synchronization in a semiclassical kuramoto model synchronization is a ubiquitous phenomenon occurring in social biological and technological systems when the internal rythms of their constituents are adapted to be in unison as a result of their coupling this natural tendency towards dynamical consensus has spurred a large body of theoretical and experimental research in recent decades the kuramoto model constitutes the most studied and paradigmatic framework in which to study synchronization in particular it shows how synchronization appears as a phase transition from a dynamically disordered state at some critical value for the coupling strength between the interacting units the critical properties of the synchronization transition of this model have been widely studied and many variants of its formulations have been considered to address different physical realizations however the kuramoto model has been studied only within the domain of classical dynamics thus neglecting its applications for the study of quantum synchronization phenomena based on a system bath approach and within the feynman path integral formalism we derive equations for the kuramoto model by taking into account the first quantum fluctuations we also analyze its critical properties the main result being the derivation of the value for the synchronization onset this critical coupling increases its value as quantumness increases as a consequence of the possibility of tunneling that quantum fluctuations provide

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

brauer groups and quotient stacks a natural question is to determine which algebraic stacks are quotient stacks in this paper we give some partial answers and relate it to the question of whether for a scheme x the natural map from the brauer group equivalence classes of azumaya algebras to the cohomological brauer group the torsion subgroup of h 2 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i m is surjective

factorization homology i higher categories abstract we construct a pairing which we call factorization homology between framed manifolds and higher categories the essential geometric notion is that of a vari framing of a stratified manifold which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata our main result constructs labeling systems on disk stratified vari framed n manifolds from n categories these n categories in contrast with the literature to date are not required to have adjoints this allows the following conceptual definition the factorization homology m c of a framed n manifold m with coefficients in an n category c is the classifying space of c labeled disk stratifications over m the core calculation underlying our main result is the following for any disk stratified manifold the space of conically smooth diffeomorphisms which preserve a vari framing is discrete

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

generalized quantum anharmonic oscillator using an operator ordering approach we construct a generalized expression for the normal ordering of a a m for integral values of m and use the result to study the quantum anharmonic oscillator problem in the heisenberg approach in particular we derive generalized expressions for energy eigenvalues and frequency shifts for the hamiltonian h x2 2 2 2 m xm we also derive a closed form first order multiscale perturbation theoretic operator solution of this hamiltonian with a view to generalize some recent results of bender and bettencourt 1996 phys rev lett 77 4114 1996 phys rev d 54 1710

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

the chen ruan cohomology of moduli of curves of genus 2 with marked points in this work we describe the chen ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points in the first half of the paper we compute the additive structure of the

the algebra of discrete torsion abstract we analyze the algebraic structures of g frobenius algebras which are the algebras associated to global group quotient objects a k a global orbifolds here g is any finite group first we show that these algebras are modules over the drinfel d double of the group ring k g and are moreover k g module algebras and k g comodule algebras we furthermore consider g frobenius algebras up to projective equivalence and define universal shifts of the multiplication and the g action preserving the projective equivalence class we show that these shifts are parameterized by z 2 g k and by h 2 g k when considering of g frobenius algebras up to isomorphism we go on to show that these shifts can be realized by the forming of tensor products with twisted group rings thus providing a group action of z 2 g k on g frobenius algebras acting transitively on the classes g frobenius related by universal shifts the multiplication is changed according to the cocycle while the g action transforms with a different cocycle derived from the cocycle defining the multiplication the values of this second cocycle also appear as a factor in front of the trace which is considered in the trace axiom of g frobenius algebras this allows us to identify the effect of our action by z 2 g k as so called discrete torsion effectively unifying all known approaches to discrete torsion for global orbifolds in one algebraic theory the new group action of discrete torsion is essentially derived from the multiplicative structure of g frobenius algebras this yields an algebraic realization of discrete torsion defined via the perturbation of the multiplication resulting from a tensor product with a twisted group ring additionally we show that this algebraic formulation of discrete torsion allows for a treatment of g frobenius algebras analogous to the theory of projective representations of groups group extensions and twisted group ring modules lastly we identify another set of discrete universal transformations among g frobenius algebras pertaining to their super structure and classified by hom g z 2 z which are essential for the application to mirror symmetry for singularities with symmetries

