stability and instability of periodic travelling wave solutions for the critical korteweg de vries and nonlinear schrodinger equations abstract in this paper we establish new results about the existence stability and instability of periodic travelling wave solutions related to the critical korteweg de vries equation u t 5 u 4 u x u x x x 0 and the critical nonlinear schrodinger equation i v t v x x v 4 v 0 the periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario the stability approach is based on the theory developed by angulo natali in j angulo f natali positivity properties of the fourier transform and the stability of periodic travelling wave solutions siam j math anal 40 2008 1123 1151 for positive periodic travelling wave solutions associated to dispersive evolution equations of korteweg de vries type the instability approach is based on an extension to the periodic setting of arguments found in bona souganidis strauss j l bona p e souganidis w a strauss stability and instability of solitary waves of korteweg de vries type proc roy soc london ser a 411 1987 395 412 regarding the critical schrodinger equation stability instability theories similar to the critical korteweg de vries equation are obtained by using the classical grillakis shatah strauss theory in m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry ii j funct anal 94 1990 308 348 m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry i j funct anal 74 1987 160 197 the arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

a bi hamiltonian structure for the integrable discrete non linear schrodinger system abstract this paper shows that the al ablowitz ladik hierarchy of integrable equations can be explicitly viewed as a hierarchy of commuting flows which a are hamiltonian with respect to both a standard local poisson operator j and a new non local skew almost poisson operator k on the appropriate space b can be recursively generated from a recursion operator r k j 1 in addition the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi hamiltonian structures for other families of discrete integrable equations

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

statistical mechanics of nonlinear wave equations 4 cubic schrodinger the cubic schrodinger equation is considered on the circle both in the de focussing and the focussing case the existence of the flow is proved together with the invariance of the appropriate gibbsian measure namely the petit canonical measure in the defocussing case and the micro canonical measure in the focussing case

an alternative to metric rescaling in viterbi decoders in the viterbi algorithm the negative log likelihood estimates accumulated distances or path metrics are unboundedly increasing functions of time for implementation all variables must be confined to a finite range the following properties of the viterbi algorithm can be exploited for this purpose 1 path selection depends only on differences of metrics an 2 the difference between metrics is bounded in the rescaling scheme at each iteration the minimum metric is subtracted from all metrics the use of two s complement arithmetic is proposed as an alternative to the rescaling method this scheme avoids any kind of rescaling subtractions obvious advantages in implementation are hardware savings and a speedup inside the metric update loop which is critical to the decoder s computational throughput

concentration of the invariant measures for the periodic zakharov kdv nls and gross piatevskii equations in 1d and 2d abstract this paper concerns gibbs measures for some nonlinear pde over the d torus t d the hamiltonian h t d u 2 t d u p has canonical equations with solutions in n u l 2 t d u 2 n this n is a parameter in quantum field theory analogous to the number of particles in a classical system for d 1 and 2 p 6 n supports the gibbs measure d u z 1 e h u x t d u x which is normalized and formally invariant under the flow generated by the pde the paper proves that n l 2 is a metric probability space of finite diameter that satisfies the logarithmic sobolev inequalities for the periodic kdv the focussing cubic nonlinear schrodinger equation and the periodic zakharov system for suitable subset of n a logarithmic sobolev inequality also holds in the critical case p 6 for d 2 the gross piatevskii equation has h t 2 u 2 t 2 v u 2 u 2 for a suitable bounded interaction potential v and the gibbs measure lies on a metric probability space h s which satisfies lsi in the above cases d is the limit in l 2 transportation distance of finite dimensional n n given by fourier sums

a complexity approach to the soliton resolution conjecture the soliton resolution conjecture is one of the most interesting open problems in the theory of nonlinear dispersive equations roughly speaking it asserts that a solution with generic initial condition converges to a finite number of solitons plus a radiative term in this paper we use the complexity of a finite object a notion introduced in algorithmic information theory to show that the soliton resolution conjecture is equivalent to the analogous of the second law of thermodynamics for the complexity of a solution of a dispersive equation

chaotic and homoclinic behavior for numerical discretizations of the nonlinear schro dinger equation abstract certain conservative discretizations of the nls can produce irregular behavior we consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum we find that irregularity sets in when two homoclinic structures are present and in this case many and continual homoclinic crossings occur throughout the irregular time series we indicate a melnikov analysis to study the consequences of this homoclinic behavior

entropy length profiles bounds on the minimal covering of bipartite graphs and trellis complexity of nonlinear codes the trellis representation of nonlinear codes is studied from a new perspective we introduce the new concept of entropy length profile elp this profile can be considered as an extension of the dimension length profile dlp to nonlinear codes this elaboration of the dlp the entropy length profiles appears to be suitable to the analysis of nonlinear codes additionally and independently we use well known information theoretic measures to derive novel bounds on the minimal covering of a bipartite graph by complete subgraphs we use these bounds in conjunction with the elp notion to derive both lower and upper bounds on the state complexity and branch complexity profiles of nonlinear block codes represented by any trellis diagram we lay down no restrictions on the trellis structure and we do not confine the scope of our results to proper or one to one trellises only the basic lower bound on the state complexity profile implies that the state complexity at any given level cannot be smaller than the mutual information between the past and the future portions of the code at this level under a uniform distribution of the codewords we also devise a different probabilistic model to prove that the minimum achievable state complexity over all possible trellises is not larger than the maximum value of the above mutual information over all possible probability distributions of the codewords this approach is pursued further to derive similar bounds on the branch complexity profile to the best of our knowledge the proposed upper bounds are the only upper bounds that address nonlinear codes the novel lower bounds are tighter than the existing bounds the new quantities and bounds reduce to well known results when applied to linear codes

an additional gibbs state for the cubic schrodinger equation on the circle an invariant gibbs state for the nonlinear schrodinger equation on the circle was constructed by bourgain 2 3 and mckean 6 out of the basic hamiltonian using a trigonometric cutoff the cubic nonlinear schrodinger equation is a completely integrable system having an infinite number of additional integrals of motion in this paper we construct the second invariant gibbs state from one of these additional integrals for the cubic nls on the circle this additional gibbs state is singular with respect to the gibbs state previously constructed from the basic hamiltonian our approach employs the ablowitz ladik system a completely integrable discretization of the cubic schrodinger equation 2001 john wiley sons inc

on minimal tree realizations of linear codes a tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree a tree decomposition can be extended to a tree realization i e a cycle free realization of the code on the underlying tree by specifying a state space at each edge of the tree and a local constraint code at each vertex of the tree the constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes a measure of the complexity of maximum likelihood ml decoding for a code is its treewidth which is the least constraint complexity of any of its tree realizations it is known that among all tree realizations of a linear code that extends a given tree decomposition there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree in this paper we give two new constructions of these minimal realizations as a by product of the first construction a generalization of the state merging procedure for trellis realizations we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree the second construction relies on certain code decomposition techniques that we develop we further observe that the treewidth of a code is related to a measure of graph complexity also called treewidth we exploit this connection to resolve a conjecture of forney s regarding the gap between the minimum trellis constraint complexity and the treewidth of a code we present a family of codes for which this gap can be arbitrarily large

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

on the trellis representation of the delsarte goethals codes in this correspondence the trellis representation of the kerdock and delsarte goethals codes is addressed it is shown that the states of a trellis representation of dg m spl delta under any bit order are either strict sense nonmerging or strict sense nonexpanding except maybe at indices within the code s distance set for spl delta spl ges 3 and for m spl ges 6 the state complexity s sub max dg m spl delta is found for all values of m and spl delta a formula for the number of states and branches of the biproper trellis diagram of dg m spl delta is given for some of the indices and upper and lower bounds are given for the remaining indices the formula and the bounds refer to the delsarte goethals codes when arranged in the standard bit order

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

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

continuity of the flow of kdv with regard to the wasserstein metrics and application to an invariant measure abstract in this paper we prove the continuity of the flow of kdv on spaces of probability measures with respect to a combination of wasserstein distances on h s s 0 and l 2 we are motivated by the existence of an invariant measure belonging to the spaces onto which these distances are defined

statistical mechanics of the periodic benjamin ono equation the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

optimal sectionalization of a trellis while the complexity of trellis decoding for a given block code is essentially a function of the number of states and branches in its trellis the decoding complexity may be often reduced by means of an appropriate sectionalization of the trellis notwithstanding the many examples of such sectionalizations for particular codes that appeared in the literature no systematic method for finding the best sectionalization of a given trellis is presently known we present a polynomial time algorithm which produces the optimal sectionalization of a given trellis t in time o n sup 2 where n is the length of the code generated by t the algorithm is developed in a general setting of certain operations and functions defined on the set of trellises it therefore applies to both linear and nonlinear codes and easily accommodates a broad range of optimality criteria the particular optimality criterion based on minimizing the total number of additions and comparisons required for maximum likelihood trellis decoding is investigated in detail several different methods for decoding a given trellis are discussed and compared in a number of examples finally analysis of the dynamical properties of certain optimal sectionalizations is presented

