legendre tau approximations for functional differential equations in this paper we consider the numerical approximation of solutions to linear retarded functional differential equations using the so called legendre tau method the functional differential equation is first eformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation the approximate solution is then represented as a truncated legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations the method is very easy to code and yields very accurate approximations convergence is established various numerical examples are presented and comparison between the latter and cubic spline approximations is made

onset of chaos in differential delay equations abstract in the present paper we study the onset of chaos in the differential delay equation x dot t x t bx t 2 1 x 10 t 2 b 0 we develop a new numerical technique and apply it to study the global behavior of the flow on the attractor using this technique we find unstable periodic orbits connections between different periodic orbits transverse homoclinic orbits and try to explain the underlying reason for chaos

positive energy representations of the loop groups of non simply connected lie groups we classify and construct all irreducible positive energy representations of the loop group of a compact connectedand simple lie group and show that they admit an intertwining action of diff s 1

groupoid equivariant prequantization in their 2005 paper c laurent gengoux and p xu define prequantization for pre hamiltonian actions of quasi presymplectic lie groupoids in terms of s1 central extensions of lie groupoids the definition requires that the quasi presymplectic structure be exact i e the closed 3 form on the unit space of the lie groupoid must be exact in the present paper we define prequantization for pre hamiltonian actions of not necessarily exact quasi presymplectic lie groupoids in terms of dixmier douady bundles the definition is a natural adaptation of e meinrenken s treatment of prequantization for quasi hamiltonian lie group actions with group valued moment map the definition given in this paper is shown to be compatible with the definition of laurent gengoux and xu when the underlying quasi preysmplectic structure is exact properties related to morita invariance and symplectic reduction are established

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

theta characteristics on algebraic curves the theory of theta characteristics is developed algebraically so that it may be applied to possibly singular and or reducible algebraic curves the configuration of theta characteristics on a curve is described in terms of its singularities with applications to the geometry of plane quartic curves and the problem of appolonius some results on gorenstein local rings are appended 0 introduction the theory of theta characteristics has been studied from different points of view and often inadvertently cf for example the 19th century literature on bitangents to smooth plane quartic curves e g 4 and 10 until recently however virtually all treatments of theta characteristics per se involved the transcendental theory of the theta function associated to a compact riemann surface then in 1971 mumford in 8 gave a completely algebraic treatment of the subject formulating the main results and proving them without reference to the transcendental theory this in turn suggested that theta characteristics could be studied in a more general context for example on possibly singular and or reducible algebraic curves in this paper we undertake such an analysis in 1 following mumford we establish some of the basic facts about theta characteristics algebraically so as to apply to more general curves in fact 1 10 i and 1 13 in their present generality are due to mumford while he assumes throughout 8 that his curve x is smooth his proof seems to apply as given the argument for 1 13 here is somewhat different in order to make the proof more elementary and selfcontained one bonus is the result 1 11 on the subvariety of the moduli space 9thg consisting of curves with large semicanonical linear series this fact seemingly can be proved only via the algebraic set up in 2 we investigate the number and configuration of odd and even theta characteristics on a singular curve the results are expressed in the two main theorems 2 12 and 2 22 giving a more or less complete answer to these questions in terms of the singularities of the curve interestingly a recurrent motif in these arguments is one introduced by mumford in 8 the behavior of the maximal isotropic subspaces for a nondegenerate quadratic form on an even dimensional vector space here the mainspring of theorems 1 13 2 12 and 2 22 is assertion 1 4 below that the sum of the dimensions of the pairwise intersections of any three such subspaces is congruent to n mod 2 received by the editors july 11 1980 and in revised form october 22 1981 1980 mathematics subject classification primary 14h99 14h30 secondary 14h20 14h40 14k25 research partially supported by nsf grant mcs78 04008 aol dl 982 american mathematical society 0002 9947 8 1 0000 0371 10 25

