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 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

the higher twisted index theorem for foliations abstract given a gerbe l on the holonomy groupoid g of the foliation m f whose pull back to m is torsion we construct a connes map from the twisted dupont sullivan bicomplex of g to the cyclic complex of the l projective leafwise smoothing operators on m f our construction allows to couple the k theory analytic indices of l projective leafwise elliptic operators with the twisted cohomology of b g producing scalar higher invariants finally by adapting the bismut quillen superconnection approach we compute these higher twisted indices as integrals over the ambient manifold of the expected twisted characteristic classes

beyond the poisson renewal process a tutorial survey after sketching the basic principles of renewal theory and recalling the classical poisson process we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of mittag leffler and of wright type we compare these three processes with each other

assessing the inherent uncertainty of one dimensional diffusions in this paper we assess the inherent uncertainty of one dimensional diffusion processes via a stochasticity classification which provides an a la mandelbrot categorization into five states of uncertainty infra mild mild borderline wild and ultra wild two settings are considered i stopped diffusions the diffusion initiates from a high level and is stopped once it first reaches a low level in this setting we analyze the inherent uncertainty of the diffusion s maximal exceedance above its initial high level ii stationary diffusions the diffusion is in dynamical statistical equilibrium in this setting we analyze the inherent uncertainty of the diffusion s equilibrium level in both settings general closed form analytic results are established and their application is exemplified by stock prices in the stopped diffusions setting and by interest rates in the stationary diffusions setting these results provide a highly implementable decision making tool for the classification of uncertainty in the context of one dimensional diffusions

relaxation time distributions for an anomalously diffusing particle abstract as well known the generalized langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle here we focus attention on some new aspects of the dynamics successively considering the memory kernel the particle s mean velocity and the scattering function all these quantities are studied from a unique angle namely the discussion of the possible existence of a distribution of relaxation times characterizing their time decay although a very popular concept a relaxation time distribution cannot be associated with any time decreasing quantity from a mathematical point of view the decay has to be described by a completely monotonic function technically we use a memory kernel decaying as a mittag leffler function the mittag leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times we show that in the case of a subdiffusive motion relaxation time distributions can be defined for the memory kernel and for the scattering function but not for the particle s mean velocity the situation is opposite in the superdiffusive case

numerical evaluation of two and three parameter mittag leffler functions the mittag leffler ml function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation in this work we present a method for the efficient computation of the ml function based on the numerical inversion of its laplace transform lt an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the lt with the aim of minimizing the computational effort and reducing the propagation of errors numerical experiments are presented to show accuracy and efficiency of the proposed approach the application to the three parameter ml also known as prabhakar function is also presented

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

the matchmaking paradox a statistical explanation medical surveys regarding the number of heterosexual partners per person yield different female and male averages a result which from a physical standpoint is impossible in this paper we term this puzzle the matchmaking paradox and establish a statistical model explaining it we consider a bipartite graph with n male and n female nodes n 1 and b bonds connecting them b 1 each node is associated a random attractiveness level and the bonds connect to the nodes randomly with probabilities which are proportionate to the nodes attractiveness levels the population s average bonds per nodes b n is estimated via a sample average calculated from a survey of size n n 1 a comprehensive statistical analysis of this model is carried out asserting that i the sample average well estimates the population average if and only if the attractiveness levels possess a finite mean ii if the attractiveness levels are governed by a fat tailed probability law then the sample average displays wild fluctuations and strong skew thus providing a statistical explanation to the matchmaking paradox

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

canonical fitness model for simple scale free graphs we consider a fitness model assumed to generate simple graphs with a power law heavy tailed degree sequence p k ensuremath propto k ensuremath 1 ensuremath ensuremath alpha with 0l ensuremath alpha l1 in which the corresponding distributions do not possess a mean we discuss the situations in which the model is used to produce a multigraph and examine what happens if the multiple edges are merged to a single one and thus a simple graph is built we give the relation between the normalized fitness parameter r and the expected degree ensuremath nu of a node and show analytically that it possesses nontrivial intermediate and final asymptotic behaviors we show that the model produces p k ensuremath propto k ensuremath 2 for large values of k independent of ensuremath alpha our analytical findings are confirmed by numerical simulations

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

on the integrability of tonelli hamiltonians in this article we discuss a weaker version of liouville s theorem on the integrability of hamiltonian systems we show that in the case of tonelli hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced moreover we prove that on the n dimensional torus this weaker condition implies classical integrability in the sense of liouville the main idea of the proof consists in relating the existence of independent integrals of motion of a tonelli hamiltonian to the size of its mather and aubry sets as a byproduct we point out the existence of non trivial common invariant sets for all hamiltonians that poisson commute with a tonelli hamiltonian

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

on commuting tonelli hamiltonians autonomous case we show that the aubry sets the mane sets and the barrier functions are the same for two commuting autonomous tonelli hamiltonians we also show the quasi linearity of functions from the viewpoint of dynamics and the existence of common c1 1 critical subsolution for their associated hamilton jacobi equations

hopf lemma for the fractional diffusion operator and its application to a fractional free boundary problem abstract we consider a one dimensional moving boundary problem for the time fractional diffusion equation where the time fractional derivative of order 0 1 is taken in the caputo sense a generalization of the hopf lemma is proved and then used to prove a monotonicity property for the free boundary when a fractional free boundary stefan problem is investigated

solving the time fractional schrodinger equation by krylov projection methods the time fractional schrodinger equation is a fundamental topic in physics and its numerical solution is still an open problem here we start from the possibility to express its solution by means of the mittag leffler function then we analyze some approaches based on the krylov projection methods to approximate this function their convergence properties are discussed together with related issues numerical tests are presented to confirm the strength of the approach under investigation

dynamics of reaction diffusion systems in a subdiffusive regime in this paper we examine the dynamics of reaction diffusion systems with fractional time derivatives it is shown that in these conditions diffusion is anomalous in the sense that the mean square displacement r 2 ensuremath sim t ensuremath gamma where ensuremath gamma l1 a situation known as subdiffusion we study the conditions for the appearance of a diffusion driven instability and show that the restrictive conditions for a turing instability are relaxed this implies that systems whose kinetics are not of the activator inhibitor kind can have a turing instability and a modulated final state we demonstrate our results with numerical calculations in two dimensions using a generic turing model

non markovian diffusion equations and processes analysis and simulations abstract in this paper we introduce and analyze a class of diffusion type equations related to certain non markovian stochastic processes we start from the forward drift equation which is made non local in time by the introduction of a suitable chosen memory kernel k t the resulting non markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l t we then consider the subordinated process y t x l t where x t is a markovian diffusion the corresponding time evolution of the marginal density function of y t is governed by a non markovian fokker planck equation which involves the memory kernel k t we develop several applications and derive the exact solutions we consider different stochastic models for the given equations providing path simulations

from shape to randomness a classification of langevin stochasticity abstract the langevin equation perhaps the most elemental stochastic differential equation in the physical sciences describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise the langevin equation is in effect a mechanism that maps the stochastic white noise input to a stochastic output a stationary steady state distribution in the case of potential wells and a transient extremum distribution in the case of potential gradients in this paper we explore the degree of randomness of the langevin equation s stochastic output and classify it a la mandelbrot into five states of randomness ranging from infra mild to ultra wild we establish closed form and highly implementable analytic results that determine the randomness of the langevin equation s stochastic output based on the shape of the langevin equation s potential field

on the extreme flights of one sided levy processes abstract we explore the statistical behavior of the order statistics of the flights of one sided levy processes olps we begin with the study of the extreme flights of general olps and then focus on the class of selfsimilar processes investigating the following issues i the inner hierarchy of the extreme flights for example how big is the 7th largest flight relative to the 2nd largest one and ii the relative contribution of the extreme flights to the entire flight aggregate for example how big is the 3rd largest flight relative to the olp s value furthermore we show that all hierarchical results obtained but not the aggregate results are explicitly extendable to the class of olps with arbitrary power law flight tails which is far larger than the selfsimilar class

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena

space time fractional zener wave equation the space time fractional zener wave equation describing viscoelastic materials obeying the time fractional zener model and the space fractional strain measure is derived and analysed this model includes waves with finite speed as well as non propagating disturbances the existence and the uniqueness of the solution to the generalized cauchy problem are proved special cases are investigated and numerical examples are presented

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

almost automorphic mild solutions to fractional differential equations abstract we introduce the concept of resolvent families to prove the existence of almost automorphic mild solutions to the differential equation d t u t a u t t n f t 1 2 n z considered in a banach space x where f r x is almost automorphic we also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t 1 2 assuming f t x is almost automorphic in t for each x x satisfies a global lipschitz condition and takes values on x finally we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t u t 1 2 under analogous conditions as in the previous case

on selfsimilar levy random probabilities abstract we explore a class of random probabilities induced by the normalization of selfsimilar levy random measures random measures whose probability laws are governed by stable one sided levy distributions various statistical properties of these random probabilities are analyzed i moment structure ii auto covariance structure iii one dimensional and multidimensional tail probabilities and iv behavior in limiting cases for the levy smirnoff case corresponding to the selfsimilarity index of order 1 2 an explicit analytic formula for the multidimensional probability density functions is derived last a comparison between the class of selfsimilar levy random probabilities and the dirichlet random probability induced by the normalization of the gamma random measure is conducted showing the former to be far more robust than the latter

chern character in twisted k theory equivariant and holomorphic cases it was argued in 25 5 that in the presence of a nontrivial b field d brane charges in type iib string theories are classified by twisted k theory in 4 it was proved that twisted k theory is canonically isomorphic to bundle gerbe k theory whose elements are ordinary hilbert bundles on a principal projective unitary bundle with an action of the bundle gerbe determined by the principal projective unitary bundle the principal projective unitary bundle is in turn determined by the twist this paper studies in detail the chern weil representative of the chern character of bundle gerbe k theory that was introduced in 4 extending the construction to the equivariant and the holomorphic cases included is a discussion of interesting examples

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 fractal characterization of paretian poisson processes abstract paretian poisson processes are poisson processes which are defined on the positive half line have maximal points and are quantified by power law intensities paretian poisson processes are elemental in statistical physics and are the bedrock of a host of power law statistics ranging from pareto s law to anomalous diffusion in this paper we establish evenness based fractal characterizations of paretian poisson processes considering an array of socioeconomic evenness based measures of statistical heterogeneity we show that amongst the realm of poisson processes which are defined on the positive half line and have maximal points paretian poisson processes are the unique class of fractal processes exhibiting scale invariance the results established in this paper are diametric to previous results asserting that the scale invariance of poisson processes with respect to physical randomness based measures of statistical heterogeneity is characterized by exponential poissonian intensities

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

on the non existence of viscosity solutions of multi time hamilton jacobi equations abstract we prove that the multi time hamilton jacobi equation in general cannot be solved in the viscosity sense in the non convex setting even when the hamiltonians are in involution

principal bundles and the dixmier douady class a systematic consideration of the problem of the reduction and lifting of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another we apply these to the study of the dixmier douady class in various contexts including string structures bundles and other examples motivated by considerations from quantum field theory

on the first passage of one sided levy motions abstract we study the first passage problem for one sided levy motions how does such a motion started at the origin cross a barrier positioned at the point x x 0 since one sided levy motions are pure jump processes they always leap over barriers rather than crossing them continuously we hence explore the following issues i first passage times fpts how long would it take the motion to cross the barrier ii first passage leapovers fpls how far would the motion leap over the barrier and iii what is dependence between the fpts and the fpls formulae for the joint laplace transforms of the fpts and fpls are derived and the scaling limits of the fpts and fpls are computed the scaling limits turn out to display dramatically different behavior when i the underlying levy motion has finite cumulants and when ii the underlying levy motion is heavy tailed special attention is devoted to the investigation of the first passage problem for self similar one sided levy motions

asymptotic periodicity for some evolution equations in banach spaces abstract this work deals with the existence and uniqueness of pseudo almost periodic and asymptotically periodic mild solutions to some evolution equations in banach spaces

characters of cycles equivariant characteristic classes and fredholm modules we derive simple explicit formula for the character of a cycle in the connes b b bicomplex of cyclic cohomology and apply it to write formulas for the equivariant chern character and characters of finitely summable bounded fredholm modules

on the structure and phase transitions of power law poissonian ensembles power law poissonian ensembles are poisson processes that are defined on the positive half line and that are governed by power law intensities power law poissonian ensembles are stochastic objects of fundamental significance they uniquely display an array of fractal features and they uniquely generate a span of important applications in this paper we apply three different methods oligarchic analysis lorenzian analysis and heterogeneity analysis to explore power law poissonian ensembles the amalgamation of these analyses combined with the topology of power law poissonian ensembles establishes a detailed and multi faceted picture of the statistical structure and the statistical phase transitions of these elemental ensembles

proportionate vs disproportionate distribution of wealth of two individuals in a tempered paretian ensemble abstract we study the distribution p of the random variable x 1 x 1 x 2 where x 1 and x 2 are the wealths of two individuals selected at random from the same tempered paretian ensemble characterized by the distribution x x x 1 where 0 is the pareto index and x is the cut off function we consider two forms of x a bounded function x 1 for l x h and zero otherwise and a smooth exponential function x exp l x x h in both cases x has moments of arbitrary order we show that for 1 p always has a unimodal form and is peaked at 1 2 so that most probably x 1 x 2 for 0 1 we observe a more complicated behavior which depends on the value of l h in particular for c a certain threshold value p has a three modal for a bounded x and a bimodal m shape for an exponential x form which signifies that in such ensembles the wealths x 1 and x 2 are disproportionately different

almost periodic and pseudo almost periodic solutions to fractional differential and integro differential equations abstract we study the existence of almost periodic resp pseudo almost periodic mild solutions for fractional differential and integro differential equations in the case when the forcing term belongs to the class of stepanov almost resp stepanov like pseudo almost periodic functions

symplectic aspects of mather theory we prove that the so called aubry and mane sets introduced by john mather in lagrangian dynamics are symplectic invariants in order to do so we introduce a barrier in phase space and propose definitions of aubry and mane sets for non convex hamiltonian systems on montre que les ensembles dits d aubry et de mane introduits par john mather en dynamique lagrangienne sont des invariants symplectiques pour ceci on introduit une barriere sur l espace des phases et on definit des ensembles d aubry et de mather pour des systemes hamiltoniens non convexes

models of anomalous diffusion the subdiffusive case abstract the paper discusses a model for anomalous diffusion processes their one point probability density functions p d f are exact solutions of fractional diffusion equations the model reflects the asymptotic behavior of a jump anomalous random walk process with random jump sizes and random inter jump time intervals with infinite means and variances which do not satisfy the law of large numbers in the case when these intervals have a fractional exponential p d f the fractional komogorov feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed finally some statistical properties of solutions of the related langevin equation are studied the subdiffusive case is explored in detail the emphasis is on a rigorous presentation which however would be accessible to the physical sciences audience

investigation of the cumulative diminution process using the fibonacci method and fractional calculus abstract in this study we investigate the cumulative diminution phenomenon for a physical quantity and a diminution process with a constant acquisition quantity in each step in a viscous medium we analyze the existence of a dynamical mechanism that underlies the success of fractional calculus compared with standard mathematics for describing stochastic processes by proposing a fibonacci approach where we assume that the complex processes evolves cumulatively in fractal space and discrete time thus when the differential integral order is attained this indicates the involvement of the viscosity of the medium in the evolving process the future value of the diminishing physical quantity is obtained in terms of the mittag leffler function mlf and two rheological laws are inferred from the asymptotic limits thus we conclude that the differential integral calculus of fractional mathematics implicitly embodies the cumulative diminution mechanism that occurs in a viscous medium

fractional dynamics of systems with long range space interaction and temporal memory abstract field equations with time and coordinate derivatives of noninteger order are derived from a stationary action principle for the cases of power law memory function and long range interaction in systems the method is applied to obtain a fractional generalization of the ginzburg landau and nonlinear schrodinger equations as another example dynamical equations for particle chains with power law interaction and memory are considered in the continuous limit the obtained fractional equations can be applied to complex media with without random parameters or processes

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

divergent series and memory of the initial condition in the long time solution of some anomalous diffusion problems we consider various anomalous d dimensional diffusion problems in the presence of an absorbing boundary with radial symmetry the motion of particles is described by a fractional diffusion equation their mean square displacement is given by r 2 t 0 1 resulting in normal diffusive motion if 1 and subdiffusive motion otherwise for the subdiffusive case in sufficiently high dimensions divergent series appear when the concentration or survival probabilities are evaluated via the method of separation of variables while the solution for normal diffusion problems is at most divergent as t 0 the emergence of such series in the long time domain is a specific feature of subdiffusion problems we present a method to regularize such series and in some cases validate the procedure by using alternative techniques laplace transform method and numerical simulations in the normal diffusion case we find that the signature of the initial condition on the approach to the steady state rapidly fades away and the solution approaches a single the main decay mode in the long time regime in remarkable contrast long time memory of the initial condition is present in the subdiffusive case as the spatial part 1 r describing the long time decay of the solution to the steady state is determined by a weighted superposition of all spatial modes characteristic of the normal diffusion problem the weight being dependent on the initial condition interestingly 1 r turns out to be independent of the anomalous diffusion exponent

subdiffusion in a bounded domain with a partially absorbing reflecting boundary the exit time of a subdiffusive process from a bounded domain with a partially absorbing reflecting boundary is considered the short time and long time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the laplace operator eigenfunctions rotation invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations the asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane in a chemical reactor filled with catalytic grains of finite reactivity or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation