k theorie equivariante des tours de bott application a la structure multiplicative de la k theorie equivariante des varietes de drapeaux

a fast galerkin finite element method for a space time fractional allen cahn equation abstract in this paper a space time fractional allen cahn equation is investigated to account for the memory effect of certain materials or the anomalous diffusion processes in heterogeneous distributed media for the mixture of two immiscible phases due to the nonlocal features of fractional derivatives conventional solvers such as the gauss elimination method require o m 2 m n memory units and o m n 2 m 3 n operations to solve the space time fractional allen cahn equation where m is the number of spatial unknowns and n is the number of time steps by exploring the special structure of the stiffness matrix and utilizing the invariant energy quadratization approach we developed an efficient linearized divide and conquer galerkin finite element method without resorting to any lossy compression this new method is efficient as it significantly reduces the memory requirement to o m n and the operation count to o m n log 2 n log m we have developed a fast galerkin finite element method with local refinement that allows accurate capture of the coarsening dynamics at the interface furthermore we have investigated the modeling capacities and energy dissipation properties of the space time fractional allen cahn equation numerical experiments are presented to demonstrate the efficiency of the new methods and the flexibility of turnable sharpness and decay behavior of the space time fractional allen cahn equation

fundamental solutions transition densities and the integration of lie symmetries abstract in this paper we present some new applications of lie symmetry analysis to problems in stochastic calculus the major focus is on using lie symmetries of parabolic pdes to obtain fundamental solutions and transition densities the method we use relies upon the fact that lie symmetries can be integrated with respect to the group parameter we obtain new results which show that for pdes with nontrivial lie symmetry algebras the lie symmetries naturally yield fourier and laplace transforms of fundamental solutions and we derive explicit formulas for such transforms in terms of the coefficients of the pde

new classes of non convolution integral equations arising from lie symmetry analysis of hyperbolic pdes abstract in this paper we consider some new classes of integral equations that arise from lie symmetry analysis specifically we consider the task of obtaining solutions of a cauchy problem for some classes of second order hyperbolic partial differential equations our analysis leads to new integral equations of non convolution type which can be solved by classical methods we derive solutions of these integral equations which in turn lead to solutions of the associated cauchy problems

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

a study of saturated tensor cone for symmetrizable kac moody algebras let mathfrak g be a symmetrizable kac moody lie algebra with the standard cartan subalgebra mathfrak h and the weyl group w let p be the set of dominant integral weights for lambda in p let l lambda be the integrable highest weight irreducible representation of mathfrak g with highest weight lambda for a positive integer s define the saturated tensor semigroup as begin aligned gamma s lambda 1 dots lambda s mu in p s 1 exists n ge 1 text with l n mu subset l n lambda 1 otimes dots otimes l n lambda s end aligned the aim of this paper is to begin a systematic study of gamma s in the infinite dimensional symmetrizable kac moody case in this paper we produce a set of necessary inequalities satisfied by gamma s these inequalities are indexed by products in h g mathrm min b mathbb z for b the standard borel subgroup where g mathrm min is the minimal kac moody group with lie algebra mathfrak g the proof relies on the kac moody analogue of the borel weil theorem and geometric invariant theory specifically the hilbert mumford index in the case that mathfrak g is affine of rank 2 we show that these inequalities are necessary and sufficient we further prove that any integer d 0 is a saturation factor for a 1 1 and 4 is a saturation factor for a 2 2

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

a divide and conquer fast finite difference method for spacetime fractional partial differential equation fractional partial differential equations fpdes provide better modeling capabilities for challenging phenomena with long range time memory and spatial interaction than integer order pdes do a conventional numerical discretization of spacetime fpdes requires o n2 mn memory and o mn3 m2n computational work where n is the number of spatial freedoms per time step and m is the number of time steps we develop a fast finite difference method fdm for spacetime fpde i we utilize the toeplitz like structure of the coefficient matrix to develop a matrix free preconditioned fast krylov subspace iterative solver to invert the coefficient matrix at each time step ii we utilize a divide and conquer strategy a recursive direct solver to handle the temporal coupling of the numerical scheme the fast method has an optimal memory requirement of o mn and an approximately linear computational complexity of o nm logn log2m without resorting to any lossy compression numerical experiments show the utility of the method

