nodal surface conjectures for the convex quantum billiard stemming from known properties of one dimensional 1 d and 2 d quantum billiards it is conjectured that the nodal surface of the first excited state of the convex 3 d quantum billiard intersects the billiard surface in a single simple closed curve examples of the validity of this conjecture are given for a number of elementary 3 d billiard configurations from these examples a second conjecture is introduced that addresses convex quantum billiards which are figures of rotation and contain one and only one plane of mirror symmetry normal to the axis of rotation two characteristic displacement parameters are defined which are labeled an axis length l and diameter a it is conjectured that a parameter 1 exists whose exact value depends on the properties of the billiard such that for l a prolatelike the nodal surface of the first excited state of a quantum billiard is a plane surface of mirror symmetry which divides the length of the billiard in half for l a oblatelike the nodal surface of the first excited state is a plane surface of mirror symmetry which contains the rotation axis and divides the diameter of the billiard in half arguments are given in support of a third conjecture which addresses the regular polyhedra quantum billiards termed spherical like it is hypothesized that the nodal surface of the first excited state for any of these billiards is any plane of reflection symmetry of the given polyhedron

symmetry preserving difference schemes for some heat transfer equations lie group analysis of differential equations is a generally recognized method which provides invariant solutions integrability conservation laws etc in this paper we present three characteristic examples of the construction of invariant difference equations and meshes where the original continuous symmetries are preserved in discrete models conservation of symmetries in difference modelling helps to retain qualitative properties of the differential equations in their difference counterparts

on the joint source channel coding error exponent for discrete memoryless systems we investigate the computation of csisza spl acute r s bounds for the joint source channel coding jscc error exponent e sub j of a communication system consisting of a discrete memoryless source and a discrete memoryless channel we provide equivalent expressions for these bounds and derive explicit formulas for the rates where the bounds are attained these equivalent representations can be readily computed for arbitrary source channel pairs via arimoto s algorithm when the channel s distribution satisfies a symmetry property the bounds admit closed form parametric expressions we then use our results to provide a systematic comparison between the jscc error exponent e sub j and the tandem coding error exponent e sub t which applies if the source and channel are separately coded it is shown that e sub t spl les e sub j spl les 2e sub t we establish conditions for which e sub j e sub t and for which e sub j 2e sub t numerical examples indicate that e sub j is close to 2e sub t for many source channel pairs this gain translates into a power saving larger than 2 db for a binary source transmitted over additive white gaussian noise awgn channels and rayleigh fading channels with finite output quantization finally we study the computation of the lossy jscc error exponent under the hamming distortion measure

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

solution of the schrodinger equation for a particle in an equilateral triangle the complete solution for the quantum mechanical problem of a particle in an equilateral triangle is derived by use of projection operators eigenfunctions belonging explicitly to each of the irreducible representations of the symmetry group c3v are constructed the most natural definition of the quantum numbers p and q includes not only integers but also nonintegers of the class 1 3 and 2 3 modulo 1 some relevant features relating to symmetry and degeneracy are also discussed

comparison of some lie symmetry based integrators lie symmetry based integrators are constructed in order to preserve the local invariance properties of the equations the geometrical methods leading to discretized equations for numerical computations involve many different concepts therefore they give rise to numerical schemes that vary in the accuracy in the computational cost and in the implementation in this paper a comparison is made between some alternative lie symmetry based methods illustrated on the example of the burgers equation the importance of the symmetry preservation is numerically highlighted

new performance bounds for turbo codes we derive a new upper bound on the word and bit error probabilities of turbo codes with maximum likelihood decoding by using the gallager bound since the derivation of the bound for a given interleaver is intractable we assume uniform interleaving as in the derivation of the standard union bound for turbo codes the result is a generalization of the transfer function bound and remains useful for a wider range of signal to noise ratios particularly for some range below the channel cutoff rate the new bound is also applicable to other linear codes

analytically solvable dynamical systems which are not integrable abstract the central two fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class toward this end information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity hence all systems exhibiting null metric entropy including ergodic only and mixing only systems have orbits which can be specified by algorithmically meaningful analytic expressions whose information content is very much less than that of the orbits they describe in consequence we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable called a integrable herein after establishing the existence of a integrability this paper turns to the problem of exposing a specific nonintegrable but nonetheless a integrable system whose algorithmically meaningful solution can be explicitly derived the billiard moving within a plane polygon having the shape of a 60 120 rhombus is shown to be a simple example indeed the whole class of billiards moving in polygons whose angles are all rational multiples of have null metric entropy and are therefore a integrable systems moreover all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by bjorn birnir because rational billiards are dense in the set of all plane billiards including chaotic ones the present paper suggests that a integrable billiards may provide a long sought analytic route to chaos finally it is pointed out that null entropied a integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer

invariant discretization schemes for the shallow water equations invariant discretization schemes are derived for the one and two dimensional shallow water equations with periodic boundary conditions while originally designed for constructing invariant finite difference schemes we extend the use of difference invariants to allow construction of invariant finite volume methods as well it is found that the classical invariant schemes converge to the lagrangian formulation of the shallow water equations these schemes require redistribution of the grid points according to the physical fluid velocity i e the mesh cannot remain fixed in the course of the numerical integration invariant eulerian discretization schemes are proposed for the shallow water equations in computational coordinates instead of using the fluid velocity as the grid velocity an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level the numerical conservation of energy mass and momentum is evaluated for both the invariant

schrodinger equation for convex plane polygons a tiling method for the derivation of eigenvalues and eigenfunctions motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem

invariantization of the crank nicolson method for burgers equation abstract in this article a geometric technique to construct numerical schemes for partial differential equations pdes that inherit lie symmetries is proposed the moving frame method enables one to adjust the numerical schemes in a geometric manner and systematically construct proper invariant versions of them to illustrate the method we study invariantization of the crank nicolson scheme for burgers equation with careful choice of normalization equations the invariantized schemes are shown to surpass the standard scheme successfully removing numerical oscillation around sharp transition layers

convecting reference frames and invariant numerical models abstract in the recent paper by bernardini et al 1 the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied in a simplified investigation carried out using the viscous burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow here we further discuss this problem and relate it to known results on invariant discretization schemes non invariant and invariant finite difference discretizations of burgers equation are proposed and compared with the discretization using the remedy proposed by bernardini et al

difference schemes with point symmetries and their numerical tests symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented they have the same three or four dimensional symmetry groups as the original differential equations the new difference schemes are tested as numerical methods the obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost for an example involving a solution with a singularity in the integration region the symmetry preserving scheme contrary to standard ones provides solutions valid beyond the singular point

non periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons this paper proves the existence of non periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons for anyn 3 there exists a corresponding convexn agon forn 3 this will be a right triangle with a small acute angle while in three dimensional space it will be a prism then agon serving as the base

circular sector quantum billiard and allied configurations the circular sector quantum billiard problem is studied numerical evaluation of the zeros of first order bessel functions finds that there is an abrupt change in the nodal line structure of the first excited state of the system equivalently second eigenstate of the laplacian at the critical sector angle c 0 354 for sector angle 0 in the domain 0 0 c the nodal curve of the first excited state is a circular arc segment for c 0 the nodal curve of the first excited state is the bisector of the sector otherwise nondegenerate first excited states become twofold degenerate at the critical angle c the ground and first excited state energies eg e1 increase monotonically as 0 decreases from its maximum value a graph of e1 vs 0 reveals an inflection point at 0 c which is attributed to the change in bessel function contribution to the development of e1 a proof is given for the existence of a common zero for two bessel functions whose respective orders differ by a noninteger applicat

spatial structure of the focusing singularity of the nonlinear schro dinger equation a geometrical analysis the cubic nonlinear schrodinger equation has solutions that become singular in finite time for spatial dimension d geq 2 numerical simulations demonstrate that in three dimensions the blowup is self similar and symmetric with the structure described by a nonlinear nonautonomous profile equation in two dimensions the blowup is again believed to be symmetric but the self similarity is weakly broken with structure described by the profile equation in the limit as d tends to 2 from above this paper gives a proof of the existence of a locally unique solution to the profile equation for d 2 and sufficiently close to 2 satisfying the boundary and global conditions associated with the blowup solution dynamical systems methods are used to transform previously derived asymptotic analysis into constructions of manifolds of solutions satisfying the relevant boundary conditions and to follow these manifolds to show that they have a transverse intersection

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

function mixing hypothesis and quantum chaos abstract billiards are studied whose boundaries comprise two or more segments which allow local but not global separation of the helmholtz equation related local solutions are labeled sep functions such billiards comprise the and sets a definition of quantum chaos for these sets of billiards is presented based on the ratio of the fluctuation length of the wave function nodal pattern to the c diameter of the billiard the function mixing hypothesis states that a sufficient condition for a billiard to be chaotic is that the billiard be an element of one of these sets it further ascribes such chaotic behavior to be due to mixing of dissimilar sep functions examples of the application of this hypothesis are described a set theoretic formalism is introduced to describe perturbation theory for infinite potentials and applied to the concave sided square billiard of sufficiently small concavity it is concluded that the adiabatic theorem of quantum mechanics does not apply to this configuration in the limit of large quantum numbers

orthogonal multiple access with correlated sources achievable region and pragmatic schemes in this paper we consider orthogonal multiple access coded schemes with correlated sources the sources are distributedly encoded and their correlation is exploited at the access point ap in particular we assume that each source uses a classical channel code to transmit through an additive white gaussian noise awgn channel its information to the ap here component decoders associated with the source encoders iteratively exchange soft information by taking into account the source correlation the key goal of this paper is to investigate the ultimate achievable performance limits in terms of a multidimensional feasible region in the space of channel parameters deriving insights on the impact of the number of sources in order to analyze the performance of given coded schemes we propose an extrinsic information transfer exit based approach which allows to determine the corresponding multi dimensional feasible regions on the basis of the proposed analytical framework the performance of pragmatic schemes based on serially concatenated convolutional codes scccs and low density parity check ldpc codes is discussed

a linear encoding approach to index assignment in lossy source channel coding we present a general scheme for the lossy transmission of a source with arbitrary statistics through a noisy channel under the mean square error fidelity criterion our approach is based on transform coding scalar quantization of the transform coefficients and linear encoding of the quantization indices entropy coding and channel coding are merged into a single linear encoding function such that the catastrophic behavior of conventional entropy coding is avoided and the full power of modern coding techniques and iterative belief propagation decoding can be exploited we show that this approach is asymptotically optimal in the limit of large block length for arbitrary source statistics and binary input output symmetric channel in the practical regime of finite block length and low decoding complexity we show through the explicit construction of codes for the lossy transmission of digital images over a binary symmetric channel that our approach yields significant improvements with respect to previously proposed channel optimized quantization schemes and also with respect to the conventional concatenation of state of the art image coding with state of the art channel coding although our constructive example focuses on a special case the approach is general and can be applied to other classes of sources of practical interest

symmetries of the discrete burgers equation a discrete cole hopf transformation is used to derive a discrete burgers equation that is linearizable to a discrete heat equation a five dimensional symmetry algebra is obtained that reduces to the lie point symmetry algebra of the usual burgers equation in the continuous limit this lie algebra is used to obtain explicit invariant solutions

symmetries and first integrals of ordinary difference equations this paper describes a new symmetry based approach to solving a given ordinary difference equation by studying the local structure of the set of solutions we derive a systematic method for determining one parameter lie groups of symmetries in closed form such groups can be used to achieve successive reductions of order if there are enough symmetries the difference equation can be completely solved several examples are used to illustrate the technique for transitive and intransitive symmetry groups it is also shown that every linear second order ordinary difference equation has a lie algebra of symmetry generators that is isomorphic to sl 3 the paper concludes with a systematic method for constructing first integrals directly which can be used even if no symmetries are known

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

the korteweg de vries equation and its symmetry preserving discretization the korteweg de vries equation is one of the most important nonlinear evolution equations in the mathematical sciences in this article invariant discretization schemes are constructed for this equation both in the lagrangian and in the eulerian form we also propose invariant schemes that preserve the momentum numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes we find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving galilean transformations which is demonstrated numerically as well

map decoding for multi antenna systems with non uniform sources exact pairwise error probability and applications we study the maximum a posteriori map decoding of memoryless non uniform sources over multiple antenna channels our model is general enough to include space time coding blast architectures and single transmit multi receive antenna systems which employ any type of channel coding we derive a closed form expression for the codeword pairwise error probability pep of general multi antenna codes using moment generating function and laplace transform arguments we then consider space time orthogonal block stob coding and prove that similar to the maximum likelihood ml decoding case detection of symbols is decoupled in map decoding we also derive the symbol pep in closed form for stob codes we apply these results in several scenarios first we design a binary antipodal signaling scheme which minimizes the system bit error rate ber under stob coding at a ber of 10 6 this constellation has a channel signal to noise ratio csnr gain of 4 7 db over conventional bpsk signaling for a binary nonuniform source with p0 delta p 0 0 9 we next design space time linear dispersion ld codes which are optimized for the source distribution under the criterion of minimizing the union upper bound on the frame error rate fer two codes are given here one outperforms v blast by 3 5 db and alamouti s code by 12 3 db at an fer of 10 2 for a binary source with p0 0 9 and the other outperforms v blast by 4 2 db at an fer of 10 3 for a uniform source these codes also outperform the ld codes of constructed under a different criteria finally the problem of bit to signal mapping is studied it is shown that for a binary source with p0 0 9 64 qam signaling and ser 10 3 a gain of 3 7 db can be achieved using a better than gray mapping for a system with one transmit and two receive antennas that uses trellis coding with 16 qam signaling a 1 8 db gain over quasi gray mapping and ml decoding is observed when map decoding is used for binary sources with p0 0 9

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

the polygon quantum billiard problem the present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates the polygon ground state theorem is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the elemental polygons defined in the text the polygon first excited state theorem is established which states that for every n sided regular polygon n equivalent first excited states exist each of which contains a nodal curve that is a line of mirror symmetry of the related polygon a vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the n sided regular polygon is spanned by any two of these n nodal diagonal eigenstates i e the first excited state is twofold degenerate at various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations thus for example correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive nonoverlapping classes of polygons stemming from the preceding conclusions a discussion is included on the possibility of employing transverse magnetic tm modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

new self similar solutions of the nonlinear schrodinger equation with moving mesh computations we study the blow up self similar solutions of the radially symmetric nonlinear schrodinger equation nls given by iut urr d 1rur u u 2 with dimension d 2 these solutions become infinite in a finite time t by a series of careful numerical computations partly supported by analytic results we demonstrate that there is a countably infinite set of blow up self similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint these solutions are characterised by the number of oscillations in their amplitude when d is close to 2 the solutions are computed as functions of d and their behaviour in the critical limit as d 2 is investigated the stability of these solutions is then studied by solving the nls by using an adaptive numerical method this method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation we demonstrate that the single humped self similar solution is globally stable whereas the multi humped solutions all appear to be unstable

on homoclinic structure and numerically induced chaos for the nonlinear schro dinger equation it has recently been demonstrated that standard discretizations of the cubic nonlinear schrodinger nls equation may lead to spurious numerical behavior in particular the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the nls equation in this paper an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated differences between an integrable discretization and standard discretizations are highlighted

the jacobi last multiplier for linear partial difference equations we present a discretization of the jacobi last multiplier with some applications to the computation of solutions of linear partial difference equations

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

transmission of nonuniform memoryless sources via nonsystematic turbo codes we investigate the joint source channel coding problem of transmitting nonuniform memoryless sources over binary phase shift keying modulated additive white gaussian noise and rayleigh fading channels via turbo codes in contrast to previous work recursive nonsystematic convolutional encoders are proposed as the constituent encoders for heavily biased sources we prove that under certain conditions and when the length of the input source sequence tends to infinity the encoder state distribution and the marginal output distribution of each constituent recursive convolutional encoder become asymptotically uniform regardless of the degree of source nonuniformity we also give a conjecture which is empirically validated on the condition for the higher order distribution of the encoder output to be asymptotically uniform irrespective of the source distribution consequently these conditions serve as design criteria for the choice of good encoder structures as a result the outputs of our selected nonsystematic turbo codes are suitably matched to the channel input since a uniformly distributed input maximizes the channel mutual information and hence achieves capacity simulation results show substantial gains by the nonsystematic codes over previously designed systematic turbo codes furthermore their performance is within 0 74 1 17 db from the shannon limit finally we compare our joint source channel coding system with two tandem schemes which employ a fourth order huffman code performing near optimal data compression and a turbo code that either gives excellent waterfall bit error rate ber performance or good error floor performance at the same overall transmission rate our system offers robust and superior performance at low bers 10 sup 4 while its complexity is lower

discretization of partial differential equations preserving their physical symmetries a procedure for obtaining a minimal discretization of a partial differential equation preserving all of its lie point symmetries is presented minimal in this case means that the differential equation is replaced by a partial difference scheme involving n difference equations where n is the number of independent and dependent variables we restrict ourselves to one scalar function of two independent variables as examples invariant discretizations of the heat burgers and korteweg de vries equations are presented some exact solutions of the discrete schemes are obtained

continuous symmetries of lagrangians and exact solutions of discrete equations one of the difficulties encountered when studying physical theories in discrete space time is that of describing the underlying continuous symmetries like lorentz or galilei invariance one of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices in a previous article we have classified ordinary difference schemes invariant under lie groups of point transformations the present article is devoted to an invariant lagrangian formalism for scalar single variable difference schemes the formalism is used to obtain first integrals and explicit exact solutions of the schemes equations invariant under two and three dimensional groups of lagrangian symmetries are considered

construction of invariant compact finite difference schemes in this paper we propose a method which is based on equivariant moving frames for development of high order accurate invariant compact finite difference schemes that preserve lie symmetries of underlying partial differential equations in this method variable transformations that are obtained from the extended symmetry groups of partial differential equations pdes are used to transform independent and dependent variables and derivative terms of compact finite difference schemes constructed for these pdes such that the resulting schemes are invariant under the chosen symmetry groups the unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames in some cases owing to selection of convenient moving frames numerical representation of invariant or symmetry preserving compact numerical schemes is found to be notably simpler than that of standard noninvariant compact numerical schemes further the accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite difference algorithms application of the proposed method is demonstrated through construction of invariant compact finite difference schemes for some common linear and nonlinear pdes including the linear advection diffusion equation in one or two dimensions the inviscid burgers equation in one dimension viscous burgers equation in one or two dimensions spherical burgers equation in one dimension and shallow water equations in two dimensions results obtained from our numerical simulations indicate that invariant compact finite difference schemes not only inherit selected symmetry properties of underlying pdes but are also comparably more accurate than the standard noninvariant base numerical schemes considered here

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

improved image decoding over noisy channels using minimum mean squared estimation and a markov mesh joint source channel jsc decoding based on residual source redundancy is a technique for providing channel robustness to quantized data previous work assumed a model equivalent to viewing the encoder noisy channel tandem as a discrete hidden markov model hmm with transmitted indices the hidden states we generalize this hmm based 1 d approach for images using the more powerful hidden markov mesh random field hmmrf model while previous state estimation methods for hmmrfs base estimates on only a causal subset of the observed data our new method uses both causal and anticausal subsets for jsc based image decoding the new method provides significant benefits over several competing techniques

