rotation numbers eigenvalues and the poincare birkhoff theorem using the relation between rotation numbers and eigenvalues we prove the existence of nontrivial t periodic solutions having precise oscillatory properties for a class of asymptotically linear second order differential equations

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

zero error capacity of duplication channels this paper is concerned with the problem of error free communication over the i i d duplication channel which acts on a transmitted sequence x 1 cdots x n by inserting a random number of copies of each symbol x i next to the original symbol the random variables representing the numbers of inserted copies at each position i are independent and take values in 0 1 ldots r where r is a fixed parameter a more general model in which blocks of ell consecutive symbols are being duplicated and which is inspired by dna based data storage systems wherein the stored molecules are subject to tandem duplication mutations is also analyzed a construction of optimal codes correcting all patterns of errors of this type is described and the zero error capacity of the duplication channel the largest rate at which information can be transmitted through it in an error free manner is determined for each ell and r

achievable rates for channels with deletions and insertions this paper considers a binary channel with deletions and insertions where each input bit is transformed in one of the following ways it is deleted with probability d or an extra bit is added after it with probability i or it is transmitted unmodified with probability 1 d i a computable lower bound on the capacity of this channel is derived the transformation of the input sequence by the channel may be viewed in terms of runs as follows some runs of the input sequence get shorter longer some runs get deleted and some new runs are added it is difficult for the decoder to synchronize the channel output sequence to the transmitted codeword mainly due to deleted runs and new inserted runs the main idea is a mutual information decomposition in terms of the rate achieved by a suboptimal decoder that determines the positions of the deleted and inserted runs in addition to decoding the transmitted codeword the mutual information between the channel input and output sequences is expressed as the sum of the rate achieved by this decoder and the rate loss due to its suboptimality obtaining computable lower bounds on each of these quantities yields a lower bound on the capacity the bounds proposed in this paper provide the first characterization of achievable rates for channels with general insertions and for channels with both deletions and insertions for the special case of the deletion channel the proposed bound improves on the previous best lower bound for deletion probabilities up to 0 3

resonance and rotation numbers for planar hamiltonian systems multiplicity results via the poincare birkhoff theorem abstract in the general setting of a planar first order system 0 1 u g t u u r 2 with g 0 t r 2 r 2 we study the relationships between some classical nonresonance conditions including the landesman lazer one at infinity and in the unforced case i e g t 0 0 at zero and the rotation numbers of large and small solutions of 0 1 respectively such estimates are then used to establish via the poincare birkhoff fixed point theorem new multiplicity results for t periodic solutions of unforced planar hamiltonian systems j u u h t u and unforced undamped scalar second order equations x g t x 0 in particular by means of the landesman lazer condition we obtain sharp conclusions when the system is resonant at infinity

nonautonomous nonlinear odes nonresonance conditions and rotation numbers abstract we deal with the t periodic problem associated with a nonlinear scalar differential equation like 1 x f t x 0 where for x 0 and x the nonlinearity f is assumed behave linearly with a time dependent coefficient we prove that the landesman lazer conditions at zero and at infinity possess a rotational effect on small and large in the phase plane solutions of 1 as a consequence we are able to generalize previous multiplicity results in the resonant case i e when the linearizations at zero and or at infinity are resonant through the use of the poincare birkhoff fixed point theorem

lee metric bch codes and their application to constrained and partial response channels shows that each code in a certain class of bch codes over gf p specified by a code length n spl les p sup m 1 and a runlength r spl les p 1 2 of consecutive roots in gf p sup m has minimum lee distance spl ges 2r for the very high rate range these codes approach the sphere packing bound on the minimum lee distance furthermore for a given r the length range of these codes is twice as large as that attainable by berlekamp s 1984 extended negacyclic codes the authors present an efficient decoding procedure based on euclid s algorithm for correcting up to r 1 errors and detecting r errors that is up to the number of lee errors guaranteed by the designed minimum lee distance 2r bounds on the minimum lee distance for r spl ges p 1 2 are provided for the reed solomon case i e when the bch code roots are in gf p the authors present two applications first lee metric bch codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and or deletion of zeros in d k constrained channels second the code construction with its decoding algorithm can be formulated over the integer ring providing an algebraic approach to correcting errors in partial response channels where matched spectral null codes are used

simulation based computation of information rates for channels with memory the information rate of finite state source channel models can be accurately estimated by sampling both a long channel input sequence and the corresponding channel output sequence followed by a forward sum product recursion on the joint source channel trellis this method is extended to compute upper and lower bounds on the information rate of very general channels with memory by means of finite state approximations further upper and lower bounds can be computed by reduced state methods

