flux and differences in action for continuous time hamiltonian systems for a time periodic hamiltonian h p q t of period t the area crossing a collection of curves at time 0 spanning two homotopic orbits of common period nt in a time t is shown to be the difference between the actions contour integral pdq hdt of the orbits similarly in an autonomous hamiltonian system of two degrees of freedom the flux of energy surface volume per unit time through a surface spanning two homotopic orbits of the same energy is given by the difference between the actions contour integral p dq of the orbits analogous results hold for pairs of orbits which converge together in both directions of time

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

exponential error rates of sdp for block models beyond grothendieck s inequality in this paper we consider the cluster estimation problem under the stochastic block model we show that the semidefinite programming sdp formulation for this problem achieves an error rate that decays exponentially in the signal to noise ratio the error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime an immediate corollary of our results yields error bounds under the censored block model moreover these error bounds are robust continuing to hold under heterogeneous edge probabilities and a form of the so called monotone attack significantly this error rate is achieved by the sdp solution itself without any further pre or post processing and improves upon existing polynomially decaying error bounds proved using the grothendieck s inequality our analysis builds on two key ingredients 1 showing that the graph has a well behaved spectrum even in the sparse regime after discounting an exponentially small number of edges and 2 an order statistics argument that governs the final error rate both arguments highlight the implicit regularization effect of the sdp formulation

minimax rates of community detection in stochastic block models in statistics as well as in computer science probability and applied mathematics community detection for the stochastic block model sbm is probably the most studied topic in network analysis many methodologies have been proposed some beautiful and signicant phase transition results are obtained in various settings in this paper we provide a general minimax theory for community detection it gives minimax rates of the mis match ratio for a wide rage of settings including homogeneous and inhomogeneous sbms dense and sparse networks nite and growing number of communities the minimax rates are exponential dierent from polynomial rates we often see in statistical literature an immediate consequence of the result is to establish threshold phenomenon for strong consistency exact recovery as well as weak consistency partial recovery we obtain the upper bound by a range of penalized likelihood type approaches the lower bound is achieved by a novel reduction from a global mis match ratio to a local clustering problem for one node through an exchangeability property

linear pdes and numerical methods that preserve a multisymplectic conservation law multisymplectic methods have recently been proposed as a generalization of symplectic ode methods to the case of hamiltonian pdes their excellent long time behavior for a variety of hamiltonian wave equations has been demonstrated in a number of numerical studies a theoretical investigation and justification of multisymplectic methods is still largely missing in this paper we study linear multisymplectic pdes and their discretization by means of numerical dispersion relations it is found that multisymplectic methods in the sense of bridges and reich phys lett a 284 2001 pp 184 193 and reich j comput phys 157 2000 pp 473 499 such as gauss legendre runge kutta methods possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity a certain cfl type restriction on delta t delta x might be required for methods higher than second order in time it is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of delta t delta x despite being multisymplectic in the sense of discrete variational mechanics j e marsden g p patrick and s shkoller commun math phys 199 1999 pp 351 395

stability of water waves we apply some general results for hamiltonian systems depending on the notion of signature of eigenvalues to determine the circumstances under which collisions of imaginary eigenvalue for the linearized problem about a travelling water wave of permanent form are avoided or lead to loss of stability up to non degeneracy assumptions a new superharmonic instability is predicted and verified

dispersive properties of multisymplectic integrators multisymplectic ms integrators i e numerical schemes which exactly preserve a discrete space time symplectic structure are a new class of structure preserving algorithms for solving hamiltonian pdes in this paper we examine the dispersive properties of ms integrators for the linear wave and sine gordon equations in particular a leapfrog in space and time scheme a member of the lobatto runge kutta family of methods and the preissman box scheme are considered we find the numerical dispersion relations are monotonic and that the sign of the group velocity is preserved the group velocity dispersion gvd is found to provide significant information and succinctly explain the qualitative differences in the numerical solutions obtained with the different schemes further the numerical dispersion relations for the linearized sine gordon equation provides information on the ability of the ms integrators to capture the sine gordon dynamics we are able to link the numerical dispersion relations to the total energy of the various methods thus providing information on the coarse grid behavior of ms integrators in the nonlinear regime

unstable eigenvalues and the linearization about solitary waves and fronts with symmetry the linear stability of solitary wave or front solutions of hamiltonian evolutionary equations which are equivariant with respect to a lie group is studied the organizing centre for the analysis is a multisymplectic formulation of hamiltonian pdes where a distinct symplectic operator is assigned for time and space this separation of symplectic structures leads to new characterizations of the following components of the analysis the states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure the momentum of the connecting orbit or shape of the solitary wave considered as a heteroclinic orbit in a phase space representation is given a new characterization as a one form on the tangent space to the heteroclinic manifold and this one form is a restriction of the temporal symplectic structure for the linear stability analysis a new symplectic characterization of the evans function and its derivatives are obtained leading to an abstract geometric proof of instability for a large class of solitary wave states of equivariant hamiltonian evolutionary pdes the theory sheds new light on several well known models the gkdv equation a boussinesq system and a nonlinear wave equation the generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed

statistical computational tradeoffs in planted problems and submatrix localization with a growing number of clusters and submatrices we consider two closely related problems planted clustering and submatrix localization in the planted clustering problem a random graph is generated based on an underlying cluster structure of the nodes the task is to recover these clusters given the graph the submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real valued random matrix of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size these formulations cover several classical models such as planted clique planted densest subgraph planted partition planted coloring and the stochastic block model which are widely used for studying community detection graph clustering and bi clustering for both problems we show that the space of the model parameters cluster submatrix size edge probabilities and the mean of the submatrices can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities 1 the impossible regime where all algorithms fail 2 the hard regime where the computationally expensive maximum likelihood estimator mle succeeds 3 the easy regime where the polynomial time convexified mle succeeds 4 the simple regime where a local counting thresholding procedure succeeds moreover we show that each of these algorithms provably fails in the harder regimes our results establish the minimax recovery limits which are tight up to universal constants and hold even with a growing number of clusters submatrices and provide order wise stronger performance guarantees for polynomial time algorithms than previously known our study demonstrates the tradeoffs between statistical and computational considerations and suggests that the minimax limits may not be achievable by polynomial time algorithms

strong consistency of spectral clustering for stochastic block models in this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models sbms we show that under some weak conditions on the minimal degree the number of communities and the eigenvalues of the probability block matrix the k means algorithm applied to the eigenvectors of the graph laplacian associated with its first few largest eigenvalues can classify all individuals into the true community uniformly correctly almost surely extensions to both regularized spectral clustering and degree corrected sbms are also considered we illustrate the performance of different methods on simulated networks

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

analysis of the error performance of trellis coded modulations in rayleigh fading channels this work presents an exact expression for the pairwise error event probability of trellis coded modulation tcm transmitted over rayleigh fading channels it includes phase shift keying psk and multilevel quadrature amplitude modulation qam codes as well as coherent and partially coherent e g differential pilot tone etc detection due to the form of the exact pairwise error event probabilities this calculation technique cannot be used with the transfer function technique to obtain an upper union bound on the overall bit error probability for this reason the authors estimate the bit error probability by considering only a small number of short error events through simulations they found that the estimation is usually very accurate at high signal to noise ratios but not as accurate at lower signal to noise ratios they study several coded modulation schemes this way among the results are the fact that tcm provides significant improvement in the error floor when detected differentially and an asymmetry in the pairwise error event probability for 16 qam

multisymplectic geometry local conservation laws and fourier pseudospectral discretization for the good boussinesq equation the multisymplectic geometry for the good boussinesq equation is presented in this paper the multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle the multisymplectic hamiltonian formulation is also presented based on the multisymplectic formulation we investigate the corresponding multisymplectic fourier pseudospectral discretization

