generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

iterative methods for large convex quadratic programs a survey in this paper we give a state of the art review of many iterative methods for solving large convex quadratic programs we attempt to classify several of the more basic methods in two categories within each of which a unified iterative scheme will be introduced and its convergence analyzed hybrid iterative methods such as the proximal point algorithm and a diagonalization scheme that make use of the more basic schemes will also be described the results of an extensive computer experimentation which is aimed at comparing the relative performance of the various methods will be reported and discussed finally several important topics which require future research will be highlighted

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

convergence rates in forward backward splitting forward backward splitting methods provide a range of approaches to solving large scale optimization problems and variational inequalities in which structures conducive to decomposition can be utilized apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out efforts at evaluating the convergence potential of such methods have so far relied on lipschitz properties and strong monotonicity or inverse strong monotonicity of the mapping involved in the forward step the perspective mainly being that of projection algorithms here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well for the first time in such a general setting global and local contraction rates are derived moreover they are derived in a form which makes it possible to determine the optimal step size relative to certain constants associated with the given problem insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected

existence and algorithm of solutions for mixed quasi variational like inclusions in banach spaces in this paper we study a class of mixed quasi variational like inclusion with a bifunction in reflexive banach spaces by applying a minimax inequality due to the first author some existence and uniqueness theorems of solutions for the mixed quasi variational like inclusions are proved next by applying the auxiliary variational inequality technique we suggest an iterative algorithm to compute the approximate solutions of the mixed quasi variational like inclusions the convergence criteria of the algorithm is studied under quite mild conditions our results represent improvement and generalization of the previously known results in literature

the extension problem of the mean curvature flow i we show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in mathbb r 3

uniqueness of blowups and łojasiewicz inequalities once one knows that singularities occur one naturally wonders what the singu larities are like for minimal varieties the first answer already known to federer fleming in 1959 is that they weakly resemble cones 1 for mean curvature flow by the combined work of huisken ilmanen and white singularities weakly resemble shrinkers unfortunately the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup but under higher magnification it might as far as anyone knows resemble a completely different blowup whether this ever happens is perhaps the most fundamental question about singularities it is this long stand ing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities

uniqueness of convex ancient solutions to mean curvature flow in mathbb r 3 r 3 a well known question of perelman concerns the classification of noncompact ancient solutions to the ricci flow in dimension 3 which have positive sectional curvature and are kappa noncollapsed in this paper we solve the analogous problem for mean curvature flow in mathbb r 3 and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in mathbb r 3 which is strictly convex and noncollapsed

asymmetric variational inequality problems over product sets applications and iterative methods in this paper we i describe how several equilibrium problems can be uniformly modelled by a finite dimensional asymmetric variational inequality defined over a cartesian product of sets and ii investigate the local and global convergence of various iterative methods for solving such a variational inequality problem because of the special cartesian product structure these iterative methods decompose the original variational inequality problem into a sequence of simpler variational inequality subproblems in lower dimensions the resulting decomposition schemes often have a natural interpretation as some adjustment processes

splitting methods for pseudomonotone mixed variational inequalities we consider and analyze some new splitting methods for solving monotone mixed variational inequalities by using the technique of updating the solution the modified methods converge for monotone and pseudomonotone continuous operators the new splitting methods differ from the existing splitting methods the new results are versatile and proof of convergence is very simple

a parallel algorithm for a class of convex programs a parallel algorithm is proposed in this paper for solving the problem min q x x in c 1 cap cdots cap c m where q is an uniformly convex function and c i are closed convex sets in r n in each iteration of the method we solve in parallel m independent subproblems each minimizing a definite quadratic function over an individual set c i the method has attractive convergence properties and can be implemented as parallel algorithms for tackling definite quadratic programs linear programs systems of linear equations and systems of generalized nonlinear inequalities

averaging schemes for variational inequalities and systems of equations we study averaging methods for solving variational inequalities whose underlying maps are nonexpansive and for solving systems of asymmetric equations our goal is to establish global convergence results using weaker assumptions than are traditional in the literature we examine averaging schemes for relaxation algorithms and for their specialization as projection and linearization methods and as cohen s auxiliary framework for solving systems of equations we consider averaging for a general class of methods that includes as a special case a generalized steepest descent method we also develop a new interpretation of a norm condition typically used for establishing convergence of relaxation schemes by associating it with a strong f monotonicity condition

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

a successive projection method it is of both theoretical and practical interest to find the projection of a point to the intersection of a finite number of closed convex sets by a sequence of projections to the individual sets successively in this paper we study such a method and analyze its convergence properties a main feature of the method is its capability to decompose the projection problem into several small ones for some structured sparse problems these small subproblems can be solved independently and the presented method has a potential use in parallel computation

the primal douglas rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem we apply the douglas rachford splitting algorithm to a class of multi valued equations consisting of the sum of two monotone mappings compared with the dual application of the same algorithm which is known as the alternating direction method of multipliers the primal application yields algorithms that seem somewhat involved however the resulting algorithms may be applied effectively to problems with certain special structure in particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

fast alternating linearization methods for minimizing the sum of two convex functions we present in this paper alternating linearization algorithms based on an alternating direction augmented lagrangian approach for minimizing the sum of two convex functions our basic methods require at most o 1 epsilon iterations to obtain an epsilon optimal solution while our accelerated i e fast versions of them require at most o 1 sqrt epsilon iterations with little change in the computational effort required at each iteration for both types of methods we present one algorithm that requires both functions to be smooth with lipschitz continuous gradients and one algorithm that needs only one of the functions to be so algorithms in this paper are gauss seidel type methods in contrast to the ones proposed by goldfarb and ma in fast multiple splitting algorithms for convex optimization columbia university 2009 where the algorithms are jacobi type methods numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms

further applications of a splitting algorithm to decomposition in variational inequalities and convex programming a classical method for solving the variational inequality problem is the projection algorithm we show that existing convergence results for this algorithm follow from one given by gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators moreover we extend the projection algorithm to solveany monotone affine variational inequality problem when applied to linear complementarity problems we obtain a matrix splitting algorithm that is simple and for linear quadratic programs massively parallelizable unlike existing matrix splitting algorithms this algorithm converges under no additional assumption on the problem when applied to generalized linear quadratic programs we obtain a decomposition method that unlike existing decomposition methods can simultaneously dualize the linear constraints and diagonalize the cost function this method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets

a unifying geometric solution framework and complexity analysis for variational inequalities in this paper we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations to establish this result we build upon insights from several algorithms for linear and nonlinear programs the ellipsoid algorithm the method of centers of gravity the method of inscribed ellipsoids and vaidya s algorithm to develop a unifying geometric framework for solving variational inequality problems the analysis rests upon the assumption of strong f monotonicity which is weaker than strict and strong monotonicity since linear programs satisfy this assumption the general framework applies to linear programs

an iterative scheme for variational inequalities in this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities this iterative scheme not only contains as special cases the projection linear approximation and relaxation methods but also induces new algorithms then we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in en induced by symmetric positive definite matrices gm thus in contrast to the above mentioned methods this technique allows the possibility of adjusting the norm at each step of the algorithm this flexibility will generally yield convergence under weaker assumptions

minimal 2 spheres in 3 spheres we prove that any manifold diffeomorphic to s3 and endowed with a generic metric contains at least two embedded minimal 2 spheres the existence of at least one minimal 2 sphere was obtained by simon and smith in 1983 our approach combines ideas from min max theory and mean curvature flow we also establish the existence of smooth mean convex foliations in 3 manifolds we apply our methods to solve a problem posed by s t yau in 1987 on whether the planar 2 spheres are the only minimal spheres in ellipsoids centered about the origin in r4 finally considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity 1 conjecture and the equidistribution of widths conjecture are in a certain sense sharp

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

computational models of mri characteristics of focal cortical dysplasia improve lesion detection abstract in many patients focal cortical dysplasia fcd is characterized by minor structural changes that may go unrecognized by standard radiological analysis we previously demonstrated that visual analysis of a composite map based on three simple models of mri features of fcd increased the sensitivity of fcd lesion detection compared to visual analysis of conventional mri here we report on the use of improved methods for characterizing fcd which improve contrast in the composite maps a laplacian based metric for measuring cortical thickness a convolutional kernel to model blurring of the gm wm interface and an operator to measure hyperintense t1 signal to validate these methods we processed the mris of 14 fcd patients with our original set of image processing operators and an improved set of image processing operators comparison of the composite maps associated with the two sets of operators revealed that contrast between lesional tissue and nonlesional cortex was significantly increased in the composite maps associated with the set of improved operators increasing this contrast is an important step toward the goal of automated fcd lesion detection

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

the nature of singularities in mean curvature flow of mean convex sets let k be a compact subset of r or more generally of an n 1 dimensional riemannian manifold we suppose that k is mean convex if the boundary of k is smooth and connected this means that the mean curvature of k is everywhere nonnegative with respect to the inward unit normal and is not identically 0 more generally it means that ft k is contained in the interior of k for t 0 where ft k is the set obtained by letting k evolve for time t under the level set mean curvature flow as k evolves it traces out a closed set k of spacetime k x t r r x ft k also there is associated to k a brakke flow m t 7 mt of rectifiable varifolds we call the pair m k a mean convex flow let x x t be a point in spacetime with t 0 suppose xi ti is a sequence of points converging to x and i is a sequence of positive numbers tending to infinity translate the pair m and k in spacetime by y 7 y xi ti and then dilate parabolically by y 7 iy i to get new flows mi and ki the sequence mi ki is called a blow up sequence at x general compactness theorems guarantee that this sequence will have subsequential limits a subsequential limit m k is called a limit flow here m t 7 m t

piecewise linear approximation methods for nonseparable convex optimization an algorithm is described for the solution of nonseparable convex optimization problems this method utilizes iterative piecewise linear approximation of the nonseparable objective function but requires function values only along a translated set of axes thereby avoiding the curse of dimensionality commonly associated with grid methods for multi dimensional problems a global convergence proof is given under the assumptions that the objective function is lipschitz continuous and differentiable and that the feasible set is convex and compact the method is well suited to linearly constrained large scale optimization since the direction finding problems reduce to linear programs of manageable size it is particularly appropriate for nonlinear networks since it preserves the network structure of the constraints in addition because the resulting objective function approximation is separable this approach permits for certain problem classes a decomposition that may be exploited for parallel computation some numerical results on the crystal multicomputer are presented to illustrate this decomposition feature

asymptotic convergence analysis of the forward backward splitting algorithm the asymptotic convergence of the forward backward splitting algorithm for solving equations of type 0 t z is analyzed where t is a multivalued maximal monotone operator in the n dimensional euclidean space when the problem has a nonempty solution set and t is split in the form t h with being maximal monotone and h being co coercive with modulus greater than convergence rates are shown under mild conditions to be linear superlinear or sublinear depending on how rapidly 1 and h 1 grow in the neighborhoods of certain specific points as a special case when both and h are polyhedral functions we get r linear convergence and 2 step q linear convergence without any further assumptions on the strict monotonicity on t or on the uniqueness of the solution as another special case when h 0 the splitting algorithm reduces to the proximal point algorithm and we get new results which complement r t rockafellar s and f j luque s earlier results on the proximal point algorithm

weak co coercivity and its applications in several algorithms for solving variational inequalities abstract in this paper we investigate several algorithms for solving variational inequalities vis and the zero inclusion problem respectively it is shown that invested algorithms are applicable in broader area in addition for the purpose of practical implementation we further modify one of the algorithms discussed by adopting an armijo like search

asymptotic behavior of flows by powers of the gaussian curvature we consider a 1 parameter family of strictly convex hypersurfaces in mathbb r n 1 moving with speed k alpha where denotes the outward pointing unit normal vector and alpha geqslant 1 n 2 for alpha gt 1 n 2 we show that the flow converges to a round sphere after rescaling in the affine invariant case alpha 1 n 2 our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling

mean curvature flow with surgery of mean convex surfaces in mathbb r 3 we define a notion of mean curvature flow with surgery for two dimensional surfaces in mathbb r 3 with positive mean curvature our construction relies on the earlier work of huisken and sinestrari in the higher dimensional case one of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author invent math 2015

a sharp bound for the inscribed radius under mean curvature flow we consider a family of embedded mean convex hypersurfaces which evolve by the mean curvature flow it follows from general results of white that the inscribed radius at each point on the hypersurface is at least frac c h where c is a positive constant that depends only on the initial data andrews recently gave a new proof of that fact using the maximum principle in this paper we show that the inscribed radius is at least frac 1 1 delta h at each point where the curvature is sufficiently large

existence and algorithm of solutions for nonlinear mixed variational like inequalities in banach spaces in this paper we study a new class of nonlinear mixed variational like inequalities in reflexive banach spaces by applying a minimax inequality due to author we first prove an existence uniqueness theorem of solutions for the nonlinear mixed variational like inequalities next we extend the auxiliary problem technique under reflexive banach space setting to suggest and analyze an innovative iterative algorithm to compute the approximate solutions of the nonlinear mixed variational like inequalities the convergence criteria of the iterative algorithm is also given these results improve and generalize many known results in recent literature

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

conditions ensuring the applicability of cutting plane methods for solving variational inequalities let vip f c denote the variational inequality problem associated with the mapping f and the closed convex set c in this paper we introduce weak conditions on the mapping f that allow the development of a convergent cutting plane framework for solving vip f c in the process we introduce in a natural way new and useful notions of generalized monotonicity for which first order characterizations are presented

a monotone skew splitting model for composite monotone inclusions in duality the principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew adjoint operator an algorithmic framework is developed for solving this generic problem in a hilbert space setting new primal dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators and convergence results are established these algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration comparisons with existing methods are made and applications to composite variational problems are demonstrated

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

segmentation of focal cortical dysplasia lesions on mri using level set evolution focal cortical dysplasia fcd is the most frequent malformation of cortical development in patients with medically intractable epilepsy on mri fcd lesions are not easily differentiable from the normal cortex and defining their spatial extent is challenging in this paper we introduce a method to segment fcd lesions on t1 weighted mri it relies on two successive three dimensional deformable models whose evolutions are based on the level set framework the first deformable model is driven by probability maps obtained from three mri features cortical thickness relative intensity and gradient these features correspond to the visual characteristics of fcd and allow discriminating lesions and normal tissues in a second stage the previous result is expanded towards the underlying and overlying cortical boundaries throughout the whole cortical section the method was quantitatively evaluated by comparison with manually traced labels in 18 patients with fcd the automated segmentations achieved a strong agreement with the manuals labels demonstrating the applicability of the method to assist the delineation of fcd lesions on mri this new approach may become a useful tool for the presurgical evaluation of patients with intractable epilepsy related to cortical dysplasia

general iterative algorithms for solving mixed quasi variational like inclusions in this paper we extend the auxiliary variational inequality technique due to ding and yao x p ding j c yao existence and algorithm of solutions for mixed quasi variational like inclusions in banach spaces comput math appl 49 2005 857 869 to develop iterative algorithms for finding the approximate solutions of a mixed quasi variational like inclusion problem in short mqvlip in the setting of banach spaces we first establish a result on the existence of a solution of the equilibrium problem by virtue of the fan kkm lemma then by using this result and a result by ding and tan x p ding k k tan a minimax inequality with applications to existence of equilibrium point and fixed point theorems colloq math 63 2 1992 233 247 we derive the existence of a unique solution of mqvlip and the existence of approximate solutions generated by the proposed algorithms moreover we also provide the new criteria for convergence of approximate solutions to the exact solution of mqvlip

groupwise retargeted least squares regression in this brief we propose a new groupwise retargeted least squares regression grelsr model for multicategory classification the main motivation behind grelsr is to utilize an additional regularization to restrict the translation values of relsr so that they should be similar within same class by analyzing the regression targets of relsr we propose a new formulation of relsr where the translation values are expressed explicitly on the basis of the new formulation discriminative least squares regression can be regarded as a special case of relsr with zero translation values moreover a groupwise constraint is added to relsr to form the new grelsr model extensive experiments on various machine leaning data sets illustrate that our method outperforms the current state of the art approaches

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

mean curvature flow of mean convex hypersurfaces in the last 15 years white and huisken sinestrari developed a far reaching structure theory for the mean curvature flow of mean convex hypersurfaces their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow in the present paper we give a new treatment of the theory of mean convex and k convex flows this includes 1 an estimate for derivatives of curvatures 2 a convexity estimate 3 a cylindrical estimate 4 a global convergence theorem 5 a structure theorem for ancient solutions and 6 a partial regularity theorem our new proofs are both more elementary and substantially shorter than the original arguments our estimates are local and universal a key ingredient in our new approach is the new noncollapsing result of andrews 2 some parts are also inspired by the work of perelman 32 33 in a forthcoming paper 17 we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper note added in may 2015 since the first version of this paper was posted on arxiv in april 2013 the estimates have been used to construct mean convex flow with surgery in 3 by brendle and huisken 5 in september 2013 and in another paper by the authors in april 2014 2016 wiley periodicals inc

a monotonicity formula for mean curvature flow with surgery abstract we prove a monotonicity formula for mean curvature flow with surgery this formula differs from huisken s monotonicity formula by an extra term involving the mean curvature as a consequence we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is in fact free of surgeries this result is used in the analysis of mean curvature flow with surgery in riemannian three manifolds cf 5