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

bounds on the capacity of discrete memoryless channels corrupted by synchronization and substitution errors we study the capacity of discrete memoryless channels with synchronization errors and additive noise we first show that with very large alphabets their capacity can be achieved by independent and identically distributed input sources and establish proven tight lower and upper capacity bounds we also derive tight numerical capacity bounds for channels where the synchronization between the input and output is partly preserved for instance using incorruptible synchronization markers such channels include channels with duplication errors channels that only insert or delete zeros and channels with bitshift errors studied in magnetic recording channels with small alphabets and corrupted by synchronization errors have an infinite memory revisiting the theoretical work of dobrushin and adapting techniques used to compute capacity bounds for finite state source channel models we compute improved numerical capacity lower bounds for discrete memoryless channels with small alphabets synchronization errors and memoryless noise an interesting and some what surprising result is that as long as the input sequences are not completely deleted the capacity of channels corrupted by discrete timing errors is always nonzero even if all the symbols are corrupted

directly lower bounding the information capacity for channels with i i d deletions and duplications in this paper we directly lower bound the information capacity for channels with independent identically distributed i i d deletions and duplications our approach differs from previous work in that we focus on the information capacity using ideas from renewal theory rather than focusing on the transmission capacity by analyzing the error probability of some randomly generated code using a combinatorial argument of course the transmission and information capacities are equal but our change of perspective allows for a much simpler analysis that gives more general theoretical results we then apply these results to the binary deletion channel to improve existing lower bounds on its capacity

an optimal parallel algorithm for the visibility of a simple polygon from a point we present a parallel algorithm for computing the visible portion of a simple polygonal chain with 7i vertices from a point in the plane the algorithm runs in o logn time using 0 nf log n processors in the crew pram computational model and hence is asymptotically optimal

reliable communication over channels with insertions deletions and substitutions a new block code is introduced which is capable of correcting multiple insertion deletion and substitution errors the code consists of nonlinear inner codes which we call watermark codes concatenated with low density parity check codes over nonbinary fields the inner code allows probabilistic resynchronization and provides soft outputs for the outer decoder which then completes decoding we present codes of rate 0 7 and transmitted length 5000 bits that can correct 30 insertion deletion errors per block we also present codes of rate 3 14 and length 4600 bits that can correct 450 insertion deletion errors per block

improved lower bounds for the capacity of i i d deletion and duplication channels this paper considers the capacity of binary deletion channels where bits are deleted independently with probability d it improves significantly upon the best previous framework used to obtain provable lower bounds on this capacity by utilizing a stronger definition of a typical output from the channel the new results give the best known provable bounds on the capacity for all values of d moreover the techniques presented here extend to yield lower bounds for channels with certain types of random insertions namely duplications or combinations of duplications and deletions to demonstrate these techniques in this context two binary channels are analyzed a channel where each transmitted bit is copied with probability nu and a channel where each transmitted bit is copied a geometrically distributed number of times

sharp analytical capacity upper bounds for sticky and related channels we study natural examples of binary channels with synchronization errors these include the duplication channel which independently outputs a given bit once or twice and geometric channels that repeat a given bit according to a geometric rule with or without the possibility of bit deletion we apply the general framework of cheraghchi journal of the acm 2019 to obtain sharp analytical upper bounds on the capacity of these channels previously upper bounds were known via numerical computations involving the computation of finite approximations of the channels by a computer and then using the obtained numerical results to upper bound the actual capacity while leading to sharp numerical results further progress on the full understanding of the channel capacity inherently remains elusive using such methods our results can be regarded as a major step toward a complete understanding of the capacity curves quantitatively our upper bounds sharply approach and in some cases surpass the bounds that were previously only known by purely numerical methods among our results we notably give a completely analytical proof that when the number of repetitions per bit is geometric supported on the non negative integers with mean growing to infinity the channel capacity remains substantially bounded away from 1

half eigenvalues of periodic sturm liouville problems we consider the nonlinear sturm liouville problem 1 pu qu au bu u in 0 2 2 u 0 u 2 pu 0 pu 2 where 1 p q l1 0 2 with p 0 a e on 0 2 a b l1 0 2 is a real parameter and u t max u t 0 for t 0 2 values of for which 1 2 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions the set of half eigenvalues will be denoted by h we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties associated with h these properties yield results on the existence and non existence of solutions of the problem 3 pu qu f t u h in 0 2 together with 2 where h l1 0 2 f 0 2 r r is a caratheodory function and the limits a t lim f t b t lim f t exist for a e t 0 2 when the functions a and b are constant the set h is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

polynomial time decodable codes for the binary deletion channel in the random deletion channel each bit is deleted independently with probability p for the random deletion channel the existence of codes of rate 1 p 9 and thus bounded away from 0 for any p has been known we give an explicit construction with polynomial time encoding and deletion correction algorithms with rate c 0 1 p for an absolute constant c 0 0

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

on the number of subsequences when deleting symbols from a string we consider the problem of finding the number of subsequences when deleting symbols from a string we present a framework to find closed form expressions for the number of subsequences and use it to prove the first explicit formulas for and the first formulas for nonbinary alphabets we also present the first efficient algorithm to compute the number of subsequences for any string number of deletions and alphabet size

markov processes asymptotically achieve the capacity of finite state intersymbol interference channels recent progress in capacity evaluation has made it possible to compute a sequence of lower bounds on the capacity of a finite state intersymbol interference isi channel by finding a sequence of optimized markov input processes with increasing order for which the state of the process is the previous input symbols in this correspondence we prove that as the order goes to infinity the sequence of optimized markov sources asymptotically achieves the capacity of the channel the conclusion is extended to two dimensional finite state isi channels the binary symmetric channel bsc with constrained inputs and general indecomposable finite state channels with a mild constraint

positive periodic solutions to an indefinite minkowski curvature equation abstract we investigate the existence non existence multiplicity of positive periodic solutions both harmonic i e t periodic and subharmonic i e kt periodic for some integer k 2 to the equation u 1 u 2 a t g u 0 where 0 is a parameter a t is a t periodic sign changing weight function and g 0 0 is a continuous function having superlinear growth at zero in particular we prove that for both g u u p with p 1 and g u u p 1 u p q with 0 q 1 p the equation has no positive t periodic solutions for close to zero and two positive t periodic solutions a small one and a large one for large enough moreover in both cases the small t periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period the proof of the existence of t periodic solutions relies on a recent extension of mawhin s coincidence degree theory for locally compact operators in product of banach spaces while subharmonic solutions are found by an application of the poincare birkhoff fixed point theorem after a careful asymptotic analysis of the t periodic solutions for

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

upper bounds on the capacity of deletion channels using channel fragmentation we study memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol drop outs or symbol insertions with particular emphasis on the binary and non binary deletion channels we offer a different look at these channels by considering equivalent models by fragmenting the input sequence where different subsequences travel through different channels the resulting output symbols are combined appropriately to come up with an equivalent input output representation of the original channel which allows for derivation of new upper bounds on the channel capacity we consider both random and deterministic types of fragmentation processes applied to binary and nonbinary deletion channels with two specific applications of this idea a random fragmentation applied to a binary deletion channel and a deterministic fragmentation process applied to a nonbinary deletion channel we prove certain inequality relations among the capacities of the original channels and those of the introduced subchannels the resulting inequalities prove useful in deriving tighter capacity upper bounds for 1 independent identically distributed i i d deletion channels when the deletion probability exceeds 0 65 and 2 nonbinary deletion channels some extensions of these results for instance to the case of deletion substitution channels are also explored

a simple lower bound for the capacity of the deletion channel we present a simple proof that the capacity of the binary independent and identically distributed i i d deletion channel where each bit is deleted independently with probability d is at least 1 d 9 by developing a correspondence between the deletion channel and an insertion deletion channel that we call a poisson repeat channel

feedback capacity of finite state machine channels we consider a finite state machine channel with a finite memory length e g finite length intersymbol interference channels with finite input alphabets also known as partial response channels for such a finite state machine channel we show that feedback dependent markov sources achieve the feedback capacity and that the required memory length of the markov process matches the memory length of the channel further we show that the whole history of feedback is summarized by the causal posterior channel state distribution which is computed by the sum product forward recursion of the bahl cocke jelinek raviv bcjr baum welch discrete time wonham filtering algorithm these results drastically reduce the space over which the optimal feedback dependent source distribution needs to be sought further the feedback capacity computation may then be formulated as an average reward per stage stochastic control problem which is solved by dynamic programming with the knowledge of the capacity achieving source distribution the value of the capacity is easily estimated using markov chain monte carlo methods when the feedback is delayed we show that the feedback capacity can be computed by similar procedures we also show that the delayed feedback capacity is a tight upper bound on the feedforward capacity by comparing it to tight existing lower bounds we demonstrate the applicability of the method by computing the feedback capacity of partial response channels and the feedback capacity of run length limited rll sequences over binary symmetric channels bscs

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

improved upper bounds and structural results on the capacity of the discrete time poisson channel new capacity upper bounds are presented for the discrete time poisson channel with no dark current and an average power constraint these bounds are a consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels previously the best known capacity upper bound in the regime where the average power constraint does not approach zero was due to martinez josa b 2007 which is re derived as a special case of the framework developed in this paper furthermore this framework is carefully instantiated in order to obtain a closed form bound that improves the result of martinez everywhere finally capacity achieving distributions for the discrete time poisson channel are studied under an average power constraint and or a peak power constraint and arbitrary dark current in particular it is shown that the support of the capacity achieving distribution under an average power constraint must only be countably infinite this settles a conjecture of shamai iee proceedings i 1990 in the affirmative previously it was only known that the support must be an unbounded set

on lower bounds for the capacity of deletion channels this correspondence considers binary deletion channels where bits are deleted independently with probability d it improves upon the framework used to analyze the capacity of binary deletion channels established by diggavi and grossglauser improving on their lower bounds diggavi and grossglauser considered codebooks with codewords generated by a first order markov chain they only consider typical outputs where an output is typical if an n bit input gives an n 1 d 1 epsi bit output the improvements in this correspondence arise from two considerations first a stronger notion of a typical output from the channel is used which yields better bounds even for the codebooks studied by diggavi and grossglauser second codewords generated by more general processes than first order markov chains are considered

capacity bounds for sticky channels the capacity of sticky channels a subclass of insertion channels where each symbol may be duplicated multiple times is considered the primary result is to provide nearly tight numerical upper and lower bounds for the independent and identically distributed i i d duplication channel

on the capacity of channels with timing synchronization errors we consider a new formulation of a class of synchronization error channels and derive analytical bounds and numerical estimates for the capacity of these channels for the binary channel with only deletions we obtain an expression for the symmetric information rate in terms of subsequence weights which reduces to a tight lower bound for small deletion probabilities we are also able to exactly characterize the markov 1 rate for the binary channel with only replications for a channel that introduces deletions as well as replications of input symbols we design approximating channels that parameterize the state space and show that the information rates of these approximate channels approach that of the deletion replication channel as the state space grows for the case of the channel where deletions and replications occur with the same probabilities a stronger result in the convergence of mutual information rates is shown the numerous advantages this new formulation presents are explored

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

on the use of morse index and rotation numbers for multiplicity results of resonant bvps abstract motivated by a recent series of papers by k li and co workers 14 15 16 17 18 we consider the problem of the existence and multiplicity of solutions for the neumann or periodic bvps associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems one considered near zero and the other near infinity we show how a rotation number approach together with the poincare birkhoff theorem and a recent variant of it allows to obtain multiplicity results in terms of the gap between the morse indexes of the referred linear systems at zero and at infinity these systems may also be resonant when the gaps are sufficiently large our multiplicity results improve the ones obtained by variational methods in the quoted papers also our approach allows a description of the solutions obtained in terms of their nodal properties

construction of fixed length insertion deletion correcting runlength limited codes an algorithm is presented for the construction of fixed length insertion deletion correcting runlength limited rll codes in this construction binary d k constrained codewords are generated by codewords of a q ary lee metric based code it is shown that this new construction always yields better codes than known constructions the use of a q ary lee 1987 metric code q odd is based on the assumption that an error insertion deletion or peak shift has maximal size q 1 2 it is shown that a decoding algorithm for the lee metric code can be extended so that it can also be applied to insertion deletion correcting rll codes furthermore such an extended algorithm can also correct some error patterns containing errors of size more than q 1 2 as a consequence if s denotes the maximal size of an error then in some cases the alphabet size of the generating code can be s 1 instead of 2 spl middot s 1

continuity in weak topology and extremal problems of eigenvalues of the p laplacian we will study the dependence of eigenvalues of the one dimensional p laplacian on potentials or weights two results are obtained one is the continuity of eigenvalues in potentials with respect to the weak topologies of l spaces 1 and the other is the continuous differentiability of eigenvalues in potentials with respect to l norms as applications we will study some extremal problems of eigenvalues by developing some analytical methods

intersection of convex objects in two and three dimensions one of the basic geometric operations involves determining whether a pair of convex objects intersect this problem is well understood in a model of computation in which the objects are given as input and their intersection is returned as output for many applications however it may be assumed that the objects already exist within the computer and that the only output desired is a single piece of data giving a common point if the objects intersect or reporting no intersection if they are disjoint for this problem none of the previous lower bounds are valid and algorithms are proposed requiring sublinear time for their solution in two and three dimensions

the existence of transverse homoclinic points in the sitnikov problem abstract using melnikov s method we are able to prove the existence of transverse homoclinic orbits and therefore the existence of a horseshoe in a special restricted three body problem this analysis is an alternative to the one described by moser stable and random motions in dynamical systems princeton univ press princeton nj 1973 based on sitnikov s original work dokl akad nauk ussr 133 no 2 1960 303 306 where the task is accomplished using a more direct construction of the horseshoe

nonuniform nonresonance of semilinear differential equations abstract consider the dirichlet problem of nonlinear differential equations with the principal part the p laplacian when the nonlinearity satisfies some semilinearity conditions the usual nonuniform nonresonance conditions are obtained by comparing nonlinear equations with the classical eigenvalues in this article we will introduce some weighted eigenvalues the nonuniform nonresonance conditions proved in this article using weighted eigenvalues will improve the usual ones significantly

cascading divide and conquer a technique for designing parallel algorithms techniques for parallel divide and conquer are presented resulting in improved parallel algorithms for a number of problems the problems for which improved algorithms are given include segment intersection detection trapezoidal decomposition and planar point location efficient parallel algorithms are algo given for fractional cascading three dimensional maxima two set dominance counting and visibility from a point all of the algorithms presented run in o log n time with either a linear or a sublinear number of processors in the crew pram model

matched information rate codes for partial response channels in this paper we design capacity approaching codes for partial response channels the codes are constructed as concatenations of inner trellis codes and outer low density parity check ldpc codes unlike previous constructions of trellis codes for partial response channels we disregard any algebraic properties e g the minimum distance or the run length limit in our design of the trellis code our design is purely probabilistic in that we construct the inner trellis code to mimic the transition probabilities of a markov process that achieves a high capacity approaching information rate hence we name it a matched information rate mir design we provide a set of five design rules for constructions of capacity approaching mir inner trellis codes we optimize the outer ldpc code using density evolution tools specially modified to fit the superchannel consisting of the inner mir trellis code concatenated with the partial response channel using this strategy we design degree sequences of irregular ldpc codes whose noise tolerance thresholds are only fractions of a decibel away from the capacity examples of code constructions are shown for channels both with and without spectral nulls

achievable rates for noisy channels with synchronization errors we develop several lower bounds on the capacity of binary input symmetric output channels with synchronization errors which also suffer from other types of impairments such as substitutions erasures additive white gaussian noise awgn etc more precisely we show that if a channel suffering from synchronization errors as well as other type of impairments can be decomposed into a cascade of two component channels where the first one is another channel with synchronization errors and the second one is a memoryless channel with no synchronization errors a lower bound on the capacity of the original channel in terms of the capacity of the component synchronization error channel can be derived a primary application of our results is that we can employ any lower bound derived on the capacity of the component synchronization error channel to find lower bounds on the capacity of the original noisy channel with synchronization errors we apply the general ideas to several specific classes of channels such as synchronization error channels with erasures and substitutions with symmetric q ary outputs and with awgn explicitly and obtain easy to compute bounds we illustrate that with our approach it is possible to derive tighter capacity lower bounds compared to the currently available bounds in the literature for certain classes of channels e g deletion substitution channels and deletion awgn channels for certain signal to noise ratio snr ranges

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity