multiple existence of sign changing solutions for coupled nonlinear schrodinger equations abstract in this paper we consider the multiple existence of sign changing solutions of coupled nonlinear schrodinger equations p u 1 u 1 u 3 v 2 u in r 3 v 2 v 2 v 3 u 2 v in r 3 where 1 2 1 2 are positive and 0

infinitely many sign changing solutions for system of p laplace equations in rn abstract in this paper we consider the system of p laplacian equations in r n p u j j x u j p 2 u j i 1 k i j u i q u j q 2 u j in r n u j x 0 as x j 1 k where 1 p n max 2 p 2 q p n p n p i j are constants i j 1 k j j 0 j 1 2 k i j 0 i j i j 1 k j j 1 k are the potential functions and satisfy suitable decay assumptions the existence of infinitely many sign changing solutions is proved by the truncation method and by the concentration compactness analysis on the approximating solutions

optimal liouville type theorems for noncooperative elliptic schrodinger systems and applications we study multi component elliptic schrodinger systems arising in nonlinear optics and bose einstein condensation phenomena we prove new liouville type nonexistence theorems as well as a priori bounds decay and singularity estimates this is shown under an optimal sobolev growth restriction on the nonlinearities thus improving on recent results of dancer et al and of tavares et al these systems are of non cooperative form and hence cannot be tackled by maximum principle methods such as moving planes instead we rely on a delicate combination of rellich pohozaev type identities sobolev and interpolation inequalities on sn 1 and feedback and measure arguments we also extend our results to a rather general class of gradient type systems

self similarity and asymptotic stability for coupled nonlinear schrodinger equations in high dimensions abstract this paper is concerned with systems of coupled schrodinger equations with polynomial nonlinearities and dimension n 1 we show the existence of global self similar solutions and prove that they are asymptotically stable in a framework based on weak l p spaces whose elements have local finite l 2 mass the radial symmetry of the solutions is also addressed

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

infinitely many solutions to a linearly coupled schrodinger system with non symmetric potentials we study a linearly coupled schrodinger system in n n 3 assume that the potentials in the system are continuous functions satisfying suitable decay assumptions but without any symmetry properties and the parameters in the system satisfy some suitable restrictions using the liapunov schmidt reduction methods two times and combining localized energy method we prove that the problem has infinitely many positive synchronized solutions which extends result theorem 1 2 about nonlinearly coupled schrodinger equations in ao and wei calculus var partial differ equations 51 761 798 2014 to our linearly coupled problem

bifurcation diagrams of coupled schrodinger equations radially symmetric solutions of many important systems of partial differential equations can be reduced to systems of special ordinary differential equations a numerical solver for initial value problems for such systems is developed based on matlab and numerical bifurcation diagrams are obtained according to the behavior of the solutions various bifurcation diagrams of coupled schrodinger equations from nonlinear physics are obtained which suggests the uniqueness of the ground state solution

multiple existence of solutions for coupled nonlinear schrodinger equations abstract in this paper we consider the multiple existence of solutions of the coupled nonlinear schrodinger equations p u 1 u u 3 v 2 u in r 3 v 2 v v 3 u 2 v in r 3 where 1 2 and 0

existence of multiple solutions to a class of nonlinear schrodinger system with external sources terms abstract we study a class of nonlinear schrodinger system with external sources terms as perturbations in order to obtain existence of multiple solutions this system arises from bose einstein condensates etc as these external sources terms are positive functions and small in some sense we use nehari manifold to get the existence of a positive ground state solution and a positive bound state solution

norm bounds for hadamard products and an arithmetic geometric mean inequality for unitarily invariant norms abstract an arithmetic geometric mean inequality for unitarily invariant norms and matrices 2 a xb aa x xbb is an immediate consequence of a basic inequality for singular values of hadamard products

solutions with a prescribed number of zeros for nonlinear schrodinger systems abstract we find an interval of some parameters where there exists a sign changing solution of nonlinear schrodinger systems having a prescribed number of zeros

multi bump bound states for a system of nonlinear schrodinger equations abstract motivated by problems arising in nonlinear optics and bose einstein condensates we consider in r n with n 3 the following system of coupled schrodinger equations u i v i x u i u i j 1 d i j u j 2 u i 0 lim x u i x 0 i 1 d where 0 is a parameter i j j i are positive constants and v i non negative given potentials we assume that the interior of i 1 d v i 1 0 admits m connected components 1 m which are of class c 1 and isolated in each v i 1 0 for each non empty j 1 m we prove that the system admits for any large a multi bump solution u r n r d which is small in r n j j j and on each j j j close in h 1 norm to a least energy solution of the limit problem u i u i j 1 d i j u j 2 i 1 d subjected to homogeneous dirichlet boundary condition an explicit condition on the matrix i j is given to ensure our solutions have at least two positive components

a minimal energy tracking method for non radially symmetric solutions of coupled nonlinear schrodinger equations we aim at developing methods to track minimal energy solutions of time independent m component coupled discrete nonlinear schrodinger dnls equations we first propose a method to find energy minimizers of the 1 component dnls equation and use it as the initial point of the m component dnls equations in a continuation scheme we then show that the change of local optimality occurs only at the bifurcation points the fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve by combining all these techniques with a parameter switching scheme we successfully compute a non radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

induced norms of the schur multiplier operator abstract fix an n by n complex matrix a and consider the operator x sa x a x on n by n complex matrices x where a x denotes the schur product of a and x we show that the induced norm of sa with respect to the numerical radius norm is at most one if and only if the matrix a admits a factorization a b wb where w is a contractive matrix and the euclidean norms of the columns of b are at most one we give other equivalent characterizations and derive as a consequence a formally similar result about the induced norm of sa with respect to the spectral norm

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

a sharp threshold of blow up for coupled nonlinear schrodinger equations this paper is concerned with the coupled supercritical nonlinear schrodinger equations which have applications in many physical problems especially in nonlinear optics two types of new invariant evolution flows are established a sharp threshold of blow up and global existence of solutions for the equations is derived it is shown that the main result obtained includes parts of those presented by fanelli and montefusco 2007 j phys a math theor 40 14139 50x

existence and asymptotic behavior of vector solutions for coupled nonlinear kirchhoff type systems abstract this paper deals with the following linearly coupled nonlinear kirchhoff type system a 1 b 1 r 3 u 2 d x u 1 u f u v in r 3 a 2 b 2 r 3 v 2 d x v 2 v g v u in r 3 u v h 1 r 3 where a i 0 b i 0 i 0 are constants for i 1 2 0 is a parameter and f g c r r under the general berestycki lions type assumptions on f and g we establish the existence of positive vector solutions and positive vector ground state solutions respectively by using variational methods we also study the asymptotic behavior of these solutions as 0

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

generalized schur complements and p complementable operators abstract let a be a selfadjoint operator and p be an orthogonal projection both operating on a hilbert space h we say that a is p complementable if a p 0 holds for some r in this case we define i p a max r a p 0 as a tool for computing ip a we introduce a natural generalization of the schur complement or shorted operator of a to s r p denoted by a p we give expressions and a characterization for ip a that generalize some known results for particular choices of p we also study some aspects of the shorted operator a p for p complementable a under the hypothesis of compatibility of the pair a s we give some applications in the finite dimensional context

special set and solutions of coupled nonlinear schrodinger equations a special set of n coupled nonlinear schrodinger equations which consists of 2n interaction types each of which is characterized by a specific array of interaction parameters is presented it is shown that every lame function of order n n is a solution for one or more components for one or more interaction types of this special set simple rules that relate interaction and solution types and interesting features of these relationships are presented

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

some further results on hermitian matrix inequalities abstract some new results on the lowner partial ordering between certain sums products and direct products of matrices are derived proofs of these results are based on a necessary and sufficient condition for the nonnegative definiteness of the difference of two hermitian matrices which follows from the criterion for the nonnegative definiteness of a partitioned hermitian matrix due to albert 1969

multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent abstract this paper deals with the following system linearly coupled by nonlinear elliptic equations u 1 u u 2 2 u v x v 2 v v 2 2 v u x u v 0 on here is a smooth bounded domain in r n n 3 1 2 1 are constants 1 is the first eigenvalue of h 0 1 2 2 n n 2 is the sobolev critical exponent and r is a coupling parameter by variational method we prove that this system has a positive ground state solution for some 0 via a perturbation argument we show that this system also admits a positive higher energy solution when is small moreover the asymptotic behaviors of the positive ground state and higher energy solutions as 0 are analyzed

stationary solitons of a three wave model generated by type ii second harmonic generation in quadratic media abstract in this paper we prove the uniqueness of stationary standing wave solutions of an optical model generated by type ii second harmonic generation shg with behaviors tending to zero at infinity under certain conditions on parameters in addition we provide the same issues for the dirichlet boundary value problems on the ball centered at the origin a classification of solutions for radial case is also established

existence and symmetry of positive ground states for a doubly critical schrodinger system n n 2 and 1 1 satisfying 2 this problem is related to coupled nonlinear schrodinger equations with critical exponent for bose einstein condensate for different ranges of n and 0 we obtain positive ground state solutions via some quite different methods which are all radially symmetric it turns out that the least energy level depends heavily on the relations among and 2 besides for sufficiently small 0 positive solutions are also obtained via a variational perturbation approach note that the palais smale condition can not hold for any positive energy level which makes the study via variational methods rather complicated

on the classification of standing wave solutions to a coupled schrodinger system in this paper we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear schrodinger equation we establish the existence and uniqueness of solutions to the dirichlet problem on the ball and entire space as the parameters within certain regions in addition a complete structure of different types of solutions for the radial case is also provided

a liouville s theorem for a fractional elliptic system with indefinite nonlinearities abstract in this paper we consider a fractional elliptic system with indefinite nonlinearities we prove that every positive bounded solution of the system is monotone increasing along x 1 direction by using the direct method of moving planes based on this monotonicity we obtain the nonexistence of positive solutions and a liouville type results for the fractional elliptic system as an application we get the liouville type results for the fractional elliptic system with cubic nonlinearities

a linear time algorithm for optimal routing around a rectangle the problem of connecting a set of terminals that lie on the sides of a rectangle to minimize the total area is discussed an o n algorithm is presented to solve this problem when the set of n terminals is initially sorted the strategy in this paper is to reduce the problem to several problems such that no matter what instance is started with at least one of these problems can be solved optimally by a greedy method

standing waves of a weakly coupled schrodinger system with distinct potential functions abstract the standing wave solutions of a weakly coupled nonlinear schrodinger system with distinct trapping potential functions in r n 1 n 3 are considered this type of system arises from models in bose einstein condensates theory and nonlinear optics the existence of a positive ground state solution is shown when the coupling constant is larger than a sharp threshold value which is explicitly defined in terms of potential functions and system parameters it is also shown that such solutions concentrate near the minimum points of potential functions and multiple positive concentration solutions exist when the topological structure of the set of minimum points satisfies certain condition variational approach is used for the existence and concentration of positive solutions

spike layer solutions to singularly perturbed semilinear systems of coupled schrodinger equations abstract let be a bounded domain in r n n 3 we are concerned with the interaction and the configuration of spikes in a double condensate by analyzing the least energy solutions of the following two couple schrodinger equations in se e 2 u u 1 u 3 u v 2 e 2 v v 2 v 3 u 2 v u 0 v 0 where 1 2 are positive constants we prove that under neumann or dirichlet boundary conditions for any e 0 when min 1 2 or max 1 2 system se has a least energy solution u e v e and when min 1 2 max 1 2 system se has no solution suppose p e q e are the local maximum points of u e v e respectively then under neumann boundary conditions as e small enough both of p e q e locate on the boundary of furthermore when 0 p e q e e 0 as e 0 and for n 2 and n 3 p e q e converge to the same point on the boundary which is the maximum point of mean curvature of the boundary however when 0 p e q e e as e 0 and suppose p e p and q e q then for n 2 and n 3 p q must be the maximum points of the mean curvature on the boundary and p q might be a same point if the mean curvature of the boundary has only one maximum point under dirichlet boundary conditions we can prove that as long as the least energy solution se exists the same asymptotic behavior of the least energy solution u e v e holds as described in lin and wei 2005 10 for 0 or for 0 thus our results are an extension of the results in lin and wei 2005 10

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

sign changing solutions for coupled nonlinear schrodinger equations with critical growth abstract we consider the following nonlinear schrodinger system with critical growth u j j u j i 1 k i j u i 2 2 u j 2 2 2 u j in u j 0 on j 1 k where is a bounded smooth domain in r n 2 2 n n 2 0 j 1 j 1 k 1 is the first eigenvalue of with zero dirichlet boundary condition we consider the repulsive case namely j j 0 j 1 k i j j i 0 i j i j 1 k the existence of infinitely many sign changing solutions as bound states is proved provided n 7 by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions

positive solutions to some systems of coupled nonlinear schrodinger equations abstract we study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time independent schrodinger equations u j j u j j u j 3 k 1 k j n j k u j u k 2 u j w 1 2 r n j 1 n where n 1 2 3 j j 0 for j 1 n the coupling parameters j k k j r for j k 1 n j k precisely we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors additionally with more restrictive conditions on the coupled parameters we show that the bound states founded are positive

sharp thresholds of blow up and global existence for the coupled nonlinear schrodinger system in this paper we establish two new types of invariant sets for the coupled nonlinear schrodinger system in the euclidean n space rn and derive two sharp thresholds of blow up and global existence for its solutions some analogous results for the nonlinear schrodinger system posed on the hyperbolic space hn and on the standard 2 sphere s2 are also presented our arguments and constructions are improvements of some previous works on this direction at the end we give some heuristic analysis about the strong instability of the solitary waves the relation between the two types of thresholds is a very interesting problem and we leave it as an open problem for further study

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

existence and nonexistence of bound state solutions for schrodinger systems with linear and nonlinear couplings abstract we study the schrodinger systems with linear and nonlinear coupling terms doubly coupled nonlinear schrodinger system for short which arise naturally in nonlinear optics and in the hartree fock theory for bose einstein condensates among other physical problems first for small linear coupling constant we get existence of a nontrivial bound state solution to the system via perturbation method furthermore we prove each component of the bound state solution is nonnegative by energy estimate second we establish a version of pohozaev nehari identity and prove a nonexistence result for the more general system when the spatial dimension n 4

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

solitary and self similar solutions of two component system of nonlinear schrodinger equations conventionally to learn wave collapse and optical turbulence one must study finite time blow up solutions of one component self focusing nonlinear schrodinger equations nlse here we consider simultaneous blow up solutions of two component systems of self focusing nlse by studying the associated self similar solutions we prove two components of solutions blow up at the same time these self similar solutions may come from solitary wave solutions with multi bumps forming abundant geometric patterns which cannot be found in one component self focusing nlse our results may provide the first step to investigate optical turbulence in two component systems of nlse

semitrivial vs fully nontrivial ground states in cooperative cubic schrodinger systems with d 3 equations abstract in this work we consider the weakly coupled schrodinger cubic system u i i u i i u i 3 u i j i b i j u j 2 u i h 1 r n r i 1 d where 1 n 3 i i 0 and b i j b j i 0 for i j this system admits semitrivial solutions that is solutions u u 1 u d with null components we provide optimal qualitative conditions on the parameters i i and b i j under which the ground state solutions have all components nontrivial or conversely are semitrivial this question had been clarified only in the d 2 equations case for d 3 equations prior to the present paper only very restrictive results were known namely when the above system was a small perturbation of the super symmetrical case i and b i j b we treat the general case uncovering in particular a much more complex and richer structure with respect to the d 2 case

index of hadamard multiplication by positive matrices ii for each n n positive semidefinite matrix a we define the minimal index i a max 0 a b b for all b 0 and for each norm n the n index in a min n a b b 0 and n b 1 where a b aijbij is the hadamard or schur product of a aij and b bij and b 0 means that b is a positive semidefinite matrix a comparison between these indexes is done for different choices of the norm n as an application we find for each bounded invertible selfadjoint operator s on a hilbert space the best constant m s such that sts s 1ts 1 m s t for all t 0

uniform holder bounds for nonlinear schrodinger systems with strong competition for the positive solutions of the gross pitaevskii system u u 1 u 3 u v 2 v v 2 v 3 u 2 v we prove that l boundedness implies c 0 boundedness for every 0 1 uniformly as moreover we prove that the limiting profile as is lipschitz continuous the proof relies upon the blowup technique and the monotonicity formulae by almgren and alt caffarelli and friedman this system arises in the hartree fock approximation theory for binary mixtures of bose einstein condensates in different hyperfine states extensions to systems with k 2 densities are given

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

positive ground state solutions and multiple nontrivial solutions for coupled critical elliptic systems in this study we consider the following coupled elliptic system with a sobolev critical exponent p u1 1u1 1u1p1 1 1u12 1 u12 2 1u22 2 x u2 2u2 2u2p2 1 2u22 1 u12 2u22 2 1 x u1 u2 0in u1 u2 0on where rn is a bounded smooth domain n 5 2 0 for j 1 2 and 1 is the first eigenvalue of with the dirichlet boundary condition we demonstrate the existence of a positive ground state solution for problem p when the coupling parameter 1 2 under some other conditions we show the nonexistence of positive solutions for p when n 3 we also construct multiple nontrivial solutions and sign changing solutions for the following system u1 1u1 1u13 u22u1 x u2 2u2 2u23 u12u2 x u1 u2 0on where rn is a bounded smooth domain and n 4

a simple approximation algorithm for two problems in circuit design this paper provides a very simple two approximation algorithm for two np hard problems that arise in electronic circuit design to our knowledge this is the best approximation bound known for these problems in addition the simplicity of the proposed algorithm makes it attractive for real time applications for similar problems in areas such as telecommunications and parallel processing

two component bose einstein condensates abstract recently coupled systems of nonlinear schrodinger equations have been used extensively to describe bose einstein condensates in this paper we study the structure of vortices of the coupled nonlinear equations for two component bose einstein condensates bec in a three dimensional space we show that vortices is 1 rectifiable set and give its mean curvature in particular we show that large interspecies scattering length causes vortices for two component bec

blowup and solitary wave solutions with ring profiles of two component nonlinear schrodinger systems abstract blowup ring profiles have been investigated by finding non vortex blowup solutions of nonlinear schrodinger equations nlses cf fibich et al 2005 7 and fibich et al 2007 8 however those solutions have infinite l 2 norm so one may not maintain the ring profile all the way up to the singularity to find h 1 non vortex blowup solutions with ring profiles we study the blowup solutions of two component systems of nlses with nonlinear coefficients and j j 1 2 when 0 and 1 2 0 the two component system can be transformed into a multi scale system with fast and slow variables which may produce h 1 blowup solutions with non vortex ring profiles we use the localized energy method with symmetry reduction to construct these solutions rigorously on the other hand these solutions may describe steady non vortex bright ring solitons various types of ring profiles including m ring and ring ring profiles are presented by numerical solutions

minimum congestion hypergraph embedding in a cycle the minimum congestion hypergraph embedding in a cycle mchec problem is to embed the n edges in an m vertex hypergraph as paths in a cycle on the same number of vertices such that congestion the maximum number of paths that use any single edge in the cycle is minimized the mchec problem has applications in electronic design automation and parallel computing in this paper it is proven that the mchec problem is np complete an o nm sup k 1 algorithm is described that computes an embedding with congestion k or determines that such an embedding does not exist finally a linear time approximation algorithm for arbitrary instances is presented that computes an embedding whose congestion is at most three times optimal

segregated and synchronized vector solutions for nonlinear schrodinger systems we consider the following nonlinear schrodinger system in mathbb r 3 left begin array ll delta u p x u mu u 2 u beta v 2u quad x in mathbb r 3 delta v q x v nu v 2 v beta u 2v quad x in mathbb r 3 end array right where p r and q r are positive radial potentials mu 0 nu 0 and beta in mathbb r is a coupling constant this type of system arises in particular in models in bose einstein condensates theory

geometry and the norms of hadamard multipliers abstract the norm of a matrix b as a hadamard multiplier is the norm of the map x x b where is the hadamard or entrywise product of matrices we give a geometric interpretation to a criterion of haagerup and use this interpretation to find an explicit formula for the hadamard multiplier norms of real 2 2 matrices for hermitian matrices with one positive eigenvalue we give necessary and sufficient conditions for the norm to be the magnitude of the largest entry

symmetric ground state solutions of m coupled nonlinear schrodinger equations abstract we prove the existence of radial and radially decreasing ground states of an m coupled nonlinear schrodinger equation with a general nonlinearity

on strongly indefinite systems involving the fractional laplacian abstract this paper is devoted to study strongly indefinite systems involving the fractional laplacian on bounded domains explicitly we obtain existence and non existence results a p r i o r i estimates of gidas spruck type and a symmetry result in addition we give a different proof for the a p r i o r i estimate for nonlinear elliptic problems with the fractional laplacian obtained in cabre and tan 2010 and tan 2013

multi peak solutions to coupled schrodinger systems with neumann boundary conditions abstract in this paper we are concerned with the following two coupled schrodinger systems in a bounded domain r n n 2 3 with neumann boundary conditions e 2 u u 1 u 3 u v 2 e 2 v v 2 v 3 u 2 v u 0 v 0 u n 0 v n 0 on suppose the mean curvature h p of the boundary has several local minimums or local maximums we obtain the existence of solutions with multi peaks to the system with all peaks being on the boundary and all peaks locate either near the local maxima or near the local minima of the mean curvature at the boundary of the domain

blow up behavior of ground states for a nonlinear schrodinger system with attractive and repulsive interactions abstract we consider a nonlinear schrodinger system arising in a two component bose einstein condensate bec with attractive intraspecies interactions and repulsive interspecies interactions in r 2 we get ground states of this system by solving a constrained minimization problem for some kinds of trapping potentials we prove that the minimization problem has a minimizer if and only if the attractive interaction strength a i i 1 2 of each component of the bec system is strictly less than a threshold a furthermore as a 1 a 2 a a the asymptotical behavior for the minimizers of the minimization problem is discussed our results show that each component of the bec system concentrates at a global minimum of the associated trapping potential

multiplicity of localized solutions of nonlinear schrodinger systems for infinite attractive case abstract in this paper we consider the following nonlinear schrodinger system 2 u v 1 x u u v 2 in r n u h 1 r n 2 v v 2 x v u 2 v in r n v h 1 r n where n 3 0 1 v i c 1 r n r 0 inf x r n v i x sup x r n v i x i 1 2 and there exists a bounded open set r n with smooth boundary such that inf x min v 1 x v 2 x 0 we prove the existence of a given number of semi positive solutions localized in here we say u v is semi positive if either u 0 or v 0

note on ground states of a nonlinear schrodinger system abstract we give a sufficient condition for the existence of positive radial ground states of the time independent schrodinger system u 1 u 1 1 u 1 3 12 u 2 2 u 1 13 u 3 2 u 1 in r n u 2 u 2 12 u 1 2 u 2 2 u 2 3 23 u 3 2 u 2 in r n u 3 u 3 13 u 1 2 u 3 23 u 2 2 u 3 3 u 3 3 in r n u 1 x 0 u 2 x 0 u 3 x 0 as x where n 1 2 3 0 j 0 and i j 0 i j for i j 1 2 3 and in some special cases our conditions are also necessary

on the invertibility of the map t sts 1 s 1ts and operator norm inequalities abstract for an invertible operator s we consider the invertibility of the map t sts 1 s 1ts when this map is not invertible the kernel is examined for a normal invertible operator n a condition on the spectrum of n is found to be equivalent to the invertibility of the map t ntn 1 n 1tn

geometric operator inequalities abstract the geometrical meaning of several well known inequalities is discussed they include the so called loewner heinz mcintosh and segal inequalities it is shown that some of them can be deduced from the others even for unitarily invariant forms some spectral properties of the elementary operators associated to the inequalities are studied