on the performance of exponential integrators for problems in magnetohydrodynamics exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems while the results of these investigations showed that exponential integrators can provide significant computational savings the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages resistive magnetohydrodynamic mhd modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas in many problems numerical solution of mhd equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest in this paper we evaluate the performance of exponential integrators on large mhd problems and compare them to a state of the art implicit time integrator both the variable and constant time step exponential methods of epirk type are used to simulate magnetic reconnection and the kevin helmholtz instability in plasma performance of these methods which are part of the epic software package is compared to the variable time step variable order bdf scheme included in the cvode part of sundials library we study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as reynolds lundquist and prandtl numbers we find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for mhd problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as mhd

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

time evolution with the density matrix renormalization group algorithm a generic implementation for strongly correlated electronic systems a detailed description of the time step targeting time evolution method within the density matrix renormalization group algorithm is presented the focus of this publication is on the implementation of the algorithm and its generic application the case of one site excitations within a hubbard model is analyzed as a test for the algorithm using open chains and two leg ladder geometries the accuracy of the procedure in the case of the recently discussed holon doublon photo excitations of mott insulators is also analyzed performance and parallelization issues are discussed in addition the full open source code is provided as supplementary material

sharp upper bounds for a variational problem with singular perturbation let be a c2 bounded open set of mathbb r 2 and consider the functionals f epsilon u int limits omega left frac 1 nabla u x 2 2 epsilon epsilon d 2 u x 2 right rm d x we prove that if u in w 1 infty omega u 1 a e and u bv then gamma lim limits epsilon downarrow0 f epsilon u frac13 int limits j nabla u nabla u 3 rm d fancyscript h 1 the new result is the lim sup inequality

reduced order modeling for time dependent optimization problems with initial value controls this paper presents a new reduced order model rom hessian approximation for linear quadratic optimal control problems where the optimal control is the initial value such problems arise in parameter identification and data assimilation where the parameters to be identified appear in the initial data and also as subproblems in multiple shooting formulations of more general optimal control problems the new rom hessians can provide a substantially better approximation than the underlying basic rom approximation and thus can substantially reduce the computing time needed to solve these optimal control problems the computation of a hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the hessian is applied followed by the solution of the second order adjoint equation projection based roms of these two linear differential equations are used to generate the hessian approximation while the objective function and gradient are compute

on entire solutions of an elliptic system modeling phase separations we study the qualitative properties of a limiting elliptic system arising in phase separation for bose einstein condensates with multiple states

symplectic and multisymplectic numerical methods for maxwell s equations in this paper we compare the behaviour of one symplectic and three multisymplectic methods for maxwell s equations in a simple medium this is a system of pdes with symplectic and multisymplectic structures we give a theoretical discussion of how some numerical methods preserve the discrete versions of the local and global conservation laws and verify this behaviour in numerical experiments we also show that these numerical methods preserve the divergence furthermore we extend the discussion on dispersion for multi symplectic methods applied to pdes with one spatial dimension to include anisotropy when applying multi symplectic methods to maxwell s equations in two spatial dimensions lastly we demonstrate how varying the courant friedrichs lewy cfl number can cause the multi symplectic methods in our comparison to behave differently which can be explained by the study of backward error analysis for pdes

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

rate of convergence of local linearization schemes for initial value problems there is a large variety of local linearization ll schemes for the numerical integration of initial value problems which differ with respect to the algorithm that is used in the numerical implementation of the local linear discretization however in contrast with the ll discretization the order of convergence of the ll schemes have not been studied so far in this paper a general result about that matter is given in addition a brief survey of the main implementations of the ll method is also presented

analyzing the positivity preservation of numerical methods for the liouville von neumann equation abstract the density matrix is a widely used tool in quantum mechanics in order to determine its evolution with respect to time the liouville von neumann equation must be solved however analytic solutions of this differential equation exist only for simple cases additionally if the equation is coupled to maxwell s equations to model light matter interaction the resulting equation set the maxwell bloch or maxwell liouville von neumann mln equations becomes nonlinear in these advanced cases numerical methods are required since the density matrix has certain mathematical properties the numerical methods applied should be designed to preserve those properties we establish the criterion that only methods that have a completely positive trace preserving cptp update map can be used in long term simulations subsequently we assess the three most widely used methods the matrix exponential me method the runge kutta rk method and the predictor corrector pc approach whether they provide this feature and demonstrate that only the update step of the matrix exponential method is a cptp map

regularity of flat level sets in phase transitions we consider local minimizers of the ginzburg landau energy functional 1 2 u 2 1 4 1 u 2 2 dx and prove that if the 0 level set is included in a flat cylinder then in the interior it is included in a flatter cylinder as a consequence we prove a conjecture of de giorgi which states that level sets of global solutions of u u 3 u such that u 1 n u 0 lim u x x n 1 x n are hyperplanes in dimension n 8

on the heteroclinic connection problem for multi well gradient systems abstract we revisit the existence problem of heteroclinic connections in r n associated with hamiltonian systems involving potentials w r n r having several global minima under very mild assumptions on w we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells where length is computed with respect to a degenerate metric having conformal factor w then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times it gives rise to a heteroclinic connection between the two wells this work improves upon the approach of 22 and represents a more geometric alternative to the approaches of e g 5 10 14 17 for finding such connections

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

numerical methods for electronic structure calculations of materials the goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems the paper is intended for a diverse scientific computing audience for this reason we assume the reader does not have an extensive background in the related physics our overview focuses on the nature of the numerical problems to be solved their origin and the methods used to solve the resulting linear algebra or nonlinear optimization problems it is common knowledge that the behavior of matter at the nanoscale is in principle entirely determined by the schrodinger equation in practice this equation in its original form is not tractable successful but approximate versions of this equation which allow one to study nontrivial systems took about five or six decades to develop in particular the last two decades saw a flurry of activity in developing effective software one of the main practical variants of the schrodinger equation is based on what is referred to as density functional theory dft the combination of dft with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials this article will emphasize pseudopotential density functional theory but other techniques will be discussed as well

transversality for equivariant exact 1 forms and gauge theory on 3 manifolds an equivariant jet transversality framework is developed for the study of critical sets of invariant functions on g manifolds techniques are developed to extend transversality results to the infinite dimensional fredholm setting as an application the generic structure of the su 4 perturbed flat moduli space of an integral homology 3 sphere is described as well as the generic structure of the parameterized moduli space for a path of perturbations a similar analysis of the u 3 moduli space for rational homology 3 spheres is also carried out

dimensional reductions for the computation of time dependent quantum expectations we consider dimension reduction techniques for the liouville von neumann equation for the evaluation of the expectation values in a mixed quantum system we describe several existing methods that have appeared in the literature showing the failure of them when the system is scaled up we introduce a new method termed direct expectation values via chebyshev based on evaluation of a trace formula combined with a direct expansion in modified chebyshev polynomials this reduction is highly efficient and does not destroy any information we demonstrate the practical application of the scheme for a nuclear spin system and compare with popular alternatives in nuclear spin dynamics the main goal for simulations is being able to simulate a system with as many spins as possible for this reason it is very important to have an efficient method that scales the least with respect to the number of particles this method may be applied to autonomous quantum problems where the desired outcome of quantum simulation rather than being a full description of the system is only the expectation value of some observables

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

a compactness result in the gradient theory of phase transitions in the plane as 0 this functional favours jraj 1 and penalizes singularities where jrraj concentrates our main result is a compactness theorem if fe a g 0 is uniformly bounded then fra g 0 is compact in l thus in the limit 0 a solves the eikonal equation jraj 1 almost everywhere our analysis uses entropy relations and the div curl lemma adopting tartar s approach to the interaction of linear di erential equations and nonlinear algebraic relations

a γ convergence result for the two gradient theory of phase transitions the generalization to gradient vector fields of the classical double well singularly perturbed functionals i e u 1w u e 2 u 2 dx where w 0 if and only if a or b and a b is a rank 1 matrix is considered under suitable constitutive and growth hypotheses on w it is shown that i e converge to k h n 1 s u if u w 1 1 r d i u u bv a b otherwise where k is the constant interfacial energy per unit area

entire solutions of semilinear elliptic equations in r 3 and a conjecture of de giorgi this paper is concerned with the study of bounded solutions of semilinear elliptic equations u f u in the whole space r under the assumption that u is monotone in one direction say nu in r n the goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes this type of symmetry question was raised by de giorgi in who made the following conjecture we quote literally page of dg

lower bound for the energy of bloch walls in micromagnetics we study a two dimensional nonconvex and nonlocal energy in micromagnetics defined over s2 valued vector fields this energy depends on two small parameters and varepsilon penalizing the divergence of the vector field and its vertical component respectively our objective is to analyze the asymptotic regime beta ll varepsilon ll 1 through the method of convergence finite energy configurations tend to become divergence free and in plane in the magnetic sample except in some small regions of typical width varepsilon called bloch walls where the magnetization connects two directions on s2 we are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions for one dimensional transition layers we show by convergence analysis that the exact line density of the energy is quadratic in the jump size we expect the same behaviour for the two dimensional model in order to prove that we investigate the concept of entropies in the prototype case of a periodic strip we establish a quadratic lower bound for the energy with a non optimal constant then we introduce and study a special class of lipschitz entropies and obtain lower bounds coinciding with the one dimensional limit in some particular cases finally we show that entropies are not appropriate in general for proving the expected sharp lower bound

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

monotonicity and 1 dimensional symmetry for solutions of an elliptic system arising in bose einstein condensation we study monotonicity and 1 dimensional symmetry for positive solutions with algebraic growth of the following elliptic system left begin array ll delta u u upsilon 2 quad rm in mathbb r n delta upsilon u 2 upsilon quad rm in mathbb r n end array right for every dimension n geqq 2 in particular we prove a gibbons type conjecture proposed by berestycki et al

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

paraexp a parallel integrator for linear initial value problems a novel parallel algorithm for the integration of linear initial value problems is proposed this algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems an overlapping time domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems and a near optimal krylov method is used for the fast exponential integration of the homogeneous subproblems we present an error analysis and discuss the parallel scaling of our algorithm the efficiency of this approach is demonstrated with numerical examples

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

an efficient algorithm based on splitting for the time integration of the schrodinger equation we present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the schrodinger equation when the hamiltonian operator is either time independent or changes slowly with time in the later case the evolution operator can be effectively approximated in a step by step manner first divide the time integration interval in sufficiently short subintervals and then successively solve a schrodinger equation with a different time independent hamiltonian operator in each of these subintervals when discretized in space the schrodinger equation with the time independent hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous hamiltonian system corresponding to a system of coupled harmonic oscillators the particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals sharp local error bounds are obtained for the solution of the linear autonomous hamiltonian system considered in each time subinterval our schemes can be considered in this setting as polynomial approximations to the matrix exponential in a similar way as methods based on chebyshev and taylor polynomials the theoretical analysis supported by numerical experiments performed for different time independent hamiltonians indicates that the new methods are more efficient than schemes based on chebyshev polynomials for all tolerances and time interval lengths the algorithm we present automatically selects for each time subinterval the most efficient splitting scheme among several new optimized splitting methods for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the hamiltonian operator

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

asymmetric domain walls of small angle in soft ferromagnetic films we focus on a special type of domain wall appearing in the landau lifshitz theory for soft ferromagnetic films these domain walls are divergence free mathbb s 2 valued transition layers that connect two directions m theta pm in mathbb s 2 differing by an angle 2 theta and minimize the dirichlet energy our main result is the rigorous derivation of the asymptotic structure and energy of such asymmetric domain walls in the limit theta downarrow 0 as an application we deduce that a supercritical bifurcation causes the transition from symmetric to asymmetric walls in the full micromagnetic model

stopping criteria for rational matrix functions of hermitian and symmetric matrices building upon earlier work by golub meurant strakos and tich we derive new a posteriori error bounds for krylov subspace approximations to f a b the action of a function f of a real symmetric or complex hermitian matrix a on a vector b to this purpose we assume that a rational function in partial fraction expansion form is used to approximate f and the krylov subspace approximations are obtained as linear combinations of galerkin approximations to the individual terms in the partial fraction expansion our error estimates come at very low computational cost in certain important special situations they can be shown to actually be lower bounds of the error our numerical results include experiments with the matrix exponential as used in exponential integrators and with the matrix sign function as used in lattice quantum chromodynamics simulations and demonstrate the accuracy of the estimates the use of our error estimates within acceleration procedures is also discussed

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

compactness in ginzburg landau energy by kinetic averaging we consider a ginzburg landau energy for two dimensional divergence free fields appearing in the gradient theory of phase transition for instance we prove that as the relaxation parameter vanishes families of such fields with finite energy are compact in l p our proof is based on a kinetic interpretation of the entropies which were introduced by desimone kohn muller and otto the so called kinetic averaging lemmas allow to generalize their compactness results also the method yields a kinetic equation for the limit where the righthand side is an unknown kinetic defect bounded measurefrom which we deduce some sobolev regularity this measure also satisfies some cancellation properties depending on its local regularity which seem to indicate several level of singularities in the limit c 2001 john wiley sons inc

error analysis of krylov methods in a nutshell error and residual bounds for the matrix iteration methods bicg qmr fom and gmres are derived in a simple and unified way

the su 3 casson invariant for integral homology 3 spheres we derive a gauge theoretic invariant of integral homology 3 spheres which counts gauge orbits of irreducible perturbed flat su 3 connections with sign given by spectral flow to compensate for the dependence of this sum on perturbations the invariant includes contributions from the reducible perturbed flat orbits our formula for the correction term generalizes that given by walker in his extension of casson s su 2 invariant to rational homology 3 spheres

self consistent field calculations using chebyshev filtered subspace iteration the power of density functional theory is often limited by the high computational demand in solving an eigenvalue problem at each self consistent field scf iteration the method presented in this paper replaces the explicit eigenvalue calculations by an approximation of the wanted invariant subspace obtained with the help of well selected chebyshev polynomial filters in this approach only the initial scf iteration requires solving an eigenvalue problem in order to provide a good initial subspace in the remaining scf iterations no iterative eigensolvers are involved instead chebyshev polynomials are used to refine the subspace the subspace iteration at each step is easily five to ten times faster than solving a corresponding eigenproblem by the most efficient eigen algorithms moreover the subspace iteration reaches self consistency within roughly the same number of steps as an eigensolver based approach this results in a significantly faster scf iteration

linear estimation of continuous discrete linear state space models with multiplicative noise this paper deals with the estimation of the state variable of continuous discrete linear state space models with multiplicative noise specifically the optimal minimum variance linear filter for that class of models is constructed moreover the solutions of the differential equations that describe the evolution of the two first conditional moments between observations are obtained and an algorithm for their numerical computation is also given the performance of the linear filter is illustrated by means of numerical experiments

on phase separation models asymptotics and qualitative properties in this paper we study bound state solutions of a class of two component nonlinear elliptic systems with a large parameter tending to infinity the large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface to obtain the profile of bound state solutions near the interface we prove the uniform lipschitz continuity of bound state solutions when the spatial dimension is n 1 furthermore we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity these unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of du and zhang discontin dynam sys 2012 when the spatial dimension is n 2 we establish the de giorgi type conjecture for the blow up nonlinear elliptic system under suitable conditions at infinity on bound state solutions these results naturally lead us to formulate de giorgi type conjectures for these types of systems in higher dimensions

on the lavrentiev phenomenon for multiple integral scalar variational problems abstract we prove the non occurrence of lavrentiev gaps between lipschitz and sobolev functions for functionals of the form i u f u u u when r n r is lipschitz and belongs to a wide class of open bounded sets in r n containing lipschitz domains the lagrangian f is assumed to be either convex in both variables or a sum of functions f s a s g b s with g convex and s a s g 0 b s satisfying a non oscillatory condition at infinity we thus derive the non occurrence of the lavrentiev phenomenon for unnecessarily convex functionals of the gradient no growth conditions are assumed

on semiiterative methods generated by faber polynomials consider a nonsingular system of linear algebraic equationsa x b or in fixed point formx tx c where the eigenvalues oft are contained in some compact subset of the complex plane using faber polynomials iteration methods of the form y m mu m 0 left c ty m 1 right mu m 1 y m 1 ldots mu m k y m k m 0 m j 1 for allm 1 are constructed which include the chebyshev iterative method as well as the stationaryk step methods when is a rectangle we also develop four step methods which yield fast convergence in cases when the speed of convergence of the optimized chebyshev method is poor

on lower semicontinuity of a defect energy obtained by a singular limit of the ginzburg landau type energy for gradient fields a defect energy j which measures jump discontinuities of a unit length gradient field is studied the number indicates the power of the jumps of the gradient fields that appear in the density of j it is shown that j for 3 is lower semicontinuous on the space of unit gradient fields belonging to bv in l 1 convergence of gradient fields a similar result holds for the modified energy which measures only a particular type of defect the result turns out to be very subtle since with 3 is not lower semicontinuous as is shown in this paper the key idea behind semicontinuity is a duality representation for j 3 and the duality representation is also important for obtaining a lower bound by using j 3 for the relaxation limit of the ginzburg landau type energy for gradient fields the lower bound obtained here agrees with the conjectured value of the relaxation limit

on de giorgi s conjecture in dimension n 9 a celebrated conjecture due to de giorgi states that any bounded solution of the equation u 1 u 2 u 0 in r n with ynu 0 must be such that its level setsfu g are all hyperplanes at least for dimension n 8 a counterexample for n 9 has long been believed to exist starting from a minimal graph which is not a hyperplane found by bombieri de giorgi and giusti in r n n 9 we prove that for any small 0 there is a bounded solution u y with yn u 0 which resembles tanh a t p 2 a where t t y denotes a choice of signed distance to the blown up minimal graph 1 this solution is a counterexample to de giorgi s conjecture for n 9

one dimensional symmetry of bounded entire solutions of some elliptic equations this paper is about one dimensional symmetry properties for some entire and bounded solutions of u f u 0 in ir we consider solutions u such that 1 u 1 and u x1 xn 1 as xn uniformly with respect to x1 xn 1 under some conditions on f we prove that the solutions only depend on the variable xn we also discuss more general elliptic operators the qualitative properties then strongly depend on the coefficients of the operator these results extend to higher dimensions and to more general operators a result of ghoussoub and gui 21 proved for n 3 ams classification 35b05 35b40 35b50 35j60

graph spectral image smoothing using the heat kernel a new method for smoothing both gray scale and color images is presented that relies on the heat diffusion equation on a graph we represent the image pixel lattice using a weighted undirected graph the edge weights of the graph are determined by the gaussian weighted distances between local neighboring windows we then compute the associated laplacian matrix the degree matrix minus the adjacency matrix anisotropic diffusion across this weighted graph structure with time is captured by the heat equation and the solution i e the heat kernel is found by exponentiating the laplacian eigensystem with time image smoothing is accomplished by convolving the heat kernel with the image and its numerical implementation is realized by using the krylov subspace technique the method has the effect of smoothing within regions but does not blur region boundaries we also demonstrate the relationship between our method standard diffusion based pdes fourier domain signal processing and spectral clustering experiments and comparisons on standard images illustrate the effectiveness of the method