the boundary field theory induced by the chern simons theory the chern simons theory defined on a three dimensional manifold with boundary is written as a two dimensional field theory defined only on the boundary of the three manifold the resulting theory is essentially the pull back to the boundary of a symplectic structure defined on the space of auxiliary fields in terms of which the connection one form of the chern simons theory is expressed when solving the condition of vanishing curvature the counting of the physical degrees of freedom living in the boundary associated with the model is performed using dirac s canonical analysis for the particular case of the gauge group su 2 the result is that the specific model has one physical local degree of freedom moreover the role of the boundary conditions on the original chern simons theory is displayed and clarified in an example which shows how the gauge content as well as the structure of the constraints of the induced boundary theory is affected

relative quantum field theory we highlight the general notion of a relative quantum field theory which occurs in several contexts one is in gauge theory based on a compact lie algebra rather than a compact lie group this is relevant to the maximal superconformal theory in six dimensions

the kauffman skein algebra of a surface at sqrt 1 we study the structure of the kauffman algebra of a surface with parameter equal to sqrt 1 we obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat su 2 connections over the surface we analyse the asymptotics of traces of curve operators in tqft in non standard regimes where the root of unity parametrizing the tqft accumulates to a root of unity we interpret the case of sqrt 1 in terms of parallel transport operators

periodic solutions and chaos in a non linear model for the delayed cellular immune response abstract we model the cellular immune response using a set of non linear delayed differential equations we observe that the stationary solution becomes unstable above a critical immune response time the exponents characterizing the approach to this bifurcation point as well as the critical slow dynamics are obtained in the periodic regime the minimum virus load is substantially reduced with respect to the stationary solution further increasing the delay time the dynamics display a series of bifurcations evolving to a chaotic regime characterized by a set of 2d portraits

transgression to loop spaces and its inverse iii gerbes and thin fusion bundles abstract we show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space these principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops the equivalence is established by a functor called regression and complements a similar equivalence for bundles and gerbes equipped with connections derived previously in part ii of this series of papers the two equivalences provide a complete loop space formulation of the geometry of gerbes functorial monoidal natural in the base manifold and consistent with passing from the setting with connections to the one without connections we discuss an application to lifting problems which provides in particular loop space formulations of spin structures complex spin structures and spin connections

bohr sommerfeld orbits in the moduli space of flat connections and the verlinde dimension formula we show how the moduli space of flatsu 2 connections on a two manifold can be quantized in the real polarization of 15 using the methods of 6 the dimension of the quantization given by the number of integral fibres of the polarization matches the verlinde formula which is known to give the dimension of the quantization of this space in a kahler polarization

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

abelian bf theory and turaev viro invariant the u 1 bf quantum field theory is revisited in the light of deligne beilinson cohomology we show how the u 1 chern simons partition function is related to the bf one and how the latter on its turn coincides with an abelian turaev viro invariant significant differences compared to the non abelian case are highlighted

the ring structure for equivariant twisted k theory we prove under some mild conditions that the equivariant twisted k theory group of a crossed module admits a ring structure if the twisting 2 cocycle is 2multiplicative we also give an explicit construction of the transgression map t1 h a h 1 n o a for any crossed module n and prove that any element in the image is multiplicative as a consequence we prove that under some mild conditions for a crossed module n and any e s that the equivariant twisted k theory group k e n admits a ring structure as an application we prove that for a compact connected and simply connected lie group g the equivariant twisted k theory group k c g g is endowed with a canonical ring structurek c g g k j d c g g k i j d c g g where d dimg and c h gog s

exponential fitting of the delayed recruitment renewal equation abstract from respiratory physiology to laser based optical devices the so called delayed recruitment renewal equation 1 e d c t d t c t f c t 1 provides the mathematical model in a diverse spectrum of practical applications here e is inversely proportional to the product of the time delay inherent in the physical system and its rate of decay when this time lag is large relative to the reciprocal of the decay rate e is small and this delay differential equation dde is singularly perturbed when this situation obtains c t can exhibit initial layers and chaotic oscillations in order to accurately capture such solution features numerically one must use an approximation technique tailored to singular perturbation problems in this work we develop such a family of exponentially fitted schemes for the numerical approximation of this fundamental dde application of this new technique is then made to a variety of interesting and important problems not the least of which is the subject of dynamical diseases

dynamics for a type of general reaction diffusion model abstract in this paper we discuss the following reaction diffusion model which is a general form of many population models u t x t u t x u t x f u t x we study the oscillatory behavior of solutions about the positive equilibrium k of system with neumann boundary conditions sufficient and necessary conditions are obtained for global attractivity of the zero solution and acceptable conditions are established for the global attractivity of k these results improve and complement existing results for system without diffusion moreover when these results are applied to the diffusive nicholson s blowflies model and the diffusive model of hematopoiesis some new results are obtained for the latter

transgressive loop group extensions a central extension of the loop group of a lie group is called transgressive if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the lie group transgressive loop group extensions are those that can be explored by finite dimensional higher categorical geometry over the lie group we show how transgressive central extensions can be characterized in a loop group theoretical way in terms of loop fusion and thin homotopy equivariance

on cobordism of manifolds with corners this work sets up a cobordism theory for manifolds with corners and gives an identication with the homotopy of a certain limit of thom spectra it thereby creates a geometrical interpretation of adams novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized as an application lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting chern numbers the work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

quadratic functions in geometry topology and m theory we describe an interpretation of the kervaire invariant of a riemannian manifold of dimension 4k 2 in terms of a holomorphic line bundle on the abelian variety h 2k 1 m otimes r z our results are inspired by work of witten on the fivebrane partition function in m theory hep th 9610234 hep th 9609122 our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics and is based on our theory of differential functions these differential functions generalize the differential characters of cheeger simons and the bulk of this paper is devoted to their study

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

homoclinic solution and chaos in xf x 0 is followed by a move in the opposite direction i t 0 however there are equations 1 2 which are closely related to applications and where the negative feedback condition 3 is only true in a certain neighbourhood of x 0 furumochi 3 studied equation

a spectral method for nonlinear wave equations abstract in this paper we introduce a new integration technique to find radially symmetric solutions of nonlinear wave equations defined on a spherical domain with dirichlet boundary conditions and we use this technique to study orbital stability of the standing waves we prove analytically that the wave of lowest energy is stable and we show by numerical computations that the standing waves of higher energy are unstable

η invariants and determinant lines the invariant of an odd dimensional manifold with boundary is investigated the natural boundary condition for this problem requires a trivialization of the kernel of the dirac operator on the boundary the dependence of the invariant on this trivialization is best encoded by the statement that the exponential of the invariant lives in the determinant line of the boundary our main results are a variational formula and a gluing law for this invariant these results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of dirac operators also known as the global anomaly formula the ideas developed here fit naturally with recent work in topological quantum field theory in which gluing which is a characteristic formal property of the path integral and the classical action is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary

existence of chaos in control systems with delayed feedback here f r r denotes some nonlinear function a a positive constant t e r is the independent variable x e r the dependent variable and 1 dx dt the investigation is motivated i by the applications eq 1 1 has found in several areas of applied sciences and ii by the observation of intensely structured periodic and aperiodic oscillations in numerical simulations for a certain class of nonlinearities f we show that these structures are not numerical artefacts but properties of the solution manifold of eq 1 1 in the applications eq 1 1 has been used to explain normal and pathological behavior of control systems in the physiology of blood cell production and respiration 7 8 10 111 and periodic 2 or irregular s activity in neural networks computer simulations to eq 1 1 may be found in 3 the mathematical analysis of this equation so far has been restricted to prove the existence of slowly oscillating simple periodic solutions 1 4 61 a periodic solution is called slowly oscillating if successive

lifting problems and transgression for non abelian gerbes abstract we discuss lifting and reduction problems for bundles and gerbes in the context of a lie 2 group we obtain a geometrical formulation and a new proof for the exactness of breen s long exact sequence in non abelian cohomology we use our geometrical formulation in order to define a transgression map in non abelian cohomology this transgression map relates the degree one non abelian cohomology of a smooth manifold represented by non abelian gerbes with the degree zero non abelian cohomology of the free loop space represented by principal bundles we prove several properties for this transgression map for instance it reduces in case of a lie 2 group with a single object to the ordinary transgression in ordinary cohomology we describe applications of our results to string manifolds first we obtain a new comparison theorem for different notions of string structures second our transgression map establishes a direct relation between string structures and spin structures on the loop space

