variable order fractional differential operators in anomalous diffusion modeling abstract the purpose of this paper is to offer a unified discussion of variable order differential operators in anomalous diffusion modeling the characteristics of the new models in contrast to constant order fractional diffusion models change with time space concentration or other independent quantities we introduced a classification of variable order fractional diffusion models based on the possible physical origins which prompt the variable order some potential applications of the variable order fractional diffusion models are also discussed

explicit combinatorial interpretation of kerov character polynomials as numbers of permutation factorizations abstract we find an explicit combinatorial interpretation of the coefficients of kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups s n in terms of free cumulants r 2 r 3 of the corresponding young diagram our interpretation is based on counting certain factorizations of a given permutation

fractional derivative for interpolation in rn and so n applications in functionally graded materials and rigid body transformations abstract an integration of fractional calculus data interpolation and lie group leads to new treatments of interpolation problems a local and centered fractional derivative is introduced which enables the fractional interpolant to create a c smooth r curve by extending integer order cost functions to fractional ones the fractional interpolant fills the gaps between the traditional linear interpolating segments cubic splines and higher order splines interpolating rigid body transformations has enormous applications in mechanics computer graphics and robotics previous approaches have provided the re parametrization methods the recursive methods e g de casteljau s algorithm and the gradient methods to approximate the correct interpolating curve on so 3 a closed form model that merges the tangent spaces of knots is proposed the model first constructs a family of candidate curves and then combines them into an improved approximation applications in modeling functionally graded materials and rigid body transformations are discussed

explicit and averaging a posteriori error estimates for adaptive finite volume methods local mesh refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady state convection diffusion reaction equations the paper establishes residual type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global h1 and l2 norms reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh refining algorithms over uniform mesh refining a discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper

lyapunov functions for riemann liouville like fractional difference equations abstract discrete memory effects are introduced by fractional difference operators asymptotic stabilities of nonlinear fractional difference equations are investigated in this paper a linear scalar fractional difference equality is utilized lyapunov second direct method is proposed for nonlinear discrete fractional systems asymptotic stability conditions are provided and some examples are given

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

analyzing signal attenuation in pfg anomalous diffusion via a modified gaussian phase distribution approximation based on fractal derivative model abstract pulsed field gradient pfg technique is a noninvasive tool and has been increasingly employed to study anomalous diffusions in nuclear magnetic resonance nmr and magnetic resonance imaging mri however the analysis of pfg anomalous diffusion is much more complicated than normal diffusion in this paper a fractal derivative model based modified gaussian phase distribution method is proposed to describe pfg anomalous diffusion by using the phase distribution obtained from the effective phase shift diffusion method based on fractal derivatives and employing some of the traditional gaussian phase distribution approximation techniques a general signal attenuation expression for free fractional diffusion is derived this expression describes a stretched exponential function based attenuation which is distinct from both the exponential attenuation for normal diffusion obtained from conventional gaussian phase distribution approximation and the mittag leffler function based attenuation for anomalous diffusion obtained from fractional derivative the obtained signal attenuation expression can analyze the finite gradient pulse width fgpw effect additionally it can generally be applied to all three types of pfg fractional diffusions classified based on time derivative order and space derivative order these three types of fractional diffusions include time fractional diffusion with 0 2 2 space fractional diffusion with 1 0 2 and general fractional diffusion with 0 2 the results in this paper are consistent with reported results based on effective phase shift diffusion equation method and instantaneous signal attenuation method this method provides a new convenient approximation formalism for analyzing pfg anomalous diffusion experiments the expression that can simultaneously interpret general fractional diffusion and fgpw effect could be especially important in pfg mri where the narrow gradient pulse limit cannot be satisfied

random generators of the symmetric group diameter mixing time and spectral gap abstract let g h be a random pair of generators of g sym n or g alt n we show that with probability tending to 1 as n a the diameter of g with respect to s g h g 1 h 1 is at most o n 2 log n c and b the mixing time of g with respect to s is at most o n 3 log n c both c and the implied constants are absolute these bounds are far lower than the strongest worst case bounds known in helfgott seress 2013 they roughly match the worst known examples we also give an improved though still non constant bound on the spectral gap our results rest on a combination of the algorithm in babai beals seress 2004 and the fact that the action of a pair of random permutations is almost certain to act as an expander on l tuples where l is an arbitrary constant friedman et al 1998

