on the l r hodge theory in complete non compact riemannian manifolds we study solutions for the hodge laplace equation delta u omega on p forms with displaystyle l r estimates for displaystyle r 1 our main hypothesis is that delta has a spectral gap in displaystyle l 2 we use this to get sl non classical displaystyle l r hodge decomposition theorems an interesting feature is that to prove these decompositions we never use the boundedness of the riesz transforms in displaystyle l s par these results are based on a generalisation of the raising steps method to complete non compact riemannian manifolds

quadratic transformations on matrices rank preservers abstract letfbe an algebraically closed field of characteristic not 2 and letx xij be then nmatrix whose entriesxijare independent indeterminates overf now letq x qij x be anothern nmatrix each of whose entriesqij x is a quadraticf polynomial in thexij the main result in this paper is forn 5 q x satisfies rank a2 rimplies rank q a r for alla fn nforr 0 1 and 2 if and only if there exist invertible matricesp1 p2infn nsuch that eitherq x p1x2p2orq x p1 x2 tp2

controllability cost of conservative systems resolvent condition and transmutation abstract this article concerns the exact controllability of unitary groups on hilbert spaces with unbounded control operator it provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality by the transmutation of controls in some time l for the corresponding second order conservative system it is proved that the cost of controls in time t for the unitary group grows at most like exp l 2 t as t tends to 0 in the application to the cost of fast controls for the schrodinger equation l is the length of the longest ray of geometric optics which does not intersect the control region this article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

discretization of riemannian manifolds applied to the hodge laplacian for kappa geq 0 and r 0 0 let bbb m n kappa r 0 be the set of all connected compact n dimensional riemannian manifolds such that k g leq kappa and inj m g geq r 0 we study the relation between the k rm th positive eigenvalue of the hodge laplacian on differential forms and the k rm th positive eigenvalue of the combinatorial laplacian associated to an open cover acting on v c ech cochains we show that for a fixed sufficiently small varepsilon 0 there exist positive constants c 1 and c 2 depending only on n kappa and varepsilon such that for any m in bbb m n kappa r 0 and for any varepsilon discretization x of m we have c 1 lambda k p x leq lambda k p m leq c 2 lambda k p x for any k leq k k depends on x moreover we find a lower bound for the spectrum of the combinatorial laplacian and a lower bound for the spectrum of the hodge laplacian

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

linear rank one preservers between spaces of matrices with zero trace abstract let x y be a pair of vector spaces over a field f associated with a bilinear form such that x y 0 for all y in y implies that x 0 let x y 0 be the subspace of x y spanned by all decomposable elements x y with x y 0 let u v be any two vector spaces over f in this note we study linear mappings from x y 0 to u v that send nonzero decomposable elements to nonzero decomposable elements and some of its consequences

keller osserman conditions for diffusion type operators on riemannian manifolds abstract in this paper we obtain essentially sharp generalized keller osserman conditions for wide classes of differential inequalities of the form l u b x f u l u and l u b x f u l u g u h u on weighted riemannian manifolds where l is a non linear diffusion type operator prototypical examples of these operators are the p laplacian and the mean curvature operator the geometry of the underlying manifold is reflected via bounds for the modified bakry emery ricci curvature by growth conditions for the functions b and l a weak maximum principle which extends and improves previous results valid for the laplacian is also obtained geometric comparison results valid even in the case of integral bounds for the modified bakry emery ricci tensor are presented

collapsing with a lower bound on the curvature operator cheeger and gromov showed that f structures are related to collapse with a double sided curvature bound we define fibered f structures and extend some of the cheeger gromov results to the setting of collapse with a lower bound on the curvature operator

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

existence and non existence of area minimizing hypersurfaces in manifolds of non negative ricci curvature we study minimal hypersurfaces in manifolds of non negative ricci curvature euclidean volume growth and quadratic curvature decay at infinity by comparison with capped spherical cones we identify a precise borderline for the ricci curvature decay above this value no complete area minimizing hypersurfaces exist below this value in contrast we construct examples

linear spaces of real matrices of constant rank abstract the largest possible dimensions of linear spaces of real n n matrices of constant rank n 1 or n 2 are determined using topological k theory and expressed in terms of hurwitz radon numbers

good wannier bases in hilbert modules associated to topological insulators for a large class of physically relevant operators on a manifold with discrete group action we prove general results on the non existence of a basis of well localized wannier functions for their spectral subspaces this turns out to be equivalent to the freeness of a certain hilbert module over the group c algebra canonically associated with the spectral subspace this brings into play k theoretic methods and justifies their importance as invariants of topological insulators in physics