nonabelian localization in equivariant k theory and riemann roch for quotients abstract we prove a localization formula in equivariant algebraic k theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space this extends to nondiagonalizable groups the localization formulas of nielsen diagonalizably linearized coherent sheaves bull soc math france 102 1974 85 97 and thomason une formule de lefschetz en k theorie equivariante algebrique duke math j 68 3 1992 447 462 as an application we give a riemann roch formula for quotients of smooth algebraic spaces by proper group actions this formula extends previous work of toen theoremes de riemann roch pour les champs de deligne mumford k theory 18 1 1999 33 76 and the authors riemann roch for quotients and todd classes of simplicial toric varieties comm algebra 31 2003 3735 3752

chern character for twisted k theory of orbifolds for an orbifold x and h 3 x z we introduce the twisted cohomology h c x and prove that the non commutative chern character of connes karoubi establishes an isomorphism between the twisted k groups k x c and the twisted cohomology h

gerbes over orbifolds and twisted k theory in this paper we construct an explicit geometric model for the group of ger bes over an orbifold x we show how from its curvature we can obtain its characteristic class in h 3 x via chern weil theory for an arbitrary gerbe l a twisting l korb x of the orbifold k theory of x is constructed and shown to generalize previous twisting by rosenberg 28 witten 35 atiyah segal 2 and bowknegt et al 4 in the smooth case and by adem ruan 1 for discrete torsion on an orbifold

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

stringy product on twisted orbifold k theory for abelian quotients in this paper we present a model to calculate the stringy product on twisted orbifold k theory of adem ruan zhang for abelian complex orbifolds in the first part we consider the non twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian lie group we give an explicit description of the obstruction bundle we explain the relation with the product defined by jarvis kaufmann kimura and via a chern character map with the chen ruan cohomology we explicitly calculate the stringy product for a weighted projective orbifold in the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology we show a decomposition formula for twisted orbifold k theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is 2 3

the caldeira leggett quantum master equation in wigner phase space continued fraction solution and application to brownian motion in periodic potentials the continued fraction method of solving classical fokker planck equations has been adapted to tackle quantum master equations of the caldeira leggett type this was done taking advantage of the phase space wigner representation of the quantum density matrix the approach differs from those in which some continued fraction expression is found for a certain quantity in that the full solution of the master equation is obtained by continued fraction methods this allows us to study in detail the effects of the environment fluctuations and dissipation on several classes of nonlinear quantum systems we apply the method to the canonical problem of quantum brownian motion in periodic potentials both for cosine and ratchet potentials lacking inversion symmetry

on the topology of birational minimal models in the study of higher dimensional algebraic geometry an important reduction step is to study certain good birational models of a given algebraic manifold this leads to the famous minimal model program initiated by mori the search for birational models with numerically effective canonical divisors and with at most terminal singularities the existence problem is still completely open in dimensions bigger than three but even worse in contrast to the two dimensional case the minimal model is not unique in higher dimensions it is then an important question to see what kind of invariants are shared by all the birationally equivalent minimal models and more generally to see what kind of invariants are preserved under certain elementary birational transformations in this paper some results in this direction are given theorem a let f x x be a birational map between two smooth complex projective varieties such that the canonical bundles are numerically effective along the exceptional loci then x and x have the same betti numbers in particular birational smooth minimal models have the same betti numbers

quantum brownian motion and generalizations of hermite polynomials abstract we introduce new families of orthogonal polynomials h d n motivated by the non equilibrium evolution of a quantum brownian particle qbp the h d n s generalize non trivially the standard hermite polynomials employed for classical brownian motion we treat several models labelled by d for a non equilibrium qbp by means of the wigner function w in the presence of a heat bath at thermal equilibrium with and without ab initio friction for long times for a suitable class of initial conditions the non equilibrium wigner function w should approach in some sense the time independent equilibrium wigner function w e q d which describes the thermal equilibrium of the qbp with the heat bath and plays a central role w e q d is chosen to be the weight function which orthogonalizes the h d n s new results on w e q d and on the h d n s are reported we justify the key role of the h d n s as follows using the h d n s moments w e q d n and w n are introduced for w e q d and w respectively at equilibrium all moments w e q d n except the lowest one w e q d 0 vanish identically off equilibrium one expects that for long times for suitable initial conditions i all non equilibrium moments w n except the lowest moment w 0 will approach zero while ii the lowest non equilibrium moment w 0 will tend to w e q d 0 0 to complete the justification we outline how the approximate long time non equilibrium theories determined by w 0 for the different models d yield smoluchowski equations and irreversible evolutions of the qbp towards thermal equilibrium

quantum smoluchowski equation a systematic study the strong friction regime at low temperatures is analyzed systematically starting from the formally exact path integral expression for the reduced dynamics this quantum smoluchowski regime allows for a type of semiclassical treatment in the inverse friction strength so that higher order quantum corrections to the original quantum smoluchowski equation j ankerhold p pechukas and h grabert phys rev lett 87 086802 2001 j ankerhold and h grabert phys rev lett 101 119903 2008 can be derived drift and diffusion coefficients are determined by the equilibrium distribution in position and are directly related to the corresponding action of extremal paths and fluctuations around them it is shown that the inclusion of higher order corrections reproduces the quantum enhancement above crossover for the decay rate out of a metastable well exactly

relaxation dynamics of a particle in the presence of an external potential exact solution in terms of matrix continued fractions exact expressions for the laplace transform of the after effect function arising from the solution of the langevin or underlying fokker planck equation for brownian motion in an external potential are obtained as a sum of products of infinite matrix continued fractions this is accomplished by reducing the scalar multiterm recurrence relations associated with the fokker planck equation to matrix three term recurrence relations the solution is illustrated by considering the problem of dielectric relaxation of a single axis rotator subjected to both constant and crystalline anisotropy fields

quantum smoluchowski equation for driven systems we consider a driven quantum harmonic oscillator strongly coupled to a heat bath starting from the exact quantum langevin equation we use a green s function approach to determine the corresponding semiclassical equation for the wigner phase space distribution in the limit of high friction and high temperature we apply brinkman s method to derive the quantum smoluchowski equation for the probability distribution in position space we further determine the range of validity of the equation and discuss the special case of a brownian parametric oscillator

delocalized chern character for stringy orbifold k theory in this paper we define a stringy product on k orb x c the orbifold k theory of any almost complex presentable orbifold x we establish that under this stringy product the delocali zed chern character chdeloc k x c h cr x after a canonical modification is a ring isomorphism here h cr x is the chen ruan cohomology of x the proof relies on an intrinsic description of the obstruc tion bundles in the construction of the chen ruan product as an application we investigate this stringy product on the equivariant k theory k g g of a finite group g with the conjugation action it turns out that the stringy pr oduct is different from the pontryagin product the latter is also called the fusion pro duct in string theory

inner products of 2 representations abstract we define and calculate inner products of 2 representations along the way we prove that the categorical trace t r of 24 sec 3 is multiplicative with respect to various notions of categorical tensor product and we identify the center of the category of equivariant objects we discuss applications ranging from schur s result about the number of irreducible projective representations to a formula for the hochschild cohomology of a global quotient orbifold

time scales and diffusion mechanisms in the kramers equation with periodic potentials i abstract noise assisted diffusion on a periodic substrate is studied solving the klein kramers equation kke by the matrix continued fraction method assuming a periodic cosine potential and a homogeneous friction the one particle dynamics is completely characterized both at short and long times calculating up to large momentum transfer the dynamic structure factor its full width at half maximum the velocity self correlation spectrum and the mean square displacement four different dynamical regimes are found as the potential strength or the friction vary at high potential barriers diffusion proceeds by single or multiple jumps at high and low friction respectively while two different kinds of continuous unactivated diffusion are found at lower barriers these dynamical features are discussed in terms of three characteristic time scales whose relations determine the diffusion mechanisms

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

duality theorems for etale gerbes on orbifolds abstract let g be a finite group and y a g gerbe over an orbifold b a disconnected orbifold y and a flat u 1 gerbe c on y is canonically constructed from y motivated by a proposal in physics we study a mathematical duality between the geometry of the g gerbe y and the geometry of y twisted by c we prove several results verifying this duality in the contexts of non commutative geometry and symplectic topology in particular we prove that the category of sheaves on y is equivalent to the category of c twisted sheaves on y when y is symplectic we show by a combination of techniques from non commutative geometry and symplectic topology that the chen ruan orbifold cohomology of y is isomorphic to the c twisted orbifold cohomology of y as graded algebras

mirror transformations of hamiltonian systems we demonstrate through various examples of hamiltonian systems that symplectic structures have been encoded into the painleve test each principal balance in the painleve test induces a mirror transformation that regularizes movable singularities moreover for finite dimensional hamiltonian systems the mirror transformations are canonical

the periodic cyclic homology of crossed products of finite type algebras we study the periodic cyclic homology groups of the cross product of a finite type algebra a by a discrete group in case a is commutative and the group is finite our results are complete and given in terms of the singular cohomology of the sets of fixed points these groups identify our cyclic homology groups with the orbifold cohomology of the underlying algebraic orbifold the proof is based on a careful study of localization at fixed points and of the resulting koszul complexes this is achieved by extending to our class of noncommutative algebras the concept of an infinitesimal neighborhood that plays such an important role in commutative algebra we provide examples of azumaya algebras for which this identification is however no longer valid as an example we discuss some affine weyl groups

cohomology theory in birational geometry this is a continuation of 9 where it was shown that k equivalent complex projective manifolds have the same betti numbers by using the theory of p adic integrals and deligne s solution to the weil conjecture the aim of this note is to show that with a little more book keeping work namely by applying faltings p adic hodge theory our p adic method also leads to the equivalence of hodge numbers a result which was previously known via motivic integration

etale stacks as prolongations abstract we give a complete and categorical characterization of etale stacks generalized orbifolds in various geometric contexts including differentiable stacks and topological stacks this characterization states that etale stacks are precisely those stacks that arise as prolongations of stacks on a site of spaces and local homeomorphisms and makes no mention of an atlas or groupoid presentation moreover we show that the bicategory of etale differentiable stacks and local diffeomorphisms is equivalent to the bicategory of stacks on the site of smooth manifolds and local diffeomorphisms an analogous statement holds for other flavors of manifolds topological c k complex super and topological spaces locally homeomorphic to a given space x a slight modification of this result also holds in an even more general context including all etale topological stacks and zariski etale stacks which are analogues of deligne mumford stacks for the zariski topology and we also sketch a proof of an analogous characterization of classical deligne mumford algebraic stacks which we prove in full in 7 by different methods we go on to characterize effective etale stacks e g reduced orbifolds as precisely those stacks arising as the prolongations of sheaves of sets rather than stacks of groupoids it follows that etale stacks and in particular orbifolds induce a small gerbe over their effective part and all gerbes over effective etale stacks arise in this way we show that well known lie groupoids from foliation theory literature can be reproduced from this framework as natural presentations for moduli stacks of certain geometric structures for example there exists a classifying stack for riemannian metrics presented by haefliger s groupoid r 13 and submersions into this stack classify riemannian foliations and similarly for symplectic structures with the role of r replaced with s p we also prove some unexpected results for example the category of smooth n manifolds and local diffeomorphisms has binary products

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

semiclassical klein kramers and smoluchowski equations for the brownian motion of a particle in an external potential the quantum brownian motion of a particle in an external potential v x is treated using the master equation for the wigner distribution function w x p t in phase space x p a heuristic method of determination of diffusion coefficients in the master equation is proposed the time evolution equation so obtained contains explicit quantum correction terms up to o 4 and in the classical limit 0 reduces to the klein kramers equation for a quantum oscillator the method yields an evolution equation for w x p t coinciding with that of agarwal 1971 phys rev a 4 739 in the non inertial regime by applying the brinkman expansion of the momentum distribution in weber functions brinkman 1956 physica 22 29 the corresponding semiclassical smoluchowski equation is derived

a new equivariant cohomology ring in this paper motivated by chen ruan s stringy orbifold theory on almost complex orbifolds we construct a new equivariant cohomology ring mathscr h g x for an equivariant almost complex pair x g where x is a compact connected almost complex manifold g is a connected compact lie group which acts on x and preserves the almost complex structure

the resolution property for schemes and stacks we prove the equivalence of two fundamental properties of algebraic stacks being a quotient stack in a strong sense and the resolution property which says that every coherent sheaf is a quotient of some vector bundle moreover we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme mathematics subject classification primary 14a20 secondary 14l30

analytical inversion of general tridiagonal matrices in this paper we give a complete analysis for general tridiagonal matrix inversion for both non block and block cases and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or toeplitz tridiagonal matrices

flops and equivalences of derived categories for threefolds with only terminal gorenstein singularities the main purpose of this paper is to show that bridgeland s moduli space of perverse point sheaves for certain flopping contractions gives the flops and the fourier mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories

the integral orbifold chow ring of toric deligne mumford stacks in this paper we study the integral chow ring of toric deligne mumford stacks we prove that the integral chow ring of a semi projective toric deligne mumford stack is isomorphic to the stanley reisner ring of the associated stacky fan the integral orbifold chow ring is also computed our results are illustrated with several examples

orbifold cohomology for global quotients let x be an orbifold that is a global quotient of a manifold y by a finite group g we construct a noncommutative ring h sp ast y g with a g action such that h sp y g sp g is the orbifold cohomology ring of x defined by w chen and y ruan cr when y s sp n with s a surface with trivial canonical class and g mathfrak s sb n we prove that a small modification of the orbifold cohomology of x is naturally isomorphic to the cohomology ring of the hilbert scheme s sp n computed by m lehn and c sorger ls2

on the role of effective representations of lie groupoids abstract in this paper we undertake the study of the tannaka duality construction for the ordinary representations of a proper lie groupoid on vector bundles we show that for each proper lie groupoid g the canonical homomorphism of g into the reconstructed groupoid t g is surjective although contrary to what happens in the case of groups it may fail to be an isomorphism we obtain necessary and sufficient conditions in order that g may be isomorphic to t g and more generally in order that t g may be a lie groupoid we show that if t g is a lie groupoid the canonical homomorphism g t g is a submersion and the two groupoids have isomorphic categories of representations

a new cohomology theory of orbifold based on the orbifold string theory model in physics we construct a new cohomology ring for any almost complex orbifold the key theorem is the associativity of this new ring some examples are computed

twisted orbifold k theory we use equivariant methods to define and study the orbifold k theory of an orbifold x adapting techniques from equivariant k theory we construct a chern character and exhibit a multiplicative decomposition for k orb x in particular showing that it is additively isomorphic to the orbifold cohomology of x a number of examples are provided we then use the theory of projective representations to define the notion of twisted orbifold k theory in the presence of discrete torsion an explicit expression for this is obtained in the case of a global quotient

higher algebraic k theory for actions of diagonalizable groups we study the k theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces our main result shows how to reconstruct the k theory ring of such an action from the k theory rings of the loci where the stabilizers have constant dimension we apply this to the calculation of the equivariant k theory of toric varieties and give conditions under which the merkurjev spectral sequence degenerates so that the equivariant k theory ring determines the ordinary k theory ring we also prove a very refined localization theorem for actions of this type

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

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

presentations of noneffective orbifolds it is well known that an effective orbifold m one for which the local stabilizer groups act effectively can be presented as a quotient of a smooth manifold p by a locally free action of a compact lie group k we use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented we also note some connections to stacks and gerbes

iterated spans and classical topological field theories we construct higher categories of iterated spans possibly equipped with extra structure in the form of higher categorical local systems and classify their fully dualizable objects by the cobordism hypothesis these give rise to framed topological quantum field theories which are the framed versions of the classical topological quantum field theories considered in the quantization programme of freed hopkins lurie teleman using this machinery we also construct an infty 1 category of symplectic derived algebraic stacks and lagrangian correspondences and show that all its objects are dualizable

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

flops motives and invariance of quantum rings for ordinary ops the correspondence dened by the graph closure is shown to give equivalence of chow motives and to preserve the poincar e pairing in the case of simple ordinary ops this correspondence preserves the big quantum cohomology ring after an analytic continuation over the

erratum master equation in phase space applied to the quantum brownian motion in a tilted periodic potential quantum effects in the brownian motion in a tilted cosine potential are treated in the high temperature and weak bath particle coupling limit via the semiclassical master equation for the time evolution of the reduced wigner function in phase space the differential recurrence relation generated from the master equation by expanding the periodic wigner function expressed as a function of wave number k in fourier series via floquet s theorem is solved for bounded periodic initial conditions using matrix continued fractions yielding both the time independent and the time dependent bounded periodic solutions just as the classical problem the non periodic solution is determined from the recurrence relation by integrating w r t k over the first brillouin zone with the initial conditions generated by the non periodic wigner distribution corresponding to zero probability current the time independent periodic locked solution is used to estimate quantum effects in the dc current voltage characteristic of a josephson junction including the capacitance while the time dependent non periodic running solution is used to determine the dynamic structure factor and ultimately via integration over the first brillouin zone the lifetime of the zero voltage state the current voltage characteristics displayed as average velocity versus tilt reproduce in the high damping limit those from the semiclassical smoluchowski equation while the decrease in the lifetime of the zero voltage state due to the barrier lowering quantum effects appears to be very sensitive to the tilt

stringy k theory and the chern character we construct two new g equivariant rings mathcal k x g called the stringy k theory of the g variety x and mathcal h x g called the stringy cohomology of the g variety x for any smooth projective variety x with an action of a finite group g for a smooth deligne mumford stack mathcal x we also construct a new ring mathsf k mathrm orb mathcal x called the full orbifold k theory of mathcal x we show that for a global quotient mathcal x x g the ring of g invariants k mathrm orb mathcal x of mathcal k x g is a subalgebra of mathsf k mathrm orb x g and is linearly isomorphic to the orbifold k theory of adem ruan ar and hence atiyah segal but carries a different quantum product which respects the natural group grading

intersection pairings in the n fold reduced product of adjoint orbits in previous work we computed the symplectic volume of the symplectic reduced space of the product of n adjoint orbits of a compact lie group in this paper we compute the intersection pairings in the same object in previous work we computed the symplectic volume of the symplectic reduced space of the product of n adjoint orbits of a compact lie group in this paper we compute the intersection pairings in the same object

a derham model for chen ruan cohomology ring of abelian orbifolds we present a derham model for chen ruan cohomology ring of abelian orbifolds we introduce the notion of twist factors so that formally the stringy cohomology ring can be defined without going through pseudo holomorphic orbifold curves thus our model can be viewed as the classical description of chen ruan cohomology for abelian orbifolds the model simplifies computation of chen ruan cohomology ring using our model we give a version of wall crossing formula

brownian motion quantum corrections and a generalization of the hermite polynomials the nonequilibrium evolution of a brownian particle in the presence of a heat bath at thermal equilibrium without imposing any friction mechanism from the outset is considered using a suitable family of orthogonal polynomials moments of the nonequilibrium probability distribution for the brownian particle are introduced which fulfill a recurrence relation we review the case of classical brownian motion in which the orthogonal polynomials are the hermite ones and the recurrence relation is a three term one after having performed a long time approximation in the recurrence relation the approximate nonequilibrium theory yields irreversible evolution of the brownian particle towards thermal equilibrium with the heat bath for quantum brownian motion which is the main subject of the present work we restrict ourselves to include the first quantum correction this leads us to introduce a new family of orthogonal polynomials which generalize the hermite ones some general properties of the new family are established the recurrence relation for the new moments of the nonequilibrium distribution including the first quantum correction turns out to be also a three term one which justifies the new family of polynomials a long time approximation on the new three term recurrence relation describes irreversible evolution towards equilibrium for the new moment of lowest order the standard smoluchowski equations for the lowest order moments are recovered consistently both classically and quantum mechanically

how to calculate the wigner function from phase space we present a method for the direct computation of the wigner function by solving a coupled system of linear partial differential equations in phase space our modified spectral method relies on chebyshev polynomials since this approach allows us to include arbitrary high orders of partial derivatives our procedure is applicable to arbitrary binding potentials we apply our scheme to wigner functions of the harmonic oscillator the morse oscillator and an asymmetric double well potential

mukai flop and ruan cohomology suppose that two compact symplectic manifolds x x are connected by a sequence of simple mukai flops in this paper we construct a ring isomorphism between cohomology rings of x and x using the localization technique we prove that the quantum corrected products on x x are the ordinary intersection products furthermore x x have isomorphic ruan cohomology i e we verify the cohomological minimal model conjecture proposed by ruan for the pair x x

continued fraction as a discrete nonlinear transform the connection between a taylor series and a continued fraction involves a nonlinear relation between the taylor coefficients an and the continued fraction coefficients bn in many instances it turns out that this nonlinear relation transforms a complicated sequence an into a very simple one bn this simplification is illustrated in the context of graph combinatorics