nonabelian bundle gerbes their differential geometry and gauge theory bundle gerbes are a higher version of line bundles we present nonabelian bundle gerbes as a higher version of principal bundles connection curving curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates these are the gauge fields needed for the construction of yang mills theories with 2 form gauge potential

cyclic cocycles on twisted convolution algebras we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids for proper etale groupoids tu and xu adv math 207 2 455 483 2006 provide a map between the periodic cyclic cohomology of a gerbe twisted convolution algebra and twisted cohomology groups which is similar to the construction of mathai and stevenson adv math 200 2 303 335 2006 when the groupoid is not proper we cannot construct an invariant connection on the gerbe therefore to study this algebra we instead develop simplicial techniques to construct a simplicial curvature 3 form representing the class of the gerbe then by using a jlo formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3 form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras

on an inequality by n trudinger and j moser and related elliptic equations it has been shown by trudinger and moser that for normalized functions u of the sobolev space w 1 n whereis a bounded domain in r n one has r exp njuj n n 1 dx cn where n is an explicit constant depending only on n and cn is a constant depending only on n and carleson and chang proved that there exists a corresponding extremal function in the case that is the unit ball in r n in this paper we give a new proof a generalization and a new interpretation of this result in particular we give an explicit sequence that is maximizing for the above integral among all normalized concentrating sequences as an application the existence of a nontrivial solution for a related elliptic equation with trudinger moser growth is proved c 2002 john wiley sons inc

strongly self absorbing c algebras say that a separable unital c algebra v c is strongly self absorbing if there exists an isomorphism v such that lx ai e approximately unitarily equivalent homomorphisms we study this class of algebras which includes the cuntz algebras 2 ooo the uhf algebras of infinite type the jiang su algebra z and tensor products of 00 with uhf algebras of infinite type given a strongly self absorbing c algebra v we characterise when a separable c algebra absorbs v tensorially i e is p stable and prove closure properties for the class of separable testable c algebras finally we compute the possible if groups and prove a number of classification results which suggest that the examples listed above are the only strongly self absorbing c algebras

on a generalised connes hochschild kostant rosenberg theorem the central result of this paper is an explicit computation of the hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds x as their spectrum more precisely the hochschild homology is identified with the space of differential forms on x and the periodic cyclic homology with the twisted de rham cohomology of x thereby generalising some fundamental results of connes and hochschild kostant rosenberg the connes chern character is also identified here with the twisted chern character

families index theorem in supersymmetric wzw model and twisted k theory the su 2 case the construction of twisted k theory classes on a compact lie group is reviewed using the supersymmetric wess zumino witten model on a cylinder the quillen superconnection is introduced for a family of supercharges parametrized by a compact lie group and the chern character is explicitly computed in the case of su 2 for large euclidean time the character form is localized on a d brane

analytic torsion for twisted de rham complexes we dene analytic torsion x e h 2 det h x e h for the twisted de rham complex consisting of the spaces of dieren tial forms on a compact oriented riemannian man ifold x valued in a at vector bundle e with a dieren tial given by r e h where r e is a at connection on e h is an odd degree closed dieren tial form on x and h x e h denotes the cohomology of this z2 graded complex the denition uses pseudo dieren tial operators and residue traces we show that when dim x is odd x e h is independent of the choice of metrics on x and e and of the representative h in the cohomology class h we explain the relation to generalized geometry when h is a 3 form we demonstrate some basic func torial properties when h is a top degree form we compute the torsion dene its simplicial counterpart and prove an analogue of the cheeger mtheorem we also establish the rela tionship of an invariant form of the analytic torsion for t dual circle bundles with 3 form uxes

a fourier mukai approach to the k theory of compact lie groups abstract let g be a compact connected simply connected lie group we use the fourier mukai transform in twisted k theory to give a new proof of the ring structure of the k theory of g

noncommutative tachyons and k theory we show that the relation between d branes and noncommutative tachyons leads very naturally to the relation between d branes and k theory we also discuss some relations between d branes and k homology provide a noncommutative generalization of the abs construction and give a simple physical interpretation of bott periodicity in addition a framework for constructing neveu schwarz fivebranes as noncommutative solitons is proposed

dynamical aspects of tsallis entropy abstract the description of the temporal evolution of some physical systems is tackled from an information theory view point in which tsallis entropy is extremalized with the constraints posed by the knowledge of the generalized expectation values of a selected set of observables the concomitant tsallis density operator is shown to be a particular solution of von neumann s equation and tsallis entropy is seen to provide one with a constant of the motion the pertinent ideas are illustrated with reference to the larmor precession

a rearrangement based proof for the existence of extremal functions for the sobolev poincare inequality on bn abstract existence of extremals for 1 p n 1 2 is shown for the sobolev poincare inequality on b n using a mean value zero preserving rearrangement

