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

stratifiable maps and topological invariants let x and y be spaces and f x y be a map between them then a large family of problems in topology and geometry involve the attempt to relate the invariants of x and y via f this attempt has been successful often for the case x and y manifolds and f the projection of a fibre bundle for many types of important invariants for example the result of chernhirzebruch serre chs asserts that for y simply connected the signature of x is the product of the signatures of y and of the fibre f 1 y in this paper we undertake the study of maps in which the fiber may vary from point to point the spaces x and y will be any whitney stratified spaces with even codimension strata e g algebraic varieties and f will be a stratified map this means essentially that for v a component of a stratum of y thus v is an open manifold f 1v is a union of strata of x and the restriction of f to f71v is a locally trivial fibre bundle for example subanalytic maps of subanalytic sets can be stratified with respect to whitney

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

karoubianness of a triangulated category abstract we prove that triangulated categories with bounded t structures are karoubian consequently for an ext finite abelian category over a commutative noetherian complete local ring its bounded derived category is krull schmidt

motivic bivariant characteristic classes let k0 v x k0 v x be the relative grothendieck group of varieties over x obj v x obj v with v vk qp resp v vcan the category of quasi projective algebraic resp compact complex analytic varieties over a base field k then we constructed the motivic hirzebruch class transformation ty k0 v x h x q y ty k0 v x h x q y in the algebraic context for k of characteristic zero with h x ch x h x ch x resp in the complex algebraic or analytic context with h x h2 bm x it unifies the well known three characteristic class transformations of singular varieties macpherson s chern class baum fulton macpherson s todd class and the l class of goresky macpherson and cappell shaneson in this paper we construct a bivariant relative grothendieck group k0 v x y k0 v x y for v vk qp resp vcan so that k0 v x pt k0 v x k0 v x pt k0 v x in the algebraic context with k of characteristic zero resp complex analytic context we also construct in the algebraic context in any characteristic two grothendieck transformations mcy ymot k0 vqp x y kalg x y z y and ty k0 vqp x y h x y q y ty k0 vqp x y h x y q y with kalg f kalg f the bivariant algebraic k theory of f perfect complexes and hh the bivariant operational chow groups or the even degree bivariant homology in the case k ck c evaluating at y 0y 0 we get a motivic lift t0t0 of fulton macpherson s bivariant riemann roch transformation kalg h q kalg h q the covariant transformations mcy k0 vqp x pt g0 x z y mcy k0 vqp x pt g0 x z y and ty k0 vqp x pt h x q y ty k0 vqp x pt h x q y agree for k of characteristic zero with our motivic chern and hirzebruch class transformations defined on k0 vqp x k0 vqp x finally evaluating at y 1y 1 for k of characteristic zero we get a motivic lift t 1t 1 of ernstrom yokura s bivariant chern class transformation f ch

approximation theorems for the propagators of higher order abstract cauchy problems in this paper we present two quite general approximation theorems for the propagators of higher order in time abstract cauchy problems which extend largely the classical trotter kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions then we apply the approximation theorems to deal with the second order dynamical boundary value problems

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

availability analysis for software system with intrusion tolerance this paper is devoted to analyzing the instantaneous availability of a typical software system with intrusion tolerance by formulating the system with a couple of ordinary differential and partial differential equations this paper describes the system as a time delay partial differential equation based on the time delay model both steady state availability and instantaneous availability are investigated the optimal policy for preventive patch management to maximize the steady state availability of the software system is obtained and its related availability criterions are also presented employing the finite difference scheme and trotter kato theorem we converted the time delay partial equation into a time delay ordinary equation as a result the instantaneous availability of the system is derived some numerical results are given to show the effectiveness of the method presented in the paper

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

the method of parameter functionalization in the hopf bifurcation problem depending on a parameter for an extensive bibliography see l the greater part of these investigations exploit information about not only the linear terms of the right hand member of the system l the linear operator fl a 0 but also information about the terms of higher orders in the taylor expansion in x of the function f a x with the help of such information one can answer the questions on the number of arising self oscillations their stability their dependence on a parameter etc proofs of corresponding assertions use as a rule an analytic technique such as varied forms of the implicit function theorem the theory of invariant manifolds and the like at the same time in problems arising during the study of complicated physical technological etc processes often the only rather complete information available is that concerning the linear terms of the right hand member of the system 1 in such circumstances it is very difficult to apply an analytic technique for studying the problem of generation of self oscillations yet as things turn out 2 in some cases the very fact of generation of small self oscillations can be picked out of information about the linear terms of the right hand member of the system 1 of course under lack of information about the high order terms in the taylor expansion in x of f a x one can say next to nothing about properties of arising self oscillations this article contains a topological proof of the hopf theorem in this proof the method of parameter functionalization 3 introduced by krasnosel skii in another situation has much significance employment of topological considerations makes it possible to throw aside usual assumptions of differentiability of the right hand member of the system 1 this provides an opportunity to investigate systems l the right hand side of which contains for example hysteresis type or relay type nonlinearities the proof presented below goes through without changes in the case of functional differential equations with lagging arguments 4 we do not present here the exact formulations of such assertions because our purpose is the demonstration of the method in s 61 the method of parameter functionalization has been applied to the investigation of bifurcation of long periodic solutions of differential equations and mappings

discontinuity induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems a rich variety of dynamical scenarios has been shown to occur when a fixed point of a non smooth map undergoes a border collision this paper concerns a closely related class of discontinuity induced bifurcations those involving equilibria of n dimensional piecewise smooth flows specifically transitions are studied which occur when a boundary equilibrium that is one lying within a discontinuity manifold is perturbed it is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios conditions to classify among the possible simplest scenarios are given for piecewise smooth continuous filippov and impacting systems also we investigate the possible birth of other attractors e g limit cycles at a boundary equilibrium bifurcation

mixed hodge complexes on algebraic varieties using the theory of mixed hodge modules we introduce the notion of mixed hodge complex on an algebraic variety and establish the relation between the filtered complex of du bois and the corresponding complex of mixed hodge modules some application to the du bois singularity is given

characteristic classes of symmetric products of complex quasi projective varieties we prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi projective variety more concretely we study homology hirzebruch classes for motivic coefficients as well as for complexes of mixed hodge modules as a special case we obtain a generating series formula for the intersection homology hirzebruch classes of symmetric products in some cases the latter yields a similar formula for twisted homology l classes generalizing results of hirzebruch zagier and moonen our methods also apply to the study of todd classes of complexes of coherent sheaves as well as chern classes of complexes of constructible sheaves generalizing to arbitrary coefficients results of moonen and resp ohmoto

the l class of non witt spaces characteristic classes for oriented pseudomanifolds can be defined using appropriate self dual complexes of sheaves on non witt spaces self dual complexes compatible to intersection homology are determined by choices of lagrangian structures at the strata of odd codimension we prove that the associated signature and l classes are independent of the choice of lagrangian structures so that singular spaces with odd codimensional strata such as e g certain compactifications of locally symmetric spaces have well defined l classes provided lagrangian structures exist we illustrate the general results with the example of the reductive borel serre compactification of a hilbert modular surface

computing twisted signatures and l classes of non witt spaces in previous joint work with cappell and shaneson we have established an atiyah lusztig meyer type multiplicative characteristic class formula for the twisted signature and more generally the twisted l class of a stratified witt space the present paper shows that these formulae hold even when the stratified space does not satisfy the witt condition it constitutes one of the first applications of signature homology

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

discretisation of an infinite delay equation in this paper a banach phase space containing bc 0 and contained in c 0 is defined with which existence of a solution and convergence of a discrete scheme are proved for an infinite delay differential equation

on the hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity the paper studies existence uniqueness and stability of large amplitude periodic cycles arising in hopf bifurcation at infinity of autonomous control systems with bounded nonlinear feedback we consider systems with functional nonlinearities of landesman lazer type and a class of systems with hysteresis nonlinearities the method is based on the technique of parameter functionalization and methods of monotone concave and convex operators c 2001 academic press

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

hodge genera of algebraic varieties ii we study the behavior of hodge genera under algebraic maps we prove thatthemotivic c y genussatisfiesthe stratifiedmultiplicativeproperty whichshows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map by considering morphisms to a curve we obtain a hodge theoretic version of the riemann hurwitz formula we also study the monodromy contributions to the y genus of a family of compact complex manifolds and prove an atiyah meyer type formula in the algebraic and analytic contexts this formula measures the deviation from multiplicativity of the y genus and expresses the correction terms as higher genera associated to the period map these higher gen era are hodge theoretic extensions of novikov higher signatures to analytic and alge braic settings characteristic class formulae of atiyah meyer type are also obtained by making use of saito s theory of mixed hodge modules

nonlinear resonance in systems with hysteresis a general approach is suggested for the analysis of nonlinear resonance in control systems including an integral linear term and a nonlinear term with hysteresis this approach uses a special property of nonlinear terms which is valid for many important classes of hysteresis operators for systems with such hysteresis nonlinearities we introduce a method to derive conditions for the existence of forced oscillations and for systems with a parameter of nonlinear resonance at infinity

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

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

characteristic classes of complex hypersurfaces abstract the milnor hirzebruch class of a locally complete intersection x in an algebraic manifold m measures the difference between the poincare dual of the hirzebruch class of the virtual tangent bundle of x and respectively the brasselet schurmann yokura homology hirzebruch class of x in this note we calculate the milnor hirzebruch class of a globally defined algebraic hypersurface x in terms of the corresponding hirzebruch invariants of vanishing cycles and singular strata in a whitney stratification of x our approach is based on schurmann s specialization property for the motivic hirzebruch class transformation of brasselet schurmann yokura the present results also yield calculations of todd chern and l type characteristic classes of hypersurfaces

characteristic classes singular embeddings and intersection homology abstract this note announces some results on the relationship between global invariants and local topological structure the first section gives a local global formula for pontrjagin classes or l classes the second section describes a corresponding decomposition theorem on the level of complexes of sheaves a final section mentions some related aspects of singular knot theory and the study of nonisolated singularities analogous equivariant analogues with local global formulas for atiyah singer classes and their relations to g signatures will be presented in a future paper

hopf bifurcations at infinity it is the purpose of this paper to study the behaviour of linear dominated systems for large x it is reasonable to assume that in this region the linear part will determine the local and perhaps global dynamics of the system under fairly weak conditions as a parameter of the system is varied a bifurcation occurs which generates arbitrarily large periodic solutions it is natural to think of this as a hopf bifurcation at infinity

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

exact stability chart of an elastic beam subjected to delayed feedback abstract the stability of an elastic beam is studied when the beam is subjected to a longitudinal force governed by a feedback loop the setup of the simplified mechanical model is motivated by a basic problem of electroacoustics the corresponding governing equation is the 1d wave equation with delayed boundary conditions by means of the d alembert solution the system can be transformed into a delay differential equation of neutral type that includes two time delays the intricate stability chart is constructed analytically in the parameter plane of the gain parameter and the ratio of the time delays the complete chart extends the numerical results of the literature while it also draws attention to the numerical difficulties of finite degree of freedom approximations and to the extreme sensitivity of the dynamics with respect to the time delay parameters of the system

the effect of frequency and clearance variations on single degree of freedom impact oscillators abstract a simple piecewise linear forced oscillator with impacts is taken as a model for more complicated vibro impacting systems the presence of impacts makes the system discontinuous and non linear and to study it effectively the theory of dynamical systems is used by a combination of analytical and numerical techniques many phenomena are identified which are characteristic of non linear dynamical systems as the system parameters are varied in particular these include chaotic regimes and multiply coexisting stable states the domains of attraction of the periodic states are plotted and compared with the strange attractors of the chaotic states in addition phenomena are observed related to the discontinuity which have no counterpart in smooth dynamical systems chatter when infinitely many impacts occur in a finite time and grazing when a stable periodic orbit encounters a discontinuity and disappears catastrophically under a smooth change in the parameters

bifurcations from a center at infinity in 3d piecewise linear systems with two zones abstract we consider continuous piecewise linear systems in r 3 with two zones under the assumption of having a linear center in the invariant manifold of the point at infinity a specific conic projection is introduced so that it is possible to analyze in a convenient way the dynamics near such a center at infinity in two qualitative different situations in the semi homogeneous case such a center is associated to the existence of a continuum of invariant semi cones sharing the vertex at the origin perturbing the configuration it is possible to detect the bifurcation of a limit cycle at infinity leading to the bifurcation of isolated invariant semi cones for the non homogeneous case the cycles at infinity does not imply invariant semi cones however the non generic case when the center at infinity is associated to the existence of invariant cylinders for one of the involved vector fields becomes rather interesting it is possible then by perturbing the other vector field to get the bifurcation of a big limit cycle from infinity without destroying the center

numerical methods for a class of singular integro differential equations based on semigroup approximation in this paper computational methods for singular integro differential equations of neutral type are discussed the study is based on a well posed state space formulation and uses semigroup theoretical framework to analyze the convergence of semi and fully discrete numerical schemes feasibility of the fully discrete scheme is demonstrated by applying it to abel volterra equations of the first kind and to a singular neutral functional differential equation

a novel route to a hopf bifurcation scenario in switched systems with dead zone abstract planar switched systems with dead zone are analyzed in particular we consider the effects of a perturbation which is applied to a linear control law and due to the perturbation the control changes from purely positional to position velocity control this type of a perturbation leads to a novel hopf like discontinuity induced bifurcation we show that this bifurcation leads to the creation of a small scale limit cycle attractor which scales as the square root of the bifurcation parameter we then investigate numerically a planar switched system with a positional feedback law dead zone and time delay in the switching function using the same parameter values as for the switched system without time delay in the switching function we show a hopf like bifurcation scenario which exhibits a qualitative and a quantitative agreement with the scenario analyzed for the non delayed system

chern numbers for singular varieties and elliptic homology a fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones intersection homology for example directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so here we use ideas from minimal model theory to define some characteristic numbers for singular varieties generalizing the chern numbers of a smooth variety this was suggested by goresky and macpherson as a next natural problem after the definition of intersection homology 11 we find that only a subspace of the chern numbers can be defined for singular varieties a convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism whereas a singular variety can at most have a fundamental class in a weaker homology theory elliptic homology we use this idea to give an algebro geometric definition of elliptic homology complex bordism modulo flops equals elliptic homology this paper was inspired by some questions asked by jack morava the descriptions of elliptic homology given by gerald hohn 13 were also an important influence thanks to dave bayer mike stillman john stembridge sheldon katz and stein stromme for their computer algebra programs macaulay sf and schubert which helped in guessing the right answer

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

computing twisted signatures and l classes of stratified spaces this paper establishes an atiyah type characteristic class formula for both the twisted signature and the twisted l classes of a singular space the result in its present form applies to coefficient systems that satisfy a transversality condition near the singularities this condition is frequently automatically satisfied e g on supernormal spaces or for local systems arising in certain geometric mapping situations we obtain applications to stratified maps by combining our formula with the cappell shaneson signature formula

stability of large cycles in a nonsmooth problem with hopf bifurcation at infinity an autonomous system which have a principal linear part at infinity and a continuous nonlinear term of sublinear growth is investigated results for systems with smooth nonlinearities are presented where accurate asymptotics of periodic cycles were introduced and cycle stability analyzed by the usual linearization of the system in a neighborhood of the cycle a new method to study both existence and stability of large cycles for systems with nonsmooth nonlinearities asymptotically homogeneous at infinity is developed the method combines the parameter functionalization technique and methods of monotone concave operators

superperverse intersection cohomology stratification in dependence within its traditional range of perversity parameters intersection cohomology is a topological invariant of pseudomanifolds this is no longer true once one allows superperversities perversities with open image in new window in this case intersection cohomology may depend on the choice of the stratification by which it is defined topological invariance also does not hold if one allows stratifications with codimension one strata nonetheless both errant situations arise in important situations the former in the cappell shaneson superduality theorem and the latter in any discussion of pseudomanifold bordism we show that while full invariance of intersection cohomology under restratification does not hold in this generality it does hold up to restratifications that fix the the top stratum

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

generalized hopf bifurcation emerged from a corner in general planar piecewise smooth systems abstract we investigate a generalized hopf bifurcation emerged from a corner located at the origin which is the intersection of n discontinuity boundaries in planar piecewise smooth dynamical systems with the jacobian matrix of each smooth subsystem having either two different nonzero real eigenvalues or a pair of complex conjugate eigenvalues we obtain a novel result that the generalized hopf bifurcation can occur even when the jacobian matrix of each smooth subsystem has two different nonzero real eigenvalues according to the eigenvalues of the jacobian matrices and the number of smooth subsystems we provide a general method and prove some generalized hopf bifurcation theorems by studying the associated poincare map

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

equivariant characteristic classes of singular complex algebraic varieties homology hirzebruch characteristic classes for singular varieties have been recently defined by brasselet schuurmann and yokura as an attempt to unify previously known characteristic class theories for singular spaces e g macpherson chern classes baum fulton macpherson todd classes and goresky macpherson l classes in this paper we define equivariant analogues of these classes for singular quasi projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology hirzebruch classes of global quotient varieties we also compute the new classes in the context of monodromy problems e g for varieties that fiber equivariantly in the complex topology over a connected algebraic manifold as another application we discuss atiyah meyer type formulae for twisted hirzebruch classes of global orbifolds 2012 wiley periodicals inc

on cappell shaneson s homology l classes of singular algebraic varieties s cappell and j shaneson stratifiable maps and topological invariants j amer math soc 4 1991 521 551 have recently developed a theory of homology l classes extending goresky macpherson s homology lclasses in this paper we show that cappell shaneson s homology l classes for compact complex possibly singular algebraic varieties can be interpreted as a unique natural transformation from a covariant cobordism function q to the q homology functor h q satisfying a certain normalization condition just like macpherson s chern classes and baum fulton macpherson s todd classes

on topological invariants of stratified maps with non witt target the cappell shaneson decomposition theorem for self dual sheaves asserts that on a space with only even codimensional strata any self dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata in sharp contrast to this result we prove that on a space with only odd codimensional strata not necessarily witt any self dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum the strata of odd codimension do not contribute terms as a consequence we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the witt space condition to prove these results we introduce a new category of super perverse sheaves which we show to be abelian finally we apply the results to the study of desingularization of non witt spaces and exhibit a singular space which admits a pl resolution in the sense of m kato but no resolution by a stratified map

pattern formation in parabolic equations containing hysteresis with diffusive thresholds abstract we consider a reaction diffusion system with discontinuous reaction terms modeled by non ideal relays the system is motivated by an epigenetic population model of evolution of two phenotype bacteria which switch phenotype in response to variations of environment we prove the formation of patterns in the phenotype space the mechanism responsible for pattern formation is based on memory hysteresis of the non ideal relays

characteristic classes of singular toric varieties in this paper we compute the motivic chern classes and homology hirzebruch characteristic classes of possibly singular toric varieties which in the context of complete toric varieties fit nicely with a generalized hirzebruch riemann roch theorem as important special cases we obtain new or recover well known formulae for the baum fulton macpherson todd or macpherson s chern classes of toric varieties as well as for the thom milnor l classes of simplicial projective toric varieties we present two different perspectives for the computation of these characteristic classes of toric varieties first we take advantage of the torus orbit decomposition and the motivic properties of the motivic chern and respectively homology hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the dual todd classes of closures of orbits this method even applies to torus invariant subspaces of a given toric variety the obtained formula is then applied to weighted lattice point counting in lattice polytopes and their subcomplexes yielding generalized pick type formulae second in the case of simplicial toric varieties we compute our characteristic classes by using the lefschetz riemann roch theorem of edidin graham in the context of the geometric quotient description of such varieties in this setting we define mock hirzebruch classes of simplicial toric varieties which specialize to the mock chern mock todd and mock l classes of such varieties and investigate the difference between the actual homology hirzebruch class and the mock hirzebruch class we show that this difference is localized on the singular locus and we obtain a formula for it in which the contribution of each singular cone is identified explicitly finally the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by cappell and shaneson in the early 1990s 2015 wiley periodicals inc

intersection homology kunneth theorems cohen goresky and ji showed that there is a kunneth theorem relating the intersection homology groups i bar p h x times y to i bar p h x and i bar p h y provided that the perversity bar p satisfies rather strict conditions we consider biperversities and prove that there is a kunneth theorem relating i bar p bar q h x times y to i bar p h x and i bar q h y for all choices of bar p and bar q furthermore we prove that the kunneth theorem still holds when the biperversity p q is loosened a little and using this we recover the kunneth theorem of cohen goresky ji

intersection homology with field coefficients k witt spaces and k witt bordism we construct geometric examples of pseudomanifolds that satisfy the witt condition for intersection homology poincare duality with respect to certain fields but not others we then compute the bordism theory of k witt spaces for an arbitrary field k extending results of siegel for k 2008 wiley periodicals inc

differentiability and convergence rates of approximating semigroups for retarded functional differential equations in this paper the averaging approximations i e semidiscrete finite difference scheme for linear retarded functional differential equations are considered in the context of the semigroups on the hilbert space r n times l 2 it is shown that the approximating semigroups are all differentiable uniformly with respect to the index n determining the mesh size this fact makes it possible to prove the existence of a uniform exponential decay or growth rate for the approximating semigroups which can be made arbitrarily close to the decay respectively growth rate of the original semigroup more importantly the uniform differentiability makes it possible to establish convergence rates that improve with time and to prove the convergence of the approximating semigroups in the uniform operator topology the dependence of the convergence rate on the initial conditions and system parameters is also established

method of lines approximations of delay differential equations we consider a special type of numerical methods for delay differential equations ddes by introducing a new independent variable an initial value problem for ddes is converted into an initial boundary value problem for the convection equation thus it is also possible to get an approximate solution to the dde problem by solving the initial boundary value problem with a suitable numerical method instead of solving the original problem in this paper we study a family of method of lines mol approximations to the problem which is obtained by applying runge kutta rk methods for space discretization and prove their convergence under the assumption that the rk methods satisfy a condition known as an algebraic characterization of a stability the result is also confirmed by numerical experiments moreover we show that the condition derives several stability properties of the mol approximations

the trotter kato theorem and approximation of pdes we present formulations of the trotter kato theorem for approximation of linear c 0 semigroups which provide very useful framework when convergence of numerical approximations to solutions of pdes are studied applicability of our results is demonstrated using a first order hyperbolic equation a wave equation and stokes equation as illustrative examples

hodge genera of algebraic varieties i the aim of this paper is to study the behavior of hodge theoretic intersection homology genera and their associated characteristic classes under proper morphisms of complex algebraic varieties we obtain formulae that relate parametrized families of global invariants of a complex algebraic variety x to such invariants of singularities of proper algebraic maps defined on x such formulae severely constrain both topologically and analytically the singularities of complex maps even between smooth varieties similar results were announced by the first and third author in 13 32 2007 wiley periodicals inc

modelling and analysis of repairable systems with preventive maintenance the time dependent solution of a kind of repairable system with preventive maintenance is investigated in the paper with total probability formula we show that the behavior of the system can be described as a group of ordinary differential equations coupled with partial differential equations which can be formulated as a time delay equation in an appropriate banach space based on the time delay equation this paper presents a difference scheme as an approximating method to solve the time dependent solution which is necessary for analyzing the instantaneous availability of the repairable system some numerical examples are shown to illustrate the effectiveness of this approach

additivity and non additivity for perverse signatures a well known property of the signature of closed oriented 4n dimensional manifolds is novikov additivity which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary wall showed that this property is not true of signatures on manifolds with boundary and that the dierence from additivity could be described as a certain maslov triple index perverse signatures are signatures dened for any stratied pseudomanifold using the intersection homology groups of goresky and macpherson in the case of witt spaces the middle perverse signature is the same as the witt signature this paper proves a generalization to perverse signatures of wall s non additivity theorem for signatures of manifolds with boundary under certain topological conditions on the dividing hypersurface novikov additivity for perverse signatures may be deduced as a corollary in particular siegel s version of novikov additivity for witt signatures is a special case of this corollary