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

fractional noether s theorem in the riesz caputo sense abstract we prove a noether s theorem for fractional variational problems with riesz caputo derivatives both lagrangian and hamiltonian formulations are obtained illustrative examples in the fractional context of the calculus of variations and optimal control are given

analysis of reaction diffusion system via a new fractional derivative with non singular kernel abstract in this paper we obtain analytical solutions for the fractional cubic isothermal auto catalytic chemical system with caputo fabrizio and atangana baleanu fractional time derivatives in liouville caputo sense we utilize the q homotopy analysis transform method to compute the approximate solutions we find the optimal values of h so we assure the convergence of the approximate solutions finally we compare our results numerically with the finite difference method and excellent agreement is found

discrete mittag leffler kernel type fractional difference initial value problems and gronwall s inequality abstract in this article we studied the caputo and riemann liouville type discrete fractional difference initial value problems with discrete mittag leffler kernels the existence and uniqueness of the solution is proved by using banach contraction principle the linear type equations are used to prove new discrete fractional versions of the gronwall s inequality the nabla discrete laplace transform is used to obtain solution representations the proven gronwall s inequality under a new defined lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions numerical examples are discussed to demonstrate the reliability of the theoretical results

fractional vector calculus for fractional advection dispersion abstract we develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient divergence and curl and a fractional divergence theorem and stokes theorem these basic tools are then applied to provide a physical explanation for the fractional advection dispersion equation for flow in heterogeneous porous media

coupling of the finite volume element method and the boundary element method an a priori convergence result the coupling of the finite volume element method and the boundary element method is an interesting approach to simulate a coupled system of a diffusion convection reaction process in an interior domain and a diffusion process in the corresponding unbounded exterior domain this discrete system maintains naturally local conservation and a possible weighted upwind scheme guarantees the stability of the discrete system also for convection dominated problems we show existence and uniqueness of the continuous system with appropriate transmission conditions on the coupling boundary provide a convergence and an a priori analysis in an energy semi norm and provide an existence and an uniqueness result for the discrete system all results are also valid for the upwind version numerical experiments show that our coupling is an efficient method for the numerical treatment of transmission problems which can also be convection dominated

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

characters of symmetric groups sharp bounds and applications we provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible it follows from our general bound that if a permutation s n has at most n o 1 cycles of length m then 1 1 m o 1 for all irreducible characters of s n this is a far reaching generalization of a result of fomin and lulov

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

space time fractional diffusion equation using a derivative with nonsingular and regular kernel abstract in this paper using the fractional operators with mittag leffler kernel in caputo and riemann liouville sense the space time fractional diffusion equation is modified the fractional equation will be examined separately with fractional spatial derivative and fractional temporal derivative for the study cases the order considered is 0 1 respectively in this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space time derivatives into the fractional diffusion equation these parameters related to equation results in a fractal space time geometry provide a new family of solutions for the diffusive processes the proposed mathematical representation can be useful to understand electrochemical phenomena propagation of energy in dissipative systems viscoelastic materials material heterogeneities and media with different scales

cutoff for conjugacy invariant random walks on the permutation group we prove a conjecture raised by the work of diaconis and shahshahani z wahrscheinlichkeitstheorie verwandte geb 57 2 159 179 1981 about the mixing time of random walks on the permutation group induced by a given conjugacy class to do this we exploit a connection with coalescence and fragmentation processes and control the kantorovich distance by using a variant of a coupling due to oded schramm as well as contractivity of the distance recasting our proof in the language of ricci curvature our proof establishes the occurrence of a phase transition which takes the following form in the case of random transpositions at time cn 2 the curvature is asymptotically zero for c le 1 and is strictly positive for c 1

relaxation and diffusion models with non singular kernels abstract anomalous relaxation and diffusion processes have been widely quantified by fractional derivative models where the definition of the fractional order derivative remains a historical debate due to its limitation in describing different kinds of non exponential decays e g stretched exponential decay meanwhile many efforts by mathematicians and engineers have been made to overcome the singularity of power function kernel in its definition this study first explores physical properties of relaxation and diffusion models where the temporal derivative was defined recently using an exponential kernel analytical analysis shows that the caputo type derivative model with an exponential kernel cannot characterize non exponential dynamics well documented in anomalous relaxation and diffusion a legitimate extension of the previous derivative is then proposed by replacing the exponential kernel with a stretched exponential kernel numerical tests show that the caputo type derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel implying the potential applicability of the new derivative in quantifying real world anomalous relaxation and diffusion processes

a nonconforming a posteriori estimator for the coupling of cell centered finite volume and boundary element methods the coupling of the cell centered finite volume method and the boundary element method is an interesting approach to solve elliptic problems on an unbounded domain where local flux conservation is important based on the piecewise constant interior finite volume solution we define a morley type interpolant built on a nonconforming finite element together with the cauchy data of the exterior boundary element solution this allows us to define a residual based a posteriori error estimator with respect to an energy norm we prove reliability and efficiency of this estimator and use its local contributions to steer an adaptive mesh refining algorithm in two examples we illustrate the effectiveness of the new adaptive coupling method and compare it with the coupling approach with a conforming morley interpolant

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

approximating the ideal free distribution via reaction diffusion advection equations abstract we consider reaction diffusion advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness where fitness is defined as a logistic local population growth rate we show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics the modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution the analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability the models are quasilinear parabolic equations with nonlinear boundary conditions

diameters of finite simple groups sharp bounds and applications let g be a finite simple group and let s be a normal subset of g we determine the diameter of the cayley graph r g s associated with g and s up to a multiplicative constant many applications follow for example we deduce that there is a constant c such that every element of g is a product of c involutions and we generalize this to elements of arbitrary order we also show that for any word w w xl xd there is a constant c c w such that for any simple group g on which w does not vanish every element of g is a product of c values of w from this we deduce that every verbal subgroup of a semisimple profinite group is closed other applications concern covering numbers expanders and random walks on finite simple groups

modeling electro magneto hydrodynamic thermo fluidic transport of biofluids with new trend of fractional derivative without singular kernel abstract this paper investigates the electro magneto hydrodynamic flow of the non newtonian behavior of biofluids with heat transfer through a cylindrical microchannel the fluid is acted by an arbitrary time dependent pressure gradient an external electric field and an external magnetic field the governing equations are considered as fractional partial differential equations based on the caputo fabrizio time fractional derivatives without singular kernel the usefulness of fractional calculus to study fluid flows or heat and mass transfer phenomena was proven several experimental measurements led to conclusion that in such problems the models described by fractional differential equations are more suitable the most common time fractional derivative used in continuum mechanics is caputo derivative however two disadvantages appear when this derivative is used first the definition kernel is a singular function and secondly the analytical expressions of the problem solutions are expressed by generalized functions mittag leffler lorenzo hartley robotnov etc which generally are not adequate to numerical calculations the new time fractional derivative caputo fabrizio without singular kernel is more suitable to solve various theoretical and practical problems which involve fractional differential equations using the caputo fabrizio derivative calculations are simpler and the obtained solutions are expressed by elementary functions analytical solutions of the biofluid velocity and thermal transport are obtained by means of the laplace and finite hankel transforms the influence of the fractional parameter eckert number and joule heating parameter on the biofluid velocity and thermal transport are numerically analyzed and graphic presented this fact can be an important in biochip technology thus making it possible to use this analysis technique extremely effective to control bioliquid samples of nanovolumes in microfluidic devices used for biological analysis and medical diagnosis

the random k cycle walk on the symmetric group we study the random walk on the symmetric group s n generated by the conjugacy class of cycles of length k we show that the convergence to uniform measure of this walk has a cut off in total variation distance after frac n k log n steps uniformly in k o n as n rightarrow infty the analysis follows from a new asymptotic estimation of the characters of the symmetric group evaluated at cycles

some uniqueness and existence results for the initial boundary value problems for the generalized time fractional diffusion equation in this paper some uniqueness and existence results for the solutions of the initial boundary value problems for the generalized time fractional diffusion equation over an open bounded domain gx 0 t email protected r n are given to establish the uniqueness of the solution a maximum principle for the generalized time fractional diffusion equation is used in turn the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is considered in the paper too another important consequence of the maximum principle is the continuous dependence of the solution on the problem data to show the existence of the solution the fourier method of the variable separation is used to construct a formal solution under certain conditions the formal solution is shown to be a generalized solution of the initial boundary value problem for the generalized time fractional diffusion equation that turns out to be a classical solution under some additional conditions

