the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

on the numerical approximation of finite speed diffusion problems lagrangian formulations for the cauchy problems for the generalized heat and porous media equations are introduced and equivalence and existence results discussed efficient interface tracking finite difference and finite element discretizations of the lagrangian formulation are discussed mixed euler lagrange formulations for mixed problems and the one phase stefan problem are presented numerical experiments are discussed

properties of eigenstates of the six vertex model with twisted and open boundary conditions we use the method proposed by izergin and korepin to discuss the bethe ansatz equations of the six vertex model with twisted and open boundary conditions let two parameters of the bethe wave functions be equal then an additional bethe ansatz equation arises for the open boundary condition case we also have discussed the dual pseudovacuum state and have written out the simplest scalar products of the bethe wave function

eventual c regularity and concavity for flows in one dimensional porous media we study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation u t eft u m ight xx inq r imes eft 0 nfty ight ueft x 0 ight u 0 eft x ight forx n r

data sparse approximation to a class of operator valued functions in earlier papers we developed a method for the data sparse approximation of the solution operators for elliptic parabolic and hyperbolic pdes based on the dunford cauchy representation to the operator valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents in the present paper we discuss how these techniques can be applied to approximate a hierarchy of the operator valued functions generated by an elliptic operator

interfaces with a corner point in one dimensional porous medium flow we prove that for a large class of initial distributions the solutions of the initial value problem for the one dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time this happens for example if the initial pressure is o x 2 as x 0 we also give sufficient conditions for the smoothness of the interface which improve previous results

the number of peaks of nonnegative solutions to some nonlinear degenerate parabolic equations abstract we consider the cauchy problems for some nonlinear degenerate parabolic equations in r n for nonzero bounded nonnegative initial data having compact support and show that the set of peaks of the nonnegative solution consists of one point after a finite time

qualitative behavior and traveling waves for flux saturated porous media equations arising in optimal mass transportation abstract this paper is devoted to the analysis of qualitative properties of porous media flux saturated type operators which combine the effects of reaction terms of fkpp type in dimension one specifically we study the existence of traveling wave profiles sub and super solutions waiting time and dynamical properties of the support the paper also deals with the connection between these minimal traveling waves with those of other related models porous media flux saturated mechanism and the heat equation

kronecker product approximations for some function related matrices abstract a new approximation tool such as sums of kronecker products is recently found to provide a superb compression property on a series of numerical examples of quite a general nature the purpose of the paper is explanation of this phenomenon in the form of existence theorems for matrix approximations of low kronecker rank for some classes of function related matrices including important specimens from potential theory this lays the grounds for development of new approximation algorithms for example in the cases when a matrix is associated with a shift invariant function on the cartesian product of nonunform grids which is of great practical interest in the solution of integral equations on plates or screens

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

structured data sparse approximation to high order tensors arising from the deterministic boltzmann equation we develop efficient data sparse representations to a class of high order tensors via a block many fold kronecker product decomposition such a decomposition is based on low separation rank approximations of the corresponding multivariate generating function we combine the sinc interpolation and a quadrature based approximation with hierarchically organised block tensor product formats different matrix and tensor operations in the generalised kronecker tensor product format including the hadamard type product can be implemented with the low cost an application to the collision integral from the deterministic boltzmann equation leads to an asymptotical cost o n 4 log n o n 5 log n in the one dimensional problem size n depending on the model kernel function which noticeably improves the complexity o n 6 log n of the full matrix representation

behaviour of the velocity of one dimensional flows in porous media the one dimensional flow of gas of density u through a porous medium obeys the equation u um x where m 1 x e r and t 0 we prove that the local velocity of the gas given by v mu 2u1 not only is bounded for t t 0 but approaches an n wave profile as t oo n waves are the typical asymptotic profiles for some first order conservation laws a class of nonlinear hiperbolic equations the case m 0 0 2 u x 0 u0 x forx e r where m is a positive constant and u0 satisfies 0 3 u0 e l r u0 0 a e u0 0 apart from its mathematical interest as a simple model of nonlinear evolution equation of possibly degenerate or singular parabolic type equation 0 1 and its n dimensional analogue appear in a number of applications cf 18 for a reference in the typical application to the flow of gas through a porous medium we have m 1 and u stands for the density of the gas writing 0 1 as a conservation law for the mass

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

concavity of solutions of the porous medium equation abstract the flow of a gas through a porous medium is governed by a degenerate quasilinear parabolic equation it is known that the nonnegative solutions to this equation possess a lower bound for the second derivative of the pressure in the spatial variables this bound plays an important role in the mathematical treatment and is related to the entropy of the flow since the solutions exhibit interfaces across which v sub x jumps positively no upper bound is possible globally for v subxx nevertheless it is proven that the concavity of v t in the region where v is positive is preserved in time this is in itself an interesting geometric property of the solution it also allows one to obtain precise information about the asymptotic behaviour of the flow

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

a fast algorithm for the energy space boson boltzmann collision operator abstract this paper introduces a fast algorithm for the energy space boson boltzmann collision operator compared to the direct o n3 calculation and the previous o n2 logn method markowich and pareschi 2005 the new algorithm runs in complexity o n log n which is optimal up to a logarithmic factor n is the number of grid points in energy space the basic idea is to partition the 3 d summation domain recursively into elementary shapes so that the summation within each shape becomes a special double convolution that can be computed efficiently by the fast fourier transform numerical examples are presented to illustrate the efficiency and accuracy of the proposed algorithm

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

homogeneous diffusion in ℝ with power like nonlinear diffusivity we study the nonnegative solutions of the initial value problem ut ur ux p 1ux x u x 0 l1 where p 0 r p 0 the local velocity of propagation of the solutions is identified as v vx vx p 1 where v cu with r p 1 p and c r p r p 1 is the nonlinear potential our main result is the a priori estimate vx vx p 1 x frac 1 2p r t

generalized t j model by parametrizing the t j model we present a new electron correlation model with one free parameter in one dimension this model is of suq 1 mod 2 symmetry the energy spectra are shown to be modulated by the free parameter in the model the solution and symmetric structures of the hilbert space as well as the bethe ansatz approach are discussed for special cases

bounded speed of propagation for solutions to radiative transfer equations the radiative transfer equation is the nonlinear transport equation partial t f frac 1 varepsilon v cdot nabla x f frac 1 varepsilon 2 sigma tilde f f tilde f 0 rte where tilde f x mathbf t ff x mathbf v t dv denotes the average off x t on the unit sphere v 1 it describes the absorption and emission of photons in a hot medium as the mean free path goes to 0 f converges to a solution of the porous medium equation t u f u withf u n u 1 since blows up atu 0 solutions to the pme propagate with finite speed specifically ifu 0 has compact support inr n so doesu t for everyt 0 and the sets t x u x t 0 t 0 form an expanding family ast increases andu t r n we show in this paper that these propagation properties hold for the solutionsf of the rte for all small moreover the growth of the support off is uniform in our proofs rely on the construction of explicit solutions of the travelling wave type and subsolutions to the rte to our knowledge this is the first example of a kinetic equation with high velocities where localized data propagate always with bounded speed for vlasov poisson equations this arises only for particular initial data

the interfaces of one dimensional flows in porous media abstract a porous media equation pme has been used as a model for a number of physical phenomena heat diffusion at high temperatures boundary layer theory spread of a thin layer of viscous material and mainly the flow of gas in a porous medium the most distinctive characteristic of the solutions to pme as compared with the linear heat equation is the finite speed of propagation in this paper the properties of the interfaces are studied in terms of the initial data sometimes the interface is stationary for a certain time and then begins to move we characterize the existence of a positive waiting time and give bounds for it

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

coordinate bethe ansatz for the one dimensional su n hubbard model with open boundary conditions the one dimensional 1d su n hubbard model with open boundary condition is solved by using the coordinate bethe ansatz method the energy and integrable boundary conditions are obtained at the same time the corresponding bethe ansatz equations are achieved by diagonalizing the inhomogeneous transfer matrix of the open su 2n 2 xxx vertex model when n 2 our result comes back to that of the 1d hubbard model given by deguchi and yue con mat 9704138

waiting time solutions of a nonlinear diffusion equation similarity solutions of the equation partial c partial t partial partial x c n partial c partial x n 0 are studied for both t 0 and t 0 and there is a stationary free boundary at x 0 and for t 0 have support in x at beta where beta 1 a 0 i e there is a moving free boundary x at beta by prescribing appropriate initial data c c 0 x x 0 at t t 0 these solutions demonstrate the existence of a waiting time t 0 before the free boundary moves a comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of waiting times for general initial data a three parameter family of solutions which exhibit blow up at t 0 is also constructed applications of these results to models for the spreading of a viscous blob on a hor

waiting time for a non newtonian polytropic filtration equation with convection abstract in this paper we study the waiting time phenomena for a class of non newtonian polytropic filtration equation with convection according to the nonlinearity in the diffusion and the convection terms some necessary and sufficient conditions on the initial data are given for the existence of waiting time

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

integrable quantum chain and the representation of the quantum group suq 2 the authors discuss the xxz spin chain with certain boundary terms and the representation of quantum group suq 2 it is shown that bethe ansatz states are highest weight states of suq 2 with a generic q they construct the irreducible representations of the quantum group for q being a root of unity they show that there are new bethe ansatz states which coincide with null states of suq 2 by taking certain limits the authors can derive the state mod b which is necessary for constructing the indecomposable but reducible representations of suq 2 and for the completeness of the state space in this case the hamiltonian may not be completely diagonalized

asymptotic behavior for the critical nonhomogeneous porous medium equation in low dimensions abstract we deal with the large time behavior for a porous medium equation posed in nonhomogeneous media with singular critical density x 2 t u x t u m x t x t r n 0 m 1 posed in dimensions n 1 and n 2 which are also interesting in applied models according to works by kamin and rosenau we deal with the cauchy problem with bounded and continuous initial data u 0 we show that in dimension n 2 the asymptotic profiles are self similar solutions that vary depending on whether u 0 0 0 or u 0 0 k 0 in dimension n 1 things are strikingly different and we find new asymptotic profiles of an unusual mixture between self similar and traveling wave forms we thus complete the study performed in previous recent works for the higher dimensions n 3

lagrangian non oscillatory and fem schemes for the porous media equation abstract interface tracking schemes derived from a lagrangian formulation of the equation are considered existence and equivalence results valid even for the non smooth interface case are given and non oscillatory schemes needed in this case are discussed mixed euler lagrange fem discretization for mixed problems and numerical experiments are presented

data sparse approximation to the operator valued functions of elliptic operator llin previous papers the arithmetic of hierarchical matrices has been described which allows us to compute the inverse for instance of finite element stiffness matrices discretising an elliptic operator l the required computing time is up to logarithmic factors linear in the dimension of the matrix in particular this technique can be used for the computation of the discrete analogue of a resolvent zi l 1 z c in the present paper we consider various operator functions the operator exponential e tl negative fractional powers l the cosine operator function cos t l l k and finally the solution operator of the lyapunov equation using the dunford cauchy representation we get integrals which can be discretised by a quadrature formula which involves the resolvents z k i l 1 mentioned above we give error estimates which are partly exponentially partly polynomially decreasing

counting solutions of the bethe equations of the quantum group invariant open xxz chain at roots of unity we consider the invariant open spin 1 2 xxz quantum spin chain of finite length n for the case that q is a root of unity we propose a formula for the number of admissible solutions of the bethe ansatz equations in terms of dimensions of irreducible representations of the temperley lieb algebra and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting modules these formulas include corrections that appear if two or more tilting modules are spectrum degenerate for the xx case we give explicit formulas for the number of admissible solutions and degeneracies we also consider the cases of generic q and the isotropic limit numerical solutions of the bethe equations up to n 8 are presented our results are consistent with the bethe ansatz solution being complete