large deviations for stochastic pde with levy noise abstract we prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small levy noise we use general large deviations theorems of varadhan and bryc coupled with the techniques of feng and kurtz 2006 15 viscosity solutions of integro partial differential equations in hilbert spaces and deterministic optimal control methods the laplace limit is identified as a viscosity solution of a hamilton jacobi bellman equation of an associated control problem we also establish exponential moment estimates for solutions of stochastic evolution equations driven by levy noise general results are applied to stochastic hyperbolic equations perturbed by subordinated wiener process

convexity of sample path performance and strong consistency of infinitesimal perturbation analysis estimates a strong consistency of infinitesimal perturbation analysis ipa estimates for a class of stochastic processes satisfying certain convex properties is established the approach is based on convex analysis of sample paths applications of the results to tandem and cyclic queueing systems are discussed possible extensions to more general systems are also illustrated

harmonic maps and the schoen conjecture we show that every quasisymmetric homeomorphism of the circle h 2 admits a harmonic quasiconformal extension to the hyperbolic plane h 2 this proves the schoen conjecture

approximating an optimal production policy in a continuous flow line recurrence and asymptotic properties this work is concerned with manufacturing systems with two failure prone tandem machines the production is regulated by a continuous version of buffer control our goal is to obtain an optimal buffer control policy to minimize a long run average cost function concentrating on threshold type of control policies our effort is devoted to parameter optimization problems for the continuous material produce to stock models we estimate the gradients of the cost function with respect to the parameter using perturbation analysis techniques and approximate the optimal value of the parameter via a constant step size stochastic approximation algorithm an analysis for error accumulation in perturbation propagation is undertaken and a sufficient condition for breaking the propagation chain is derived in addition we show that the event of breaking the perturbation propagation chain is recurrent if the system has sufficient capacity derive the consistency of the gradient estimators and establish the convergence of the iterative algorithm we also treat non markovian models with the machine repair time following an erlang distribution and provide numerical examples to illustrate the proposed algorithm

angular derivatives of quasiconformal harmonic maps on the poincare disk let s 1 be the boundary of the open unit disk mathbb d and let h be a quasisymmetric homeomorphism from the unit circle s 1 onto itself let h be the quasiconformal harmonic extension of h to mathbb d with respect to the poincare metric in this paper it is shown that if h zeta alpha ne 0 at zeta in s 1 then when z rightarrow zeta in mathbb d non tangentially begin aligned lim z rightarrow zeta frac h z h zeta z zeta alpha end aligned and the complex derivatives h z z and h bar z z approach alpha and 0 respectively i e h has an angular derivative alpha at zeta conversely if h has a non tangential derivative alpha ne 0 at zeta then h zeta alpha and hence h has an angular derivative alpha at zeta

structural properties of optimal production controllers in failure prone manufacturing systems a failure prone manufacturing system with one part type and multiple machine states is considered each machine state has a given production capacity but is associated with several possible failure stages although transition times between failure stages are assumed exponential the existence of multiple stages within each machine state allows failure times to have a distribution more general than the exponential distribution for such a system it is shown that the well known hedging point policy minimizes inventory and backlog costs for a constant demand rate the value of the optimal hedging points is now a function of failure stage as well as machine state by ordering machine states according to increasing distance from the zero capacity state and failure stages according to distance from the capacity reduction transition we show that hedging points increase monotonically when the machine state number decreases or the stage number increases this can be intuitively interpreted as follows the closer to the zero capacity state the system is the larger the hedging point should be this structural property of the optimal control is very useful in searching for the optimal control or designing near optimal controls

on the quasi isometries of harmonic quasiconformal mappings abstract we prove versions of the ahlfors schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the poincare metric

a schwarz pick inequality for harmonic quasiconformal mappings and its applications abstract the main result of this paper is the sharp generalized schwarz pick inequality for euclidean harmonic quasiconformal mappings with convex ranges which generalizes a result given by mateljevic as its applications we obtain the property of quasi isometry with respect to the poincare distance and an analogue of the koebe theorem for this class of mappings

harmonic diffeomorphisms between hadamard manifolds in this paper we study the dirichlet problem at infinity for harmonic maps between complete hyperbolic hadamard surfaces we will address the existence and uniqueness questions relating to the problem in particular we generalize results in the work of li tam and wan 0 introduction in a series of papers l t 1 l t 2 and l t 3 the first two authors considered the existence uniqueness and boundary regularity theory for the dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces if we denote em to be ther m dimensional hyperbolic space with constant 1 sectional curvature then using the poincare disk model we can represent hm by the unit m disk d m endowed with the poincare metric ds2m 4 ds2 where p denotes the euclidean distance to the origin and ds2 is the euclidean metric the geometric compactification is given by the natural compactification of ledm by the unit m 1 sphere sm 1 using this identification a map from sm 1 to s 1 can be viewed as a map from infinity of h m to infinity of hn let us summarize some of the results in l t 1 l t 2 and l t 3 as follows theorem li tam let q s m 1 n 1 be a c1 map which has nowhere vanishing energy density then there exists a unique harmonic extension h hm h hn of x which is in ci lrd m edn also if x is in ck a s m l s n 1 for some 1 k m and some 0 a then h is in ck y ld d ledn for some o y a we would like to mention that the existence part of this theorem for the special case when m 2 n and for c3 boundary maps which have nowhere vanishing energy density was independently proved by akutagawa a using a different point of view wan w studied harmonic diffeomorphisms between hyperbolic 2 spaces via their associated hopf differentials by using the fact that c t the gauss map of a constant mean curvature cut in a minkowski space time is a harmonic map into hyperbolic space wan established that there is a one to one correspondence between the set of equivalent classes of constant mean curvature cuts and the equivalent classes of quadratic differentials in h2 received by the editors december 12 1994 1991 mathematics subject classification primary 58e20 the authors were partially supported by nsf grant dms9300422 c 1995 american mathematical society

harmonic maps and constant mean curvature surfaces in h 2 x r we introduce a it hyperbolic gauss map into the poincar e disk for any surface in bbb h 2 times bbb r with regular vertical projection and prove that if the surface has constant mean curvature h 1 2 this hyperbolic gauss map is harmonic conversely we show that every nowhere conformal harmonic map from an open simply connected riemann surface sigma into the poincar e disk is the hyperbolic gauss map of a two parameter family of such surfaces as an application we obtain that any holomorphic quadratic differential on sigma can be realized as the abresch rosenberg holomorphic differential of some and generically infinitely many complete surfaces with h 1 2 in bbb h 2 times bbb r a similar result applies to minimal surfaces in the heisenberg group rm nil 3 finally we classify all complete minimal vertical graphs in bbb h 2 times bbb r

production control of manufacturing systems with production rate dependent failure rates it is known that for single part type production systems with homogeneous markovian machine failure rates special single threshold feedback policies called hedging point policies are optimal and the stationary probability distribution of the part type surplus for given tentative hedging point values can be obtained analytically this approach is extended to multiple threshold policies with production rate dependent machine failure rates it is shown that the stationary distribution of the part type surplus can be obtained under the extended policy and in the presence of production rate dependent failure rates the advantage of multiple threshold policies is that they can provide a piecewise constant step function approximation of any feedback policy it is observed that hedging point policies are not always optimal and in fact feedback policies are not always optimal either

holomorphic representation of constant mean curvature surfaces in minkowski space consequences of non compactness in loop group methods abstract we give an infinite dimensional generalized weierstrass representation for spacelike constant mean curvature cmc surfaces in minkowski 3 space r 2 1 the formulation is analogous to that given by dorfmeister pedit and wu for cmc surfaces in euclidean space replacing the group su 2 with su 1 1 the non compactness of the latter group however means that the iwasawa decomposition of the loop group used to construct the surfaces is not global we prove that it is defined on an open dense subset after doubling the size of the real form su 1 1 and prove several results concerning the behavior of the surface as the boundary of this open set is encountered we then use the generalized weierstrass representation to create and classify new examples of spacelike cmc surfaces in r 2 1 in particular we classify surfaces of revolution and surfaces with screw motion symmetry as well as studying another class of surfaces for which the metric is rotationally invariant

strong solutions for a stochastic model of two dimensional second grade fluids driven by levy noise abstract in this paper we consider stochastic second grade fluids driven by levy noise on a bounded domain of r 2 global existence and uniqueness of strong probabilistic solutions are established which improves the existing corresponding results for martingale solutions in literature

scheduling multiple part types in an unreliable single machine manufacturing system quadratic approximations to the differential cost to go function which yield linear switching curves have been extensively studied in the literature in this paper we provide solutions to the partial differential equations associated with the components of the steady state probability density function of the buffer levels for two part type single machine flexible manufacturing systems under a linear switching curve lsc policy when there are more than two part types we derive the probability density function under a prioritized hedging point php policy by decomposing the multiple part type problem into a sequence of two part type problems the analytic expression for the steady state probability density function is independent of the cost function therefore for average cost functions we can compute the optimal php policy or the more general optimal lsc policy for two part type problems

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

perturbation analysis of discrete event systems concepts algorithms and applications abstract perturbation analysis of discrete event systems is a technique that provides estimates of performance sensitivities with respect to system parameters by analyzing a single sample path of a discrete event system this technique saves a great amount of computation in simulation and can be applied to on line performance optimization of real world systems where changing the values of parameters is infeasible this paper gives an overview of the main concepts general methodologies applications and further directions of the area

harmonic degree 1 maps are diffeomorphisms lewy s theorem for curved metrics in 1936 h lewy showed that the jacobian determinant of a harmonic homeomorphism between planar domains does not vanish and thus the map is a diffeomorphism this built on the earlier existence

a note on harmonic quasiconformal mappings abstract in this note we show that a harmonic quasiconformal mapping f u i v with respect to the poincare metric of the upper half plane r 2 onto itself such that v x y v y or u x y u x is a conformal mapping

stochastic tamed 3d navier stokes equations existence uniqueness and ergodicity in this paper we prove the existence of a unique strong solution to a stochastic tamed 3d navier stokes equation in the whole space as well as in the periodic boundary case then we also study the feller property of solutions and prove the existence of invariant measures for the corresponding feller semigroup in the case of periodic conditions moreover in the case of periodic boundary and degenerated additive noise using the notion of asymptotic strong feller property proposed by hairer and mattingly ann math 164 993 1032 2006 we prove the uniqueness of invariant measures for the corresponding transition semigroup

optimal average cost manufacturing flow controllers convexity and differentiability the authors consider the control of a production facility consisting of a single workstation with multiple failure modes and part types using a continuous flow control model technical issues concerning the convexity and differentiability of the differential cost function are investigated it is proven that under an optimal control policy the differential cost is c sup 1 on attractive control switching boundaries

structural conditions for perturbation analysis derivative estimation finite time performance indices in recent years there has been a surge of research into methods for estimating derivatives of performance measures from sample paths of stochastic systems in the case of queueing systems typical performance measures are mean queue lengths throughputs etc and the derivatives estimated are with respect to system parameters such as parameters of service and interarrival time distributions derivative estimates potentially offer a general means of optimizing performance and are useful in sensitivity analysis this paper concerns one approach to derivative estimation known as infinitesimal perturbation analysis we first develop a general framework for these types of estimates then give simple sufficient conditions for them to be unbiased the key to our results is identifying conditions under which certain finite horizon performance measures are almost surely continuous functions of the parameter of differentiation throughout an interval the sufficient conditions we introduce are formulated in the setting of generalized semi markov processes but translate into readily verifiable conditions for queueing systems these results substantially extend the domain of problems in which infinitesimal perturbation analysis is provably applicable

existence of global solutions and invariant measures for stochastic differential equations driven by poisson type noise with non lipschitz coefficients the purpose of this paper is twofold firstly we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi martingales secondly we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent

2d stochastic navier stokes equations driven by jump noise abstract in the paper we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of levy type our result is formulated in an abstract setting this type of equation covers the stochastic 2d navier stokes equations the 2d stochastic magneto hydrodynamic equations the 2d stochastic boussinesq model for the benard convection the 2d stochastic magnetic bernard problem the 3d stochastic leray model for the navier stokes equations and several stochastic shell models of turbulence

a manufacturing system with general stationary failure process stability and ipa of hedging control policies this paper concerns a new methodology for the adaptive optimization of piecewise deterministic non markovian systems via a simple example of interest in manufacturing this methodology takes into account the fact that piecewise deterministic systems are rarely markovian and that classical control theory based on dynamic programming of markovian systems cannot provide quantitative answers in most realistic situations the example considered is that of a single machine single part production system from the point of view of the control theory by kimemia and gershwin 1983 we obtain stochastic gradient estimates via the perturbation analysis of ho and cao 1991 by the regenerative method of konstantopoulos and zazanis 1992 in view of stochastic optimization its mathematical justification requires a careful study of the regenerative structure of the process in particular we give the necessary and sufficient conditions of stability of this system via the method of loynes 1962

generalized dpw method and an application to isometric immersions of space forms let g be a complex lie group and g denote the group of maps from the unit circle mathbb s 1 into g of a suitable class a differentiable map f from a manifold m into g is said to be of connection order a b if the fourier expansion in the loop parameter of the mathbb s 1 family of maurer cartan forms for f namely f lambda 1 rm d f lambda is of the form sum i a b alpha i lambda i most integrable systems in geometry are associated to such a map roughly speaking the dpw method used a birkhoff type splitting to reduce a harmonic map into a symmetric space which can be represented by a certain order 1 1 map into a pair of simpler maps of order 1 1 and 1 1 respectively conversely one could construct such a harmonic map from any pair of 1 1 and 1 1 maps this allowed a weierstrass type description of harmonic maps into symmetric spaces we extend this method to show that for a large class of loop groups a connection order a b map for a 0 b splits uniquely into a pair of a 1 and 1 b maps as an application we show that constant non zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds reducing the problem at least locally to the flat case to extend the dpw method sufficiently to handle this problem requires a more general iwasawa type splitting of the loop group which we prove always holds at least locally

optimal control of production rate in a failure prone manufacturing system we address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost where certain cost rates are specified for both positive and negative inventories and there is a constant demand rate for the commodity produced the underlying theoretical problem is the optimal control of a continuous time system with jump markov disturbances with an infinite horizon discounted cost criterion we use two complementary approaches first proceeding informally and using a combination of stochastic coupling linear system arguments stable and unstable eigenspaces renewal theory parametric optimization etc we arrive at a conjecture for the optimal policy then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides singularity of the optimal control problem smoothness and validity of the dynamic programming equation etc to give a rigorous proof of optimality of the conjectured policy it is hoped that both approaches will find uses in other such problems also we obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number which we call the optimal inventory level if the current inventory level exceeds the optimal one should not produce at all if less one should produce at the maximum rate while if exactly equal one should produce exactly enough to meet demand we also give a simple explicit formula for the optimal inventory level

global l2 solutions of stochastic navier stokes equations this paper concerns the cauchy problem in r d for the stochastic navier stokes equation 1 u u u u p f u u p g u o w u 0 u 0 div u 0 driven by white noise w under minimal assumptions on regularity of the coefficients and random forces the existence of a global weak martingale solution of the stochastic navier stokes equation is proved in the two dimensional case the existence and pathwise uniqueness of a global strong solution is shown a wiener chaos based criterion for the existence and uniqueness of a strong global solution of the navier stokes equations is established

dynamics and design of a class of parameterized manufacturing flow controllers the optimal production control of failure prone flexible manufacturing systems has received considerable attention analytically intractable optimality conditions render near optimal controller design the only available option for realistic size systems this paper deals with a class of suboptimal feedback control policies that are parameterized over a finite set in depth analysis of the dynamics of the controlled system is undertaken and important properties of these dynamics are proven for the first time for multiple part type systems these properties provide the theoretical justification of an infinitesimal perturbation analysis based controller parameter optimization technique whose validity has been previously supported by empirical evidence

harmonic diffeomorphisms between the annuli with rotational symmetry abstract we give some necessary and sufficient conditions for the existence of rotationally symmetric harmonic diffeomorphism between the annuli with poincare metric or euclidean metric on the target respectively

perturbation analysis for the design of flexible manufacturing system flow controllers dynamic allocation of stochastic capacity among competing activities in a just in time manufacturing environment is addressed by optimal flow control optimal policies are characterized by generally intractable bellman equations a near optimal controller design technique is proposed it provides an approximate numerical solution to the bellman equation a tight lower bound for the optimality gap of tractable near optimal controller designs and a building block for improved near optimal controller designs that rely on the decomposition of a multiple part type problem to smaller two or three part type problems computational experience is reported for two and three part type problems

harmonic maps between 3 dimensional hyperbolic spaces we prove that a quasiconformal map of the sphere s 2 admits a harmonic quasi isometric extension to the hyperbolic space h 3 thus confirming the well known schoen conjecture in dimension 3

production control of a manufacturing system with multiple machine states the production control of a single product manufacturing system with arbitrary number of machine states failure modes is discussed the objective is to find a production policy that would meet the demand for the product with minimum average inventory or backlog cost the optimal production policy has a special structure and is called a hedging point policy if the hedging points are known the optimal production rate is readily specified assuming a set of tentative hedging points the simple structure of the optimal policy is utilized to find the steady state probability distribution of the surplus inventory or backlog once this function is determined the average surplus cost is easily calculated in terms of the values of the hedging points the average cost is then minimized to find the optimum hedging points

3d tamed navier stokes equations driven by multiplicative levy noise existence uniqueness and large deviations abstract in this paper we show the existence and uniqueness of a strong solution to stochastic 3d tamed navier stokes equations driven by multiplicative levy noise with periodic boundary conditions then we establish the large deviation principles of the strong solution on the state space d 0 t h 1 where the weak convergence approach plays a key role

optimal release times in a single server an optimal control perspective the paper is concerned with the basic structure of optimal control of discrete event dynamic processes defined over a max plus algebra only a simple system is considered namely a single server processing a given sequence of jobs but the structural conditions that are discovered may lead to extensions to more general systems the problem in question is how to optimally control the completion output times of the jobs by assigning their release input times so as to minimize a measure of the discrepancy between the completion times and given desired due dates the concept of the costate is applied to the discrete dynamics to identify structural optimality conditions and in the case of quadratic cost measures the optimal control is shown to be computable by a state feedback law that is linear in the max plus algebra

production control of manufacturing systems a multiple time scale approach in this note a stochastic production model containing processes with different time scales is developed it is shown that if the time scales of the processes are very different some hierarchical algorithms that are much more efficient than the standard policy iteration method can be developed to find the optimal production control moreover if the time scales fall far apart the optimal control of a deterministic limiting problem depending only on the mean characteristics of the processes can be used to approximate the optimal control of the original problem the limiting problem has much lower dimension than its original counterpart and thus is much easier to solve a numerical example is used to illustrate the potential of the proposed approach

on quasiconformal self mappings of the unit disk satisfying poisson s equation let qc k g be a family of k quasiconformal mappings of the open unit disk onto itself satisfying the pde w g g c u w 0 0 it is proved that qc k g is a uniformly lipschitz family moreover if g is small enough then the family is uniformly bi lipschitz the estimations are asymptotically sharp as k 1 and g 0 so w qc k g behaves almost like a rotation for sufficiently small k and g

spacelike hypersurfaces with prescribed boundary values and mean curvature we consider the boundary value problem for the mean curvature operator in minkowski space and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions our main results hold for non constant mean curvature and make no assumptions about the smoothness of the boundary or boundary data

harmonic maps and classical surface theory in minkowski 3 space harmonic maps from a surface s with nondegenerate prescribed and induced metrics are characterized showing that holomorphic quadratic differentials play the same role for harmonic maps from a surface with indefinite prescribed metric as they do in the riemannian case moreover holomorphic quadratic differentials are shown to arise as naturallv on surfaces of constant h or k in m3 as on their counterparts in e3 the connection between the sine gordon sinh gordon and cosh gordon equations and harmonic maps is explained various local and global results are established for surfaces in m3 with constant h or constant k 0 in particular the gauss map of a spacelike or timelike surface in m3 is shown to be harmonic if and only if h is constant also k is shown to assume values arbitrarilv close to h2 on any entire spacelike surface in m3 with constant h except on a hyperbolic cylinder

perturbation realization potentials and sensitivity analysis of markov processes two fundamental concepts and quantities realization factors and performance potentials are introduced for markov processes the relations among these two quantities and the group inverse of the infinitesimal generator are studied it is shown that the sensitivity of the steady state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a markov process based on these results algorithms for estimating performance sensitivities on a single sample path of a markov process can be proposed the potentials in this paper are defined through realization factors and are shown to be the same as those defined by poisson equations the results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis ipa and non ipa approaches to the sensitivity analysis of steady state performance they also provide a theoretical background for the pa algorithms developed in recent years

a numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system abstract the simultaneous planning of the production and the maintenance in a flexible manufacturing system is considered in this paper the manufacturing system is composed of one machine that produces a single product there is a preventive maintenance plan to reduce the failure rate of the machine this paper is different from the previous researches in this area in two separate ways first the failure rate of the machine is supposed to be a function of its age second we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product the objective is to maximize the expected discounted total profit of the firm over an infinite time horizon in the process of finding a solution to the problem we first characterize an optimal control by introducing a set of hamilton jacobi bellman partial differential equations then we realize that under practical assumptions this set of equations can not be solved analytically thus to find a suboptimal control we approximate the original stochastic optimal control model by a discrete time deterministic optimal control problem then proposing a numerical method to solve the steady state riccati equation we approximate a suboptimal solution to the problem

holomorphic quadratic differentials and the bernstein problem in heisenberg space we classify the entire minimal vertical graphs in the heisenberg group nil 3 endowed with a riemannian left invariant metric this classification which provides a solution to the bernstein problem in nil 3 is given in terms of the abresch rosenberg holomorphic differential for minimal surfaces in nil 3

stochastic 3d tamed navier stokes equations existence uniqueness and small time large deviation principles abstract in this paper we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3d tamed navier stokes equations in case of dirichlet boundary conditions for the stokes laplacian then we prove a small time large deviation principle for the solutions

stochastic dynamic job shops and hierarchical production planning this paper presents an asymptotic analysis of hierarchical production planning in a general manufacturing system consisting of a network of unreliable machines producing a variety of products the concept of a dynamic job shop is introduced by interpreting the system as a directed graph and the structure of the system dynamics is characterized for its use in the asymptotic analysis the optimal control problem for the system is a state constrained problem since the number of parts in any buffer between any two machines must remain nonnegative a limiting problem is introduced in which the stochastic machine capacities are replaced by corresponding equilibrium mean capacities as the rate of change in machine states approaches infinity the value function of the original problem is shown to converge to that of the limiting problem and the convergence rate is obtained furthermore near optimal controls for the original problem are constructed from near optimal controls of the limiting problem and an error estimate is obtained on the near optimality of the constructed controls

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness