weak stability of ricci expanders with positive curvature operator we prove the l infty stability of expanding gradient ricci solitons with positive curvature operator and quadratic curvature decay at infinity

schauder theorems for ornstein uhlenbeck equations in infinite dimension abstract we prove schauder type estimates for stationary and evolution equations driven by the classical ornstein uhlenbeck operator in a separable banach space endowed with a centered gaussian measure

l p uniqueness for elliptic operators with unbounded coefficients in r n let a be an elliptic operator with unbounded and sufficiently smooth coefficients and let be a sub invariant measure of the operator a in this paper we give sufficient conditions guaranteeing that the closure of the operator a cc rn generates a sub markovian strongly continuous semigroup of contractions in lp rn applications are given in the case when a is a generalized schrodinger operator

convergent sdp relaxations in polynomial optimization with sparsity we consider a polynomial programming problem p on a compact basic semialgebraic set k subset r n described by m polynomial inequalities g j x geq0 and with criterion f in r x we propose a hierarchy of semidefinite relaxations in the spirit of those of waki e siam j optim 17 2006 pp 218 242 in particular the sdp relaxation of order r has the following two features a the number of variables is o kappa 2r where kappa max kappa 1 kappa 2 with kappa 1 resp kappa 2 being the maximum number of variables appearing in the monomials of f resp appearing in a single constraint g j x geq0 b the largest size of the linear matrix inequalities lmis is o kappa r this is to compare with the respective number of variables o n 2r and lmi size o n r in the original sdp relaxations defined in j b lasserre siam j optim 11 2001 pp 796 817 therefore great computational savings are expected in case of sparsity in the data g j f i e when k is small a frequent case in practical applications of interest the novelty with respect to h waki s kim m kojima and m maramatsu siam j optim 17 2006 pp 218 242 is that we prove convergence to the global optimum of p when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications and known as the running intersection property in graph theory in such cases and as a by product we also obtain a new representation result for polynomials positive on a compact basic semialgebraic set a sparse version of putinar s positivstellensatz m putinar indiana univ math j 42 1993 pp 969 984

gk dimension of birationally commutative surfaces let k be an algebraically closed field let k k be a finitely generated field extension of transcendence degree 2 let aut k k and let a q k t be an n graded subalgebra with dim k a n 0 then if a is big enough in q in an appropriate sense we prove that gk a 3 4 5 or oo with the exact value depending only on the geometric properties of the proof uses techniques in the birational geometry of surfaces which are of independent interest

on the ornstein uhlenbeck operator in spaces of continuous functions we study the realization of the ornstein uhlenbeck operator a in the space of the uniformly continuous and bounded functions in rn we prove that it generates a semigroup which is neither strongly continuous nor analytic but enjoys nice smoothing properties such properties are used to prove a schauder type theorem for the elliptic equation u au f

generalized lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems sequences of generalized lagrangian duals and their sums of squares sos of polynomials relaxations for a polynomial optimization problem pop are introduced the sparsity of polynomials in the pop is used to reduce the sizes of the lagrangian duals and their sos relaxations it is proved that the optimal values of the lagrangian duals in the sequence converge to the optimal value of the pop using a method from the penalty function approach the sequence of sos relaxations is transformed into a sequence of semidefinite programing sdp relaxations of the pop which correspond to duals of modification and generalization of sdp relaxations given by lasserre for the pop

canonical primal dual algorithm for solving fourth order polynomial minimization problems this paper focuses on implementation of a general canonical primal dual algorithm for solving a class of fourth order polynomial minimization problems a critical issue in the canonical duality theory has been addressed i e in the case that the canonical dual problem has no interior critical point in its feasible space s a a quadratic perturbation method is introduced to recover the global solution through a primal dual iterative approach and a gradient based method is further used to refine the solution a series of test problems including the benchmark polynomials and several instances of the sensor network localization problems have been used to testify the effectiveness of the proposed algorithm

strong feller property and irreducibility for diffusions on hilbert spaces it is shown that the transition semigroup p t t 0 corresponding to a nonlinear stochastic evolution equation is strong feller and irreducible provided the nonlinearities are lipschitz continuous and the diffusion term is nondegenerate this result ensures the uniqueness of the invariant measure for p t t 0

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

the universal enveloping algebra of the witt algebra is not noetherian abstract this work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional lie algebra to be noetherian to address this problem we answer a 23 year old question of carolyn dean and lance small namely we prove that the universal enveloping algebra of the witt or centerless virasoro algebra is not noetherian to show this we prove our main result the universal enveloping algebra of the positive part of the witt algebra is not noetherian we employ algebro geometric techniques from the first author s classification of noncommutative birationally commutative projective surfaces as a consequence of our main result we also show that the enveloping algebras of many other infinite dimensional lie algebras are not noetherian these lie algebras include the virasoro algebra and all infinite dimensional z graded simple lie algebras of polynomial growth

convergent relaxations of polynomial optimization problems with noncommuting variables we consider optimization problems with polynomial inequality constraints in noncommuting variables these noncommuting variables are viewed as bounded operators on a hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints such problems arise naturally in quantum theory and quantum information science to solve them we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution we also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem

multi scale modelling and canonical dual finite element method in phase transitions of solids this paper presents a multi scale model in phase transitions of solid materials with both macro and micro effects this model is governed by a semi linear nonconvex partial differential equation which can be converted into a coupled quadratic mixed variational problem by the canonical dual transformation method the extremality conditions of this variational problem are controlled by a triality theory which reveals the multi scale effects in phase transitions therefore a potentially useful canonical dual finite element method is proposed for the first time to solve the nonconvex variational problems in multi scale phase transitions of solids applications are illustrated results shown that the canonical duality theory developed by the first author in nonconvex mechanics can be used to model complicated physical phenomena and to solve certain difficult nonconvex variational problems in an easy way the canonical dual finite element method brings some new insights into computational mechanics