achieving the bayes error rate in synchronization and block models by sdp robustly we study the statistical performance of semidefinite programming sdp relaxations for clustering under random graph models under the mathbb z 2 synchronization model censored block model cbm and stochastic block model sbm we show that sdp achieves an error rate of the form exp 1 o 1 bar n i here bar n is an appropriate multiple of the number of nodes and i is an information theoretic measure of the signal to noise ratio we provide matching lower bounds on the bayes error for each model and therefore demonstrate that the sdp approach is bayes optimal as a corollary our results imply that sdp achieves the optimal exact recovery threshold under each model furthermore we show that sdp is robust the above bound remains valid under semirandom versions of the models in which the observed graph is modified by a monotone adversary our proof is based on a novel primal dual analysis of sdp under a unified framework for all three models and the analysis shows that sdp tightly approximates a joint majority voting procedure

whitham modulation equations coalescing characteristics and dispersive boussinesq dynamics abstract whitham modulation theory with degeneracy in wave action is considered the case where all components of the wave action conservation law when evaluated on a family of periodic travelling waves have vanishing derivative with respect to wavenumber is considered it is shown that whitham modulation equations morph on a slower time scale into the two way boussinesq equation both the 1 1 and 2 1 cases are considered the resulting boussinesq equation arises in a universal form in that the coefficients are determined from the abstract properties of the lagrangian and do not depend on particular equations one curious by product of the analysis is that the theory can be used to confirm that the two way boussinesq equation is not a valid model in shallow water hydrodynamics modulation of nonlinear travelling waves of the complex klein gordon equation is used to illustrate the theory

constellation labeling for linear encoders this paper investigates optimal constellation labeling in the context of the edge profile a constellation s edge profile lists the minimum distance edge for each binary symbol error the paper introduces the symmetric ultracomposite su labeling structure and shows that this structure provides undominated edge profiles for 2 sup n psk 2 sup n pam and 2 sup 2n point square qam the su structure is a generalization of the commonly used reflected binary gray code with the proper choice of basis vectors su labeling can support either set partition or gray code labeling of 2 sup n psk 2 sup n pam and 2 sup 2n point square qam notably there are gray code and set partition labelings that do not have the su structure these labelings yield inferior edge profiles the su structure does not apply to cross constellations however for any standard cross constellation with 32 or more points a quasi su labeling structure can approximate the su structure with the correct choice of basis quasi su labelings produce quasi gray labelings however the quasi su structure cannot support set partition labeling in fact the quasi su structure provides a better edge profile than standard set partition labeling thus for cross constellations there is a choice between edge profile optimality and the group structure provided by set partitioning here the correct choice depends on whether the encoder trellis has parallel branches

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

energy dissipation preserving birkhoffian multi symplectic methods for maxwell s equations with dissipation terms in this paper we propose two new energy dissipation preserving birkhoffian multi symplectic methods birkhoffian and birkhoffian box for maxwell s equations with dissipation terms after investigating the non autonomous and autonomous birkhoffian formalism for maxwell s equations with dissipation terms we first apply a novel generating functional theory to the non autonomous birkhoffian formalism to propose our birkhoffian scheme and then implement a central box method to the autonomous birkhoffian formalism to derive the birkhoffian box scheme we have obtained four formal local conservation laws and three formal energy global conservation laws we have also proved that both of our derived schemes preserve the discrete version of the global local conservation laws furthermore the stability dissipation and dispersion relations are also investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable dissipation preserving for maxwell s equations in a perfectly matched layer pml medium and have second order accuracy in both time and space numerical experiments for problems with exact theoretical results are given to demonstrate that the birkhoffian multi symplectic schemes are much more accurate in preserving energy than both the exponential finite difference time domain fdtd method and traditional hamiltonian scheme we also solve the electromagnetic pulse emp propagation problem and the numerical results show that the birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation

co clustering of nonsmooth graphons performance bounds are given for exploratory co clustering blockmodeling of bipartite graph data where we assume the rows and columns of the data matrix are samples from an arbitrary population this is equivalent to assuming that the data is generated from a nonsmooth graphon it is shown that co clusters found by any method can be extended to the row and column populations or equivalently that the estimated blockmodel approximates a blocked version of the generative graphon with estimation error bounded by o p n 1 2 analogous performance bounds are also given for degree corrected blockmodels and random dot product graphs with error rates depending on the dimensionality of the latent variable space

on semidefinite relaxations for the block model the stochastic block model sbm is a popular tool for community detection in networks but fitting it by maximum likelihood mle involves a computationally infeasible optimization problem we propose a new semidefinite programming sdp solution to the problem of fitting the sbm derived as a relaxation of the mle we put ours and previously proposed sdps in a unified framework as relaxations of the mle over various subclasses of the sbm which also reveals a connection to the well known problem of sparse pca our main relaxation which we call sdp 1 is tighter than other recently proposed sdp relaxations and thus previously established theoretical guarantees carry over however we show that sdp 1 exactly recovers true communities over a wider class of sbms than those covered by current results in particular the assumption of strong assortativity of the sbm implicit in consistency conditions for previously proposed sdps can be relaxed to weak assortativity for our approach thus significantly broadening the class of sbms covered by the consistency results we also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed sdp approaches and not an artifact of the proofs our analysis of sdps is based on primal dual witness constructions which provides some insight into the nature of the solutions of various sdps in particular we show how to combine features from sdp 1 and already available sdps to achieve the most flexibility in terms of both assortativity and block size constraints as our relaxation has the tendency to produce communities of similar sizes this tendency makes it the ideal tool for fitting network histograms a method gaining popularity in the graphon estimation literature as we illustrate on an example of a social networks of dolphins we also provide empirical evidence that sdps outperform spectral methods for fitting sbms with a large number of blocks

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

geometry and dynamics of stable and unstable cylinders in hamiltonian systems abstract stable and unstable manifolds in phase space with an s 1 r 1 cylindrical geometry are shown to exist for certain two degree of freedom hamiltonian systems specific attention is given to hamiltonian systems with potential barriers although the concepts developed are more general the existence of these cylinders is independent of the nature of the hamiltonian dynamics i e regular or chaotic a detailed discussion is given where we show that appropriate poincare sections of the cylinders yield a map structure which we denote as reactive islands that is distinct from the usual homoclinic tangle the cylinders have the physical property that all motion across a barrier must occur through the interior of these surfaces the cylinders thus mediate the reaction dynamics a kinetic mechanism based upon the properties of the cylinders is developed and tested against several numerical simulations of the reaction dynamics of a model hamiltonian system the threshold limiting form of the standard theory of microcanonical reaction rates is derived

bit interleaved coded modulation revisited a mismatched decoding perspective we revisit the information theoretic analysis of bit interleaved coded modulation bicm by modeling the bicm decoder as a mismatched decoder the mismatched decoding model is well defined for finite yet arbitrary block lengths and naturally captures the channel memory among the bits belonging to the same symbol we give two independent proofs of the achievability of the bicm capacity calculated by caire et al where bicm was modeled as a set of independent parallel binary input channels whose output is the bitwise log likelihood ratio our first achievability proof uses typical sequences and shows that due to the random coding construction the interleaver is not required the second proof is based on the random coding error exponents with mismatched decoding where the largest achievable rate is the generalized mutual information moreover the generalized mutual information of the mismatched decoder coincides with the infinite interleaver bicm capacity we show that the error exponent and hence the cutoff rate of the bicm mismatched decoder is upper bounded by that of coded modulation and may thus be lower than in the infinite interleaved model for binary reflected gray mapping in gaussian channels the loss in error exponent is small we also consider the mutual information appearing in the analysis of iterative decoding of bicm with extrinsic information transfer exit charts if the symbol metric has knowledge of the transmitted symbol exit mutual information admits a representation as a pseudo generalized mutual information which is in general not achievable a different symbol decoding metric for which the extrinsic side information refers to the hypothesized symbol induces a generalized mutual information lower than the coded modulation capacity in this case perfect extrinsic side information turns the mismatched decoder error exponent into that of coded modulation

vorticity and symplecticity in lagrangian fluid dynamics the relationship between potential vorticity pv and the symplectic form is explored for the shallow water equations governing lagrangian particle paths starting with the symplectic form the pv is found by the pull back operation to the reference space at first sight the encoding of pv in the symplectic form appears to be independent of the particle relabelling symmetry the analysis is carried a step further in two ways using the conservation of symplecticity as a starting point the fluxes of symplecticity arise as differential forms and a complete pull back of the flux forms leads to a geometric description of pv conservation secondly symmetry methods are used to give a rigorous connection between particle relabelling symplecticity and pv conservation generalizations of these issues to semi geostrophic flow and three dimensional lagrangian fluid flows and the connection with ertel s theorem are also discussed

multisymplectic conservation laws for differential and differential difference equations many well known partial differential equations can be written as multisymplectic systems such systems have a structural conservation law from which scalar conservation laws can be derived these c

a geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities action symplecticity signature and complex instability are fundamental concepts in hamiltonian dynamics which can be characterized in terms of the symplectic structure in this paper hamiltonian pdes on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two form is assigned to each space direction and time this leads to a new geometric formulation of the conservation of wave action for linear and nonlinear hamiltonian pdes and via stokes s theorem a conservation law for symplecticity each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode the first invariant in this family is classical krein signature or energy sign when the energy is time independent and the other spatial signatures are energy flux signs leading to a classification of instabilities that includes information about directional spatial spreading of an instability the theory is applied to several examples the boussinesq equation the water wave equations linearized about an arbitrary stokes s wave rotating shallow water flow and flow past a compliant surface some implications for non conservative systems are also discussed

stability theory for dissipatively perturbed hamiltonian systems it is shown that for an appropriate class of dissipatively perturbed hamiltonian systems the number of unstable modes of the dynamics linearized at a nondegenerate equilibrium is determined solely by the index of the equilibrium regarded as a critical point of the hamiltonian in addition the movement of the associated eigenvalues in the limit of vanishing dissipation is analyzed 1995 john wiley sons inc

a variational approach to m 1 dimensional n order field we present a method to construct the multisymplectic formulation for m 1 dimensional n order field based on the variational principle then we derive the multisymplectic conservation law and the local energy and momentum conservation laws for m 1 dimensional n order field the centred box discretization of the multisymplectic formulation for m 1 dimensional n order field is proved to be a multisymplectic scheme as an example the process of constructing the multisymplectic formulation for the kp equation is given

on bicm receivers for tcm transmission recent results have shown that the performance of bit interleaved coded modulation bicm using convolutional codes in nonfading channels can be significantly improved when the interleaver takes a trivial form bicm t i e when it does not interleave the bits at all in this paper we give a formal explanation for these results and show that bicm t is in fact the combination of a tcm transmitter and a bicm receiver to predict the performance of bicm t a new type of distance spectrum for convolutional codes is introduced analytical bounds based on this spectrum are developed and asymptotic approximations are presented it is shown that the free hamming distance of the code is not the relevant optimization criterion for bicm t asymptotically optimal convolutional codes for different constraint lengths are tabulated and bicm t is shown to offer asymptotic gains of about 2 db over traditional bicm designs based on random interleavers the asymptotic gains over uncoded transmission are found to be the same as those obtained by ungerboeck s one dimensional trellis coded modulation 1d tcm and therefore in nonfading channels bicm t is shown to be as good as 1d tcm

a survey on symplectic and multi symplectic algorithms we simply review symplectic and multi symplectic algorithms in the paper these algorithms have many superiorities to general non symplectic algorithms such as they have long time behavior they can preserve the symplectic geometry structure of the hamiltonian system exactly and they can simulate original physical phenomena well some commonly used methods to construct symplectic and multi symplectic schemes are simply surveyed numerical examples are also enumerated to illustrate their performances

recovering a single community with side information we study the effect of the quality and quantity of side information on the recovery of a hidden community of size k o n in a graph of size n side information for each node in the graph is modeled by a random vector in which either the vector dimension or the llr of each component with respect to node labels is independent of n these two models represent the variation in quality and quantity of side information under maximum likelihood detection we calculate tight necessary and sufficient conditions for exact recovery of the labels we demonstrate how side information needs to evolve with n in terms of either its quantity or quality to improve the exact recovery threshold a similar set of results are obtained for weak recovery under belief propagation tight necessary and sufficient conditions for weak recovery are calculated when the llrs are constant and sufficient conditions when the llrs vary with n moreover we design and analyze a local voting procedure using side information that can achieve exact recovery when applied after belief propagation

community detection in sparse networks via grothendieck s inequality we present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs the method is based on grothendieck s inequality unlike the previous uses of this inequality that lead to constant relative accuracy we achieve any given relative accuracy by leveraging randomness we illustrate the method with the problem of community detection in sparse networks those with bounded average degrees we demonstrate that even in this regime various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices the method is general it can be applied to a variety of stochastic models of networks and semidefinite programs

multi symplectic formulation of near local hamiltonian balanced models we transform near local hamiltonian balanced models hbms describing nearly geostrophic fluid motion with constant coriolis parameter into multi symplectic ms systems this allows us to determine conservation of lagrangian momentum energy and potential vorticity for salmon s l1 dynamics a similar approach works for other near local balanced models such as the model the ms approach also enables us to determine a class of systems that have a contact structure similar to that of the semigeostrophic model the contact structure yields a contact transformation that makes the problem of front formation tractable the new class includes the first local model with a variable coriolis parameter that preserves all of the most useful geometric features of the semigeostrophic model

on the asymptotic performance of bit wise decoders for coded modulation two decoder structures for coded modulation over the gaussian channel are studied 1 the maximum likelihood symbol wise decoder and 2 the suboptimal bit wise decoder based on the bit interleaved coded modulation paradigm we consider a 16 ary quadrature amplitude constellation labeled with a gray labeling it is shown that the asymptotic loss in terms of pairwise error probability for any two codewords caused by the bit wise decoder is bounded by 1 25 db the analysis also shows that the asymptotic loss is zero for a wide range of linear codes including all rate 1 2 convolutional codes

the symplectic evans matrix and the instability of solitary waves and fronts hamiltonian evolution equations which are equivariant with respect to the action of a lie group are models for physical phenomena such as oceanographic flows optical fibres and atmospheric flows and such systems often have a wide variety of solitary wave or front solutions in this paper we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts at the heart of the analysis is a multi symplectic formulation of hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions with a third symplectic structure with two form denoted by omega associated with a coordinate frame moving at the speed of the wave this leads to a geometric decomposition and symplectification of the evans function formulation for the linearization about solitary waves and fronts we introduce the concept of the symplectic evans matrix a matrix consisting of restricted omega symplectic forms by applying hodge duality to the exterior algebra formulation of the evans function we find that the zeros of the evans function correspond to zeros of the determinant of the symplectic evans matrix based on this formulation we prove several new properties of the evans function restricting the spectral parameter lambda to the real axis we obtain rigorous results on the derivatives of the evans function near the origin based solely on the abstract geometry of the equations and results for the large lambda behaviour which use primarily the symplectic structure but also extend to the non symplectic case the lie group symmetry affects the evans function by generating zero eigenvalues of large multiplicity in the so called systems at infinity we present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation by combining all these results a new rigorous sufficient condition for instability of solitary waves and fronts is obtained the theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as x tends to infinity to illustrate the theory it is applied to three examples a boussinesq model from oceanography a class of nonlinear schrodinger equations from optics and a nonlinear klein gordon equation from atmospheric dynamics

singularities on the boundary of the stability domain near 1 1 resonance abstract we study the linear differential equation x l x in 1 1 resonance that is x r 4 and l is 4 4 matrix with a semi simple double pair of imaginary eigenvalues i i i i we wish to find all perturbations of this linear system such that the perturbed system is stable since linear differential equations are in one to one correspondence with linear maps we translate this problem to gl 4 r in this setting our aim is to determine the stability domain and the singularities of its boundary the dimension of gl 4 r is 16 therefore we first reduce the dimension as far as possible here we use a versal unfolding of l i e a transverse section of the orbit of l under the adjoint action of gl 4 r repeating a similar procedure in the versal unfolding we are able to reduce the dimension to 4 a 3 sphere in this 4 dimensional space contains all information about the neighborhood of l in gl 4 r considering the 3 sphere as two 3 discs glued smoothly along their common boundary we find that the boundary of the stability domain is contained in two right conoids one in each 3 disc the singularities of this surface are transverse self intersections whitney umbrellas and an intersection of self intersections where the surface has a self tangency a whitney stratification of the 3 sphere such that the eigenvalue configurations of corresponding matrices are constant on strata allows us to describe the neighborhood of l and in particular identify the stability domain

conformal conservation laws and geometric integration for damped hamiltonian pdes conformal conservation laws are defined and derived for a class of multi symplectic equations with added dissipation in particular the conservation laws of energy and momentum are considered along with those that arise from linear symmetries numerical methods that preserve these conformal conservation laws are presented in detail providing a framework for proving a numerical method exactly preserves the dissipative properties considered the conformal methods are compared analytically and numerically to standard conservative methods which includes a thorough inspection of numerical solution behavior for linear equations damped klein gordon and sine gordon equations and a damped nonlinear schrodinger equation are used as examples to demonstrate the results

gray coding for multilevel constellations in gaussian noise the problem of finding the optimal labeling bit to symbol mapping of multilevel coherent phase shift keying psk pulse amplitude modulation pam and quadrature amplitude modulation qam constellations with respect to minimizing the bit error probability bep over a gaussian channel is addressed we show that using the binary reflected gray code brgc to label the signal constellation results in the lowest possible bep for high enough signal energy to noise ratios and analyze what is high enough in this sense it turns out that the brgc is optimal for psk and pam systems whenever the target bep is at most a few percent which covers most systems of practical interest new and simple closed form expressions are presented for the bep of psk pam and qam using the brgc

some extended results on the search for good convolutional codes we provide useful results on two classes of convolutional codes binary codes and nonbinary codes the best codes or the best known codes for these two classes of convolutional codes are found by computer search some of them are better than those found in the past we specify these codes by their transfer function matrices distance spectra and information weight spectra furthermore we derive an upper bound on the free distances of binary to m ary codes and q ary to m ary codes numerical values of this bound closely fit the computer searched values

rational rate punctured convolutional codes for soft decision viterbi decoding we present rational rate k n punctured convolutional codes n up to 8 k 1 spl middot spl middot spl middot n 1 and constraint length spl nu up to 8 with good performance many of these codes improve the free distance and or weight spectra over previously reported codes with the same parameters the tabulated codes are found by an exhaustive or a random search

on wave action and phase in the non canonical hamiltonian formulation the long timeevolution of disturbances to slowlyvarying solutions of partial differential equations is subject to the adiabatic invariance of the wave action generally this approximate conservati

on optimal tcm encoders an asymptotically optimal trellis coded modulation tcm encoder requires the joint design of the encoder and the binary labeling of the constellation since analytical approaches are unknown the only available solution is to perform an exhaustive search over the encoder and the labeling for large constellation sizes and or many encoder states however an exhaustive search is unfeasible traditional tcm designs overcome this problem by using a labeling that follows the set partitioning principle and by performing an exhaustive search over the encoders in this paper we study binary labelings for tcm and show how they can be grouped into classes which considerably reduces the search space in a joint design for 8 ary constellations the number of different binary labelings that must be tested is reduced from 8 40320 to 240 for the particular case of an 8 ary pulse amplitude modulation constellation this number is further reduced to 120 and for 8 ary phase shift keying to only 30 an algorithm to generate one labeling in each class is also introduced asymptotically optimal tcm encoders are tabulated which are up to 0 3 db better than the previously best known encoders

a universal form for the emergence of the korteweg de vries equation a new perspective on the emergence of the kortewegde vries kdv equation is presented the conventional view is that the kdv equation arises as a model when the dispersion relation of the lineariz

universal behaviour of two coupled circle maps symmetric coupling of two critical circle maps near the golden mean rotation number is considered on the basis of a renormalization group method the three universal types of interaction are found the theoretical scaling predictions are confirmed by numerical calculations

on the optimality of the binary reflected gray code this paper concerns the problem of selecting a binary labeling for the signal constellation in m psk m pam and m qam communication systems gray labelings are discussed and the original work by frank gray is analyzed as is noted the number of distinct gray labelings that result in different bit error probability grows rapidly with increasing constellation size by introducing a recursive gray labeling construction method called expansion the paper answers the natural question of what labeling among all possible constellation labelings will give the lowest possible average probability of bit errors for the considered constellations under certain assumptions on the channel the answer is that the labeling proposed by gray the binary reflected gray code is the optimal labeling for all three constellations which has surprisingly never been proved before

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free