geometry of differential operators on weyl manifolds we define natural operators of laplace type for a weyl manifold which transform conformally we use the asymptotics of the heat equation for these operators to construct global invariants in weyl geometry

exponential bounds for discrete time singularly perturbed markov chains abstract this paper develops exponential type upper bounds for scaled occupation measures of singularly perturbed markov chains in discrete time by considering two time scale in the markov chains asymptotic analysis is carried out the cases of the fast changing transition probability matrix is irreducible and that are divisible into l ergodic classes are examined first the upper bounds of a sequence of scaled occupation measures are derived then extensions to markov chains involving transient states and or nonhomogeneous transition probabilities are dealt with the results enable us to further our understanding of the underlying markov chains and related dynamic systems which is essential for solving many control and optimization problems

exotic smooth structures on small 4 manifolds let m be either mathbb cp 2 3 overline mathbb cp 2 or 3 mathbb cp 2 5 overline mathbb cp 2 we construct the first example of a simply connected irreducible symplectic 4 manifold that is homeomorphic but not diffeomorphic to m

generalized petrov galerkin methods for the numerical solution of burgers equation generalized petrov galerkin methods are applied in two different ways for the numerical solution of burgers equation in the first place it is used to produce upwinding in the way described by mitchell and griffiths 9 10 the main objective of this paper is in the second place to use it as a moving finite element method in particular it is shown that the methods of miller and co workers4 6 and herbst et al 7 8 can be incorporated in the more general approach presented in this paper several choices for the test functions are discussed and numerical results are given

cesaro limits of analytically perturbed stochastic matrices abstract let p 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let p e p 0 a e be an analytic perturbation which is still stochastic for small enough positive e we determine the limit lim e 0 1 n e l 1 n e p l e when n e at various rates

determination of a riemannian metric from the first variation of its spectrum on determine la metrique g par une etude du comportement d ordre 1 du spectre sous perturbations de la metrique inconnue g par des 2 tenseurs covariants symetriques connus

integral kahler invariants and the bergman kernel asymptotics for line bundles abstract on a compact kahler manifold one can define global invariants by integrating local invariants of the metric assume that a global invariant thus obtained depends only on the kahler class then we show that the integrand can be decomposed into a chern polynomial the integrand of a chern number and divergences of one forms which do not contribute to the integral we apply this decomposition formula to describe the asymptotic expansion of the bergman kernel for positive line bundles and to show that the cr q curvature on a sasakian manifold is a divergence

efficient numerical techniques for burgers equation this paper presents new efficient numerical techniques for solving one dimensional quasi linear burgers equation burgers equation is used as a model problem in the study of turbulence boundary layer behavior shock waves convection dominated diffusion phenomena gas dynamics acoustic attenuation in fog and continuum traffic simulation using a non linear cole hopf transformation the burgers equation is reduced to one dimensional diffusion equation the linearized diffusion equation is semi discretized by using method of lines mol which leads to a system of ordinary differential equations in time resulting system of ordinary differential equations is solved by backward differentiation formulas bdf of order one two and three and the analysis of numerical errors are presented numerical results for modest values of kinematic viscosity are compared with the exact solution to demonstrate the efficiency of proposed numerical methods

control of a hybrid stochastic system abstract we consider in this paper a continuous time stochastic hybrid control system with a finite time horizon the objective is to minimize a linear function of the expected state trajectory the state evolves according to a linear dynamics however the parameters of the state evolution equation may change at discrete times according to a controlled markov chain which has finite state and action spaces we use a procedure similar in form to the maximum principle this determines a control strategy which is asymptotically optimal as the number of transitions during the finite time horizon grows to infinity

a moving petrov galerkin method for transport equations a time dependent transport equation is solved numerically by using a petrov galerkin technique on a time dependent grid a piecewise linear trial solution and piecewise hermite cubic test functions are used the same method is also applied to a time independent problem in which case it automatically generates a grid in a way which bears some resemblance to certain existing grid generation techniques several numerical examples are given

the donaldson witten function for gauge groups of rank larger than one we study correlation functions in topologically twisted d 4 supersymmetric yang mills theory for gauge groups of rank larger than one on compact four manifolds x we find that the topological invariance of the generator of correlation functions of brst invariant observables is not spoiled by noncompactness of field space we show how to express the correlators on simply connected manifolds of b 2 x 0 in terms of seiberg witten invariants and the classical cohomology ring of x for manifolds x of simple type and gauge group su n we give explicit expressions of the correlators as a sum over 1 vacua we describe two applications of our expressions one to superconformal field theory and one to large n expansions of su n d 4 supersymmetric yang mills theory

from ginzburg landau to hilbert einstein via yamabe abstract in this paper based on some mathematical results obtained by yamabe osgood phillips and sarnak we demonstrate that in dimensions three and higher the famous ginzburg landau equations used in the theory of phase transitions can be obtained without any approximations by minimization of the riemannian type hilbert einstein action functional for pure gravity in the presence of cosmological term we use this observation in order to bring to completion the work by lifshitz done in 1941 on group theoretical refinements of the landau theory of phase transitions in addition this observation allows us to develop a systematic extension to higher dimensions of known string theoretic path integral methods developed for calculation of observables in two dimensional conformal field theories

logarithmic singularity of the szego kernel and a global invariant of strictly pseudoconvex domains this paper is a continuation of fefferman s program 7 for studying the geometry and analysis of strictly pseudoconvex domains the key idea of the program is to consider the bergman and szeg kernels of the domains as analogs of the heat kernel of riemannian manifolds in riemannian or confor mai geometry the coefficients of the asymptotic expansion of the heat kernel can be expressed in terms of the curvature of the metric by integrating the co efficients one obtains index theorems in various settings for the bergman and szeg kernels there has been much progress made on the description of their asymptotic expansions based on invariant theory 7 1 15 we now seek for invariants that arise from the integral of the coefficients of the expansions we here prove that the integral of the coefficient of the logarithmic sin gularity of the szeg kernel gives a biholomorphic invariant of a domain ft or a cr invariant of the boundary dft and moreover that the invariant is un changed under perturbations of the domain theorem 1 we also show that the same invariant appears as the coefficient of the logarithmic term of the volume expansion of the domain with respect to the bergman volume element theorem 2 this second result is an analogue of the derivation of a conformai invariant from the volume expansion of conformally compact einstein mani folds which arises in the ads cft correspondence see 10 for a discussion and references the proofs of these results are based on kashiwara s microlocal analysis of the bergman kernel in 17 where he showed that the reproducing prop erty of the bergman kernel on holomorphic functions can be quantized to a reproducing property of the microdifferential operators i e classical ana lytic pseudodifferential operators this provides a system of microdifferential equations that characterizes the singularity of the bergman kernel which can be formulated as a microfunction up to a constant multiple such an argument

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

the geometric β function in curved space time under operator regularization in this paper i compare the generators of the renormalization group flow or the geometric functions for dimensional regularization and operator regularization i then extend the analysis to show that the geometric function for a scalar field theory on a closed compact riemannian manifold is defined on the entire manifold i then extend the analysis to find the generator of the renormalization group flow to conformally coupled scalar field theories on the same manifolds the geometric function in this case is not defined

invariants of conformal laplacians the conformal laplacian d d d n 2 s 4 n 1 acting on functions on a riemannian manifold mn with scalar curvature s is a conformally invariant operator in this paper we will use d to construct new conformal invariants one of these is a pointwise invariant one is the integral of a local expression and one is a nonlocal spectral invariant derived from functional determinants we begin in 1 by describing the laplacian d and its green function in the context of conformal geometry we then derive a basic formula giving the variation in the heat kernel of d this formula is strikingly simpler than the corresponding formula for the ordinary laplacian given by ray and singer 15 the heat kernel of d has an asymptotic expansion k t x x 4 t n 2 ak x tk in 2 we prove that a n 2 2 is a pointwise conformal invariant of weight 2 i e it satisfies a n 2 2 x 2g 2a n 2 2 x g where g is the metric and is any smooth positive function in particular this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold the key to the proof is to consider the parametrix of the green s function which is obtained from the heat kernel by an integral transform one finds that a n 2 2 occurs as the coefficient of the first log term in this parametrix and its conformal invariance then follows directly from the conformal invariance of the green s function in 3 we show that an 2 is a global conformal invariant the calculations in 4 show that it is not a pointwise invariant the proof is a direct calculation of the invariant of an 2 using equation 1 10

on the orders of periodic diffeomorphisms of 4 manifolds this paper initiated an investigation on the following question sup pose a smooth 4 manifold does not admit any smooth circle actions does there exist a constant c 0 such that the manifold support no smooth zp actions of prime order for p c we gave affirmative results to this question for the case of holomorphic and symplectic actions with an interesting finding that the constant c in the holomorphic case is topological in nature while in the symplectic case it involves also the smooth structure of the manifold

singularly perturbed discrete time markov chains originating from a wide range of applications in optimization and control of large scale systems such as telecommunications queueing networks and manufacturing systems this work is devoted to a class of singularly perturbed discrete time markov chains the states of the markov chain are naturally decomposable into recurrent and transient classes such that within each class the interactions are strong and among different classes the interactions are weak our study focuses on the difference equations representing the probabilityvector and aims at deriving matched asymptotic expansions of the solutions justification and error analysis are also provided both time homogeneous and time inhomogeneous models are examined

a novel numerical scheme for solving burgers equation abstract in this paper we propose a novel numerical scheme for solving burgers equation the scheme is based on a cubic spline quasi interpolant and multi node higher order expansion which make the algorithm simple and easy to implement the numerical experiments show that the proposed method produces high accurate results

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

the geometry of the yang mills moduli space for definite manifolds on etudie la geometrie riemannienne de l espace des modules des connexions auto duales su 2 k 1 sur une classe tres generale de 4 varietes des 4 varietes compactes orientees simplement connexe m a forme d intersection definie positive

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

simple time and space adaptation in one dimensional evolutionary partial differential equations a simple technique for time and space adaptation in one dimensional evolutionary partial differential equations suggested by sanz serna and christie 1 is tested it is found that the equidistribution of hu xi greatly improves on the equidistribution of arc length used by those authors the time step control is found to perform poorly in the integration of rough solutions and the reasons for this behaviour are analysed

smooth s cobordisms of elliptic 3 manifolds the main result of this paper states that a symplectic s cobordism of elliptic 3 manifolds is diffeomorphic to a product assuming a canonical contact structure on the boundary based on this theorem we conjecture that a smooth s cobordism of elliptic 3 manifolds is smoothly a product if its universal cover is smoothly a product we explain how the conjecture fits naturally into the program of taubes of constructing symplectic structures on an oriented smooth 4 manifold with b 1 from generic self dual harmonic forms the paper also contains an auxiliary result of independent interest which generalizes taubes theorem sw gr to the case of symplectic 4 orbifolds

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

p version least squares finite element formulation of burgers equation a p version least squares finite element formulation for non linear problems is presented and applied to the steady state one dimensional burgers equation the second order equation is recast as a set of first order equations which permit the use of c0 elements the primary and auxiliary variables are approximated using equal order p version hierarchical approximation functions the system of non linear simultaneous algebraic equations resulting from the least squares process is solved using newton s method with a line search the use of exact and reduced quadrature rules is investigated and the results are compared the formulation is found to produce excellent results when the exact integration rule is used the combination of least squares finite element formulation and p version works extremely well for burgers equation and appears to have great potential in fluid dynamics problems

risk sensitive production planning of stochastic manufacturing systems a singular perturbation approach this paper is concerned with robust production planning of a stochastic manufacturing system in which the rate of machine breakdown and repair is much larger than the rate of fluctuation in demand it is shown that the risk sensitive production planning problem can be approximated by a limiting problem in which the stochastic machine availability process is replaced by its mean availability the value function of the limiting problem is shown to be the only viscosity solution to the isaacs equation associated with a zero sum two player differential game near optimal production plans are constructed from near optimal controls of the limiting problem finally these results are extended to problems with state constraints

anomalies associated to the polar decomposition of gl n c let d be a selfadjoint elliptic differential operator on a hermitian bundle over a compact manifold for positive d the variation of the functional determinant of d under positive definite hermitian gauge transformations is calculated this corresponds to computing a gauge anomaly in the nonunitary directions of the polar decomposition of the frame bundle gl e the variation of the eta invariant for general d is also calculated if d is not selfadjoint the integrand in the heat equation proof of the atiyah singer index theorem is interpreted as an anomaly for d d

small lefschetz fibrations and exotic 4 manifolds we explicitly construct genus 2 lefschetz fibrations whose total spaces are minimal symplectic 4 manifolds homeomorphic to complex rational surfaces mathbb cp 2 p overline mathbb cp 2 for p 7 8 9 and to 3 mathbb cp 2 q overline mathbb cp 2 for q 12 ldots 19 complementarily we prove that there are no minimal genus 2 lefschetz fibrations whose total spaces are homeomorphic to any other simply connected 4 manifold with b le 3 with one possible exception when b 3 meanwhile we produce positive dehn twist factorizations for several new genus 2 lefschetz fibrations with small number of critical points including the smallest possible example which follow from a reverse engineering procedure we introduce for this setting we also derive exotic minimal symplectic 4 manifolds in the homeomorphism classes of mathbb cp 2 4 overline mathbb cp 2 and 3 mathbb cp 2 6 overline mathbb cp 2 from small lefschetz fibrations over surfaces of higher genera

an implementation of a moving finite element method abstract the moving finite element method an adaptive gridding procedure for systems of partial differential equations whose solutions contain steep gradients has been implemented in a very straightforward way accurate results can be obtained using fewer nodes than that required by fixed mesh solution methods performance of the method using some minor changes to a standard ordinary differential equation algorithm is illustrated with solutions to burger s equation

algorithms for singularly perturbed limiting average markov control problems a singularly perturbed markov decision process with the limiting average reward criterion is considered it is assumed that the underlying process is composed of n separate irreducible processes and that the small perturbation is such that it unites these processes into a single irreducible process two algorithms for the solution of the underlying limit markov control problem are presented the first of these is a linear program possessing the wolfe dantzig structure inherited from the ergodic nearly decomposable assumption in the model the second is an aggregation disaggregation policy improvement algorithm

the variation of the de rham zeta function calcul de la variation des valeurs speciales de la fonction zeta de de rham par rapport a la variation de la metrique etablissement de l equation d euler lagrange pour les variations conformes

critical metrics for the determinant of the laplacian in odd dimensions let m be a closed compact n dimensional manifold with n odd we calculate the first and second variations of the zeta regularized determinants det a and det l as the metric on m varies where a denotes the laplacian on functions and l denotes the conformal laplacian we see that the behavior of these functionals depends on the dimension indeed every critical metric for 1 n 1 2det a or 1 n 1 21 det l has finite index consequently there are no local maxima if n 4rm 1 and no local minima if n 4m 3 we show that the standard 3 sphere is a local maximum for det a while the standard 4m 3 sphere with m 1 2 is a saddle point by contrast for all odd n the standard n sphere is a local extremal for det l an important tool in our work is the canonical trace on odd class operators in odd dimensions this trace is related to the determinant by the formula det q tr log q and we prove some basic results on how to calculate this trace

differentiable classification of 4 manifolds with singular riemannian foliations in this paper we first prove that any closed simply connected 4 manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4 manifolds mathbb s 4 mathbb cp 2 mathbb s 2 times mathbb s 2 or mathbb cp 2 pm mathbb cp 2 as an application we prove that any closed simply connected 4 manifold admitting a nontrivial singular riemannian foliation is diffeomorphic to a connected sum of copies of standard mathbb s 4 pm mathbb cp 2 and mathbb s 2 times mathbb s 2 a classification of singular riemannian foliations of codimension 1 on all closed simply connected 4 manifolds is obtained as a byproduct in particular there are exactly 3 non homogeneous singular riemannian foliations of codimension 1 complementing the list of cohomogeneity one 4 manifolds

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

heat kernel expansions ambient metrics and conformal invariants abstract the conformal powers of the laplacian of a riemannian metric which are known as the gjms operators admit a combinatorial description in terms of the taylor coefficients of a natural second order one parameter family h r g of self adjoint elliptic differential operators h r g is a non laplace type perturbation of the conformal laplacian p 2 g h 0 g it is defined in terms of the metric g and covariant derivatives of the curvature of g we study the heat kernel coefficients a 2 k r g of h r g on closed manifolds we prove general structural results for the heat kernel coefficients a 2 k r g and derive explicit formulas for a 0 r g and a 2 r g in terms of renormalized volume coefficients the taylor coefficients of a 2 k r g as functions of r interpolate between the renormalized volume coefficients of a metric g k 0 and the heat kernel coefficients of the conformal laplacian of g r 0 although h r g is not conformally covariant there is a beautiful formula for the conformal variation of the trace of its heat kernel its proof rests on the combinatorial relations between taylor coefficients of h r g and gjms operators as a consequence we give a heat equation proof of the conformal transformation law of the integrated renormalized volume coefficients by refining these arguments we also give a heat equation proof of the conformal transformation law of the renormalized volume coefficients itself the taylor coefficients of a 2 r define a sequence of higher order riemannian curvature functionals with extremal properties at einstein metrics which are analogous to those of integrated renormalized volume coefficients among the various additional results the reader finds a polyakov type formula for the renormalized volume of a poincare einstein metric in terms of q curvature of its conformal infinity and additional holographic terms the present study of relations between spectral theoretic heat kernel coefficients and geometric quantities like renormalized volume coefficients is stimulated by the holographic perspective of the ads cft duality proposing relations between functional determinants and renormalized volumes

surgery in cusp neighborhoods and the geography of irreducible 4 manifolds since its inception the theory of smooth 4 manifolds has relied upon complex surface theory to provide its basic examples this is especially true for simply connected 4 manifolds a longstanding conjecture held that every smooth simply connected closed 4 manifold is the connected sum of manifolds each admitting a complex structure with one of its orientations for this conjecture the 4 sphere was considered as the empty connected sum quite recently this was disproved by gompf and mrowka gm1 who produced infinite families of examples of 4 manifolds homeomorphic to the k3 surface and which admit no complex structure with either orientation the discovery of these dramatic counterexamples to the h cobordism theorem in dimension 4 was the most recent success in a long line of applications of donaldson invariants to the study of smooth structures on 4 manifolds beginning with donaldson s d4 and followed by fm1 ko ov1 ov2 which depended on donaldson s f invariant or a suitable generalization and fmm dk e2 eo1 mn2 sa using donaldson s polynomial invariant a common thread running through many of these counterexamples to the h cobordism conjecture is the process of logarithmic transformation which introduces multiple fibres in an elliptic surface in controlled situations this process will change the diffeomorphism type of a given elliptic surface while keeping its homotopy type fixed taking advantage of the complex structure on an elliptic surface this change can often be detected by using the relationship between anti self dual connections over a complex surface and stable holomorphic bundles d2 an alternative and more differential geometric

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

semiclassical yang mills theory i instantons the partition functions of quantum yang mills theory have an expansion in powers of the coupling constant the leading order term in this expansion is called the semiclassical approximation we study the semiclassical approximation for yang mills theory on a compact riemannian 4 manifold using geometric techniques and do explicit calculations for the case when the manifold is the 4 sphere this involves calculating the riemannian measure and certain functional determinants on the moduli space of self dual connections the main result is that the contribution to the semiclassical partition functions coming from thek 1 connections on the 4 sphere isfinite andcalculable we also discuss a renormalization procedure in which the radius of the 4 sphere is allowed to tend to infinity

nonlocal invariants in index theory in its original form the atiyah singer index theorem equates two global quantities of a closed manifold one analytic the index of an elliptic operator and one topological a characteristic number because it relates invariants from different branches of mathematics the index theorem has many applications and extensions to differential geometry k theory mathematical physics and other fields this report focuses on advances in geometric aspects of index theory for operators naturally associated to a riemannian metric on a closed manifold the topological side of the index theorem can often be expressed as the integral of local i e pointwise curvature expression we will first discuss these local refinements in 1 which arise naturally in heat equation proofs of the index theorem in 2 3 we discuss further developments in index theory which lead to spectral invariants the eta invariant and the determinant of an elliptic operator that are definitely nonlocal finally in 4 we point out some recent connections among these nonlocal invariants and classical index theory