhoff equation involving double critical nonlinearities a b r 3 u 2 u u 2 s 1 1 x s 1 u 2 s 2 1 x s 2 in r 3 0 u d 1 2 r 3 u 0 in r 3 where a b are two positive constants 2 s 2 3 s and 0 s 1 s 2 2 a natural strategy is to search the weak solutions of the equation as critical points of a suitable functional via variational methods but by dilation invariance we see that the functional doesn t satisfy p s condition we will make use of several techniques to recover compactness our results show that the higher order term dominates lower order term and there exist multi scale characteristics in this nonlinear problem

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

the digital morphological sampling theorem morphological sampling reduces processing time and cost and yet produces results sufficiently close to the result of full processing a morphological sampling theorem is described which states 1 how a digital image must be morphologically filtered before sampling in order to preserve the relevant information after sampling 2 to what precision an appropriate morphologically filtered image can be reconstructed after sampling and 3 the relationship between morphologically operating before sampling and the more computationally efficient scheme of morphologically operating on the sampled image with a sampled structuring element the digital sampling theorem is developed first for the case of binary morphology and then it is extended to gray scale morphology through the use of the umbra homomorphism theorems

solutions in spectral gaps for a nonlinear equation of schrodinger type abstract in this paper we study the existence of a nontrivial h2 r n solution for an equation of the form u x p x u x f x u x u x x r n r where p l r n is a periodic function we assume that the operator p h2 r n l2 r n l2 r n is strongly indefinite and invertible and that x r r is odd and satisfies some superlinear but subcritical growth conditions we extend the class of nonlinearities which has been studied up to now in particular under standard technical restrictions the existence of a solution is derived when lim x f x s f s 0 exists for all s r if we assume that f x s f s for all s r and a e on r n

characterization of linear differential systems in several variables abstract we show that a set of smooth trajectories is the solution set of a linear constant coefficient partial differential equation if and only if it is linear shift invariant and complete by completeness we mean exactly what willems called jet completeness in his automatica paper in 1986

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

strong consistency of the lad l sub 1 estimator of parameters of stationary autoregressive processes with zero mean strong consistency almost sure convergence to the true parameters of the lad least absolute deviations ar autoregressive parameter estimator has been proven by s gross and w l steiger 1979 under the condition that i i d noise driving a stationary autoregressive process has zero median this work extends their proof to include the case when the driving noise has zero mean thus when the noise pdf probability density function is asymmetric with distinct mean and median the lad estimator will be strongly consistent with the pdf centered with either mean or median at the origin the results of this work extend computer simulations which further indicate that under these conditions the lad estimator is ms consistent mean squared convergence to the true parameters the importance of these results in lad signal processing applications is discussed

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

characterising solution sets of lti differential equations we prove that a set of smooth trajectories is ltid i e the solution set of a constant coefficient linear differential equation if and only if it is the direct sum of a linear time invariant closed controllable part and a linear time invariant finite dimensional part this characterisation does not directly involve derivatives in its formulation it solves a problem opened by willems 1991 we also characterise morphisms between ltid sets as linear time invariant maps which do not increase support

two positive solutions for the kirchhoff type elliptic problem with critical nonlinearity in high dimension abstract we investigate the kirchhoff type elliptic problem with critical nonlinearity 1 u 2 d x u u q u 2 1 u 0 in u 0 on where n 5 r n is a bounded domain with smooth boundary 0 r 2 2 n n 2 and q 1 2 1 we prove the existence of two solutions of it via the variational method since n 5 and 0 the uniqueness assertion for the associated limiting problem may fail this causes serious difficulties in controlling concentrating palais smale sequences we overcome these by introducing new techniques for a mountain pass type solution we utilize the limit function of the fibering maps of the concentrating palais smale sequence this tool is based on our careful setting of nehari type sets on the other hand a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution this is the first work which proves the multiplicity of positive solutions of the kirchhoff type critical problem in high dimension

bifurcation in lp rn for a semilinear elliptic equation keywords lsp p bifurcation constrained minimization sobolev spaces nonlinear eigenvalue problem semilinear spectrum linearisation variational structure reference ana article 1988 001doi 10 1112 plms s3 57 3 511 record created on 2008 12 10 modified on 2016 08 08