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

symmetry groups of partial differential equations separation of variables and direct integral theory abstract we study unitary symmetries of linear partial differential equations by applying representation theory we show how to decompose a hilbert space of solutions to certain classes of equations as direct integrals

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

a preconditioned fast quadratic spline collocation method for two sided space fractional partial differential equations abstract quadratic spline collocation methods for the one and two dimensional fractional diffusion equations are proposed by carefully exploring the mathematical structure of the coefficient matrix we propose a matrix free fast krylov subspace iterative solver for the corresponding quadratic spline collocation scheme and then preconditioning technique is applied to further accelerate the convergence of the fast krylov subspace iterative method it shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from o n 2 to o n log n per iteration and memory requirement from o n 2 to o n while still well approximates the fractional diffusion equations without any accuracy lost numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method

tangent cones to schubert varieties in types an bn and cn abstract we study tangent cones to schubert subvarieties of the flag variety of a complex reductive group g let t be a maximal torus of g b be a borel subgroup of g containing t be the root system of g with respect to t w be the weyl group of and f g b be the flag variety we prove that if every irreducible component of is of type b n or c n and w 1 w 2 are two distinct involutions in w then the tangent cones at the point p e b to the corresponding schubert subvarieties x w 1 x w 2 of f do not coincide as subschemes of the tangent space t p f we also show that if every irreducible component of is of type a n or c n then the reduced tangent cones to x w 1 and x w 2 do not coincide as subvarieties of t p f

permutation representations on schubert varieties this paper defines and studies permutation representations on the equivariant cohomology of schubert varieties as representations both over bbb c and over bbb c t 1 t 2 ldots t n we show these group actions are the same as an action of simple transpositions studied geometrically by m brion and give topological meaning to the divided difference operators of berstein gelfand gelfand demazure kostant kumar and others we analyze these representations using the combinatorial approach to equivariant cohomology introduced by goresky kottwitz macpherson we find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial though the equivariant representation is not

on the equivalence of lie symmetries and group representations abstract we consider families of linear parabolic pdes in n dimensions which possess lie symmetry groups of dimension at least four we identify the lie symmetry groups of these equations with the 2 n 1 dimensional heisenberg group and sl 2 r we then show that for pdes of this type the lie symmetries may be regarded as global projective representations of the symmetry group we construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetry groups banach algebras of symmetries are introduced

reduction and reconstruction of stochastic differential equations via symmetries an algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction inspired by the classical reconstruction by quadratures is proposed as a side result the well known solution formula for linear one dimensional stochastic differential equations is obtained within this symmetry approach the complete procedure is applied to several examples with both theoretical and applied relevance

a conjectural peterson isomorphism in k theory abstract we state a precise conjectural isomorphism between localizations of the equivariant quantum k theory ring of a flag variety and the equivariant k homology ring of the affine grassmannian in particular relating their schubert bases and structure constants this generalizes peterson s isomorphism in co homology we prove a formula for the pontryagin structure constants in the k homology ring and we use it to check our conjecture in few situations

the symmetry groups of linear partial differential equations and representation theory 1 abstract we study the symmetry groups of three closely related pdes it is shown that the symmetry groups for these equations are actually global lie groups and that the symmetry operations arise from standard group representation theory via intertwining operators derived from the fundamental solutions of the equations

spectral asymptotics for sturm liouville equations with indefinite weight the sturm liouville equation py qy ary on 0 l is considered subject to the boundary conditions y 0 cos py 0 sin y l cos py l sin we assume that p is positive and that pr is piecewise continuous and changes sign at its discontinuities we give asymptotic approximations up to o 2 n for n or equivalently up to o n for n the eigenvalues of the above boundary value problem

multigrid method for fractional diffusion equations the fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted grunwald formula the scheme is unconditionally stable and the coefficient matrix possesses the toeplitz like structure a multigrid method is proposed to solve the resulting system meanwhile the fast toeplitz matrix vector multiplication is utilized to lower the computational cost with only o nlogn complexity where n is the number of the grid points numerical experiments are given to demonstrate the efficiency of the method

a finite volume method for two dimensional riemann liouville space fractional diffusion equation and its efficient implementation abstract we develop a finite volume method based on crank nicolson time discretization for the two dimensional nonsymmetric riemann liouville space fractional diffusion equation stability and convergence are then carefully discussed we prove that the finite volume scheme is unconditionally stable and convergent with second order accuracy in time and min 1 1 order in space with respect to a weighted discrete norm here 0 1 are the space fractional order indexes in x and y directions respectively furthermore we rewrite the finite volume scheme into a matrix form and develop a matrix free preconditioned fast krylov subspace iterative method which only requires storage of o n and computational cost of o n log n per iteration without losing any accuracy compared to the direct gaussian elimination method here n is the total number of spatial unknowns consequently the fast finite volume method is particularly suitable for large scale modeling and simulation numerical experiments verify the theoretical results and show strong potential of the fast method

equivalence and symmetries for variable coefficient linear heat type equations i a systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations pdes is presented the complete symmetry group classification is re performed in the current context a useful criterion which is necessary and sufficient for being reducible to the standard heat equation by point transformations is established a similar criterion is also valid for the equations to have a four or six dimensional symmetry group nontrivial symmetry groups in this situation the basis elements are listed in terms of coefficients a number of illustrative examples are given in particular some applications from the recent literature are re examined in our new approach multidimensional parabolic pdes of heat and schrodinger type are also considered

a fast finite volume method for conservative space time fractional diffusion equations discretized on space time locally refined meshes abstract a fast finite volume method was developed for conservative space time fractional diffusion partial differential equations in two space dimensions in the method a locally refined composite spatial mesh is used to resolve the boundary layers of the solutions and a graded temporal mesh is used to resolve the singularity of the solutions near the initial time a fast numerical method was developed by fully exploring the structure of the stiffness matrices numerical results show the utility of the method

mhd flow and heat transfer analysis of fractional oldroyd b nanofluid between two coaxial cylinders abstract in previous study on the flow of fractional viscoelastic fluid between two coaxial cylinders due to the complexity of computation researchers commonly assume that the flow area is a rectangular pipe this paper studies two dimensional magnetohydrodynamic mhd flow and heat transfer of fractional oldroyd b nanofluid between two coaxial cylinders with variable pressure the boundary layer fractional cylindrical governing equations are firstly formulated and derived meanwhile the numerical solutions are obtained by using the newly developed finite difference method combined with l1 algorithm whose validity is confirmed by the comparison with the exact solution constructed results show that the fractional derivative parameters 1 and exponent parameter related to the wall temperature have significant effects on velocity and temperature fields specifically the velocity decreases with the augment of 1 but increases as rises up the temperature profiles intersect each other for different meanwhile increasing leads to an increase in temperature near the wall of inner tube but a decline near the outer cylinder which demonstrates increasing exponent parameter can enhance heat transfer

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

preconditioned iterative methods for space time fractional advection diffusion equations in this paper we propose practical numerical methods for solving a class of initial boundary value problems of space time fractional advection diffusion equations first we propose an implicit method based on two sided grunwald formulae and discuss its stability and consistency then we develop the preconditioned generalized minimal residual preconditioned gmres method and preconditioned conjugate gradient normal residual preconditioned cgnr method with easily constructed preconditioners importantly because resulting systems are toeplitz like fast fourier transform can be applied to significantly reduce the computational cost we perform numerical experiments to demonstrate the efficiency of our preconditioners even in cases with variable coefficients

the schubert calculus braid relations and generalized cohomology let x be the flag variety of a compact lie group and let h be a complex oriented generalized cohomology theory we introduce operators on h x which generalize operators introduced by bernstein gel fand and gel fand for rational cohomology and by demazure for k theory using the becker gottlieb transfer we give a formula for these operators which enables us to prove that they satisfy braid relations only for the two classical cases thereby giving a topological interpretation of a theorem proved by the authors and extended by gutkin one of the central issues in lie theory is the geometry of the flag variety associated to a compact lie group g an important problem concerning the flag variety is the schubert calculus which studies the ring structure of the cohomology of the flag variety work initiated by borel bott and kostant which culminated in a paper by bernstein gel fand and gel fand bgg gave a complete solution to the problem demazure studied the same problem for k theory moreover these techniques have been generalized to the kac moody situation by kac peterson kostant kumar and others this work has focussed on algebro geometric properties of the flag variety here on the other hand we study the flag variety from the point of view of algebraic topology as a consequence not only do we recover the classical results described above but we extend these results to a certain class of cohomology theories those which are complex oriented examples of complex oriented theories include ordinary cohomology k theory complex cobordism and elliptic cohomology since the context we have chosen in very general the proofs are universal and are often simpler than the classical arguments in the work of bgg a crucial role is played by operators ai associated to each simple reflection si of the weyl group of g defined by demazure in k theory these operators ai satisfy the braid relations which are the relations between pairs of simple reflections in this paper we generalize the a to give operators d acting on h g t for any complex oriented theory h we prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of k theory theorem received by the editors june 21 1988 1980 mathematics subject classification 1985 revision primary 55n20 57t15 the second author was supported by an nsf graduate fellowship 3 1990 american mathematical society 0002 9947 90 1 00 25 perpage

symmetries of the fokker planck equation with a constant diffusion matrix in 2 1 dimensions we completely classify the symmetries of the fokker planck equation in two spatial dimensions with a constant positive definite diffusion matrix we apply these results to construct group invariant solutions for a physically interesting family of fokker planck equations

residue construction of hecke algebras abstract hecke algebras are usually defined algebraically via generators and relations we give a new algebra geometric construction of affine and double affine hecke algebras the former is known as the iwahori hecke algebra and the latter was introduced by cherednik in terms of residues more generally to any generalized cartan matrixaand a pointqin a one dimensional complex algebraic group c we associate an associate algebrahq ifais of finite type and c c the algebrahqis theaffinehecke algebra of the corresponding finite root system ifais of affine type and c c thenhqis essentially the cherednik algebra the case c c corresponds to degenerate counterparts of the above objects considered by drinfeld and lusztig finally taking c to be an elliptic curve one gets some new elliptic analogues of the affine hecke algebra

equivariant k theory of affine flag manifolds and affine grothendieck polynomials we study the equivariant k group of the affine flag manifold with respect to the borel group action we prove that the structure sheaf of the infinite dimensional schubert variety in the k group is represented by a unique polynomial which we call the affine grothendieck polynomial

equivalence and symmetries for variable coefficient linear heat type equations ii fundamental solutions we present a comparative study of fundamental solutions heat kernels of variable coefficient heat type partial differential equations based on lie symmetry group methods and equivalence transformations discussed in the work of gungor j math phys 59 051507 2018 applications will include both one and two dimensional equations we present a comparative study of fundamental solutions heat kernels of variable coefficient heat type partial differential equations based on lie symmetry group methods and equivalence transformations discussed in the work of gungor j math phys 59 051507 2018 applications will include both one and two dimensional equations

a fast direct method for block triangular toeplitz like with tri diagonal block systems from time fractional partial differential equations in this paper we study the block lower triangular toeplitz like with tri diagonal blocks system which arises from the time fractional partial differential equation existing fast numerical solver e g fast approximate inversion method cannot handle such linear system as the main diagonal blocks are different the main contribution of this paper is to propose a fast direct method for solving this linear system and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system our idea is based on the divide and conquer strategy and together with the fast fourier transforms for calculating toeplitz matrix vector multiplication the complexity needs o m n log 2 m arithmetic operations where m is the number of blocks the number of time steps in the system and n is the size number of spatial grid points of each block numerical examples from the finite difference discretization of time fractional partial differential equations are also given to demonstrate the efficiency of the proposed method

uniform estimates for eulerian lagrangian methods for singularly perturbed time dependent problems we prove a priori optimal order error estimates in a weighted energy norm for several eulerian lagrangian methods for singularly perturbed time dependent convection diffusion equations with full regularity the estimates depend only on certain sobolev norms of the initial and right hand side data but not on varepsilon or any norm of the true solution and so hold uniformly with respect to varepsilon we use the interpolation of spaces and stability estimates to derive an varepsilon uniform estimate for problems with minimal or intermediate regularity where the convergence rates are proportional to certain besov norms of the initial and right hand side data

fourier type transforms on lie symmetry groups in this paper we construct operators on a lie symmetry group which may be regarded as fourier transforms essentially we integrate solutions generated by lie symmetries against suitable test functions we show that this idea leads to a powerful method for solving cauchy problems for parabolic and hyperbolic equations in two and higher dimensions we also discuss applications to the elliptic case

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved with gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

a preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh numerical methods for fractional differential equations generate full stiffness matrices which were traditionally solved via gaussian type direct solvers that require o n 3 of computational work and o n 2 of memory to store where n is the number of spatial grid points in the discretization we develop a preconditioned fast krylov subspace iterative method for the efficient and faithful solution of finite volume schemes defined on a locally refined composite mesh for fractional differential equations to resolve boundary layers of the solutions numerical results are presented to show the utility of the method

motivic decompositions of twisted flag varieties and representations of hecke type algebras motivated by the motivic galois group and the kostant kumar results on equivariant cohomology of flag varieties we provide a uniform de scription of motivic direct sum decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil hecke algebra h more generally we establish an equivalence between the h motivic subcat egory generated by the motive of e b and the category of projective modules of the associated rational algebra d of push pull operators where e is a tor sor for a split semisimple linear algebraic group g over a field k b is a borel subgroup of g h is an algebraic oriented cohomology theory in the sense of levine morel e g chow ring ch or an algebraic cobordism the algebra d can be think of as an integer analogue of the hopf algebra of the h motivic galois group of e b as an application taking h ch and specializing the coefficients to the finite field fp we obtain that p modular projective representations of d h are generated by an irreducible h module corresponding to the generalized rost voevodsky motive for g p

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

lie group symmetries as integral transforms of fundamental solutions abstract we obtain fundamental solutions for pdes of the form u t x u x x f x u x x r u by showing that if the symmetry group of the pde is nontrivial it contains a standard integral transform of the fundamental solution we show that in this case the problem of finding a fundamental solution can be reduced to inverting a laplace transform or some other classical transform

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

schubert calculus in complex cobordism we study the structure of the complex cobordism ring of the flag variety of a compact connected lie group an explicit procedure for determining products of basis elements is obtained generalizing the work of bernstein gel fand gel fand on ordinary cohomology and of kostant kumar on k theory bott samelson resolutions are used to replace the classical basis of schubert cells

quantum integrability and generalised quantum schubert calculus abstract we introduce and study a new mathematical structure in the generalised quantum cohomology theory for grassmannians namely we relate the schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six vertex model our approach offers a new perspective on already established and well studied special cases for example equivariant k theory and in addition allows us to formulate a conjecture on the so far unknown case of quantum equivariant k theory

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

lie symmetry methods for multi dimensional parabolic pdes and diffusions abstract in this paper we introduce new methods based upon integrating lie symmetries for the construction of explicit fundamental solutions of multi dimensional second order parabolic pdes we present applications to the problem of finding transition probability densities for multi dimensional diffusions and to representation theory many explicit examples are given to illustrate the techniques

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

k theoretic analogues of factorial schur p and q functions abstract we introduce two families of symmetric functions generalizing the factorial schur p and q functions due to ivanov we call them k theoretic analogues of factorial schur p and q functions we prove various combinatorial expressions for these functions e g as a ratio of pfaffians a sum over set valued shifted tableaux and a sum over excited young diagrams as a geometric application we show that these functions represent the schubert classes in the k theory of torus equivariant coherent sheaves on the maximal isotropic grassmannians of symplectic and orthogonal types this generalizes a corresponding result for the equivariant cohomology given by the authors we also discuss a remarkable property enjoyed by these functions which we call the k theoretic q cancellation property we prove that the k theoretic p functions form a formal basis of the ring of functions with the k theoretic q cancellation property

the nil hecke ring and singularity of schubert varieties let g be a semi simple simply connected complex algebraic group and t b a maximal torus and a borel subgroup respectively let lie t be the cartan subalgebra of the lie algebra lie g and w n t t the weyl group associated to the pair g t where n t is the normalizer of t in g we can view any element omega overline omega mod t i w as the element denoted by the corresponding german character xo of g b denned as m overline omega b for any w i w there is associated the schubert variety rm x omega overline b omega b b subset g b and the t fixed points of x w under the canonical left action are precisely rm i omega left o upsilon in wand upsilon leqslant omega right

a fast finite element method for space fractional dispersion equations on bounded domains in mathbb r 2 we develop a fast and accurate finite element method for space fractional dispersion equations in two space dimensions which are expressed in terms of fractional directional derivatives in all the directions that are integrated with respect to a probability measure on the unit circle the fast method significantly reduces the computational work of solving the discrete linear algebraic systems from o n 3 by a direct solver to o n log n per iteration and a memory requirement from o n 2 to o n furthermore the fast method reduces the evaluation of the stiffness matrix which often constitutes a large portion of the cpu time from o n 2 to o n the developed preconditioned fast krylov subspace iterative solver significantly reduces the number of iterations in a krylov subspace iterative method and may improve the convergence behavior of the solver numerical results show the utility of the method

a superfast preconditioned iterative method for steady state space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the classical second order diffusion equations because of the nonlocal property of fractional differential operators the corresponding numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 for a problem of size n we develop a superfast preconditioned conjugate gradient squared method for the efficient solution of steady state space fractional diffusion equations the method reduces the computational work from o n 2 to o nlogn per iteration and reduces the memory requirement from o n 2 to o n furthermore the method significantly reduces the number of iterations to be mesh size independent preliminary numerical experiments for a one dimensional steady state diffusion equation with 2 1 3 nodes show that the fast method reduces the overall cpu time from 3h and 27min for the gaussian elimination to 0 39s for the fast method while retaining the accuracy of gaussian elimination in contrast the regular conjugate gradient squared method diverges after 2days of simulations and more than 20 000 iterations

double grothendieck polynomials for symplectic and odd orthogonal grassmannians abstract we study the double grothendieck polynomials of kirillov naruse for the symplectic and odd orthogonal grassmannians these functions are explicitly written as pfaffian sum form and are identified with the stable limits of fundamental classes of the schubert varieties in torus equivariant connective k theory of these isotropic grassmannians we also provide a combinatorial description of the ring formally spanned be the double grothendieck polynomials

a circulant preconditioner for fractional diffusion equations the implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable is employed to discretize fractional diffusion equations the resulting systems are toeplitz like and then the fast fourier transform can be used to reduce the computational cost of the matrix vector multiplication the preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant hence the convergence rate of the proposed iterative algorithm is superlinear numerical experiments are carried out to demonstrate that our circulant preconditioner works very well even though for cases of variable diffusion coefficients

a grobner basis for kazhdan lusztig ideals it kazhdan lusztig ideals a family of generalized determinantal ideals investigated in woo yong 08 provide an explicit choice of coordinates and equations encoding a neighborhood of a torus fixed point of a schubert variety on a type a flag variety our main result is a gr o bner basis for these ideals this provides a single geometric setting to transparently explain the naturality of pipe dreams on the it rothe diagram of a permutation and their appearance in begin itemize item combinatorial formulas fomin kirillov 94 for schubert and grothendieck polynomialsof lascoux sch u tzenberger 82 item the equivariant k theory specialization formula of buch rim a nyi 04 and item a positive combinatorial formula for multiplicities of schubert varieties in good cases including those for which the associated kazhdan lusztig ideal is homogeneous under the standard grading end itemize our results generalize with alternate proofs knutson miller 05 s gr o bner basis theorem for schubert determinantal ideals and their geometric interpretation of the monomial positivity of schubert polynomials we also complement recent work of knutson 08 09 on degenerations of kazhdan lusztig varieties in general lie type as well as work of goldin 01 on equivariant localization and of lakshmibai weyman 90 rosenthal zelevinsky 01 and krattenthaler 01 on grassmannian multiplicity formulas

positivity in the grothendieck group of complex flag varieties abstract we prove a conjecture of a s buch concerning the structure constants of the grothendieck ring of a flag variety with respect to its basis of schubert structure sheaves for this we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs equivalently the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of schubert varieties

fast iterative solvers for linear systems arising from time dependent space fractional diffusion equations in this paper we study the linear systems arising from the discretization of time dependent space fractional diffusion equations by using a finite difference discretization scheme for the time derivative and a finite volume discretization scheme for the space fractional derivative toeplitz like linear systems are obtained we propose using the approximate inverse circulant preconditioner to deal with such toeplitz like matrices and we show that the spectra of the corresponding preconditioned matrices are clustered around 1 experimental results on time dependent and space fractional diffusion equations are presented to demonstrate that the preconditioned krylov subspace methods converge very quickly

geometric representations of the formal affine hecke algebra abstract for any formal group law there is a formal affine hecke algebra defined by hoffnung malagon lopez savage zainoulline coming from this formal group law there is also an oriented cohomology theory we identify the formal affine hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the steinberg variety as a consequence this algebra acts on the corresponding cohomology of the springer fibers this generalizes the action of classical affine hecke algebra on the k theory of the springer fibers constructed by lusztig we also give a residue interpretation of the formal affine hecke algebra which generalizes the residue construction of ginzburg kapranov vasserot when the formal group law comes from a 1 dimensional algebraic group

on certain hecke rings abstract we examine rings that embed into the smash product of the group algebra of the weyl group with the field of meromorphic functions on the cartan subalgebra and are generated by elements that satisfy braid relations we prove that every such ring is isomorphic to either the hecke algebra the nil hecke ring or the group algebra of the weyl group

representation theory over tropical semifield and langlands duality recently we propose a class of infinite dimensional integral representations of classical mathfrak gl ell 1 whittaker functions and local archimedean local l factors using two dimensional topological field theory framework the local archimedean langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories in this note we introduce elementary analogs of the whittaker functions and the archimedean l factors given by ul 1 equivariant symplectic volumes of appropriate kahler ul 1 manifolds we demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids gl ell 1 mathcal r mathcal r being the tropical semifield we also show that the elementary whittaker functions can be obtained from the non archimedean whittaker functions over mathbb q p by taking the formal limit p 1 hence the elementary special functions constructed in this way might be considered as functions over the mysterious field mathbb q 1 the existence of two representations for the elementary whittaker functions one as an equivariant volume and the other as a matrix element should be considered as a manifestation of a hypothetical elementary analog of the local langlands duality for number fields we would like to note that the elementary local l factors coincide with l factors introduced previously by kurokawa

equivariant quantum cohomology of the odd symplectic grassmannian the odd symplectic grassmannian mathrm ig mathrm ig k 2n 1 parametrizes k dimensional subspaces of mathbb c 2n 1 which are isotropic with respect to a general necessarily degenerate symplectic form the odd symplectic group acts on mathrm ig with two orbits and mathrm ig is itself a smooth schubert variety in the submaximal isotropic grassmannian mathrm ig k 2n 2 we use the technique of curve neighborhoods to prove a chevalley formula in the equivariant quantum cohomology of mathrm ig i e a formula to multiply a schubert class by the schubert divisor class this generalizes a formula of pech in the case k 2 and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring

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

a matroid invariant via the k theory of the grassmannian abstract let g d n denote the grassmannian of d planes in c n and let t be the torus c n diag c which acts on g d n let x be a point of g d n and let t x be the closure of the t orbit through x then the class of the structure sheaf of t x in the k theory of g d n depends only on which plucker coordinates of x are nonzero combinatorial data known as the matroid of x in this paper we will define a certain map of additive groups from k g d n to z t letting g x t denote the image of 1 n dim t x o t x g x behaves nicely under the standard constructions of matroid theory such as direct sum two sum duality and series and parallel extensions we use this invariant to prove bounds on the complexity of kapranov s lie complexes m kapranov chow quotients of grassmannians i adv soviet math 16 2 1993 29 110 hacking keel and tevelev s very stable pairs p hacking s keel e tevelev compactification of the moduli space of hyperplane arrangements j algebraic geom 15 2006 657 680 and the author s tropical linear spaces when they are realizable in characteristic zero d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 namely in characteristic zero a lie complex or the underlying d 1 dimensional scheme of a very stable pair can have at most n i 1 d i n d i i 1 strata of dimensions n i and d i respectively this prove the author s f vector conjecture from d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 in the case of a tropical linear space realizable in characteristic 0

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

a coproduct structure on the formal affine demazure algebra in the present paper we generalize the coproduct structure on nil hecke rings introduced and studied by kostant and kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law we then construct an algebraic model of the t equivariant oriented cohomology of the variety of complete flags

combinatorics of the k theory of affine grassmannians abstract we introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize grothendieck polynomials and k schur functions we prove that the polynomials drawn from these tableaux are the affine grothendieck polynomials and k k schur functions schubert representatives for the k theory of affine grassmannians and their dual in the nil hecke ring we prove a number of combinatorial properties including pieri rules

noncommutative schubert calculus and grothendieck polynomials abstract in this paper we extend the work of fomin and greene on noncommutative schur functions by defining noncommutative analogs of schubert polynomials if the variables satisfy certain relations essentially the same as those needed in the theory of noncommutative schur functions we prove a pieri type formula and a cauchy identity for our noncommutative polynomials our results imply the conjecture of fomin and kirillov concerning the expansion of an arbitrary grothendieck polynomial in the basis of schubert polynomials we also present a combinatorial interpretation for the coefficients of the expansion we conclude with some open problems related to it