multiple positive solutions for a quasilinear elliptic equation involving singular potential and critical sobolev exponent in this paper we are concerned with a class of quasilinear elliptic equation in which the potential is singular at the origin and the nonlinearity is p sublinear near the origin by the ekeland variational principle and mountain pass theorem we prove that there exist at least two positive solutions under suitable assumptions on the potential and the nonlinearity

blind constant modulus equalization via convex optimization in this paper we formulate the problem of blind equalization of constant modulus cm signals as a convex optimization problem the convex formulation is obtained by performing an algebraic transformation on the direct formulation of the cm equalization problem using this transformation the original nonconvex cm equalization formulation is turned into a convex semidefinite program sdp that can be efficiently solved using interior point methods our sdp formulation is applicable to baud spaced equalization as well as fractionally spaced equalization performance analysis shows that the expected distance between the equalizer obtained by the sdp approach and the optimal equalizer in the noise free case converges to zero exponentially as the signal to noise ratio snr increases in addition simulations suggest that our method performs better than standard methods while requiring significantly fewer data samples

a class of p q laplacian type equation with concave convex nonlinearities in bounded domain abstract in this paper our main purpose is to establish the existence of multiple solutions of a class of p q laplacian equation involving concave convex nonlinearities p u q u v x u r 2 u u p 2 u f x u x u 0 x where is a bounded domain in r n 0 1 r q p n and p n p n p is the critical sobolev exponent s u div u s 2 u is the s laplacian of u we prove that for any 0 0 is a constant there is a 0 such that for every 0 the above problem possesses infinitely many weak solutions we also obtain some results for the case 1 q p r p the existence results of solutions are obtained by variational methods

deterministic cca based algorithms for blind equalization of fir mimo channels in this paper a new deterministic technique for blind equalization of multiple input multiple output mimo channels is presented the proposed method relies on the channel order diversity of the finite impulse response mimo channel and it does not make any assumption about the spectra or the finite alphabet property of the sources assuming that the lengths of the single input multiple output channels are known or estimated a priori it is able to extract the sources up to a scale and rotation matrix ambiguity which only affects to those sources associated to single input multiple output simo channels with equal lengths unlike previously proposed techniques the method described in this paper obtains the equalizers and the best linear combination of their outputs in a single step and thus optimally the reformulation of the proposed method as a set of nested canonical correlation analysis problems is exploited to obtain efficient batch and adaptive equalization algorithms finally the performance of the proposed algorithms is evaluated by means of some simulation examples

a novel multiset integrated canonical correlation analysis framework and its application in feature fusion multiset canonical correlation analysis mcca is difficult to effectively express the integrated correlation among multiple feature vectors in feature fusion thus this paper firstly presents a novel multiset integrated canonical correlation analysis micca framework the micca establishes a discriminant correlation criterion function of multi group variables based on generalized correlation coefficient the criterion function can clearly depict the integrated correlation among multiple feature vectors then the paper presents a multiple feature fusion theory and algorithm using the micca method the detailed process of the algorithm is as follows firstly extract multiple feature vectors from the same patterns by using different feature extraction methods then extract multiset integrated canonical correlation features using micca finally form effective discriminant feature vectors through two given feature fusion strategies for pattern classification the multi group feature fusion method based on micca not only achieves the aim of feature fusion but also removes the redundancy between features the experiment results on cenparmi handwritten arabic numerals and uci multiple features database show that the micca method has better recognition rates and robustness than the fusion methods based on canonical correlation analysis cca and mcca

the brezis nirenberg problem for the laplacian with a singular drift in rn and sn abstract we consider the brezis nirenberg problem for the laplacian with a singular drift for a geodesic ball in both r n and s n 3 n 5 the singular drift we consider derives from a potential which is symmetric around the center of the geodesic ball here the potential is given by a parameter say times the logarithm of the distance to the center of the ball in both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions the parameter space is characterized by the geodesic radius of the ball and the coupling constant of the linear term of the brezis nirenberg problem

existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents abstract in this paper we consider the existence of a non trivial weak solution to a quasilinear elliptic equation with singular weights and multiple critical exponents in the whole space firstly we get the existence of a local palais smale sequence by verifying the geometric conditions of the mountain pass lemma secondly we study the concentration properties of the palais smale sequence of a zero weak limit thirdly we deduce by contradiction to eliminate the possibility of a zero weak limit case lastly applying a monotonic inequality we shall prove that the nontrivial weak limit of the palais smale sequence is indeed a weak solution