oscillation and global attractivity in hematopoiesis model with periodic coefficients in this paper we shall consider the nonlinear delay differential equation p t email protected t p m email protected 1 p n email protected email protected t p t where k is a positive integer b t and c t are positive periodic functions of period w in the nondelay case we shall show that has a unique positive periodic solution email protected t and we will study the global attractivity of email protected t in the delay case we shall establish some sufficient conditions for oscillation of all positive solutions of about email protected t and establish some sufficient conditions for the global attractivity of email protected t our results in this paper extend as well as improve the results in the autonomous case

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

classical bv theories on manifolds with boundary in this paper we extend the classical bv framework to gauge theories on spacetime manifolds with boundary in particular we connect the bv construction in the bulk with the bfv construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with corners we present several examples including electrodynamics yang mills theory and topological field theories coming from the aksz construction in particular the chern simons theory the bf theory and the poisson sigma model this paper is the first step towards developing the perturbative quantization of such theories on manifolds with boundary in a way consistent with gluing

gerbes simplicial forms and invariants for families of foliated bundles the notion of smooth deligne cohomology is conveniently reformulated in terms of the simplicial derham complex in particular the usual chern weil and chern simons theory is well adapted to this framework and rather easily gives rise to characteristic deligne cohomology classes associated to families of bundles and connections in turn this gives invariants for families of foliated bundles the construction provides representing cocycles in the usual cech derham model for smooth deligne cohomology called gerbes with connection as they generalize usual hermitian line bundles with connection a special case is the quillen line bundle associated to families of flat su 2 bundles

higher degree analogs of the determinant line bundle in the first part of this paper given a smooth family of dirac type operators on an odd dimensional closed manifold we construct an abelian gerbe with connection whose curvature is the three form component of the atiyah singer families index the orem in the second part of the paper given a smooth family of dirac type operators whose index lies in the subspace b of the reduced k theory of the parametrizing space we construct a set of deligne cohomology classes of degree i whose curvatures are the i form component of the atiyah singer families index theorem

twisted k theory and k theory of bundle gerbes in this note we introduce the notion of bundle gerbe k theory and investigate the relation to twisted k theory we provide some examples possible applications of bundle gerbe k theory to the classification of k brane charges in nontrivial backgrounds are briefly discussed

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

higgs fields bundle gerbes and string structures we use bundle gerbes and their connections and curvings to obtain an explicit formula for a de rham representative of the string class of a loop group bundle this is related to earlier work on calorons

approximating chaotic saddles for delay differential equations chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos they play a key role in mediating transport processes involving scattering and chaotic transients here we present evidence long chaotic transients and fractal basins of attraction of transient chaos in a logistic delay differential equation we adapt an existing method stagger and step to numerically construct the chaotic saddle for this system this is the first such analysis of transient chaos in an infinite dimensional dynamical system and in delay differential equations in particular using poincar e section techniques we illustrate approaches to visualizing the saddle set and confirm that the saddle has the cantor like fractal structure consistent with a chaotic saddle generated by horseshoe type dynamics

some comments on chern simons gauge theory following m f atiyah and r bott ab and e witten w we consider the space of flat connections on the trivialsu 2 bundle over a surfacem modulo the space of gauge transformations we describe on this quotient space a natural hermitian line bundle with connection and prove that if the surfacem is now endowed with a complex structure this line bundle is isomorphic to the determinant bundle we show heuristically how path integral quantisation of the chern simons action yields holomorphic sections of this bundle

bundle gerbes for chern simons and wess zumino witten theories we develop the theory of chern simons bundle 2 gerbes and multiplicative bundle gerbes associated to any principal g bundle with connection and a class in h4 bg for a compact semi simple lie group g the chern simons bundle 2 gerbe realises differential geometrically the cheeger simons invariant we apply these notions to refine the dijkgraaf witten correspondence between three dimensional chern simons functionals and wess zumino witten models associated to the group g we do this by introducing a lifting to the level of bundle gerbes of the natural map from h4 bg to h3 g the notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non simply connected lie groups the implications for wess zumino witten models are also discussed

