the transverse index theorem for proper cocompact actions of lie groupoids given a proper cocompact action of a lie groupoid we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes we prove a cohomological index formula for this pairing by applying the van est map and algebraic index theory finally we discuss in examples the meaning of the index pairing and our index formula

α continuity properties of one dimensional quasicrystals we apply the jitomirskaya last extension of the gilbert pearson theory to discrete one dimensional schrodinger operators with potentials arising from generalized fibonacci sequences we prove for certain rotation numbers that for every value of the coupling constant there exists an 0 such that the corresponding operator has purely continuous spectrum this result follows from uniform upper and lower bounds for the l norm of the solutions corresponding to energies from the spectrum of the operator

the atiyah singer index theorem as passage to the classical limit in quantum mechanics a new approach to the atiyah singer index theorem is described using the technique of continuous fields ofc algebras the proof is given in the case of elliptic pseudodifferential operators on n

analytical and numerical studies of the variable coefficient gardner equation the gardner or extended korteweg de vries equation with the variable sign cubic non linear term is studied this equation occurred in the theory of interfacial waves in the stratified shear flow at some certain conditions on the medium stratification an asymptotic method is applied to study the solitary wave transformation when the coefficient of the cubic non linear term is varied slowly it is found the one branch of the solitary waves destroys in the vicinity of the bifurcation point where the exponential soliton transforms into the breather through algebraic soliton other solitary waves change their amplitude accordingly to the cubic non linear term variation the structure of the tail formed behind the solitary wave is determined direct numerical simulation of the gardner equation is performed to analyze the transformation and breaking of the solitary wave it confirms most of the theoretical results the calculated dynamics of the table top soliton due to the large values of the mass and momentum integrals differs from the asymptotic prediction because the characteristic time of the variation of the cubic non linear term has the same order as the duration of table top soliton

interaction of a solitary wave with an external force abstract using the forced korteweg de vries equation as a model we investigate the interaction of a solitary wave with an isolated moving external force of small amplitude we use both analysis and numerical calculations our analysis is based on a multi scale perturbation asymptotic analysis and is carried through to either the first or second order depending on whether the first or more accurate second order expression for the solitary wave speed is used the perturbation theory leads to a pair of autonomous equations for the solitary wave amplitude and position and these are analyzed in detail for the two extreme cases when the lengthscale of the external force is either much greater or much less than that of the solitary wave the theory predicts that the main regimes are passage when the solitary wave passes through the external force with some amplitude modulation repulsion when the solitary wave is reflected by the external force with a significant amplitude change and trapping when the solitary wave remains either partially or totally trapped at the location of the external force these theoretical predictions are confirmed in our numerical simulations of the forced korteweg de vries equation

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

local index theory over foliation groupoids we give a local proof of an index theorem for a dirac type operator that is invariant with respect to the action of a foliation groupoid g i fm denotes the space of units of g then the input is a g equivariant fiber bundle p m along with a g invariant fiberwise dirac type operator d on p the index theorem is a formula for the pairing of the index of d a s an element of a certain k theory group with a closed graded trace on a certain noncommutative de rham algebra b associated to g the proof is by means of superconnections in the framework of noncommutative geometry

soliton train dynamics in a weakly nonlocal non kerr nonlinear medium we analyze chainlike n soliton dynamics in a weakly nonlocal essentially nonintegrable system described by the cubic quintic nonlinear schrodinger equation quintic nonlinearity is not assumed to be small this system is reduced to a generalized complex toda chain model numerical simulations demonstrate adverse action of both cubic and quintic nonlocal responses in their own right on the quasi equidistant train propagation with a development of a chaotic regime from the toda chain model we predict a possibility of mutually compensating both types of nonlocality induced distortion restoring thereby a deterministic mode of the train propagation in a weakly nonlocal medium analytical predictions corroborate well with numerical results

interactions of solitons in nonintegrable systems direct perturbation method and applications abstract problems of approximate description of soliton interaction in nonintegrable systems are discussed it is shown that in an arbitrary lagrangian system in the first nontrivial order of the small parameter asymptotic scheme for a system of solitons with close velocities a lagrangian may be introduced analogous to that for classical particles with a pair interaction potential a classification of possible types of soliton interaction is given and criteria of the existence of bound states are formulated a possibility of generalization of approximated description of soliton interaction in the multidimensional case is noted

the fibonacci hamiltonian we consider the fibonacci hamiltonian the central model in the study of electronic properties of one dimensional quasicrystals and establish relations between its spectrum and spectral characteristics namely the optimal holder exponent of the integrated density of states the dimension of the density of states measure the dimension of the spectrum and the upper transport exponent and the dynamical properties of the fibonacci trace map such as dimensional characteristics of the non wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures topological entropy multipliers of periodic orbits we also exhibit a connection between the spectral quantities and the thermodynamic pressure function as a result a detailed description of the spectral properties for all values of the coupling constant is obtained in contrast to all previous quantitative results which could be established only in the regime of small or large coupling in particular we show that the spectrum of this operator is a dynamically defined cantor set and that the density of states measure is exact dimensional this implies that all standard fractal dimensions coincide in each case we show that all the gaps of the spectrum allowed by the gap labeling theorem are open for all values of the coupling constant also we establish strict inequalities between the four spectral characteristics in question and provide the exact large coupling asymptotics of the dimension of the density of states measure for the other three quantities the large coupling asymptotics were known before

stable particle like solutions of multidimensional nonlinear fields abstract stable localized structures of two and three dimensional fields are investigated a theory of the elementary particle interaction the formation of their bound states and dynamics under the action of external perturbations is considered for scalar and complex fields

the spectrum of a quasiperiodic schrodinger operator the spectrum h of the tight binding fibonacci hamiltonian hmn m n 1 m 1 n m n v n v n chi omega 3 omega 2 n 1 1 is the golden number is shown to coincide with the dynamical spectrum the set on which an infinite subsequence of traces of transfer matrices is bounded the point spectrum is absent for any and h is a cantor set for 4 combining this with casdagli s earlier result one finds that the spectrum is singular continuous for 16

a homology theory for etale groupoids etale groupoids arise naturally as models for leaf spaces of foliations for orbifolds and for orbit spaces of discrete group actions in this paper we introduce a sheaf homology theory for etale groupoids we prove its invariance under morita equivalence as well as verdier duality between haeiger cohomology and this homology we also discuss the relation to the cyclic and hochschild homologies of connes convolution algebra of the groupoid and derive some spectral sequences which serve as a tool for the computation of these homologies

fedosov differentials and catalan numbers the aim of the paper is to establish a non recursive formula for the general solution of fedosov s quadratic fixed point equation fedosov 1994 j diff geom 40 213 38 fedosov s geometrical fixed point equation for a differential is rewritten in a form similar to the functional equation for the generating function of catalan numbers this allows us to guess the solution an adapted example for kaehler manifolds of constant sectional curvature is considered in detail also for every connection on a manifold a familiar classical differential will be introduced

random pining of localized states and the birth of deterministic disorder within gradient models the birth of spatial disorder from almost regular initial conditions within the swift hohenberg model equation with subcritical bifurcation is considered the complexity of the space series measured by the spatialk2 entropy grows with time and reaches a stationary value depending on the period of the initial regular disturbance a qualitative model is suggested describing the process via the birth of localized structures and its subsequent disordering due to weak interaction

integrability of lie brackets in this paper we present the solution to a longstanding problem of differential geometry lie s third theorem for lie algebroids we show that the integrability problem is controlled by two computable obstructions as applications we derive explain and improve the known integrability results we establish integrability by local lie groupoids we clarify the smoothness of the poisson sigma model for poisson manifolds and we describe other geometrical applications

formality and kontsevich duflo type theorems for lie pairs newcommand poly operatorname poly bullet newcommand td operatorname td l a nabla frac 1 2 newcommand cx 1 operatorname tot big gamma lambda bullet a vee otimes r mathcal 1 poly big newcommand cy 1 mathbb h bullet operatorname ce a mathcal 1 poly kontsevich s formality theorem states that there exists an l infty quasi isomorphism from the dgla t poly m of polyvector fields on a smooth manifold m to the dgla d poly m of polydifferential operators on m which extends the classical hochschild kostant rosenberg map in this paper we extend kontsevich s formality theorem to lie pairs a framework which includes a range of diverse geometric contexts such as complex manifolds foliations and mathfrak g manifolds the spaces cx t and cx d associated with a lie pair l a each carry an l infty algebra structure canonical up to l infty isomorphism these two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators respectively their corresponding cohomology groups cy t and cy d admit canonical gerstenhaber algebra structures we establish the following formality theorem for lie pairs there exists an l infty quasi isomorphism from cx t to cx d whose first taylor coefficient is equal to operatorname hkr circ td here td acts on cx t by contraction furthermore we prove a kontsevich duflo type theorem for lie pairs the hochschild kostant rosenberg map twisted by the square root of the todd class of the lie pair l a is an isomorphism of gerstenhaber algebras from cy t to cy d as applications we establish formality theorems and kontsevich duflo type theorems for complex manifolds foliations and mathfrak g manifolds in the case of complex manifolds we recover the kontsevich duflo theorem of complex geometry

eigenvalue spacings and dynamical upper bounds for discrete one dimensional schrodinger operators we prove dynamical upper bounds for discrete one dimensional schrodinger operators in terms of various spacing properties of the eigenvalues of finite volume approximations we demonstrate the applicability of our approach by a study of the fibonacci hamiltonian

renormalization group theory for perturbed evolution equations we show that proto rg renormaliation group theory can be used to give a systematic description of the evolution of soltion of perturbed equations the equations describing the deformation of its shape as the effect of perturbation are proto rg equations the rg approach may be simpler than inverse scattering theory ist and another approaches because it dose not rely on any knowledge of ist it is very concise and easy to understand

geometric structures as deformed infinitesimal symmetries a general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries specifically this model consists of a lie algebroid equipped with an affine connection compatible with the lie algebroid structure the curvature of this connection vanishes precisely when the structure is locally symmetric this model generalizes cartan geometries a substantial class to the intransitive case simple examples are surveyed and corresponding local obstructions to symmetry are identified these examples include foliations riemannian structures infinitesimal g structures symplectic and poisson structures

dynamical upper bounds on wavepacket spreading we derive a general upper bound on the spreading rate of wavepackets in the framework of schrodinger time evolution our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent schrodinger equation we also derive a new lower bound on the spreading rate which is strongly connected with our upper bound we apply these new bounds to the fibonacci hamiltonian the most studied one dimensional model of quasicrystals as a result we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization the bounds have the same qualitative behavior in the limit of large coupling

quantum dynamics via complex analysis methods general upper bounds without time averaging and tight lower bounds for the strongly coupled fibonacci hamiltonian abstract we develop further the approach to upper and lower bounds in quantum dynamics via complex analysis methods which was introduced by us in a sequence of earlier papers here we derive upper bounds for non time averaged outside probabilities and moments of the position operator from lower bounds for transfer matrices at complex energies moreover for the time averaged transport exponents we present improved lower bounds in the special case of the fibonacci hamiltonian these bounds lead to an optimal description of the time averaged spreading rate of the fast part of the wavepacket in the large coupling limit this provides the first example which demonstrates that the time averaged spreading rates may exceed the upper box counting dimension of the spectrum

a universal separatrix map for weak interactions of solitary waves in generalized nonlinear schrodinger equations abstract it is known that weak interactions of two solitary waves in generalized nonlinear schrodinger nls equations exhibit fractal dependence on initial conditions and the dynamics of these interactions is governed by a universal two degree of freedom ode system y zhu j yang universal fractal structures in the weak interaction of solitary waves in generalized nonlinear schrodinger equations phys rev e 75 2007 036605 in this paper this ode system is analyzed comprehensively using asymptotic methods along separatrix orbits a simple second order map is derived this map does not have any free parameters after variable rescalings and thus is universal for all weak interactions of solitary waves in generalized nls equations comparison between this map s predictions and direct simulations of the ode system shows that the map can capture the fractal scattering phenomenon of the ode system very well both qualitatively and quantitatively

direct perturbation theory for a nearly integrable nonlinear equation with application to dark soliton solutions since the linearized equation for a nearly integrable nonlinear equation is essentially a linear equation the direct perturbation theory is now developed in the frame of green s function theory of linear differential equations the perturbed nls equation with normal dispersion is treated as an example some obscure points in previous works about this equation are clarified

local index theory over etale groupoids we give a superconnection proof of connes index theorem for proper cocompact actions of etale groupoids this includes connes general foliation index theo rem for foliations with hausdorholonomy groupoid

on co morphisms of lie pseudoalgebras and groupoids we give a unified description of morphisms and comorphisms of lie pseudoalgebras showing that the both types of morphisms can be regarded as subalgebras of a lie pseudoalgebra called the sum we also provide similar descriptions for morphisms and comorphisms of lie algebroids and groupoids

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

on integrability of infinitesimal actions we use foliations and connections on principal lie groupoid bundles to prove various integrability results for lie algebroids in particular we show under quite general assumptions that the semi direct product associated to an infinitesimal action of one integrable lie algebroid on another is integrable this generalizes recent results of dazord and nistor

lie algebroid structures on double vector bundles and representation theory of lie algebroids abstract a vb algebroid is essentially defined as a lie algebroid object in the category of vector bundles there is a one to one correspondence between vb algebroids and certain flat lie algebroid superconnections up to a natural notion of equivalence in this setting we are able to construct characteristic classes which in special cases reproduce characteristic classes constructed by crainic and fernandes we give a complete classification of regular vb algebroids and in the process we obtain another characteristic class of lie algebroids that does not appear in the ordinary representation theory of lie algebroids

algebraic constructions in the category of lie algebroids the aim of this paper is to give the first general and abstract treatment of the algebraic properties of lie algebroids the concept of lie algebroid was introduced in 1967 by pradines ls as the basic infinitesimal invariant of a differentiable groupoid the construction of the lie algebroid of a differentiable groupoid is a direct generalization of the construction of the lie algebra of a lie group and 183 described a full lie theory for differentiable groupoids and lie algebroids encompassing many phenomena in the foundations of differential geometry a more detailed account and further references are given in 13 however at this stage the algebraic properties of lie algebroids were not pursued in 13 iii section 2 iv section 11 one of us gave a fairly detailed account of the abstract algebra of lie algebroids over a fixed base lie algebroids are vector bundles with several additional structures and a category of lie algebroids on a given base and morphisms which preserve that base has properties similar to those of the category of lie algebras using this algebra it was demonstrated in 13 that the lie theory of a category of locally trivial differentiable lie groupoids over a fixed base and the corresponding category of transitive lie algebroids over that base is coextensive with the standard theory of connections in principal bundles we now propose to show that the lie functor from the category of all differentiable groupoids over arbitrary bases and arbitrary smooth morphisms to the category of all lie algebroids preserves the basic algebraic constructions known to be possible in the category of differentiable groupoids the main work is to give abstract formulations of the corresponding lie algebroid concepts the groupoid results with which we

dynamics of domain walls governed by the convective cahn hilliard equation the convective cahn hilliard cch equation ut uxx u u 3 xx d 2 u 2 x 0 has been suggested recently for the description of several physical phenomena including spinodal decomposition of driven phase separating systems in an external field instability of steps moving on a crystal surface and faceting of growing thermodynamically unstable surfaces in this paper the dynamics of domain walls kinks governed by the convective cahn hilliard equation is studied by means of asymptotic and numerical methods a special attention is paid to the dynamics of kink pairs and triplets that play crucial role in the coarsening ostwald ripening process for the driving parameter d d 0 2 3 and large distance between the kinks l 1 analytical formulas are found that describe the motion of the kink pairs and triplets the analytical formulas are in excellent agreement with the results of direct numerical simulations of cch equation they explain the logarithmically slow coarsening observed for the kink dynamics governed by cch equation when the distances between the kinks become large unlike the fast coarsening for moderate distances between the kinks at intermediate stages 2005 elsevier b v all rights reserved

representations up to homotopy of lie algebroids we introduce and study the notion of representation up to homotopy of a lie algebroid paying special attention to examples we use representations up to homotopy to define the adjoint representation of a lie algebroid and show that the resulting cohomology controls the deformations of the structure the weil algebra of a lie algebroid is defined and shown to coincide with kalkman s brst model for equivariant cohomology in the case of group actions the relation of this algebra with the integration of poisson and dirac structures is explained in arias abad crainic ann inst fourier

chaotic behaviour of soliton lattice in an unstable dissipative dispersive nonlinear system abstract dynamical behaviours of a sequence of asymmetric soliton like pulses arising in an unstable dissipative dispersive non linear wave system are investigated in terms of a nonlinear lattice model it is found in a three pulse periodic system in which either periodic or chaotic motions which are consistent with the results of the numerical initial value problem occur when the individual interacting pulse has an asymmetric oscillatory tail structure the existence of oscillatory structure is found to be responsible for chaotic motions several general properties of the asymmetric lattice are also discussed in comparison with the symmetric lattices for the korteweg de vries kdv and the fifth order kdv solitons

dynamics of spiral waves in non equilibrium media in this paper the spiral waves in two dimensional nonequilibrium media are studied on the basis of the two dimensional ginzburg landau equation in the general case the perturbation of origin spiral waves leads to the spiral rotation and to the motion of spiral cores motion in media with hard and soft excitation is analysed separately

an algebraic index theorem for orbifolds abstract using the concept of a twisted trace density on a cyclic groupoid a trace is constructed on a formal deformation quantization of a symplectic orbifold an algebraic index theorem for orbifolds follows as a consequence of a local riemann roch theorem for such densities in the case of a reduced orbifold this proves a conjecture by fedosov schulze and tarkhanov finally it is shown how the kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem

uniform spectral properties of one dimensional quasicrystals i absence of eigenvalues we consider discrete one dimensional schrodinger operators with sturmian potentials for a full measure set of rotation numbers including the fibonacci case we prove absence of eigenvalues for all elements in the hull

lie algebroid foliations and script e1 m dirac structures we prove some general results about the relation between the 1 cocycles of an arbitrary lie algebroid a over m and the leaves of the lie algebroid foliation on m associated with a using these results we show that a 1 m dirac structure l induces on every leaf f of its characteristic foliation a 1 f dirac structure lf which comes from a precontact structure or from a locally conformal presymplectic structure on f in addition we prove that a dirac structure on m can be obtained from l and we discuss the relation between the leaves of the characteristic foliations of l and

transport exponents of sturmian hamiltonians 2015 elsevier inc we consider discrete schrodinger operators with sturmian potentials and study the transport exponents associated with them under suitable assumptions on the frequency we establish upper and lower bounds for the upper transport exponents as an application of these bounds we identify the large coupling asymptotics of the upper transport exponents for frequencies of constant type we also bound the large coupling asymptotics uniformly from above for lebesgue typical frequency a particular consequence of these results is that for most frequencies of constant type transport is faster than for lebesgue almost every frequency we also show quasi ballistic transport for all coupling constants generic frequencies and suitable phases

on the homology of algebras of whitney functions over subanalytic sets in this article we study several homology theories of the algebra e x of whitney functions over a subanalytic set x r n with a view towards noncommutative geometry using a localization method going back to teleman we prove a hochschild kostant rosenberg type theorem for e x when x is a regular subset of r n having regularly situated diagonals this includes the case of subanalytic x we also compute the hochschild cohomology of e x for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories it is shown that the periodic cyclic homology coincides with the de rham cohomology thus generalizing a result of feigin tsygan motivated by the algebraic de rham theory of grothendieck we finally prove that for subanalytic sets the de rham cohomology of e x coincides with the singular cohomology for the proof of this result we introduce the notion of a bimeromorphic subanalytic triangulation and show that every bounded subanalytic set admits such a triangulation contents

the localized longitudinal index theorem for lie groupoids and the van est map abstract we define the localized index of longitudinal elliptic operators on lie groupoids associated with lie algebroid cohomology classes we derive a topological expression for these numbers using the algebraic index theorem for poisson manifolds on the dual of the lie algebroid underlying the definition and computation of the localized index is an action of the hopf algebroid of jets around the unit space and the characteristic map it induces on lie algebroid cohomology this map can be globalized to differentiable groupoid cohomology giving a definition of the global index that can be computed by localization this correspondence between the global and localized index is given by the van est map for lie groupoids

lagrangian reduction by stages for non holonomic systems in a lie algebroid framework the lagrange d alembert equations of a non holonomic system with symmetry can be reduced to the lagrange d alembert poincarequations in a previous contribution we have shown that both sets of equations fall in the category of the so called lagrangian systems on a subbundle of a lie algebroid in this paper we investigate the special case when the reduced system is again invariant under a new symmetry group and so forth via lie algebroid theory we develop a geometric context in which successive reduction can be performed in an intrinsic way we prove that at each stage of the reduction the reduced systems are part of the above mentioned category and that the lie algebroid structure in each new step is the quotient lie algebroid of the previous step we further show that that reduction in two stages is equivalent with direct reduction

hochschild lefschetz class for mathcal d modules we introduce a notion of hochschild lefschetz class for a good coherent mathcal d module on a compact complex manifold and prove that this class is compatible with the direct image functor we prove an orbifold riemann roch formula for a mathcal d module on a compact complex orbifold

uniform spectral properties of one dimensional quasicrystals iii α continuity we study the spectral properties of one dimensional whole line schrodinger operators especially those with sturmian potentials building upon the jitomirskaya last extension of the gilbert pearson theory of subordinacy we demonstrate how to establish continuity of a whole line operator from power law bounds on the solutions on a half line however we require that these bounds hold uniformly in the boundary condition

an equivariant index for proper actions i abstract equivariant indices taking values in group theoretic objects have previously been defined in cases where either the group acting or the orbit space of the action is compact in this paper we define an equivariant index without assuming the group or the orbit space to be compact this allows us to generalise an index of deformed dirac operators defined for compact groups by braverman in parts ii and iii of this series we explore some properties and applications of this index

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

riemann roch theorems via deformation quantization i abstract we deduce the riemann roch type formula expressing the microlocal euler class of a perfect complex of d modules in terms of the chern character of the associated symbol complex and the todd class of the manifold from the riemann roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization the proof of the latter is contained in the sequel to this paper

higher rho invariants and the moduli space of positive scalar curvature metrics abstract given a closed smooth manifold m which carries a positive scalar curvature metric one can associate an abelian group p m to the space of positive scalar curvature metrics on this manifold the group of all diffeomorphisms of the manifold naturally acts on p m the moduli group p m of positive scalar curvature metrics is defined to be the quotient abelian group of this action i e the coinvariant of the action p m measures the size of the moduli space of positive scalar curvature metrics on m in this paper we use the higher rho invariant and the finite part of the k theory of the group c algebra of 1 m to give a lower bound of the rank of the moduli group p m the key idea of our proof is the use of higher rho invariant this is a secondary invariant associated to dirac operators and in particular depends on the choice of riemannian metric we show that the higher rho invariant remains unchanged in a certain k theory group under the action of the diffeomorphism group allowing us to distinguish elements in p m

cyclic cocycles on deformation quantizations and higher index theorems abstract we construct a nontrivial cyclic cocycle on the weyl algebra of a symplectic vector space using this cyclic cocycle we construct an explicit local quasi isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof we give a new proof of nest tsygan s algebraic higher index theorem by computing the pairing between such cyclic cocycles and the k theory of the formal deformation quantization furthermore we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold as an application we obtain the analytic higher index theorem of connes moscovici and its extension to orbifolds

universal fractal structures in the weak interaction of solitary waves in generalized nonlinear schrodinger equations weak interactions of solitary waves in the generalized nonlinear schr odinger equations are studied it is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities then by using the karpman solov ev method a universal system of dynamical equations is derived for the velocities amplitudes positions and phases of interacting solitary waves these dynamical equations contain a single parameter which accounts for the different forms of nonlinearity when this parameter is zero these dynamical equations are integrable and the exact analytical solutions are derived when this parameter is nonzero the dynamical equations exhibit fractal structures which match those in the original wave equations both qualitatively and quantitatively thus the universal nature of fractal structures in the weak interaction of solitary waves is analytically established the origin of these fractal structures is also explored it is shown that these structures bifurcate from the initial conditions where the solutions of the integrable dynamical equations develop finite time singularities based on this observation an analytical criterion for the existence and locations of fractal structures is obtained lastly these analytical results are applied to the generalized nonlinear schr odinger equations with various nonlinearities such as the saturable nonlinearity and predictions on their weak interactions of solitary waves are made

perturbation theory for kinks and its application for multisoliton interactions in hydrodynamics using an integrable gardner equation as an example a perturbation theory is developed for systems in which limiting amplitude solitons exist in the form of a pair of distanced kinks approximate equations describing multisoliton interactions are derived and further used for modeling the evolution of an arbitrary set of solitons the results are compared with an exact solution and numerical results the theory is applied to data from observation of a train of strongly nonlinear internal waves in the ocean

dimensions of multitype moran sets with lower limit of the contractions being zero abstract multitype moran sets were studied by liu and wen 2005 9 under the assumption of primitivity they proved that for any multitype moran set e with a positive lower bounded condition on contracting ratios dim h e s s dim p e dim b e where s and s are the lower and upper pre dimension according to the natural coverings in this paper we permit that the lower limit of contraction ratios is zero and study the two classes of them for one of the two by assuming some stronger condition than primitivity we show that the above formula still holds for the other under no assumption of primitivity we prove that dim h e s resp dim b e s by assuming a mild condition and show that the above formula also holds provided primitivity and these two mild conditions are satisfied

nonlinearity saturation as a singular perturbation of the nonlinear schroźdinger equation saturation of a kerr type nonlinearity in the nonlinear schro dinger equation nlse can be regarded as a singular perturbation which regularizes the well known blow up phenomenon in the cubic nlse an asymptotic expansion is proposed which takes into account multiple scale behavior both in the longitudinal and transverse directions in one dimension this leads to a free boundary problem reduction where the solitary wave acts solely to reflect impinging waves and is accelerated by an elastic transfer of momentum in two dimensions we find that interaction of a solitary wave and an adjacent wave field is governed by behavior of certain eigenfunctions of the linearized fast scale operator this leads to an outer solution with a free logarithmic singularity whose position evolves by virtue of a large transfer of momentum between the ambient and solitary waves however for a certain value of wave power we find there is essentially no interaction and the solitary wave is asymptotically transparent to the ambient field we test our results by numerical simulation of both the full equation and free boundary reductions

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

cyclic cohomology of the weyl algebra abstract we give an explicit formula for sp 2 n basic representatives of the cyclic cohomology of the weyl algebra h c a 2 n as an application we prove a generalization of a theorem of nest and tsygan concerning the relation of the todd class and the cyclic cohomology of the differential operators on a complex manifold

vector bundles over lie groupoids and algebroids abstract we study vb groupoids and vb algebroids which are vector bundles in the realm of lie groupoids and lie algebroids through a suitable reformulation of their definitions we elucidate the lie theory relating these objects i e their relation via differentiation and integration we also show how to extend our techniques to describe the more general lie theory underlying double lie algebroids and la groupoids

integration of twisted dirac brackets given a lie groupoid g over a manifold m we show that multiplicative 2 forms on g relatively closed with respect to a closed 3 form phi on m correspond to maps from the lie algebroid of g into t m satisfying an algebraic condition and a differential condition with respect to the phi twisted courant bracket this correspondence describes as a special case the global objects associated to phi twisted dirac structures as applications we relate our results to equivariant cohomology and foliation theory and we give a new description of quasi hamiltonian spaces and group valued momentum maps

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

reduction of a symplectic like lie algebroid with momentum map and its application to fiberwise linear poisson structures this paper addresses the problem of developing an extension of the marsden weinstein reduction process to symplectic like lie algebroids and in particular to the case of the canonical cover of a fiberwise linear poisson structure whose reduction process is the analog to cotangent bundle reduction in the context of lie algebroids dedicated to the memory of jerrold e marsden

absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian author s damanik d gorodetski a solomyak b abstract 2015 we prove for the square fibonacci hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants this is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact dimensional measures

quantum k theory i the chern character we construct a cocycle on an infinite dimensional generalization of ap summable fredholm module our framework is related to connes cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds thep summability condition is characteristic of dimensiono p we replace this assumption by the requirement that there exists an underlying heat kernel which is trace class then we use the heat kernel to regularize states in dimension independent fashion our cocycle may be interpreted as an infinite dimensional chern character

multi pulse embedded solitons as bound states of quasi solitons abstract embedded solitons ess is a name given to localised solutions of multi component dispersive wave models that occur in resonance with the linear spectrum they exist at isolated codimension one parameter values within the steady state equations which have linearisation with eigenvalues and i at the same time quasi solitons qss or generalised solitary waves with non decaying radiation tails are known to be endemic we consider the general question of when two qss may be glued together to form a two humped bound state bs es in the limit 0 a generalisation of the method of gorshkov and ostrovsky is used within the framework of general normal forms for hamiltonian reversible systems a simple asymptotic formula is derived that governs the existence symmetry and accumulation rate of the bss this generalises earlier ad hoc calculations for specific examples the formula depends on few simple ingredients the eigenvalues of the equilibrium an asymptotic estimate for the tail amplitude of the qss the birkhoff signature and symmetry properties of the hamiltonian only the tail amplitude estimate is difficult to calculate requiring exponential asymptotics but it is shown that this only affects the third order term in the asymptotic formula the calculation is worked out in detail for the steady states of several example pde systems taken from nonlinear optics and fluid mechanics and excellent agreement with numerical results is found

dynamical upper bounds for one dimensional quasicrystals abstract following the killip kiselev last method we prove quantum dynamical upper bounds for discrete one dimensional schrodinger operators with sturmian potentials these bounds hold for sufficiently large coupling almost every rotation number and every phase