linear block codes over cyclic groups the main building block for the construction of a geometrically uniform coded modulation scheme is a subgroup of g sup i where g is a group generating a low dimensional signal constellation and i is an index set in this paper we study the properties of these subgroups when g is cyclic we exploit the fact that any cyclic group of q elements is isomorphic to the additive group of z sub q the ring of integers modulo q so that we can make use of concepts related to linearity our attention is focused mainly on indecomposable cyclic groups i e of prime power order since they are the elementary building blocks of any abelian group in analogy with the usual construction of linear codes over fields we define a generator matrix and a parity check matrix trellis construction and bounds on the minimum euclidean distance are also investigated some examples of coded modulation schemes based on this theory are also exhibited and their performance evaluated

dispersion and group velocity in numerical schemes for three dimensional hydrodynamic equations we investigate dispersion and group velocity relations of numerical schemes for the three dimensional hydrodynamic equations by using the transform function method the numerical methods are developed for two of arakawa s spatial grid types namely b grid and c grid using a spectral method in the vertical direction one of our results is that if the horizontal scale length is greater than the rossby radius of deformation then the b grid is more appropriate than the commonly used c grid

symmetry properties of multilevel coded modulation we consider the geometrical uniformity properties of signal space codes generated by multilevel coded modulation first we examine multilevel schemes based on 2 sup l ary psk signal sets and we derive conditions on the binary component codes which lead to geometrically uniform signal space codes next we study the geometrical uniformity of multilevel schemes based on signal sets having a generating group isomorphic to a direct product finally we provide an easy to test criterion for finding if a given signal set or signal space code obtained as the union of two orbits of the same generating group has the uniform distance property

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

boosting the accuracy of finite difference schemes via optimal time step selection and non iterative defect correction abstract in this article we present a unified analysis of the simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations pdes by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non iterative manner the power of the technique which is applicable to time dependent semilinear scalar pdes where the leading order spatial derivative has a constant coefficient is its ability to increase the accuracy of formally low order finite difference schemes without major modification to the basic numerical algorithm through straightforward numerical analysis arguments we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method we demonstrate the utility of optimal time step ots selection combined with non iterative defect correction nidc on several different types of finite difference schemes for a wide array of classical linear and semilinear pdes in one and more space dimensions on both regular and irregular domains

on linear structure and phase rotation invariant properties of block m psk modulation codes two important structural properties of block m 2 sup ary psk modulation codes linear structure and phase symmetry are investigated an m ary modulation code is first represented as a code with symbols from the integer group s sub m psk 0 1 2 m 1 under modulo m addition then the linear structure of block m psk modulation codes over s sub m psk with respect to modulo m vector addition is defined and conditions are derived under which a block m psk modulation code is linear once the linear structure is developed the phase symmetry of block m psk modulation codes is studied in particular a necessary and sufficient condition for a block m psk modulation code that is linear as a binary code to be invariant under 2 sup h 180 degrees m phase rotation for 1

three level techniques for one dimensional parabolic equation with nonlinear initial condition in this paper a boundary value problem for one dimensional heat equation is considered under the constraint of a nonlocal initial condition in place of the classical specification of initial data we impose the nonlocal initial condition efficient higher order algorithms are developed for solving this parabolic partial differential equation with nonstandard initial condition several three level finite difference schemes are presented these schemes are based on the modification of the centred time centred space explicit formula the three level 1 3 1 explicit technique the sixth order 1 3 3 explicit method and the dufort frankel finite difference scheme numerical computations combined with a simple iteration procedure of some examples are given to demonstrate the efficiency and accuracy of the new algorithms

signal sets matched to groups recently linear codes over z sub m the ring of integers mod m have been presented that are matched to m ary phase modulation the general problem of matching signal sets to generalized linear algebraic codes is addressed based on these codes a definition is given for the notion of matching it is shown that any signal set in n dimensional euclidean space that is matched to an abstract group is essentially what d slepian 1968 called a group code for the gaussian channel if the group is commutative this further implies that any such signal set is equivalent to coded phase modulation with linear codes over z sub m some further results on such signal sets are presented and the signal sets matched to noncommutative groups and the linear codes over such groups are discussed

compact finite difference schemes for ocean models 1 ocean waves abstract a general framework is derived which leads to generic expressions of discrete dispersion relationships for inertia gravity and rossby waves valid for every finite difference scheme and every type of grid these relationships are used to investigate the performance of fourth order and sixth order compact implicit schemes on arakawa grids a e it is shown that the use of compact schemes leads to very clear improvement in approximating frequency and group velocity of inertia gravity waves on the other hand increasing the order of the schemes does not necessarily improve the accuracy of the discrete dispersion relationship in the case of rossby waves globally the fourth order family is found to be a good compromise which improves significantly the quality of the approximation of the dispersion properties with regard to conventional second order centered schemes

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

construction of linear ring codes for 6 psk the algebraic construction of linear codes over zm for m psk is considered for the case that m is the product of two distinct primes by means of the chinese remainder theorem crt such codes can be constructed from linear codes over the corresponding prime fields it is shown that for given component codes the crt composition yields psk codes that are at least as good and often much better than the codes obtained from the standard coset coding technique a lower bound on the minimum euclidean distance of 6 psk codes is derived that is based on the hamming weight distribution of the component codes

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

numerical solution of a nonlinear dissipative system using a pseudospectral method and inertial manifolds we consider the numerical solution of nonlinear dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds coarse and fine grids are employed with a nonlinear mapping to relate the solutions computed on these grids the approach is illustrated by consideration of the kuramoto sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions numerical results are presented on the ability of the method to mimic the time evolution of the infinite dimensional dynamical system the results demonstrate that the coarse fine grid method is more accurate than the standard pseudospectral method on the same coarse grid however time integration studies suggest that the increased accuracy is bought at an increase in computational cost to achieve a specified accuracy the computational cost of the two grid method exceeds that of a standard pseudospectral method for this spatially periodic problem

enslaved finite difference approximations for quasigeostrophic shallow flows abstract we describe a procedure to improve both the accuracy and computational efficiency of finite difference schemes used to simulate the nonlinear pdes that govern barotropic two dimensional geophysical fluid dynamics our underlying strategy is to reduce the truncation error of a given difference scheme in such a way that the time step restrictions are not changed to accomplish this reduction we use information from the governing equations in the case of vanishing time derivatives our method is based on a change of variables from fine to coarse grids which allows us to order the various terms that appear and justify further approximations these approximations lead to algebraic closures for the new higher order variables and finally to a new enslaved scheme we demonstrate the utility of the procedure for the shallow water equations in both periodic and closed basins in the latter case we present results that demonstrate the ability of the enslaved scheme to capture dynamics on scales smaller than those resolved by the original scheme

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

grading functions and mesh redistribution we consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two point boundary value problems by finite difference or finite element methods an analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary value problems in one dimension error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria

grid adaption for problems in fluid dynamics analyse et application d une technique de grille adaptative aux problemes fortement non lineaires en mecanique des fluides et transfert de chaleur resolution de problemes d ecoulement externe avec decollement de propagation de flamme et d interaction choc couche limite

an approximate inertial manifold for computing burgers equation abstract we present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals our algorithm is motivated from the dynamical systems point of view in particular we adapt the methodology of approximate inertial manifolds to a finite difference scheme this leads to a differential treatment in which the higher i e unresolved modes are expressed in terms of the lower modes as a particular example we derive an approximate inertial manifold for burgers equation and develop a numerical algorithm suitable for computing we perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme for all values of the parameters which are the coefficient of viscosity and the cell size we obtain a decrease in the numerical error by at least a factor of 2 0 with the modified scheme the decrease in error is substantially greater over large regions of the parameters space

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

reduced order based feedback control of the kuramoto sivashinsky equation in this paper we consider the kuramoto sivashinsky equation kse which describes the long wave motions of a thin film over a vertical plane solution procedures for the kse often yield a large or infinite dimensional nonlinear system we first discuss two reduced order methods the approximate inertial manifold and the proper orthogonal decomposition and show that these methods can be used to obtain a reduced order system that can accurately describe the dynamics of the kse moreover from this resulting reduced order system the feedback controller can readily be designed and synthesized for our control techniques we use the linear and nonlinear quadratic regulator methods which are the first and second order approximated solutions of the hamilton jacobi bellman equation respectively numerical simulations comparing the performance of the reduced order based linear and nonlinear controllers are presented

strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with arbitrary functionals abstract in l byszewski z angew math mech 70 1990 3 202 206 l byszewski j appl math stochastic ann 3 1990 5 65 79 the author studied strong maximum principles for nonlinear parabolic problems with nonlocal inequalities together with particular functionals our purpose here is to extend results in l byszewski z angew math mech 70 1990 3 202 206 l byszewski j appl math stochastic ann 3 1990 5 65 79 to strong maximum principles for nonlinear parabolic problems with nonlocal inequalities together with arbitrary functionals the results obtained in this paper can be applied in the theories of diffusion and heat conduction

multilevel construction of block and trellis group codes the theory of group codes has been shown to be a useful starting point for the construction of good geometrically uniform codes in this paper we study the problem of building multilevel group codes i e codes obtained combining separate coding at different levels in such a way that the resulting code is a group code a construction leading to multilevel group codes for semi direct and direct products is illustrated the codes that can be obtained in this way are identified new geometrically uniform euclidean space codes obtained from multilevel codes over abelian and nonabelian groups are presented

accuracy and nonoscillatory properties of enslaved difference schemes we describe a general method for modifying a given finite difference scheme by representing the effects of the subgrid scales in terms of the resolved scales through enslaving the new scheme is more accurate in certain parameter regimes while retaining the time step stability of the original scheme we consider two general enslaving relations an approximate enslaving based on truncation analysis and an exact enslaving based on the dynamics of the governing equation we find that the modified schemes based on the exact enslaving eliminate unphysical oscillations producing monotone solutions even when the original difference schemes do not have this property we offer a truncation analysis to justify this property we apply our enslaving technique to advection diffusion equations in both one and two spatial dimensions

the two level finite difference schemes for the heat equation with nonlocal initial condition abstract in this paper the two level finite difference schemes for the one dimensional heat equation with a nonlocal initial condition are analyzed as the main result we obtain conditions for the numerical stability of the schemes in addition we revise the stability conditions obtained in 21 for the crank nicolson scheme we present several numerical examples that confirm the theoretical results within linear as well as nonlinear problems in some particular cases it is shown that for small regions of the time step size values the explicit ftcs scheme is stable while certain implicit methods such as crank nicolson scheme are not

qpsk block modulation codes for unequal error protection unequal error protection uep codes find applications in broadcast channels as well as in other digital communication systems where messages have different degrees of importance binary linear uep luep codes combined with a gray mapped qpsk signal set are used to obtain new efficient qpsk block modulation codes for unequal error protection several examples of qpsk modulation codes that have the same minimum squared euclidean distance as the best qpsk modulation codes of the same rate and length are given in the new constructions of qpsk block modulation codes even length binary luep codes are used good even length binary luep codes are obtained when shorter binary linear codes are combined using either the well known u u v construction or the so called construction x both constructions have the advantage of resulting in optimal or near optimal binary luep codes of short to moderate lengths using very simple linear codes and may be used as constituent codes in the new constructions luep codes lend themselves quite naturally to multistage decoding up to their minimum distance using the decoding of component subcodes a new suboptimal two stage soft decision decoding of luep codes is presented and its application to qpsk block modulation codes for uep illustrated

efficient methods using high accuracy approximate inertial manifolds summary we extend the idea of the post processing galerkin method in the context of dissipative evolution equations to the nonlinear galerkin the filtered galerkin and the filtered nonlinear galerkin methods in general the post processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear galerkin method and galerkin method for approximating smooth bounded solutions is much smaller than the total error of the method in each case an improvement in accuracy is obtained by post processing these more accurate lower modes with an appropriately chosen highly accurate approximate inertial manifold aim we present numerical experiments that support the theoretical improvements in accuracy both the theory and computations are presented in the framework of a two dimensional reaction diffusion system with polynomial nonlinearity however the algorithm is very general and can be implemented for other dissipative evolution systems the computations clearly show the post processed filtered galerkin method to be the most efficient method

minimization of the truncation error by grid adaptation a new grid adaptation strategy which minimizes the truncation error of a pth order finite difference approximation is proposed the main idea of the method is based on the observation that the local truncation error associated with discretization on nonuniform meshes can be minimized if the interior grid points are redistributed in an optimal sequence the method does not explicitly require the truncation error estimate and at the same time it allows one to increase the design order of approximation by one globally so that the same finite difference operator reveals superconvergence properties on the optimal grid another very important characteristic of the method is that if the differential operator and the metric coefficients are evaluated identically by some hybrid approximation the single optimal grid generator can be employed in the entire computational domain independently of points where the hybrid discretization switches from one approximation to another generalization of the present method to multiple dimensions is presented numerical calculations of several one dimensional and one two dimensional test examples demonstrate the performance of the method and corroborate the theoretical results

multistage decoding of multilevel block m psk modulation codes and its performance analysis multistage decoding of multilevel block multilevel phase shift keying m psk modulation codes for the additive white gaussian noise awgn channel is investigated several types of multistage decoding including a suboptimum soft decision decoding scheme are devised and analyzed upper bounds on the probability of an incorrect decoding of a code are derived for the proposed multistage decoding schemes error probabilities of some specific multilevel block 8 psk modulation codes are evaluated and simulated the computation and simulation results for these codes show that with multistage decoding significant coding gains can be achieved with large reduction in decoding complexity in one example it is shown that the difference in performance between the proposed suboptimum multistage soft decision decoding and the single stage optimum decoding is small only a fraction of a db loss in snr at the block error probability of 10 sup 6

gilbert varshamov bound for euclidean space codes over distance uniform signal sets in this correspondence in an extension of piret s bound for codes over phase shift keying psk signal sets we investigate the application of the gilbert varshamov gv bound to a variety of distance uniform du signal sets in euclidean space it is shown that four dimensional signal sets matched to binary tetrahedral binary octahedral and binary icosahedral groups lead to better bounds compared to the bounds for signal sets matched to dicyclic groups with the same number of signal points and comparable symmetric psk signal sets

algebraic characterization of mds group codes over cyclic groups an n k group code over a group g is a subset of g sup n which forms a group under componentwise group operation and can be defined in terms of n k homomorphisms from g sup k to g the set of homomorphisms which define maximum distance separable mds group codes defined over cyclic groups are characterized each defining homomorphism can be specified by a set of k endomorphisms of g a matrix is associated with the k n k defining endomorphisms of the code and necessary and sufficient conditions for this matrix to define an mds code over cyclic groups is proved using this matrix characterization it is proved that over a cyclic group with m elements where m p sub 1 sup d 1 p sub 2 sup d 2 spl middot spl middot spl middot p sub m sup d m k s k mds group codes for all s k spl ges 2 do not exist if max s k spl ges min p sub 1 p sub 2 spl middot spl middot p sub m finally it is shown that the dual code of an mds group code over c sub m a cyclic group with m elements is also an mds group code

high order non uniform grid schemes for numerical simulation of hypersonic boundary layer stability and transition the direct numerical simulation of receptivity instability and transition of hypersonic boundary layers requires high order accurate schemes because lower order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields the main limiting factor in the application of high order schemes to practical boundary layer flow problems is the numerical instability of high order boundary closure schemes on the wall this paper presents a family of high order non uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary layer transition by using an appropriate grid stretching and clustering grid points near the boundary high order schemes with stable boundary closures can be obtained the order of the schemes ranges from first order at the lowest to the global spectral collocation method at the highest the accuracy and stability of the new high order numerical schemes is tested by numerical simulations of the linear wave equation and two dimensional incompressible flat plate boundary layer flows the high order non uniform grid schemes up to the 11th order are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge the steady and unsteady results show that the new high order schemes are stable and are able to produce high accuracy for computations of the nonlinear two dimensional navier stokes equations for the wall bounded supersonic flow

controllability of systems with a general nonlocal condition abstract in this work we are concerned with the controllability of first order distributed systems with a general nonlocal condition specifically by a general nonlocal condition we understand a nonlocal condition modeled in terms of an inclusion in this context assuming that the underlying linear system is approximately controllable we present three types of conditions under which the nonlinear system is also approximately controllable

modified truncation finite difference schemes a method is presented for modifying the truncation error of a given finite difference scheme approximating a nonlinear evolution equation the new scheme has several advantages over the original it is higher order in the absence of time derivatives has the same time step requirements it removes nonphysical oscillations and it is not less accurate than the original scheme the idea applied to a finite difference scheme approximating a geophysical flow produces a scheme consistent with the accuracy of the original scheme but on a mesh three times more refined

a new adi technique for two dimensional parabolic equation with an integral condition abstract nonclassical parabolic initial boundary value problems arise in the study of several important physical phenomena this paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions a new fourth order finite difference technique based upon the noye and hayman n h alternating direction implicit adi scheme is used as the basis to solve the two dimensional time dependent diffusion equation with an integral condition replacing one boundary condition this scheme uses less central processor time cpu than a second order fully implicit scheme based on the classical backward time centered space btcs method for two dimensional diffusion it also has a larger range of stability than a second order fully explicit scheme based on the classical forward time centered space ftcs method the basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyeet this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods the results of numerical experiments for the new method are presented the central processor times needed are also reported error estimates derived in the maximum norm are tabulated

on multilevel block modulation codes the multilevel technique for combining block coding and modulation is investigated a general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures a specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed given a multilevel block modulation code c with no interdependency among the binary component codes the proposed method gives a multilevel block modulation code c that has the same rate as c a minimum squared euclidean distance not less than that of c a trellis diagram with the same number of states as that of c and a smaller number of nearest neighbor codewords than that of c finally a technique is presented for analyzing the error performance of block modulation codes for an additive white gaussian noise awgn channel based on soft decision maximum likelihood decoding error probabilities of some specific codes are evaluated by simulation and upper bounds based on their euclidean weight distributions

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

labelings and encoders with the uniform bit error property with applications to serially concatenated trellis codes the well known uniform error property for signal constellations and codes is extended to encompass information bits we introduce a class of binary labelings for signal constellations called bit geometrically uniform bgu labelings for which the uniform bit error property holds i e the bit error probability does not depend on the transmitted signal strong connections between the symmetries of constellations and binary hamming spaces are involved for block coded modulation bcm and trellis coded modulation tcm euclidean space codes bgu encoders are introduced and studied the properties of bgu encoders prove quite useful for the analysis and design of codes aimed at minimizing the bit rather than symbol error probability applications to the analysis and the design of serially concatenated trellis codes are presented together with a case study which realizes a spectral efficiency of 2 b s hz

construction of inertial manifolds by elliptic regularization abstract in many cases an inertial manifold 2 m for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function from a finite dimensional hilbert space hp to another hilbert space hq the invariance property of m means that can be written as the solution of a first order partial differential equation d p g1 p p a p g2 p p 0 over hp where g1 and g2 are nonlinear functions which depend on the original dynamical system and a is a suitably stable linear operator in this paper we use a method introduced by sacker r j sacker a new approach to the perturbation theory of invariant surface comm pure appl math 18 1965 717 732 for the study of finite dimensional dynamical systems to find inertial manifolds in the infinite dimensional setting this method involves replacing the first order equation for by the regularized elliptic equation e d p g1 p p a p g2 p p with suitable boundary conditions it is shown that if a satisfies a spectral gap condition then the solutions e of the elliptic equation converge to a weak solution of 0 as e 0 furthermore m graph is an invariant manifold for the given dynamical system

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

on invariance properties of the wave equation a complete group classification is given of both the wave equation c2 x uxx utt 0 i and its equivalent system vt ux c2 x vx ut ii when the wave speed c x const equations i and ii admit either a two or four parameter group for the exceptional case c x ax b 2 equation i admits an infinite group equations i and ii do not always admit the same group for a given c x the group for i can have more parameters or fewer parameters than that for ii moreover the groups can be different with the same number of parameters separately for i and ii all possible c x that admit a four parameter group are found explicitly the corresponding invariant similarity solutions are considered some of these wave speeds have realistic physical properties c x varies monotonically from one positive constant to another positive constant as x goes from to

a fully multidimensional positive definite advection transport algorithm with small implicit diffusion in numerical modeling of physical phenomena it is often necessary to solve the advective transport equation for positive definite scalar functions numerical schemes of second or higher order accuracy can produce negative values in the solution due to the dispersive ripples lower order schemes such as the donor cell or lax friedrichs or higher order schemes with zeroth order diffusion added produce no ripples but suffer from excessive implicit diffusion in the last ten years a possible resolution of this dilemma has been developed in the form of hybrid schemes in which the advective fluxes are given as a weighted average of a first order positive definite scheme s fluxes and a higher order scheme s fluxes the difference in determination of the weights in the calculation of the average advective fluxes has led to different hybrid schemes two main hybrid type schemes have been developed one the so called flux corrected transport fct method was originated by boris and book i 31 and generalized by zalesak 141 the other was developed by harten and zwas 6 9 in the form of the self adjusting hybrid schemes sahs method both methods were constructed to deal effectively with shocks and contact discontinuities solutions of the advection transport equation obtained by using fct or sahs maintain positive definiteness of the initial condition and as can be seen from presented tests zalesak 141 harten 6 be very accurate unfortunately application of these methods to the modeling of complex multidimensional hydrodynamical systems like atmospheric phenomena is rather limited due to the excessive computer time required furthermore in many hydrodynamical systems

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

block coded psk modulation using two level group codes over dihedral groups a length n group code over a group g is a subgroup of g sup n under component wise group operation group codes over dihedral groups d sub m with 2m elements that are two level constructible using a binary code and a code over z sub m residue class integer ring modulo m as component codes are studied for arbitrary m a set of necessary and sufficient conditions on the component codes for the two level construction to result in a group code over d sub m are obtained the conditions differ for m odd and even using two level group codes over d sub m as label codes the performance of a block coded modulation scheme is discussed under all possible matched labelings of 2m apsk and 2m spsk asymmetric and symmetric psk signal sets in terms of the minimum squared euclidean distance matched labelings that lead to automorphic euclidean distance equivalent codes are identified it is shown that depending upon the ratio of hamming distances of the component codes some labelings perform better than others the best labeling is identified under a set of restrictive conditions finally conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed

finite difference solutions for parabolic equations with the time weighting initial conditions abstract a class of nonlinear parabolic boundary value problems with time weighting initial condition is considered instead of studying the existence uniqueness and continuous dependence of the solution upon the data we give a brief review of the known results and then concentrate on the performance of the backward euler and crank nicolson finite difference schemes 1 the results of some numerical examples are presented which demonstrate the efficiency and rapid convergence

grid adaptation and non iterative defect correction for improved accuracy of numerical solutions of pdes in this work we present a computational approach for improving the order of accuracy of a given finite difference method for solution of linear and nonlinear hyperbolic partial differential equations the methodology consists of analysis of leading order terms in the discretization error of any given finite difference method leading to a modified version of the original partial differential equation singular perturbations of this modified equation are regularized using an adaptive grid distribution and a non iterative defect correction method is used to eliminate the leading order regular perturbation terms in the modified equation implementation of this approach on a low order finite difference scheme not only results in an increase in its order of accuracy but also results in an improvement in its numerical stability due to the regularization of singular perturbations the proposed approach is applied to four different canonical problems including the numerical solution of 1 liouville equation 2 inviscid burgers equation 3 nonlinear reaction advection equation and 4 a system of hyperbolic pdes when compared to exact solutions the numerical results demonstrate the ability of this method in both boosting the accuracy of finite difference schemes up to the desired order and also providing a fully stable numerical solution

integral deferred correction methods constructed with high order runge kutta integrators spectral deferred correction sdc methods for solving ordinary differential equations odes were introduced by dutt greengard and rokhlin 2000 it was shown in that paper that sdc methods can achieve arbitrary high order accuracy and possess nice stability properties their sdc methods are constructed with low order integrators such as forward euler or backward euler and are able to handle stiff and non stiff terms in the odes in this paper we use high order runge kutta rk integrators to construct a family of related methods which we refer to as integral deferred correction idc methods the distribution of quadrature nodes is assumed to be uniform and the corresponding local error analysis is given the smoothness of the error vector associated with an idc method measured by the discrete sobolev norm is a crucial tool in our analysis the expected order of accuracy is demonstrated through several numerical examples superior numerical stability and accuracy regions are observed when high order rk integrators are used to construct idc methods

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

uniqueness of solutions of parabolic semilinear nonlocal boundary problems abstract the purpose of the paper is to give two theorems about the uniqueness of solutions of parabolic semilinear nonlocal boundary problems the paper is a continuation of previous papers by byszewski and the generalization of some results from r rabczuk introduction to differential inequalities pwn warsaw 1976 polish j chabrowski on nonlocal problems for parabolic equations nagoya math j 93 1984 109 131 the theorems obtained in this paper can be applied in the theories of diffusion and heat conduction with better effects than the analogous theorems about parabolic initial boundary problems and than the analogous theorems about parabolic periodic boundary problems