deco polyominoes permutations and random generation abstract in this paper we introduce a class of polyominoes called deco polyominoes in bijection with the set of permutations of the first k integers we evaluate some typical parameters for this class of polyominoes and define a linear algorithm for randomly generating them moreover we define the polyominoes corresponding to the directed animals on the hexagonal and triangular lattices and examine these polyominoes deco class as far as deco polyominoes corresponding to animals on the hexagonal lattice are concerned we find a bijection with a class of permutations finally we give linear algorithms for randomly generating both hexagonal and triangular deco polyominoes

a new two variable generating function for convex self avoiding polygons the width of a self avoiding polygon on the square lattice is defined as the minimal horizontal or vertical distance between two of its parallel edges if the polygons are convex this distance is internal the perimeter generating functions for such convex polygons whose widths exceed a threshold can be given explicitly from these expressions a two variable width and perimeter generating function can be constructed the corresponding phase diagram shows two types of critical behaviour which meet at a tricritical point

convex polyominoes and heaps of segments builds a one to one correspondence between some polyominoes called parallelogram polyominoes and some heaps of segments then the author derives explicit expressions for the generating function of convex polyominoes according to their height width and area the author also enumerates two subsets of convex polyominoes namely the directed and convex polyominoes and the parallelogram polyominoes according to these three parameters generating functions of polyominoes having a fixed width are also studied

transverse vibration of simply supported elliptical and circular plates using boundary characteristic orthogonal polynomials in two variables abstract the first four frequencies and mode shapes have been computed for elliptical and circular plates of uniform thickness with simply supported edges by employing orthogonal polynomials in the rayleigh ritz method in general this is found to give better results than traditional methods successive approximations have been worked out to ensure convergence comparisons have been made with existing results in some cases calculations have been done for several values of b a i e the ratio of the semi major and semi minor axes mode shapes have been used to obtain three dimensional plots of plates in displaced configurations

stability and non normality of the k e equations abstract the analytical and numerical solutions of the equations of the k e turbulence model are analyzed under certain conditions on the boundary values and the interior values of k and e the analytical and numerical solutions are bounded if the steady state solution is obtained numerically by a runge kutta time stepping method then severe constraints on the time step and the non normality of the jacobian matrix make the convergence very slow the simplifications and conclusions are supported by data from a numerical solution of flow over a flat plate

the complexity of computing the number of strings of given length in context free languages abstract computing the number of strings of given length contained in a language is related to classical problems of combinatorics formal languages and computational complexity here we study the complexity of this problem in the case of context free languages it is shown that for unambiguous context free languages such a computation is easy and can be carried out by efficient parallel algorithms on the contrary for some context free languages of ambiguity degree two the problem becomes intractable these results are related to other classical subjects concerning counting problems exponential time recognizable languages and sparse sets

vibration of tapered thickness annular sector plates by spline element method abstract this paper deals with the determination of the vibrations of thin annular sector plates of tapered thickness by using the spline element method to demonstrate the accuracy of the present method several examples are solved and results are compared with those obtained by analytical methods and other numerical methods good accuracy is obtained frequencies of tapered annular sector plates with several boundary conditions are analyzed for the different sector angles and thickness parameters

a phase error analysis of multigrid methods for hyperbolic equations in this paper we study the effects of the coarse grid correction process on multigrid convergence for hyperbolic problems in one and two dimensions we approach this from the perspective of phase error which allows us to exploit the hyperbolic nature of the underlying pde in particular we consider three combinations of coarse grid operators and coarse grid solution approaches 1 inexact coarse grid solve with direct discretization 2 exact coarse grid solve with direct discretization and 3 exact coarse grid solve with galerkin coarse grid operator for all these approaches we show that the convergence behavior of multigrid can be precisely described by the phase error analysis of the coarse grid correction matrix and we verify our results by numerical examples in one and two dimensions

a tiling system for the class of l convex polyominoes a polyomino is said to be l convex if any two of its cells can be connected by a path entirely contained in the polyomino and having at most one change of direction in this paper answering a problem posed by castiglione and vaglica 6 we prove that the class of l convex polyominoes is tiling recognizable to reach this goal first we express the l convexity constraint in terms of a set of independent properties then we show that each class of convex polyominoes having one of these properties is tiling recognizable

rigorous derivation of the perimeter generating function for the mean squared radius of gyration of convex polygons we have rigorously derived the perimeter generating function for the mean squared radius of gyration of convex polygons this function was first conjectured by jensen his nonrigorous result is based on the analysis of the long series expansions

a new multigrid approach to convection problems abstract a new multigrid method for convection problems is presented it is designed to overcome the problem of alignment the flow being aligned with the grid the technique employs semi coarsening in several directions simultaneously and gives rise to multiple coarser grids with the same total number of points per grid level but with different sizes in each co ordinate direction the amount of work per multigrid cycle is still o n as an example the method is applied to the nonlinear upwind differenced euler equations of gas dynamics in two dimensions convergence rates are estimated by two level fourier analysis for the linearised equations numerical experiments on the nonlinear equations confirm these estimates

on the exhaustive generation of k convex polyominoes the degree of convexity of a convex polyomino p is the smallest integer k such that any two cells of p can be joined by a monotone path inside p with at most k changes of direction in this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates given an integer k all k convex polyominoes of area n in constant amortized time using space o n furthermore by applying few changes we are able to generate all convex polyominoes whose degree of convexity is exactly k

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

directed compact percolation near a wall iii exact results for the mean length and number of contacts existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall the results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below the results for the cluster length satisfy previously conjectured differential equations a closed expression is conjectured for the mean size above in terms of a hypergeometric function

grid independent convergence of the multigrid method for first order equations the multigrid method for numerical solution of systems of first order partial differential equations is analyzed the smoothing iterations are of polynomial type and are used on all grids under certain conditions the convergence of the iterations is independent of the grid size this is explained by the propagation of smooth error modes out through the boundary and the damping of oscillatory modes an upper bound is derived on the number of multigrid v cycles needed to solve a two dimensional scalar equation numerical experiments with a simple scalar equation and the euler equations confirm the theoretical results

transverse vibrations of a trapezoidal cantilever plate of variable thickness n atural frequencies of a trapezoidal cantilever plate of variable thickness which model an aircraft wing structure approximately are determined using energy techniques characteristic orthogonal polynomials in two variables are constructed to describe the structural deflections which are used in the rayleigh ritz method to obtain the natural frequencies the fundamental natural frequencies for different parameter values are also obtained using a deflection function containing an optimized exponent the first technique can be used to obtain the natural frequencies and corresponding mode shapes for the preliminary design of aircraft wing structures

generating functions for classes of self avoiding polygons on the square lattice abstract a transfer matrix technique is presented which allows the explicit calculation of the generating functions for those self avoiding polygons on the square lattice which have a perimeter equal to or two larger than the perimeter of their bounding rectangles

unidirectional convex polygons on the honeycomb lattice the authors obtain the exact generating function for a class of self avoiding polygons on the honeycomb lattice which are convex in one of the three lattice directions it is shown that the critical exponent alpha of these unidirectional convex polygons is 3 2

convex polyominoes and algebraic languages uses the so called dsv methodology that links some enumeration problems to the theory of algebraic languages to get a system of q difference equations involving the generating functions of convex polyominoes of convex and directed polyominoes and of parallelogram polyominoes according to their height width and area then the author shows various applications of this system

enumeration of 4 stack polyominoes in this paper we consider the class of 4 stack polyominoes i e polyominoes which can be decomposed into a central rectangle supporting four stack polyominoes one on each side of the rectangle this class of objects recently introduced by marc noy extends the class of centered convex polyominoes and is included into the class of z convex polyominoes using an inclusion exclusion approach we obtain the enumeration of 4 stack polyominoes according to the number of rows and columns moreover we solve some problems posed by marc noy proving that the generating function of 4 stack polyominoes according to the semi perimeter is algebraic and that their asymptotic behavior is n4 n which is immediately smaller than the asymptotic behavior of z convex polyominoes which is n4 n as a corollary of our result we find the generating function of bi centered polyominoes i e convex polyominoes which are both horizontally and vertically centered

on an involution on dyck paths les derniers resultats concernant l enumeration des polyamines parallelogrammes 4 5 8 9 11 14 selon l aire et le nombre de colonnes font apparaitre des fonctions generatrices s exprimant comme rapport de q analogues de fonctions de bessel dans 13 deux interpretations combinatoires de ces memes fonctions sont donnees d une part en termes de multiparcours d arbres et d autre part en termes de multichaines de chemins de dyck a l aide d une involution sur l ensemble d des chemins de dyck d une bijection entre chemins et multiparcours d arbres et d une bijection entre chemins et multichaines nous donnons dans ce papier une preuve bijective de l equivalence de ces deux interpretations

walks on the slit plane in the first part of this paper we enumerate exactly walks on the square lattice that start from the origin but otherwise avoid the half line we call them walks on the slit plane we count them by their length and by the coordinates of their endpoint the corresponding three variable is algebraic of degree 8 moreover for any point i j the length for walks of this type ending at i j is also algebraic of degree 2 or 4 and involves the famous catalan numbers

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

context free grammars differential operators and formal power series abstract in this paper we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substitution rules we call such a set of rules a context free grammar because these rules act like a context free grammar in the sense of a formal language given a context free grammar we can associate each formal function with an exponential formal power series in this way we obtain grammatical interpretations of addition multiplication and functional composition of formal power series a surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation we apply this method to obtain simple demonstrations of faa di bruno s formula and some identities concerning bell polynomials stirling numbers and symmetric functions in particular the lagrange inversion formula has a simple grammatical representation from this point of view one sees that cayley s formula on labeled trees is equivalent to the lagrange inversion formula

two generalizations of column convex polygons column convex polygons were first counted by area several decades ago and the result was found to be a simple rational generating function in this work we generalize that result let a p column polyomino be a polyomino whose columns can have 1 2 p connected components then column convex polygons are equivalent to 1 convex polyominoes the area generating function of even the simplest generalization namely 2 column polyominoes is unlikely to be solvable we therefore define two classes of polyominoes which interpolate between column convex polygons and 2 column polyominoes we derive the area generating functions of those two classes using extensions of existing algorithms the growth constants of both classes are greater than the growth constant of column convex polyominoes rather tight lower bounds on the growth constants complement a comprehensive asymptotic analysis

object grammars and bijections a new systematic approach for the specification of bijections between sets of combinatorial objects is presented it is based on the notion of object grammars object grammars give recursive descriptions of objects and generalize context free grammars the study of a particular substitution in these object grammars confirms once more the key role of dyck words in the domain of enumerative and bijective combinatorics

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

convergence analysis of iterative methods by pseudodifference operators abstract the convergence of iterative methods to solve linear partial differential equations numerically is analyzed by the theory of pseudodifference operators approximate inverses are determined from the symbol of the iterative operator so that implicit and preconditioned methods are covered the behavior of the symbol for higher wave numbers describes the convergence rate for the corresponding error modes the results are applied to krylov subspace methods stationary iterations of gauss seidel type the multigrid algorithm and time stepping methods for flow problems

statistics of collapsed lattice animals rigorous results and monte carlo simulations in this paper a two dimensional model of collapsed lattice animals with nearest neighbour contact interactions is introduced and the limiting free energy for the model is proved to exist we also establish that the partition function for collapsed lattice animals is related to the reliability polynomial for the m m grid graph and to the generating function for spanning forests on the m m grid graph from these relationships and using results about directed animals we are able to obtain rigorous upper and lower bounds on the free energy for collapsed lattice animals in addition we present monte carlo evidence that the heat capacity associated with this model remains bounded in the thermodynamic limit in fact we do not find any evidence for a second order phase transition for collapsed lattice animals

numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non uniform thickness abstract a comparison of natural frequencies of transverse vibration of thin rectangular plates of non uniform thickness with different combinations of boundary conditions is presented in this paper as obtained by using the following methodologies 1 the rayleigh ritz method with characteristic orthogonal polynomial shape functions 2 the rayleigh ritz method with a shape function which includes two exponents that are to be determined by minimizing the fundamental frequency coefficient 3 the optimized kantorovich method which has been proposed rather recently 4 the finite element method the results are presented in tabular form and discussed

fourier analysis of multigrid methods for general systems of pdes most iteration methods for solving boundary value problems can be viewed as approximations of a time dependent differential equation in this paper we show that the multigrid method has the effect of increasing the timestep for the smooth part of the solution leading back to an increase of the convergence rate for the nonsmooth part the convergence is an effect of damping fourier analysis is used to find the relation between the convergence rate for multigrid methods and singlegrid methods the analysis is performed for general partial differential equations and an arbitrary number of grids the difference in the behavior of the iterations between firstand second order equations is discussed the theoretical results are confirmed in simple numerical experiments

the perimeter generating functions of three choice imperfect and one punctured staircase polygons we consider the isotropic perimeter generating functions of three choice imperfect and one punctured staircase polygons whose 8th order linear fuchsian odes are previously known we derive simple relationships between the three generating functions and show that all three generating functions are joint solutions of a common 12th order fuchsian linear ode we find that the 8th order differential operators can each be rewritten as a direct sum of a direct product with operators no larger than 3rd order we give closed form expressions for all the solutions of these operators in terms of 2 f 1 hypergeometric functions with rational and algebraic arguments the solutions of these linear differential operators can in fact be expressed in terms of two modular forms since these 2 f 1 hypergeometric functions can be expressed with two rational or algebraic pullbacks

flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables a set of characteristic orthogonal polynomials in two variables is used as deflection functions to obtain the natural frequencies and mode shapes of polygonal plates by the rayleigh ritz method the set of orthogonal polynomials is constructed by employing the gram schmidt orthogonalization procedure the first six natural frequencies are numerically evaluated for triangular plates of different configurations by using these orthogonal functions and the corresponding mode shapes are plotted results are compared with those obtained previously by other methods