geometric algebras on projective surfaces abstract let x be a projective surface let aut x and let l be a ample invertible sheaf on x we study the properties of a family of subrings parameterized by geometric data of the twisted homogeneous coordinate ring b x l in particular we find necessary and sufficient conditions for these subrings to be noetherian we also study their homological properties their associated noncommutative projective schemes and when they are maximal orders in the process we produce new examples of maximal orders these are graded and have the property that no veronese subring is generated in degree 1 our results are used in the companion paper s j sierra 2009 sie09 to give defining data for a large class of noncommutative projective surfaces

a new proof of doobʼs theorem abstract we prove that every bounded positive irreducible stochastically continuous semigroup on the space of bounded measurable functions which is strong feller consists of kernel operators and possesses an invariant measure converges pointwise this differs from doob s theorem in that we do not require the semigroup to be markovian and request a fairly weak kind of irreducibility in addition we elaborate on the various notions of kernel operators in this context show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domains the proofs are based on the theory of positive semigroups and do not use probability theory

semidefinite approximations for global unconstrained polynomial optimization we consider the problem of minimizing a polynomial function on mathbb r n known to be hard even for degree 4 polynomials therefore approximation algorithms are of interest lasserre siam j optim 11 2001 pp 796 817 and parrilo math program 96 2003 pp 293 320 have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations we propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming the method is applied to several examples

generic noncommutative surfaces abstract we study a class of noncommutative surfaces and their higher dimensional analogs which come from generic subalgebras of twisted homogeneous coordinate rings of projective space such rings provide answers to several open questions in noncommutative projective geometry specifically these rings r are the first known graded algebras over a field k which are noetherian but not strongly noetherian in other words r k b is not noetherian for some choice of commutative noetherian extension ring b this answers a question of artin small and zhang the rings r are also maximal orders but they do not satisfy all of the conditions of artin and zhang in particular they satisfy the 1 condition but not i for i 2 answering a question of stafford and zhang and a question of stafford and van den bergh finally we show that the noncommutative scheme r proj has finite global dimension

long time behavior of heat kernels of operators with unbounded drift terms abstract given a second order elliptic operator on r d with bounded diffusion coefficients and unbounded drift which is the generator of a strongly continuous semigroup on l 2 r d represented by an integral we study the time behavior of the integral kernel and prove estimates on its decay at infinity if the diffusion coefficients are symmetric a local lower estimate is also proved

generalised witt algebras and idealizers abstract let k be an algebraically closed field of characteristic zero and let be an additive subgroup of k results of kaplansky santharoubane and su classify intermediate series representations of the generalised witt algebra w in terms of three families one parameterised by a 2 and two by p 1 in this note we use the first family to construct a homomorphism from the enveloping algebra u w to a skew extension k a 2 of the coordinate ring of a 2 we show that the image of is contained in a double idealizer subring of this skew extension and that the representation theory of idealizers explains the three families we further show that the image of u w under is not left or right noetherian giving a new proof that u w is not noetherian we construct as an application of a general technique to create ring homomorphisms from shift invariant families of modules let g be an arbitrary group and let a be a g graded ring a graded a module m is an intermediate series module if m g is one dimensional for all g g given a shift invariant family of intermediate series a modules parametrised by a scheme x we construct a homomorphism from a to a skew extension of k x the kernel of consists of those elements which annihilate all modules in x

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

an existence theorem for the navier stokes flow in the exterior of a rotating obstacle we consider the three dimensional navier stokes initial value problem in the exterior of a rotating obstacle it is proved that a unique solution exists locally in time if the initial velocity possesses the regularity l 1 2 this regularity assumption is the same as that in the famous paper of fujita kato an essential step for the proof is the deduction of a certain smoothing property together with estimates near t 0 of the semigroup which is not an analytic one generated by the operator in the space l 2 where stands for the angular velocity of the rotating obstacle and p denotes the projection associated with the helmholtz decomposition

geometric idealizer rings let b b x be the twisted homogeneous coordinate ring of a projective variety x over an algebraically closed field k we construct and investigate a large class of interesting and highly noncommutative noetherian subrings of b specifically let z be a closed subscheme of x and let i b be the corresponding right ideal of b we study the subalgebra r k i of b under mild conditions on z and aut k x r is the idealizer of i in b the maximal subring of b in which i is a two sided ideal our main result gives geometric conditions on z and that determine the algebraic properties of r we say that n z is critically transverse if for any closed subscheme y of z for n 0 the sub schemes y and n z are homologically transverse we show that if n z is critically transverse then r is left and right noetherian has finite left and right cohomological dimension is strongly right noetherian but not strongly left noetherian and satisfies right d where d codim z but fails left 1 this generalizes results of rogalski in the case that z is a point in d we also give an example of a right noetherian ring with infinite right cohomological dimension partially answering a question of stafford and van den bergh further we study the geometry of critical transversality and show that it is often generic behavior in a sense that we make precise

second order cone programming relaxation of sensor network localization the sensor network localization problem has been much studied recently biswas and ye proposed a semidefinite programming sdp relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed here we study a second order cone programming socp relaxation of this problem motivated by its simpler structure and its potential to be solved faster than sdp we show that the socp relaxation though weaker than the sdp relaxation has nice properties that make it useful as a problem preprocessor in particular sensors that are uniquely positioned among interior solutions of the socp relaxation are accurate up to the square root of the distance error thus these sensors which are easily identified are accurately positioned in our numerical simulation the interior solution found can accurately position up to 80 90p of the sensors we also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interior point method