discriminant learning through multiple principal angles for visual recognition canonical correlation has been prevalent for multiset based pairwise subspace analysis as an extension discriminant canonical correlations dccs have been developed for classification purpose by learning a global subspace based on fisher discriminant modeling of pairwise subspaces however the discriminative power of dccs is not optimal as it only measures the local canonical correlations within subspace pairs which lacks the global measurement among all the subspaces in this paper we propose a multiset discriminant canonical correlation method i e multiple principal angle mpa it jointly considers both local and global canonical correlations by iteratively learning multiple subspaces one for each set as well as a global discriminative subspace on which the angle among multiple subspaces of the same class is minimized while that of different classes is maximized the proposed computational solution is guaranteed to be convergent with much faster converging speed than dcc extensive experiments on pattern recognition applications demonstrate the superior performance of mpa compared to existing subspace learning methods

on an elliptic problem with critical exponent and hardy potential in this paper we study the following elliptic problem with critical exponent and a hardy potential u x 2u u u 2 2u u h01 where is a smooth open bounded domain in rn n 3 which contains the origin and 2 is the critical sobolev exponent we show that if n 5 and 0 n 22 2 n 2n 2 this problem has a ground state solution for each fixed 0 moreover we give energy estimates from below and bounds on the number of nodal domains for these ground state solutions if n 7 and 0 n 22 2 4 this problem has infinitely many sign changing solutions for each fixed 0

on the quasilinear elliptic problems with critical sobolev hardy exponents and hardy terms abstract let r n be a smooth bounded domain such that 0 n 3 in this paper we study the critical quasilinear elliptic problems p u u p 2 u x p u p t 2 x t u u q 2 x s u u w 0 1 p with dirichlet boundary condition where p u div u p 2 u 1 p n 0 n p p p 0 0 s t p p q p s p n s n p p t p n t n p p s and p t are the critical sobolev hardy exponents via variational methods we deal with the conditions that ensure the existence of positive solutions for the equation the results depend crucially on the parameters p q s and

extremal functions and best constants to an inequality involving hardy potential and critical sobolev exponent abstract in this paper we study the asymptotic behavior of radial extremal functions to an inequality involving hardy potential and critical sobolev exponent based on the asymptotic behavior at the origin and the infinity we shall deduce a strict inequality between two best constants finally as an application of this strict inequality we consider the existence of a nontrivial solution of a quasilinear brezis nirenberg type problem with hardy potential and critical sobolev exponent

remarks on functional canonical variates alternating least squares methods and ace we discuss properties of some data analytic methods which are intimately related to each other alternating least squares als correspondence analysis and more recently breiman and friedman s ace algorithm the application of these methods to regression produces nonparametric estimators of nonlinear transformations both of the response and the predictors we point out some anomalies as well as some curiosities in the mathematics of these methods and we relate them to some areas in computer aided tomography projection pursuit regression and nonlinear devices in the theory of noise

solutions of the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term abstract in the present paper the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions to the problem is obtained

a new local and global optimization method for mixed integer quadratic programming problems in this paper a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable

solutions of a quasilinear elliptic problem involving a critical sobolev exponent and multiple hardy type terms in the present paper a quasilinear elliptic problem with a critical sobolev exponent and multiple hardy type terms is considered by means of a variational method the existence of positive solutions of the problem is obtained

fractional order embedding canonical correlation analysis and its applications to multi view dimensionality reduction and recognition due to the noise disturbance and limited number of training samples within set and between set sample covariance matrices in canonical correlation analysis cca usually deviate from the true ones in this paper we re estimate within set and between set covariance matrices to reduce the negative effect of this deviation specifically we use the idea of fractional order to respectively correct the eigenvalues and singular values in the corresponding sample covariance matrices and then construct fractional order within set and between set scatter matrices which can obviously alleviate the problem of the deviation on this basis a new approach is proposed to reduce the dimensionality of multi view data for classification tasks called fractional order embedding canonical correlation analysis fecca the proposed method is evaluated on various handwritten numeral face and object recognition problems extensive experimental results on the cenparmi uci at t ar and coil 20 databases show that fecca is very effective and obviously outperforms the existing joint dimensionality reduction or feature extraction methods in terms of classification accuracy moreover its improvements for recognition rates are statistically significant on most cases below the significance level 0 05 a new method namely fecca is presented for multi view dimensionality reduction fractional order within set and between set scatter matrices are constructed by sample spectrum modeling the extracted features by fecca have strong discriminant power for recognition experimental results show fecca has better recognition rates than the existing joint feature extraction methods

schrodinger equations with critical nonlinearity singular potential and a ground state we study semilinear elliptic equations in a generally unbounded domain rn when the pertinent quadratic form is nonnegative and the potential is generally singular typically a homogeneous function of degree 2 we prove solvability results based on the asymptotic behavior of the potential with respect to unbounded translations and dilations while the nonlinearity is a perturbation of a self similar possibly oscillating term f of critical growth satisfying f js n 2n 2jf s j z s r this paper focuses on two qualitatively different cases of this problem one when the quadratic form has a generalized ground state and another where the presence of potential does not change the energy space in the latter case we allow nonlinearities with oscillatory critical growth an important example of such quadratic form is the one on rn with the radial hardy potential x 2 with in the first case in the second case where n 2 24 is the largest constant for which the energy form remains nonnegative

nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains abstract in this paper we consider the semilinear elliptic problem u u x 2 u k x u 2 2 u in u 0 on where is a smooth bounded domain in r n n 4 0 n 2 2 4 2 2 n n 2 is the critical sobolev exponent k x is a continuous function when and k x are invariant under a group of orthogonal transformations we prove the existence of nodal and positive solutions for 0 1 where 1 is the first dirichlet eigenvalue of x 2 on

large correlation analysis in this paper a novel supervised dimensionality reduction method is developed based on both the correlation analysis and the idea of large margin learning the method aims to maximize the minimal correlation between each dimensionality reduced instance and its class label thus named as large correlation analysis lca unlike most existing correlation analysis methods such as cca ccas and cda which all maximize the total or ensemble correlation over all training instances lca devotes to maximizing the individual correlations between given instances and its associated labels and is established by solving a relaxed quadratic programming with box constraints experimental results on real world datasets from both uci and usps show its effectiveness compared to the existing canonical correlation analysis methods

on a neumann problem with critical exponents and hardy potentials abstract in this paper we consider the neumann problem of the potential elliptic system with hardy terms and critical exponents on a bounded domain using the critical point theory we first find infinitely many solutions for the corresponding perturbed subcritical problem analyzing concentration points in the bubbles carefully which appear in the approximate solutions we establish the concentration estimates and then obtain infinitely many solutions with positive energy by passing to a limit

multiple positive solutions for semilinear elliptic equations involving multi singular inverse square potentials and concave convex nonlinearities abstract in this paper we deal with the existence and multiplicity of positive solutions for the multi singular semilinear elliptic equations u i 1 k i x a i 2 u u 2 2 u u q 2 u x where r n n 3 is a smooth bounded domain such that the points a i i 1 2 k k 2 are different 0 i n 2 2 2 0 1 q 2 and 2 2 n n 2 the results depend crucially on the parameters q and i for i 1 2 k

multiple positive solutions for a semilinear elliptic equation with critical sobolev exponent in this paper we study a class of semilinear elliptic equations with hardy potential and critical sobolev exponent by means of the ekeland variational principle and mountain pass theorem multiple positive solutions are obtained

existence results for singular critical elliptic problems in rn abstract in the paper we consider the following semilinear elliptic problems with critical sobolev hardy exponents p s u x 2 u v x u k x u 2 s 2 x s u in r n lim x u x 0 where n 3 0 n 2 2 2 0 0 s 2 2 s 2 n s n 2 under suitable conditions we prove that p s has at least one nontrivial solution by means of linking theorem also via the pseudo index theory we consider the multiplicity of nontrivial solutions for p s

on the existence and multiplicity of solutions for a class of singular elliptic problems abstract in this paper we prove the existence and multiplicity of solutions for a class of elliptic problems with singular coefficients we also prove the existence of a sign changing solution for a brezis nirenberg type problem with hardy potential and singular coefficients our proofs are based on the krasnoselskii genus the concentration compactness principle of lions and the linking theorem

multiset canonical correlations using globality preserving projections with applications to feature extraction and recognition multiset features extracted from the same patterns always represent different characteristics of data thus it is very valuable to perform the extraction on multiple feature sets this paper addresses the issue of multiset correlation feature extraction mcfe in multiple feature representations a novel method is proposed to carry out the mcfe for classification called multiset canonical correlations using globality preserving projections mcc gps which can perform joint dimensionality reduction for high dimensional data mcc gp integrates correlational characteristics of feature pairs and global geometric information of data in the transformed low dimensional space this makes mcc gps have better discriminant ability than a previous method proposed by the authors called multiset integrated canonical correlation analysis micca which only considers correlations for recognition tasks furthermore mcc gp can subsume two popular feature extraction methods into its framework under some constraints this also provides a new insight for these two methods the proposed method is applied to pattern recognition and examined using the coil 100 and eth 80 object databases and ar cmu pie and yale face databases extensive experimental results show that mcc gp outperforms micca and multiset canonical correlation analysis in terms of classification accuracy and efficiency

existence of positive solutions for the p laplacian with p gradient term abstract in this paper we study the existence of positive solutions for the p laplacian involving a p gradient term due to the non variational structure and the fact that the nonlinearity may be critical or supercritical the variational method is no longer valid taking advantage of global c 1 estimates and the liouville type theorems we employ the blow up argument to obtain the a priori estimates on solutions and finally obtain the existence result based on the krasnoselskii fixed point theorem

solutions to critical elliptic equations with multi singular inverse square potentials abstract let be an open bounded domain in r n n 3 with smooth boundary we are concerned with the multi singular critical elliptic problem u i 1 k i u x a i 2 q x u 2 2 u u h 0 1 where i r 2 2 n n 2 a i 1 i k and q x is a positive bounded function on using moser iteration we prove the asymptotic behavior of solutions for at points a i 1 i k by exploiting the effect of the coefficient of the critical nonlinearity we by means of a variational method establish the existence of positive and sign changing solutions for problem

existence of solutions for a nonlinear pde with an inverse square potential abstract via linking theorem and delicate energy estimates the existence of nontrivial solutions for a nonlinear pde with an inverse square potential and critical sobolev exponent is proved this result gives a partial positive answer to an open problem proposed in ferrero and gazzola j differential equations 177 2001 494

existence of solution for a singular critical elliptic equation in this paper a singular semilinear elliptic problem involving the critical sobolev exponent is studied by variational method the existence of a solution is proved under certain conditions the hardy inequality is used and plays an important role in the discussion

ground states of nonlinear schrodinger equations with sum of periodic and inverse square potentials abstract we study the existence of solutions of the following nonlinear schrodinger equation u v x x 2 u f x u for x r n 0 where v r n r and f r n r r are periodic in x r n we assume that 0 does not lie in the spectrum of v and n 2 2 4 n 3 the superlinear and subcritical term f satisfies a weak monotonicity condition for sufficiently small 0 we find a ground state solution as a minimizer of the energy functional on a natural constraint if 0 and 0 lies below the spectrum of v then ground state solutions do not exist

multiset canonical correlations analysis and multispectral truly multitemporal remote sensing data this paper describes two and multiset canonical correlations analysis cca for data fusion multisource multiset or multitemporal exploratory data analysis these techniques transform multivariate multiset data into new orthogonal variables called canonical variates cvs which when applied in remote sensing exhibit ever decreasing similarity as expressed by correlation measures over sets consisting of 1 spectral variables at fixed points in time r mode analysis or 2 temporal variables with fixed wavelengths t mode analysis the cvs are invariant to linear and affine transformations of the original variables within sets which means for example that the r mode cvs are insensitive to changes over time in offset and gain in a measuring device in a case study cvs are calculated from landsat thematic mapper tm data with six spectral bands over six consecutive years both rand t mode cvs clearly exhibit the desired characteristic they show maximum similarity for the low order canonical variates and minimum similarity for the high order canonical variates these characteristics are seen both visually and in objective measures the results from the multiset cca r and t mode analyses are very different this difference is ascribed to the noise structure in the data the cca methods are related to partial least squares pls methods this paper very briefly describes multiset cca based multiset pls also the cca methods can be applied as multivariate extensions to empirical orthogonal functions eof techniques multiset cca is well suited for inclusion in geographical information systems gis

optimality conditions for the minimization of a quadratic with two quadratic constraints the trust region method has been proven to be very successful in both unconstrained and constrained optimization it requires the global minimum of a general quadratic function subject to ellipsoid constraints in this paper we generalize the trust region subproblem by allowing two general quadratic constraints conditions and properties of its solution are discussed

derivative constrained optimum broad band antenna arrays the optimization problem corresponding to the design of a presteered broadband processor with maximally flat second order spatial response in the look direction is discussed the constraints corresponding to necessary and sufficient conditions for second order flatness are presented the nonlinearly constrained optimization approach is unable to consistently locate the global minimum an approach to solving this problem that essentially converts the nonconvex optimization problem into a parameterized set of convex problems is presented in the case of 2 d scenarios the global optimum is determined by finding the roots of a cubic function the characteristics of the constraints including the minimum number required and the dependence on the choice of coordinate systems are examined

elliptic problems with critical exponents and hardy potentials this paper is devoted to the existence of positive solutions of a dirichlet problem with critical exponent and a singular potential under various assumption on the domain omega which include some kinds of unbounded domains we prove the existence of ground states and of symmetric solutions c 2002 elsevier science usa all rights reserved

existence of solutions for singular elliptic systems with critical exponents abstract let 0 be an open bounded domain in r n n 3 and 2 2 n n 2 we consider the following elliptic system of two equations in h 0 1 h 0 1 u t u x 2 2 u 2 u v u v t v x 2 2 u v 2 v v where 0 and 1 satisfy 2 using moser iteration we prove the asymptotic behavior of solutions for at the origin by exploiting the mountain pass theorem we establish the existence of solutions to we also discuss sign changing solutions for in this paper

on the singular critical quasilinear problems in rn abstract in this paper we study the following quasilinear problem p u u p 2 u u p 2 u x p u p 2 u f u u w r 1 p r n where n 3 2 p n p n p n p and f u satisfies some kinds of assumptions by employing the variational methods and analytic techniques we obtain the existence of a nontrivial solution to the problem under some assumptions for the parameters p n and at last we manage to prove that the problem has at least a pair of nodal solutions u k for each k n and both u k possess exactly k nodes

on schrodinger operators with multipolar inverse square potentials abstract positivity essential self adjointness and spectral properties of a class of schrodinger operators with multipolar inverse square potentials are discussed in particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established

weak eigenfunctions for the linearization of extremal elliptic problems abstract we consider the semilinear elliptic problem formula where is a nonnegative parameter and g is a positive nondecreasing convex nonlinearity there exists a value of the parameter which is extremal in terms of existence of solution we study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter in some cases this linearized problem has discrete and positive h 1 0 spectrum however we prove that there always exists a positive weak eigenfunction in l 1 with eigenvalue zero for this linearized problem the zero l 1 eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for finally we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when is the unit ball and g u e u or g u 1 u p

a global optimization algorithm for concave quadratic programming problems using a global optimality criterion for concave quadratic problems due to hiriart urruty and lemarechal the authors present an algorithm which manages to escape from a local solution and lead towards the global one in addition this procedure recognizes the unsolvability of the problem due to unboundedness of the objective function on the feasible region and also generates an improving feasible direction even if the starting point does not satisfy the karush kuhn tucker conditions as a key subroutine a recursive procedure will be used which determines whether or not a given symmetric n times n matrix is copositive i e yields a quadratic form that is positive on a given polyhedral cone both this subroutine and the main body of the algorithm frequently employ the simplex method while all other operations are elementary

a statistical view of some chemometrics regression tools chemometrics is a field of chemistry that studies the application of statistical methods to chemical data analysis in addition to borrowing many techniques from the statistics and engineering literatures chemometrics itself has given rise to several new data analytical methods this article examines two methods commonly used in chemometrics for predictive modeling partial least squares and principal components regression from a statistical perspective the goal is to try to understand their apparent successes and in what situations they can be expected to work well and to compare them with other statistical methods intended for those situations these methods include ordinary least squares variable subset selection and ridge regression

non convex quadratic minimization problems with quadratic constraints global optimality conditions in this paper we first examine how global optimality of non convex constrained optimization problems is related to lagrange multiplier conditions we then establish lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints we also obtain necessary global optimality conditions which are different from the lagrange multiplier conditions for special classes of quadratic optimization problems these classes include weighted least squares with ellipsoidal constraints and quadratic minimization with binary constraints we discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi extremal non convex quadratic optimization problems

necessary global optimality conditions for nonlinear programming problems with polynomial constraints in this paper we develop necessary conditions for global optimality that apply to non linear programming problems with polynomial constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete optimization in particular we show that our optimality conditions readily apply to problems where the objective function is the difference of polynomial and convex functions over polynomial constraints and to classes of fractional programming problems our necessary conditions become also sufficient for global optimality for polynomial programming problems our approach makes use of polynomial over estimators and a polynomial version of a theorem of the alternative which is a variant of the positivstellensatz in semi algebraic geometry we discuss numerical examples to illustrate the significance of our optimality conditions