new conditions on nonlinearity for a periodic schrodinger equation having zero as spectrum abstract we consider the semilinear schrodinger equation u v x u f x u for x r n u x 0 as x where f is a superlinear and subcritical nonlinearity we mainly study the case when both v and f are periodic in x and 0 is a boundary point of a spectral gap of v we extend a linking theorem of kryszewski and szulkin 15 and establish a new variational setting which is more suitable to the above case we obtain two theorems on the existence of ground state solutions with mild assumptions on f

partial difference operators on weighted graphs for image processing on surfaces and point clouds partial difference equations pdes and variational methods for image processing on euclidean domains spaces are very well established because they permit to solve a large range of real computer vision problems with the recent advent of many 3d sensors there is a growing interest in transposing and solving pdes on surfaces and point clouds in this paper we propose a simple method to solve such pdes using the framework of pdes on graphs this latter approach enables us to transcribe for surfaces and point clouds many models and algorithms designed for image processing to illustrate our proposal three problems are considered 1 p laplacian restoration and inpainting 2 pdes mathematical morphology and 3 active contours segmentation

nonstationary homoclinic orbits for an infinite dimensional hamiltonian system we study the following system of diffusion equations on r rn tu xu v x u g t x v tv xv v x v f t x u where z u v r rn r2 v c rn r g t x v f t x u are periodic in t x and superlinear at infinity by a variant linking theorem and concentration compactness arguments we establish the existence of nonstationary homoclinic type solutions of least energy for this systems

infinitely many solutions for schrodinger maxwell equations with indefinite sign subquadratic potentials in this paper we deal with multiplicity of solutions of a class of sublinear schrodinger maxwell equations du v x u fu f x u inr 3 d f u 2 lim x f x 0 inr 3 under more relaxed assumptions on v and f we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical theory

existence of positive ground state solutions for kirchhoff type problems abstract in this paper we study the existence of positive ground state solutions for the nonlinear kirchhoff type problem a b r 3 u 2 u v x u f u in r 3 u h 1 r 3 u 0 in r 3 where a b 0 are constants f c r r is subcritical near infinity and superlinear near zero and satisfies the berestycki lions condition by using an abstract critical point theorem established by jeanjean and a new global compactness lemma we show that the above problem has at least a positive ground state solution our result generalizes the results of li and ye 2014 concerning the nonlinearity f u u p 1 u with p 2 5

when are linear differentiation invariant spaces differential it is shown that a linear differentiation invariant subspace of a c trajectory space is differential i e can be represented as the kernel of a linear constant coefficient differential operator if and only if its mcmillan degree is finite

existence of solutions for nonperiodic superquadratic hamiltonian elliptic systems abstract this paper is concerned with the following nonperiodic hamiltonian elliptic system u v x u r v x u v in r n v v x v r u x u v in r n u x 0 and v x 0 as x where r x z is superquadratic in z as z with z u v by applying a critical point theorem for strongly indefinite functionals we prove the existence of solution for the above system

gradient watersheds in morphological scale space after introducing several results relating to the modification of the homotopy of gradient functions based on extrema in the base image and building on earlier results in morphological scale space we introduce a scale space monotonicity theorem for regions of an image defined by watersheds of a gradient function modified to retain only the local minima or maxima of its smoothed parent image we then illustrate the theorem with an example of the scale space extraction of texture features from the nuclei of cervical cells

optimal morphological pattern restoration from noisy binary images a theoretical analysis of morphological filters for the optimal restoration of noisy binary images is presented the problem is formulated in a general form and an optimal solution is obtained by using fundamental tools from mathematical morphology and decision theory consideration is given to the set difference distance function as a measure of comparison between images this function is used to introduce the mean difference function as a quantitative measure of the degree of geometrical and topological distortion introduced by morphological filtering it is proved that the class of alternating sequential filters is a set of parametric smoothing morphological filters that best preserve the crucial structure of input images in the least mean difference sense

on a superquadratic elliptic system with strongly indefinite structure abstract in this paper we consider the elliptic system u g x v in v f x u in u v 0 on where is a bounded smooth domain in r n and f and g satisfy a general superquadratic condition by using variational methods we prove the existence of infinitely many solutions our argument relies on the application of a generalized variant fountain theorem for strongly indefinite functionals previous results in the topic are improved

on ground state solutions for quasilinear elliptic equations with a general nonlinearity in the critical growth abstract in this paper we consider the following problem p u v u p 2 u f u u w 1 p r n where 1 p n and f has critical growth the purpose of this paper is to study the existence of the ground state solution for a class of quasilinear elliptic equations some recent results are extended

compactness results for schrodinger equations with asymptotically linear terms abstract we study the nonlinear problem u v x f x u x r n lim x u x 0 where the schrodinger operator v is semibounded and the nonlinearity f is linearly bounded we establish compactness of palais smale sequences and cerami sequences for the associated energy functional under general spectral theoretic assumptions applying these results we obtain existence of three nontrivial solutions if the energy functional has a mountain pass geometry

critical point theory for indefinite functionals with symmetries letxbe a hilbert space and c1 x r be strongly indefinite assume in addition that a compact lie groupgacts orthogonally onxand that is invariant in order to find critical points of we develop the limit relative category of fournieret al in the equivariant context we use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem in the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension also other conditions of the classical symmetric mountain pass theorem forg z 2 due to ambrosetti and rabinowitz can be weakened considerably for example we are able to deal with infinite dimensional fixed point spaces the proofs consist of a direct reduction to a relative borsuk ulam type theorem this provides a new proof even for the classical mountain pass theorem the abstract critical point theorems are applied to an elliptic system with dirichlet boundary conditions we only need a weak version of the usual superquadraticity condition the linking theorem can be applied to asymptotically linear hamiltonian systems which are symmetric with respect to a generalized symplectic group action

a general methodology for the determination of 2d bodies elastic deformation invariants application to the automatic identification of parasites a novel methodology is introduced here that exploits 2d images of arbitrary elastic body deformation instances so as to quantify mechanoelastic characteristics that are deformation invariant determination of such characteristics allows for developing methods offering an image of the undeformed body general assumptions about the mechanoelastic properties of the bodies are stated which lead to two different approaches for obtaining bodies deformation invariants one was developed to spot a deformed body s neutral line and its cross sections while the other solves deformation pdes by performing a set of equivalent image operations on the deformed body images both of these processes may furnish a body undeformed version from its deformed image this was confirmed by obtaining the undeformed shape of deformed parasites cells protozoa fibers and human lips in addition the method has been applied to the important problem of parasite automatic classification from their microscopic images to achieve this we first apply the previous method to straighten the highly deformed parasites and then apply a dedicated curve classification method to the straightened parasite contours it is demonstrated that essentially different deformations of the same parasite give rise to practically the same undeformed shape thus confirming the consistency of the introduced methodology finally the developed pattern recognition method classifies the unwrapped parasites into six families with an accuracy rate of 97 6 percent

a note on continuous behavior homomorphisms the purpose of this note is the characterization of the class of behavior homomorphisms that have nice polynomial representations the key to this is a judicious application of duality theory together with additional continuity requirements

strongly indefinite functionals and multiple solutions of elliptic systems we study existence and multiplicity of solutions of the elliptic system left begin array 20 l updelta u h u x u v hfill text in upomega hfill updelta v h v x u v hfill text in upomega quad u x v x 0 quad text on partial upomega hfill end array right where upomega subset mathbb r n n ge 3 is a smooth bounded domain and h in mathcal c 1 overline upomega times mathbb r 2 mathbb r we assume that the nonlinear term h x u v sim left u right p left v right q r x u v text with mathop lim limits left u v right to infty frac r x u v left u right p left v right q 0 where p in 1 2 2 2n n 2 text and q in 1 infty so some supercritical systems are included nontrivial solutions are obtained when h x u v is even in u v we show that the system possesses a sequence of solutions associated with a sequence of positive energies resp negative energies going toward infinity resp zero if p 2 resp p 2 all results are proved using variational methods some new critical point theorems for strongly indefinite functionals are proved