solving the 2 rooted mini max spanning forest problem by branch and bound the 2 rooted mini max spanning forest problem is to find a spanning forest with two given root nodes on an undirected graph such that the maximum tree cost in this forest is minimized we introduce a branch and bound algorithm based on selecting nodes on test instances up to 50 nodes the algorithm gives much better computational results than a known algorithm that is based on selecting edges furthermore the new algorithm can easily solve problem instances up to 80 nodes we consider some alternative and polynomial criteria finally we discuss some generalizations e g the problem without given root nodes i e the root nodes have to be chosen

existence and uniqueness of weak solutions to ginzburg landau equation with external noise and stochastic perturbation abstract in this paper the ginzburg landau and fractional ginzburg landau equations with external noise and stochastic perturbations are studied and we obtain the global existence and uniqueness of weak l 1 solutions our proof is supported by the generalized ito formula and a careful analysis of the nonlinear transport equation after founding the well posedness for the nonlinear transport problem in the class of l p 1 p we also gain the well posedness of the ginzburg landau and fractional ginzburg landau equations in the same spaces correspondingly

bounded solutions for an ordinary differential system from the ginzburg landau theory in this paper we look at a linear system of ordinary differential equations as derived from the two dimensional ginzburg landau equation in two cases it is known that this system admits bounded solutions coming from the invariance of the ginzburg landau equation by translations and rotations the specific contribution of our work is to prove that in the other cases the system does not admit any bounded solutions we show that this bounded solution problem is related to an eigenvalue problem ams classification 34b40 ordinary differential equations boundary value problems on infinite intervals 35j60 nonlinear pde of elliptic type 35p15 estimation of eigenvalues upper and lower bound

variational reduction for ginzburg landau vortices let be a bounded domain with smooth boundary in r2 we construct non constant solutions to the complex valued ginzburg landau equation e2 u 1 u 2 u 0 in as e 0 both under zero neumann and dirichlet boundary conditions we reduce the problem of finding solutions having isolated zeros vortices with degrees 1 to that of finding critical points of a small c1 perturbation of the associated renormalized energy this reduction yields general existence results for vortex solutions in particular for the neumann problem we find that if is not simply connected then for any k 1 a solution with exactly k vortices of degree one exists

submodular functions from discrete to continous domains submodular set functions have many applications in combinatorial optimization as they can be minimized and approximately maximized in polynomial time a key element in many of the algorithms and analyses is the possibility of extending the submodular set function to a convex function which opens up tools from convex optimization submodularity goes beyond set functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains where it corresponds essentially to cross second derivatives being nonpositive in this paper we show that most results relating submodularity and convexity for set functions can be extended to all submodular functions in particular a we naturally define a continuous extension in a set of probability measures b show that the extension is convex if and only if the original function is submodular c prove that the problem of minimizing a submodular function is equivalent to a typically non smooth convex optimization problem and d propose another convex optimization problem with better computational properties e g a smooth dual problem most of these extensions from the set function situation are obtained by drawing links with the theory of multi marginal optimal transport which provides also a new interpretation of existing results for set functions we then provide practical algorithms to minimize generic submodular functions on discrete domains with associated convergence rates

submodular function minimization submodular functions often arise in various fields of operations research including discrete optimization game theory queueing theory and information theory in this survey paper we give overview on the fundamental properties of submodular functions and recent algorithmic devolopments of their minimization

maximizing the smallest eigenvalue of a symmetric matrix a submodular optimization approach this paper studies the problem of selecting a submatrix of a positive definite matrix in order to achieve a desired bound on the smallest eigenvalue of the submatrix maximizing this smallest eigenvalue has applications to selecting input nodes in order to guarantee consensus of networks with negative edges as well as maximizing the convergence rate of distributed systems we develop a submodular optimization approach to maximizing the smallest eigenvalue by first proving that positivity of the eigenvalues of a submatrix can be characterized using the probability distribution of the quadratic form induced by the submatrix we then exploit that connection to prove that positive definiteness of a submatrix can be expressed as a constraint on a submodular function we prove that our approach results in polynomial time algorithms with provable bounds on the size of the submatrix we also present generalizations to non symmetric matrices alternative sufficient conditions for the smallest eigenvalue to exceed a desired bound that are valid for laplacian matrices and a numerical evaluation

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

numerical study of quantized vortex interaction in complex ginzburg landau equation on bounded domains in this paper we study numerically quantized vortex dynamics and their interaction in the two dimensional complex ginzburg landau equation cgle with a dimensionless parameter e 0 on bounded domains under either dirichlet or homogeneous neumann boundary condition we begin with a review of the reduced dynamical laws rdls for time evolution of quantized vortex centers in cgle and show how to solve these nonlinear ordinary differential equations numerically then we present efficient and accurate numerical methods for solving the cgle on either a rectangular or a disk domain under either dirichlet or homogeneous neumann boundary condition based on these efficient and accurate numerical methods for cgle and the rdls we explore rich and complicated quantized vortex dynamics and interaction of cgle with different e and under different initial physical setups including single vortex vortex pair vortex dipole and vortex lattice compare them with those obtained from the corresponding rdls and identify the cases where the rdls agree qualitatively and or quantitatively as well as fail to agree with those from cgle on vortex interaction finally we also obtain numerically different patterns of the steady states for quantized vortex lattices in the cgle dynamics on bounded domains

a branch and bound algorithm for the mini max spanning forest problem the mini max spanning forest problem requires to find a spanning forest of an undirected graph that minimizes the maximum of the costs of constituent trees in a previous work we proved this problem np hard in the current paper we present three lower bounds for this problem and develop a branch and bound algorithm to solve the problem exactly the algorithm is implemented and numerical experiments are conducted on a series of test problems the experiments compare the performances of the proposed bounds and search strategies in the algorithm as well as investigate the effects of instance characteristics on the behavior of the algorithm also extension of the problem to the case of more than two root vertices as well as to the problem of determining the root locations are discussed

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

optimality cuts and a branch and cut algorithm for the k rooted mini max spanning forest problem abstract let g v e be an undirected graph with costs associated with its edges and k pre specified root vertices the k rooted mini max spanning forest problem asks for a spanning forest of g defined by exactly k mutually disjoint trees each tree must contain a different root vertex and the cost of the most expensive tree must be minimum this paper introduces a branch and cut algorithm for the problem it involves a multi start linear programming heuristic and the separation of some new optimality cuts extensive computational tests indicate that the new algorithm significantly improves on the results available in the literature improvements being reflected by lower cpu times smaller enumeration trees and optimality certificates for previously unattainable k 2 instances with as many as 200 vertices furthermore for the first time instances of the problem with k 3 4 are solved to proven optimality

revisiting the two dimensional defect free azimuthal nematic equilibrium on an annulus we study the azimuthal defect free nematic state on a two dimensional annulus within a simplified and reduced two dimensional landau de gennes model for nematic liquid crystals we perform a detailed asymptotic analysis of the instabilities of the defect free state in terms of a dimensionless material and temperature dependent variable and the annular aspect ratio the asymptotic analysis is accompanied by a rigorous local stability result again in terms of a dimensionless material and temperature dependent parameter and annular aspect ratio in contrast to oseen frank predictions the defect free state can be unstable in this model with elastic isotropy and strong anchoring for a range of macroscopically relevant annular aspect ratios

submodular spectral functions of principal submatrices of a hermitian matrix extensions and applications we extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions we show that if f is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix a then the function i mapsto rm tr f a i is supermodular meaning that rm tr f a i rm tr f a j leq rm tr f a i cup j rm tr f a i cap j where a i denotes the i times i principal submatrix of a we discuss extensions to self adjoint operators on infinite dimensional hilbert space and to m matrices we discuss an application to cur approximation of nonnegative hermitian matrices

upper and lower bounding procedures for minimum rooted k subtree problem abstract given an undirected graph and a fixed root node on it the minimum rooted k subtree problem is the problem of finding a minimum cost connected subtree with exactly k edges that includes the root node applications of this problem appear in several fields the problem is proved to be np hard due to the existence of a root node we are able to derive a greedy procedure that gives a stronger lower bound to this problem than those in earlier studies heuristics are also discussed to obtain an upper bound computational results indicate that the proposed lower bounding procedure is effective when k is small especially for grid type graphs we observe that the upper and lower bounds frequently coincide with each other

radial minimizer of p ginzburg landau functional with nonvanishing dirichlet boundary condition abstract the author studies the asymptotic behavior of the radial minimizer of the p ginzburg landau functional with nonvanishing dirichlet boundary condition in the case p 2 the location of the zeros of the radial minimizer is discussed moreover the w loc 1 p convergence of the radial minimizer is proved

stability of symmetric vortices for two component ginzburg landau systems abstract we study ginzburg landau equations for a complex vector order parameter c 2 we consider the dirichlet problem in the disk in r 2 with a symmetric degree one boundary condition and study its stability in the sense of the spectrum of the second variation of the energy we find that the stability of the degree one equivariant solution depends on the ginzburg landau parameter as well as the sign of the interaction term in the energy

radial minimizers of p ginzburg landau type with p n 1 n abstract the author studies the asymptotic behavior of the radial minimizer of the p ginzburg landau functional with non vanishing dirichlet boundary condition in the case of p n 1 n where n is the dimension the location of the zeros of the radial minimizer is discussed moreover the c 1 convergence of the radial minimizer is proved

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

norm inequalities related to operator monotone functions let a b be positive semidefinite matrices and cdot any unitarily invariant norm on the space of matrices we show f a f b geq f a b for any non negative operator monotone function f t on 0 infty and g a g b leq g a b for non negative increasing function g t on 0 infty with g 0 0 and g infty infty whose inverse function is operator monotone

some remarks on the linearized operator about the radial solution for the ginzburg landau equation abstract we consider the linearized operators denoted l d 1 of the ginzburg landau operator u u 1 u 2 in r2 about the radial solutions ud 1 x fd r eid for all d 1 we state the correspondence between the real vector space of the bounded solutions of the equation l d 1 w 0 and the eigenvalues of the linearized operators of the equations u 1 e2u 1 u 2 0 in b 0 1 about the radial solutions ud e x fd r e eid that tend to 0 as e tends to 0

on the dynamical law of the ginzburg landau vortices on the plane we study the ginzburg landau equation on the plane with initial data being the product of n well separated 1 vortices and spatially decaying perturbations if the separation distances are o e 1 e 1 we prove that the n vortices do not move on the time scale o e 2 e e o log 1 e instead they move on the time scale o e 2 log 1 according to the law x j x jw w l6 j logjxl x jj x j j j 2 r 2 the location of the j th vortex the main ingredients of our proof consist of estimating the large space behavior of solutions a monotonicity inequality for the energy density of solutions and energy comparisons combining these we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law c 1999 john wiley sons inc

an exact algorithm for the knapsack sharing problem with common items we are concerned with a variation of the knapsack problem as well as of the knapsack sharing problem where we are given a set of n items and a knapsack of a fixed capacity as usual each item is associated with its profit and weight and the problem is to determine the subset of items to be packed into the knapsack however in the problem there are s players and the items are divided into s 1 disjoint groups nk k 0 1 s the player k is concerned only with the items in n0 nk where n0 is the set of common items while nk represents the set of his own items the problem is to maximize the minimum of the profits of all the players an algorithm is developed to solve this problem to optimality and through a series of computational experiments we evaluate the performance of the developed algorithm 2004 elsevier b v all rights reserved

a heuristic algorithm for the mini max spanning forest problem abstract in this paper we formulate the mini max spanning forest problem for undirected graphs as a generalization of the minimum spanning tree problem we prove that the former problem is np hard then we develop a heuristic algorithm to solve this problem approximately and give an upper bound for relative errors through a series of numerical tests we examine the performance of the developed algorithm

radial solutions for the ginzburg landau equation with applied magnetic field abstract the purpose of this work is a systematic study of symmetric vortices for the ginzburg landau model of superconductivity along a cylinder with applied magnetic field parallel to its axis the ginzburg landau constant of the material and the degree d of the vortex are fixed for any given parameters r the radius of the cylinder and h the intensity of the applied magnetic field one can find a symmetric vortex a which satisfies the boundary conditions b r e i d and curl a b r he 3 it is then shown that symmetric vortices form a family depending continuously on two real parameters and c which describe the behaviour at the center of the vortex as the boundary conditions depend smoothly on those parameters one can distinguish two main connected domains of vortices the first one defined by the boundary conditions b r e i d and ra r b r is a zone of stability where remains increasing the second one defined by the boundary conditions b r e i d and ra r b r d is a zone where some instability can occur attention is focused on the smooth branch of vortices which separates those two domains it is indexed by the parameter running from some l 0 to and the limit as decreases to l corresponds to a berger and chen type vortex

stability of axisymmetric chiral skyrmions abstract we examine topological solitons in a minimal variational model for a chiral magnet so called chiral skyrmions in the regime of large background fields we prove linear stability of axisymmetric chiral skyrmions under arbitrary perturbations in the energy space a long standing open question in physics literature moreover we show strict local minimality of axisymmetric chiral skyrmions and nearby existence of moving soliton solutions for the landau lifshitz gilbert equation driven by small spin transfer torques

uniqueness of symmetric vortex solutions in the ginzburg landau model of superconductivity abstract symmetric vortices are finite energy solutions a to the ginzburg landau equations of superconductivity with the form f r e id a s r r 2 y x the existence regularity and asymptotic form of the solutions f r s r for any d z 0 have been established by plohr and by burger and chen in this paper we prove the uniqueness of these solutions when the ginzburg landau parameter satisfies 2 2 d 2 for any fixed d z 0 to do this we show that any such solution is a non degenerate relative minimizer of the free energy functional constrained to a convex set then use a version of the mountain pass theorem to derive a contradiction should there be more than one solution

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

asymptotic analysis for the radial minimizer of a second order energy functional the convergence for the radial minimizers of a second order energy functional when the parameter tends to 0 is studied and the location of the zeros of the radial minimizers of this functional is presented based on this result the uniqueness of the radial minimizer is discussed

the hasegawa petz mean properties and inequalities abstract we study a family of means introduced by h hasegawa and d petz 1996 13 properties with respect to the parameter such as monotonicity and logarithmic concavity further monotonicity and concavity in the mean variables are shown besides the comparison between the hasegawa petz mean and the geometric mean is completely solved the connection to earlier results on operator monotonicity and some applications are also discussed

vortex structure in p wave superconductors we study vortices in p wave superconductors in a ginzburg landau setting the state of the superconductor is described by a pair of complex wave functions and the p wave symmetric energy functional couples these in both the kinetic gradient and potential energy terms giving rise to systems of partial differential equations which are nonlinear and coupled in their second derivative terms we prove the existence of energy minimizing solutions in bounded domains 2 and consider the existence and qualitative properties such as the asymptotic behavior of equivariant solutions defined in all of 2 the coupling of the equations at highest order changes the nature of the solutions and many of the usual properties of classical ginzburg landau vortices either do not hold for the p wave solutions or are not immediately evident