regularization methods for sdp relaxations in large scale polynomial optimization we study how to solve semidefinite programming sdp relaxations for large scale polynomial optimization when interior point methods are used typically only small or moderately large problems could be solved this paper studies regularization methods for solving polynomial optimization problems we describe these methods for semidefinite optimization with block structures and then apply them to solve large scale polynomial optimization problems the performance is tested on various numerical examples with regularization methods significantly bigger problems could be solved on a regular computer which is almost impossible with interior point methods

second order cone programming relaxation of sensor network localization the sensor network localization problem has been much studied recently biswas and ye proposed a semidefinite programming sdp relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed here we study a second order cone programming socp relaxation of this problem motivated by its simpler structure and its potential to be solved faster than sdp we show that the socp relaxation though weaker than the sdp relaxation has nice properties that make it useful as a problem preprocessor in particular sensors that are uniquely positioned among interior solutions of the socp relaxation are accurate up to the square root of the distance error thus these sensors which are easily identified are accurately positioned in our numerical simulation the interior solution found can accurately position up to 80 90p of the sensors we also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interior point method

existence of positive solutions of some semilinear elliptic equations with singular coefficients in this paper we consider the semilinear elliptic problem in a bounded domain omega subseteq r n delta u frac mu x alpha u 2 ast alpha 1 f x g u in omega u 0 in omega u 0 on partial omega where mu geq 0 0 leq alpha leq 2 2 asterik alpha 2 n alpha n 2 f omega rightarrow r is measurable f 0 a e having a lower order singularity than x 2 at the origin and g r rightarrow r is either linear or superlinear for 1 p n we characterize a class of singular functions im p for which the embedding w 0 1 p omega hookrightarrow l p omega f is compact when p 2 alpha 2 f in im 2 and 0 leq mu frac 1 2 n 2 2 we prove that the linear problem has h 0 1 discrete spectrum by improving the hardy inequality we show that for f belonging to a certain subclass of im 2 the first eigenvalue goes to a positive number as mu approaches frac 1 2 n 2 2 furthermore when g is superlinear we show that for the same subclass of im 2 the functional corresponding to the differential equation satisfies the palais smale condition if alpha 2 and a brezis nirenberg type of phenomenon occurs for the case 0 leq alpha 2

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

positive ground state solutions for an elliptic system with hardy sobolev critical exponent growth this paper considers a nonlinear elliptic system with hardy sobolev critical exponent growth several existence results on the positive ground state solutions to the system are established under proper conditions this paper considers a nonlinear elliptic system with hardy sobolev critical exponent growth several existence results on the positive ground state solutions to the system are established under proper conditions

existence of positive solutions for a class of degenerate or singular equations abstract in this paper we consider the class of equations div x 2 a u a x 2 a u u x b p x r n 0 where a b r n 2 a r 0 and 0 p 2 n n 2 1 a b in various range of the parameters involved we obtain both results of existence and nonexistence of positive solutions by combining the mountain pass lemma and the pohozaev type identity

optimality conditions and optimization methods for quartic polynomial optimization abstract in this paper multivariate quartic polynomial optimization program qpop is considered quartic optimization problems arise in various practical applications and are proved to be np hard we discuss necessary global optimality conditions for quartic problem qpop and then we present a new strongly or e strongly local optimization method according to necessary global optimality conditions which may escape and improve some kkt points finally we design a global optimization method for problem qpop by combining the new strongly or e strongly local optimization method and an auxiliary function numerical examples show that our algorithms are efficient and stable

solutions for a singular critical growth problem with a weight in this paper we consider some semilinear elliptic equations with hardy potential by using linking theorem in p rabinowitz minimax methods in critical points theory with applications to differential equations cbms reg conf ser math vol 65 amer math soc providence ri 1986 and analyzing the effect of nonlinearities we establish the existence of nontrivial solutions

a semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints we present a general semidefinite relaxation scheme for general n variate quartic polynomial optimization under homogeneous quadratic constraints unlike the existing sum of squares approach which relaxes the quartic optimization problems to a sequence of typically large linear semidefinite programs sdp our relaxation scheme leads to a possibly nonconvex quadratic optimization problem with linear constraints over the semidefinite matrix cone in mathbb r n times n it is shown that each alpha factor approximate solution of the relaxed quadratic sdp can be used to generate in randomized polynomial time an o alpha factor approximate solution for the original quartic optimization problem where the constant in o cdot depends only on problem dimension in the case where only one positive definite quadratic constraint is present in the quartic optimization problem we present a randomized polynomial time approximation algorithm which can provide a guaranteed relative approximation ratio of 1 o n 2

np hardness of deciding convexity of quartic polynomials and related problems we show that unless p np there exists no polynomial time or even pseudo polynomial time algorithm that can decide whether a multivariate polynomial of degree four or higher even degree is globally convex this solves a problem that has been open since 1992 when n z shor asked for the complexity of deciding convexity for quartic polynomials we also prove that deciding strict convexity strong convexity quasiconvexity and pseudoconvexity of polynomials of even degree four or higher is strongly np hard by contrast we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time

existence of multiple positive solutions to singular elliptic systems involving critical exponents abstract this paper is devoted to investigate multiple positive solutions to a singular elliptic system where the nonlinearity involves a combination of concave and convex terms by exploiting the effect of the coefficient of the critical nonlinearity and a variational method we establish the main result which is based on the argument of the compactness

existence of solutions for elliptic equations with critical sobolev hardy exponents let omega subset of rn be a smooth bounded domain such that 0 is an element of omega n greater than or equal to 3 0 less than or equal to s 2 2 s 2 n s n 2 we prove the existence of nontrival solutions for the singular critical problem

existence of sign changing solutions for semilinear singular elliptic equations with critical sobolev exponents we consider the semilinear elliptic problem u u x 2 f x u k x u 2 2u in u 0 on where 0 is a smooth bounded domain in rn n 4 0 0 is continuous we prove the existence of sign changing solutions under different assumptions when is a usual domain and a symmetric domain respectively

global minimization of normal quartic polynomials based on global descent directions a normal quartic polynomial is a quartic polynomial whose fourth degree term coefficient tensor is positive definite its minimization problem is one of the simplest cases of nonconvex global optimization and has engineering applications we call a direction a global descent direction of a function at a point if there is another point with a lower function value along this direction for a normal quartic polynomial we present a criterion to find a global descent direction at a noncritical point a saddle point or a local maximizer we give sufficient conditions to judge whether a local minimizer is global and give a method for finding a global descent direction at a local but not global minimizer we also give a formula at a critical point and a method at a noncritical point to find a one dimensional global minimizer along a global descent direction based upon these we propose a global descent algorithm for finding a global minimizer of a normal quartic polynomial when n 2 for the case n ge 3 we propose an algorithm for finding an epsilon global minimizer at each iteration of a second algorithm a system of constrained nonlinear equations is solved numerical tests show that these two algorithms are promising

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

conopt a grg code for large sparse dynamic nonlinear optimization problems the paper presents conopt an optimization system for static and dynamic large scale nonlinearly constrained optimization problems the system is based on the grg algorithm all computations involving the jacobian of the constraints use sparse matrix algorithm from linear programming modified to take optimal advantage of the nonlinearity the paper presents the main features of the system especially the inversion routines and their data structures the dynamic setting of tolerances in newton s algorithm and the user features in the overall packaging computational experience with some medium to large models is presented

existence and asymptotic behavior of ground states for schrodinger systems with hardy potential abstract in this paper we study the following schrodinger systems with hardy potential u u x 2 v f x z v x r n v v x 2 u f x z u x r n where z u v r 2 and r is a positive parameter this problem is related to coupled nonlinear schrodinger equations for bose einstein condensate under some suitable conditions on the parameter and nonlinearity f we first prove the existence exponential decay and convergence of ground state solutions via variational methods moreover we prove the monotonicity and convergence property of the energy of ground state solutions finally we also give the asymptotic behavior of ground state solutions as parameter tends to 0

global optimality conditions for quadratic optimization problems with binary constraints we consider nonconvex quadratic optimization problems with binary constraints our main result identifies a class of quadratic problems for which a given feasible point is global optimal we also establish a necessary global optimality condition these conditions are expressed in a simple way in terms of the problem s data we also study the relations between optimal solutions of the nonconvex binary quadratic problem versus the associated relaxed and convex problem defined over the l infty norm our approach uses elementary arguments based on convex duality

the existence of solutions for quasilinear elliptic problems with combined critical sobolev hardy terms abstract in this paper we investigate the quasilinear elliptic equations with dirichlet boundary conditions and combined critical sobolev hardy terms on bounded smooth domains r n n 3 and prove the existence and multiplicity of solutions by employing ekeland s variational principle

multiple positive solutions for semilinear elliptic equations with critical weighted hardy sobolev exponents abstract the existence and multiplicity of positive solutions are obtained for a class of semilinear elliptic equations with critical weighted hardy sobolev exponents and the concave convex nonlinearity by variational methods and some analysis techniques

multiplicity result for a class of elliptic equations with singular term abstract we consider the existence of nontrivial solutions of the equation u x 2 u u 2 2 u x 2 u f x u x 0 u h 0 1 where 0 is a smooth bounded domain in r n n 3 by variational methods and nehari set techniques we show that this problem under some additional hypotheses on 0 0 0 0 1 and f l has four nontrivial solutions in h 0 1 and that least one of them is sign changing

quasilinear elliptic problems with critical exponents and hardy terms abstract let 0 be an open bounded domain r n n p 2 we are concerned with the multiplicity of positive solutions of p u u p 2 u x p u p 2 u q x u p 2 u u w 0 1 p where p u div u p 2 u 1 p n p np n p 0 n p p p 0 and q x is a nonnegative function on by investigating the effect of the coefficient of the critical nonlinearity we by means of variational method prove the existence of multiple positive solutions

the regularized iteratively reweighted mad method for change detection in multi and hyperspectral data this paper describes new extensions to the previously published multivariate alteration detection mad method for change detection in bi temporal multi and hypervariate data such as remote sensing imagery much like boosting methods often applied in data mining work the iteratively reweighted ir mad method in a series of iterations places increasing focus on difficult observations here observations whose change status over time is uncertain the mad method is based on the established technique of canonical correlation analysis for the multivariate data acquired at two points in time and covering the same geographical region we calculate the canonical variates and subtract them from each other these orthogonal differences contain maximum information on joint change in all variables spectral bands the change detected in this fashion is invariant to separate linear affine transformations in the originally measured variables at the two points in time such as 1 changes in gain and offset in the measuring device used to acquire the data 2 data normalization or calibration schemes that are linear affine in the gray values of the original variables or 3 orthogonal or other affine transformations such as principal component pc or maximum autocorrelation factor maf transformations the ir mad method first calculates ordinary canonical and original mad variates in the following iterations we apply different weights to the observations large weights being assigned to observations that show little change i e for which the sum of squared standardized mad variates is small and small weights being assigned to observations for which the sum is large like the original mad method the iterative extension is invariant to linear affine transformations of the original variables to stabilize solutions to the ir mad problem some form of regularization may be needed this is especially useful for work on hyperspectral data this paper describes ordinary two set canonical correlation analysis the mad transformation the iterative extension and three regularization schemes a simple case with real landsat thematic mapper tm data at one point in time and partly constructed data at the other point in time that demonstrates the superiority of the iterative scheme over the original mad method is shown also examples with spot high resolution visible data from an agricultural region in kenya and hyperspectral airborne hymap data from a small rural area in southeastern germany are given the latter case demonstrates the need for regularization

existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving hardy term and hardy sobolev critical exponents abstract existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with hardy term hardy sobolev critical exponents and sublinear nonlinearity by variational methods and some analysis techniques

existence and multiplicity of solutions to a singular elliptic system with critical sobolev hardy exponents and concave convex nonlinearities abstract this paper is concerned with a singular elliptic system which involves the caffarelli kohn nirenberg inequality and critical sobolev hardy exponents the existence and multiplicity results of positive solutions are obtained by variational methods

multiple solutions for nonhomogeneous elliptic equations involving critical sobolev exponent we consider the following problem where is continuous on r n and h x 0 by using ekeland s variational principle and the mountain pass theorem without ps conditions through a careful inspection of the energy balance for the approximated solutions we show that the probelm has at least two solutions for some 0 and 0 in particular if p 2 in a different way we prove that problem with 1 and h x 0 has at least two positive solutions as

describing complicated objects by implicit polynomials this paper introduces and focuses on two problems first is the representation power of closed implicit polynomials of modest degree for curves in 2 d images and surfaces in 3 d range data super quadrics are a small subset of object boundaries that are well fitted by these polynomials the second problem is the stable computationally efficient fitting of noisy data by closed implicit polynomial curves and surfaces the attractive features of these polynomials for vision is discussed

using copositivity for global optimality criteria in concave quadratic programming problems in this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems using this condition it follows that from the perspective of worst case complexity of concave quadratic problems the difference between local and global optimality conditions is not as large as in general as an essential ingredient we here use thee subdifferential calculus via an approach of hiriart urruty and lemarechal 1990

nontrivial solutions for some singular critical growth semilinear elliptic equations abstract let be a bounded domain in r n n 5 with smooth boundary and the origin 0 n 2 2 2 2 2 n n 2 k x is a bounded positive function on we prove the existence results for nontrivial solutions to the dirichlet problem u u x 2 k x u 2 2 u in u 0 on for suitable numbers and