the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

analytic torsion of a bounded generalized cone we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of m spreafico and using the symmetry in the de rham complex as established by m lesch we evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over s 1 since the standard cone over the sphere is simply a flat disc

signatures and higher signatures of s 1 quotients we define and study the signature widehat a genus and higher signatures of the quotient space of an s 1 action on a closed oriented manifold we give applications to questions of positive scalar curvature and to an equivariant novikov conjecture

on the maslov index in this paper we give four definitions of maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other moreover relationships of several symplectic and differential geometric analytic and topological invariants including triple maslov indices eta invariants spectral flow and signatures of quadratic forms to the maslov index are developed and formulae relating them are given the broad presentation is designed with a view to applications both in geometry and in analysis 1994 john wiley sons inc

on the curvature of piecewise flat spaces we consider analogs of the lipschitz killing curvatures of smooth riemannian manifolds for piecewise flat spaces in the special case of scalar curvature the definition is due to t regge considerations in this spirit date back to j steiner we show that if a piecewise flat space approximates a smooth space in a suitable sense then the corresponding curvatures are close in the sense of measures

comparison between two analytic torsions on orbifolds in this paper we establish an equality between the analytic torsion introduced by dar math z 194 2 193 216 1987 and the orbifold analytic torsion defined by ma trans am math soc 357 6 2205 2233 2005 on an even dimensional manifold with isolated conical singularities which in addition has an orbifold structure we assume the orbifold flat vector bundle is an honest vector bundle although the metric on the flat bundle may not be flat

intersection r torsion and analytic torsion for pseudomanifolds pour une pseudovariete x n on definit un invariant i p la r torsion intersection parametrisee par la perversite p cet invariant est analogue a la torsion de reidemeister et est un invariant combinatoire

cech cocycles for characteristic classes we give general formulae for explicit cech cocycles representing characteristic classes of real and complex vector bundles as well as for cocycles representing chern simons classes of bundles with arbitrary connections our formulae involve integrating differential forms over moving simplices inside homogeneous spaces an important feature of our cocycles is that they take integer values as opposed to real or rational values we find in particular a formula for the instanton number of a connection over a closed four manifold with arbitrary structure group for flat connections our formulae recover and generalize those of cheeger and simons the methods of this paper apply also to the purely geometric construction of the quillen line bundle with its metric

the recursive reduced order numerical solution of the singularly perturbed matrix differential riccati equation under stability observability conditions imposed on a singularly perturbed system an efficient numerical method for solving the corresponding matrix differential riccati equation is obtained in terms of the reduced order problems the order reduction is achieved via the use of the chang transformation applied to the hamiltonian matrix of a singularly perturbed linear quadratic control problem an efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the chang transformation

local geometry of singular real analytic surfaces let v r n be a compact real analytic surface with isolated singularities and assume its smooth part v 0 is equipped with a riemannian metric that is induced from some analytic riemannian metric on r n we prove 1 each point of v has a neighborhood which is quasi isometric naturally and almost isometrically to a union of metric cones and horns glued at their tips 2 a full asymptotic expansion for any p v of the length of v n q dist q p r as r 0 0 3 a gauss bonnet theorem saying that each singular point contributes 1 l 2 where l is the coefficient of the linear term in the expansion of 2 4 the l 2 stokes theorem selfadjointness and discreteness of the laplace beltrami operator on vo an estimate on the heat kernel and a gauss bonnet theorem for the l 2 euler characteristic as a central tool we use resolution of singularities

zeta regularized determinants for conic manifolds we study relative zeta regularized determinants of laplace type operators on compact conic manifolds we establish gluing formulae for relative zeta regularized determinants for arbitrary self adjoint extensions of the laplace beltrami operator we express the relative determinants for these as a ratio of the determinants of certain finite matrices for the self adjoint extensions corresponding to dirichlet and neumann conditions the formula is particularly simple and elegant

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

brownian motion in cones we study the asymptotic behavior of brownian motion and its conditioned process in cones using an infinite series representation of its transition density a concise probabilistic interpretation of this series in terms of the skew product decomposition of brownian motion is derived and used to show properties of the transition density

decoupling and order reduction via the riccati transformation a program that dates back to riccati 1724 and that seeks to replace a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use

witt spaces a geometric cycle theory for ko homology at odd primes introduction this paper studies a class of stratified piecewiselinear pseudomanifolds which we call witt spaces characterized by natural local intersection homology conditions 11 we compute the cobordism groups by introducing an invariant taking values in the witt group of symmetric bilinear forms over the rationals w q these pseudomanifolds solve a problem posed by d sullivan 25 to construct a class of p l cycles with signature which represent the connected version of ko homology at odd primes ko 0d z 1 2 the result was one of the first applications of the intersection homology theory of goresky and macpherson 11 specifically the rational intersection homology groups of witt spaces satisfy a poincare duality theorem to each witt space x we associate a p l invariant w x with values in w q this invariant generalizes the signature of manifolds and satisfies cobordism invariance additivity and a product formula adapting classical surgery spherical modification to this setting we prove that w x determines the cobordism class of x and obtain an explicit description of the witt cobordism groups the only nontrivial groups occur in dimensions 4k and for k 0 they are w q sullivan 25 27 has shown that a class of p l pseudomanifolds equipped with such an invariant possesses canonical orientations in ko 0 z 1 2 the orientations induce a natural transformation it of homology theories from witt space bordism to ko 0 z 1 2 the structure of w q is known 14 and we conclude that it is an equivalence at odd primes in this paper we study the cobordism theory of a class of stratified p l pseudomanifolds by means of the recent intersection homology theory of goresky and macpherson 11 goresky and macpherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the hirzebruch l class to the setting of manifolds with

an index formula for the intersection euler characteristic of an infinite cone the aim of this article is to establish an index formula for the intersection euler characteristic of a cone the main actor is the model witten laplacian on the infinite cone first we study its spectral properties and establish a mckean singer type formula we also give an explicit formula for the zeta function of the model witten laplacian in a second step we apply local index techniques to the model witten laplacian by combining these two steps we express the absolute and relative intersection euler characteristic of the cone as a sum of two terms a term which is local and a second term which is the cheeger invariant

a new version of the chang transformation a new decoupling transformation is proposed for singularly perturbed linear systems this transformation has the advantage over the previous one in that in order to find such a transformation only one algorithm is required and the computation can be done in parallel because the transformation matrices are obtained from two independent equations that are identical in form

the spectral zeta function for laplace operators on warped product manifolds of the type i f n in this work we study the spectral zeta function associated with the laplace operator acting on scalar functions defined on a warped product of manifolds of the type i f n where i is an interval of the real line and n is a compact d dimensional riemannian manifold either with or without boundary starting from an integral representation of the spectral zeta function we find its analytic continuation by exploiting the wkb asymptotic expansion of the eigenfunctions of the laplace operator on m for which a detailed analysis is presented we apply the obtained results to the explicit computation of the zeta regularized functional determinant and the coefficients of the heat kernel asymptotic expansion

spectral functions for the schrodinger operator on r with a singular potential in this article we analyze the spectral zeta function the heat kernel and the resolvent of the operator d2 dr2 r2 r2 over the interval 0 for 1 4 depending on the self adjoint extension chosen nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed in particular for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s k k n0 do occur this implies that the small t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk ln t l 1 for k l n0 the corresponding statements for the resolvent are also shown furthermore we evaluate the zeta determinant of the operator for all values of and any self adjoint extension

lipschitz killing curvatures of angular partially ordered sets abstract the present paper is motivated by recent work of j cheeger w muller and r schrader on the lipschitz killing curvatures of piecewise flat spaces it is proved that all basic properties of lipschitz killing curvatures can be derived by combinatorial methods using only the gram sommerville and mcmullen equations angular partially ordered sets yield the appropriate framework for proving the main results by means of the rota calculus in the incidence algebra of cell complexes

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

the homotopy type of the matroid grassmannian stiefel 26 and whitney 28 and with integral coefficients by pontrjagin 24 make up the primary source of first order invariants of smooth manifolds when their utility was first recognized it became an obvious goal to study the ways in which they admitted extensions to other categories such as the categories of topological or pl manifolds perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques modulo 2 this hope was realized rather quickly it is not hard to see that the stiefel whitney classes are pl invariants moreover whitney was able to produce a simple explicit formula for the class in codimension i in terms of the i skeleton of the barycentric subdivision of a triangulated manifold for a proof of this result see 13 one would like to find an analogue of these results for the pontrjagin classes however such a naive goal is entirely out of reach indeed milnor s use of the pontrjagin classes to construct an invariant which distinguishes between nondiffeomorphic manifolds which are homeomorphic and pl isomorphic

linearizability of the perturbed burgers equation inst de fis teorica universidade estadual paulista rua pamplona 145 01405 900 sao paulo

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

on the tangent cone of kahler manifolds with ricci curvature lower bound let x be the gromov hausdorff limit of a sequence of pointed complete kahler manifolds m n i p i satisfying ric m i ge n 1 and the volume is noncollapsed we prove that there exists a lie group isomorphic to mathbb r acting isometrically on the tangent cone at each point of x moreover the action is locally free on the cross section this generalizes the metric cone theorem of cheeger colding to the kahler case we also discuss some applications to complete kahler manifolds with nonnegative bisectional curvature

on the asymptotics of determinant of laplacian at the principal boundary of the principal stratum of the moduli space of abelian differentials let x be a translation surface of genus g 1 with 2g 2 conical points of angle 4 and let be two homologous saddle connections of length s joining two conical points of x and bounding two surfaces s and s with boundaries s and s gluing the opposite sides of the boundary of each surface s s one gets two closed translation surfaces x x of genera g g g g g let and be the friedrichs extensions of the laplacians corresponding to the flat conical metrics on x x and x respectively we study the asymptotical behavior of the modified i e with zero modes excluded zeta regularized determinant det as and shrink we find the asymptotics

bounds on the optimum quadratic cost of structure constrained controllers in this note lower and upper bounds are derived on the quadratic cost of optimal output feedback controllers subject to constraints on the controller structure the use of these bounds in forming a comparative assessment of alternative controller confugurations is described and an illustrative example is given the principal advantage of these bounds arises from the fact that they can save considerable computational effort

aps boundary conditions eta invariants and adiabatic limits we prove an adiabatic limit formula for the eta invariant of a manifold with boundary the eta invariant is defined using the atiyah patodisinger boundary condition and the underlying manifold is fibered over a manifold with boundary our result extends the work of bismut cheeger to manifolds with boundary

wave interactions and the analysis of the perturbed burgers equation in multiple front solutions of the burgers equation all the fronts except for two are generated through the inelastic interaction of exponential wave solutions of the lax pair associated with the equation the inelastically generated fronts are the source of two interrelated difficulties encountered in the standard normal form expansion of the approximate solution of the perturbed burgers equation when the zero order term is a multiple front solution i the higher order terms in the expansion are not bounded ii the normal form equation obeyed by the zero order approximation is not asymptotically integrable its solutions lose the simple wave structure of the solutions of the unperturbed equation the freedom inherent in the normal form method allows a simple modification of the expansion procedure making it possible to overcome both problems in more than one way the loss of asymptotic integrability is shifted from the normal form to the higher order terms part of which has to be computed numerically in the expansion of the solution the front velocity update is different from the one obtained in the standard analysis

novel results for a class of singular perturbed slow fast system in this paper an iterative method is illustrated for solving a class of singular perturbed slow fast system including two or three differential equations by some techniques we not only give an error accuracy by differential inequality but also present an error estimate formula at last some typical examples are given to illustrate the new iterative method is a useful tool

extended two dimensional equation for the description of nonlinear waves in gas liquid mixture we derive two dimensional equation describing waves in a gas liquid mixture we investigate integrability of this equation using the painleve approach we construct some exact solutions of the equation derived we perform numerical investigation of the waves described by the equation we consider a system of equations for the description of nonlinear waves in a liquid with gas bubbles taking into account high order terms with respect to a small parameter we derive a new nonlinear partial differential equation for the description of density perturbations of mixture in the two dimensional case we investigate integrability of this equation using the painleve approach we show that traveling wave reduction of the equation is integrable under some conditions on parameters some exact solutions of the equation derived are constructed we also perform numerical investigation of the nonlinear waves described by the derived equation

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

near optimal h sup spl infin control of linear singularly perturbed systems we consider the singularly perturbed h sup spl infin control problem under perfect state measurements for both finite and infinite horizons we suggest a construction of high order approximations to a controller that guarantees a desired performance level on the basis of the exact decomposition of the full order riccati equations to the reduced order slow and fast equations this leads to effective asymptotic and numerical algorithms we show that the high order accuracy controller improves the performance

an extension of a theorem by cheeger and muller to spaces with isolated conical singularities the aim of this article is to extend a theorem by cheeger and muller to spaces with isolated conical singularities by generalizing the proof of bismut and zhang to the singular setting the main tools in this approach are the witten deformation and local index techniques

generalized ray singer conjecture i a manifold with a smooth boundary this paper is devoted to a proof of a generalized ray singer conjecture for a manifold with boundary the dirichlet and the neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary the ray singer conjecture rs claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal for a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed some new general properties of the ray singer analytic torsion are found the proof does not use any computation of eigenvalues and their asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

the exact slow fast decomposition of the algebraic ricatti equation of singularly perturbed systems the algebraic riccati equation for singularly perturbed control systems is completely and exactly decomposed into two reduced order algebraic riccati equations corresponding to the slow and fast time scales the pure slow and pure fast algebraic riccati equations are asymmetric ones but their o epsilon perturbations are symmetric it is shown that the newton method is very efficient for solving the obtained asymmetric algebraic riccati equations the method presented is very suitable for parallel computations due to the complete and exact decomposition of the riccati equation this procedure might produce new insight into the two time scale optimal filtering and control problems

strichartz estimates and wave equation in a conic singular space consider the metric cone x c y 0 infty r times y with metric g dr 2 r 2h where the cross section y is a compact n 1 dimensional riemannian manifold y h let delta g be the positive friedrichs extension laplacian on x and let delta h be the positive laplacian on y and consider the operator mathcal l v delta g v 0 r 2 where v 0 in mathcal c infty y such that delta h v 0 n 2 2 4 is a strictly positive operator on l 2 y in this paper we prove global in time strichartz estimates without loss regularity for the wave equation associated with the operator mathcal l v it verifies a conjecture in wang remark 2 4 in ann inst fourier 56 1903 1945 2006 for wave equation the range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of delta h v 0 n 2 2 4 to prove the result we show a sobolev inequality and a boundedness of a generalized riesz transform in this setting in addition as an application we study the well posed theory and scattering theory for energy critical wave equation with small data on this setting of dimension n ge 3

dynamical analysis and perturbation solution of an seir epidemic model abstract in this paper we present a kind of technique to obtain the asymptotical solution of an seir where s is the susceptible population e is the exposed population i is the infectious population and r is the recovered population epidemic model by employing the method of perturbation at first we investigate the epidemic model by analysing its dynamical behavior then we use the method of perturbation to obtain the analytical solution of the model we assign values for parameters and draw figures to observe the magnitude of error of the perturbation method in a macro view finally we analyse macroscopically the two comparison chart on analytical solution and the exact solution and know that it is feasible to analyse the solution of the epidemic model by using the perturbation method

exact slow fast decomposition of the singularly perturbed matrix differential riccati equation in this paper the hamiltonian matrix formulation of the riccati equation is used to derive the reduced order pure slow and pure fast matrix differential riccati equations of singularly perturbed systems these pure slow and pure fast matrix differential riccati equations are obtained by decoupling the singularly perturbed matrix differential riccati equation of dimension n 1 n 2 into the pure slow regular matrix differential riccati equation of dimension n 1 and the pure fast stiff matrix differential riccati equation of dimension n 2 a formula is derived that produces the solution of the original singularly perturbed matrix differential riccati equation in terms of solutions of the pure slow and pure fast reduced order matrix differential riccati equations and solutions of two reduced order initial value problems in addition to its theoretical importance the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time invariant systems independently in pure slow and pure fast time scales

foliated stratified spaces and a de rham complex describing intersection space cohomology the method of intersection spaces associates cell complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that poincare duality holds between the ordinary rational cohomology groups of the cell complexes associated to complementary perversities the cohomology of these intersection spaces defines a cohomology theory hi for singular spaces which is not isomorphic to intersection cohomology ih mirror symmetry tends to interchange ih and hi the theory ih can be tied to type iia string theory while hi can be tied to iib theory for pseudomanifolds with stratification depth 1 and flat link bundles the present paper provides a de rham theoretic description of the theory hi by a complex of global smooth differential forms on the top stratum we prove that the wedge product of forms introduces a perversity internal cup product on hi for every perversity flat link bundles arise for example in foliated stratified spaces and in reductive borel serre compactifications of locally symmetric spaces a precise topological definition of the notion of a stratified foliation is given

fredholm theory for elliptic operators on quasi asymptotically conical spaces we consider the mapping properties of generalized laplace type operators l r on the class of quasi asymptotically conical qac spaces which provide a riemannian generalization of the qale manifolds considered by joyce joy00 our main result gives conditions under which such operators are fredholm when between

the dirac operator on spaces with conical singularities and positive scalar curvatures on obtient un theoreme d annulation de type bochner et on demontre un theoreme d indice dans le cas singulier on etudie la relation avec des varietes a bord

zeta determinant for laplace operators on riemann caps the goal of this paper is to compute the zeta function determinant for the massive laplacian on riemann caps or spherical suspensions these manifolds are defined as compact and boundaryless d dimensional manifolds deformed by a singular riemannian structure the deformed spheres considered previously in the literature belong to this class after presenting the geometry and discussing the spectrum of the laplacian we illustrate a method to compute its zeta regularized determinant the special case of the deformed sphere is recovered as a limit of our general formulas

an embedding technique for the solution of reaction diffusion equations on algebraic surfaces with isolated singularities abstract in this paper we construct a parametrization free embedding technique for numerically evolving reaction diffusion pdes defined on algebraic curves that possess an isolated singularity in our approach we first desingularize the curve by appealing to techniques from algebraic geometry we create a family of smooth curves in higher dimensional space that correspond to the original curve by projection following this we pose the analogous reaction diffusion pde on each member of this family and show that the solutions their projection onto the original domain approximate the solution of the original problem finally we compute these approximants numerically by applying the closest point method which is an embedding technique for solving pdes on smooth surfaces of arbitrary dimension or codimension and is thus suitable for our situation in addition we discuss the potential to generalize the techniques presented for higher dimensional surfaces with multiple singularities

brief paper eigenvector approach for order reduction of singularly perturbed linear quadratic optimal control problems in this paper we show how to decompose the singularly perturbed algebraic riccati equation and the corresponding linear quadratic optimal control problem at steady state in terms of reduced order pure slow and pure fast problems by using the eigenvector approach the eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small in such cases the decomposition methods based on series expansions fixed point iterations subspace iterations and newton iterations either fail to produce solutions of the corresponding algebraic equations or display very slow convergence

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series