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

eigenvalues estimates on complete noncompact riemannian manifolds and applications we obtain some sharp estimates on the first eigenvalues of complete noncom pact riemannian manifolds under assumptions of volume growth using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures

local index theory over etale groupoids we give a superconnection proof of connes index theorem for proper cocompact actions of etale groupoids this includes connes general foliation index theo rem for foliations with hausdorholonomy groupoid

twistor spaces and the general adiabatic expansions abstract we investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of dirac operator on a twistor space when the metric is blown up in the base space direction such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants most of whose cores consist of some derivatives of the fundamental solution

sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature in this paper we consider complete noncompact riemannian manifolds m g with nonnegative ricci curvature and euclidean volume growth of dimension n ge 3 for every bounded open subset omega subset m with smooth boundary we prove that begin aligned int limits partial omega left frac mathrm h n 1 right n 1 mathrm d sigma ge mathrm avr g big mathbb s n 1 big end aligned where mathrm h is the mean curvature of partial omega and mathrm avr g is the asymptotic volume ratio of m g moreover the equality holds true if and only if m setminus omega g is isometric to a truncated cone over partial omega an optimal version of huisken s isoperimetric inequality for 3 manifolds is obtained using this result finally exploiting a natural extension of our techniques to the case of parabolic manifolds we also deduce an enhanced version of kasue s non existence result for closed minimal hypersurfaces in manifolds with nonnegative ricci curvature

the calderon zygmund inequality and sobolev spaces on noncompact riemannian manifolds abstract we introduce the concept of calderon zygmund inequalities on riemannian manifolds for 1 p these are inequalities of the form hess u l p c 1 u l p c 2 u l p valid a priori for all smooth functions u with compact support and constants c 1 0 c 2 0 such an inequality can hold or fail depending on the underlying riemannian geometry after establishing some generally valid facts and consequences of the calderon zygmund inequality like new denseness results for second order l p sobolev spaces and gradient estimates we establish sufficient geometric criteria for the validity of these inequalities on possibly noncompact riemannian manifolds these results in particular apply to many noncompact hypersurfaces of constant mean curvature

linear transformations between matrix spaces that map one rank specific set into another we characterize in several instances those linear transformations from the linear space of m n matrices into the linear space of p q matrices that map the set of matrices having a fixed rank into the set of matrices having a fixed rank examples are given showing that in contrast with the case of linear transformations on the linear space of m n matrices mapping a rank specific set into itself in the more general case of linear transformations between two full matrix spaces often one cannot expect neat and predictable results

stability of fluid queueing systems with parallel servers and stochastic capacities this note introduces a piecewise deterministic queueing pdq model to study the stability of traffic queues in parallel link transportation systems facing stochastic capacity fluctuations the saturation rate capacity of the pdq model switches between a finite set of modes according to a markov chain and link inflows are controlled by a state feedback policy a pdq system is stable only if a lower bound on the time average link inflows does not exceed the corresponding time average saturation rate furthermore a pdq system is stable if the following two conditions hold the nominal mode s saturation rate is high enough that all queues vanish in this mode and a bilinear matrix inequality involving an underestimate of the discharge rates of the pdq in individual modes is feasible the stability conditions can be strengthened for two mode pdqs these results can be used for design of routing policies that guarantee stability of traffic queues under stochastic capacity fluctuations

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

perturbation analysis of feedback controlled stochastic flow systems stochastic flow systems arise naturally or as abstractions of discrete event systems dess referred to as stochastic flow models sfms in this paper we consider such systems operating with a feedback control mechanism building on earlier work that has studied such sfms without any feedback using infinitesimal perturbation analysis we derive gradient estimators for loss and workload related performance metrics with respect to threshold parameters used for buffer control these estimators are shown to be unbiased they are also shown to depend only on data observable from a sample path of the actual des this renders them computable in on line environments and easily implementable for control and performance optimization purposes in the case of linear feedback we further show that the estimators are nonparametric finally we illustrate the use of these estimators in network control by combining them with standard gradient based stochastic optimization schemes and providing several simulation based examples

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

bilinear transformations on matrices rank preservers let a be any element in an algebraically closed field f of characteristic not 2 and let m f x f 4 f be a bilinear map on the algebra f of iz x n matrices over f we prove for n 5 that if ab aba 0 implies m a b 0 and rank ab aba 1 implies rank m a b 1 for all matrices a b e f then there exist invertible matrices p q e f such that either m x y p xy ayx q or m x y p xy ayx q 0 elsevier science inc 1997 throughout f is a field and fnx is the algebra of n x n matrices over f during the past 35 years a substantial effort has been made to solve preserver problems for linear maps on f an entire issue of linear and multilinear algebra 3 consists of recent survey articles on every aspect of the subject the linear preserver problem is to characterize all linear maps on fnxn that preserve some function of a matrix or some subset of matrices one such problem is to characterize all linear maps l such that

properties of the probability density function of the non central chi squared distribution abstract in this paper we consider the probability density function pdf of a non central 2 distribution with arbitrary number of degrees of freedom for this function we prove that can be represented as a finite sum and we deduce a partial derivative formula moreover we show that the pdf is log concave when the degrees of freedom is greater or equal than 2 at the end of this paper we present some turan type inequalities for this function and an elegant application of the monotone form of l hospital s rule in probability theory is given

coherence invariant mappings on block triangular matrix spaces abstract in this paper we classify bijective mappings on the space of block upper triangular matrices such that both and 1 preserve pairs of matrices whose differences have rank one applications to rank one preservers and semi isomorphisms are considered

boundedness of spectral multipliers of generalized laplacians on compact manifolds with boundary consider a second order strongly elliptic negative semidefinite differential operator l may be a system on a compact riemannian manifold overline m with smooth boundary where the domain of l is defined by a coercive boundary condition classically known results and also recent work in duong et al j funct anal 196 443 485 2002 and duong and mcintosh rev math iberoam 15 233 265 1999 establish sufficient conditions for l infty text bmo l continuity of varphi sqrt l where varphi in s 0 1 mathbb r and l is a suitable elliptic operator using a variant of the cheeger gromov taylor functional calculus due to mauceri et al math res lett 16 861 879 2009 and short time upper bounds on the integral kernel of e tl due to greiner arch ration mech anal 41 168 218 1971 we prove that a variant of such sufficient conditions holds for our operator l

a fluid model for systems with random disruptions motivated by modeling manufacturing systems in which job arrivals and processing times are essentially deterministic but the environment is typically random we develop a fluid model with random disruptions the quality and relevance of such a model are supported by the following facts which we establish in this study the fluid model is more susceptible to analysis its dynamical sample paths are continuous and piecewise linear and its stationary behavior can be studied using standard approaches in random walk the model is a limit of corresponding queueing systems with random disruptions that are usually difficult to analyze there exist pathwise bounds between the fluid model and d d 1 queues with random disruptions and the bounds are simple and tight

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

additive mappings that preserve rank one nilpotent operators let b x be the algebra of bounded operators on a complex infinite dimensional banach space x and f x be the subalgebra of all finite rank operators in b x a characterization of additive mappings on f x which preserve rank one nilpotent operators in both directions is given as applications of this result some additive preservers are described

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

a dynamic network flow problem with uncertain arc capacities formulation and problem structure we consider a dynamic network flow problem where the arc capacities are random variables this gives a multistage stochastic linear program we describe the randomness using a multi scenario approach because of the multilayered structure there are several ways to decompose the linear program we classify different decomposition schemes and we develop a scheme calledcompath decomposition which is derived from path decomposition for network flows we give a polynomial time algorithm to find a cheapest compath that can solve the subproblems resulting from compath decomposition finally compath decomposition is extended to multicommodity flow problems