the anisotropic total variation flow we prove existence and uniqueness of solutions of the anisotropic total variation flow when the initial data is an l2 function and we give a characterization of such solutions that allows us to find explicit evolutions of sets in the presence of an anisotropy

existence and uniqueness for anisotropic and crystalline mean curvature flows an existence and uniqueness result up to fattening for crystalline mean curvature flows with forcing and arbitrary convex mobilities is proven this is achieved by introducing a new notion of solution to the corresponding level set formulation such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the approach accounts for the possible presence of a time dependent bounded forcing term with spatial lipschitz continuity as a by product of the analysis the problem of the convergence of the almgren taylor wang minimizing movements scheme to a unique up to fattening flat flow in the case of general possibly crystalline anisotropies is settled

convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature approaches based on the object oriented paradigm motion by weighted mean curvature is a geometric evolution law for surfaces and represents steepest descent with respect to anisotropic surface energy it has been proposed that this motion could be computed numerically by using a crystalline approximation to the surface energy in the evolution law in this paper we prove the convergence of this numerical method for the case of simple closed convex curves in the plane

stochastic evolution of 2d crystals we study a crystalline version of the modified stefan problem in the plane the feature of our approach is that we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon the existence of solution to our stochastic system is established galerkin s method is one of the main tools used in the proof of our assertion

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

inequalities and identities for generalized matrix functions abstract let be a character on the symmetric group s n and let a a ij be an n by n matrix the function d a s n n t 1 a t t is called a generalized matrix function if is an irreducible character then d is called an immanent for example if is the alternating character then d is the determinant and if 1 then d is called the permanent denoted per suppose that a is positive semidefinite hermitian we prove that the inequality 1 id d a per a holds for a variety of characters including the irreducible ones corresponding to the partitions n 1 1 and n 2 1 1 of n the main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents these expressions for the immanents come from analogous expressions for schur polynomials by means of a correspondence of d e littlewood

bethe ansatz for the gaudin model and its relation with knizhnik zamolodchikov equations we recall the construction of the common eigenvectors of gaudin hamiltonians based on the bethe ansatz in the case of an arbitrary lie algebra this construction can be done either recursively or explicitly and we prove the equivalence of the two methods we also prove that bethe vectors are singular only if the bethe equations are satisfied in each eigenspace of the spin operator we construct additional common eigenvectors having the same eigenvalue as the vacuum vector and which can not be obtained by the bethe ansatz these eigenvectors are not singular we also recall the connection between bethe vectors and integral solutions of the kz equation in an analogous way the additional vectors lead to solutions of kz equation which are not singular vectors and do not have an integral representation

existence and uniqueness for a crystalline mean curvature flow an existence and uniqueness result up to fattening for a class of crystalline mean curvature flows with natural mobility is proved the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the comparison principle is obtained by means of a suitable weak formulation of the flow while the existence of a global in time solution follows via a minimizing movement approach 2016 wiley periodicals inc

convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature v k alpha a finite difference scheme is constructed for the evolution of a plane curve driven by a power of curvature v k alpha alpha 0 where v and k are the normal velocity and the curvature respectively here we assume that the curve is closed strictly convex and immersed in the plane r2 this curve evolution is discretized using a crystalline approximation which is a kind of finite difference scheme for the nonlinear evolution equation we prove convergence of this scheme and show the rate of convergence moreover some numerical examples are presented

message passing algorithms for counting short cycles in a graph a message passing algorithm for counting short cycles in a graph is presented for bipartite graphs which are of particular interest in coding the algorithm is capable of counting cycles of length g g 2 2g 2 where g is the girth of the graph for a general non bipartite graph cycles of length g g 1 2g 1 can be counted the algorithm is based on performing integer additions and subtractions in the nodes of the graph and passing extrinsic messages to adjacent nodes the complexity of the proposed algorithm grows as o g e 2 where e is the number of edges in the graph for sparse graphs the proposed algorithm significantly outperforms the existing algorithms tailored for counting em short cycles in terms of computational complexity and memory requirements we also discuss a more generic and basic approach of counting short cycles which is based on matrix multiplication and provide a message passing interpretation for such an approach we then demonstrate that an efficient implementation of the matrix multiplication approach has essentially the same complexity as the proposed message passing algorithm

