two grid method for the two dimensional time dependent schrodinger equation by the finite element method abstract in this paper we construct a backward euler full discrete two grid finite element scheme for the two dimensional time dependent schrodinger equation with this method the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two poisson equations on the same fine grid we analyze the error estimate of the standard finite element solution and the two grid solution in the h 1 norm it is shown that the two grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h k k 1 finally a numerical experiment indicates that our two grid algorithm is more efficient than the standard finite element method

liapunov schmidt reduction and continuation for nonlinear schro dinger equations we study the bifurcation scenario of nonlinear schro dinger equations nls the liapunov schmidt reduction is applied to show that the simple bifurcations of a single nls are pitchfork the pitchfork bifurcation can be subcritical or supercritical depending on the coefficient of the cubic term we choose we also describe numerical methods so that the liapunov schmidt reduction can effectively handle a corank 2 bifurcation point next we apply numerical continuation methods to trace solution curves and surfaces of the nls where the system is discretized by the centered difference approximations numerical results on two and three dimensional m coupled nls are reported where the physical properties such as the effect of trapping potentials isotropic and nonisotropic trapping potentials mass conservation constraints and strong and weak repulsive interactions are considered in our numerical experiments

on transitive algebras containing a standard finite von neumann subalgebra abstract let m be a finite von neumann algebra acting on a hilbert space h and a be a transitive algebra containing m in this paper we prove that if a is 2 fold transitive then a is strongly dense in b h this implies that if a transitive algebra containing a standard finite von neumann algebra in the sense of u haagerup the standard form of von neumann algebras math scand 37 1975 271 283 is 2 fold transitive then a is strongly dense in b h non selfadjoint algebras related to free products of finite von neumann algebras e g l f n and m 2 c 1 2 tr m 2 c 1 2 tr are studied brown measures of certain operators in m 2 c 1 2 tr m 2 c 1 2 tr are explicitly computed

the non microstates free entropy dimension of dt operators abstract dykema and haagerup introduced the class of dt operators amer j math 126 2004 121 189 and also showed that every dt operator generate l f 2 j funct anal 209 2004 332 366 the von neumann algebra generated by the free group on two generators in this paper we prove that voiculescu s non microstates free entropy dimension is 2 for all dt operators

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

transition probabilities for symmetric jump processes we consider symmetric markov chains on the integer lattice in d dimensions where a 0 2 and the conductance between x and y is comparable to x y d we establish upper and lower bounds for the transition probabilities that are sharp up to constants

resolvents of r diagonal operators we consider the resolvent a 1 of any diagonal operator a in a ii 1 factor our main theorem theorem 1 1 gives a universal asymptotic formula for the norm of such a resolvent en route to its proof we calculate the transform of the operator c 2 where c is voiculescu s circular operator and we give an asymptotic formula for the negative moments of a 2 for any diagonal a we use a mixture of complex analytic and combinatorial techniques each giving finer information where the other can give only coarse detail in particular we introduce partition structure diagrams in section 4 a new combinatorial structure arising in free probability

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

harnack inequality for some classes of markov processes in this paper we establish a harnack inequality for nonnegative harmonic functions of some classes of markov processes with jumps

r cyclic families of matrices in free probability we introduce the concept of r cyclic family of matrices with entries in a noncommutative probability space the definition consists in asking that only the cyclic noncrossing cumulants of the entries of the matrices are allowed to be nonzero let a1 as be an r cyclic family of d d matrices over a noncommutative probability space a we prove a convolution type formula for the explicit computation of the joint distribution of a1 as considered in md a with the natural state in terms of the joint distribution considered in the original space a of the entries of the s matrices several important situations of families of matrices with tractable joint distributions arise by application of this formula moreover let a1 as be a family of d d matrices over a noncommutative probability space a let d md a denote the algebra of scalar diagonal matrices and let c be the subalgebra of md a generated by a1 as d we prove that the r cyclicity of a1 as is equivalent to a property of c namely that c is free from md c with amalgamation over d

two grid method for two dimensional nonlinear schrodinger equation by mixed finite element method abstract a conservative two grid mixed finite element scheme is presented for two dimensional nonlinear schrodinger equation one newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse grid solution as the initial guess moreover error estimates are conducted for the two grid method it is shown that the coarse space can be extremely coarse with no loss in the order of accuracy and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 2 in the two grid method the numerical results show that this method is very effective

numerical solution of two dimensional nonlinear schrodinger equation using a new two grid finite element method abstract a new two grid finite element scheme is presented for two dimensional nonlinear schrodinger equation one newton iteration is applied on the fine grid to linearize the nonlinear system using the coarse grid solution as the initial guess and furthermore one more linear system on the coarse space is solved the error estimations of the two grid solution in the l 2 and h 1 norm are given it is shown both theoretically and numerically that the coarse space can be extremely coarse and two grid algorithm still achieves optimal approximation as long as the mesh sizes satisfy h o h 1 3

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

lyapunov exponents of free operators abstract lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system this paper offers a definition of lyapunov exponents for a sequence of free linear operators the definition is based on the concept of the extended fuglede kadison determinant we establish the existence of lyapunov exponents derive formulas for their calculation and show that lyapunov exponents of free variables are additive with respect to operator product we illustrate these results using an example of free operators whose singular values are distributed by the marchenko pastur law and relate this example to c m newman s triangle law for the distribution of lyapunov exponents of large random matrices with independent gaussian entries as an interesting by product of our results we derive a relation between the extended fuglede kadison determinant and voiculescu s s transform

multiple bifurcations in a reaction diffusion problem abstract we study multiple bifurcations in a system of reaction diffusion equations defined on the unit square first we investigate linear stability of the system at the uniform steady state solution then we discuss necessary conditions for mode interactions in particular we show that a corank four bifurcation point of the system can be easily generated by the mode interactions of two double bifurcation points we also study the possibilities of preserving the multiplicity of bifurcation points in the discrete system a continuation bcg algorithm proposed by chien et al is exploited to trace the solution branches finally sample numerical results are reported

asymptotic moments of some random vandermonde matrices abstract appropriately normalized square random vandermonde matrices based on independent random variables with uniform distribution on the unit circle are studied it is shown that as the matrix sizes increases without bound with respect to the expectation of the trace there is an asymptotic distribution equal to that of a c 0 1 valued r diagonal element

dt operators and decomposability of voiculescu s circular operator the dt operators are introduced one for every pair c consisting of a compactly supported borel probability measure on the complex plane and a constant c 0 these are operators on hilbert space that are defined as limits in moments of certain upper triangular random matrices the dt operators include voiculescu s circular operator and elliptic deformations of it as well as the circular free poisson operators we show that every dt operator is strongly decomposable we also show that a dt operator generates a ii1 factor whose isomorphism class depends only on the number and sizes of atoms of those dt operators that are also r diagonal are identified for a quasi nilpotent dt operator t we find the distribution of t t and a recursion formula for general moments of t

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

continuous family of invariant subspaces for r diagonal operators we show that every r diagonal operator x has a continuous family of invariant subspaces relative to the von neumann algebra generated by x this allows us to find the brown measure of x and to find a new conceptual proof that voiculescu s s transform is multiplicative our considerations base on a new concept of r diagonality with amalgamation for which we give several equivalent characterizations

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

uniqueness in law for pure jump markov processes let a be the operator defined on c2 functions by af left x right smallint left f left x h right f left x right f left x right h 1 1 1 left h right right v left x dh right sufficient conditions are given for existence and uniqueness for the martingale problem associated with a in the case of stable like processes where v x dh is equal to the levy measure for the stable symmetric process of index x for each x the conditions reduce to x continuous for existence and x dini continuous for uniqueness

a two grid discretization scheme for semilinear elliptic eigenvalue problems we present a two grid finite element discretization scheme with a two loop continuation algorithm for tracing solution branches of semilinear elliptic eigenvalue problems first we use the predictor corrector continuation method to compute an approximating point for the solution curve on the coarse grid then we use this approximating point as a predicted point for the solution curve on the fine grid in the corrector step we solve the first and the second order approximations of the nonlinear pde to obtain corrections for the state variable on the fine grid and the coarse grid respectively the continuation parameter is updated by computing the rayleigh quotient on the fine space to guarantee the approximating point we just obtained lies on the solution curve we perform newton s method we repeat the process described above until the solution curve on the fine space is obtained we show how the singular points such as folds and bifurcation points can be well approximated comprehensive numerical experiments show that the two grid finite element discretization scheme with a two loop continuation algorithm is efficient and robust for solving second order semilinear elliptic eigenvalue problems

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

rectangular random matrices related convolution we characterize asymptotic collective behavior of rectangular random matrices the sizes of which tend to infinity at different rates it appears that one can compute the limits of all noncommutative moments thus all spectral properties of the random matrices we consider because when embedded in a space of larger square matrices independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra therefore we can define a rectangular free convolution which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values this convolution is linearized by cumulants and by an analytic integral transform that we called the rectangular r transform

a spectral galerkin continuation method for numerical solutions of the gross pitaevskii equation abstract we study spectral galerkin methods sgm for nonlinear eigenvalue problems where the legendre polynomials are used as the basis functions for the trial function space the sgm is applied to find the ground state solution of the gross pitaevskii equation gpe and the gpe in a periodic potential when the sgm is incorporated in the context of continuation methods for curve tracking a manifest advantage is that the target points are very close to the bifurcation points thus it takes less continuation steps to reach the target point when compared with the centered difference methods cdm or finite element methods fem we also implement the spectral collocation method scm so that the computational cost for the numerical integration in the sgm can be saved comprehensive numerical experiments on the gpe using various numerical methods described in this paper are reported

r diagonal dilation semigroups this paper addresses extensions of the complex ornstein uhlenbeck semigroup to operator algebras in free probability theory if a 1 a k are free fancyscript r diagonal operators in a ii1 factor then d t a i 1 cdots a i n e nt a i 1 cdots a i n defines a dilation semigroup on the non self adjoint operator algebra generated by a 1 a k we show that d t extends in two different ways to a semigroup of completely positive maps on the von neumann algebra generated by a 1 a k moreover we show that d t satisfies an optimal ultracontractive property d t colon l 2 to l infty sim t 1 for small t 0

two grid discretization schemes for nonlinear schrodinger equations we study efficient two grid discretization schemes with two loop continuation algorithms for computing wave functions of two coupled nonlinear schrodinger equations defined on the unit square and the unit disk both linear and quadratic approximations of the operator equations are exploited to derive the schemes the centered difference approximations the six node triangular elements and the adini elements are used to discretize the pdes defined on the unit square the proposed schemes also can compute stationary solutions of parameter dependent reaction diffusion systems our numerical results show that it is unnecessary to perform quadratic approximations

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

a two grid method for expanded mixed finite element solution of semilinear reaction diffusion equations we present a scheme for solving two dimensional semilinear reaction diffusion equations using an expanded mixed finite element method to linearize the mixed method equations we use a two grid algorithm based on the newton iteration method the solution of a non linear system on the fine space is reduced to the solution of two small one linear and one non linear systems on the coarse space and a linear system on the fine space it is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h1 3 as a result solving such a large class of non linear equation will not be much more difficult than solving one single linearized equation copyright 2003 john wiley sons ltd

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models