euler characteristics of algebraic varieties this note studies the behavior of euler characteristics and of intersection homology euler characteristics under proper morphisms of algebraic respectively analytic varieties the methods also yield for algebraic respectively analytic varieties formulae comparing these two kinds of euler characteristics the main results are direct consequences of the calculus of constructible functions and grothendieck groups of constructible sheaves similar formulae for hodgetheoretic invariants of algebraic varieties under morphisms were announced by the first and third authors in 5 14 c 2007 wiley periodicals inc

rene thom s work on geometric homology and bordism by the early 1950 s algebraic topology had reached great heights with serre s thesis and the calculations in the seminar of henri cartan of the cohomology of spaces with one nonzero homotopy group in terms of steenrod operations there was also the appearance of the new characteristic classes of vector bundles pontryagin classes z coefficients and chern weil classes coefficients in r joining those of stiefel whitney z mod 2 coefficients rene thom absorbed all this structure made vigorous use of it and added a geometric perspective that combined to revolutionize topology manifold theory and algebraic geometry for the unexpected and fertile results in bordism closed manifolds mod boundaries of manifolds of the 1954 paper 1 thom received the fields medal at edinburgh in 1958 many more applications of thom s ideas came even later

generalized hopf bifurcation emanated from a corner for piecewise smooth planar systems abstract in this paper we study the existence of periodic orbits bifurcating from a corner in a piecewise smooth planar dynamical system this phenomenon is interpreted as generalized hopf bifurcation philos trans r soc london ser a 359 1789 2001 2483 2496 hopf bifurcation for nonsmooth planar systems universitat zu koln 2000 northeast math j 17 4 2001 383 386 northeast math j 17 3 2001 261 264 generalized hopf bifurcation for non smooth planar dynamical systems university cologne 2003 in the case of smoothness hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis this characterization does not apply to a piecewise smooth system due to the lack of linearization in fact the generalized hopf bifurcation is determined by interactions between the geometrical structure of the corner and the eigenstructure of each smooth subsystem we combine a geometrical method and an analytical method to investigate the generalized hopf bifurcation the bifurcating periodic orbits are obtained by studying the fixed points of a return map

singular spaces characteristic classes and intersection homology global topological invariant to be studied will be the l classes li x e hi x q x a stratified pseudomanifold with even codimension strata for manifolds these characteristic classes are the poincare duals of the hirzebruch polynomials in the pontrjagin classes goresky macpherson extended the definition to stratified spaces with even codimension strata using intersection homology gm1 cheeger ch defined l classes analytically using his development of l2cohomology theory of singular riemannian spaces see also s csw the importance of these classes is illustrated by the famous theorem of browder and novikov the homeomorphism type of a compact simply connected manifold of dimension at least five is determined by its homotopy type and its l classes up to a finite number of possibilities for 4 manifolds the homotopy type alone suffices f by recent work of cappell weinberger see we similar results hold for stratified spaces with simply connected even codimension strata and simply connected links with respect to isovariant homotopy type and l classes of the space and its strata the study of l classes will take place in the context of a stratified pseudomanifold x7 of real dimension n embedded say piecewise linearly in a manifold mm of dimension m n 2 for example x might be a hypersurface in a projective complex algebraic variety for x a smoothly or pl locally flatly embedded submanifold the l classes are given by the classical formula l x x n i p m u 1 x f the total hirzebruch l polynomial p m e h m the total pontrjagin class of m i the inclusion and x the poincare dual of i x

boundary controllability of a linear semi discrete 1 d wave equation derived from a mixed finite element method in this article one discusses the controllability of a semi discrete system obtained by discretizing in space the linear 1 d wave equation with a boundary control at one extremity it is known that the semi discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero see 8 here we introduce a new semi discrete model based on a mixed finite element method with two different basis functions for the position and velocity we show that the controls obtained with these semi discrete systems can be chosen uniformly bounded in l 2 0 t and in such a way that they converge to the hum control of the continuous wave equation i e the minimal l 2 norm control we illustrate the mathematical results with several numerical experiments