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

a new cohomology theory of orbifold based on the orbifold string theory model in physics we construct a new cohomology ring for any almost complex orbifold the key theorem is the associativity of this new ring some examples are computed

twisted orbifold k theory we use equivariant methods to define and study the orbifold k theory of an orbifold x adapting techniques from equivariant k theory we construct a chern character and exhibit a multiplicative decomposition for k orb x in particular showing that it is additively isomorphic to the orbifold cohomology of x a number of examples are provided we then use the theory of projective representations to define the notion of twisted orbifold k theory in the presence of discrete torsion an explicit expression for this is obtained in the case of a global quotient

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

codes on graphs constraint complexity of cycle free realizations of linear codes cycle free graphical realizations of linear codes generalize trellis realizations given a linear code c and a cycle free graph topology there exists a well defined minimal realization for c on that graph in which each constraint is a linear code with a well defined length and dimension the constraint complexity of the realization is defined as maximum dimension of any constraint code there exists a graph that minimizes constraint complexity in which all internal nodes have degree 3 and all interface nodes have degree 2 and which moreover can be put in the form of a tree structured trellis realization the constraint complexity of a general cycle free graph realization can be less than that of any conventional trellis realization but not by very much such realizations can yield reductions in decoding complexity even when they do not reduce constraint complexity

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

minimum weight codewords as generators of generalized reed muller codes we establish the range of values of spl rho where 0 spl les spl rho spl les m q 1 for which the generalized reed muller code r sub fq spl rho m of length q sup m over the field f sub q of order q is spanned by its minimum weight vectors

the extraction and complexity limits of graphical models for linear codes two broad classes of graphical modeling problems for codes can be identified in the literature constructive and extractive problems the former class of problems concern the construction of a graphical model in order to define a new code the latter class of problems concern the extraction of a graphical model for a fixed given code the design of a new low density parity check code for some given criteria e g target block length and code rate is an example of a constructive problem the determination of a graphical model for a classical linear block code that implies a decoding algorithm with desired performance and complexity characteristics is an example of an extractive problem this work focuses on extractive graphical model problems and aims to lay out some of the foundations of the theory of such problems for linear codes the primary focus of this work is a study of the space of all graphical models for a fixed given code the tradeoff between cyclic topology and complexity in this space is characterized via the introduction of a new bound the forest inducing cut set bound fi csb the proposed bound provides a more precise characterization of this tradeoff than that which can be obtained using existing tools e g the csb and can be viewed as a generalization of the square root bound for tail biting trellises to graphical models with arbitrary cyclic topologies searching the space of graphical models for a given code is then enabled by introducing a set of basic graphical model transformation operations that are shown to span this space finally heuristics for extracting novel graphical models for linear block codes using these transformations are investigated

conditionally cycle free graphical models for coset codes conditionally cycle free graphical models i e cyclic graphical models which become cycle free after conditioning on a subset of the hidden variables are constructed for coset codes following the description of a general construction procedure examples of a number of families of codes including first order reed muller rm and the delsarte goethals codes are provided for which the proposed procedure yields optimal soft in soft out siso decoding algorithms that are less complex than the best known trellis based algorithms in the case of the first order rm codes which have a recursive coset construction the optimal siso decoding algorithm that results when the proposed construction is applied repeatedly is denoted recursive coset representation rcr decoding connections are made between rcr decoding and existing algorithms that exploit fast hadamard transforms finally the utility of the proposed decoding algorithms are supported by a practically motivated application the construction of serially concatenated codes that have high rates and low error floors extended hamming codes are proposed as outer codes in such constructions with efficient decoding employing the siso algorithms developed herein

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

simple map decoding of first order reed muller and hamming codes a maximum a posteriori map probability decoder of a block code minimizes the probability of error for each transmitted symbol separately the standard way of implementing map decoding of a linear code is the bahl cocke jelinek raviv bcjr algorithm which is based on a trellis representation of the code the complexity of the bcjr algorithm for the first order reed muller rm 1 codes and hamming codes is proportional to n sup 2 where n is the code s length in this correspondence we present new map decoding algorithms for binary and nonbinary rm 1 and hamming codes the proposed algorithms have complexities proportional to q sup 2 n log sub q n where q is the alphabet size in particular for the binary codes this yields complexity of order n log n

a statistical approach to the asymptotic behavior of a class of generalized nonlinear schrodinger equations a statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear schrodinger type which govern non integrable non singular dynamics in a bounded domain the solutions of these equations have been shown numerically to tend in the long time limit toward a gibbsian statistical equilibrium state consisting of a ground state solitary wave on the large scales and gaussian fluctuations on the small scales the main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit the large deviation principle pertains to a process governed by a gibbs ensemble that is canonical in energy and microcanonical in particle number some supporting monte carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation especially the high frequency growth of the dispersion relation the large deviation principle for the process governed by the gibbs ensemble is based on a large deviation principle for gaussian processes for which two independent proofs are given

fast ml decoding algorithm for the nordstrom robinson code corresp the extended nordstrom robinson code is an optimum nonlinear double error correcting code of length 16 with considerable practical importance this code is also useful as a rate 1 2 vector quantizer for random waveforms such as speech linear predictive coding lpc residual a fast decoding algorithm is described for maximum likelihood decoding or nearest neighbor search in the squared error sense with 304 additions and 128 comparisons

invariant measures for the periodic derivative nonlinear schrodinger equation we construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear schrodinger equation dnls for small data in l 2 and we show these measures to be absolutely continuous with respect to the gaussian measure the key ingredient of the proof is the analysis of the gauge group of transformations associated to dnls as an intermediate step for our main result we prove quasi invariance with respect to the gauge maps of the gaussian measure on l 2 with covariance mathbb i delta k 1 for any k geqslant 2

brownian motion with restoring drift the petit and micro canonical ensembles letf q be odd and positive near then the non linear wave equation 2q t2 2q x2 f q 0 considered on the circle 0 x l can be written in hamiltonian formq h p p h q with p q cdot and h tfrac 1 2 mathop smallint limits 0 l q 2 mathop smallint limits 0 l f q tfrac 1 2 mathop smallint limits 0 l p 2 the corresponding flow preserves the suitably interpreted petit ensemble e h d qd p and forl q settles down to the stationary diffusion with infinitesimal operator 1 2 2 q2 m q q m being the logarithmic derivative of the ground state of d2 dq2 f q this diffusion is the brownian motion with restoring drift see mckean vaninsky 1993 1 for reasons suggested by the paper of lebowitz rose speer 1988 on nls it is interesting to study the micro canonical ensemble obtained by restricting to the sphere int limits 0 l q 2 n and makingl with fixedd n l now forf q q2 the same type of diffusion appears but with drift arising from the modified potentialf q cq2 c being chosen so that the mean ofq2 is the assigned numberd the proof employs doblin s method of loops 1937 and steepest descent the same is true forf q m2q2 only now the proof is elementary the outcome is also the same iff q q2 0 providedd is smaller than the petit canonical mean ofq2 ford larger than this mean the matter is more subtle and the outcome is unknown

constraint complexity of realizations of linear codes on arbitrary graphs a graphical realization of a linear code c consists of an assignment of the coordinates of c to the vertices of a graph along with a specification of linear state spaces and linear local constraint codes to be associated with the edges and vertices respectively of the graph the kappa complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes kappa complexity is a reasonable measure of the computational complexity of a sum product decoding algorithm specified by a graphical realization the main focus of this paper is on the following problem given a linear code c and a graph g how small can the kappa complexity of a realization of c on g be as useful tools for attacking this problem we introduce the vertex cut bound and the notion of vc treewidth for a graph which is closely related to the well known graph theoretic notion of treewidth using these tools we derive tight lower bounds on the kappa complexity of any realization of c on g our bounds enable us to conclude that good error correcting codes can have low complexity realizations only on graphs with large vc treewidth along the way we also prove the interesting result that the ratio of the kappa complexity of the best conventional trellis realization of a length n code c to the kappa complexity of the best cycle free realization of c grows at most logarithmically with code length n such a logarithmic growth rate is in fact achievable

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