polymer melt dynamics microscopic roots of fractional viscoelasticity the rheological properties of polymer melts and other complex macromolecular fluids are often successfully modeled by phenomenological constitutive equations containing fractional differential operators we suggest a molecular basis for such fractional equations in terms of the generalized langevin equation gle that underlies the renormalized rouse model developed by schweizer j chem phys 91 5802 1989 the gle describes the dynamics of the segments of a tagged chain under the action of random forces originating in the fast fluctuations of the surrounding polymer matrix by representing these random forces as fractional gaussian noise and transforming the gle into an equivalent diffusion equation for the density of the tagged chain segments we obtain an analytical expression for the dynamic shear relaxation modulus g t which we then show decays as a power law in time this power law relaxation is the root of fractional viscoelastic behavior

on the homotopy of the stable mapping class group by considering all surfaces and their mapping class groups at once it is shown that the classifying space of the stable mapping class group after plus construction b has the homotopy type of an infinite loop space the main new tool is a generalized group completion theorem for simplicial categories the first deloop of b coincides with that of miller m induced by the pairs of pants multiplication the classical representation of the mapping class group onto siegel s modular group is shown to induce a map of infinite loop spaces from b to k theory it is then a direct consequence of a theorem by charney and cohen cc that there is a space y such that b im j 1 2 y where im j 1 2 is the image of j localized away from the prime 2

fractional diffusion waiting time distributions and cattaneo type equations we discuss a generalized diffusion equation resulting from an additive two state process in combination with an asymptotically fractal asymptotic power law waiting time distribution the obtained equation is an extension to previously discussed fractional diffusion equations our description leads to a mean squared displacement which describes enhanced subballistic transport for long times the short time behavior however is of a ballistic nature this separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting time distribution this is also mirrored by the observation that for small times our generalized diffusion equation reduces to the standard cattaneo equation the asymptotic probability density is of compressed gaussian type and thus differs from the l evy tail generally found for these kinds of processes

homological stability for automorphism groups abstract given a family of groups admitting a braided monoidal structure satisfying mild assumptions we construct a family of spaces on which the groups act and whose connectivity yields via a classical argument of quillen homological stability for the family of groups we show that stability also holds with both polynomial and abelian twisted coefficients with no further assumptions this new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups such as the symmetric groups general linear groups and mapping class groups by making systematic the proofs of classical stability results we show that they all hold with the same type of coefficient systems obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups braid groups automorphisms of free groups unitary groups mapping class groups of non orientable surfaces and mapping class groups of 3 manifolds our construction can also be applied to families of groups not considered before in the context of homological stability as a byproduct of our work we construct the braided analogue of the category fi of finite sets and injections relevant to the present context and define polynomiality for functors in the context of pre braided monoidal categories

stochastic renormalization group approach to random point processes a fractal generalization of poisson statistics a generalization of the shlesinger hughes stochastic renormalization method is suggested for random point processes a decimation procedure is introduced in terms of two parameters the probability that a decimation step takes place and the probability that during a decimation step a random dot is removed from the process at each step a random number of dots are removed by this procedure a chain of point processes is generated the renormalized point process is a superposition of the intermediate processes attached to the different steps for a statistical ensemble the fraction lq of systems for which q decimation steps occur is a power function of the fraction q of points which survive q decimation steps lq q 1 df where df 1 ln ln 1 is a fractal exponent smaller than unity df 1 if the random points are initially independent then a complete analysis is possible in this case explicit expressions for the renormalized janossy densities the joint densities and the generating functional of

analogies between colored levy noise and random channel approach to disordered kinetics we point out some interesting analogies between colored levy noise and the random channel approach to disordered kinetics these analogies are due to the fact that the probability density of the levy noise source plays a similar role as the probability density of rate coefficients in disordered kinetics although the equations for the two approaches are not identical the analogies can be used for deriving new useful results for both problems the random channel approach makes it possible to generalize the fractional uhlenbeck ornstein processes fuo for space and time dependent colored noise we describe the properties of colored noise in terms of characteristic functionals which are evaluated by using a generalization of huber s approach to complex relaxation phys rev b 31 6070 1985 we start out by investigating the properties of symmetrical white noise and then define the levy colored noise in terms of a langevin equation with a levy white noise source we derive exact analytical expressions for the various characteristic functionals which characterize the noise and a functional fractional fokker planck equation for the probability density functional of the noise at a given moment in time second by making an analogy between the theory of colored noise and the random channel approach to disordered kinetics we derive fractional equations for the evolution of the probability densities of the random rate coefficients in disordered kinetics these equations serve as a basis for developing methods for the evaluation of the statistical properties of the random rate coefficients from experimental data special attention is paid to the analysis of systems for which the observed kinetic curves can be described by linear or nonlinear stretched exponential kinetics

translational flows of an oldroyd b fluid with fractional derivatives the objective of this paper is to study the unsteady flow of an oldroyd b fluid with fractional derivative model between two infinite coaxial circular cylinders using laplace and finite hankel transforms the motion of the fluid is produced by the inner cylinder that at time t 0 applies a time dependent longitudinal shear stress to the fluid velocity field and the adequate shear stress are presented in series form in terms of the generalized g and r functions the solutions that have been obtained satisfy all imposed initial and boundary conditions the corresponding solutions for ordinary oldroyd b fractional maxwell ordinary maxwell fractional second grade ordinary second grade and newtonian fluids performing the same motion are obtained as limiting cases of general solutions in particular the existing solutions for ordinary oldroyd b and second grade fluids are compared with the present solutions finally the influence of the pertinent parameters on the fluid motion as well as a comparison between models is underlined by graphical illustrations

a fractional order maxwell model for non newtonian fluids abstract this work considers an extension of the fractional order maxwell arrangement to incorporate a relaxation process with non newtonian viscosity behavior the resulting model becomes a fractional order nonlinear differential equation with stable solution converging asymptotically to a unique equilibrium point expressions for the corresponding storage and loss moduli as function of strain frequency and amplitude are computed via a first harmonic analysis of the differential equation some distinctive features and their relationship to the classical and fractional order linear maxwell models are discussed three examples are used to illustrate the ability of the fractional order maxwell model to describe experimental data

differential geometry of viscoelastic models with fractional order derivatives viscoelastic materials with memory effect are studied based on the fractional rheonomic geometry the geometric objects are regarded as basic quantities of fractional viscoelastic models i e the metric tensor and torsion tensor are interpreted as the strain and the fractional strain rate respectively the generalized viscoelastic equations are expressed by the geometric objects especially the basic constitutive equations such as voigt and maxwell models can be derived geometrically from the generalized equation this leads to the fact that various viscoelastic models can be unified into one geometric expression

higher operations in string topology of classifying spaces examples of non trivial higher string topology operations have been regrettably rare in the literature in this paper working in the context of string topology of classifying spaces we provide explicit calculations of a wealth of non trivial higher string topology operations associated to a number of different lie groups as an application of these calculations we obtain an abundance of interesting homology classes in the twisted homology groups of automorphism groups of free groups the ordinary homology groups of holomorphs of free groups and the ordinary homology groups of affine groups over the integers and the field of two elements

multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions abstract in this paper we study the existence of multiple solutions for the integral boundary value problems of fractional differential equations by the method of upper and lower solutions and leray schauder degree theory the sufficient conditions about the existence of at least three solutions are obtained moreover it is proved that the integral boundary value problem has at least three positive solutions under the conditions of m 0 and f is nonnegative by given two upper and lower solutions which can be easily obtained through our methods we can present the existence theorem of at least three solutions two examples are also included to illustrate the effectiveness of the proposed results

on the relationship among three theories of relaxation in disordered systems abstract empirical evidence has accumulated showing the wide occurrence of a stretched exponential relaxation decay law for many diverse condensed matter systems several theories based on different physical mechanisms have been successful in deriving the stretched exponential decay law in a natural way three of these theories the direct transfer model the hierarchically constrained dynamics model and the defect diffusion model are shown here to have an underlying common mathematical structure

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

the homotopy type of the cobordism category the embedded cobordism category under study in this paper generalizes the category of conformal surfaces introduced by g segal in s2 in order to formalize the concept of field theories our main result identifies the homotopy type of the classifying space of the embedded d dimensional cobordism category for all d for d 2 our results lead to a new proof of the generalized mumford conjecture somewhat different in spirit from the original one presented in mw

fractal ladder models and power law wave equations extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power law dependence on frequency to describe this power law behavior in tissue a hierarchical fractal network model is proposed the viscoelastic and self similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite ladder topology involving alternating springs and dashpots in the low frequency limit this ladder network yields a stress strain constitutive equation with a time fractional derivative by combining this constitutive equation with linearized conservation principles and an adiabatic equation of state the fractional partial differential equations previously proposed by m caputo linear models of dissipation whose q is almost frequency independent ii geophys j r astron soc 13 529 539 1967 and mg wismer finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation j acoust soc am 120 3493 3502 2006 are derived the resulting attenuation coefficient is a power law with exponents ranging between 1 and 2 while the phase velocity is in agreement with the kramers kronig relations the power law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry

homological stability for the mapping class groups of non orientable surfaces we prove that the homology of the mapping class groups of non orientable surfaces stabilizes with the genus of the surface combining our result with recent work of madsen and weiss we obtain that the classifying space of the stable mapping class group of non orientable surfaces up to homology isomorphism is the infinite loop space of a thom spectrum built from the canonical bundle over the grassmannians of 2 planes in n 2 in particular we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i this is the non oriented analogue of the mumford conjecture

the free vibration of a type of tapered beams this paper studies the vibrational characteristics of tapered beams with continuously varying rectangular cross section of depth and breadth proportional to xs and xt respectively where both s and t are arbitrary real numbers for a truncated beam and arbitrary positive numbers for a sharp ended beam and x is the axial co ordinate measured from the sharp end of the beam the bernoulli euler theory of bending is used to describe the motion of the beam a new set of beam functions are developed as the admissible functions which are the complete solution of the tapered beam under an arbitrary static load expanded into a taylor series the eigenfrequency equation is obtained by the rayleigh ritz method the accuracy is assured from the convergency and comparison studies the effect of the location of the taylor series expanding point on the convergency is discussed the analysis shows that the present approach is convergent for arbitrary truncation factor by taking the midpoint of the beam as the expanding point of the taylor series numerical results are tabulated for three different tapered beams with various boundary conditions and truncation factors it is shown that the eigenfrequencies can be obtained with high accuracy by using only a small number of terms of the static beam functions 2000 elsevier science s a all rights reserved

free vibration analysis of a type of tapered beams by using adomian decomposition method abstract the adomian decomposition method adm is employed in this paper to investigate the free vibrations of tapered euler bernoulli beams with a continuously exponential variation of width and a constant thickness along the length under various boundary conditions based on adm the governing differential equation for the tapered beam becomes a recursive algebraic equation by using the boundary condition equations the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously the computed results for different boundary conditions and non uniformity ratios are presented the accuracy is assured from the convergence and comparison published results

a stable category of lagrangian cobordisms abstract given an exact symplectic manifold m and a support lagrangian m we construct an category lag m which we conjecture to be equivalent after specialization of the coefficients to the partially wrapped fukaya category of m relative to roughly speaking the objects of lag m are lagrangian branes inside of m t r n for large n and the morphisms are lagrangian cobordisms that are non characteristic with respect to the main theorem of this paper is that lag m is a stable category and in particular its homotopy category is triangulated with mapping cones given by an elementary construction the shift functor is equivalent to the familiar shift of grading for lagrangian branes

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

mass action law versus local contagion dynamics a mean field statistical approach with application to the theory of epidemics a mean field approach for epidemic processes with high migration is suggested by analogy with non equilibrium statistical mechanics for large systems a limit of the thermodynamic type is introduced for which both the total size of the system and the total number of individuals tend to infinity but the population density remains constant in the thermodynamic limit the infection rate is proportional to the product of the proportion of individuals susceptible to infection and the average probability of infection the limit form of the average probability of infection is insensitive to the detailed behaviour of the fluctuations of the number of infectious individuals and may belong to two universality classes 1 if the fluctuation of the number of infectives is non intermittent it increases with the increase of the partial density of infectives and approaches exponentially the asymptotic value one for large densities 2 for intermittent fluctuations obeying a power law scaling the average probability of infection also displays a saturation effect for large densities of infectives but the asymptotic value one is approached according to a power law rather than exponentially for low densities of infectives both expressions for the average probability of infection are linear functions of the proportion of infectives and the infection rate is given by the mass action law

an interpolation between the wave and diffusion equations through the fractional evolution equations dirac like through fractional calculus and following the method used by dirac to obtain his well known equation from the klein gordon equation we analyze a possible interpolation between the dirac and the diffusion equations in one space dimension we study the transition between the hyperbolic and parabolic behaviors by means of the generalization of the d alembert formula for the classical wave equation and the invariance under space and time inversions of the interpolating fractional evolution equations dirac like such invariance depends on the values of the fractional index and is related to the nonlocal property of the time fractional differential operator for this system of fractional evolution equations we also find an associated conserved quantity analogous to the hamiltonian for the classical dirac case

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

invariant analysis and conservation laws for the time fractional 2 1 dimensional zakharov kuznetsov modified equal width equation using lie group analysis abstract in this paper the invariance properties of the time fractional 2 1 dimensional zakharov kuznetsov modified equal width zk mew equation have been investigated using the lie group analysis method lie point symmetries of the time fractional 2 1 dimensional zk mew equation have been derived by using the lie group analysis method of fractional differential equations using the lie symmetry analysis the vector fields and the symmetry reduction of this equation are obtained it is shown that the time fractional 2 1 dimensional zk mew equation can be transformed to an equation with erdelyi kober fractional derivative finally using new conservation theorem with formal lagrangian the new conserved vectors are well constructed with a detailed derivation which constitutes the conservation analysis for the time fractional 2 1 dimensional zk mew equation

response theory for random channel kinetics in complex systems application to lifetime distributions of active intermediates we present a response theory for random channel kinetics based on the method of characteristic functionals we assume that the uctuations of the rate coecients are generated by the contributions of a large number of reaction channels and can be described in terms of a set of grand canonical probability density functionals we consider linear as well as nonlinear responses for systems with static or dynamic disorder for all cases considered we derive analytical expressions for the ensemble average of the response of the system to an external perturbation the theory is applied for developing a new approach to the analysis of the kinetics in complex chemical systems based on a random channel approach we assume that the total transition rate from one state of the system to another is made up of the contributions of a large number of random reaction pathways which can be described in terms of a stochastic point process we study the response of such a system to an external perturbation and show that it can be expressed in terms of a set of lifetime distributions of reaction intermediates we examine the possibilities of applying the technique to the study of large chemical systems by means of tracer experiments and suggest methods for computing rate distributions from measured lifetime distributions we generalize the method of spectral kinetic analysis for the interpretation of measured lifetime distributions we investigate the relationships between spectral kinetics analysis and random channel statistics c 2000 published by elsevier science b v all rights reserved

nonequilibrium fluctuation dissipation relations for independent random rate processes with dynamical disorder a class of rate processes with dynamical disorder is investigated based on the two following assumptions a the system is composed of a random number of particles or quasiparticles which decay according to a first order kinetic law b the rate coefficient of the process is a random function of time with known stochastic properties the formalism of characteristic functionals is used for the direct computation of the dynamical averages the suggested approach is more general than the other approaches used in the literature it is not limited to a particular type of stochastic process and can be applied to any type of random evolution of the rate coefficient we derive an infinity of exact fluctuation dissipation relations which establish connections among the moments of the survival function and the moments of the number of surviving particles the analysis of these fluctuation dissipation relations leads to the unexpected result that in the thermodynamic limit the fluctuations of the number of partic

a fractional order model on new experiments of linear viscoelastic creep of hami melon this paper describes our experimental testing of linear viscoelastic creep behaviours in hami melon experimental data shows that hami melon has complex viscoelastic property which cannot be well described by the standard model consequently this study develops a fractional derivative model to describe such complex viscoelastic creep behaviours of hami melon the analytical creep function of the proposed fractional linear viscoelastic models is derived via the boltzmann superposition principle and discrete inverse laplace transform and then such analytical solutions are used to fit the experimental viscoelastic data of hami melon our study shows that the present fractional linear viscoelastic model with merely three parameters is more efficient and accurate than the generalised kelvin viscoelastic model of six parameters to describe the stress strain constitutive relations of hami melon it is noted that the present fractional model with adjustable parameters can also be used to describe creep damage

application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives two nonlinear anelastic models with fractional derivatives describing the properties of a series of materials as polymers and polycrystalline materials are presented in this paper these models are studied analytically using a variational iteration method the paper clarifies the different ways in which the fractional differentiation operator can be defined a volterra series method of model parameters identification from the experimental data is also presented

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

l1 fourier spectral methods for a class of generalized two dimensional time fractional nonlinear anomalous diffusion equations abstract in this paper l1 fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two dimensional time fractional nonlinear anomalous diffusion equations involving caputo fractional derivative firstly we establish the l1 fourier galerkin full discrete and l1 fourier collocation schemes with fourier spectral discretization in spatial direction and l1 difference method in temporal direction secondly stability and convergence for both galerkin and collocation approximations are proved it is shown that the proposed methods are convergent with spectral accuracy in space and 2 order accuracy in time for implementation the equivalence between pseudospectral method and collocation method is discussed furthermore a numerical algorithm of fourier pseudospectral method is developed based on two dimensional fast fourier transform fft2 technique finally numerical examples are provided to test the theoretical claims as is shown in the numerical experiments fourier spectral methods are powerful enough with excellent efficiency and accuracy

fractional diffusion exact representations of spectral functions for all the relevant transformed spaces i e fourier laplace and fourier laplace we present exact solutions of a fractional diffusion equation describing random transport on fractals the potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes in contrast to the well known asymptotic results the exact representations provide a much broader basis for comparison with data

justification for power laws and fractional models in previous chapters it has been shown that the standard linear viscoelastic models can be generalized to their fractional counterparts it has also been demonstrated that the fractional zener kelvin voigt maxwell and newtonian models all satisfy physical criteria such as passivity and causality these models have temporal responses that are characterized by power laws either exactly or asymptotically and this corresponds to observed temporal characteristics in many media the character of the fractional derivative operator is also such that it describes temporal memory in the media

vibration in the two principal planes of a non uniform beam of rectangular cross section one side of which varies as the square root of the axial co ordinate abstract the transverse vibration of a type of slender beam of continuously varying rectangular cross section is considered in this paper one side of the cross section is constant while the other side is proportional to the square root of the axial co ordinate it is assumed that the beam dimensions are such that application of the euler bernoulli theory of bending is valid thus the vibration in the two principal planes will be uncoupled the two mode shape equations are fourth order linear differential equations with variable coefficients and with regular singularity analytical solutions of the two mode shape equations are derived based on the method of frobenius combinations of ideal clamped pinned sliding and free boundary conditions are considered the sharp end of a complete beam cannot sustain a bending moment or shearing force and hence must always remain free the large end may be clamped pinned sliding or free the frequency equations for a complete beam is the determinant of a 2 2 matrix equated to zero for a truncated beam any combinations of the four boundary conditions is possible and the frequency equations are the determinant of a 4 4 matrix equated to zero the first three dimensionless natural frequencies for vibration in the two principal planes are tabulated for 16 combinations of the four boundary conditions and truncation factors from 0 05 to 0 7 to preserve the accuracy of the frequencies to six figures the results are presented in tabular form it is hoped that they will serve as benchmarks

unbounded complex modulus of viscoelastic materials and the kramers kronig relations abstract the kramers kronig k k dispersion relations developed for the complex modulus of elasticity of solid viscoelastic materials connect the frequency dependences of the dynamic modulus and loss modulus whether the boundedness of the complex modulus at high infinite frequency is required or not for the applicability of the k k relations is investigated in this paper the derivation of the k k relations developed for the complex modulus is presented by examining the physical background of the relations it is shown that the k k relations can be applied even if the complex modulus is unbounded at high frequencies the fractional derivative kelvin model is used to demonstrate the application of the k k relations for a class of unbounded complex modulus

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

stochastic renormalization group approach to space dependent supercritical branched chain processes a model for physicochemical and biological space dependent branched chain processes is suggested based on the assumption that the number and the positions of the direct offspring of a particle are random variables characterized by a given point process the generating functional of the random point process characterizing the number and the position of the total offspring from a given generation may be computed by repeated iteration of the generating functional attached to the offspring of a particle a stochastic renormalization transformation is introduced by assuming that for each generation there is a constant probability that the growth of the population stops and that the actual state of the population is an average of the contributions of different generations a detailed analysis is performed for supercritical self similar branching processes for which the distance of a particle from its direct ancestor is a random variable selected from a symmetric jump probability density in this case all janossy and product densities of the population can be computed exactly the fluctuations of particle concentrations for a given generation have an intermittent behavior the asymptotic behavior of the total number n of particles is independent of the spatial distribution the probabilty of n has a long tail of the inverse power law type modulated by a periodic function of lnn in contrast the spatial distribution of particles depends both on the jump probability density and on the total number of particles if the moments of the jump probability density are finite then the positions mathbf r 1 mathbf r mathit n of n particles are gaussian random variables with a correlation matrix obeying a logarithmic scaling law mathbf r mathit i ensuremath cdot mathbf r mathit j ensuremath sim mathrm ensuremath delta mathit i mathit j lnn as n ensuremath rightarrow ensuremath infty if the jump probability density has a long tail then the probability density of mathbf r 1 mathbf r mathit n for n ensuremath rightarrow ensuremath infty is given by a separable l evy law with a dimensional parameter given by a second scaling law b ensuremath sim lnn 1 mathrm ensuremath varepsilon as n ensuremath rightarrow ensuremath infty where ensuremath varepsilon is the fractal exponent of the l evy law

axioms for bundle transfers and traces in topology it is often profitable to characterize the algebraic invariants of spaces or morphisms by axioms of course the most basic examples along these lines are the eilenberg steenrod axioms for homology and cohomology eis and in many cases there are axiomatic characterizations for more refined algebraic structures that exist in homology and cohomology for example axioms for the steenrod cohomology operations appear in eps and axioms for various sorts of transfers known in the nineteen sixties were described in unpublished work of j m boardman bo in this paper we shall describe axioms for the bundle transfers considered by the first named author and d h gottlieb in bg2 3 and from a different perspective by a dold m clapp and d puppe in cl d1 3 dp previous work of f roush rou and l g lewis l describes axioms for the special case of finite coverings and bundles whose fibers are compact lie groups that act smoothly on the fibers another axiomatic characterization of finite covering transfers appears in bes sect 13 the characterization in this paper is valid for topological fiber bundles for which the fibers and the bases are finite dimensional compact anr s and finite cw complexes respectively we shall also describe axioms for the homomorphisms in generalized homology and cohomology theories that are determined by bundle transfers

hierarchical analogues to fractional relaxation equations recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena such as stress strain relationships in polymeric materials in the rheological constitutive equations this implies the replacement of the first order derivatives by fractional derivatives here we show that such procedures have hierarchical mechanical analogues we focus on the generalized dashpot and the generalized maxwell model and display the corresponding arrangements our models allow a transparent interpretation of the parameters which enter the fractional equations and reveal that the internal dynamics are hierarchically constrained

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

flutter analysis of a flag of fractional viscoelastic material we develop a two dimensional model to study the effects of the material viscoelasticity on the dynamics of a flag in flow two periodic states of an elastic flag are firstly identified with different dimensionless bending stiffness a lower frequency state and a higher frequency state the scott blair model and the fractional kelvin voigt model are further used to represent the viscoelasticity of the flag material when the scott blair model is used with the increase of the fractional derivative order the flag flapping frequency of the higher frequency state decreases abruptly and that of the lower frequency state also shows a downward trend when the system parameters are in a certain range an interesting phenomenon is observed where the time needed to achieve the periodic steady state initially increases and then decreases with increasing the phenomenon implies that the flag has a higher energy harvesting speed when approaches 1 when the fractional kelvin voigt model is used the increasing also causes the transition from the higher frequency state to the lower frequency state and quasi periodic states are observed during the transition the fractional kelvin voigt type viscoelasticity produces complex effects on the lower frequency state

on the dynamic behaviour of polymers under finite strains constitutive modelling and identification of parameters abstract in this paper we apply a model of finite viscoelasticity and propose an identification technique to represent the dynamic properties of polymers the model is based on a multiplicative split of the deformation gradient into a thermal and a mechanical part the latter being decomposed further into elastic and viscous parts in order to formulate the constitutive equations we transfer the concept of discrete relaxation spectra to finite strains specify the free energy as a function of elastic strain tensors and evaluate the dissipation principle of thermodynamics in the form of the clausius duhem inequality then we investigate the dynamic moduli of a polyethylene melt under harmonic shear deformations and determine the material parameters to this end we linearise the constitutive model and calculate the analytical solution of the evolution equations in addition we formulate a second model which represents the experimental data on the basis of a fairly small number of fitting parameters this so called substitute model is based on the fractional calculus and corresponds to a continuous relaxation spectrum in order to identify the material constants of the finite strain model we are looking for we proceed as follows we determine the parameters of the substitute model calculate the so called cumulative relaxation spectrum and approximate it by means of a series of step functions the height of the steps corresponds to the stiffness parameters of the finite strain model and their locations to the inverse relaxation times

fractional differential equations and related exact mechanical models the aim of the paper is the description of fractional order differential equations in terms of exact mechanical models this result will be archived in the paper for the case of linear multiphase fractional hereditariness involving linear combinations of power laws in relaxation creep functions the mechanical model corresponding to fractional order differential equations is the extension of a recently introduced exact mechanical representation di paola and zingales 2012 33 and di paola et al 2012 34 of fractional order integrals and derivatives some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots

spaces of graphs and surfaces on the work of soren galatius galatius most striking result is easy enough to state let n be the symmetric group on n letters and fn be the free non abelian group with n generators the symmetric group n acts naturally by permutation on the n generators of fn and every permutation gives thus rise to an automorphism of the free group galatius proves that the map n autfn in homology induces an isomorphism in degrees less than n 1 2 the homology of the symmetric groups in these ranges is well understood in particular in common with all finite groups it has no non trivial rational homology by galatius theorem in low degrees this then is also true for autfn h autfn q 0 for 0 2n 3

parameter estimation for the generalized fractional element network zener model based on the bayesian method abstract in the present study an inverse problem to estimate parameters in the zener model of viscoelasticity based on the generalized fractional element gfe network is studied the bayesian method is proposed to obtain the optimal estimation of the viscoelastic parameters three examples are performed to certify the validity of the method all numerical results lead to an excellent fitting between the calculative results and experimental data it is shown that the bayesian method is feasible in the inverse problem of parameter estimation for the fractional constitutive equation and the gfe network zener model is efficient in the modeling of the viscoelastic behavior

anomalous diffusion modeling by fractal and fractional derivatives this paper makes an attempt to develop a fractal derivative model of anomalous diffusion we also derive the fundamental solution of the fractal derivative equation for anomalous diffusion which characterizes a clear power law this new model is compared with the corresponding fractional derivative model in terms of computational efficiency diffusion velocity and heavy tail property the merits and distinctions of these two models of anomalous diffusion are then summarized

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

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

generating functional approach to multichannel parallel relaxation with application to the problem of direct energy transfer in fractal systems with dynamic disorder a model for multichannel parallel relaxation is suggested based on the following assumptions a an individual channel is characterized by a set of continuous state variables the corresponding relaxation rate is a function of the state variables as well as of the time interval for which the channel is open b the number of channels is a random variable described by a correlated point process defined in the space of state parameters of an individual channel analytical expressions for the generating functional of the overall relaxation rate and for the average survival function are derived in terms of the generating functional of the point process the general formalism is applied to the problem of direct energy transfer from excited donors to acceptors in fractal systems with dynamic disorder it is assumed that the number of acceptors obeys a poissonian distribution law with a constant average density in a df dimensional fractal structure embedded in a ds dimensional euclidean space ds 1 2 3 and th

time domain analysis of the fractional order weighted distributed parameter maxwell model in this paper we derive in time domain the fundamental solution and relevant properties of the fractional order weighted distributed parameter maxwell model fowdpmm the weight function is replaced by the truncated fourier series which is leading to three basic fractional order distributed parameter elements the inverse laplace transforms of the distributed parameter operators are derived by cutting the complex plane and computing the complex path integral along the hankel path the asymptotic property and boundary problem are discussed by using the inverse laplace transform the energy of the weight function and the band width of the fourier series the relaxation modulus of fowdpmm is solved as well which is closely related to some other viscoelastic phenomena as creep precondition and hysteresis a number of novel characteristics of fowdpmm such as power law decay and intermediate phenomenon are concluded as well several illustrated examples are provided to validate the concepts

universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder dynamical generalizations of stretched exponential the asymptotic behavior of multichannel parallel relaxation processes for systems with dynamical disorder is investigated in the limit of a very large number of channels an individual channel is characterized by a state vector x which due to dynamical disorder is a random function of time a limit of the thermodynamic type in the x space is introduced for which both the volume available and the average number of channels tend to infinity but the average volume density of channels remains constant scaling arguments combined with a stochastic renormalization group approach lead to the identification of two different types of universal behavior of the relaxation function corresponding to nonintermittent and intermittent fluctuations respectively for nonintermittent fluctuations a dynamical generalization of the static huber s relaxation equation is derived which depends only on the average functional density of channels w t d w t the channels being classified according to their different relaxation rates w w t which are random functions of time for intermittent fluctuations a more complicated relaxation equation is derived which in addition to the average density of channels w t d w t depends also on a positive fractal exponent h which characterizes the fluctuations of the density of channels the general theory is applied for constructing dynamical analogs of the stretched exponential relaxation function for nonintermittent fluctuations the type of relaxation is determined by the regression dynamics of the fluctuations of the relaxation rate if the regression process is fast and described by an exponential attenuation function then after an initial stretched exponential behavior the relaxation process slows down and it is not fully completed even in the limit of very large times for self similar regression obeying a negative power law the relaxation process is less sensitive to the influence of dynamical disorder both for small and large times the relaxation process is described by stretched exponentials with the same fractal exponent as for systems with static disorder for large times the efficiency of the relaxation process is also slowed down by fluctuations similar patterns are found for intermittent fluctuations with the difference that for very large times and a slow regression process a crossover from a stretched exponential to a self similar algebraic relaxation function occurs some implications of the results for the study of relaxation processes in

the stable moduli space of riemann surfaces mumford s conjecture d mumford conjectured in 30 that the rational cohomology of the stable moduli space of riemann surfaces is a polynomial algebra generated by certain classes i of di mension 2i for the purpose of calculating rational cohomology one may replace the stable moduli space of riemann surfaces by b 1 where 1 is the group of isotopy classes of automorphisms of a smooth oriented connected surface of large genus tillmann s insight 41 that the plus construction makes b 1 into an infinite loop space led to a stable homo topy version of mumford s conjecture stronger than the original 22 we prove the stronger version relying on harer s stability theorem 15 vassiliev s theorem concerning spaces of functions with moderate singularities 43 42 and methods from homotopy theory

bi asymptotic fractals fractals between lower and upper bounds extending the idea of rigaut s asymptotic fractals issuing a turnover from a constant to a power law behaviour towards smaller scales we extend this idea to asymptotic fractals with both lower and upper turnover points i e the fractal region is terminated towards larger scales by another constant value this behaviour is typical for natural fractals such biological cell boundaries here we present a new analytic function describing this bi asymptotic behaviour the introduced parameters can be directly interpreted we show the advantage of this function in fitting processed images of natural and mathematical fractals in comparison with standard procedures the determined dimension is significantly more accurate a program package with this new function is available

on levy or stable distributions and the williams watts model of dielectric relaxation this paper is concerned with the levy or stable distribution function defined by the fourier transform q alpha left z right frac 1 2 pi int infty infty exp left izu left u right alpha right du with 0 alpha leqslant 2 when 2 it becomes the gauss distribution function and when 1 the cauchy distribution when 2 the distribution has a long inverse power tail q alpha left z right sim frac gamma left 1 alpha right sin tfrac 1 2 pi alpha pi left z right 1 alpha in the regime of small if logz 1 the distribution is mimicked by a log normal distribution we have derived rapidly converging algorithms for the numerical calculation ofq z for various in the range 0 1 the functionq z appears naturally in the williams watts model of dielectric relaxation in that model one expresses the normalized dielectric parameter as in n left omega right equiv in n left omega right i in n left omega right int 0 infty e i omega t left d phi left t right mathord left vphantom d phi left t right dt right kern nulldelimiterspace dt right dt with phi left t right exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha it has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form of t e n is shown to be directly related toq z it is also shown that if the williams watts exponential is expressed as a weighted average of exponential relaxation functions exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha int 0 infty g left lambda alpha right e lambda t dt the weight functiong is expressible as a stable distribution some suggestions are made about physical models that might lead to the williams watts form of t

analogs of renormalization group transformations in random processes abstract we review the properties of a real space renormalization group transformation of the free energy including the existence of oscillatory terms multiplying the non analytic part of the free energy we then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation the essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary these random processes are shown to be generated directly from cantor sets in a spatial representation the ensuing random process exhibits a transition between gaussian and fractal behavior in the fractal regime the trajectories will in an average sense form self similar clusters in a temporal representation the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior the fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses

a fractional model with parallel fractional maxwell elements for amorphous thermoplastics abstract we develop a fractional model to describe the thermomechanical behavior of amorphous thermoplastics the fractional model is composed of two parallel fractional maxwell elements the first fractional maxwell model is used to describe the glass transition while the second component is aimed at describing the viscous flow we further derive the analytical solutions for the stress relaxation modulus and complex modulus through laplace transform we then demonstrate the model is able to describe the master curves of the stress relaxation modulus storage modulus and loss modulus which all show two distinct transition regions the obtained parameters show that the modulus of the two fractional maxwell elements differs in 2 3 orders of magnitude while the relaxation time differs in 7 9 orders of magnitude finally we apply the model to describe the stress response of constant strain rate tests the model together with the parameters obtained from fitting the master curve of stress relaxation modulus can accurately predict the temperature and strain rate dependent stress response

the stable mapping class group and q ℂp in t2 it was shown that the classifying space of the stable mapping class groups after plus construction b has an infinite loop space structure this result and the tools developed in bm to analyse transfer maps are used here to show the following splitting theorem let p p e 0 e p 2 be the adams splitting of the p completed suspension spectrum of p then for some infinite loop space w p b p e 0 e p 3 w p where e i denotes the infinite loop space associated to the spectrum e i the homology of e i is known and as a corollary one obtains large families of torsion classes in the homology of the stable mapping class group this splitting also detects all the miller morita mumford classes our results suggest a homotopy theoretic refinement of the mumford conjecture the above p adic splitting uses a certain infinite loop map b p 1 that induces an isomorphims in rational cohomology precisely if the mumford conjecture is true we suggest that might be a homotopy equivalence

fractional visco elastic timoshenko beam deflection via single equation summary this paper deals with the response determination of a visco elastic timoshenko beam under static loading condition and taking into account fractional calculus in particular the fractional derivative terms arise from representing constitutive behavior of the visco elastic material further taking advantages of the mellin transform method recently developed for the solution of fractional differential equation the problem of fractional timoshenko beam model is assessed in time domain without invoking the laplace transforms as usual further solution provided by the mellin transform procedure will be compared with classical central difference scheme one based on the grunwald letnikov approximation of the fractional derivative moreover timoshenko beam response is generally evaluated by solving a couple of differential equations in this paper expressing the equation of the elastic curve just through a single relation a more general procedure which allows the determination of the beam response for any load condition and type of constraints is developed copyright 2015 john wiley sons ltd

fractional differential forms a generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders that is a fractional exterior derivative is defined this is found to generate new vector spaces of finite and infinite dimension fractional differential form spaces the definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case coordinate transformation rules are also computed the transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative the metric for the fractional form spaces is given based on the coordinate transformation rules all results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one

geometry of surfaces with caputo fractional derivatives and applications to incompressible two dimensional flows geometric structures of surfaces are formulated based on caputo fractional derivatives the gauss frame of a surface with fractional order is introduced then the non locality of the fractional derivative characterizes the asymmetric second fundamental form the mean and gaussian curvatures of the surface are defined in the case of fractional order based on the fractional curvatures incompressible two dimensional flows are discussed the stream functions are obtained from a fractional continuity equation the asymmetric second fundamental form of stream function surface is related to the path dependence of flux moreover the fractional curvatures are calculated for the stream function surfaces of uniform and corner flows the uniform flow with fractional order is characterized by the non vanishing mean curvature the non locality of corner flow is expressed by the mean and gaussian curvatures with fractional order in particular the fractional order within the stream function of corner flow determines the change of sign of gaussian curvature therefore the non local property of incompressible flows can be investigated by the fractional curvatures

fractional derivative modeling for axisymmetric consolidation of multilayered cross anisotropic viscoelastic porous media abstract this paper presents a fractional derivative modeling for axisymmetric consolidation of multilayered cross anisotropic viscoelastic porous media the elastic solution for multilayered porous media is acquired by the extended precise integration method the flexibility coefficient of the fractional merchant viscoelastic model is deduced in the laplace transformed domain and the viscoelastic solution of the problem is further obtained by the elastic viscoelastic correspondence principle the numerical results are compared with those of published literatures to verify the proposed method and the influences of fractional derivative order viscoelasticity of solid skeleton and layering of materials on the axisymmetric consolidation of multilayered viscoelastic porous media are investigated by several examples

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

fractional systems and fractional bogoliubov hierarchy equations we consider the fractional generalizations of the phase volume volume element and poisson brackets these generalizations lead us to the fractional analog of the phase space we consider systems on this fractional phase space and fractional analogs of the hamilton equations the fractional generalization of the average value is suggested the fractional analogs of the bogoliubov hierarchy equations are derived from the fractional liouville equation we define the fractional reduced distribution functions the fractional analogs of the vlasov equation and the debye radius are considered

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

on the edge of the stable range we prove a general homological stability theorem for certain families of groups equipped with product maps followed by two theorems of a new kind that give information about the last two homology groups outside the stable range these last two unstable groups are the edge in our title applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range a description of the last unstable group up to a single ambiguity and a lower bound on the rank of the penultimate unstable group we give similar applications to the general linear groups of the integers and of the field of order 2 this time recovering the known stability range the results can also be applied to general linear groups of arbitrary principal ideal domains symmetric groups and braid groups our methods require us to use field coefficients throughout

a renormalization approach to a class of exponential random processes with application to the bronchial tree abstract a new stochastic renormalization approach for multi step decay phenomena is developed a simple physical interpretation of the renormalization method is suggested it consists in grouping successions of variable numbers of decay events into blocks the law of probability multiplication leads to an exponential random process y yx1 x2 0 where both the exponents x1 x2 and the basis y0 are random variables the physically consistent solution of the model corresponds to a super strong renormalization regime the dependence of the moments of the decay parameter on the number of decay steps q can be exactly determined it consists in a linear superposition of inverse power laws in q modulated by periodic functions in in q having different periods the theory is applied to the renormalization of the bronchial tree our computation shows that the lung structure is very tolerant to fluctuations this result supports the mechanism of morphogenesis of fractal biological organs suggested by west ann biomed eng 18 1990 135 the possibilities of application to the scattering phenomena in one dimensional disordered media are also investigated