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

elliptic pdes with constant coefficients on convex polyhedra via the unified method abstract we provide a new method to study the classical dirichlet problem for constant coefficient second order elliptic pdes on convex polyhedrons our approach is heavily motivated by fokas unified method for boundary value problems and can be interpreted as the fourier analogue to the classical boundary integral equations the central object in this approach is the global relation an integral equation which couples the known boundary data and the unknown boundary values this integral equation depends holomorphically on two complex parameters and the resulting analysis takes place on a banach space of complex analytic functions closely related to the classical paley wiener space we write the global relation in the form of an operator equation and show that the analysis can be reduced to the case of laplace s equation from which the more general problem turns out to be a compact perturbation we give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realisation of the fundamental principle of ehrenpreis for all constant coefficient elliptic pdes on convex polyhedra

string structures and trivialisations of a pfaffian line bundle the present paper is a contribution to categorial index theory its main result is the calculation of the pfaffian line bundle of a certain family of real dirac operators as an object in the category of line bundles furthermore it is shown how string structures give rise to trivialisations of that pfaffian

asymptotic faithfulness of the quantum su n representations of the mapping class groups we prove that the sequence of projective quantum su n representations of the mapping class group of a closed oriented surface obtained from the projective flat su n verlinde bundles over teichm uller space is asymptotically faithful that is the intersection over all levels of the kernels of these representations is trivial whenever the genus is at least 3 for the genus 2 case this intersection is exactly the order 2 subgroup generated by the hyper elliptic involution in the case of even degree and n 2 otherwise the intersection is also trivial in the genus 2 case

mackey glass equation with variable coefficients the mackey glass equation dndt r t n g t 1 n g t c b t n t is considered with variable coefficients and a nonconstant delay under rather natural assumptions all solutions are positive and bounded persistence and extinction conditions are presented for this equation in the case when there exists a constant positive equilibrium local asymptotic stability of the constant solution and oscillation about this equilibrium are analyzed the results are illustrated by numerical examples in particular it is demonstrated that with delay in both terms a solution with positive initial conditions may become negative

fractional analytic index for a finite rank projective bundle over a compact manifold so associated to a torsion dixmier douady 3 class w on the manifold we define the ring of dierential operators acting on sections of the projective bundle in a formal sense in particular any oriented even dimensional manifold carries a projective spin dirac operator in this sense more generally the correspond ing space of pseudodierential operators is defined with supports suciently close to the diagonal i e the identity relation for such elliptic operators we define the numerical index in an essentially analytic way as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant using the heat kernel method for the twisted projective spin dirac operator we show that this index is given by the usual formula now in terms of the twisted chern character of the symbol which in this case defines an element of k theory twisted by w hence the index is a rational number but in general it is not an integer a genus is a rational number but not necessarily an integer in this paper we show that in the oriented even dimensional case one can nevertheless define a projective spin dirac operator with an analytic index valued in the rational numbers and prove the analogue of the atiyah singer index theorem for this operator twisted by a general projective bundle in fact we establish the analogue of the atiyah singer index theorem for a general projective elliptic pseudodierential operator in a subsequent paper the families case will be discussed for a compact manifold z and vector bundles e and f over z the schwartz ker nel theorem gives a one to one correspondence between continuous linear operators 1

the witten reshetikhin turaev invariant for links in finite order mapping tori i abstract we state asymptotic expansion and growth rate conjectures for the witten reshetikhin turaev invariants of arbitrary framed links in 3 manifolds and we prove these conjectures for the natural links in mapping tori of finite order automorphisms of marked surfaces our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface which we show coincides with the construction of the witten reshetikhin turaev invariants using conformal field theory as was recently completed by andersen and ueno

sign choices for orientifolds we analyse the problem of assigning sign choices to o planes in orientifolds of type ii string theory we show that there exists a sequence of invariant p gerbes with p ge 1 which give rise to sign choices and are related by coboundary maps we prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations concrete calculations for spherical and toroidal orientifolds are carried out and in particular we exhibit a four dimensional orientifold where not every sign choice is geometrically attainable we elucidate how the k theory groups associated with invariant p gerbes for p 1 0 1 interact with the coboundary maps this allows us to interpret a notion of k theory due to gao and hori as a special case of twisted kr theory which consequently implies the homotopy invariance and fredholm module description of their construction