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

symmetry preserving discretization of ordinary differential equations large symmetry groups and higher order equations ordinary differential equations odes and ordinary difference systems o ss invariant under the actions of the lie groups and of projective transformations of the independent variables x and dependent variables y are constructed the odes are continuous limits of the o ss or conversely the o ss are invariant discretizations of the odes the invariant o ss are used to calculate numerical solutions of the invariant odes of order up to five the solutions of the invariant numerical schemes are compared to numerical solutions obtained by standard runge kutta methods and to exact solutions when available the invariant method performs at least as well as standard ones and much better in the vicinity of singularities of solutions

high order accurate finite difference schemes based on symmetry preservation abstract in this paper we present a mathematical approach that is based on modified equations and the method of equivariant moving frames for construction of high order accurate invariant finite difference schemes that preserve lie symmetry groups of underlying partial differential equations pdes in the proposed approach invariant or symmetry preserving numerical schemes with a desired or fixed order of accuracy are constructed from some non invariant base numerical schemes modified forms of pdes are used to improve the order of accuracy of existing schemes and these modified forms are obtained through addition of defect correction terms to the original forms of pdes these defect correction terms of modified pdes that are noted from truncation error analysis are either completely removed from schemes or their representation is significantly simplified by considering convenient moving frames this feature of the proposed method can especially be useful to avoid cumbersome numerical representations when high order schemes are developed from low order ones via the method of modified equations the proposed method is demonstrated via construction of invariant numerical schemes with fixed and higher order of accuracy for some common linear and nonlinear problems including the linear advection diffusion equation in 1d and 2d inviscid burgers equation and viscous burgers equation and the performance of these invariant numerical schemes is further evaluated our results indicate that such invariant numerical schemes obtained from existing base numerical schemes have the potential to significantly improve the quality of results not only in terms of desired higher order accuracy but also in the context of preservation of appropriate symmetry properties of underlying pdes

construction of partial difference schemes i the clairaut schwarz young theorem on the lattice difference operators in several variables do not always commute their commutation depends on the lattice equations in this paper we study the conditions over a lattice to allow commutativity of these difference operators and their implications for the discretization of partial differential equations

the discretized boussinesq equation and its conditional symmetry reduction in this article we show that we can carry out the symmetry preserving discretization of the boussinesq equation with respect to three of its more significant conditional symmetries we perform the symmetry reduction of the obtained nonlinear discrete schemes with respect to the conditional symmetries and obtain the reduced discrete equations which unlike in the continuous case are not always reducible to second order difference equations a numerical comparison with the exact continuous solution given by weierstrass elliptic functions is carried out

the numerical study of blowup with application to a nonlinear schrodinger equation abstract we discuss the use of numerical methods in the study of the solutions of evolution problems which exhibit finite time unbounded growth we first examine a naive approach in which the growth rate of the numerical solution is accepted as an approximation of the true growth rate as we shall demonstrate for a radial nonlinear schrodinger equation this approach is inadequate since different discretizations exhibit different growth rates the spurious behaviour of discretizations in the neighbourhood of the singularity is discussed a reliable procedure for the estimation of the blowup parameters is considered which eliminates the discrepancies between different numerical methods

unveiling turbo codes some results on parallel concatenated coding schemes a parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver the input bits to the first encoder are scrambled by the interleaver before entering the second encoder the codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders this construction can be generalized to any number of constituent codes parallel concatenated schemes employing two convolutional codes as constituent codes in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes have been previously shown to yield remarkable coding gains close to theoretical limits they have been named and are known as turbo codes we propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length the analytical bounding technique is then used to shed some light on some crucial questions which have been floating around in the communications community since the proposal of turbo codes

symmetry adapted moving mesh schemes for the nonlinear schrodinger equation in this paper we consider symmetry preserving difference schemes for the nonlinear schrodinger equation where n is the number of space dimensions this equation describes one dimensional waves in n space dimensions in many physical situations including phenomena in plasma physics and nonlinear optics we will consider the nonintegrable case n 2 for which the equation admits solutions that blow up in a finite time and construct discretizations based upon moving mesh schemes that have the same lie group properties and lagrangian structures as the continuous counterpart

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces