constructions and decoding of cyclic codes over b symbol read channels symbol pair read channels in which the outputs of the read process are pairs of consecutive symbols were recently studied by cassuto and blaum this new paradigm is motivated by the limitations of the reading process in high density data storage systems they studied error correction in this new paradigm specifically the relationship between the minimum hamming distance of an error correcting code and the minimum pair distance which is the minimum hamming distance between symbol pair vectors derived from codewords of the code it was proved that for a linear cyclic code with minimum hamming distance d h the corresponding minimum pair distance is at least d h 3 in this paper we show that for a given linear cyclic code with a minimum hamming distance d h the minimum pair distance is at least d h left lceil d h 2 right rceil we then describe a decoding algorithm based upon a bounded distance decoder for the cyclic code whose symbol pair error correcting capabilities reflect the larger minimum pair distance finally we consider the case where the read channel output is a larger number b geqslant 3 of consecutive symbols and we provide extensions of several concepts results and code constructions to this setting

a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

reconstruction techniques for classical inverse sturm liouville problems this paper gives constructive algorithms for the classical inverse sturm liouville problem it is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator two methods of solving this latter problem are then provided and numerical examples are presented

dependence of eigenvalues of sturm liouville problems with eigenparameter dependent boundary conditions abstract this paper is concerned with regular sturm liouville problems with eigenparameter dependent boundary conditions we obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem in particular eigenparameter dependent boundary condition matrix moreover the differential expression of the eigenvalues with respect to the data is given the dependence of the nth eigenvalue as a function of the parameters of the problem is also investigated by pr u fer transformation and the inequalities of eigenvalues

more results on the complexity of identifying problems in graphs we investigate the complexity of several problems linked with identification in graphs for instance given an integer r 1 and a graph g v e the existence of or search for optimal r identifying codes in g or optimal r identifying codes in g containing a subset of vertices x v we locate these problems in the complexity classes of the polynomial hierarchy

a proof of the local borg marchenko theorem in 2 5 a local version of a basic uniqueness theorem in inverse spectral theory is given this paper gives a simple proof of this theorem

connected identifying codes we consider the problem of generating a connected identifying code for an arbitrary graph after a brief motivation we show that the decision problem regarding the existence of such a code is np complete and we propose a novel polynomial time approximation connectid that transforms any identifying code into a connected version of at most twice the size thus leading to an asymptotically optimal approximation bound when the input identifying code to is robust to graph distortions we show that the size of the resulting connected code is related to the best error correcting code of a given minimum distance permitting the use of known coding bounds in addition we show that the size of the input and output codes converge for increasing robustness meaning that highly robust identifying codes are almost connected finally we evaluate the performance connectid of on various random graphs simulations for erdos renyi random graphs show that the connected codes generated are actually at most 25 larger than their unconnected counterparts while simulations with robust input identifying codes confirm that robustness often provides connectivity for free

efficient reconstruction of sequences we introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type these erroneous versions are considered as outputs of repeated transmissions over a channel either a combinatorial channel defined by the maximum number of permissible errors of a given type or a discrete memoryless channel we are interested in the smallest n such that n erroneous versions always suffice to reconstruct a sequence of length n either exactly or with a preset accuracy and or with a given probability we are also interested in simple reconstruction algorithms complete solutions for combinatorial channels with some types of errors of interest in coding theory namely substitutions transpositions deletions and insertions of symbols are given for these cases simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found in general for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius a certain sufficient condition for solution of this problem is presented for a discrete memoryless channel the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within hamming distance d with error probability spl epsiv is found when d n and spl epsiv tend to 0 as n spl rarr spl infin a similar result for the continuous channel with discrete time and additive gaussian noise is also obtained

integrality gaps of 2 o 1 for vertex cover sdps in the lovasz schrijver hierarchy linear and semidefinite programming are highly successful approaches for obtaining good approximations for np hard optimization problems for example breakthrough approximation algorithms for max cut and sparsest cut use semidefinite programming perhaps the most prominent np hard problem whose exact approximation factor is still unresolved is vertex cover probabilistically checkable proof pcp based techniques of dinur and safra ann of math 2 162 2005 pp 439 486 show that it is not possible to achieve a factor better than 1 36 on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm there is a widespread belief that semidefinite programming sdp techniques are the most promising methods available for improving upon this factor of 2 following a line of study initiated by arora et al theory comput 2 2006 pp 19 51 our aim is to show that a large family of linear programming lp and sdp based algorithms fail to produce an approximation for vertex cover better than 2 lovasz and schrijver siam j optim 1 1991 pp 166 190 introduced the systems ls and ls for systematically tightening lp and sdp relaxations respectively over many rounds these systems naturally capture large classes of lp and sdp relaxations indeed ls captures the celebrated sdp based algorithms for max cut and sparsest cut mentioned above we rule out polynomial time sdp based 2 omega 1 approximations for vertex cover using ls in particular for every epsilon 0 we prove an integrality gap of 2 epsilon for vertex cover sdps obtained by tightening the standard lp relaxation with omega sqrt log n log log n rounds of ls while tight integrality gaps were known for vertex cover in the weaker ls system g schoenebeck l trevisan and m tulsiani proceedings of the 39th annual acm symposium on theory of computing acm press new york 2007 pp 302 310 previous results did not rule out a 2 omega 1 approximation after even two rounds of ls

eigenvalue ratios for sturm liouville operators in this paper we prove various optimal bounds for eigenvalue ratios for the sturm liouville equation p x y q x y w x y and certain specializations our results primarily concern the regular case with dirichlet boundary conditions though various extensions and generalizations to other situations are possible our results here extend the result m 1 m2 obtained in a previous paper for the one dimensional schrodinger equation y q x y y on a finite interval with dirichlet boundary conditions and nonnegative potential q 0 in particular we obtain m 1 km2 k where the constants k k satisfy 0 k p x w x k for all x if q 0 lower bounds can also be obtained our methods involve a slight modification of the prufer variable techniques employed in the schrodinger case we also examine the consequences of our recent proof of the payne polya weinberger conjecture in the one dimensional sturm liouville setting finally we compare our general bounds to the detailed analyses of keller and of mahar and willner for the special case of the inhomogeneous stretched string

equivalence of inverse sturm liouville problems with boundary conditions rationally dependent on the eigenparameter abstract three inverse problems for a sturm liouville boundary value problem y qy y y 0 cos y 0 sin and y 1 f y 1 are considered for rational f it is shown that the weyl m function uniquely determines f and q and is in turn uniquely determined by either two spectra from different values of or by the prufer angle for this it is necessary to produce direct results of independent interest on asymptotics and oscillation

recovery of the m function from spectral data for generalized sturm liouville problems the sturm liouville problem y qy y y 0 cos y 0 sin y y 1 h g is studied where h and g are real polynomials generalized norming constants p n k associated with eigenvalues n are defined and formulae are given for recovering the m function from these constants this leads to a uniqueness theorem for the associated inverse problem

zeros of the wronskian and renormalized oscillation theory for general sturm liouville operators with separated boundary conditions we prove the following if e 1 2 r and if u 1 2 solve the differential equation hu j e ju j j 1 2 and respectively satisfy the boundary condition on the left right then the dimension of the spectral projection p e 1 e 2 h of h equals the number of zeros of the wronskian of u 1 and u 2

codes for information retrieval with small uncertainty in a recent paper by yaakobi and bruck the problem of information retrieval in associative memories has been considered in an associative memory each memory entry is associated to the neighboring entries when searching information a fixed number of input clues are given and the output set is formed by the entries associated to all the input clues the maximum size of an output set is called the uncertainty of the associative memory in this paper we study the problem of information retrieval in associative memories with small uncertainty in particular we concentrate on the cases where the memory entries and their associations form a binary hamming space or an infinite square grid particularly we focus on minimizing the number of input clues needed to retrieve information with small uncertainty and present good constructions some of which are optimal i e use the smallest possible number of clues

on t revealing codes in binary hamming spaces abstract in this paper we introduce t revealing codes in the binary hamming space f n let c f n be a code and denote by i t c x the set of elements of c which are within hamming distance t from a word x f n a code c is t revealing if the majority voting on the coordinates of the words in i t c x gives unambiguously x these codes have applications for instance to the list decoding problem of the levenshtein s channel model where the decoder provides a list based on several different outputs of the channel with the same input and to the information retrieval problem of the yaakobi bruck model of associative memories we give t revealing codes which improve some of the key parameters for these applications compared to earlier code constructions

on a new class of codes for identifying vertices in graphs we investigate a new class of codes for the optimal covering of vertices in an undirected graph g such that any vertex in g can be uniquely identified by examining the vertices that cover it we define a ball of radius t centered on a vertex spl upsi to be the set of vertices in g that are at distance at most t from spl upsi the vertex spl upsi is then said to cover itself and every other vertex in the ball with center spl upsi our formal problem statement is as follows given an undirected graph g and an integer t spl ges 1 find a minimal set c of vertices such that every vertex in g belongs to a unique set of balls of radius t centered at the vertices in c the set of vertices thus obtained constitutes a code for vertex identification we first develop topology independent bounds on the size of c we then develop methods for constructing c for several specific topologies such as binary cubes nonbinary cubes and trees we also describe the identification of sets of vertices using covering codes that uniquely identify single vertices we develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords finally we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems

information retrieval with varying number of input clues information retrieval in associative memories was studied in a recent paper by yaakobi and bruck 2012 associations between memory entries give us the t neighbourhood of an entry in their model an information unit is retrieved from the memory with the aid of input clues which are chosen from a reference set in this paper we consider the situation where the information unit is found unambiguously using the associated t neighbourhoods of the input clues a varying number of input clues are allowed but a limit m u on the maximum number of them is imposed of course we would like m u to be as small as possible we consider the problem over the binary hamming space mathbb f n and focus on the minimum of m u denoted by nu n t using linear reference sets we show that nu n 2 leq 5 for any n geq 9 we also give infinite families of reference sets which provide good bounds on nu n t for t 3 in addition efficient methods are given to obtain bounds on nu n t for any t from known reference sets we also discuss the applications of this model to the levenshtein s sequence reconstruction problem and the sensor network monitoring

on the uniqueness of inverse spectral problems associated with incomplete spectral data abstract the inverse spectral problem for a sturm liouville problem which consists of a sturm liouville equation defined on the interval 0 a and two separated boundary conditions at the endpoints 0 and a one of which is involved function f is considered in this paper when f is the type of herglotz functions and is known as a priori we prove that the potential on 0 a and the other boundary condition can be uniquely determined in terms of appropriate partial information on the spectrum and partial information on the set of normalizing constants

information retrieval with unambiguous output the main problem in information storage has previously been how large amounts of data can be stored however the technological development over the years has been able to give rather satisfactory answers to this problem recently the focus has shifted towards determining how stored information can be efficiently retrieved this problem is addressed in an article by e yaakobi and j bruck 2012 where information retrieval in associative memories is studied in this paper we focus on the case where the retrieved information unit is unambiguous in particular we present characterizations and study various extremal properties of such associative memories moreover the algorithmic complexity of certain naturally rising problems is considered

inverse spectral problem for non selfadjoint dirac operator with boundary and jump conditions dependent on the spectral parameter the inverse spectral problem for non selfadjoint dirac operator with boundary and jump conditions dependent on the spectral parameter is considered by introducing the generalized norming constants the uniqueness theorems are proved and the constructive procedures for solving the inverse problem are given