design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

map estimation via agreement on trees message passing and linear programming we develop and analyze methods for computing provably optimal maximum a posteriori probability map configurations for a subclass of markov random fields defined on graphs with cycles by decomposing the original distribution into a convex combination of tree structured distributions we obtain an upper bound on the optimal value of the original problem i e the log probability of the map assignment in terms of the combined optimal values of the tree problems we prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common an important implication is that any such shared configuration must also be a map configuration for the original distribution next we develop two approaches to attempting to obtain tight upper bounds a a tree relaxed linear program lp which is derived from the lagrangian dual of the upper bounds and b a tree reweighted max product message passing algorithm that is related to but distinct from the max product algorithm in this way we establish a connection between a certain lp relaxation of the mode finding problem and a reweighted form of the max product min sum message passing algorithm

on expander codes sipser and spielman see ibid vol 42 p 1717 22 nov 1996 have introduced a constructive family of asymptotically good linear error correcting codes expander codes together with a simple parallel algorithm that will always remove a constant fraction of errors we introduce a variation on their decoding algorithm that with no extra cost in complexity provably corrects up to 12 times more errors

opening nodes on horosphere packings we use bryant representation to construct constant mean curvature 1 surfaces in hyperbolic space by desingularisation of a horosphere packing

the phase plane picture for a class of fourth order conservative differential equations we study the bounded solutions of a class of fourth order equations u u f u 0 0 we show that when is not too large then the paths in the u u plane of two bounded solutions do not cross moreover the conserved quantity associated with the equation puts an ordering on the bounded solutions in the phase plane and a continuation theorem shows that they fill up part of the phase plane we apply these results to the extended fisher kolmogorov efk equation a fourth order model equation for bi stable systems the uniqueness and ordering results imply that as long as the stable equilibrium points are real saddles the bounded solutions of the stationary efk equation correspond exactly to those of the classical second order fisher kolmogorov equation besides we establish the asymptotic stability of the heteroclinic solution of the efk equation

improved nearly mds expander codes a construction of expander codes is presented with the following three properties i the codes lie close to the singleton bound ii they can be encoded in time complexity that is linear in their code length and iii they have a linear time bounded distance decoder by using a version of the decoder that corrects also erasures the codes can replace maximum distance separable mds outer codes in concatenated constructions thus resulting in linear time encodable and decodable codes that approach the zyablov bound or the capacity of memoryless channels the presented construction improves on an earlier result by guruswami and indyk in that any rate and relative minimum distance that lies below the singleton bound is attainable for a significantly smaller alphabet size

periodic solutions and their connecting orbits of hamiltonian systems abstract this article deals with second order periodic hamiltonian systems we apply variational methods to obtain non constant periodic solutions for the time reversible hamiltonian systems there exist connecting orbits joining pairs of periodic solutions our methods can also be used to treat heteroclinic orbits connecting an equilibrium to a periodic solution

existence and multiplicity results for heteroclinic orbits of second order hamiltonian systems abstract connecting orbits of nonlinear differential equations have long been studied in the dynamical systems literature generally in a setting involving perturbations and using a melnikov function in this article we consider a class of second order hamiltonian systems which possess infinitely many or finite number of equilibria using variational arguments and penalization methods we obtain the existence of multiple heteroclinic orbits joining pairs of equilibria

fast decoding of expander codes expander codes are tanner codes defined on sparse graphs that have good expansion properties sipser and spielman 1996 showed that there is a linear time decoding algorithm for expander codes when the vertex expansion is at least 3 4 and the number of errors corrected is a constant fraction of the code length later feldman et al 2007 gave a decoding algorithm that allows the expansion to be 2 3 1 3c where c is the left degree of the underlying bipartite graph at the expense of polynomial time decoding complexity recently viderman 2013 further improved the expansion parameter to 2 3 1 6c and the decoding algorithm runs in linear time these results are for expander codes whose inner codes are parity check codes by using stronger inner codes chilappagari et al 2010 showed that there is a linear time decoding algorithm for every vertex expansion greater than 1 2 in this paper it is shown that for every vertex expansion there is a linear time decoding algorithm for expander codes using inner codes with minimum distance depending on the vertex expansion and that the number of errors corrected is a constant fraction of the code length

uniqueness of the stationary wave for the extended fisher kolmogorov equation abstract the extended fisher kolmogorov equation ut uxxxx uxx u u3 0 models a binary system near the lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria 1 for the classical case 0 the heteroclinic is u x tanh x 2 and is unique up to the obvious symmetries we prove the conjecture that the uniqueness persists all the way to 1 8 where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur our methods are non perturbative and employ a global cross section to the hamiltonian flow of the stationary fourth order equation on the energy level of 1 we also prove uniform a priori bounds on all bounded stationary solutions valid for any 0

shooting methods and topological transversality we show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds the interest of such results is twofold first these orbits persist under perturbations which destroy the structure allowing the shooting method and second topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics we shall focus on heteroclinic solutions in the extended fisher kolmogorov equation

