the intersection of the similarity and conjunctivity equivalence classes abstract let a be an n n complex matrix let sim a denote the similarity equivalence class of a let conj a denote the conjunctivity equivalence class of a and let u a denote the unitary similarity equivalence class of a define cs a sim a conj a we seek to classify the matrices that have cs a u a and show by an example that this is not true in general but we show that it is true when a is hermitian or is a scalar multiple of a hermitian for n 3 we reduce the general n n case to the case when a is non singular and not a multiple of a hermitian matrix we completely classify the 2 2 case and find that cs a u a when a is non singular or normal or both

a polynomial expression for the hilbert series of the quotient ring of diagonal coinvariants a special case of haiman s identity invent math 149 2002 pp 371 407 for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded hilbert series as a sum of rational functions in q t in this paper we show how a summation identity of garsia and zabrocki for macdonald polynomial pieri coefficients can be used to transform haiman s formula for the hilbert series into an explicit polynomial in q t with integer coefficients we also provide an equivalent formula for the hilbert series as the constant term in a multivariate laurent series

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

convergence properties of numerical discretizations and regridding methods abstract in this paper attention is focused on convergence properties on nonuniform grids of numerical discretizations of first and second order spatial derivatives which occur for example in the transport and heat equations the direct study of the discretization error equations lead us to the establishment of expressions for the global discretization error which depend on the grid properties from these expressions we conclude the convergence of the discretizations and also outline a grid selection strategy

on simple moving grid methods for one dimensional evolutionary partial differential equations abstract two moving grid algorithms for the numerical integration of evolutionary one dimensional partial differential equations are considered both algorithms discretize the equation on trapezoidal space time elements and combine a prediction step with a remeshing technique in order to determine the orientation of the sides of the trapezoids joining nodes at consecutive time levels the behavior of the discretizations employed is forecast by means of the modified equation technique and then studied in a series of numerical experiments

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

on supraconvergence phenomenon for second order centered finite differences on non uniform grids in the present note we consider an example of a boundary value problem for a simple second order ordinary differential equation which may exhibit a boundary layer phenomenon we show that usual central finite differences which are second order accurate on a uniform grid can be substantially upgraded to the fourth order by a suitable choice of the underlying non uniform grid this example is quite pedagogical and may give some ideas for more complex problems

on the characterization of finite differences optimal meshes abstract regridding methods has become an important tool in the integration of pde systems whose solutions exhibit sharp transitions in spatial derivatives this paper improves the results presented in an earlier contribution of the author and f oliveira 1988 theoretical justifications of finite differences regridding criteria for the transport and heat equations are presented the nonuniform meshes in the physical space are generated by the use of coordinate transforms which map them into uniform meshes in the computational space after the two mesh systems have been generated two approaches are used for solving the pde to construct the approximations on the uniform mesh in the computational space or to construct the finite difference approximations on the nonuniform mesh in the physical space in this paper we are concerned with the question of the relationship between the two approaches namely the characterization of the mesh density coordinate transform which improves the spatial accuracy of the approximation in the physical computational space

the optimization of convergence for chebyshev polynomial methods in an unbounded domain by using the method of steepest descent i have compared the suitability of three different methods for solving problems in a semi infinite or infinite domain using chebyshev polynomials exponential mappings are uniformyly bad domain truncation and algebraic mapping both work well but each is superior for a different category of problems when the solution is an entire function then domain truncation is best it is always simpler to apply than algebraic mapping and always at least as accurate much more accurate for functions which decay very rapidly for singular functions on the other hand algebraic mapping is better because it is less sensitive to the mapping scale factor l permitting a better compromise in resolving both the singularity and the exponential decay for both types of problems i give simple explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy for the model entire function exp azk on a semi infinite interval 0 the optimum domain size is l 0 896 na 1k and the smallest attainable error is roughly e n 0 896 k where n is the number of chebyshev polynomals used similar formulas for singular functions handled via algebraic mapping are given in 3 41 to 3 45 below

an adaptive moving grid method for one dimensional systems of partial differential equations we describe a fully adaptive moving grid method for solving initial boundary value problems for systems of one space dimensional partial differential equations whose solutions exhibit rapid variations in space and time the method based on finite differences is of the lagrangian type and has been derived through a co ordinate transformation which leads to equidistribution in space of the second derivative our technique is intermediate between static regridding methods where nodes remain fixed for intervals of time and continuously moving grid methods where the node movement and the pde integration are fully coupled in our approach the computation of the moving grids and the solution on these grids are carried out separately while the nodes are moved at each time step two error monitors have been implemented one to govern the time step selection and the other to eventually adapt the number of moving nodes the method allows the use of different moving grids for different components in the pde system numerical experiments are presented for a set of five sample problems from the literature including two problems from combustion

on the convergence on nonrectangular grids the convergence properties of a full discrete approximations to the convection diffusion equation is the subject of this paper the full discrete scheme considered is of lagrangian type euler implicit on time and centered finite difference on space and is defined using nonrectangular grids we analyse this scheme under smoothness conditions on nonrectangular space time grid the main result establish the convergence of the approximations and we prove that the assumptions on the discrete spatial nodes movement are achieved if we consider the equidistribution principle

a continuous space time finite element method for the wave equation we consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time thereby giving a unified treatment of the spatial and temporal discretizations our analysis uses primarily energy arguments which are quite common for spatial discretizations but not for time we present a priori nodal in time superconvergence error estimates without any special time step restrictions our method is based on tensor product spaces for the full discretization

conjectured statistics for the q t catalan numbers abstract we introduce the distribution function f n q t of a pair of statistics on catalan words and conjecture f n q t equals garsia and haiman s q t catalan sequence c n q t which they defined as a sum of rational functions we show that f n s q t defined as the sum of these statistics restricted to catalan words ending in s ones satisfies a recurrence relation as a corollary we are able to verify that f n q t c n q t when t 1 q we also show the partial symmetry relation f n q 1 f n 1 q by modifying a proof of haiman of a q lagrange inversion formula based on results of garsia and gessel we obtain a q analogue of the general lagrange inversion formula which involves catalan words grouped according to the number of ones at the end of the word

an adaptive method of lines with error control for parabolic equations of the reaction diffusion type abstract a piecewise linear finite element based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction diffusion processes in one space dimension the vertial lines emanating from the space nodes in this method change at automatically selected times when in order to control a norm of the space discretization error adaptive spatial regridding occurs the regridding algorithm is an extension of one described previously by the authors 7 and is implemented in the program femol 1 which uses the lsodi package 14 15 of hindmarsh and painter to integrate the ordinary differential equations in time along the vertical lines computational results show that the method is efficient that a posteriori estimates of the space discretization error are accurate and that the adaptive procedure reliably controls the space discretization error

a new adaptive mesh refinement strategy for numerically solving evolutionary pde s a graph based implementation of quadtree meshes for dealing with adaptive mesh refinement amr in the numerical solution of evolutionary partial differential equations is discussed using finite volume methods the technique displays a plug in feature that allows replacement of a group of cells in any region of interest for another one with arbitrary refinement and with only local changes occurring in the data structure the data structure is also specially designed to minimize the number of operations needed in the amr implementation of the new scheme allows flexibility in the levels of refinement of adjacent regions moreover storage requirements and computational cost compare competitively with mesh refinement schemes based on hierarchical trees low storage is achieved for only the children nodes are stored when a refinement takes place these nodes become part of a graph structure thus motivating the denomination autonomous leaves graph alg for the new scheme neighbors can then be reached without accessing their parent nodes additionally linear system solvers based on the minimization of functionals can be easily employed alg was not conceived with any particular problem or geometry in mind and can thus be applied to the study of several phenomena some test problems are used to illustrate the effectiveness of the technique

a finite difference formula for the discretization of d 3 dx 3 on nonuniform grids we analyze the use of a five point difference formula for the dis cretization of the third derivative operator on nonuniform grids the formula was derived so as to coincide with the standard five point formula on regular grids and to lead to skew symmetric schemes it is shown that under periodic boundary conditions the formula is supraconvergent in the sense that in spite of being inconsistent it gives rise to schemes with second order convergence however such a supraconvergence only takes place when the number of points in the grid is odd for grids with an even number of points the inconsistency of the formula results in lack of convergence both stationary and evolutionary problems are considered and the analysis is backed by numerical experiments

unitary similarity classes within the cospectral congruence class of a matrix abstract matrix b m n c is c s equivalent resp c e equivalent to a m n c if b is both congruent and similar to resp cospectral with a we are concerned with the number typically one or infinitely many of unitary similarity classes in the c s resp c e equivalence class of a given matrix the case n 2 and the general normal case are fully understood for c s equivalence also the singular case may generally be reduced to the nonsingular case the present work includes four main results 1 if 0 lies in the interior of the field of values of a nonsingular a m n n 3 then the c e equivalence class contains infinitely many unitary similarity classes 2 when 0 is not in the interior general sufficient conditions are given for the c e class and thus the c s class to contain only one unitary class 3 when n 3 these conditions are also necessary and a classification of all c e and c s classes is given 4 for n 3 it is shown that the matrices for which the c s class contains infinitely many unitary similarity classes are dense among all matrices

the representation of the symmetric group on m tamari intervals an m ballot path of size n is a path on the square grid consisting of north and east unit steps starting at 0 0 ending at mn n and never going below the line fx myg the set of these paths can be equipped with a lattice structure called them tamari lattice and denoted by t m n which generalizes the usual tamari lattice tn obtained when m 1 this lattice was introduced by f bergeron in connection with the study of diagonally coinvariant spaces in three sets ofn variables the representation of sn on these spaces is conjectured to be closely related to the natural representation of sn on labelled intervals of them tamari lattice studied in this paper an interval p q of t m n is labelled if the north steps of q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps the symmetric group sn acts on labelled intervals of t m n by permutation of the labels we prove an explicit formula conjectured by f bergeron and the third author for the character of the associated representation of sn in particular the dimension of the representation that is the number of labelledm tamari intervals of sizen is found to be m 1 n mn 1 n 2 these results are new even whenm 1 the form of these numbers suggests a connection with parking functions but our proof is not bijective the starting point is a recursive description of m tamari intervals it yields an equation for an associated generating function which is a refined version of the frobenius series of the representation the form of this equation is highly non standard it involves two additional variables x and y a derivative with respect to y and iterated divided differences with respect tox the hardest part of the proof consists in solving it and we develop original techniques to do so r

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

cambrian hopf algebras abstract cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the classical conditions for binary search trees similar to binary trees for the tamari lattice cambrian trees provide convenient combinatorial models for n reading s cambrian lattices of type a based on a natural surjection from signed permutations to cambrian trees we define the cambrian hopf algebra extending j l loday and m ronco s algebra on binary trees we describe combinatorially the products and coproducts of both the cambrian algebra and its dual in terms of operations on cambrian trees we then construct the baxter cambrian algebra which extends s law and n reading s baxter hopf algebra on rectangulations and s giraudo s equivalent hopf algebra on twin binary trees

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes