measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

onset of chaos in differential delay equations abstract in the present paper we study the onset of chaos in the differential delay equation x dot t x t bx t 2 1 x 10 t 2 b 0 we develop a new numerical technique and apply it to study the global behavior of the flow on the attractor using this technique we find unstable periodic orbits connections between different periodic orbits transverse homoclinic orbits and try to explain the underlying reason for chaos

combinatorial bounds for list decoding informally an error correcting code has nice list decodability properties if every hamming ball of large radius has a small number of codewords in it we report linear codes with nontrivial list decodability i e codes of large rate that are nicely list decodable and codes of large distance that are not nicely list decodable specifically on the positive side we show that there exist codes of rate r and block length n that have at most c codewords in every hamming ball of radius h sup 1 1 r 1 c spl middot n this answers the main open question from the work of elias 1957 this result also has consequences for the construction of concatenated codes of good rate that are list decodable from a large fraction of errors improving previous results of guruswami and sudan see ieee trans inform theory vol 45 p 1757 67 sept 1999 and proc 32nd acm symp theory of computing stoc portland or p 181 190 may 2000 in this vein specifically for every spl epsi 0 we present a polynomial time constructible asymptotically good family of binary codes of rate spl omega spl epsi sup 4 that can be list decoded in polynomial time from up to a fraction 1 2 spl epsi of errors using lists of size o spl epsi sup 2 on the negative side we show that for every spl delta and c there exists spl tau 0 and an infinite family of linear codes c sub i sub i such that if n sub i denotes the block length of c sub i then c sub i has minimum distance at least spl delta spl middot n sub i and contains more than c sub 1 spl middot n sub i sup c codewords in some hamming ball of radius spl tau spl middot n sub i while this result is still far from known bounds on the list decodability of linear codes it is the first to bound the radius for list decodability by a polynomial sized list away from the minimum distance of the code

a class of stochastic optimization problems with one quadratic several linear objective functions and extended portfolio selection model in this paper a class of stochastic multiple objective programming problems with one quadratic several linear objective functions and linear constraints has been introduced the former model is transformed into a deterministic multiple objective nonlinear programming model by means of the introduction of random variables expectation the reference direction approach is used to deal with linear objectives and results in a linear parametric optimization formula with a single linear objective function this objective function is combined with the quadratic function using the weighted sums the quadratic problem is transformed into a linear parametric complementary problem the basic formula for the proposed approach the sufficient and necessary conditions for properly weakly efficient solutions and some construction characteristics of weakly efficient solution sets are obtained an interactive algorithm is proposed based on reference direction and weighted sums varying the parameter vector on the right hand side of the model the dm can freely search the efficient frontier with the model an extended portfolio selection model is formed when liquidity is considered as another objective to be optimized besides expectation and risk the interactive approach is illustrated with a practical example

correlated algebraic geometric codes improved list decoding over bounded alphabets we define a new family of error correcting codes based on algebraic curves over finite fields and develop efficient list decoding algorithms for them our codes extend the class of algebraic geometric ag codes via a nonobvious generalization of the approach in the recent breakthrough work of parvaresh and vardy 2005 our work shows that the pv framework applies to fairly general settings by elucidating the key algebraic concepts underlying it also more importantly ag codes of arbitrary block length exist over fixed alphabets thus enabling us to establish new trade offs between the list decoding radius and rate over a bounded alphabet size the work of parvaresh and vardy 2005 was extended in guruswami and rudra 2006 to give explicit codes that achieve the list decoding capacity optimal trade off between rate and fraction of errors corrected over large alphabets a similar extension of this work along the lines of guruswami and rudra could have substantial impact indeed it could give better trade offs than currently known over a fixed alphabet say gf 2 12 which in turn upon concatenation with a fixed well understood binary code could take us closer to the list decoding capacity for binary codes this may also be a promising way to address the significant complexity drawback of the result of guruswami and rudra and to enable approaching capacity with bounded list size independent of the block length the list size and decoding complexity in their work are both n 1 e where e is the distance to capacity similar to algorithms for ag codes from guruswami and sudan 1999 and 2001 our encoding decoding algorithms run in polynomial time assuming a natural polynomial size representation of the code for codes based on a specific optimal algebraic curve we also present an expected polynomial time algorithm to construct the requisite representation this in turn fills an important void in the literature by presenting an efficient construction of the representation often assumed in the list decoding algorithms for ag codes

robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices randomness is a vital resource for modern day information processing especially for cryptography a wide range of applications critically rely on abundant high quality random numbers generated securely here we show how to expand a random seed at an exponential rate without trusting the underlying quantum devices our approach is secure against the most general adversaries and has the following new features cryptographic level of security tolerating a constant level of imprecision in devices requiring only unit size quantum memory for each device component in an honest implementation and allowing a large natural class of constructions for the protocol in conjunction with a recent work by chung et al l2014r it also leads to robust unbounded expansion using just 2 multipart devices when adapted for distributing cryptographic keys our method achieves for the first time exponential expansion combined with cryptographic security and noise tolerance the proof proceeds by showing that the renyi divergence of the outputs of the protocol for a specific bounding operator decreases linearly as the protocol iterates at the heart of the proof are a new uncertainty principle on quantum measurements and a method for simulating trusted measurements with untrusted devices

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

decoding interleaved reed solomon codes over noisy channels we consider error correction over the non binary symmetric channel nbsc which is a natural probabilistic extension of the binary symmetric channel bsc we propose a new decoding algorithm for interleaved reed solomon codes that attempts to correct all interleaved codewords simultaneously in particular interleaved encoding gives rise to multi dimensional curves and more specifically to a variation of the polynomial reconstruction problem which we call simultaneous polynomial reconstruction we present and analyze a novel probabilistic algorithm that solves this problem our construction yields a decoding algorithm for interleaved rs codes that allows efficient transmission arbitrarily close to the channel capacity in the nbsc model

change of the attractor structure for x t f x t 1 when f changes from monotone to non monotone negative feedback abstract it is known that for bounded f with monotone negative feedback the scalar delay equation x t f x t 1 has an attractor a within the slowly oscillating solution class that is a two dimensional graph a slowly oscillating periodic orbit divides the surface a into an interior part containing zero and an exterior part we describe deformations of f to functions with non monotone negative feedback which preserve the interior part but make orbits from the exterior part converge to zero as t thus the graph structure of a is lost in such deformations of f

domain decomposition method for the global dynamics of delay differential equations with unimodal feedback x t zkmx t c f x tkt with unimodal feedback is studied the existence of the global attractor is shown and bounds of the attractor are given we find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f 0 is negative with respect to a positive equilibrium so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions in particular the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established numerical examples using nicholson s blowflies equation and the mackey glass equation are provided to illustrate the main results

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

a new class of rank metric codes and their list decoding beyond the unique decoding radius compared with classical block codes list decoding rank metric codes efficiently seems more difficult the evidences to support this view include 1 so far the only known efficient list decoding of rank metric codes mathcal c gives decoding radius beyond 1 r 2 with positive rate r when the ratio of the number of rows over the number of columns is extremely small 2 the johnson bound for rank metric codes does not exist as opposed to classical codes and 3 the gabidulin codes of square matrices cannot be list decoded beyond half of the minimum distance although the list decodability of random rank metric codes and the limits to the list decodability have been determined completely little work on efficient list decoding of rank metric codes has been done the only known efficient list decoding of rank metric codes mathcal c gives decoding radius up to the singleton bound 1 r varepsilon with positive rate r when rho mathcal c is extremely small i e o varepsilon 2 where rho mathcal c denotes the ratio of the number of rows over the number of columns of mathcal c it is commonly believed that it is difficult to list decode rank metric codes mathcal c with the ratio rho mathcal c close to 1 the main purpose of this paper is to explicitly construct a class of rank metric codes mathcal c with the ratio rho mathcal c up to 1 2 and efficiently list decode these codes beyond unique decoding radius 1 r 2 furthermore encoding and list decoding algorithms run in polynomial time mathrm poly n exp 1 varepsilon the list size can be reduced to o 1 varepsilon by randomizing the algorithm our key idea is to employ bivariate polynomials f x y where f is linearized in variable y and the variable x is used to fold the code in other words the rows are used to correct rank errors and the columns are used to fold the code to enlarge the decoding radius apart from the above algebraic technique we have to prune down the list the algebraic idea enables us to pin down the messages into a structured subspace whose dimension is linear in the number n of columns this periodic structure allows us to pre encode the messages to prune down the list more precisely we use subspace design introduced by guruswami and xing to obtain a deterministic algorithm with a larger constant list size and employ hierarchical subspace evasive sets introduced by guruswami et al to obtain a randomized algorithm with a smaller constant list size

erasure list decodable codes from random and algebraic geometry codes erasure list decoding was introduced to correct a larger number of erasures by outputting a list of possible candidates in this paper we consider both random linear codes and algebraic geometry codes for list decoding from erasures the contributions of this paper are twofold first for arbitrary 0 and epsilon 0 r and epsilon are independent we show that with high probability a q ary random linear code of rate r is an erasure list decodable code with constant list size q o 1 epsilon that can correct a fraction 1 r epsilon of erasures i e a random linear code achieves the information theoretic optimal tradeoff between information rate and fraction of erasures second we show that algebraic geometry codes are good erasure list decodable codes precisely speaking a q ary algebraic geometry code of rate r from the garcia stichtenoth tower can correct 1 r 1 sqrt q 1 1 q epsilon fraction of erasures with list size o 1 epsilon this improves the johnson bound for erasures applied to algebraic geometry codes furthermore list decoding of these algebraic geometry codes can be implemented in polynomial time note that the code alphabet size q in this paper is constant and independent of epsilon

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

algebraic list decoding of subspace codes subspace codes are collections of subspaces of a certain ambient vector space over a finite field koetter and kschischang introduced subspace codes in order to correct errors and erasures in noncoherent random linear network coding they have also studied a remarkable family of subspace codes obtained by evaluating certain linearized polynomials the koetter kschischang subspace codes are widely regarded as the counterpart of reed solomon codes in the domain of network error correction koetter and kschischang have furthermore devised an algebraic decoding algorithm for these codes analogous to the berlekamp welch decoding algorithm for reed solomon codes in this paper we develop list decoding algorithms for subspace codes thereby providing for certain code parameters a better tradeoff between rate and error correction capability than that of the koetter kschischang codes in a sense we are able to achieve for these codes what sudan was able to achieve for reed solomon codes in order to do so we have to modify and generalize the original koetter kschischang construction in many important respects the end result is this for any integer our list l decoder guarantees successful recovery of the message subspace provided that the normalized dimension of the error satisfies l l l 1 2 r where r is the packet rate which is a new parameter introduced in this paper as in the case of sudan s list decoding algorithm this exceeds the previously best known error correction radius 1 r established by koetter and kschischang only for low rates r

randomly generating portfolio selection covariance matrices with specified distributional characteristics abstract in portfolio selection there is often the need for procedures to generate realistic covariance matrices for security returns for example to test and benchmark optimization algorithms for application in portfolio optimization such a procedure should allow the entries in the matrices to have distributional characteristics which we would consider realistic for security returns deriving motivation from the fact that a covariance matrix can be viewed as stemming from a matrix of factor loadings a procedure is developed for the random generation of covariance matrices a whose off diagonal covariance entries possess a pre specified expected value and standard deviation and b whose main diagonal variance entries possess a likely different pre specified expected value and standard deviation the paper concludes with a discussion about the futility one would likely encounter if one simply tried to invent a valid covariance matrix in the absence of a procedure such as in this paper

list decoding from erasures bounds and code constructions we consider the problem of list decoding from erasures we establish lower and upper bounds on the rate of a binary linear code that can be list decoded with list size l when up to a fraction p of its symbols are adversarially erased such bounds already exist in the literature albeit under the label of generalized hamming weights and we make their connection to list decoding from erasures explicit our bounds show that in the limit of large l the rate of such a code approaches the capacity 1 p of the erasure channel such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate spl omega spl epsiv sup 2 log 1 spl epsiv that can be efficiently list decoded using lists of size o 1 spl epsiv when an adversarially chosen 1 spl epsiv fraction of symbols are erased for arbitrary spl epsiv 0 this improves previous results in this vein which achieved a rate of spl omega spl epsiv sup 3 log 1 spl epsiv

simple extractors for all min entropies and a new pseudorandom generator a randomness extractor is an algorithm that given a sample from a distribution with sufficiently high min entropy and a short random seed produces an output that is statistically indistinguishable from uniform min entropy is a measure of the amount of randomness in a distribution we present a simple self contained extractor construction that produces good extractors for all min entropies our construction is algebraic and builds on a new polynomial based approach introduced by ta shma et al 2001b using our improvements we obtain for example an extractor with output length m k log n o 1 and seed length 1 p log n for an arbitrary 0 1 where n is the input length and k is the min entropy of the input distribution a pseudorandom generator is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform our technique also gives a new way to construct pseudorandom generators from functions that require large circuits our pseudorandom generator construction is not based on the nisan wigderson generator nisan and wigderson 1994 and turns worst case hardness directly into pseudorandomness the parameters of our generator match those in impagliazzo and wigderson 1997 and sudan et al 2001 and in particular are strong enough to obtain a new proof that p bpp if e requires exponential size circuits our construction also gives the following improvements over previous work we construct an optimal hitting set generator that stretches o log n random bits into s 1 pseudorandom bits when given a function on log n bits that requires circuits of size s this yields a quantitatively optimal hardness versus randomness tradeoff for both rp and bpp and solves an open problem raised in impagliazzo et al 1999 we give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits this technique also give a quantitatively optimal hardness versus randomness tradeoff for am and the first hardness amplification result for nondeterministic circuits

homoclinic solution and chaos in xf x 0 is followed by a move in the opposite direction i t 0 however there are equations 1 2 which are closely related to applications and where the negative feedback condition 3 is only true in a certain neighbourhood of x 0 furumochi 3 studied equation

extractor codes we study error correcting codes for highly noisy channels for example every received signal in the channel may originate from some half of the symbols in the alphabet our main conceptual contribution is an equivalence between error correcting codes for such channels and extractors our main technical contribution is a new explicit error correcting code based on trevisan s extractor that can handle such channels and even noisier ones our new code has polynomial time encoding and polynomial time soft decision decoding we note that reed solomon codes cannot handle such channels and our study exposes some limitations on list decoding of reed solomon codes another advantage of our equivalence is that when the johnson bound is restated in terms of extractors it becomes the well known leftover hash lemma this yields a new proof of the johnson bound which applies to large alphabets and soft decoding our explicit codes are useful in several applications first they yield algorithms to extract many hardcore bits using few auxiliary random bits second they are the key tool in a recent scheme to compactly store a set of elements in a way that membership in the set can be determined by looking at only one bit of the representation finally they are the basis for the recent construction of high noise almost optimal rate list decodable codes over large alphabets

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

explicit list decodable rank metric and subspace codes via subspace designs we construct an explicit family of mathbb f h linear rank metric codes over any field mathbb f h that enables efficient list decoding up to a fraction rho of errors in the rank metric with a rate of 1 rho varepsilon for any desired rho in 0 1 and varepsilon 0 this is the first explicit construction of positive rate rank metric codes for efficient list decoding beyond the unique decoding radius our codes are explicit subcodes of the well known gabidulin codes which encode linearized polynomials of low degree via their values at a collection of linearly independent points the subcode is picked by restricting the message polynomials to an mathbb f h subspace that evades the structured subspaces over an extension field mathbb f h t that arise in our linear algebraic list decoder for gabidulin codes this subspace is obtained by combining subspace designs constructed by guruswami and kopparty focs 13 with subspace evasive varieties due to dvir and lovett stoc 12 we establish a similar result for subspace codes which have received much attention recently in the context of network coding we also give explicit subcodes of folded reed solomon rs codes with small folding order which are list decodable in the hamming metric with optimal redundancy motivated by the fact that list decoding rs codes reduces to list decoding such folded rs codes however as we only list decode a subcode of these codes the johnson radius continues to be the best known error fraction for list decoding rs codes

explicit codes achieving list decoding capacity error correction with optimal redundancy in this paper we present error correcting codes that achieve the information theoretically best possible tradeoff between the rate and error correction radius specifically for every 0 r 1 and epsiv 0 we present an explicit construction of error correcting codes of rate that can be list decoded in polynomial time up to a fraction 1 r epsiv of worst case errors at least theoretically this meets one of the central challenges in algorithmic coding theory our codes are simple to describe they are folded reed solomon codes which are in fact exactly reed solomon rs codes but viewed as a code over a larger alphabet by careful bundling of codeword symbols given the ubiquity of rs codes this is an appealing feature of our result and in fact our methods directly yield better decoding algorithms for rs codes when errors occur in phased bursts the alphabet size of these folded rs codes is polynomial in the block length we are able to reduce this to a constant depending on epsiv using existing ideas concerning ldquolist recoveryrdquo and expander based codes concatenating the folded rs codes with suitable inner codes we get binary codes that can be efficiently decoded up to twice the radius achieved by the standard gmd decoding

large scale mv efficient frontier computation via a procedure of parametric quadratic programming abstract despite the volume of research conducted on efficient frontiers in many cases it is still not the easiest thing to compute a mean variance mv efficient frontier even when all constraints are linear this is particularly true of large scale problems having dense covariance matrices and hence they are the focus in this paper because standard approaches for constructing an efficient frontier one point at a time tend to bog down on dense covariance matrix problems with many more than about 500 securities we propose as an alternative a procedure of parametric quadratic programming for more effective usage on large scale applications with the proposed procedure we demonstrate through computational results on problems in the 1000 3000 security range that the efficient frontiers of dense covariance matrix problems in this range are now not only solvable but can actually be computed in quite reasonable time

existence of chaos in control systems with delayed feedback here f r r denotes some nonlinear function a a positive constant t e r is the independent variable x e r the dependent variable and 1 dx dt the investigation is motivated i by the applications eq 1 1 has found in several areas of applied sciences and ii by the observation of intensely structured periodic and aperiodic oscillations in numerical simulations for a certain class of nonlinearities f we show that these structures are not numerical artefacts but properties of the solution manifold of eq 1 1 in the applications eq 1 1 has been used to explain normal and pathological behavior of control systems in the physiology of blood cell production and respiration 7 8 10 111 and periodic 2 or irregular s activity in neural networks computer simulations to eq 1 1 may be found in 3 the mathematical analysis of this equation so far has been restricted to prove the existence of slowly oscillating simple periodic solutions 1 4 61 a periodic solution is called slowly oscillating if successive

morse decompositions for delay differential equations abstract the attractor for the scalar delay differential equation x t f x t x t 1 is shown to possess a morse decomposition under the hypothesis of a negative feedback condition in the delay term using this decomposition we prove results on the asymptotic rate of oscillation of solutions of the initial value problem existence of both slowly and rapidly oscillating periodic solutions is also proved

on the list decodability of burst errors burst errors are a type of distortion in many data communications and data storage channels in this paper we consider the list decodability of codes for single burst error case and phased burst error case independently firstly we analyze the list decodability of random codes and we show that the burst list decoding radius and the rate of random codes achieve the singleton bound and the gilbert varshamov bound for single case and phased case respectively second we illustrate that cyclic codes and algebraic geometry codes are good burst list decodable codes for single case and phased case respectively

combinatorial limitations of average radius list decoding we study certain combinatorial aspects of list decoding motivated by the exponential gap between the known upper bound of o 1 and lower bound of p log 1 for the list size needed to list decode up to error fraction p with rate away from capacity i e 1 h p here p in 0 frac 1 2 and 0 our main result is the following

instabilities in threshold diffusion equations with delay the introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio temporal chaos in this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria periodic orbits and travelling waves importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback direct numerical simulations are used to confirm our bifurcation analysis and to probe some of the rich behaviour possible for mixed feedback

deletion codes in the high noise and high rate regimes the noise model of deletions poses significant challenges in coding theory with basic questions like the capacity of the binary deletion channel still being open in this paper we study the harder model of worst case deletions with a focus on constructing efficiently decodable codes for the two extreme regimes of high noise and high rate specifically we construct polynomial time decodable codes with the following tradeoffs for any varepsilon 0 1 codes that can correct a fraction 1 varepsilon of deletions with rate mathop mathrm poly nolimits varepsilon over an alphabet of size mathop mathrm poly nolimits 1 varepsilon 2 binary codes of rate 1 tilde o sqrt varepsilon that can correct a fraction varepsilon of deletions and 3 binary codes that can be list decoded from a fraction 1 2 varepsilon of deletions with rate mathop mathrm poly nolimits varepsilon this paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet the above mentioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors

optimal error correction for computationally bounded noise for adversarial but computationally bounded models of error we construct appealingly simple and efficient cryptographic encoding and unique decoding schemes whose error correction capability is much greater than classically possible in particular 1 for binary alphabets we construct positive rate coding schemes that are uniquely decodable under a 1 2 error rate for any constant 0 2 for large alphabets we construct coding schemes that are uniquely decodable under a 1 r error rate for any information rate r 0 our results for large alphabets are actually optimal since the computationally bounded but adversarial channel can simulate the behavior of the q ary symmetric channel where q denotes the size of the alphabet the capacity of which is known to be upper bounded by 1 r our results hold under minimal assumptions on the communication infrastructure namely 1 we allow the channel to be more powerful than the receiver and 2 we only assume that some information about the sender a public key is known in particular we do not require any shared secret key or joint local state between sender and receivers

linear algebraic list decoding for variants of reed solomon codes folded reed solomon rs codes are an explicit family of codes that achieve the optimal tradeoff between rate and list error correction capability specifically for any e 0 guruswami and rudra presented an no 1 e time algorithm to list decode appropriate folded rs codes of rate r from a fraction 1 r e of errors the algorithm is based on multivariate polynomial interpolation and root finding over extension fields it was noted by vadhan that interpolating a linear polynomial suffices for a statement of the above form here we give a simple linear algebra based analysis of this variant that eliminates the need for the computationally expensive root finding step over extension fields and indeed any mention of extension fields the entire list decoding algorithm is linear algebraic solving one linear system for the interpolation step and another linear system to find a small subspace of candidate solutions except for the step of pruning this subspace the algorithm can be implemented to run in quadratic time we also consider a closely related family of codes called order m derivative codes and defined over fields of large characteristic which consist of the evaluations of f as well as its first m 1 formal derivatives at n distinct field elements we show how our linear algebraic methods for folded rs codes can be used to show that derivative codes can also achieve the above optimal tradeoff the theoretical drawback of our analysis for folded rs codes and derivative codes is that both the decoding complexity and proven worst case list size bound are n 1 e by combining the above idea with a pseudorandom subset of all polynomials as messages we get a monte carlo construction achieving a list size bound of o 1 e2 which is quite close to the existential o 1 e bound however the decoding complexity remains n 1 e our work highlights that constructing an explicit subspace evasive subset that has small intersection with low dimensional subspaces an interesting problem in pseudorandomness in its own right could lead to explicit codes with better list decoding guarantees

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

better binary list decodable codes via multilevel concatenation a polynomial time construction of binary codes with the currently best known tradeoff between rate and error correction radius is given specifically linear codes over fixed alphabets are constructed that can be list decoded in polynomial time up to the so called blokh zyablov bound the work builds upon earlier work by the authors where codes list decodable up to the zyablov bound the standard product bound on distance of concatenated codes were constructed the new codes are constructed via a known generalization of code concatenation called multilevel code concatenation a probabilistic argument which is also derandomized via conditional expectations is used to show the existence of inner codes with a certain nested list decodability property that is appropriate for use in multilevel concatenated codes a ldquolevel by levelrdquo decoding algorithm which crucially uses the list recovery algorithm for the outer folded reed solomon codes enables list decoding up to the designed distance bound aka the blokh zyablov bound for multilevel concatenated codes

a reference direction approach to multiple objective quadratic linear programming in this paper we propose an interactive procedure for solving multiple criteria problems with one quadratic objective several linear objectives and a set of linear constraints the procedure is based on the use of reference directions and weighted sums reference directions for the linear functions and weighted sums for combining the quadratic function with the linear ones are used as parameters to implement the free search of nondominated solutions the idea leads to the parametric linear complementarity problem formulation an approach to deal with this type of problems is given as well the approach is illustrated with a numerical example

approximate list decoding of direct product codes and uniform hardness amplification given a message msg in 0 1 n its k wise direct product encoding is the sequence of k tuples msg i 1 dots msg i k over all possible k tuples of indices i 1 dots i k in 1 dots n k we give an efficient randomized algorithm for approximate local list decoding of direct product codes that is given oracle access to a word which agrees with a k wise direct product encoding of some message msg in 0 1 n in at least epsilon geqslant poly 1 k fraction of positions our algorithm outputs a list of poly 1 epsilon strings that contains at least one string msg which is equal to msg in all but at most k omega 1 fraction of positions the decoding is local in that our algorithm outputs a list of boolean circuits so that the j th bit of the i th output string can be computed by running the i th circuit on input j the running time of the algorithm is polynomial in log n and 1 epsilon in general when epsilon e k alpha for a sufficiently small constant alpha 0 we get a randomized approximate list decoding algorithm that runs in time quasi polynomial in 1 epsilon i e 1 epsilon poly log1 epsilon as an application of our decoding algorithm we get uniform hardness amplification for p np parallel the class of languages reducible to np through one round of parallel oracle queries if there is a language in p np parallel that cannot be decided by any bpp algorithm on more than 1 1 n omega 1 fraction of inputs then there is another language in p np parallel that cannot be decided by any bpp algorithm on more than 1 2 1 n omega 1 fraction of inputs

on the global attractivity controversy for a delay model of hematopoiesis abstract recently particular counterexamples were constructed to some theorems of a previous paper concerning the global attractivity of the positive equilibrium for the delay equation p t p m t 1 p n t p t the purpose of this note is to explore the underlying phenomenon from a more general point of view and to give an explanation of the situation a theorem is proved regarding attractivity properties of the equilibrium zero

limits to list decoding of random codes it has been known since zyablov and pinsker 1982 that a random q ary code of rate 1 hq e where 0 0 is small enough and hq is the g ary entropy function with high probability is a 1 e list decodable code that is every hamming ball of radius at most pn has at most 1 e codewords in it in this paper the converse result is proven in particular it is proven that for every 0 0 blinovsky 1986 2005 2008 guruswami and vadhan 2005 a lower bound is known for all constant 0 1 1 q independent of e though the lower bound is asymptotically weaker than our bound blinovsky 1986 2005 2008 these results however are not subsumed by ours as these other results hold for arbitrary codes of rate 1 hq e

chaotic behaviour in simple dynamical systems in this paper a description is given of the chaotic behaviour generated by a transversal homoclinic point of a plane map a proof of smale s theorem via the shadowing property of hyperbolic sets is provided the result is related to certain plane periodic systems of ode s like the periodically perturbed pendulum equation to this end the so called method of melnikov is derived

a lower bound on list size for list decoding a q ary error correcting code c 1 2 q n is said to be list decodable to radius with list size l if every hamming ball of radius contains at most l codewords of c we prove that in order for a q ary code to be list decodable up to radius 1 1 q 1 e n we must have l 1 e2 specifically we prove that there exists a constant cq 0 and a function fq such that for small enough e 0 if c is list decodable to radius 1 1 q 1 e n with list size cq e2 then c has at most fq e codewords independent of n this result is asymptotically tight treating q as a constant since such codes with an exponential in n number of codewords are known for list size l o 1 e2 a result similar to ours is implicit in blinovsky problems of information transmission 1986 for the binary q 2 case our proof is simpler and works for all alphabet sizes and provides more intuition for why the lower bound arises

high rate codes with sublinear time decoding locally decodable codes are error correcting codes that admit efficient decoding algorithms any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword the tradeoff between the rate of a code and the locality efficiency of its decoding algorithms has been well studied and it has widely been suspected that nontrivial locality must come at the price of low rate a particular setting of potential interest in practice is codes of constant rate for such codes decoding algorithms with locality o k were known only for codes of rate 1 where k is the length of the message furthermore for codes of rate 1 2 no nontrivial locality had been achieved in this article we construct a new family of locally decodable codes that have very efficient local decoding algorithms and at the same time have rate approaching 1 we show that for every 0 and 0 for infinitely many k there exists a code c which encodes messages of length k with rate 1 and is locally decodable from a constant fraction of errors using o k queries and time these codes which we call multiplicity codes are based on evaluating multivariate polynomials and their derivatives multiplicity codes extend traditional multivariate polynomial codes they inherit the local decodability of these codes and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance

reed solomon codes for correcting phased error bursts a code structure is introduced that represents a reed solomon rs code in two dimensional format based on this structure a novel approach to multiple error burst correction using rs codes is proposed for a model of phased error bursts where each burst can affect one of the columns in a two dimensional transmitted word it is shown that the bursts can be corrected using a known multisequence shift register synthesis algorithm it is further shown that the resulting codes posses nearly optimal burst correction capability under certain probability of decoding failure finally low complexity systematic encoding and syndrome computation algorithms for these codes are discussed the proposed scheme may also find use in decoding of different coding schemes based on rs codes such as product or concatenated codes

examples of transverse homoclinic orbits in delay equations on presente deux exemples specifiques d equations aux differences differentielles scalaires avec un retard de la forme x t x t f x t 1 0 pour lesquelles il y a une orbite periodique avec une orbite homocline transversale q asymptotique a

neimark sacker bifurcation for periodic delay differential equations in this paper we study the delay differential equation x t a t x t f t x t 1 where is a real parameter the functions a t f t are c 4 smooth and periodic in the variable t with period 1 varying the parameter eigenvalues of the monodromy operator the derivative of the time one map at the equilibrium 0 cross the unit circle and bifurcation of an invariant curve occurs to detect the critical parameter values we use floquet theory we give an explicit formula to compute the coefficient that determines the direction of the bifurcation we extend the center manifold projection method to our infinite dimensional banach space using spectral projection represented by a riesz dunford integral the neimark sacker bifurcation theorem implies the appearance of an invariant torus in the space c s 1 we apply our results to an equation used in neural network theory

periodic solutions for an n dimensional cyclic feedback system with delay abstract we study models for n cyclically coupled variables e g neuron activities with overall negative delayed feedback and without symmetry or monotonicity properties our aim is to extract the common parts of similar approaches that are known in dimensions one two and three so far to exhibit how these parts work for general dimension n and to show how this framework includes previous as well as new results we provide a fixed point theorem and a related theorem on periodic orbits for semiflows on banach spaces which then yield periodic solutions of cyclic delayed negative feedback systems for general n we also give criteria for the global asymptotic stability in the same systems which are derived by relating the systems to interval maps