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

quantum corrections for general partition functions abstract the classical limit for quite general partition functions is obtained using coherent states in this framework a general procedure is presented to obtain all the quantum corrections to the classical limit in particular the first and second order corrections are worked out explicitly and the classical limit for the tsallis thermostatistics is determined the results of this work generalize the ones obtained by wigner phys rev 40 1932 749 for usual statistical mechanics

extremals for moser inequalities this paper shows that extremals exist for four moser inequalities involving exponential integrals the results are for sobolev spaces defined on the spheres s2 and s3 on the disk d2 and on rp2 the method that carleson and chang used for the n ball with boundary value zero can be modified and applied to functions with mean value zero on these domains this approach also provides an elementary proof of the moser inequality for sn n 2 which is a special case of a result of fontana

replacing model categories with simplicial ones in this paper we show that model categories of a very broad class can be replaced up to quillen equivalence by simplicial model categories

regularity of solutions to axisymmetric navier stokes equations with a slightly supercritical condition abstract consider an axisymmetric suitable weak solution of 3d incompressible navier stokes equations with nontrivial swirl v v r e r v e v z e z let z denote the axis of symmetry and r be the distance to the z axis if the solution satisfies a slightly supercritical assumption that is v c ln ln r r for 0 0 028 when r is small then we prove that v is regular this extends the results in 6 16 18 where regularities under critical assumptions such as v c r were proven as a useful tool in the proof of our main result an upper bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper

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

asymptotic behavior of solutions to elliptic and parabolic equations with unbounded coefficients of the second order in unbounded domains we study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain under the assumption that the solution belongs to l q with q in 2 infty we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients as a corollary we obtain the liouville type theorem in the case when the coefficients may grow at the spacial infinity we also study a corresponding parabolic problem in the n dimensional whole space and discuss the energy identity for solutions in l q as a corollary we show also the liouville type theorem for both forward and ancient solutions

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

a strong regularity result for parabolic equations we consider a parabolic equation with a drift term u b u ut 0 under the condition divb 0 we prove that solutions possess dramatically better regularity than those provided by standard theory for example we prove continuity of solutions when not even boundedness is expected

projective elliptic genera and elliptic pseudodifferential genera abstract in this paper we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle such projective elliptic genera are rational q series that have topological definition and also have analytic interpretation via the fractional index theorem in 25 without requiring spin condition we prove the modularity properties of these projective elliptic genera as an application we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator this suggests the existence of putative s 1 equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera

thermodynamics from first principles correlations and nonextensivity the standard formulation of thermostatistics being based on the boltzmann gibbs distribution and logarithmic shannon entropy describes idealized uncorrelated systems with extensive energies and short range interactions in this rapid communication we use the fundamental principles of ergodicity via liouville s theorem the self similarity of correlations and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long range interacting systems significantly our formalism provides a justification for the well studied nonextensive thermostatistics characterized by the tsallis distribution which it includes as a special case we also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the gibbs shannon entropy the consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

a set of stationary non maxwellian distributions abstract we propose a set of distribution functions stationary solutions of a fokker planck equation the frictional and diffusion coefficients having been expanded in a polynomial series with positive coefficients these distributions show tails depleted or increased with respect to the maxwellian distribution which can in any case be recovered if the coefficients of the expansion satisfy particular conditions the stationary non maxwellian distributions derived in this paper can be used in the calculations of rates in many different fields of physics where problems still have to be solved

upper bounds for fundamental solutions to non local diffusion equations with divergence free drift we consider some non local diffusion equations in the presence of a divergence free drift term where the diffusion operators are related to certain dirichlet forms of jump type we derive pointwise upper bounds for fundamental solutions of the equations under weak assumptions for the velocity of the drift term our class of the velocity includes functions with the scale critical regularity and some growing functions at spatial infinity

form boundedness of the general second order differential operator we give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator i j ln aij i j j 1nbj j c with real or complex valued distributional coefficients aij bj and c acting from the sobolev space w1 2 n to its dual w 1 2 n this enables us to obtain analytic criteria for the fundamental notions of relative form boundedness compactness and infinitesimal form boundedness of c with respect to the laplacian on l 2 n in particular we establish a complete characterization of the form boundedness of the schrodinger operator i a 2 q with magnetic vector potential a lloc2 n n and q d n 2006 wiley periodicals inc

on divergence free drifts we investigate the validity and failure of liouville theorems and harnack inequalities for parabolic and elliptic operators with low regularity coefficients we are particularly interested in operators of the form t b resp b with a divergence free drift b we prove the liouville theorem and harnack inequality when b l bmo 1 resp b bmo 1 and provide a counterexample demonstrating sharpness of our conditions on the drift our results generalize to divergence form operators with an elliptic symmetric part and a bmo skew symmetric part we also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions depending on the non scale invariant norm b l1 in three dimensions on the other hand bounded solutions with l1 drifts may be discontinuous

polynomial expansion of diffusion and drift coefficients for classical and quantum statistics abstract we study by means of a kinetic approach recently proposed phys rev e 49 5103 1994 both classical and quantum statistics stationary solutions of a non linear fokker planck equation the non linear term in this equation has origin in a generalized exclusion inclusion principle which takes into account the quantum effects due to the bosonic fermionic or intermediate behaviour of the particles if the drift and diffusion coefficients are polynomials in the velocity variable of degree 2m 1 and 2n respectively only one well defined statistics can be derived for any particular pair of values m n in the case of classical particles in addition to the macroscopic fokker planck equation we derive the associated microscopic langevin equation and demonstrate that the pairs 0 0 1 0 and 0 1 correspond respectively to maxwell boltzmann druyvenstein and tsallis statistical distributions we study also the distribution related to the pair 1 1 which is the simplest generalization of the tsallis 1 one finally the above statistics are extended to the quantum case of particles obeying the exclusion inclusion principle

comparison of some master equation descriptions of relaxation in complex systems several models of relaxation based on master equation approaches have obtained the kohlrausch fractional exponential form t exp t 1 n 0 n 1 or its equivalent for the relaxation function in complex systems representative models include i the cohen grest free volume theory ii the work of dhar and barma and skinner based on glauber s kinetic ising model iii the theory of de dominicis et al based on a random energy model for the spin glass iv the ogielski stein theory based on dynamics in an ultrametric space and v ngai s theory of time dependent transition rates in view of the different nature of these models and because of the claims that they are applicable outside of their original contexts it is useful to make an intercomparison of these models and their consequences a presentation of these models is here given based on a unified master equation approach by experiment many real systems have been shown to exhibit not only the kohlrausch form but two additional related properties which are not encompassed in model types i iv only models that include time dependent transition rates have so far been shown to be consistent with the experimental observations of the three empirical relations

d brane charges in gepner models we construct gepner models in terms of coset conformal field theories and compute their twisted equivariant k theories these classify the d brane charges on the associated geometric backgrounds and therefore agree with the topological k theories we show this agreement for various cases in particular the fermat quintic

diffusion in turbulence we prove long time diffusive behavior homogenization for convection diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix simple shear flow examples show that this result is sharp for flows that have stationary stream matrices

regularity of solutions of quasi linear elliptic equations with llogml coefficients abstract let d be an bounded region in r n the regularity of solutions of a family of quasilinear elliptic partial differential equations is studied one example being n u v u n 1 the coefficients are assumed to be in the space l log m l d for m n 1 using a moser iteration argument coupled with the moser trudinger inequality a local l bound on the solution u is proven a harnack type inequality is then proven these results are shown to be sharp with respect to m then essential continuity of u is proven and away from the boundary a bound on the modulus of continuity

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

stratified fiber bundles quinn homology and brane stability of hyperbolic orbifolds we revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and k homology we propose a definition of type ii string theory on such backgrounds that further carry stratified systems of fiber bundles which generalize the more conventional orbifold and symmetric string backgrounds together with a classification of wrapped branes by a suitable generalized homology theory for spaces stratified fibered over hyperbolic orbifolds we use the algebraic k theory of their fundamental groups and quinn homology to derive criteria for brane stability in terms of an atiyah hirzebruch type spectral sequence with its lift to k homology stable d branes in this setting carry stratified charges which induce new additive structures on the corresponding k homology groups we extend these considerations to backgrounds which support h flux where we use k groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with k amenable isometry groups and derive stability conditions for branes wrapping the fibers of an eilenberg maclane spectrum functor

t duality as a duality of loop group bundles representing the data of a string compactified on a circle in the background of h flux in terms of the geometric data of a principal loop group bundle we show that t duality in type ii string theory can be understood as the interchange of the momentum and winding homomorphisms of the principal loop group bundle thus giving rise to a new interpretation of t duality

twisted k theory and finite dimensional approximation we provide a finite dimensional model of the twisted k group twisted by any degree three integral cohomology class of a cw complex one key to the model is furuta s generalized vector bundle and the other is a finite dimensional approximation of fredholm operators

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

generalized distribution functions and an alternative approach to generalized planck radiation law abstract in this study recently introduced generalized distribution functions are summarized and by using one of these distribution functions namely generalized planck distribution an alternative approach to the generalized planck law for the blackbody radiation has been tackled the expression obtained is compared with the expression given by c tsallis et al phys rev e 52 1995 1447 and it is found that this approximate scheme provides bounds to the exact result depending on the values of q index

on fundamental solutions for non local parabolic equations with divergence free drift abstract we are concerned with non local parabolic equations in the presence of a divergence free drift term by using the classical nash approach we show the existence of fundamental solutions together with continuity estimates under weak regularity assumptions on the kernel of the non local term and the velocity of the drift term as an application we give an alternative proof of global regularity for the two dimensional dissipative quasi geostrophic equations in the critical case

regularity gradient estimates for weak solutions of singular quasi linear parabolic equations abstract this paper studies the sobolev regularity for weak solutions of a class of singular quasi linear parabolic problems of the form u t div a x t u u div f with homogeneous dirichlet boundary conditions over bounded spatial domains our main focus is on the case that the vector coefficients a are discontinuous and singular in x t variables and dependent on the solution u global and interior weighted w 1 p t regularity estimates are established for weak solutions of these equations where is a weight function in some muckenhoupt class of weights the results obtained are even new for linear equations and for 1 because of the singularity of the coefficients in x t variables

chern weil construction for twisted k theory we give a finite dimensional and geometric construction of a chern character for twisted k theory introducing a notion of connection on a twisted vectorial bundle which can be considered as a finite dimensional approximation of a twisted family of fredholm operators our construction is applicable to the case of any classes giving the twisting and agrees with the chern character of bundle gerbe modules in the case of torsion classes

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

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

on the loss of continuity for super critical drift diffusion equations we show that there exist solutions of drift diffusion equations in two dimensions with divergence free super critical drifts that become discontinuous in finite time we consider classical as well as fractional diffusion however in the case of classical diffusion and time independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local l 1 norm of the drift which is a super critical quantity

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

variational method in generalized statistical mechanics concavity properties of a recently generalized not necessarily extensive entropy enable among others the generalization of the bogolyubov inequality hence of the variational method in equilibrium statistical mechanics

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

convergence in reaction diffusion systems an information theory approach abstract we have applied an information theory approach in order to study the problem of convergence to point like or extended attractors in reaction diffusion systems a distance between two states based on the kullback leibler relative information was defined different forms of the probability distribution some of them based on the knowledge of the nonequilibrium potential when accesible give the possibility to look for a faster and or more accurate convergence this approach offers the chance to estimate the attraction basins of the different attractors as well as detecting limit circles together with an easy evaluation of their periods

homotopical and operator algebraic twisted k theory using the framework for multiplicative parametrized homotopy theory introduced in joint work with c schlichtkrull we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted k theory both in the real and complex case we also improve several comparison results about twisted k theory of c algebras to include multiplicative structures our results can also be interpreted in the infty categorical setup for parametrized spectra

