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

a note on multiobjective second order symmetric duality abstract in this paper we establish a strong duality theorem for a pair of multiobjective second order symmetric dual programs this removes an omission in an earlier result by yang et al x m yang x q yang k l teo s h hou multiobjective second order symmetric duality with f convexity euro j oper res 165 2005 585 591

ill posedness of the stationary navier stokes equations in besov spaces abstract the solutions of the stationary navier stokes equations in r n for n 3 in the scaling invariant besov spaces are investigated it is proved that a sequence of bounded smooth external forces whose b 1 3 norms converges to zero can produce a sequence of bounded smooth solutions whose b 1 norms never converges to zero such norm inflation phenomena are shown by constructing the sequence of external forces as similar to those of initial data proposed by bourgain pavlovic in the non stationary problem

higher order duality in multiobjective fractional programming with support functions abstract in this paper a new class of higher order f type i functions for a multiobjective programming problem is introduced which subsumes several known studied classes higher order mond weir and schaible type dual programs are formulated for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support functions of compact convex sets in r n weak and strong duality results are studied in both the cases assuming the involved functions to be higher order f type i a number of previously studied problems appear as special cases

inequalities of the brunn minkowski type for gaussian measures let m 2 be an integer let be the standard gaussian measure on rn and let t t exp s2 2 ds sqrt 2 small t le given 1 l m 0 this paper gives a necessary and sufficient condition such that the inequality 1 1 a1 m a m 1 1 a 1 m 1 a m is true for all borel sets a 1 a m in hbfrn of strictly positive measure or all convex borel sets a 1 a m in bfrn of strictly positive measure respectively in particular the paper exhibits inequalities of the brunn minkowski type for which are true for all convex sets but not for all measurable sets

second order symmetric duality in multiobjective programming involving generalized cone invex functions abstract in this paper cone second order pseudo invex and strongly cone second order pseudo invex functions are defined a pair of mond weir type second order symmetric dual multiobjective programs is formulated over arbitrary cones weak strong and converse duality theorems are established under aforesaid generalized invexity assumptions a second self duality theorem is also given by assuming the functions involved to be skew symmetric

submanifolds isoperimetric inequalities and optimal transportation abstract the aim of this paper is to prove isoperimetric inequalities on submanifolds of the euclidean space using mass transportation methods we obtain a sharp weighted isoperimetric inequality and a nonsharp classical inequality similar to the one obtained in michael and simon 1973 19 the proof relies on the description of a solution of the problem of monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension

on second order symmetric duality in nondifferentiable programming this article is concerned with a pair of second order symmetric dual non differentiable programs and second order f pseudo convexity we establish the weak and the strong duality theorems for the new pair of dual models under the f pseudo convexity assumption several known results including mond and schechter as well as others are obtained as special cases

second order multiobjective symmetric duality with cone constraints abstract in this paper we formulate wolfe and mond weir type second order multiobjective symmetric dual problems over arbitrary cones weak strong and converse duality theorems are established under bonvexity pseudobonvexity assumptions this work also removes several omissions in definitions models and proofs for wolfe type problems studied in mishra 9 moreover self duality theorems for these pairs are obtained assuming the function involved to be skew symmetric

ill posedness of the 3d navier stokes equations in a generalized besov space near bmo 1 abstract the ill posedness of the 3d navier stokes equations in a generalized besov space which is smaller than b q 1 q 2 is considered in 2008 bourgain pavlovic proved that the 3d navier stokes equation is ill posed in b 1 by showing norm inflation phenomena of the solution for some initial data on the other hand in 2008 germain proved that the flow map is not c 2 in the space b q 1 for q 2 however he did not treat ill posed problem in such spaces thus our result is an extension of these previous results

cone directed higher order generalized adjacent derivative and higher order mond weir duality in this paper cone directed higher order generalized adjacent derivative for a set valued map is defined by virtue of the cone directed higher order generalized adjacent derivative a higher order mond weir type dual problem for a constrained set valued optimiza tion problem is introduced and its weak duality strong duality and converse duality properties are established the results obtained improve the corresponding results in the literatures

second order symmetric duality in non differentiable multiobjective programming with f convexity abstract this paper is concerned with a pair of mond weir type second order symmetric dual non differentiable multiobjective programming problems we establish the weak and strong duality theorems for the new pair of dual models under second order f convexity assumptions several results including many recent works are obtained as special cases

wolfe type second order symmetric duality in nondifferentiable programming abstract a pair of wolfe type second order symmetric dual programs involving nondifferentiable functions is considered and appropriate duality theorems are established under 1 bonvexity 2 boncavity several known results including that of mond and gulati et al are obtained as special cases

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

second order symmetric duality for nonlinear minimax mixed integer programs abstract wolfe type second order minimax mixed integer dual programs are formulated and a symmetric duality theorem is e established under separability and bonvexity boncavity of the kernel function k x y mond weir type symmetric duality is also discussed under weaker bonvexity assumptions moreover self duality theorems for these pairs are obtained assuming k x y to be skew symmetric

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

ill posedness of the navier stokes equations in a critical space in 3d abstract we prove that the cauchy problem for the three dimensional navier stokes equations is ill posed in b 1 in the sense that a norm inflation happens in finite time more precisely we show that initial data in the schwartz class s that are arbitrarily small in b 1 can produce solutions arbitrarily large in b 1 after an arbitrarily short time such a result implies that the solution map itself is discontinuous in b 1 at the origin

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

non differentiable second order symmetric duality in mathematical programming with f convexity abstract a pair of wolfe type non differentiable second order symmetric primal and dual problems in mathematical programming is formulated the weak and strong duality theorems are then established under second order f convexity assumptions symmetric minimax mixed integer primal and dual problems are also investigated

second order symmetric duality in mathematical programming with f convexity abstract under second order f convexity f concavity and second order f pseudoconvexity f pseudoconcavity appropriate second order duality results for pair of wolfe and mond weir type second order symmetric dual nonlinear programming problems are established these second order duality results are then used to formulate wolfe type and mond weir type second order minimax mixed integer dual programs and symmetric duality theorem is established under separability and second order f convexity f concavity of the kernel function f x y second order symmetric dual fractional mixed integer programs are studied using the above programs moreover second order self duality theorems for the above pairs are obtained assuming f x y to be skew symmetric

higher order nondifferentiable symmetric duality with generalized f convexity a pair of nondifferentiable higher order wolfe type symmetric dual models is formulated and usual duality theorems are established under higher order f convexity assumption symmetric minimax mixed integer primal and dual problems are also discussed

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data