framed m branes corners and topological invariants we uncover and highlight relations between the m branes in m theory and various topological invariants the hopf invariant over q z and z2 the kervaire invariant the f invariant and the invariant this requires either a framing or a corner structure the canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies we characterize the flux quantization condition on the c field and the topological action of the m5 brane via the hopf invariant and the dual of the c field as a refinement of an element of hopf invariant two in the signature formulation the contribution to the m brane effective action is given by the maslov index of the corner the kervaire invariant implies that the effective action of the m5 brane is quadratic our study leads to viewing the self dual string which is the boundary of the m2 brane on the m5 brane worldvolume as a string theory in the sense of cobordism of manifolds with corners we show that the dynamics of the c field and its dual are encoded in a unified way in the 4 sphere which suggests the corresponding spectrum as the generalized cohomology theory describing the fields the effective action of the corner is captured by the f invariant which is an invariant at chromatic level two finally considering m theory on manifolds with g2 holonomy we show that the canonical g2 structure minimizes the topological part of the m5 brane action this is done via the invariant and a variant that we introduce related to the one loop polynomial we uncover and highlight relations between the m branes in m theory and various topological invariants the hopf invariant over q z and z2 the kervaire invariant the f invariant and the invariant this requires either a framing or a corner structure the canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies we characterize the flux quantization condition on the c field and the topological action of the m5 brane via the hopf invariant and the dual of the c field as a refinement of an element of hopf invariant two in the signature formulation the contribution to the m brane effective action is given by the maslov index of the corner the kervaire invariant implies that the effective action of the m5 brane is quadratic our study leads to viewing the self dual string which is the boundary of the m2 brane on the m5 brane worldvolume as a string theory in the sense of cobordism of manifolds with corners

oscillation and global attractivity in hematopoiesis model with delay time in this paper we shall consider the nonlinear delay differential equations with delay time p t bp m t t 1 p n t t cp t that is proposed as a model of hematopoiesis blood cell production where p t denotes the density of mature cells in blood circulation and the time delay t is the time between the production of immature cells in the bone marrow and their maturation for release in the circulating bloodstreams our aim is to give sufficient condition for oscillation of all positive solutions of about the positive steady state and obtain some sufficient conditions for the global attractivity our results extend and improve the well known oscillation results of when m 0

asymptotic properties of the quantum representations of the mapping class group we establish two results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat su n bundle with hitchin s connection first we prove that these projective representations lift to asymptotic representations second we show that under an infinitesimal rigidity assumption the characters of these representations have an asymptotic expansion the leading order term of this expansion agrees with the formula derived heuristically by e witten in quantum field theory and the jones polynomial

on the rigorous foundations of the fokas method for linear elliptic partial differential equations in this paper we address some of the rigorous foundations of the fokas method confining attention to boundary value problems for linear elliptic partial differential equations on bounded convex domains the central object in the method is the global relation which is an integral equation in the spectral fourier space that couples the given boundary data with the unknown boundary values using techniques from complex analysis of several variables we prove that a solution to the global relation provides a solution to the corresponding boundary value problem and that the solution to the global relation is unique the result holds for any number of spatial dimensions and for a variety of boundary value problems

laplace s equation on convex polyhedra via the unified method we provide a new method to study the classical dirichlet problem for laplace s equation on a convex polyhedron this new approach was motivated by fokas unified method for boundary value problems the central object in this approach is the global relation an integral equation which couples the known boundary data and the unknown boundary values this integral equation depends holomorphically on two complex parameters and the resulting analysis takes place on a banach space of complex analytic functions closely related to the classical paley wiener space we write the global relation in the form of an operator equation and prove that the relevant operator is bounded below using some novel integral identities we give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realization of the fundamental principle of ehrenpreis

primary operations in differential cohomology abstract we characterize primary operations in differential cohomology via stacks and illustrate by differentially refining steenrod squares and steenrod powers explicitly this requires a delicate interplay between integral rational and mod p cohomology as well as cohomology with u 1 coefficients and differential forms along the way we develop computational techniques in differential cohomology including a kunneth decomposition that should also be useful in their own right and point to applications to higher geometry and mathematical physics

abelian chern simons theory i a topological quantum field theory we give a construction of the abelian chern simons gauge theory from the point of view of a 2 1 dimensional topological quantum field theory the definition of the quantum theory relies on geometric quantization ideas that have been previously explored in connection to the non abelian chern simons theory j diff geom 33 787 902 1991 topology 32 509 529 1993 we formulate the topological quantum field theory in terms of the category of extended 2 and 3 manifolds introduced in a preprint by walker in 1991 and prove that it satisfies the axioms of unitary topological quantum field theories formulated by atiyah publ math inst hautes etudes sci pans 68 175 186 1989

bundle 2 gerbes we make the category textbf bgrb m of bundle gerbes on a manifold m into a 2 category by providing 2 cells in the form of transformations of bundle gerbe morphisms this description of textbf bgrb m as a 2 category is used to define the notion of a bundle 2 gerbe to every bundle 2 gerbe on m is associated a class in h 4 m mathbb z we define the notion of a bundle 2 gerbe connection and show how this leads to a closed integral differential 4 form on m which represents the image in real cohomology of the class in h 4 m mathbb z some examples of bundle 2 gerbes are discussed including the bundle 2 gerbe associated to a principal g bundle p to m it is shown that the class in h 4 m mathbb z associated to this bundle 2 gerbe coincides with the first pontryagin class of p this example was previously considered from the point of view of 2 gerbes by brylinski and mclaughlin

examples of transverse homoclinic orbits in delay equations on presente deux exemples specifiques d equations aux differences differentielles scalaires avec un retard de la forme x t x t f x t 1 0 pour lesquelles il y a une orbite periodique avec une orbite homocline transversale q asymptotique a

quantum charges and spacetime topology the emergence of new superselection sectors a new form of superselection sectors of topological origin is developed by that it is meant a new investigation that includes several extensions of the traditional framework of doplicher haag and roberts in local quantum theories at first we generalize the notion of representations of nets of c algebras then we provide a brand new view on selection criteria by adopting one with a strong topological flavour we prove that it is coherent with the older point of view hence a clue to a genuine extension in this light we extend roberts cohomological analysis to the case where 1 cocycles bear non trivial unitary representations of the fundamental group of the spacetime equivalently of its cauchy surface in the case of global hyperbolicity a crucial tool is a notion of group von neumann algebras generated by the 1 cocycles evaluated on loops over fixed regions one proves that these group von neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type i all that amounts to a new invariant in a topological sense which can be defined as the dimension of the factor we prove that any 1 cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored this second part much resembles what in literature is known as geometric phases indeed by the very geometrical origin of the 1 cocycles that we discuss in the paper they are essential tools in the theory of net bundles and the topological part is related to their holonomy content at the end we prove the existence of net representations

exponentially growing solutions for a delay diffusion equation with negative feedback abstract the scalar partial delay differential equation u t u u t 1 u t is investigated it was known that in one space dimension all nonnegative solutions of the initial boundary value problem stay bounded as t surprisingly this ceases to remain true in two or more dimensions if the delay is large and the diffusion coefficient small there exists a large set of trajectories whose total mass tends exponentially to infinity as t these trajectories escape the negative feedback of the equation by transporting their mass around along periodic paths in space where the paths can be prescribed by the initial data

variational principles for locally variational forms we present the theory of higher order local variational principles in fibered manifolds in which the fundamental global concept is a locally variational dynamical form any two lepage forms defining a local variational principle for this form differ on intersection of their domains by a variationally trivial form in this sense but in a different geometric setting the local variational principles satisfy analogous properties as the variational functionals of the chern simons type the resulting theory of extremals and symmetries extends the first order theories of the lagrange souriau form presented by grigore and popp and closed equivalents of the first order euler lagrange forms of hakova and krupkova conceptually our approach differs from prieto who uses the poincare cartan forms which do not have higher order global analogues