global regular and singular solutions for a model of gravitating particles the existence of solutions of a nonlinear parabolic equation describing the gravitational interaction of particles is studied for the initial data in spaces of generalized pseudomeasures this approach permits us to relax regularity assumptions on the initial conditions and to prove asymptotic stability results for the above problem

mathematics of granular materials this is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media intended for physicists and for mathematicians outside the field

biset transformations of tambara functors abstract if we are given an h g biset u for finite groups g and h then any mackey functor on g can be transformed by u into a mackey functor on h in this article we show that the biset transformation is also applicable to tambara functors when u is right free and in fact forms a functor between the category of tambara functors on g and h this biset transformation functor is compatible with some algebraic operations on tambara functors such as ideal quotients or fractions in the latter part we also construct the left adjoint of the biset transformation

the spectrum of the burnside tambara functor on a finite cyclic p group abstract for a finite group g a tambara functor on g is regarded as a g bivariant analog of a commutative ring in this analogy previously we have defined an ideal of a tambara functor in this article we calculate the prime spectrum of the burnside tambara functor when g is a finite cyclic p group for a prime integer p

the hochschild cohomology ring of a group algebra there is a standard additive decomposition of the hochschild cohomology ring of the group algebra of a finite group g as the direct sum of the cohomology rings of the centralizers of representatives of the conjugacy classes of g a special case of our main result describes the cup product in terms of this decomposition as applications we determine presentations for the hochschild cohomology rings of 1 the mod 3 group algebra of the symmetric group s 3 2 the mod 2 group algebra of the alternating group a 4 and 3 the mod 2 group algebras of the dihedral 2 groups

tambarization of a mackey functor and its application to the witt burnside construction abstract for an arbitrary group g a semi mackey functor is a pair of covariant and contravariant functors from the category of g sets and is regarded as a g bivariant analog of a commutative semi group in this view a g bivariant analog of a semi ring should be a semi tambara functor a tambara functor is firstly defined by tambara which he called a tnr functor when g is finite as shown by brun a tambara functor plays a natural role in the witt burnside construction it will be a natural question if there exist sufficiently many examples of tambara functors compared to the wide range of mackey functors in the first part of this article we give a general construction of a tambara functor from any mackey functor on an arbitrary group g in fact we construct a functor from the category of semi mackey functors to the category of tambara functors this functor gives a left adjoint to the forgetful functor and can be regarded as a g bivariant analog of the monoid ring functor in the latter part when g is finite we investigate relations with other mackey functorial constructions crossed burnside ring elliott s ring of g strings jacobson s f burnside ring all these lead to the study of the witt burnside construction

scale invariant solutions to partial differential equations of fractional order with a moving boundary condition in this paper we give similarity solutions of partial differential equations of fractional order with a moving boundary condition the solutions are given in terms of a generalized wright function the time fractional caputo derivative and two types of space fractional derivatives are considered the scale invariant variable and the form of the solution of the moving boundary are obtained by the lie group analysis a comparison between the solutions corresponding to two types of fractional derivative is also given

fibred permutation sets and the idempotents and units of monomial burnside rings we study the units of monomial burnside rings and the idempotents of monomial burnside algebras introducing a tenduction map we realise the unit group and the torsion unit group as a mackey functor

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function

infinite energy solutions to the homogeneous boltzmann equation the goal of this work is to present an approach to the homogeneous boltzmann equation for maxwellian molecules with a physical collision kernel that allows us to construct unique solutions to the initial value problem in a space of probability measures defined via the fourier transform in that space the second moment of a measure is not assumed to be finite so infinite energy solutions are not a priori excluded from our considerations moreover we study the large time asymptotics of solutions and in a particular case we give an elementary proof of the asymptotic stability of self similar solutions obtained by a v bobylev and c cercignani j stat phys 106 2002 1039 1071 2009 wiley periodicals inc

ideals of tambara functors abstract for a finite group g a tambara functor on g is regarded as a g bivariant analog of a commutative ring in this article we consider a g bivariant analog of the ideal theory for tambara functors

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

renewal processes and the hurst effect consider a pure recurrent positive renewal process generated by some inter arrival waiting time if the waiting time has power law fall off exponent i e tail index in 1 2 and if the jump s amplitude has non zero but finite mean and finite variance the cumulative amplitude process is long range dependent with hurst exponent in 1 2 1 results in this direction have been obtained by daley under the sole assumptions that the waiting time has moment index in 1 2 if the amplitude has zero mean up to a brownian trend the cumulative amplitude process exhibits a negative dependence property with hurst exponent in 0 1 2 the case of delayed stationary renewal processes is also investigated together with two classes of limiting renewal processes the compound exponential and the levy classes these are of some interest when the average time between consecutive jumps tends to zero jointly with the probability mass of the jumps height concentrating about zero in some precise way under suitable hypothesis the hurst effect is maintained in the limit

continuous time random walks in finite domains and general boundary conditions some formal considerations the present work studies continuous time random walks ctrws in a finite domain a broad class of boundary conditions of which absorbing and reflecting boundaries are particular cases is considered it is shown how any ctrw in this class can be mapped to a ctrw in an infinite domain this may allow applying well known techniques for infinite ctrws to the problem of obtaining the fluid limit for finite domain ctrws where the fluid limit or hydrodynamic limit refers to the partial differential equation describing the long time and large distance behaviour of the system as an illustration the fluid limit equation and its propagator are obtained explicitly in the case of purely reflecting boundaries we also derive the modification of the riemann liouville fractional differential operators implementing the reflecting boundary conditions

a stochastic representation for backward incompressible navier stokes equations by reversing the time variable we derive a stochastic representation for backward incompressible navier stokes equations in terms of stochastic lagrangian paths which is similar to constantin and iyer s forward formulations in constantin and iyer comm pure appl math lxi 330 345 2008 using this representation a self contained proof of local existence of solutions in sobolev spaces are provided for incompressible navier stokes equations in the whole space in two dimensions or large viscosity an alternative proof to the global existence is also given moreover a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero

stochastic cascades and 3 dimensional navier stokes equations in this article we study the incompressible navier stokes equations in 3 the non linear integral equation satisfied by the fourier transform of the laplacian of the velocity field can be interpreted in terms of a branching process and a composition rule along the associated tree we derive from this representation new classes where global existence and uniqueness can be proven

bounded fractional diffusion in geological media definition and lagrangian approximation spatiotemporal fractional derivative models fdms have been increasingly used to simulate non fickian diffusion but methods have not been available to define boundary conditions for fdms in bounded domains this study defines boundary conditions and then develops a lagrangian solver to approximate bounded one dimensional fractional diffusion both the zero value and non zero value dirichlet neumann and mixed robin boundary conditions are defined where the sign of riemann liouville fractional derivative capturing non zero value spatial nonlocal boundary conditions with directional super diffusion remains consistent with the sign of the fractional diffusive flux term in the fdms new lagrangian schemes are then proposed to track solute particles moving in bounded domains where the solutions are checked against analytical or eularian solutions available for simplified fdms numerical experiments show that the particle tracking algorithm for non fickian diffusion differs from fickian diffusion in relocating the particle position around the reflective boundary likely due to the nonlocal and non symmetric fractional diffusion for a non zero value neumann or robin boundary a source cell with a reflective face can be applied to define the release rate of random walking particles at the specified flux boundary mathematical definitions of physically meaningful nonlocal boundaries combined with bounded lagrangian solvers in this study may provide the only viable techniques at present to quantify the impact of boundaries on anomalous diffusion expanding the applicability of fdms from infinite domains to those with any size and boundary conditions this article is protected by copyright all rights reserved

a stochastic approach to the solution of magnetohydrodynamic equations the construction of stochastic solutions is a powerful method to obtain localized solutions in configuration or fourier space and for parallel computation with domain decomposition here a stochastic solution is obtained for the magnetohydrodynamics equations some details are given concerning the numerical implementation of the solution which is illustrated by an example of generation of long range magnetic fields by a velocity source