solutions of the space time fractional cattaneo diffusion equation abstract the object of this paper is to present the exact solution of the fractional cattaneo equation for describing anomalous diffusion the classical cattaneo model has been generalised to the space time fractional cattaneo model the method of the joint laplace and fourier transform is used in deriving the solution the solutions of the fractional cattaneo equation are obtained under integral and series forms in terms of the h functions finally the fractional order moments are also investigated

on the coupled bem and fem for a nonlinear exterior dirichlet problem in r2 in this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior dirichlet problem in the plane specifically the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region and the laplace equation in the corresponding unbounded exterior region in addition to appropriate boundary and transmission conditions the main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation we provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence uniqueness and approximation results are established then we consider the case where the corresponding operator is strongly monotone and lipschitz continuous and derive asymptotic error estimates for a boundary finite element solution we prove the unique solvability of the discrete operator equations and based on a strang type abstract error estimate we show the strong convergence of the approximated solutions moreover under additional regularity assumptions on the solution of the continous operator equation the asymptotic rate of convergenceo h is obtained

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

lattice fractional diffusion equation in terms of a riesz caputo difference abstract a fractional difference is defined by the use of the right and the left caputo fractional differences the definition is a two sided operator of riesz type and introduces back and forward memory effects in space difference then a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains a lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders the result shows that the riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media

some identities and asymptotics for characters of the symmetric group w this paper is motivated by the problem arising from quantum gravity 9 x 2 of counting combinatorial types of triangulations s of a riemann surface x with given degrees of vertices let us color 2 simplexes of the barycentric subdivision s9 in black and white so that adjacent simplexes are of different color and consider a mapping p x a p onto the riemann sphere p which send white simplexes onto the northern hemisphere black simplexes onto the southern hemisphere and centers k simplexes of s into an equatorial point y k s 0 1 2 then deg p s 3n n is k the number of 2 simplexes in s and p is unramified outside y y y it 0 1 2 is easy to see that points of x over y and y have ramification index 2 1 2 respectively 3 while rmaification indices of points over y are just 0 degrees of vertices of s so the problem of counting triangulations reduces to the problem of counting ramified coverings p x a y of riemann surfaces with given ramification indices in section 2 we recall a connection between ramified coverings p x a y of a riemann surface y of genus g and irreducible characters x of the y symmetric group s n s deg p the starting point is the following forn w x w x mula essentially owing to hurwitz 6 see also 10 4 2

the validity of johnson nedelec s bem fem coupling on polygonal interfaces in this short article we prove that the classical one equation or johnson nedelec coupling of finite and boundary elements can be applied with a lipschitz coupling interface because of the way it was originally approached from the analytical standpoint this bem fem scheme required smooth boundaries and hence produced a consistency error in the finite element part with a variational argument we prove that this requirement is not needed and that stability holds for all pairs of discrete space as it inherits the underlying ellipticity of the problem

a second order crank nicolson scheme for fractional cattaneo equation based on new fractional derivative abstract recently caputo and fabrizio introduce a new derivative with fractional order which has the ability to describe the material heterogeneities and the fluctuations of different scales in this article a crank nicolson finite difference scheme to solve fractional cattaneo equation based on the new fractional derivative is introduced and analyzed some a priori estimates of discrete l l2 errors with optimal order of convergence rate o 2 h 2 are established on uniform partition moreover the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis

asymptotics of characters of symmetric groups related to stanley character formula we prove an upper bound for characters of the symmetric groups in particular we show that there exists a constant a 0 with a property that for every young diagram with n boxes r rows and c columns tr tr e i amax a r n c n j j n aoj j where j j is the minimal number of factors needed to write 2 sn as a product of transpositions we also give uniform estimates for the error term in the vershik kerov s and biane s character formulas and give a new formula for free cumulants of the transition measure

