regularity of harmonic functions for a class of singular stable like processes we consider the system of stochastic differential equations dx t a x t dz t where zt1 zdt are independent one dimensional symmetric stable processes of order and the matrix valued function a is bounded continuous and everywhere non degenerate we show that bounded harmonic functions associated with x are holder continuous but a harnack inequality need not hold the levy measure associated with the vector valued process z is highly singular

a norm relaxed method of feasible directions for finely discretized problems from semi infinite programming abstract in this paper a class of finely discretized semi infinite programming sip problems is discussed combining the idea of the norm relaxed method of feasible directions mfd and the technique of updating discretization index set we present a new algorithm for solving the discretized semi infinite dsi problems from sip at each iteration the iteration point is feasible for the discretized problem and an improved search direction is computed by solving only one direction finding subproblem i e a quadratic program and some appropriate constraints are chosen to reduce the computational cost a high order correction direction can be obtained by solving another quadratic programming subproblem with only equality constraints under weak conditions such as mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses weak global convergence moreover the superlinear convergence is obtained under linearly independent constraint qualification licq and other assumptions in the end some elementary numerical experiments are reported

on overdetermined problems for a general class of nonlocal operators abstract we study the overdetermined problem for a large family of non local operators given by generators of subordinate brownian motions in particular this family includes the fractional laplacian relativistic stable operators etc we consider these problems in bounded domains exterior domains and in annular domains and we show that under suitable conditions the domains and solutions are both radially symmetric our method uses both analytic and probabilistic tools

an efficient sequential quadratic programming algorithm for nonlinear programming in this paper a new feasible sequential quadratic programming fsqp algorithm is proposed to solve the nonlinear programming where a feasible descent direction is obtained by solving only one qp subproblem in order to avoid maratos effect a high order revised direction is computed by solving a linear system with involving some active constraints the theoretical analysis shows that global and superlinear convergence can be deduced

new bounds in some transference theorems in the geometry of numbers the aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in r to formulate the problem we have to introduce some notation and terminology we shall treat ir as an n dimensional euclidean space with the norm ii ii and metric d the inner product of vectors u v will be denoted by uv we shall usually write u 2 instead of uu the closed and open unit balls in f will be denoted by b and b respectively if a c ir then span a and a z denote the linear subspace spanned over a and the orthogonal complement of a in p a lattice in ir is an additive subgroup of ir generated by n linearly independent vectors the family of all lattices in ir will be denoted by a given a lattice l e a we define the polar dual reciprocal lattice l in the usual way l u u v z for each v l

a new norm relaxed method of strongly sub feasible direction for inequality constrained optimization combining the norm relaxed method of feasible direction mfd with the idea of strongly sub feasible direction method we present a new convergent algorithm with arbitrary initial point for inequality constrained optimization at each iteration the new algorithm solves one direction finding subproblem dfs which always possesses a solution some good properties of the new algorithm are that it can unify automatically the operations of initialization phase i and optimization phase ii and the number of the functions satisfying the inequality constrains is nondecreasing particularly a feasible direction of descent can be obtained by solving dfs whenever the iteration point gets into the feasible set under some mild assumptions without the linear independence the global and strong convergence of the algorithm can be obtained

a fast convergent sequential linear equation method for inequality constrained optimization without strict complementarity abstract in this paper a type of smooth nonlinear optimization problems with inequality constraints is considered and a new sequential linear equation algorithm is proposed by introducing a new constructing technique for the system of linear equations sle which depends on the perturbation of the constraints gradients at each iteration of the proposed algorithm one or more sles need to be solved specially only one sle is required to be solved when the iterates are sufficiently close to the solution i e after a finite number of iterations which decreases the amount of computations the search technique is a combination of the line armijo type and newton step size under mild assumptions without the strict complementarity it is shown that the proposed algorithm enjoys the properties of global and superlinear convergence finally some preliminary numerical tests are reported and the numerical results show that the proposed algorithm is promising

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

obstacle problem for a class of parabolic equations of generalized p laplacian type abstract we study nonlinear parabolic pdes with orlicz type growth conditions the main result gives the existence of a unique solution to the obstacle problem related to these equations to achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution moreover we prove that if the obstacle is continuous so is the solution

affine orlicz polya szego principle for log concave functions abstract the affine l p polya szego principle significantly strengthens the classical polya szego principle it is an open problem whether there exists an affine orlicz polya szego principle which includes the affine l p polya szego principle as special case in this paper an affine orlicz polya szego principle for log concave functions is established by using functional steiner symmetrizations

on lp estimates for a class of non local elliptic equations abstract we consider non local elliptic operators with kernel k y a y y d where 0 2 is a constant and a is a bounded measurable function by using a purely analytic method we prove the continuity of the non local operator l from the bessel potential space h p to l p and the unique strong solvability of the corresponding non local elliptic equations in l p spaces as a byproduct we also obtain interior l p estimates the novelty of our results is that the function a is not necessarily to be homogeneous regular or symmetric an application of our result is the uniqueness for the martingale problem associated to the operator l

an efficient feasible directions algorithm for design synthesis a nonlinear optimization algorithm is described that combines the best features of the method of feasible directions and the generalized reduced gradient method the algorithm uses the direction finding subproblem from the method of feasible directions to find a search direction that is equivalent to that of the generalized reduced gradient method but without the need to add a large number of slack variables associated with inequality constraints this leads to a core efficient algorithm for the solution of optimization problems with a large number of inequality constraints also during the one dimensional search it is not necessary to separate the design space into dependent and independent variables using the present method the concept of infrequent gradient calculations is introduced as a means of gaining further optimization efficiency finally it is shown that using the basic direction finding algorithm contained in this method the sensitivity of the optimum design with respect to problem parameters can be obtained without the need for second derivatives or lagrange multipliers the optimization algorithm and sensitivity analysis is demonstrated by numerical example

aleksandrov bakelman pucci maximum principles for a class of uniformly elliptic and parabolic integro pde abstract we prove generalized aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes of hamilton jacobi bellman isaacs types whose pde parts are either uniformly elliptic or uniformly parabolic the proofs of these results are based on the classical aleksandrov bakelman pucci maximum principles for the elliptic and parabolic pdes and an iteration procedure using solutions of pucci extremal equations we also provide proofs of nonlocal versions of the classical aleksandrov bakelman pucci maximum principles for elliptic and parabolic integro pdes

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

regularity theory for general stable operators abstract we establish sharp regularity estimates for solutions to l u f in r n l being the generator of any stable and symmetric levy process such nonlocal operators l depend on a finite measure on s n 1 called the spectral measure first we study the interior regularity of solutions to l u f in b 1 we prove that if f is c then u belong to c 2 s whenever 2 s is not an integer in case f l we show that the solution u is c 2 s when s 1 2 and c 2 s for all 0 when s 1 2 then we study the boundary regularity of solutions to l u f in u 0 in r n in c 1 1 domains we show that solutions u satisfy u d s c s for all 0 where d is the distance to finally we show that our results are sharp by constructing two counterexamples

on the differentiability of the solution to the hamilton jacobi equation with critical fractional diffusion abstract we prove that the hamilton jacobi equation for an arbitrary hamiltonian h locally lipschitz but not necessarily convex and fractional diffusion of order one critical has classical c 1 solutions the proof is achieved using a new holder estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

anisotropic fractional sobolev norms abstract bourgain brezis mironescu showed that with suitable scaling the fractional sobolev s seminorm of a function f w 1 p r n converges to the sobolev seminorm of f as s 1 the anisotropic s seminorms of f defined by a norm on r n with unit ball k are shown to converge to the anisotropic sobolev seminorm of f defined by the norm with unit ball z p k the polar l p moment body of k the limiting behavior for s 0 is also determined extending results by maz ya shaposhnikova

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

a strongly convergent norm relaxed method of strongly sub feasible direction for optimization with nonlinear equality and inequality constraints in this paper a class of optimization problems with nonlinear equality and inequality constraints is discussed firstly the original problem is transformed to an associated simpler auxiliary optimization problem with only inequality constraints and a penalty parameter and the later problem is showed to be equivalent to the original problem if the parameter is large enough but finite then combining the norm relaxed method of feasible direction mfd with the idea of method of strongly sub feasible direction mssfd we present an algorithm with arbitrary initial point for the original problem at each iteration of the auxiliary problem an improved search direction is obtained by solving one direction finding subproblem dfs i e a quadratic program which always possesses a solution in the process of iteration the feasibility of the iteration points is monotone increasing furthermore whenever an iteration point enters the feasible set the proposed algorithm reduces to a feasible and decent method for the auxiliary problem under some suitable assumptions the global and strong convergence of the proposed algorithm can be obtained finally some elementary numerical experiments are reported

a simple feasible sqp algorithm for inequality constrained optimization in this paper an sqp method generating feasible iterates is presented to solve the nonlinear inequality constrained optimization problems a new modified method which ensures the global and superlinear convergence is proposed by solving systems of linear equation instead of qp subproblems and linear squares problems here the search direction is a suitable combination of a descent direction a feasible direction and a second order revised direction the theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

semilinear fractional elliptic equations with gradient nonlinearity involving measures we study the existence of solutions to the fractional elliptic equation e1 delta alpha u epsilon g nabla u nu in a bounded regular domain omega of r n n ge2 subject to the condition e2 u 0 in omega c where epsilon 1 or 1 delta alpha denotes the fractional laplacian with alpha in 1 2 1 nu is a radon measure and g r mapsto r is a continuous function we prove the existence of weak solutions for problem e1 e2 when g is subcritical furthermore the asymptotic behavior and uniqueness of solutions are described when nu is dirac mass g s s p p geq 1 and epsilon 1

a new superlinearly convergent norm relaxed method of strongly sub feasible direction for inequality constrained optimization method of feasible directions mfd is an important method for solving nonlinearly constrained optimization however various types of mfd all need an initial feasible point which can not be found easily in generally in addition the computational cost of some mfd with superlinearly convergent property is rather high on the other hand the strongly sub feasible direction method does not need an initial feasible point but most of the proposed algorithm do not have the superlinearly convergent property and can not guarantee that the iteration point is feasible after finite iterations in this paper we present a new superlinearly convergent algorithm with arbitrary initial point at each iteration a master direction is obtained by solving one direction finding subproblem dfs and an auxiliary direction is yielded by an explicit formula after finite iterations the iteration point goes into the feasible set and the master direction is a feasible direction of descent since a new generalized projection technique is contained in the auxiliary direction formula under some mild assumptions without the strict complementarity the global convergence and superlinear convergence of the algorithm can be obtained

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

harnack inequality for some classes of markov processes in this paper we establish a harnack inequality for nonnegative harmonic functions of some classes of markov processes with jumps

a superlinearly convergent method of feasible directions method of feasible directions mfd is one of the most important methods for solving engineering optimization problems there exist two types of mfd one has at most a linear convergence rate and the other has a superlinear convergence rate the former type includes zoutendijk s mfd topkis veinott s mfd pironneau polak s mfd cawood kostreva s norm relaxed mfd and a modified mfd the latter includes panier tits fsqp approaches however the price of the superlinearly convergent mfd for achieving superlinearity is rather high computational cost per single iteration this paper presents an algorithm based on the modified mfd the theoretical analysis shows that this algorithm is locally superlinearly convergent this algorithm has attained the same convergence rate as fsqp while it only needs to solve two subproblems instead of three subproblems in fsqp hence this algorithm should be computationally more efficient than fsqp

symmetric properties of positive solutions for fully nonlinear nonlocal system abstract in this paper we obtain symmetry and monotonicity of positive solutions for the systems involving fully nonlinear nonlocal operators in a domain bounded or unbounded in r n via using a direct method of moving planes this extends the results in wang and niu 2017 and also is the first result of the symmetry for a fully nonlinear nonlocal system containing gradient terms with different order

the dirichlet problem for nonlocal operators with singular kernels convex and nonconvex domains abstract we study the interior regularity of solutions to the dirichlet problem l u g in u 0 in r n for anisotropic operators of fractional type l u x 0 d s n 1 d a 2 u x u x u x 1 2 s here a is any measure on s n 1 a prototype example for l is given by the sum of one dimensional fractional laplacians in fixed given directions when a c s n 1 and g is c solutions are known to be c inside but not up to the boundary however when a is a general measure or even when a is l s n 1 solutions are only known to be c 3 s inside we prove here that for general measures a solutions are c 1 3 s inside for all 0 whenever is convex when a l s n 1 we show that the same holds in all c 1 1 domains in particular solutions always possess a classical first derivative the assumptions on the domain are sharp since if the domain is not convex and the measure a is singular we construct an explicit counterexample for which u is not c 3 s for any 0 even if g and are c

on combining feasibility descent and superlinear convergence in inequality constrained optimization extension of quasi newton techniques from unconstrained to constrained optimization via sequential quadratic programming sqp presents several difficulties among these are the possible inconsistency away from the solution of first order approximations to the constraints resulting in infeasibility of the quadratic programs and the task of selecting a suitable merit function to induce global convergence in ths case of inequality constrained optimization both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints and that yield monotonically decreasing objective function values feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not well defined outside the feasible set it has been recently shown that this can be achieved while preserving local two step superlinear convergence in this note the essential ingredients for an sqp based method exhibiting the desired properties are highlighted correspondingly a class of such algorithms is described and analyzed tests performed with an efficient implementation are discussed

a superlinearly convergent norm relaxed method of quasi strongly sub feasible direction for inequality constrained minimax problems in this paper nonlinear minimax problems with inequality constraints are discussed combined the norm relaxed sqp method with the idea of strongly sub feasible directions method a new method of quasi strongly sub feasible directions mqssfd with arbitrary initial point for the discussed problems is presented at each iteration of the proposed algorithm an improved search direction is obtained by solving a quadratic program qp which always has a solution and a high order correction direction is yielded via a system of linear equations sle to avoid the maratos effect after finite iterations the iteration point always get into the feasible set by introducing a new non monotone curve search under some mild conditions including the weak mangasarian fromovitz constraint qualification mfcq the proposed algorithm possesses global convergence and the superlinear convergence is obtained without the strict complementarity finally some elementary numerical experiments are implemented and reported

systems of equations driven by stable processes let zjt j 1 d be independent one dimensional symmetric stable processes of index 0 2 we consider the system of stochastic differential equations

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

fractional sobolev norms and bv functions on manifolds abstract the bounded variation seminorm and the sobolev seminorm on compact manifolds are represented as a limit of fractional sobolev seminorms this establishes a characterization of functions of bounded variation and of sobolev functions on compact manifolds as an application the special case of sets of finite perimeter is considered

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

the evans krylov theorem for nonlocal fully nonlinear equations we prove a regularity result for solutions of a purely integro differential bellman equation this regularity is enough for the solutions to be understood in the classical sense if we let the order of the equation approach two we recover the theorem of evans and krylov about the regularity of solutions to concave uniformly elliptic partial differential equations

periodic solutions for the one dimensional fractional laplacian abstract in this paper we are concerned with the construction of periodic solutions of the nonlocal problem s u f u in r where s stands for the s laplacian s 0 1 we introduce a suitable framework which allows by means of regularity to link the searching of such solutions into the existence of the ones of a semilinear problem in a suitable hilbert space then by a bifurcation theory from eigenvalues of odd multiplicity and also variational method that avoid the constant solutions we get existence theorems which are lately enlightened with the analysis of some examples in particular multiplicity results for generalized benjamin ono equation are obtained

improvements in closest point search based on dual hkz bases in this paper we review the technique to solve the cvp based on dual hkz bases by j blomer 4 the technique is based on the transference theorems given by banaszczyk 3 which imply some necessary conditions on the coefficients of the closest vectors with respect to a basis whose dual is hkz reduced recursively starting with the last coefficient intervals of length i can be derived for the i th coefficient of any closest vector this leads to n candidates for closest vectors in this paper we refine the necessary conditions derived from the transference theorems giving an exponential reduction of the number of candidates the improvement is due to the fact that the lengths of the intervals are not independent in the original algorithm the candidates for a coefficient pair v i v i 1 correspond to the integer points in a rectangle of volume email protected i 1 in our analysis we show that the candidates for v i v i 1 in fact lie in an ellipse with transverse and conjugate diameter i 1 respectively i this reduces the expected number of points to be enumerated by an exponential factor of about 0 886 n we further show how a choice of the coefficients v n v i 1 influences the interval from which v i can be chosen numerical computations show that these considerations allow to bound the expected number of points to be enumerated by n 0 7 5 n for 10

a superlinearly convergent feasible method for the solution of inequality constrained optimization problems when iteratively solving optimization problems arising from engineering design applications it is sometimes crucial that all iterates satisfy a given set of hard inequality constraints and generally desirable that the objective function value improve at each iteration in this paper we propose an algorithm of the successive quadratic programming sqp type which unlike other algorithms of this type does enjoy such properties under mild assumptions the new algorithm is shown to converge from any initial point locally superlinearly numerically tested it has proven to be competitive with the most successful currently available nonlinear programming algorithms while the latter do not exhibit the desired properties

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

anisotropic fractional perimeters the anisotropic fractional s perimeter with respect to a convex body k in rn is shown to converge as s 1 to the anisotropic perimeter with respect to the moment body of k for anisotropic fractional s seminorms on bv rn the corresponding result is established generalizing results of bourgain brezis mironescu and davila minimizers of the anisotropic fractional isoperimetric inequality with respect to k are shown to converge to the moment body of k as s 1 anisotropic fractional sobolev inequalities are established for a borel set e r and 0 s 1 the fractional s perimeter of e is given by ps e