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

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

traveling salesman problem and tsallis statistics a generalization of the stochastic method of simulated annealing algorithm based on tsallis statistics is proposed this algorithm is considerably faster than the traditional ones in solving the traveling salesman problem acceptable solutions are found in fewer steps and higher temperatures than both the classical and the fast simulated annealings recent developments in solving np complete problems can be incorporated and improve the performance even more

abelian and non abelian branes in wzw models and gerbes we discuss how gerbes may be used to set up a consistent lagrangian approach to the wzw models with boundary the approach permits to study in detail possible boundary conditions that restrict the values of the fields on the worldsheet boundary to brane submanifolds in the target group such submanifolds are equipped with an additional geometric structure that is summarized in the notion of a gerbe module and includes a twisted chan paton gauge field using the geometric approach we present a complete classification of the branes that conserve the diagonal current algebra symmetry in the wzw models with simple compact but not necessarily simply connected target groups such symmetric branes are supported by a discrete series of conjugacy classes in the target group and may carry abelian or non abelian twisted gauge fields the latter situation occurs for the conjugacy classes with fundamental group 2 2 in so 4n 2 the branes supported by such conjugacy classes have to be equipped with a projectively flat twisted u 2 gauge field in one of the two possible wzw models differing by discrete torsion we show how the geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator product coefficients in the wzw models with non simply connected target groups

k homology of c categories and symmetric spectra representing k homology we define k homology groups k for small c categories in terms of hilbert modules over the c category we also define a functor a f from the category of small c categories into the category of c algebras and show that there is a natural isomorphism k cal c cong k a f cal c in addition we give an easy construction of a functor mathbb k from the category of c algebras into the category of symmetric spectra which represents k homology i e we show that the functor mathbb k comes with a natural isomorphism k a cong pi mathbb k a for c algebras a it then follows that the composition mathbb k a f provides a functor that can be used in the davis luck approach for constructing the baum connes map

markov processes hurst exponents and nonlinear diffusion equations with application to finance abstract we show by explicit closed form calculations that a hurst exponent h 1 2 does not necessarily imply long time correlations like those found in fractional brownian motion fbm we construct a large set of scaling solutions of fokker planck partial differential equations pdes where h 1 2 thus markov processes which by construction have no long time correlations can have h 1 2 if a markov process scales with hurst exponent h 1 2 then it simply means that the process has nonstationary increments for the scaling solutions we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient d x t is specified as an example we generate a class of student t like densities from the class of quadratic diffusion coefficients notably the tsallis density is one member of that large class the tsallis density is usually thought to result from a nonlinear diffusion equation but instead we explicitly show that it follows from a markov process generated by a linear fokker planck equation and therefore from a corresponding langevin equation having a tsallis density with h 1 2 therefore does not imply dynamics with correlated signals e g like those of fbm a short review of the requirements for fbm is given for clarity and we explain why the usual simple argument that h 1 2 implies correlations fails for markov processes with scaling solutions finally we discuss the question of scaling of the full green function g x t x t of the fokker planck pdes

brst anti brst and their geometry we continue the comparison between the field theoretical and geometrical approaches to the gauge field theories of various types by deriving their becchi rouet stora tyutin brst and anti brst transformation properties and comparing them with the geometrical properties of the bundles and gerbes in particular we provide the geometrical interpretation of the so called curci ferrari conditions that are invoked for the absolute anticommutativity of the brst and anti brst symmetry transformations in the context of non abelian one form gauge theories as well as the abelian gauge theory that incorporates a two form gauge field we also carry out the explicit construction of the three form gauge fields and compare it with the geometry of 2 gerbes

student s t and r distributions unified derivation from an entropic variational principle abstract the optimization of the recently generalized entropy of the form s q 1 dx p x q q 1 with the constraints dx p x 1 and x 2 q dx x 2 p x q 1 yields the student s t distribution for q 1 and the r distribution for q

moser inequalities in gauss space the sharp constants in a family of exponential sobolev type inequalities in gauss space are exhibited they constitute the gaussian analogues of the moser inequality in the borderline case of the sobolev embedding in the euclidean space interestingly the gaussian results have features in common with the euclidean ones but also reveal marked diversities

on inviscid limits for the stochastic navier stokes equations and related models we study inviscid limits of invariant measures for the 2d stochastic navier stokes equations as shown by kuksin j stat phys 115 1 2 469 492 2004 the noise scaling sqrt nu is the only one which leads to non trivial limiting measures which are invariant for the 2d euler equations using a moser type iteration for stochastic drift diffusion equations we show that any limiting measure mu 0 is in fact supported on bounded vorticities relationships of mu 0 to the long term dynamics of 2d euler in l infty with the weak topology are discussed we also obtain a drift independent modulus of continuity for a stationary deterministic model problem which leads us to conjecture that in fact mu 0 is supported on c 0 moreover in view of the batchelor krainchnan 2d turbulence theory we consider inviscid limits for a weakly damped stochastic navier stokes equation in this setting we show that only an order zero noise scaling with respect to leads to a nontrivial limiting measure in the inviscid limit

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

kramers escape rate in overdamped systems with the power law distribution abstract kramers escape rate in the overdamped systems is restudied for the power law distribution by using the mean first passage time we derive the escape rate with the power law distribution and obtain the kramers infinite barrier escape rate in this case we show that the escape rate with the power law distribution extends the kramers overdamped result to the relatively low barrier furthermore we apply the escape rate with the power law distribution to the unfolding of titin and show a better agreement with the experimental rate than the traditional escape rate

doubling properties of second order parabolic equations we prove the doubling property of l caloric measure corresponding to the second order parabolic equation in the whole space and in lipschitz domains for parabolic equations in the divergence form a weaker form of the doubling property follows easily from a recent result the backward harnack inequality and known estimates of green s function our method works for both the divergence and nondivergence cases moreover the backward harnack inequality and estimates of green s function are not needed in the course of proof

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

a dixmier douady theory for strongly self absorbing c algebras we show that the dixmier douady theory of continuous fields of c algebras with compact operators

are power law distributions an equilibrium distribution or a stationary nonequilibrium distribution abstract we examine whether the principle of detailed balance holds for the power law distributions that are generated from the well known two variable langevin equation and the associated fokker planck equations with the detailed balance and the generalized fluctuation dissipation relation we derive analytically the stationary power law distribution from the ito s stratonovich s and zwanzig s fokker planck equations and conclude that the power law distributions can either be a stationary nonequilibrium distribution or an equilibrium distribution which depend on information about the form of the diffusion coefficient function and the existence and uniqueness of an equilibrium state

d branes rr fields and duality on noncommutative manifolds we develop some of the ingredients needed for string theory on noncommutative spacetimes proposing an axiomatic formulation of t duality as well as establishing a very general formula for d brane charges this formula is closely related to a noncommutative grothendieck riemann roch theorem that is proved here our approach relies on a very general form of poincare duality which is studied here in detail among the technical tools employed are calculations with iterated products in bivariant mathsf k theory and cyclic theory which are simplified using a novel diagram calculus reminiscent of feynman diagrams

diagram spaces and symmetric spectra abstract we present a general homotopical analysis of structured diagram spaces and discuss the relation to symmetric spectra the main motivating examples are the i spaces which are diagrams indexed by finite sets and injections and j spaces which are diagrams indexed by quillen s localization construction 1 on the category of finite sets and bijections we show that the category of i spaces provides a convenient model for the homotopy category of spaces in which every e space can be rectified to a strictly commutative monoid similarly the commutative monoids in the category of j spaces model graded e spaces using the theory of j spaces we introduce the graded units of a symmetric ring spectrum the graded units detect periodicity phenomena in stable homotopy and we show how this can be applied to the theory of topological logarithmic structures

escape rate for the power law distribution in low to intermediate damping abstract escape rate in the low to intermediate damping connecting the low damping with the intermediate damping is established for the power law distribution on the basis of flux over population theory we extend the escape rate in the low damping to the low to intermediate damping and get an expression for the power law distribution then we apply the escape rate for the power law distribution to the experimental study of the excited state isomerization and show a good agreement with the experimental value the extra current and the improvement of the absorbing boundary condition are discussed

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

regularity theorems for parabolic equations abstract we study regularity properties of solutions of a parabolic equation t a b b u 0 in r r d d 3 under fairly general conditions on the drift term coefficients the results are already new for the case a i b 0 b b x and div b 0