on riemann and caputo fractional differences in this paper we define left and right caputo fractional sums and differences study some of their properties and then relate them to riemann liouville ones studied before by miller k s and ross b atici f m and eloe p w abdeljawad t and baleanu d and a few others also the discrete version of the q operator is used to relate the left and right caputo fractional differences a caputo fractional difference equation is solved the solution proposes discrete versions of mittag leffler functions

fractional differential equations of caputo katugampola type and numerical solutions abstract this paper is concerned with a numerical method for solving generalized fractional differential equation of caputo katugampola derivative a corresponding discretization technique is proposed numerical solutions are obtained and convergence of numerical formulae is discussed the convergence speed arrives at o t 1 numerical examples are given to test the accuracy

averaging techniques for the effective numerical solution of symm s integral equation of the first kind averaging techniques for finite element error control occasionally called zz estimators for the gradient recovery enjoy a high popularity in engineering because of their striking simplicity and universality one does not even require a pde to apply the nonexpensive post processing routines recently averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method this paper establishes a class of averaging error estimators for boundary integral methods symm s integral equation of the first kind with a nonlocal single layer integral operator serves as a model equation studied both theoretically and numerically we introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order the higher order terms depend on the regularity of the exact solution several numerical experiments illustrate the theoretical results and show that the normally unknown error is sharply estimated by the proposed estimators i e error and estimators almost coincide

character theory of symmetric groups subgroup growth of fuchsian groups and random walks we develop a number of statistical aspects of symmetric groups mostly dealing with the distribution of cycles in various subsets of sn asymptotic properties of ordinary characters of symmetric groups and estimates for the multiplicities of root number functions of these groups as main applications we present an estimate for the subgroup growth of an arbitrary fuchsian group a finiteness result for the number of fuchsian presentations of such a group resolving a long standing problem of roger lyndon as well as a proof of a well known conjecture of roichman concerning the mixing time of random walks on symmetric groups

character degrees and random walks in finite groups of lie type for a finite group h let irr h denote the set of irreducible characters of h and define the zeta function zeta h t sum chi in irr h chi 1 t for real t 0 we study the asymptotic behaviour of zeta h t for finite simple groups h of lie type and also of a corresponding zeta function defined in terms of conjugacy classes applications are given to the study of random walks on simple groups and on base sizes of primitive permutation groups

a posteriori error estimates and adaptive mesh refinement for the coupling of the finite volume method and the boundary element method we consider the coupling of the finite volume method and the boundary element method of erath siam j numer anal 50 2012 pp 574 594 in two and three dimensions this method can be used for example to approximate a solution of the transport of a concentration in a fluid where no boundary conditions are available we derive residual based a posteriori estimates also for an upwind version these upper and lower bounds measure the error in an energy semi norm between the exact solution and the numerical solution the upper bound is robust in the sense that it does not depend on the variation of the model data the lower bound however depends additionally on the local peclet number the local contributions of the a posteriori estimates are used to steer an adaptive mesh refining algorithm this strategy turns out to be very suitable for the numerical treatment of transmission problems which have singularities or boundary internal layers several numerical examples illustrate the effectivenes

limitations of quantum coset states for graph isomorphism it has been known for some time that graph isomorphism reduces to the hidden subgroup problem hsp what is more most exponential speedups in quantum computation are obtained by solving instances of the hsp a common feature of the resulting algorithms is the use of quantum coset states which encode the hidden subgroup an open question has been how hard it is to use these states to solve graph isomorphism it was recently shown by moore et al 2005 that only an exponentially small amount of information is available from one or a pair of coset states a potential source of power to exploit are entangled quantum measurements that act jointly on many states at once we show that entangled quantum measurements on at least n log n coset states are necessary to get useful information for the case of graph isomorphism matching an information theoretic upper bound this may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement our main theorem is very general and also rules out using joint measurements on few coset states for some other groups such as gl n fpm and gn where g is finite and satisfies a suitable property

modeling diffusive transport with a fractional derivative without singular kernel abstract in this paper we present an alternative representation of the diffusion equation and the diffusion advection equation using the fractional calculus approach the spatial time derivatives are approximated using the fractional definition recently introduced by caputo and fabrizio in the range 0 2 for the space and time domain respectively in this representation two auxiliary parameters x and t are introduced these parameters related to equation results in a fractal space time geometry provide an entire new family of solutions for the diffusion processes the numerical results showed different behaviors when compared with classical model solutions in the range 0 1 the concentration exhibits the non markovian levy flights and the subdiffusion phenomena when 1 the classical case is recovered when 1 2 the concentration exhibits the markovian levy flights and the superdiffusion phenomena finally when 2 the concentration is anomalous dispersive and we found ballistic diffusion

space time fractional diffusion on bounded domains abstract fractional diffusion equations replace the integer order derivatives in space and time by their fractional order analogues they are used in physics to model anomalous diffusion this paper develops strong solutions of space time fractional diffusion equations on bounded domains as well as probabilistic representations of these solutions which are useful for particle tracking codes

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

a posteriori error estimates for boundary element methods this paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples namely symm s integral equation an integral equation with a hypersingular operator and a boundary integral equation for a transmission problem based on these estimates an analog of eriksson and johnson s adaptive finite element method is proposed for the h version of the galerkin boundary element method for integral equations of the first kind the efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities the construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations and we refer to the pioneering work of babuska and miller 3 and eriksson and johnson 10 11 whereas the main features ofadaptivity for finite element methods now seem to be visible and the door is open to implementation 15 comparably little is known for boundary element methods for integral equations see e g 1 13 18 19 24 in this paper a new adaptive version of the galerkin discretization for the boundary element method is presented based on a posteriori error estimates a general framework for these a posteriori error estimates is derived in 2 and three examples are discussed in 3 5 involving the dirichlet problem the neumann problem for a closed and an open surface and a transmission problem for the laplacian leading to integral equations with strongly elliptic pseudodifferential operators even for smooth data the lack of regularity of the solution near corners of a polygonal domain q leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used in practical problems such information is missing e g when we have singular or nearly singular data and the main problem is how to balance a graded mesh refinement towards singularities and a global

fuchsian groups coverings of riemann surfaces subgroup growth random quotients and random walks abstract fuchsian groups acting as isometries of the hyperbolic plane occur naturally in geometry combinatorial group theory and other contexts we use character theoretic and probabilistic methods to study the spaces of homomorphisms from fuchsian groups to symmetric groups we obtain a wide variety of applications ranging from counting branched coverings of riemann surfaces to subgroup growth and random finite quotients of fuchsian groups as well as random walks on symmetric groups in particular we show that in some sense almost all homomorphisms from a fuchsian group to alternating groups an are surjective and this implies higman s conjecture that every fuchsian group surjects onto all large enough alternating groups as a very special case we obtain a random hurwitz generation of an namely random generation by two elements of orders 2 and 3 whose product has order 7 we also establish the analogue of higman s conjecture for symmetric groups we apply these results to branched coverings of riemann surfaces showing that under some assumptions on the ramification types their monodromy group is almost always sn or an another application concerns subgroup growth we show that a fuchsian group has n o 1 index n subgroups where is the measure of and derive similar estimates for so called eisenstein numbers of coverings of riemann surfaces a final application concerns random walks on alternating and symmetric groups we give necessary and sufficient conditions for a collection of almost homogeneous conjugacy classes in an to have product equal to an almost uniformly pointwise our methods involve some new asymptotic results for degrees and values of irreducible characters of symmetric groups

a recursive rule for kazhdan lusztig characters abstract a formula for computing kazhdan lusztig characters in hecke algebras of coxeter groups is given this formula holds for products of coxeter elements of parabolic subgroups in an arbitrary coxeter group where the parabolic subgroups are of type a and the kazhdan lusztig representations are of finite dimension the formula yields new combinatorial interpretations of murnaghan nakayama type to these characters and to other related characters

exact solution to fractional logistic equation abstract the logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time the solution to the fractional logistic equation fle is obtained using the carleman embedding technique that allows the nonlinear equation to be replaced by an infinite order set of linear equations which we then solve exactly the formal series expansion for the initial value solution of the fle is shown to be expressed in terms of a series of weighted mittag leffler functions that reduces to the well known analytic solution in the limit where the fractional index for the derivative approaches unity the numerical integration to the fle provides an excellent fit to the analytic solution we propose this approach as a general technique for solving a class of nonlinear fractional differential equations

a new derivative with normal distribution kernel theory methods and applications abstract new approach of fractional derivative with a new local kernel is suggested in this paper the kernel introduced in this work is the well known normal distribution that is a very common continuous probability distribution this distribution is very important in statistics and also highly used in natural science and social sciences to portray real valued random variables whose distributions are not known two definitions are suggested namely atangana gomez averaging in liouville caputo and riemann liouville sense we presented some relationship with existing integrals transform operators numerical approximations for first and second order approximation are derived in detail some applications of the new mathematical tools to describe some real world problems are presented in detail this is a new door opened the field of statistics natural and socials sciences

a new conservative numerical scheme for flow problems on unstructured grids and unbounded domains abstract we consider the prototype of a flow and transport problem in two dimensions in particular in many fluid dynamics problems the boundary conditions may be unknown or the domain may be unbounded furthermore only numerical schemes which conserve mass and are stable with respect to dominating convection are appropriate to solve these problems therefore we develop a new discrete scheme to address these issues more precisely we couple the cell centered finite volume method and the boundary element method and introduce suitable transmission conditions also semi robust a posteriori estimates of residual type are developed which allow us to use an adaptive mesh refinement algorithm to treat efficiently problems which have singularities or boundary internal layers several examples illustrate the effectiveness of our new adaptive coupling method

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

fractional discrete time diffusion equation with uncertainty applications of fuzzy discrete fractional calculus abstract this study provides some basics of fuzzy discrete fractional calculus as well as applications to fuzzy fractional discrete time equations with theories of r cut set fuzzy caputo and riemann liouville fractional differences are defined on a isolated time scale discrete leibniz integral law is given by use of w monotonicity conditions furthermore equivalent fractional sum equations are established fuzzy discrete mittag leffler functions are obtained by the picard approximation finally fractional discrete time diffusion equations with uncertainty is investigated and exact solutions are expressed in form of two kinds of fuzzy discrete mittag leffler functions this paper suggests a discrete time tool for modeling discrete fractional systems with uncertainty

upper bound on the characters of the symmetric groups for balanced young diagrams and a generalized frobenius formula we study asymptotics of an irreducible representation of the symmetric group sn corresponding to a balanced young diagram a young diagram with at most rows and columns for some fixed constant c in the limit as n tends to infinity we show that there exists a constant d which depends only on c with a property that where p denotes the length of a permutation the minimal number of factors necessary to write p as a product of transpositions our main tool is an analogue of the frobenius character formula which holds true not only for cycles but for arbitrary permutations

variable separation method for a nonlinear time fractional partial differential equation with forcing term abstract in this paper a variable coefficient nonlinear time fractional partial differential equation pde with initial and boundary conditions is solved by using the variable separation method as a result some new explicit and exact solutions of the time fractional pde are obtained including airy function solution hyperbolic function solution trigonometric function solution and rational solution it is shown that the variable separation method can provide a useful mathematical tool for solving some other nonlinear time fractional pdes in science and engineering

mixing times for random k cycles and coalescence fragmentation chains let sn be the permutation group on n elements and consider a random walk on sn whose step distribution is uniform on k cycles we prove a well known conjecture that the mixing time of this process is 1 k nlog n with threshold of width linear in n our proofs are elementary and purely probabilistic and do not appeal to the representation theory of sn

the symmetric group defies strong fourier sampling the dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of fourier sampling introduced by bernstein and vazirani and developed by simon and shor into an approach to the hidden subgroup problem this approach has proved successful for abelian groups leading to efficient algorithms for factoring extracting discrete logarithms and other number theoretic problems we show however that this method cannot resolve the hidden subgroup problem in the symmetric groups even in the weakest information theoretic sense in particular we show that the graph isomorphism problem cannot be solved by this approach our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states

high order unconditionally stable difference schemes for the riesz space fractional telegraph equation in this paper a class of unconditionally stable difference schemes based on the pade approximation is presented for the riesz space fractional telegraph equation firstly we introduce a new variable to transform the original differential equation to an equivalent differential equation system then we apply a second order fractional central difference scheme to discretize the riesz space fractional operator finally we use 1 1 2 2 and 3 3 pade approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations matrix analysis is used to show the unconditional stability of the proposed algorithms two examples with known exact solutions are chosen to assess the proposed difference schemes numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space fractional hyperbolic equation