schrodinger poisson system with concave convex nonlinearities we consider a class of the schrodinger poisson system with concave convex nonlinearities under some suitable assumptions we prove the existence of nontrivial solutions and infinitely many negative energy solutions by using the variational method and making accurate analysis on the combined effect of parameters and our results extend that in the related works we consider a class of the schrodinger poisson system with concave convex nonlinearities under some suitable assumptions we prove the existence of nontrivial solutions and infinitely many negative energy solutions by using the variational method and making accurate analysis on the combined effect of parameters and our results extend that in the related works

left vs right representations for solving weighted low rank approximation problems the weighted low rank approximation problem in general has no analytical solution in terms of the singular value decomposition and is solved numerically using optimization methods four representations of the rank constraint that turn the abstract problem formulation into parameter optimization problems are presented the parameter optimization problem is partially solved analytically which results in an equivalent quadratically constrained problem a commonly used re parameterization avoids the quadratic constraint and makes the equivalent problem a nonlinear least squares problem however it might be necessary to change this re parameterization during the iteration process it is shown how the cost function can be computed efficiently in two special cases row wise and column wise weighting

multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition abstract in this paper we study the nonlinear boundary value problem u u f u u v in v v f v u v in u n g u u v v n g v u v on where is a bounded domain in r n with smooth boundary and the pairs of f u f v and g u g v are subcritical nonlinearities with the help of the nehari manifold we show existence of at least two positive solutions when one of the pairs of f u f v and g u g v is sublinear and the other one superlinear

structured total least squares and l2 approximation problems abstract it is shown how structured and weighted total least squares and l 2 approximation problems lead to a nonlinear generalized singular value decomposition an inverse iteration scheme to find a local minimum is proposed the emphasis of the paper is not on the convergence analysis of the algorithm rather the purpose is to illustrate its use in numerous applications in systems and control including total least squares with relative errors and or fixed elements inverse singular value problems an errors in variables variant of the kalman filter impulse response realization from noisy data h 2 model reduction h 2 system identification and calculating the largest stability radius of uncertain linear systems several numerical examples are given

the nehari manifold and application to a semilinear elliptic system abstract in this paper we study the nehari manifold and its application to the following semilinear elliptic system u u f x u q 2 u x v v g x v q 2 v x u n h x u 2 u v x v n h x u v 2 v x where is a bounded domain in r n with smooth boundary we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

existence and concentration of solutions with compact support for elliptic problems with critical growth in rn abstract for a continuous and sign changing function a x we study the existence and asymptotic behavior of compactly supported solutions of the following elliptic problem in r n with n 3 u a x u q u p u 0 and u d 1 2 r n where 0 0 q 1 p n 2 n 2 and a a or a a a when is small both cases admit two solutions v and v where v is a local minimizer and v is a minimizing sequence for the sobolev embedding from d 1 2 r n to l p 1 r n as 0 in the case a a a v blows up in the l norm and concentrates at one point as 0 consequently we show that v cannot be radially symmetric when a 0 0 and is small enough

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

multiple positive solutions of a nonlinear boundary value problem involving a sign changing weight abstract in this paper we study the multiplicity of positive solutions for the following elliptic equation u u 0 in r n u a x u q 2 u b x u p 2 u on r n where 1 q 2 p 2 2 2 n 1 n 2 if n 3 2 if n 2 r n x x n r n 1 r x n 0 is an upper half space in r n 0 and the functions a and b are satisfying some suitable conditions using the decomposition of the nehari manifold we prove that the equation has at least two positive solutions provided is sufficiently small

multiple nontrivial solutions for critical growth quasilinear elliptic systems abstract in the present paper we consider the following critical growth quasilinear elliptic system p u i 1 p d u i f u i u i q 2 u i in i 1 k u i 0 on i 1 k by exploiting the nehari manifold and variational methods we obtain the existence and multiplicity results of nontrivial solutions for the system

an exact combinatorial algorithm for minimum graph bisection we present a novel exact algorithm for the minimum graph bisection problem whose goal is to partition a graph into two equally sized cells while minimizing the number of edges between them our algorithm is based on the branch and bound framework and unlike most previous approaches it is fully combinatorial we introduce novel lower bounds based on packing trees as well as a new decomposition technique that contracts entire regions of the graph while preserving optimality guarantees our algorithm works particularly well on graphs with relatively small minimum bisections solving to optimality several large real world instances with up to millions of vertices for the first time