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

a superlinearly convergent norm relaxed sqp method of strongly sub feasible directions for constrained optimization without strict complementarity in this paper a kind of optimization problems with nonlinear inequality constraints is discussed combined the ideas of norm relaxed sqp method and strongly sub feasible direction method as well as a pivoting operation a new fast algorithm with arbitrary initial point for the discussed problem is presented at each iteration of the algorithm an improved direction is obtained by solving only one direction finding subproblem which possesses small scale and always has an optimal solution and to avoid the maratos effect another correction direction is yielded by a simple explicit formula since the line search technique can automatically combine the initialization and optimization processes after finite iterations the iteration points always get into the feasible set the proposed algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity finally some numerical tests are reported

a generalization of the norm relaxed method of feasible directions this paper generalizes the norm relaxed method of feasible directions which has roots in the classical method of feasible directions mfd this generalized algorithm introduces some parameters which can be adjusted to speed up the convergence an investigation of its global convergence is included and the numerical performance is shown on 15 test problems its global convergence is guaranteed under a rather mild and standard assumption the numerical tests show that this new algorithm converges faster than the previous norm relaxed mfd for most test problems by selecting a set of suitable parameters even though it only has the same guaranteed convergence rate as the previous norm relaxed mfd

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

a variant of sqp method for inequality constrained optimization and its global convergence in this paper a variant of sqp method for solving inequality constrained optimization is presented this method uses a modified qp subproblem to generate a descent direction as each iteration and can overcome the possible difficulties that the qp subproblem of the standard sqp method is inconsistency furthermore the method can start with an infeasible initial point under mild conditions we prove that the algorithm either terminates as kkt point within finite steps or generates an infinite sequence whose accumulation point is a kkt point or satisfies certain first order necessary condition finally preliminary numerical results are reported

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

regularity of solutions to anisotropic nonlocal equations we study harmonic functions associated to systems of stochastic differential equations of the form dx t i a i1 x t dz t 1 cdots a id x t dz t d i in 1 dots d where z t j are independent one dimensional symmetric stable processes of order alpha j in 0 2 j in 1 dots d in this article we prove holder regularity of bounded harmonic functions with respect to solutions to such systems

regularity properties of nonlocal minimal surfaces via limiting arguments abstract we prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter s when s 1 as a consequence we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and s is close to 1

nonexistence of positive solutions for an indefinite fractional p laplacian equation abstract in this paper we consider the following fractional p laplacian equation with indefinite nonlinearities p s u x x 1 u q x x r n where 0 s 1 p q 1 2 we use the direct method of moving planes to prove the solution of above fractional p laplacian equation is monotone increasing along x 1 direction then obtain that the above equation possess no positive bounded solution on the whole space there is no decay condition at infinity on solution for the fractional p laplacian and the traditional approaches of translation and taking limit there no longer work for the nonlocal nonlinear fractional p laplacian here we also cannot find a suitable auxiliary function for this fractional p laplacian to circumvent these difficulties we use the following two estimate to study the property of solution i we estimate the singular integrals defining the fractional p laplacian along a sequence of approximate maxima ii we estimate the lower and upper bound of the fractional p laplacian equation

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

a new sqp method of feasible directions for nonlinear programming in this paper a new sequential quadratic programming sqp method of feasible directions is proposed and analyzed for nonlinear programming where a feasible direction of descent can be derived from solving only one qp subproblem in particular this method can produce automatically a revised direction with the explicit expression which can avoid maratos effect without solving qp subproblem the theoretical analysis shows that global and superlinear convergence can be induced in the end numerical experiment is given to illustrate the effectiveness of the method

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

sharp stability theorems for the anisotropic sobolev and log sobolev inequalities on functions of bounded variation abstract combining rearrangement techniques with gromov s proof via optimal mass transportation of the 1 sobolev inequality we prove a sharp quantitative version of the anisotropic sobolev inequality on b v r n we also deduce as a corollary of this result a sharp stability estimate for the anisotropic 1 log sobolev inequality

a method combining norm relaxed qp subproblems with systems of linear equations for constrained optimization based on the ideas of norm relaxed sequential quadratic programming sqp method and the strongly sub feasible direction method we propose a new sqp algorithm for the solution of nonlinear inequality constrained optimization unlike the previous work at each iteration the norm relaxed quadratic programming subproblem nrqps in our algorithm only consists of the constraints corresponding to an estimate of the active set and the high order correction direction used to avoid the maratos effect is obtained by solving a system of linear equations sle which also only consists of such a subset of constraints and gradients moreover the line search technique can effectively combine the initialization process with the optimization process and therefore if the starting point is not feasible the iteration points always get into the feasible set after a finite number of iterations the global convergence is proved under the mangasarian fromovitz constraint qualification mfcq and the superlinear convergence is obtained without assuming the strict complementarity finally the numerical experiments show that the proposed algorithm is effective and promising for the test problems

relative entropy of cone measures and lp centroid bodies let k be a convex body in r n we introduce a new affine invariant which we call k t hat can be found in three different ways a as a limit of normalized lp affine surface areas b as the relative entropy of the cone measure of k and the cone measure of k c as the limit of the volume difference of k and lp centroid bodies we investigate properties of k and of related new invariant quantities in particular we show new affine isoperimetric inequalities and we show an information inequality for convex bodies

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

improvements in the analysis of kannan s cvp algorithm finding a closest lattice vector to a given target vector is a widely studied problem in computational mathematics and computer science the classical enumeration algorithm was given by kannan in 1983 together with an algorithm for the shortest vector problem the time complexity of the closest vector enumeration algorithm based on hkz reduced bases was proven to within 2 o n n n 2 by hanrot and stehle in 2007 using their point counting method the constant hidden in o n is strictly bigger than 1 9 in this paper we give a new counting method which we call sequence counting method to bound the number of loop iterations and show that the time complexity is actually upper bounded by 2 0 401 n o n n n 2 our paper revisits kannan s cvp algorithm and points out a new method sequence counting to upper bound the time complexity by 2 0 401 n o n n n 2 we connects the loops of the enumeration algorithm with lattice points by a one to one mapping using the sequence as a tool we use different methods to upper bound the number of lattice points near a target vector according to the dimension of the lattice and then combine them to conclude our result

integration by parts and pohozaev identities for space dependent fractional order operators abstract consider a classical elliptic pseudodifferential operator p on r n of order 2a 0 a 1 with even symbol for example p a x d a where a x d is a second order strongly elliptic differential operator the fractional laplacian a is a particular case for solutions u of the dirichlet problem on a bounded smooth subset r n we show an integration by parts formula with a boundary integral involving d a u where d x dist x this extends recent results of ros oton serra and valdinoci to operators that are x dependent nonsymmetric and have lower order parts we also generalize their formula of pohozaev type that can be used to prove unique continuation properties and nonexistence of nontrivial solutions of semilinear problems an illustration is given with p m 2 a the basic step in our analysis is a factorization of p p p p where we set up a calculus for the generalized pseudodifferential operators p that come out of the construction

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

asymptotic mean value properties for fractional anisotropic operators abstract we obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator furthermore we get a bourgain brezis mironescu type limit formula for a related class of anisotropic nonlocal norms

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

a sequential equality constrained quadratic programming algorithm for inequality constrained optimization in this paper the feasible type sqp method is improved a new sqp algorithm is presented to solve the nonlinear inequality constrained optimization as compared with the existing sqp methods per single iteration in order to obtain the search direction it is only necessary to solve equality constrained quadratic programming subproblems and systems of linear equations under some suitable conditions the global and superlinear convergence can be induced

a qp free globally convergent locally superlinearly convergent algorithm for inequality constrained optimization an algorithm is proposed for the minimization of a smooth function subject to smooth inequality constraints unlike sequential quadratic programming type methods this algorithm does not involve the solution of quadratic programs but merely that of linear systems of equations locally the iteration can be viewed as a perturbation of a quasi newton iteration on both the primal and dual variables for the solution of the equalities in the kuhn tucker first order conditions of optimality it is observed that provided the current iterate is feasible and the current multiplier estimates are strictly positive the primal component of the quasi newton direction is a direction of descent for the objective function this fact is used to induce global convergence without the need of a surrogate merit function a careful bending of the search direction prevents any maratos like effect and local superlinear convergence is proven while the algorithm requires that an initial feasible point be available the succe

new characterizations of sobolev metric spaces abstract we provide new characterizations of sobolev ad bv spaces in doubling and poincare metric spaces in the spirit of the bourgain brezis mironescu and nguyen limit formulas holding in domains of r n