raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

representations of the discrete inhomogeneous lorentz group and dirac wave equation on the lattice we propose the fundamental and two dimensional representation of the lorentz groups on a 3 1 dimensional hypercubic lattice from which representations of higher dimensions can be constructed for the unitary representation of the discrete translation group we use the kernel of the fourier transform from the dirac representation of the lorentz group including reflections we derive in a natural way the wave equation on the lattice for spin 1 2 particles finally the induced representation of the discrete inhomogeneous lorentz group is constructed by standard methods and its connection with the continuous case is discussed

properties of harmonic measures in the dirichlet problem for nilpotent lie groups of heisenberg type in groups of heisenberg type we introduce a large class of domains which we call adp admissible for the dirichlet problem and we prove that on the boundary of such domains harmonic measure ordinary surface measure and the perimeter measure are mutually absolutely continuous we also establish the solvability of the dirichlet problem when the boundary datum belongs to l p 1 p with respect to the ordinary surface measure here the harmonic measure is that relative to a sub laplacian associated with a basis of the first layer of the lie algebra a domain is called adp if it is a nontangentially accessible domain and it satisfies an intrinsic outer ball condition

diffusions conditionnelles i hypoellipticite partielle this paper is devoted to the proof of certain results on conditional diffusions by using the malliavin calculus of variations it is shown that under minimal conditions on the vectors fields which define a two component diffusion xt zt and their lie brackets a s for any t 0 the conditional law of xt t 0 given b zs s t a c density

finite element simulation and efficient algorithm for fractional cahn hilliard equation abstract this article is devoted to designing efficient linear finite element algorithm for the fractional cahn hilliard equation an important and newly proposed phase field model combining the advantages of the classic bicg algorithm and the toeplitz like structure of the coefficient matrix we develop a fast bicg fbicg algorithm for the linearized scheme to compute numerically the fractional cahn hilliard equation our theoretical analysis and numerical experiments demonstrate that the proposed fbicg reduces the computation cost and the storage to o m log m and o m possesses the same convergence rates as newton s algorithm does in space and time and preserves the energy dissipation and equality laws the numerical experiments also demonstrate that the fbicg is almost mass conserved recognizes accurately the phase separation by a very clear coarse graining process and the influences of different indices r and s of fractional laplacian and different coefficients k and a on the width of the interfaces

raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non homogeneous lattices we complete the construction of raising and lowering operators given in a previous work for the orthogonal polynomials of hypergeometric type on a non homogeneous lattice we extend these operators to the generalized orthogonal polynomials namely those difference orthogonal polynomials that satisfy a similar difference equation of hypergeometric type

a reduced order energy stability preserving finite difference iterative scheme based on pod for the allen cahn equation abstract we focus on proposing and analyzing a reduced order finite difference rofd iterative scheme based on a proper orthogonal decomposition pod technique for allen cahn equations with a small perturbation parameter and strong nonlinearity we prove the discrete maximum principle dmp preserving and discrete energy stability des preserving of the rofd iterative solutions under a reasonable restriction on the time step size we also analyze the convergence of the rofd iterative solutions for allen cahn equations numerical tests are presented to verify our proposed rofd method such as error estimates convergence rates dmp preserving and des preserving moreover it is shown that the computing time of rofd iterative schemes with very few unknowns is much less than that of usual finite difference fd iterative schemes

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

the poisson kernel for certain degenerate elliptic operators abstract let l 1 2 k 1 d v k 2 v 0 be a smooth second order differential operator on r n written in hormander form and g be a bounded open set with smooth noncharacteristic boundary under a global condition that ensures that the dirichlet problem is well posed for l on g and a nondegeneracy condition at the boundary precisely the lie algebra generated by the vector fields v0 v1 vd is of full rank on the boundary then the harmonic measure for l starting at any point in g has a smooth density with respect to the natural boundary measure estimates on the derivatives of this density the poisson kernel similar to the classical estimates for the poisson kernel for the laplacian on a half space are given

last exit decompositions and regularity at the boundary of transition probabilities the purpose of this paper is to give a probabilistic approach to studying the regularity at the boundary of the transition probabilities of certain hypoelliptic diffusions with boundary conditions the main tools are last exit decompositions of brownian motion the malliavin calculus the theory of excursions and the calculus of variations on brownian excursions

pointwise error estimates of discontinuous galerkin methods with penalty for second order elliptic problems in this paper discontinuous galerkin methods with penalty for solving second order elliptic problems are considered error estimates are studied for these methods in particular optimal localized pointwise error estimates are established on quasi uniform grids in rn n ge 2

convergence of the cahn hilliard equation to the mullins sekerka problem in spherical symmetry abstract we show that as e 0 the solution of the cahn hilliard equation formula converges to a solution of the mullins sekerka problem formula where denotes a normal v the normal velocity and k the sum of principal curvatures of the interface provided the solutions are radially symmetric we use energy type estimates to show that the solution of the cahn hilliard equation can be approximated by the well known stationary wave solution that corresponds to the potential w

on lie algebras built from so 3 tensor operators concerning the lie algebras generated by certain linear combinations of so 3 tensor operators two theorems are given which disprove some widely accepted results in the standard literature

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

an energy stable linear diffusive crank nicolson scheme for the cahn hilliard gradient flow abstract we propose and analyze a linearly stabilized semi implicit diffusive crank nicolson scheme for the cahn hilliard gradient flow in this scheme the nonlinear bulk force is treated explicitly with two second order stabilization terms this treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation rigorous error analysis is carried out for the fully discrete scheme when the time step size and the space step size are small enough second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of 1 e here e is the thickness of the interface numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

a diffuse interface model and semi implicit energy stable finite element method for two phase magnetohydrodynamic flows abstract in this paper we propose a diffuse interface model and finite element approximation for two phase magnetohydrodynamic mhd flows with different viscosities and electric conductivities an energy stable scheme which is based on the finite element method for the spatial discretization and first order semi implicit scheme combined with convex splitting method for the temporal discretization is proposed to solve this new model the numerical scheme is proved to be mass conservative and energy law preserving by leray schauder fixed point theorem the existence of solutions to the numerical scheme is shown the uniqueness of the numerical solutions is obtained utilizing the stability of the numerical scheme and the compactness method the existence of the weak solutions to the two phase mhd model is established as well furthermore given more regularity on the weak solution the convergence of the numerical scheme is derived finally numerical experiments are provided to verify the theoretical results and validate the proposed model

energy decaying extrapolated rk sav methods for the allen cahn and cahn hilliard equations we construct and analyze a class of extrapolated and linearized runge kutta rk methods which can be of arbitrarily high order for the time discretization of the allen cahn and cahn hilliard

nontrivial zeros of the wigner 3 j and racah 6 j coefficients i linear solutions some formuals for nontrivial zeros in the 3 j and 6 j symbols have been found some formuals for nontrivial zeros in the 3 j and 6 j symbols have been found

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

the dirichlet problem for the kohn laplacian on the heisenberg group ii for x y t rn rn r denote xj xj 2yj t yj yj 2xj t and l 14 j 1nxj2 yj2 t when n 2q la represents the action of the kohn laplacian b on q forms on the heisenberg group for n n we construct a parametrix for the dirichlet problem in smooth domains d near non characteristic points of d a point w of d is non characteristic if one of x1 xn y1 yn is transverse to d at w this yields sharp local estimates in the dirichlet problem in the appropriate non isotropic lipschitz classes the main new tool is a convolution calculus of pseudo differential operators that can be applied to the relevant layer potentials for which the usual asymptotic composition formula is false characteristic points are treated in part ii

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

stratonovich s signatures of brownian motion determine brownian sample paths the signature of brownian motion in mathbb r d over a running time interval 0 t is the collection of all iterated stratonovich path integrals along the brownian motion we show that in dimension d ge 2 almost all brownian motion sample paths running up to time t are determined by their signature over 0 t

nontrivial zeros of the racah quadrupole invariant it is shown that a class of nontrivial zeros of the racah quadrupole invariant operator is given by two orbits of the group action of an infinite discrete subgroup of the proper two dimensional lorentz group so 1 1 on the hyperbola 4x2 3y2 11 4

a new parametrization and all integral realizations of the lorentz group a simple linear transformation of the biquaternionic parameters and of the group sl 2 c which is two to one homomorphic to the restricted lorentz group l is used to express each element of sl 2 c and l in terms of the first column of an element of l and the quaternionic parameters this parametrization is shown to bring the determination of the integral realizations of the restricted lorentz group under the purview of classical diophantine analysis involving the expression of a given pair of integers respectively as a sum of three squares and a sum of four squares these solutions are further constrained by the condition that a set of four integers which is linear in the solution of the three squares problem and in the solution of the four squares problem have a common factor these are all classical problems addressed and solved by euclid euler fermat gauss jacobi and others the corresponding realizations of sl 2 c fall into three distinct classes those in which the elements of

tensor product of group representations and structural zeros of racah coefficients it is shown that structural or nontrivial zeros of racah coefficients 6 j symbols can be related to decompositions of tensor products of group representations reduced to so 3 this extends a previous relation between structural zeros of 6 j symbols and boson realizations of exceptional lie groups in the present case both classical and exceptional groups may appear and examples are given for su 3 so 7 so 9 and f4

the double torus as a 2d cosmos groups geometry and closed geodesics the double torus provides a relativistic model for a closed 2d cosmos with topology of genus 2 and constant negative curvature its unfolding into an octagon extends to an octagonal tessellation of its universal covering the hyperbolic space h2 the tessellation is analysed with tools from hyperbolic crystallography actions on h2 of groups subgroups are identified for su 1 1 for a hyperbolic coxeter group acting also on su 1 1 and for the homotopy group 2 whose extension is normal in the coxeter group closed geodesics arise from links on h2 between octagon centres the direction and length of the shortest closed geodesics is computed dedicated to marcos moshinsky on the occasion of his 80th birthday

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

analysis of mixed interior penalty discontinuous galerkin methods for the cahn hilliard equation and the hele shaw flow this paper proposes and analyzes two fully discrete mixed interior penalty discontinuous galerkin dg methods for the fourth order nonlinear cahn hilliard equation both methods use the backward euler method for time discretization and interior penalty dg methods for spatial discretization they differ from each other on how the nonlinear term is treated one of them is based on fully implicit time stepping and the other uses energy splitting time stepping the primary goal of the paper is to prove the convergence of the numerical interfaces of the dg methods to the interface of the hele shaw flow this is achieved by establishing error estimates that depend on epsilon 1 only in some low polynomial orders instead of exponential orders similar to x feng and a prohl numer math 74 2004 pp 47 84 the crux is to prove a discrete spectrum estimate in the discontinuous galerkin finite element space however the validity of such a result is not obvious because the dg space is not a subspa