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

signal space characterization of iterative decoding by tracing the flow of computations in the iterative decoders for low density parity check codes we formulate a signal space view for a finite number of iterations in a finite length code on a gaussian channel maximum a posteriori map codeword decoding or maximum likelihood decoding decodes to the codeword signal that is closest to the channel output in euclidean distance in contrast we show that iterative decoding decodes to the pseudosignal that has highest correlation with the channel output the set of pseudosignals corresponds to pseudocodewords only a vanishingly small number of which correspond to codewords we show that some pseudocodewords cause decoding errors but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords

the admm penalized decoder for ldpc codes linear programming lp decoding for low density parity check codes was introduced by feldman et al and has been shown to have theoretical guarantees in several regimes furthermore it has been reported in the literature via simulation and via instanton analysis that lp decoding displays better error rate performance at high signal to noise ratios snr than does belief propagation bp decoding however at low snrs lp decoding is observed to have worse performance than bp in this paper we seek to improve lp decoding at low snrs while maintaining lp decoding s high snr performance our main contribution is a new class of decoders obtained by applying the alternating direction method of multipliers admm algorithm to a set of non convex optimization problems these non convex problems are constructed by adding a penalty term to the objective of lp decoding the goal of the penalty is to make pseudocodewords which are non integer vertices of the lp relaxation more costly we name this class of decoders admm penalized decoders for low and moderate snrs we simulate admm penalized decoding with ell 1 and ell 2 penalties we find that these decoders can outperform both bp and lp decoding for high snrs where it is difficult to obtain data via simulation we use an instanton analysis and find that asymptotically admm penalized decoding performs better than bp but not as well as lp unfortunately since admm penalized decoding is not a convex program we have not been successful in developing theoretical guarantees however the non convex program can be approximated using a sequence of linear programs an approach that yields a reweighted lp decoder we show that a two round reweighted lp decoder has an improved theoretical recovery threshold when compared with lp decoding in addition we find via simulation that reweighted lp decoding significantly attains lower error rates than lp decoding at low snrs

braided block codes a new class of binary iteratively decodable codes with good decoding performance is presented these codes called braided block codes bbcs operate on continuous data streams and are constructed by interconnection of two component block codes bbcs can be considered as convolutional or sliding version of either elias product codes or expander codes in this paper we define bbcs describe methods of their construction analyze code properties and study asymptotic iterative decoding performance

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

a parametrization of the weierstrass formulae and perturbation of complete minimal surfaces in r3 into the hyperbolic 3 space

travelling fronts and entire solutions of the fisher kpp equation in ℝn this paper is devoted to time global solutions of the fisher kpp equation in n where f is a c 2 concave function on 0 1 such that f 0 f 1 0 and f 0 on 0 1 it is well known that this equation admits a finite dimensional manifold of planar travelling fronts solutions by considering the mixing of any density of travelling fronts we prove the existence of an infinite dimensional manifold of solutions in particular there are infinite dimensional manifolds of nonplanar travelling fronts and radial solutions furthermore up to an additional assumption a given solution u can be represented in terms of such a mixing of travelling fronts

cascades of reversible homoclinic orbits to a saddle focus equilibrium abstract in a reversible system we consider a homoclinic orbit being bi asymptotic to a saddle focus equilibrium as was proved by devaney 1977 there exists a one parameter family of periodic orbits accumulating onto this homoclinic orbit in the present paper we show that for any n 2 there exist infinitely many n homoclinic orbits in a neighborhood of the primary homoclinic orbit each of them is accompanied by one or more families of periodic orbits moreover we indicate how these families of periodic orbits correspond to branches of subharmonic periodic orbits

harmonic maps and constant mean curvature surfaces in h 2 x r we introduce a it hyperbolic gauss map into the poincar e disk for any surface in bbb h 2 times bbb r with regular vertical projection and prove that if the surface has constant mean curvature h 1 2 this hyperbolic gauss map is harmonic conversely we show that every nowhere conformal harmonic map from an open simply connected riemann surface sigma into the poincar e disk is the hyperbolic gauss map of a two parameter family of such surfaces as an application we obtain that any holomorphic quadratic differential on sigma can be realized as the abresch rosenberg holomorphic differential of some and generically infinitely many complete surfaces with h 1 2 in bbb h 2 times bbb r a similar result applies to minimal surfaces in the heisenberg group rm nil 3 finally we classify all complete minimal vertical graphs in bbb h 2 times bbb r

linear time encodable decodable codes with near optimal rate we present an explicit construction of linear time encodable and decodable codes of rate r which can correct a fraction 1 r spl epsiv 2 of errors over an alphabet of constant size depending only on spl epsiv for every 0 0 the error correction performance of these codes is optimal as seen by the singleton bound these are near mds codes such near mds linear time codes were known for the decoding from erasures our construction generalizes this to handle errors as well concatenating these codes with good constant sized binary codes gives a construction of linear time binary codes which meet the zyablov bound and also the more general blokh zyablov bound by resorting to multilevel concatenation our work also yields linear time encodable decodable codes which match forney s error exponent for concatenated codes for communication over the binary symmetric channel the encoding decoding complexity was quadratic in forney s result and forney s bound has remained the best constructive error exponent for almost 40 years now in summary our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding but in addition have linear time encoding and decoding algorithms

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

oscillatory solutions of fourth order conservative systems via the conley index in this paper we investigate periodic solutions of second order lagrangian systems which oscillate around equilibrium points of center type the main ingredients are the discretization of second order lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed by ghrist et al 2003 5 the problem with applying this topological theory directly is that the braid types in our analysis are so called improper this implies that the braid invariants do not entirely depend on the topology the relevant braid classes are non isolating neighborhoods of the flow so that their conley index is not universal in first part of this paper we develop the theory of the braid invariant for improper braid classes and in the second part this theory is applied to second order lagrangian system and in particular to the swift hohenberg equation

global dynamics of stationary solutions of the extended fisher kolmogorov equation in this paper we study the fourth order differential equation d4udt4 qd2udt2 u3 u 0 which arises from the study of stationary solutions of the extended fisher kolmogorov equation denoting x u y dudt z d2udt2 v d3udt3 this equation becomes equivalent to the polynomial system x y y z z v v x qz x3 with x y z v r4 and q r as usual the dot denotes the derivative with respect to the time t since the system has a first integral we can reduce our analysis to a family of systems on r3 we provide the global phase portrait of these systems in the poincare ball i e in the compactification of r3 with the sphere s2 of the infinity

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

correcting a fraction of errors in nonbinary expander codes with linear programming a linear programming decoder for nonbinary expander codes is presented it is shown that the proposed decoder has the nearest neighbor certificate properties it is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1 4 a b where a and b are the relative minimum distances of the constituent codes

on trapping sets and guaranteed error correction capability of ldpc codes and gldpc codes the relation between the girth and the guaranteed error correction capability of left regular low density parity check ldpc codes when decoded using the bit flipping serial and parallel algorithms is investigated a lower bound on the size of variable node sets which expand by a factor of at least 3 4 is found based on the moore bound this bound combined with the well known expander based arguments leads to a lower bound on the guaranteed error correction capability the decoding failures of the bit flipping algorithms are characterized using the notions of trapping sets and fixed sets the relation between fixed sets and a class of graphs known as cage graphs is studied upper bounds on the guaranteed error correction capability are then established based on the order of cage graphs the results are extended to left regular and right uniform generalized ldpc codes it is shown that this class of generalized ldpc codes can correct a linear number of worst case errors in the code length under the parallel bit flipping algorithm when the underlying tanner graph is a good expander a lower bound on the size of variable node sets which have the required expansion is established

heteroclinic solutions for the extended fisher kolmogorov equation abstract we study the extended fisher kolmogorov efk equation and its variants by variational approach we show that for every global minimum of the potential function v there is a pair of heteroclinic solutions one emanating from and the other terminating at we require neither superquadratic growth of v nor the presence of saddle focus equilibria moreover v is allowed to approach its minimal level near infinity

on the existence and non existence of bounded solutions for a fourth order ode abstract we prove that if f c 1 r is coercive and f 0 is discrete then the efk equation 1 u c 2 u f u 0 possesses l r solutions if and only if f changes sign at least twice as a corollary we prove that if u n solves u n c n 2 u n f u n 0 then u n if c n 0 provided f has a unique local minimum its only minimum is nondegenerate and int f 0 finally we give criteria ensuring existence and non existence of t periodic solutions to 1 when f has multiple wells

holomorphic differentials and laguerre deformation of surfaces a laguerre geometric local characterization is given of l minimal surfaces and laguerre deformations t transforms of l minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential this is used to prove that via their l gauss maps the t transforms of l minimal isothermic surfaces have constant mean curvature h r in some translate of hyperbolic 3 space mathbb h 3 r 2 subset mathbb r 4 1 de sitter 3 space mathbb s 3 1 r 2 subset mathbb r 4 1 or have mean curvature h 0 in some translate of a time oriented lightcone in mathbb r 4 1 as an application we show that various instances of the lawson isometric correspondence can be viewed as special cases of the t transformation of l isothermic surfaces with holomorphic quartic differential

existence and stability of traveling fronts in the extended fisher kolmogorov equation abstract we study traveling wave solutions to a general fourth order differential equation that is a singular perturbation of the fisher kolmogorov equation we apply the geometric method for singularly perturbed systems to show that for every positive wavespeed there exists a traveling wave also we find that there exists a critical wavespeed c which divides these solutions into monotonic c c and oscillatory c c solutions we show that the monotonic fronts are locally stable under perturbations in appropriate weighted sobolev spaces by using various energy functionals

the minimum distance of turbo like codes worst case upper bounds are derived on the minimum distance of parallel concatenated turbo codes serially concatenated convolutional codes repeat accumulate codes repeat convolute codes and generalizations of these codes obtained by allowing nonlinear and large memory constituent codes it is shown that parallel concatenated turbo codes and repeat convolute codes with sub linear memory are asymptotically bad it is also shown that depth two serially concatenated codes with constant memory outer codes and sublinear memory inner codes are asymptotically bad most of these upper bounds hold even when the convolutional encoders are replaced by general finite state automata encoders in contrast it is proven that depth three serially concatenated codes obtained by concatenating a repetition code with two accumulator codes through random permutations can be asymptotically good

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

instabilities induced by a weak breaking of a strong spatial resonance abstract through multiple scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady state pattern forming instabilities in the case that the wavelengths of the instabilities are nearly in the ratio 1 2 in the case of exact 1 2 resonance the amplitude equations are odes here they are pdes we discuss the stability of spatially periodic solutions to long wavelength disturbances by including these modulational effects we are able to explore the relevance of the exact 1 2 results to spatially extended physical systems for parameter values near to this codimension two bifurcation point these new instabilities can be described in terms of reduced normal form pdes near various secondary codimension two points the robust heteroclinic cycle in the odes is destabilised by long wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle an analytic expression giving the approximate period of this orbit is derived

explicit estimates on the torus for the sup norm and the dissipative length scale of solutions of the swift hohenberg equation in one and two space dimensions abstract in this work we have obtained explicit and accurate estimates of the sup norm for solutions of the swift hohenberg equation she in one and two space dimensions by using the best so far available estimates of the embedding constants which appear in the classical functional interpolation inequalities used in the study of solutions of dissipative partial differential equations we have evaluated in an explicit manner the values of the sup norm of the solutions of the she in addition we have calculated the so called time averaged dissipative length scale associated to the above solutions

an analysis of the block error probability performance of iterative decoding asymptotic iterative decoding performance is analyzed for several classes of iteratively decodable codes when the block length of the codes n and the number of iterations i go to infinity three classes of codes are considered these are gallager s regular low density parity check ldpc codes tanner s generalized ldpc gldpc codes and the turbo codes due to berrou et al it is proved that there exist codes in these classes and iterative decoding algorithms for these codes for which not only the bit error probability p sub b but also the block frame error probability p sub b goes to zero as n and i go to infinity

decoding of expander codes at rates close to capacity the decoding error probability of codes is studied as a function of their block length it is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability specifically it is assumed that there exists a family of codes of length n and rate r 1 epsiv c c is a capacity of a binary symmetric channel whose decoding probability decreases inverse polynomially in n it is shown that if the decoding probability decreases sufficiently fast but still only inverse polynomially fast in n then there exists another such family of codes whose decoding error probability decreases exponentially fast in n moreover if the decoding time complexity of the assumed family of codes is polynomial in n and 1 epsiv then the decoding time complexity of the presented family is linear in n and polynomial in 1 epsiv these codes are compared to the recently presented codes of barg and zemor error exponents of expander codes ieee transactions on information theory 2002 and concatenated codes serial and parallel ieee transactions on information theory 2005 it is shown that the latter families cannot be tuned to have exponentially decaying in n error probability and at the same time to have decoding time complexity linear in n and polynomial in 1 epsiv

generalized and doubly generalized ldpc codes with random component codes for the binary erasure channel in this paper a method for the asymptotic analysis of generalized low density parity check gldpc codes and doubly generalized low density parity check d gldpc codes over the binary erasure channel bec based on extrinsic information transfer exit chart is described this method overcomes the problem consisting of the impossibility to evaluate the exit function for the check or variable component codes in situations where the information functions or split information functions for component codes are unknown according to the proposed technique gldpc codes and d gldpc codes where the generalized check and variable component codes are random codes with minimum distance at least 2 are considered a technique is then developed which finds the exit chart for the overall gldpc or d gldpc code by evaluating the expected exit function for each check and variable component code this technique is finally combined with the differential evolution algorithm in order to generate some good gldpc and d gldpc edge distributions numerical results of long random codes are presented which confirm the effectiveness of the proposed approach they also reveal that d gldpc codes can outperform standard ldpc codes and gldpc codes in terms of both waterfall performance and error floor

lp decoding of regular ldpc codes in memoryless channels we study error bounds for linear programming decoding of regular low density parity check ldpc codes for memoryless binary input output symmetric channels we prove bounds on the word error probability that are inverse doubly exponential in the girth of the factor graph for memoryless binary input awgn channel we prove lower bounds on the threshold for regular ldpc codes whose factor graphs have logarithmic girth under lp decoding specifically we prove a lower bound of 0 735 upper bound of eb n0 2 67 db on the threshold of 3 6 regular ldpc codes whose factor graphs have logarithmic girth our proof is an extension of a recent paper of arora daskalakis and steurer stoc 2009 who presented a novel probabilistic analysis of lp decoding over a binary symmetric channel their analysis is based on the primal lp representation and has an explicit connection to message passing algorithms we extend this analysis to any mbios channel

on the optimality of solutions of the max product belief propagation algorithm in arbitrary graphs graphical models such as bayesian networks and markov random fields mrfs represent statistical dependencies of variables by a graph the max product belief propagation algorithm is a local message passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tree furthermore when the graph is a tree the assignment based on the fixed point yields the most probable values of the unobserved variables given the observed ones good empirical performance has been obtained by running the max product algorithm or the equivalent min sum algorithm on graphs with loops for applications including the decoding of turbo codes except for two simple graphs cycle codes and single loop graphs there has been little theoretical understanding of the max product algorithm on graphs with loops here we prove a result on the fixed points of max product on a graph with arbitrary topology and with arbitrary probability distributions discrete or continuous valued nodes we show that the assignment based on a fixed point is a neighborhood maximum of the posterior probability the posterior probability of the max product assignment is guaranteed to be greater than all other assignments in a particular large region around that assignment the region includes all assignments that differ from the max product assignment in any subset of nodes that form no more than a single loop in the graph in some graphs this neighborhood is exponentially large we illustrate the analysis with examples

improved bounds on the word error probability of ra 2 codes with linear programming based decoding this paper deals with the linear programming based decoding algorithm of feldman and karger for repeat accumulate turbo like codes we present a new structural characterization that captures the event that decoding fails based on this structural characterization we develop polynomial algorithms that given an ra 2 code compute upper and lower bounds on the word error probability p sub w for the binary symmetric and the additive white gaussian noise awgn channels our experiments with an implementation of these algorithms for bounding p sub w demonstrate in many interesting cases an improvement in the upper bound on the word error probability by a factor of over 1000 compared to the bounds by feldman et al the experiments also indicate that the improvement in upper bound increases as the codeword length increases and the channel noise decreases the computed lower bounds on the word error probability in our experiments are roughly ten times smaller than the upper bound

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

the cauchy problem for the liouville equation and bryant surfaces abstract we give a construction that connects the cauchy problem for the 2 dimensional elliptic liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3 space h 3 and solve both of them we construct the unique mean curvature one surface in h 3 that passes through a given curve with a given unit normal along it and provide diverse applications in particular topics such as period problems symmetries finite total curvature planar geodesics rigidity etc are treated for these surfaces

counting in graph covers a combinatorial characterization of the bethe entropy function we present a combinatorial characterization of the bethe entropy function of a factor graph such a characterization being in contrast to the original analytical definition of this function we achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph analogously we give a combinatorial characterization of the bethe partition function whose original definition was also of an analytical nature as we point out our approach has similarities to the replica method but also stark differences the above findings are a natural backdrop for introducing a decoder for graph based codes that we will call symbolwise graph cover decoding a decoder that extends our earlier work on blockwise graph cover decoding both graph cover decoders are theoretical tools that help toward a better understanding of message passing iterative decoding namely blockwise graph cover decoding links max product min sum algorithm decoding with linear programming decoding and symbolwise graph cover decoding links sum product algorithm decoding with bethe free energy function minimization at temperature one in contrast to the gibbs entropy function which is a concave function the bethe entropy function is in general not concave everywhere in particular we show that every code picked from an ensemble of regular low density parity check codes with minimum hamming distance growing with high probability linearly with the block length has a bethe entropy function that is convex in certain regions of its domain

second order lagrangian twist systems simple closed characteristics we consider a special class of lagrangians that play a fundamental role in the theory of second order lagrangian systems twist systems this subclass of lagrangian systems is defined via a convenient monotonicity property that such systems share this monotonicity property twist property allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels this reduction has some similarities with the method of broken geodesics for the geodesic variational problem on riemannian manifolds on the other hand the monotonicity property can be related to the existence of local twist maps in the associated hamiltonian flow the finite dimensional reduction gives rise to a second order monotone recurrence relation we study these recurrence relations to find simple closed characteristics for the lagrangian system more complicated closed characteristics will be dealt with in future work furthermore we give conditions on the lagrangian that guarantee the twist property

a fully discrete c0 interior penalty galerkin approximation of the extended fisher kolmogorov equation a fully discrete c 0 interior penalty finite element method is proposed and analyzed for the extended fisher kolmogorov efk equation u email protected d email protected u 3 u 0 with appropriate initial and boundary conditions where c is a positive constant we derive a regularity estimate for the solution u of the efk equation that is explicit in c and as a consequence we derive a priori error estimates that are robust in c

complete surfaces of constant mean curvature 1 in the hyperbolic 3 space in the study of minimal surfaces in the euclidean 3 space the weierstrass representation plays an important role bryant br showed that an analogue of the weierstrass representation formula holds for surfaces of constant mean curvature i in the hyperbolic 3 space x3 in this article we abbreviate the term constant mean curvature i as cmc 1 like minimal surfaces in the euclidean space the hyperbolic gauss map of cmc 1 surfaces is defined as a holomorphic map to c u oo however in contrast to the euclidean case the hyperbolic gauss map of a cmc 1 surface may not be extended across the ends even if the total gaussian curvature is finite we call a complete cmc 1 surface whose gauss map can be extended across all of its ends a cmc 1 surface of regular ends in this article we produce an explicit tool to construct cmc 1 surfaces of regular ends in section 2 we show that such surfaces are constructed by solving some ordinary differential equations with regular singularity in our cmc 1 category ossermann s inequality is not expected and the cohn vossen inequality is the best possible one we show in section 4 that the equality of the cohn vossen inequality never holds for complete cmc 1 surfaces in xh3 in section 5 we give a necessary and sufficient condition that a regular end of a cmc 1 surface be embedded in section 6 we classify complete cmc 1 surfaces of genus 0 with two regular ends our classification contains new examples furthermore in section 7 we construct several new cmc 1 surfaces with regular embedded ends each of these examples has a nontrivial deformation which is mentioned in section 3 it should be remarked that our construction does not work for surfaces with irregular ends but there is another construction by perturbing minimal surfaces in the euclidean 3 space the authors constructed cmc 1 surfaces all of whose ends are irregular see uy1

analysis of the min sum algorithm for packing and covering problems via linear programming message passing algorithms based on belief propagation bp are successfully used in many applications including decoding error correcting codes and solving constraint satisfaction and inference problems the bp based algorithms operate over graph representations called factor graphs that are used to model the input although in many cases the bp based algorithms exhibit impressive empirical results not much has been proved when the factor graphs have cycles this paper deals with packing and covering integer programs in which the constraint matrix is zero one the constraint vector is integral and the variables are subject to box constraints we study the performance of the min sum algorithm when applied to the corresponding factor graph models of packing and covering linear programmings lps we compare the solutions computed by the min sum algorithm for packing and covering problems to the optimal solutions of the corresponding lp relaxations in particular we prove that if the lp has an optimal fractional solution then for each fractional component the min sum algorithm either computes multiple solutions or the solution oscillates below and above the fraction this implies that the min sum algorithm computes the optimal integral solution only if the lp has a unique optimal solution that is integral the converse is not true in general for a special case of packing and covering problems we prove that if the lp has a unique optimal solution that is integral and on the boundary of the box constraints then the min sum algorithm computes the optimal solution in pseudopolynomial time our results unify and extend recent results for the maximum weight matching problem and for the maximum weight independent set problem

on decoding irregular tanner codes with local optimality guarantees we consider decoding of binary linear tanner codes using message passing iterative decoding and linear programming lp decoding in memoryless binary input output symmetric mbios channels we present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular tanner codes with respect to any mbios channel this characterization is a generalization of arora proc acm symp theory of computing 2009 and vontobel proc inf theory and appl workshop 2010 and is based on a conical combination of normalized weighted subtrees in the computation trees of the tanner graph these subtrees may have any finite height h even equal or greater than half of the girth of the tanner graph in addition the degrees of local code nodes in these subtrees are not restricted to two i e these subtrees are not restricted to skinny trees we prove that local optimality in this new characterization implies maximum likelihood ml optimality and lp optimality and show that a certificate can be computed efficiently we also present a new message passing iterative decoding algorithm called normalized weighted min sum nwms nwms decoding is a belief propagation bp type algorithm that applies to any irregular binary tanner code with single parity check local codes e g low density and high density parity check codes we prove that if a locally optimal codeword with respect to height parameter h exists whereby notably h is not limited by the girth of the tanner graph then nwms decoding finds this codeword in h iterations the decoding guarantee of the nwms decoding algorithm applies whenever there exists a locally optimal codeword because local optimality of a codeword implies that it is the unique ml codeword the decoding guarantee also provides an ml certificate for this codeword finally we apply the new local optimality characterization to regular tanner codes and prove lower bounds on the noise thresholds of lp decoding in mbios channels when the noise is below these lower bounds the probability that lp decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the tanner graph

minimal heteroclinics for a class of fourth order o d e systems abstract we prove the existence of minimal heteroclinic orbits for a class of fourth order o d e systems with variational structure in our general set up the set of equilibria of these systems is a union of manifolds and the heteroclinic orbits connect two disjoint components of this set

a separation algorithm for improved lp decoding of linear block codes maximum likelihood ml decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an integer programming ip problem feldman relaxed this ip problem and presented linear programming lp based decoding in this paper we propose a new separation algorithm to improve the error correcting performance of lp decoding for binary linear block codes we use an ip formulation with indicator variables that help in detecting the violated parity checks we derive gomory cuts from the ip and use them in our separation algorithm an efficient method of finding cuts induced by redundant parity checks rpc is also proposed under certain circumstances we can guarantee that these rpc cuts are valid and cut off the fractional optimal solutions of lp decoding it is demonstrated on three ldpc codes and two bch codes that our separation algorithm performs significantly better than lp decoding and belief propagation bp decoding

complete linear weingarten surfaces of bryant type a plateau problem at infinity in this paper we study a large class of weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3 space we show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the euclidean 3 space and we use it to study their completeness we also establish some existence and uniqueness theorems by studing the plateau problem at infinity when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family and how many embedded solutions are there

expander graph arguments for message passing algorithms we show how expander based arguments may be used to prove that message passing algorithms can correct a linear number of erroneous messages the implication of this result is that when the block length is sufficiently large once a message passing algorithm has corrected a sufficiently large fraction of the errors it will eventually correct all errors this result is then combined with known results on the ability of message passing algorithms to reduce the number of errors to an arbitrarily small fraction for relatively high transmission rates the results hold for various message passing algorithms including gallager s hard decision and soft decision with clipping decoding algorithms our results assume low density parity check ldpc codes based on an irregular bipartite graph

stable patterns for fourth order parabolic equations we consider fourth order parabolic equations of gradient type for the sake of simplicity the analysis is carried out for the specific equation u

lp decoding corrects a constant fraction of errors we show that for low density parity check ldpc codes whose tanner graphs have sufficient expansion the linear programming lp decoder of feldman karger and wainwright can correct a constant fraction of errors a random graph will have sufficient expansion with high probability and recent work shows that such graphs can be constructed efficiently a key element of our method is the use of a dual witness a zero valued dual solution to the decoding linear program whose existence proves decoding success we show that as long as no more than a certain constant fraction of the bits are flipped by the channel we can find a dual witness this new method can be used for proving bounds on the performance of any lp decoder even in a probabilistic setting our result implies that the word error rate of the lp decoder decreases exponentially in the code length under the binary symmetric channel bsc this is the first such error bound for ldpc codes using an analysis based on pseudocodewords recent work by koetter and vontobel shows that lp decoding and min sum decoding of ldpc codes are closely related by the graph cover structure of their pseudocodewords in their terminology our result implies that that there exist families of ldpc codes where the minimum bsc pseudoweight grows linearly in the block length

positivity and the attractor dimension in a fourth order reaction diffusion equation in this paper we investigate the semilinear partial differential equation ut fuxxxx uxx u 1 u2 with a view particularly to obtaining some insight into how one might establish positivity preservation results for equations containing fourth order spatial derivatives the maximum principle cannot be applied to such equations however progress can be made by employing some very recent best possible interpolation inequalities due to the third named author in which the interpolation constants are both explicitly known and sharp these are used to estimate the lx distance between u and 1 during the evolution a positivity preservation result can be obtained under certain restrictions on the initial datum we also establish an explicit two sided estimate for the fractal dimension of the attractor which is sharp in terms of the physical parameters

convergence of weighted min sum decoding via dynamic programming on trees applying the max product and sum product algorithms to loopy graphs is now quite popular for best assignment problems this is largely due to their low computational complexity and impressive performance in practice still there is no general understanding of the conditions required for convergence or optimality of converged solutions or both this paper presents an analysis of both attenuated max product decoding and weighted min sum decoding for low density parity check ldpc codes which guarantees convergence to a fixed point when a weight factor is sufficiently small it also shows that if the fixed point satisfies some consistency conditions then it must be both a linear programming lp and maximum likelihood ml decoding solution for dv dc regular ldpc codes the weight factor must satisfy dv 1 1 are given finally connections are explored with recent work on the threshold of lp decoding

homotopy classes for stable connections between hamiltonian saddle focus equilibria for a class of hamiltonian systems in 4 the set of homoclinic and heteroclinic orbits which connect saddle focus equilibria is studied using a variational approach the oscillatory properties of a saddle focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle foci this variational approach does not require any assumptions on the intersections of stable and unstable manifolds such as transversality moreover these connections are shown to be local minimizers of an associated action functional this result has applications to spatial pattern formation in a class of fourth order bistable evolution equations

entire solutions of the kpp equation this paper deals with the solutions defined for all time of the kpp equation ut uxx f u 0 0 f 0 1 0i n 0 1 and f 0 s f 0 0 in 0 1 this equation admits infinitely many traveling wave type solutions increasing or decreasing in x i t also admits solutions that depend only on t in this paper we build four other manifolds of solutions one is 5 dimensional one is 4 dimensional and two are 3 dimensional some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix furthermore the traveling wave solutions are on the boundary of these four manifolds c 1999 john wiley sons inc

local correctability of expander codes in this work we present the first local decoding algorithm for expander codes this yields a new family of constant rate codes that can recover from a constant fraction of errors in the codeword symbols and where any symbol of the codeword can be recovered with high probability by reading n e symbols from the corrupted codeword where n is the block length of the code expander codes introduced by sipser and spielman are formed from an expander graph g v e of degree d and an inner code of block length d over an alphabet each edge of the expander graph is associated with a symbol in a string in e will be a codeword if for each vertex in v the symbols on the adjacent edges form a codeword in the inner code we show that if the inner code has a smooth reconstruction algorithm in the noiseless setting then the corresponding expander code has an efficient local correction algorithm in the noisy setting instantiating our construction with inner codes based on finite geometries we obtain novel locally decodable codes with rate approaching one this provides an alternative to the multiplicity codes of kopparty saraf and yekhanin stoc 11 and the lifted codes of guo kopparty and sudan itcs 13

an analogue of minimal surface theory in we shall discuss the class of surfaces with holomorphic right gauss maps in non compact duals of compact semi simple lie groups e g sl n c su n which contains minimal surfaces in r n and constant mean curvature 1 surfaces in h 3 a weierstrass type representation formula and a chern osserman type inequality for such surfaces are given

reduced complexity decoding of ldpc codes various log likelihood ratio based belief propagation llr bp decoding algorithms and their reduced complexity derivatives for low density parity check ldpc codes are presented numerically accurate representations of the check node update computation used in llr bp decoding are described furthermore approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check node update or symbol node update or both in particular two main approaches for simplified check node updates are presented that are based on the so called min sum approximation coupled with either a normalization term or an additive offset term density evolution is used to analyze the performance of these decoding algorithms to determine the optimum values of the key parameters and to evaluate finite quantization effects simulation results show that these reduced complexity decoding algorithms for ldpc codes achieve a performance very close to that of the bp algorithm the unified treatment of decoding techniques for ldpc codes presented here provides flexibility in selecting the appropriate scheme from performance latency computational complexity and memory requirement perspectives

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

distance properties of expander codes the minimum distance of some families of expander codes is studied as well as some related families of codes defined on bipartite graphs the weight spectrum and the minimum distance of a random ensemble of such codes are computed and it is shown that it sometimes meets the gilbert varshamov gv bound a lower bound on the minimum distances of constructive families of expander codes is derived the relative minimum distance of the expander code is shown to exceed the product bound i e the quantity spl delta sub 0 spl delta sub 1 where spl delta sub 0 and spl delta sub 1 are the minimum relative distances of the constituent codes as a consequence of this a polynomially constructible family of expander codes is obtained whose relative distance exceeds the zyablov bound on the distance of serial concatenations