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

asymptotic distributions of continuous time random walks a probabilistic approach we provide a systematic analysis of the possible asymptotic distributions o one dimensional continuous time random walks ctrws by applying the limit theorems of probability theory biased and unbiased walks of coupled and decoupled memory are considered in contrast to previous work concerning decoupled memory and levy walks we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean obtaining asymmetric levy stable limits suprisingly it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution we discuss interesting properties of walks with an infinite mean waiting time between successive jumps

governing equations and solutions of anomalous random walk limits fachbereich mathematik university of dortmund 44221 dortmund germany received 5 june 2002 published 30 december 2002 continuous time random walks model anomalous diffusion coupling allows the magnitude of particlejumps to depend on the waiting time between jumps governing equations for the long time scaling limits ofthese models are found to have fractional powers of coupled space and time differential operators explicitsolutions and scaling properties are presented for these equations which can be used to model ow in porousmedia and other physical systems doi 10 1103 physreve 66 060102 pacs number s 05 40 2a 02 50 cw 05 60 2k 47 55 mhi introduction

crossed burnside rings and bouc s construction of green functors abstract we use a formula for primitive idempotents of the crossed burnside ring given by f oda and t yoshida and a result from the theory of green functors obtained by s bouc to prove an induction theorem for the drinfel d double for a finite group over the complex field we obtain artin s induction theorem for group algebras as a corollary of the theorem

continuous time random walks and the fractional diffusion equation the average probability density p r t of random walks on fractals is revisited within the continuous time random walks formalism corrections to the accepted asymptotic stretched gaussian decay of p r t of the form ralpha are discussed it is shown that p r t obeys a diffusion equation with a fractional time derivative asymptotically and predictions about the value of alpha are presented

a stochastic lagrangian representation of the three dimensional incompressible navier stokes equations in this paper we derive a probabilistic representation of the deterministic three dimensional navier stokes equations based on stochastic lagrangian paths the particle trajectories obey sdes driven by a uniform wiener process the inviscid weber formula for the euler equations of ideal fluids is used to recover the velocity field this method admits a self contained proof of local existence for the nonlinear stochastic system and can be extended to formulate stochastic representations of related hydrodynamic type equations including viscous burgers equations and lagrangian averaged navier stokes alpha models 2007 wiley periodicals inc

existence and uniqueness of stationary solutions for 3d navier stokes system with small random forcing via stochastic cascades we consider the 3d navier stokes system in the fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition we explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with fourier transform majorized by a certain function h moreover we prove the following one force one solution principle the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t our explicit construction of the solution is based upon the stochastic cascade representation

a forward backward probabilistic algorithm for the incompressible navier stokes equations abstract a novel probabilistic scheme for solving the incompressible navier stokes equations is studied in which we approximate a generalized nonlinear feyman kac formula the velocity field is interpreted as the mean value of a stochastic process ruled by forward backward stochastic differential equations fbsdes driven by brownian motion following an approach by delbaen qiu and tang introduced in 2015 the pressure term is obtained from the velocity by solving a poisson problem as computing the expectation of an integral functional associated to an extra bsde the fbsdes components are numerically solved by using a forward backward algorithm based on euler type schemes for the local time integration and the quantization of the increments of brownian motion following the algorithm proposed by delarue and menozzi in 2006 numerical results are reported on spatially periodic analytic solutions of the navier stokes equations for incompressible fluids we illustrate the proposed algorithm on a two dimensional taylor green vortex and three dimensional beltrami flows

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

fluid limit of nonintegrable continuous time random walks in terms of fractional differential equations the fluid limit of a recently introduced family of nonintegrable nonlinear continuous time random walks is derived in terms of fractional differential equations in this limit it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales for this reason the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems ranging from the study of self organized criticality to the modeling of turbulent transport in fluids and plasmas

crossed burnside rings i the fundamental theorem abstract let g be a finite group and s a finite g monoid a crossed g set over s is a finite g set equipped with a g map into s called a weight function a crossed burnside ring x g s is the grothendieck ring of the category of crossed g sets with respect to disjoint unions and tensor products in this paper we prove the fundamental theorem of crossed burnside rings and an idempotent formula at characteristic 0

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

generalised burnside rings g categories and module categories this note describes an application of the theory of generalised burnside rings to algebraic representation theory tables of marks are given explicitly for the groups s4 and s5 which are of particular interest in the context of reductive algebraic groups as an application the base sets for the nilpotent element f4 a3 are computed

continuous time random walks in periodic systems fluid limit and fractional differential equations on the circle in this paper the continuous time random walk on the circle is studied we derive the corresponding generalized master equation and discuss the effects of topology especially important when levy flights are allowed then we work out the fluid limit equation formulated in terms of the periodic version of the fractional riemann liouville operators for which we provide explicit expressions finally we compute the propagator in some simple cases the analysis presented herein should be relevant when investigating anomalous transport phenomena in systems with periodic dimensions

some insights in superdiffusive transport abstract in a wide range of systems the relaxation in response to an initial pulse has been experimentally found to follow a nonlinear relationship for the mean squared displacement of the kind x 2 t t where may be greater or smaller than 1 such phenomena have been described under the generic term of anomalous diffusion levy flights stochastic processes lead to superdiffusive behaviour 1 2 and have been recently proposed to model among the others the subsurface contaminant spread in highly heterogeneous media under the effects of water flow in this paper within the continuous time random walk ctrw approach to anomalous diffusion we compare the analytical solution of the approximated fractional diffusion equation fde with the monte carlo one obtained by simulating the superdiffusive behaviour of an ensemble of particle in a medium we show that the two are neatly different as the process approaches the standard diffusive behaviour we argue that this is due to a truncation in the fourier space expansion introduced by the fde approach we propose a second order correction to this expansion and numerically solve the ctrw model under this hypothesis the accuracy of the results thus obtained is validated through monte carlo simulation over all the superdiffusive range the same kind of discrepancy is shown to occur also in the derivation of the fractional moments of the distribution analogous corrections are proposed and validated through the monte carlo approach

space fractional advection diffusion and reflective boundary condition anomalous diffusive transport arises in a large diversity of disordered media stochastic formulations in terms of continuous time random walks ctrws with transition probability densities showing space and or time diverging moments were developed to account for anomalous behaviors a broad class of ctrws was shown to correspond on the macroscopic scale to advection diffusion equations involving derivatives of noninteger order in particular ctrws with l evy distribution of jumps and finite mean waiting time lead to a space fractional equation that accounts for superdiffusion and involves a nonlocal integral differential operator within this framework we analyze the evolution of particles performing symmetric l evy flights with respect to a fluid moving at uniform speed v the particles are restricted to a semi infinite domain limited by a reflective barrier we show that the introduction of the boundary condition induces a modification in the kernel of the nonlocal operator thus the macroscopic space fractional advection diffusion equation obtained is different from that in an infinite medium

smooth or singular solutions to the navier stokes system abstract the existence of singular solutions of the incompressible navier stokes system with singular external forces the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions including stationary ones and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type

the crossed burnside rings iii the dress construction for a tambara functor abstract we discuss the dress construction for a tambara functor associated to a finite commutative monoid with action of a finite group

on the andre quillen homology of tambara functors abstract we lift to equivariant algebra three closely related classical algebraic concepts abelian group objects in augmented commutative algebras derivations and kahler differentials we define mackey functor objects in the category of tambara functors augmented to a fixed tambara functor r and we show that the usual square zero extension gives an equivalence of categories between these mackey functor objects and ordinary modules over r we then describe the natural generalization to tambara functors of a derivation building on the intuition that a tambara functor has products twisted by arbitrary finite g sets and we connect this to square zero extensions in the expected way finally we show that there is an appropriate form of kahler differentials which satisfy the classical relation that derivations out of r are the same as maps out of the kahler differentials

domain decomposition solution of nonlinear two dimensional parabolic problems by random trees a domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension based on the probabilistic representation of solutions as an average of suitable multiplicative functionals such a direct probabilistic representation requires generating a number of random trees whose role is that of the realizations of stochastic processes used in the linear problems first only few values of the sought solution inside the space time domain are computed by a monte carlo method on the trees an interpolation is then carried out in order to approximate interfacial values of the solution inside the domain thus a fully decoupled set of sub problems is obtained the algorithm is suited to massively parallel implementation enjoying arbitrary scalability and fault tolerance properties pruning the trees is shown to increase appreciably the efficiency of the algorithm numerical examples conducted in 2d including some for the kpp equation are given

reprint of boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

numerical approximation of levy feller diffusion equation and its probability interpretation in this paper we consider the levy feller fractional diffusion equation which is obtained from the standard diffusion equation by replacing the second order space derivative with a riesz feller derivative of order email protected 0 2 a 1 and skewness q q

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

levy flights from a continuous time process l evy flight dynamics can stem from simple random walks in a system whose operational time number of steps n typically grows superlinearly with physical time t thus this process is a kind of continuous time random walk ctrw dual to the typical scher montroll model in which n grows sublinearly with t models in which l evy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force which is shown to be nonlinear on the other hand the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern and leads to a normal boltzmann distribution mixed models describing normal ctrw s in superlinear operational time and l evy flights under the operational time of subdiffusive ctrw s lead to a paradoxical diffusive behavior similar to the one found in transport on polymer chains the relaxation to the boltzmann distribution in such models is slow and asymptotically follows a power law

induction formulae for mackey functors with applications to representations of the twisted quantum double of a finite group abstract in the theory of canonical induction formulae for mackey functors boltje 4 demonstrated that the plus constructions together with the mark morphism are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups in this paper we present a short exact sequence for the plus constructions derived from cauchy frobenius lemma and apply it to the proof of boltje s integrality result for canonical induction formulae the methods appearing in boltje s theory combined with the dress construction for mackey functors are applicable to induction theorems on representations of the twisted quantum double of a finite group as a sequel to such a research we describe canonical versions of two induction theorems whose origins are artin s induction theorem and brauer s induction theorem on c characters of a finite group

crossed burnside rings ii the dress construction of a green functor abstract we present some results about the dress construction of green functor associated to a g monoid in particular we see that the crossed burnside rings the representation rings of quantum double and hochschild cohomology rings are constructed by the dress construction of some green functors

a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation

effects of fractional mass transfer and chemical reaction on mhd flow in a heterogeneous porous medium abstract this paper presents a study on space fractional anomalous convective diffusion and chemical reaction in the magneto hydrodynamic fluid over an unsteady stretching sheet the fractional diffusion model is derived from decoupled continuous time random walks in a heterogeneous porous medium a novel transformation which features time finite difference is introduced to reduce the governing equations into ordinary differential ones in each time level numerical solutions are established by an implicit finite difference scheme the stability and convergence of the method are analyzed results show that increasing fractional derivative parameter enhances concentration near the surface while an opposite phenomenon occurs far away from the wall there is a reduction of mass transfer rate on the sheet with an increase in the fractional derivative parameter moreover the numerical solutions are compared with exact solutions and good agreement has been observed

generalized burnside rings and group cohomology we define the cohomological burnside ring b n g m of a finite group g with coefficients in a zgmodule m as the grothendieck ring of the isomorphism classes of pairs x u where x is a g set and u is a cohomology class in a cohomology group h n x g m the cohomology groups h x g m are defined in such a way that h x g m i h hi m when x is the disjoint union of transitive g sets g hi i fa is an abelian group with trivial action then b 1 g a is the same as the monomial burnside ring over a and when m is taken as a g monoid then b 0 g m is equal to the crossed burnside ring b c g m we discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological burnside rings we also give an interpretation of b 2 g m in terms of twisted

stochastic explosion and non uniqueness for α riccati equation abstract we consider the problem of global in time existence and uniqueness for the initial value problems u t u t u 2 t u 0 u 0 0 for scaling parameters 0 and a large class of initial data u 0 much of the focus is on the super critical case 1 where by using the multiplicative stochastic cascade techniques we prove global existence for small initial data and finite time blow up for large initial data however while uniqueness holds for solutions satisfying a growth condition in general it fails even for arbitrary small initial data we demonstrate that this lack of uniqueness is directly connected to the stochastic explosion of an associated multiplicative stochastic cascade process the key tool in establishing the above mentioned results is an iterative algorithm that allows one to exploit the stochastic explosion of the underlying multiplicative cascade to establish both existence and lack of uniqueness of solutions

multiple burnside rings and brauer induction formulae abstract let g be a finite group and suppose that g is an operator group of a finite group a define s g a h h s g and z 1 h a where s g is the set of subgroups of g and z 1 h a is the set of crossed homomorphisms from h to a we view g as an operator group of the opposite group a of a and make s g a into a left a g set the ring g a is defined to be a commutative ring consisting of all formal z linear combinations of a g orbits in s g a idempotent formulae for q z g a not only imply a generalization of dress induction theorem but bring in the case where z 1 g a is the set of linear c characters of g boltje s explicit formula for brauer s induction theorem and its hyperelementary version

witt vectors and tambara functors abstract we give a combinatorial description of the ring of g witt vectors on a polynomial algebra over the integers for every finite group g using this description we show that the functor which takes a commutative ring with trivial action of g to its ring of witt vectors coincides with the left adjoint of the algebraic functor from the category of g tambara functors to the category of commutative rings with an action of g

a rate of convergence for the lansα regularization of navier stokes equations abstract a rate of convergence of the solutions of the lans equations with periodic boundary to the solutions of the navier stokes equations as 0 is obtained in a mixed l 1 l 2 time space norm for small initial data in besov type function spaces in which global existence and uniqueness of solutions can also be established

on the computation of the trace form of some galois extensions abstract we investigate the trace form trl k l k x trl kx2of a finite galois extensionl k in particular we study 2 extensions of degree 16 using some reduction theorems these results yield a classification of nearly all trace forms of galois extensions of degree 31 finally we study the trace form of a cyclotomic extension and of its maximal real subfield

continuous time random walk as a guide to fractional schrodinger equation we argue that the continuous time random walk approach may be a useful guide to extend the schrodinger equation in order to incorporate nonlocal effects avoiding the inconsistencies raised by jeng et al j math phys 51 062102 2010 as an application we work out a free particle in a half space obtaining the time dependent solution by considering an arbitrary initial condition

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

a stochastic model for thermal transport of nanofluid in porous media derivation and applications abstract in this paper a stochastic thermal transport model is developed for nanofluid flowing through porous media this model incorporates the influences of nanoparticle migration on convective heat transfer of the colloidal solution we show that levy flight movement patterns of nanoparticles result in the derived model using fractional derivative for the diffusion term the new thermal transport model is then applied to the mixed convective problem which is solved using finite difference method numerical results show that the smaller values of levy index lead to larger nusselt numbers thus the occurrence of long jumps for nanoparticles increases the heat transport of nanofluids the effects of other involved physical parameters are also presented and discussed

stochastic lagrangian particle approach to fractal navier stokes equations in this article we study the fractal navier stokes equations by using the stochastic lagrangian particle path approach in constantin and iyer comm pure appl math lxi 330 345 2008 more precisely a stochastic representation for the fractal navier stokes equations is given in terms of stochastic differential equations driven by levy processes based on this representation a self contained proof for the existence of a local unique solution for the fractal navier stokes equation with initial data in mathbb w 1 p is provided and in the case of two dimensions or large viscosity the existence of global solutions is also obtained in order to obtain the global existence in any dimensions for large viscosity the gradient estimates for levy processes with time dependent and discontinuous drifts are proved

moments for tempered fractional advection diffusion equations this paper develops moment formulas for exponentially tempered fractional advection diffusion equations tfades that transition from anomalous to asymptotic diffusion limits over time exact analytical expressions or series representations for spatial moments up to the fourth order are derived by integral transform or asymptotic expansion approach a fully lagrangian solver cross verified by an implicit eulerian approach is also developed to calculate numerically the complete evolution of moments for the tfades with complex initial and boundary conditions moment analysis identifies the diffusion equation that attracts the tempered anomalous diffusion in the long time limit fitting of moments measured at two end members of alluvial systems checks the applicability of moment analysis in understanding real diffusion

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

constructing witt burnside rings abstract the witt burnside ring of a profinite group g over a commutative ring a generalizes both the burnside ring of virtual g sets and the rings of universal and p typical witt vectors over a the witt burnside ring of g over the monoid ring z m where m is a commutative monoid is proved isomorphic to the grothendieck ring of a category whose objects are almost finite g sets equipped with a map to m that is constant on g orbits in particular if a is a commutative ring and a denotes the set a as a monoid under multiplication then the witt burnside ring of g over z a is isomorphic to graham s ring of virtual g strings with coefficients in a this result forms the basis for a new construction of witt burnside rings and provides an important missing link between the constructions of dress and siebeneicher adv in math 70 1988 87 132 and graham adv in math 99 1993 248 263 with this approach the usual truncation frobenius verschiebung and teichmuller maps readily generalize to maps between witt burnside rings

galois equivariance and stable motivic homotopy theory for a finite galois extension of fields l k with galois group g we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence we show that after completing at a prime and the motivic hopf map this results in a full and faithful embedding whenever k is real closed and l k i it is a full and faithful embedding after completion if a motivic version of serre s finiteness theorem is valid we produce strong necessary conditions on the field extension l k for this functor to be full and faithful along the way we produce several results on the stable c2 equivariant betti realization functor and prove convergence theorems for the p primary c2 equivariant adams spectral sequence

majorizing kernels and stochastic cascades with applications to incompressible navier stokes equations a general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d navier stokes equations the basic idea generalizes a probabilistic approach introduced by lejan and sznitman 1997 to obtain weak solutions whose fourier transform may be represented by an expected value of a stochastic cascade a functional analytic framework is also developed which partially connects stochastic iterations and certain picard iterates some local existence and uniqueness results are also obtained by contractive mapping conditions on the picard iteration

random walks with infinite spatial and temporal moments the continuous time random walk of montroll and weiss has been modified by scher and lax to include a coupled spatial temporal memory we treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial and temporal moments of the memory are infinite the asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated the resulting behavior includes localized diffusive wavelike and levy s stable laws for the appropriate scaled variable we show that an infinite mean waiting time can lead to long time diffusive behavior while a finite mean waiting time is not sufficient to ensure the same

a stochastic perturbation of inviscid flows we prove existence and regularity of the stochastic flows used in the stochastic lagrangian formulation of the incompressible navier stokes equations with periodic boundary conditions and consequently obtain a ck local existence result for the navier stokes equations our estimates are independent of viscosity allowing us to consider the inviscid limit we show that as 0 solutions of the stochastic lagrangian formulation with periodic boundary conditions converge to solutions of the euler equations at the rate of o sqrt nu t

lattice boltzmann method for the fractional advection diffusion equation mass transport such as movement of phosphorus in soils and solutes in rivers is a natural phenomenon and its study plays an important role in science and engineering it is found that there are numerous practical diffusion phenomena that do not obey the classical advection diffusion equation ade such diffusion is called abnormal or superdiffusion and it is well described using a fractional advection diffusion equation fade the fade finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems however solution to the fade is difficult and the existing numerical methods are complicated and inefficient in this study a fresh lattice boltzmann method is developed for solving the fractional advection diffusion equation labfade the fade is transformed into an equation similar to an advection diffusion equation and solved using the lattice boltzmann method the labfade has all the advantages of the conventional lattice boltzmann method and avoids a complex solution procedure unlike other existing numerical methods the method has been validated through simulations of several benchmark tests a point source diffusion a boundary value problem of steady diffusion and an initial boundary value problem of unsteady diffusion with the coexistence of source and sink terms in addition by including the effects of the skewness the fractional order and the single relaxation time the accuracy and convergence of the method have been assessed the numerical predictions are compared with the analytical solutions and they indicate that the method is second order accurate the method presented will allow the fade to be more widely applied to complex mass transport problems in science and engineering

the representation ring of the quantum double of a finite group abstract we present some results about the representation ring of the quantum double of a finite group over fields of arbitrary characteristic we give a direct sum decomposition of this representation ring into ideals involving green rings of subgroups given characters of such green rings we construct characters of the representation ring of the quantum double and show that all characters arise in this fashion we prove that the characters thereby obtained from brauer characters separate modules of the quantum double up to composition factors

a generalization of the dress construction for a tambara functor and its relation to polynomial tambara functors abstract for a finite group g semi mackey functors and semi tambara functors are regarded as g bivariant analogs of semi groups and semi rings respectively in fact if g is trivial they agree with the ordinary semi groups and semi rings and many naive algebraic properties concerning rings and groups have been extended to these g bivariant analogous notions in this article we investigate a g bivariant analog of the semi group rings with coefficients just as a coefficient ring r and a monoid q yield the semi group ring r q our construction enables us to make a tambara functor t m out of a semi mackey functor m and a coefficient tambara functor t this construction is a composant of the tambarization and the dress construction as expected this construction is the one uniquely determined by the righteous adjoint property besides in analogy with the trivial group case if m is a mackey functor then t m is equipped with a natural hopf structure moreover as an application of the above construction we also obtain some g bivariant analogs of the polynomial rings

global representation rings abstract in this paper we combine the concepts of burnside ring and character ring in what we call the global representation ring of a finite group we compute all ring homomorphisms from this ring to the complex numbers we compute its spectrum its connected components and determine the primitive idempotents

levy walk approach to anomalous diffusion abstract the transport properties of levy walks are discussed in the framework of continuous time random walks ctrw with coupled memories this type of walks may lead to anomalous diffusion where the mean squared displacement r2 t t with 1 we focus on the enhanced diffusion limit 1 in one dimension and present our results on r2 t the mean number of distinct sites visited s t and p r t the probability of being at position r at time t

the tambara structure of the trace ideal for cyclic extensions abstract this paper explores the tambara functor structure of the trace ideal of a galois extension in the case of a pro cyclic extension we are able to explicitly determine the generators of the ideal furthermore we show that the absolute trace ideal of a cyclic group is strongly principal when viewed as an ideal of the burnside tambara functor applying our results we calculate the trace ideal for extensions of finite fields the appendix determines a formula for the norm of a quadratic form over an arbitrary finite extension of a finite field

on the fractions of semi mackey and tambara functors for a finite group g a semi mackey resp tambara functor is regarded as a g bivariant analog of a commutative monoid resp ring as such some naive algebraic constructions are generalized to this g bivariant setting in this article as a g bivariant analog of the fraction of a ring we consider fraction of a tambara and a semi mackey functor by a multiplicative semi mackey subfunctor

multiplicative cascades applied to pdes two numerical examples numerical approximations to the fourier transformed solution of partial differential equations are obtained via monte carlo simulation of certain random multiplicative cascades two particular equations are considered linear diffusion equation and viscous burgers equation the algorithms proposed exploit the structure of the branching random walks in which the multiplicative cascades are defined the results show initial numerical approximations with errors less than 5 in the leading fourier coefficients of the solution this approximation is then improved substantially using a picard iteration scheme on the integral equation associated with the representation of the respective pde in fourier space

the burnside ring of profinite groups and the witt vector construction abstract the classical construction of witt vectors can be understood as a special case of a construction which can be defined relative to any given profinite group g in terms of its completed burnside ring lde g it specializes to the classical situation in case g is infinite cyclic

a probabilistic representation of solutions of the incompressible navier stokes equations in r3 a new probabilistic representation is presented for solutions of the incompressible navier stokes equations in r3 with given forcing and initial velocity this representation expresses solutions as scaled conditional expectations of functionals of a markov process indexed by the nodes of a binary tree it gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data these conditions involve comparison of the forcing and initial data with majorizing kernels