solving variational inequality and fixed point problems by line searches and potential optimization we introduce a general adaptive line search framework for solving fixed point and variational inequality problems our goals are to develop iterative schemes that i compute solutions when the underlying map satisfies properties weaker than contractiveness for example weaker forms of nonexpansiveness ii are more efficient than the classical methods even when the underlying map is contractive and iii unify and extend several convergence results from the fixed point and variational inequality literatures to achieve these goals we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions as a special case we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature the square of the jacobian matrix is positive definite since the line searches we propose might be difficult to perform exactly we also consider inexact line searches

alternating projection proximal methods for convex programming and variational inequalities we consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem we propose a solution method for this problem that alternates between a proximal step for the maximal monotone operator part and a projection type step for the monotone variational inequality part and analyze its convergence and rate of convergence this method extends a decomposition method of chen and teboulle math programming 64 1994 pp 81 101 for convex programming and yields as a by product new decomposition methods

inverse variational inequalities with projection based solution methods abstract an inverse variational inequality is defined as to find a vector u r n such that f u v f u t u 0 v if an inverse function u f 1 x exists the above inverse variational inequality could be transformed as a regular variational inequality however in reality it is not uncommon that the inverse function of f 1 x does not have explicit form although its functional values can be observed existing line search algorithms cannot be applied directly to solve such inverse variational inequalities in this paper we propose two projection based methods using the co coercivity of mapping f a self adaptive strategy is developed to determine the step sizes efficiently when the co coercivity modulus is unknown the convergence of the proposed methods is proved rigorously

mean curvature flow with surgeries of two convex hypersurfaces we consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow it is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded the purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure compared with the notions of weak solutions existing in the literature the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold our construction is inspired by the procedure originally introduced by hamilton for the ricci flow and then employed by perelman in the proof of thurston s geometrization conjecture in this paper we consider initial hypersurfaces which have dimension at least three and are two convex that is such that the sum of the two smallest principal curvatures is nonnegative everywhere under these assumptions we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology as a corollary we obtain a classification up to diffeomorphism of the hypersurfaces under consideration

magnetic resonance image tissue classification using a partial volume model abstract we describe a sequence of low level operations to isolate and classify brain tissue within t1 weighted magnetic resonance images mri our method first removes nonbrain tissue using a combination of anisotropic diffusion filtering edge detection and mathematical morphology we compensate for image nonuniformities due to magnetic field inhomogeneities by fitting a tricubic b spline gain field to local estimates of the image nonuniformity spaced throughout the mri volume the local estimates are computed by fitting a partial volume tissue measurement model to histograms of neighborhoods about each estimate point the measurement model uses mean tissue intensity and noise variance values computed from the global image and a multiplicative bias parameter that is estimated for each region during the histogram fit voxels in the intensity normalized image are then classified into six tissue types using a maximum a posteriori classifier this classifier combines the partial volume tissue measurement model with a gibbs prior that models the spatial properties of the brain we validate each stage of our algorithm on real and phantom data using data from the 20 normal mri brain data sets of the internet brain segmentation repository our method achieved average indices of 0 746 0 114 for gray matter gm and 0 798 0 089 for white matter wm compared to expert labeled data our method achieved average indices 0 893 0 041 for gm and 0 928 0 039 for wm compared to the ground truth labeling on 12 volumes from the montreal neurological institute s brainweb phantom

mean curvature flow with surgery we give a new proof for the existence of mean curvature flow with surgery of 2 convex hypersurfaces in r as announced in hk13a our proof works for all n 3 including mean convex surfaces in r we also derive a priori estimates for a more general class of flows in a local and flexible setting

existence and algorithm of solutions for generalized strongly nonlinear mixed variational like inequalities in banach spaces in this paper we introduce and study a class of generalized strongly nonlinear mixed variational like inequalities in reflexive banach spaces the auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for generalized strongly nonlinear mixed variational like inequalities first the existence of solutions of the auxiliary problems for the generalized strongly nonlinear mixed variational like inequalities is shown second an iterative algorithm for solving the generalized strongly nonlinear mixed variational like inequalities is constructed by using this existence result finally not only the existence of solutions of the generalized strongly nonlinear mixed variational like inequalities is shown but also the convergence of iterative sequences generated by the algorithm is also proven