the node capacitated graph partitioning problem a computational study in this paper we consider the problem ofk partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition and the objective of minimizing the sum of the costs of the edges between the subsets of the partition based on a study of valid inequalities we present a variety of separation heuristics for cycle cycle with ears knapsack tree and path block cycle inequalities among others the separation heuristics plus primal heuristics have been implemented in a branch and cut routine using a formulation including variables for the edges with nonzero costs and node partition variables results are presented for three classes of problems equipartitioning problems arising in finite element methods and partitioning problems associated with electronic circuit layout and compiler design 1998 the mathematical programming society inc published by elsevier science b v

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

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

the nehari manifold for a degenerate elliptic equation involving a sign changing weight function abstract this paper is concerned with a degenerate elliptic equation of the form div x 2 a u x 2 b u p 1 u f x x 2 c u q 1 u in u 0 on where r n n 3 is an open bounded domain with smooth boundary 0 0 a n 2 2 a b c a 1 and b c p 1 q 1 f c be a sign changing weight function using the nehari manifold and a fibering map the existence of multiple nonnegative solutions is established

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

solving max cut to optimality by intersecting semidefinite and polyhedral relaxations we present a method for finding exact solutions of max cut the problem of finding a cut of maximum weight in a weighted graph we use a branch and bound setting that applies a dynamic version of the bundle method as bounding procedure this approach uses lagrangian duality to obtain a nearly optimal solution of the basic semidefinite max cut relaxation strengthened by triangle inequalities the expensive part of our bounding procedure is solving the basic semidefinite relaxation of the max cut problem which has to be done several times during the bounding process we review other solution approaches and compare the numerical results with our method we also extend our experiments to instances of unconstrained quadratic 0 1 optimization and to instances of the graph equipartition problem the experiments show that our method nearly always outperforms all other approaches in particular for dense graphs where linear programming based methods fail our method performs very well exact solutions are obtained in a reasonable time for any instance of size up to n 100 independent of the density for some problems of special structure we can solve even larger problem classes we could prove optimality for several problems of the literature where to the best of our knowledge no other method is able to do so

existence and multiplicity of positive solutions to elliptic systems involving critical exponents abstract in this paper a singular elliptic system involving multiple critical exponents and the caffarelli kohn nirenberg inequality is investigated by using the extremals of the best hardy sobolev constants the existence and multiplicity of positive solutions to the system are established

the geometry of weighted low rank approximations the low rank approximation problem is to approximate optimally with respect to some norm a matrix by one of the same dimension but smaller rank it is known that under the frobenius norm the best low rank approximation can be found by using the singular value decomposition svd although this is no longer true under weighted norms in general it is demonstrated here that the weighted low rank approximation problem can be solved by finding the subspace that minimizes a particular cost function a number of advantages of this parameterization over the traditional parameterization are elucidated finding the minimizing subspace is equivalent to minimizing a cost function on the grassmann manifold a general framework for constructing optimization algorithms on manifolds is presented and it is shown that existing algorithms in the literature are special cases of this framework within this framework two novel algorithms a steepest descent algorithm and a newton like algorithm are derived for solving the weighted low rank approximation problem they are compared with other algorithms for low rank approximation as well as with other algorithms for minimizing a cost function on a grassmann manifold

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

multiplicity and asymptotic behavior of nonnegative solutions for elliptic problems involving nonlinearities indefinite in sign abstract let p r s be positive numbers with 1 r s p the multiplicity of nonnegative solutions for the problem p u u s 1 u r 1 in u 0 is investigated we prove that there exist two positive constants 2 r s r s with 2 r s r s such that the above problem has at least two solutions at least one solution or no solutions according to whether p r p s 2 r s p r p s 2 r s r s or p r p s r s we also study the asymptotic behavior of the solutions as 0

elliptic equations with indefinite concave nonlinearities near the origin in this note existence of infinitely many solutions is proved for an elliptic equation with indefinite concave nonlinearities

multiple solutions for a class of biharmonic elliptic systems with sobolev critical exponent abstract in this paper we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic systems with sobolev critical exponent in a bounded domain by using the variational method and the nehari manifold we obtain the existence and multiplicity results of nontrivial solutions for the systems

three positive solutions for semilinear elliptic problems involving concave and convex nonlinearities we study the existence and multiplicity of positive solutions for the dirichlet problem where 0 1 q p 2 2 n n 2 0 e n n 3 is a bounded domain with smooth boundary and f is a non negative continuous function on assuming that f satisfies some hypothesis we prove that the equation admits at least three positive solutions for sufficiently small

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

exact multiplicity for semilinear elliptic dirichlet problems involving concave and convex nonlinearities let b be the unit ball in r n n 3 let 0 p q n 2 n 2 in 1994 ambrosetti et al found that the semilinear elliptic dirichlet problem admits at least two solutions for small 0 and no solution for large in this paper we prove that there is a critical number 0 such that this problem has exactly two solutions for 0 exactly one solution for and no solution for

the nehari manifold for a kirchhoff type problem involving sign changing weight functions abstract this paper examines a class of kirchhoff type equations that involve sign changing weight functions using nehari manifold and fibering map the existence of multiple positive solutions is established

existence of multiple solutions for singular elliptic problems with nonlinear boundary conditions in this paper the existence and multiplicity results of solutions for the following quasilinear elliptic problem 0 1 div x ap u p 2 u h x u p 2u g x u r 2u x x ap u p 2 u f x u q 2u x are established where is an exterior domain in rn with the compact and smooth boundary by the variational methods we prove that the problem 0 1 has at least two positive solutions at the last part of the paper we also consider the critical case and prove the existence of solution by concentration compactness principle

a new approach to the minimum cut problem this paper present a new approach to finding minimum cuts in undirected graphs the fundamental principle is simple the edges in a graph s minimum cut form an extremely small fraction of the graph s edges using this idea we give a randomized strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability the algorithm runs in o n 2 log 3 n time a significant improvement over the previous o mn time bounds based on maximum flows it is simple and intuitive and uses no complex data structures our algorithm can be parallelized to run in rnc with n 2 processors this gives the first proof that the minimum cut problem can be solved in rnc the algorithm does more than find a single minimum cut it finds all of them with minor modifications our algorithm solves two other problems of interest our algorithm finds all cuts with value within a multiplicative factor of of the minimum cut s in expected o n 2 time or in rnc with n 2 processors the problem of finding a minimum multiway cut of graph into r pieces is solved in expected o n 2 r 1 time or in rnc with n 2 r 1 processors the trace of the algorithm s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph this data structure can be efficiently transformed into the more standard cactus representing for minimum cuts

a remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system abstract in this work motivated by t f wu the nehari manifold for a semilinear elliptic system involving sign changing weight function nonlinear anal 68 2008 1733 1745 and using recent ideas from brown and wu k j brown t f wu a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function j math anal appl 337 2008 1326 1336 we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system p u m x u p 2 u u 2 u c x u 2 u v x p v m x v p 2 v v 2 v c x u v 2 v x u 0 v x here p denotes the p laplacian operator defined by p z div z p 2 z p 2 r n is a bounded domain with smooth boundary 1 1 2 p p p p n n p if n p p if n p n is the outer normal derivative r 2 0 0 the weight m x is a bounded function with m 0 and c x is a continuous function which changes sign in

multiple positive solutions for a critical quasilinear elliptic system with concave convex nonlinearities abstract in this paper we consider a quasilinear elliptic system with both concave convex nonlinearities and critical growth terms in bounded domains the existence and multiplicity results of positive solutions are obtained by variational methods

the nehari manifold and the existence of multiple solutions for a singular quasilinear elliptic equation abstract in this paper we are concerned with the existence of multiple positive solutions for the singular quasilinear elliptic problem div x a p u p 2 u h x u m 2 u h x u n 2 u x u x 0 x where r n n 3 is a bounded domain with smooth boundary 0 1 p n 0 a n p p 1 m p n p n n 1 a p 0 h x h x are lebesgue measurable functions which may change sign on we prove that there exist at least two positive solutions by using the nehari manifold and the fibrering maps associated with the energy functional for this problem

positive solutions for nonhomogeneous elliptic equations involving critical sobolev exponent abstract in this paper we study the existence and multiplicity of positive solutions for the dirichlet problem u f z u q 2 u g z u 2 2 u in where 0 1 q 2 2 2 n n 2 0 r n n 3 is a bounded domain with smooth boundary and f g are nonnegative continuous functions on

the nehari manifold for a fractional critical choquard equation involving sign changing weight functions abstract in this paper we prove the existence and multiplicity of nontrivial solutions for the following fractional choquard equation with critical exponent s u f x u q 2 u g x u 2 s x y d y u 2 s 2 u in u 0 in r n where is a bounded domain in r n with smooth boundary s 0 1 0 n n 2 s 1 q 2 and 2 s 2 n n 2 s is the critical exponent in the sense of hardy littlewood sobolev inequality 0 is a parameter and f g r are continuous functions but may change sign on

multiple positive solutions for dirichlet problems involving concave and convex nonlinearities abstract in this paper we study the multiplicity of positive solutions for the following dirichlet problems involving concave and convex nonlinearities u u f x u q 2 u h x u p 2 u in a u h 0 1 a where 1 q 2 p 2 2 2 n n 2 if n 3 2 if n 2 0 a r is an infinite strip in r n is a bounded domain in r n 1 and f x h x satisfy suitable conditions

a probe based heuristic for graph partitioning a new heuristic algorithm probe ba which is based on the recently introduced metaheuristic paradigm population reinforced optimization based exploration probe is proposed for solving the graph partitioning problem the exploration part of probe ba is implemented by using the differential greedy algorithm of battiti and bertossi and a modification of the kernighan lin algorithm at the heart of bui and moon s genetic algorithm bfs gba experiments are used to investigate properties of probe and show that probe ba compares favorably with other solution methods based on genetic algorithms randomized reactive tabu search or more specialized multilevel partitioning techniques in addition probe ba finds new best cut values for 10 of the 34 instances in walshaw s graph partitioning archive

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

an existence result for a class of pp laplacian elliptic systems involving homogeneous nonlinearities in rn abstract in the present paper we study the existence of nontrivial solutions for a class of p laplacian elliptic systems in r n a new existence result for nontrivial solutions is obtained by means of variational methods

n kirchhoff choquard equations with exponential nonlinearity abstract this article deals with the study of the following kirchhoff equation with exponential nonlinearity of choquard type see k c below we use the variational method in the light of moser trudinger inequality to show the existence of weak solutions to k c moreover analyzing the fibering maps and minimizing the energy functional over suitable subsets of the nehari manifold we prove existence and multiplicity of weak solutions to convex concave problem p m below

quadratic 0 1 optimization and a decomposition approach for the placement of electronic circuits the placement problem in the layout design of electronic circuits consists of finding a nonoverlapping assignment of rectangular cells to positions on the chip so that wireability is guaranteed and certain technical constraints are met this problem can be modelled as a quadratic 0 1 program subject to linear constraints we will present a decomposition approach to the placement problem and give results above np hardness and the existence ofe approximative algorithms for the involved optimization problems a graph theoretic formulation of these problems will enable us to develop approximative algorithms finally we will present details of the implementation of our approach and compare it to industrial state of the art placement routines

an optimal graph theoretic approach to data clustering theory and its application to image segmentation a novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated the data to be clustered are represented by an undirected adjacency graph g with arc capacities assigned to reflect the similarity between the linked vertices clustering is achieved by removing arcs of g to form mutually exclusive subgraphs such that the largest inter subgraph maximum flow is minimized for graphs of moderate size approximately 2000 vertices the optimal solution is obtained through partitioning a flow and cut equivalent tree of g which can be efficiently constructed using the gomory hu algorithm 1961 however for larger graphs this approach is impractical new theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size this algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices the new clustering algorithm is applied to the image segmentation problem the segmentation is achieved by effectively searching for closed contours of edge elements equivalent to minimum cuts in g which consist mostly of strong edges while rejecting contours containing isolated strong edges this method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours

polyharmonic kirchhoff problems involving exponential non linearity of choquard type with singular weights abstract in this work we study the higher order kirchhoff type choquard equation k c involving a critical exponential non linearity and singular weights we prove the existence of solution to k c using mountain pass lemma in light of moser trudinger and singular adams moser inequalities in the second part of the paper using the nehari manifold technique and minimization over its suitable subsets we prove the existence of at least two solutions to the kirchhoff type choquard equation p m involving convex concave type non linearity

multiplicity of positive solutions for a semilinear p laplacian system with sobolev critical exponent abstract in this paper we study the semilinear p laplacian system with critical growth terms in bounded domains by using the nehari manifold and variational methods we prove that the system has at least two positive solutions when the pair of the parameters belongs to a certain subset of r 2

the nehari manifold for indefinite semilinear elliptic systems involving critical exponent abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system e involving critical exponents and sign changing weight functions using nehari manifold the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

a branch and cut algorithm for the equicut problem we describe an algorithm for solving the equicut problem on complete graphs the core of the algorithm is a cutting plane procedure that exploits a subset of the linear inequalities defining the convex hull of the incidence vectors of the edge sets that define an equicut the cuts are generated by several separation procedures that will be described in the paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a branch and cut strategy we also describe the implementation of the algorithm and the interface with the lp solver finally we report on computational results on dense instances with sizes up to 100 nodes

on elliptic problems involving critical hardy sobolev exponents and sign changing function abstract in this paper we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical hardy sobolev exponent and a sign changing function some existence results are obtained by splitting the nehari manifold and by exploring some properties of the best hardy sobolev constant together with an approach developed by brezis and nirenberg

multiple positive solutions for indefinite semilinear elliptic problems involving a critical sobolev exponent in this paper we study the decomposition of the nehari manifold by exploiting the combination of concave and convex nonlinearities the result is subsequently used in conjunction with the ljusternik schnirelmann category and variational methods to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical sobolev exponent

variational approach to semilinear elliptic equation with nonlinear boundary abstract we consider the existence of positive solutions for a certain class of semilinear elliptic equation with nonlinear boundary our approach is based on the variational methods

the existence and multiplicity of classical positive solutions for a singular nonlinear elliptic problem with any growth exponents a b s t r a c t in this paper we consider a class of singular elliptic boundary problems u u p x u u 0 in u 0 on with 1 0 1 and p c 0 0 1 we emphasize that there is no restriction on the boundedness of from above by means of the approximating variational arguments and truncation two classical positive solutions are obtained 2014 the authors published by elsevier ltd this is an open access article under the cc by nc nd license http creativecommons org licenses by nc nd 3 0

rank constrained optimization and its applications abstract this paper investigates an iterative approach to solve the general rank constrained optimization problems rcops defined to optimize a convex objective function subject to a set of convex constraints and rank constraints on unknown rectangular matrices in addition rank minimization problems rmps are introduced and equivalently transformed into rcops by introducing a quadratic matrix equality constraint the rank function is discontinuous and nonconvex thus the general rcops are classified as np hard in most of the cases an iterative rank minimization irm method with convex formulation at each iteration is proposed to gradually approach the constrained rank the proposed irm method aims at solving rcops with rank inequalities constrained by upper or lower bounds as well as rank equality constraints proof of the convergence to a local minimizer with at least a sublinear convergence rate is provided four representative applications of rcops and rmps including system identification output feedback stabilization and structured h 2 controller design problems are presented with comparative simulation results to verify the feasibility and improved performance of the proposed irm method

radial solutions of dirichlet problems with concave convex nonlinearities abstract we prove the existence of a double infinite sequence of radial solutions for a dirichlet concave convex problem associated with an elliptic equation in a ball of r n we are interested in relaxing the classical positivity condition on the weights by allowing the weights to vanish the idea is to develop a topological method and to use the concept of rotation number the solutions are characterized by their nodal properties

polynomial time approximation schemes for max bisection on planar and geometric graphs the max bisection and min bisection problems are to find a partition of the vertices of a graph into two equal size subsets that respectively maximizes or minimizes the number of edges with endpoints in both subsets we design the first polynomial time approximation scheme for the max bisection problem on arbitrary planar graphs solving a long standing open problem the method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs in particular max and min bisection which could be of independent interest using a similar method we design also the first polynomial time approximation scheme for max bisection on unit disk graphs which could also be easily extended to other geometrically defined graphs

min cut clustering we describe a decomposition framework and a column generation scheme for solving a min cut clustering problem the subproblem to generate additional columns is itself an np hard mixed integer programming problem we discuss strong valid inequalities for the subproblem and describe some efficient solution strategies computational results on compiler construction problems are reported

infinitely many solutions for a class of sublinear schrodinger equations with indefinite potentials in this paper we are concerned with qualitative properties of entire solutions to a schrodinger equation with sublinear nonlinearity and sign changing potentials our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions

multiple solutions with compact support for a semilinear elliptic problem with critical growth abstract for a sign changing function a x we consider solutions of the following semilinear elliptic problem in r n with n 3 u a a u q u p u 0 and u d r n where 0 0 q 1 and p n 2 n 2 we show that all solutions are compactly supported if l i m i n f x a x 0 when x r n a x 0 has several connected components we prove that there exists an interval on in which two solutions exist and are positive in moreover one solution blows up as 0 a uniqueness result for solution with small l norm is also given

the nehari manifold for a class of concave convex elliptic systems involving the p laplacian and nonlinear boundary condition abstract the existence and multiplicity of weak solutions is established for a class of concave convex elliptic systems of the form p u m x u p 2 u a x u 2 u x p v m x v p 2 v b x v 2 v x u p 2 u n u 2 u v v p 2 v n u v 2 v x here p denotes the p laplacian operator defined by p z div z p 2 z p 2 r n is a bounded domain with smooth boundary 1 1 2 p p p p n n p if n p p if n p n is the outer normal derivative r 2 0 0 the weight m x is a positive bounded function and a x b x c are functions which change sign in our technical approach is based on the nehari manifold which is similar to the fibering method of drabek and pohozaev 1997 29 together with the recent idea from brown and wu 2008 10

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

an exact algorithm for graph partitioning an exact algorithm is presented for solving edge weighted graph partitioning problems the algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate it is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set the concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function hessian or by a diagonal shift obtained by solving a semidefinite programming problem numerical results show that the proposed algorithm is competitive with state of the art graph partitioning codes

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

existence of multiple solutions for an elliptic system with sign changing weight functions abstract in this paper we will consider a class of quasilinear elliptic problem of the form d i v x a p u p 2 u g 1 x u p 2 u h x u 2 u v h 1 x u n 2 u d i v x a p v p 2 v g 2 x v p 2 v h x v 2 v u h 2 x v n 2 v u x 0 v x 0 x r n where 0 1 p n 1 n p p n p n p d 0 a n p p a b a 1 d a 1 b 0 the weight g 1 x g 2 x are bounded and nonnegative functions and h x h 1 x h 2 x are continuous functions which change sign in r n we will prove that the problem has at least two positive solutions by using the nehari manifold and the fibering maps associated with the euler function for this problem

multiple positive solutions for p laplace elliptic equations involving concave convex nonlinearities and a hardy type term abstract in this paper the following problem is considered p u u p 2 u x p f x u q 2 u g x u p 2 u x u 0 x where r n is a bounded domain such that 0 1 q p 0 f and g are nonnegative functions n p p p is the best hardy constant and p n p n p is the critical sobolev exponent by extracting the palais smale sequence in the nehari manifold the existence of multiple positive solutions to this equation is verified

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

existence of multiple solutions for a singular quasilinear elliptic equation in rn in this paper we are concerned with the existence of multiple positive solutions for the singular quasilinear elliptic problem in r n div x a p u p email protected u g x u p 2u h x u m email protected x u n 2u email protected r nu x 0 email protected r n where l 0 is a real parameter and 1 3 10 the weight function g x is a bounded nonnegative function with email protected 0 and h x h x are continuous functions which change sign in r n we prove that there admits at least two positive solutions by using the nehari manifold and the fibrering maps associated with the euler functional for this problem

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

four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation with the help of the nehari manifold and the center mass function we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole

links between multiplicity automata observable operator models and predictive state representations a unified learning framework stochastic multiplicity automata sma are weighted finite automata that generalize probabilistic automata they have been used in the context of probabilistic grammatical inference observable operator models ooms are a generalization of hidden markov models which in turn are models for discrete valued stochastic processes and are used ubiquitously in the context of speech recognition and bio sequence modeling predictive state representations psrs extend ooms to stochastic input output systems and are employed in the context of agent modeling and planning we present sma ooms and psrs under the common framework of sequential systems which are an algebraic characterization of multiplicity automata and examine the precise relationships between them furthermore we establish a unified approach to learning such models from data many of the learning algorithms that have been proposed can be understood as variations of this basic learning scheme and several turn out to be closely related to each other or even equivalent