comparison principles for infinity laplace equations in finsler metrics abstract in this paper we study comparison principles for normalized finsler infinity laplace operators with nonhomogeneous terms that depend on solutions and their gradients this is achieved by using a combination of sup convolution methods to approximate viscosity solutions by semiconvex functions and finite difference approximation schemes characterizations of viscosity subsolutions and supersolutions of equations with constant nonhomogeneous terms through comparison with quadratic finsler cone are also discussed

decomposition of images by the anisotropic rudin osher fatemi model the total variation based image denoising model of rudin osher and fatemi can be generalized in a natural way to favor certain edge directions we consider the resulting anisotropic energies and study properties of their minimizers 2004 wiley periodicals inc

crystalline mean curvature flow of convex sets we prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in open image in new window this theorem can handle the facet breaking bending phenomena and can be generalized to any anisotropic mean curvature flow the method provides also a generalized geometric evolution starting from any compact convex set existing up to the extinction time satisfying a comparison principle and defining a continuous semigroup in time we prove that when the initial set is convex our evolution coincides with the flat curvature flow in the sense of almgren taylor wang as a by product it turns out that the flat curvature flow starting from a compact convex set is unique

generalized motion by nonlocal curvature in the plane this paper considers the level set equation for a general planar anisotropic curvature flow equation when the interfacial energy is very singular so that the anisotropic curvature effect is nonlocal a new notion of solutions is introduced to establish an analytic foundation of the level set method including a comparison principle and stability results the main idea behind the proofs is to convert the level sets of solutions into graph like functions this new procedure is called slicing and it is not limited to nonlocal curvature flow equations our theory is useful for establishing the convergence of a crystalline algorithm as well as for justifying the crystalline flow as a limit of anisotropic curvature flow with smooth interfacial energy

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a crystalline motion uniqueness and geometric properties the author studies a model of crystalline motion in the plane existence and uniqueness of local in time solutions are shown geometric properties of the flow are studied assuming that the wulff shape is a regular n sided polygon the author shows that a small convex grain of ice immersed in the cold melt shrinks provided that the flow does not overly deform the initial interface the author is able to verify the last condition only if the initial interface is a scaled wulff shape in this case the free boundary will remain a scaled wulff shape at later times this is shown by using an isoperimetric inequality

a crystalline motion of spiral shaped curves with symmetry we study the evolution of spiral shaped polygonal curves by crystalline curvature crystalline curvature is known to extend the notion of ordinary curvature to the special class of nonsmooth curves with the assumption of symmetry we show that the motion of our spirals can be analyzed up to the rest state possibly beyond singularities

facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term abstract we study crystalline driven curvature flow with spatially nonuniform driving force term we assume special monotonicity properties of the driving term which are motivated by our previous work on berg s effect we consider special initial data which we call bent rectangles we prove existence of solutions for a generic forcing term as well as generic subclass of bent rectangles we show the initially flat facets may begin to bend provided loosely speaking they are too large moreover depending on the initial configuration we notice instantaneous loss of regularity of the moving curve

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

evolution of 3 d crystals from supersaturated vapor with modified stefan condition galerkin method approach abstract we examine the evolution of crystals in three dimensions we assume that the wulff shape is a prism with a hexagonal base we include the gibbs thomson law on the crystal surface and the so called stefan condition we show local in time existence of solutions assuming that the initial crystal has admissible shape

bent rectangles as viscosity solutions over a circle abstract we study the motion of the so called bent rectangles by the singular weighted mean curvature we are interested in the curves which can be rendered as graphs over a smooth one dimensional reference manifold we establish a sufficient condition for that once we deal with graphs we can have the tools of the viscosity theory available like the comparison principle with its help we establish uniqueness of variational solutions constructed by the authors giga et al 2013 in addition we establish a criterion for the mobility coefficient guaranteeing vertex preservation

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

total variation and cheeger sets in gauss space abstract the aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the gauss space in both the finite and infinite dimensional case we first study the finite dimensional case proving the existence of a maximal cheeger set which is convex inside any bounded convex set we also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set then we extend these results in the context of the abstract wiener space and for that we study the total variation denoising problem in this context

an inverse model for a free boundary problem with a contact line steady case this paper reformulates the two phase solidification problem i e the stefan problem as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to pde constraints an advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase it is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated we describe an efficient iterative solution method for the stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional performance of the proposed approach is illustrated with sample computations concerning 2d steady solidification phenomena

convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature motion by weighted mean curvature is a geometric evolution law for surfaces representing steepest descent with respect to an isotropic surface energy it has been proposed that this motion could be computed by solving the analogous evolution law using a crystalline approximation to the surface energy we present the first convergence analysis for a numerical scheme of this type our treatment is restricted to one dimensional surfaces curves in the plane which are graphs in this context the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension

a comparison principle for singular diffusion equations with spatially inhomogeneous driving force for graphs we introduce the notions of viscosity super and subsolutions suitable for singular diffusion equations of non divergence type with a general spatially inhomogeneous driving term in particular the viscosity super and subsolutions support facets and allow a possible facet bending we prove a comparison principle by a modified doubling variables technique finally we present examples of viscosity solutions our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph like curve

curvature driven flows a variational approach this paper introduces a new mathematical approach to the study of time evolutions of solids k t in n space whose boundaries move with velocity equal to the weighted mean curvature derived from the boundary surface energy phi partial k t these flat phi curvature flows are limits of sequences of solutions to variational problems in which a sum of surface and bulk energy is minimized the construction works equally well for smooth elliptic phi s for nondifferentiable crystalline phi s and for anything in between the flows agree with classical smooth flows when the data is smooth and elliptic in any dimension and coincide with motion by crystalline curvature for polyhedral curves in the plane

evolving graphs by singular weighted curvature a new notion of solutions is introduced to study degenerate nonlinear parabolic equations in one space dimension whose diffusion effect is so strong at particular slopes of the unknowns that the equation is no longer a partial differential equation by extending the theory of viscosity solutions a comparison principle is established for periodic continuous initial data a unique global continuous solution periodic in space is constructed the theory applies to motion of interfacial curves by crystalline energy or more generally by anisotropic interfacial energy with corners when the curves are the graphs of functions even if the driving force term homogeneous in space exists the initial value problem is solvable for general nonadmissible continuous periodic initial data

the number of n cycles in a graph abstract recently chang and fu y c chang h l fu the number of 6 cycles in a graph bull inst combin appl 39 2003 27 30 derived an exact expression based on powers of the adjacency matrix for the number of 6 cycles in a graph here i demonstrate a method for obtaining the number of n cycles in a graph from the immanants of the adjacency matrix the method is applicable to cycles of all sizes and to sets of disjoint cycles of any sizes and the cycles in the set need not be the same size

on a crystalline variational problem part ii bv regularity and structure of minimizers on facets for a nonsmooth positively one homogeneous convex function n 0 it is possible to introduce the class n of smooth boundaries with respect to to define their mean curvature and to prove that for e n l e 9 based on these results we continue the analysis on the structure of e and on the regularity properties of we prove that a facet f of e is lipschitz up to negligible sets and that has bounded variation on f further properties of the jump set of are inspected in particular in three space dimensions we relate the sublevel sets of on f to the geometry of the wulff shape 1

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

singular and nonsingular eigenvectors for the gaudin model we present a method to construct a basis of singular and nonsingular common eigenvectors for gaudin hamiltonians in a tensor product module of the lie algebra sl 2 the subset of singular vectors is completely described by analogy with covariant differential operators the relation between singular eigenvectors and the bethe ansatz is discussed in each weight subspace the set of singular eigenvectors is completed to a basis by a family of nonsingular eigenvectors we discuss also the generalization of this method to the case of an arbitrary lie algebra

on a crystalline variational problem part i first variation and global l regularity let n 0 be a given positively one homogeneous convex function and let 1 pursuing our interest in motion by crystalline mean curvature in three dimensions we introduce and study the class n of smooth boundaries in the relative geometry induced by the ambient banach space n it can be seen that even when is a polytope n cannot be reduced to the class of polyhedral boundaries locally resembling curved portions must be necessarily included and this fact as well as the nonsmoothness of is the source of several technical difficulties related to the geometry of lipschitz manifolds given a boundary e in the class n we rigorously compute the first variation of the corresponding anisotropic perimeter which leads to a variational problem on vector fields defined on e it turns out that the minimizers have a uniquely determined intrinsic tangential divergence on e we define such a divergence to be the mean curvature of e the function is expected to be the initial velocity of e whenever e is considered as the initial datum for the corresponding anisotropic mean curvature flow we prove that is bounded on e and that its sublevel sets are characterized through a variational inequality

the crystalline dynamics of spiral shaped curves we study the motion of spiral shaped polygonal curves by crystalline curvature we describe this dynamics by the corresponding infinitely dimensional system of ordinary differential equations and show that the considered model is uniquely solvable banach s contraction mapping theorem and the bellman gronwall inequality are the main tools applied in our proof