permutation arrays under the chebyshev distance an n d permutation array pa is a subset of sn with the property that the distance under some metric between any two permutations in the array is at least d they became popular recently for communication over power lines motivated by an application to flash memories in this paper the metric used is the chebyshev metric a number of different constructions are given as well as bounds on the size of such pa

inapproximability results for the weight problems of subgroup permutation codes a subgroup permutation code is a set of permutations on n symbols with the property that its elements are closed under the operation of composition in this paper we give inapproximability results for the minimum and maximum weight problems of subgroup permutation codes under several well known metrics based on previous works we prove that under hamming lee cayley kendall s tau ulam s and lp distance metrics 1 there is no polynomial time 2log1 en approximation algorithm for the minimum weight problem for any constant e 0 unless np dtime 2polylog n quasi polynomial time and 2 there is no polynomial time r approximation algorithm for the minimum weight problem for any constant r 1 unless p np under l metric we prove that it is np hard to approximate the minimum weight problem within factor 2 e for any constant e 0 we also prove that for any constant e 0 it is np hard to approximate the maximum weight within p 3 2 e under lp distance metric and within 3 2 e under hamming lee cayley kendall s tau and ulam s distance metrics

representing periodic structures efficiently as substructures abstract periodic structures structures comprising a number of identical bays arranged sequentially and having identical connection properties between all pairs of adjacent bays have certain properties which can be used to advantage in the computation and understanding of their behaviour in various dynamical situations it is proposed in this paper that a free ended periodic structure can be represented very efficiently as a substructure in a larger edifice by making full use of the similarity between it and a rotationally periodic structure it is shown that this concept can provide a very significant acceleration of analysis in virtually all dynamic substructuring analyses regardless of which method of composite structure assembly method is used it also transpires that by using this approach the resonance frequencies and natural modes of the general periodic structure can be computed much more easily than previously

lp decodable multipermutation codes in this paper we introduce a new way of constructing and decoding multipermutation codes multipermutations are the permutations of a multiset that generally consist of duplicate entries we first introduce a class of binary matrices called multipermutation matrices each of which corresponds to a unique and distinct multipermutation by enforcing a set of linear constraints on these matrices we define a new class of codes that we term linear program lp decodable multipermutation codes in order to decode these codes using an lp thereby enabling soft decoding we characterize the convex hull of multipermutation matrices this characterization allows us to relax the coding constraints to a polytope and to derive two lp decoding problems these two problems are respectively formulated by relaxing the maximum likelihood decoding problem and the minimum chebyshev distance decoding problem because these codes are non linear we also study efficient encoding and decoding algorithms we first describe an algorithm that maps consecutive integers one by one to an ordered list of multipermutations based on this algorithm we develop an encoding algorithm for a code proposed by shieh and tsai a code that falls into our class of lp decodable multipermutation codes regarding decoding algorithms we propose an efficient distributed decoding algorithm based on the alternating direction method of multipliers finally we observe from the simulation results that the soft decoding techniques we introduce can significantly outperform hard decoding techniques that are based on quantized channel outputs

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

pole zero cancellation in structures repeated roots abstract the conditions for the creation of nodes of normal modes of vibration from the cancellation of poles and zeros are established when either the poles or the zeros or both appear as repeated eigenvalues the analysis is illustrated by numerical examples including the case of a pole zero cancellation at every co ordinate at the same frequency which is shown to occur whenever there are repeated poles if there are repeated poles and repeated zeros at the same frequency then the number of poles must be either one more one less or equal to the number of zeros

the decoupling of damped linear systems in configuration and state spaces abstract it has recently been reported that any viscously damped linear system can be decoupled in the configuration space by a real nonlinear time dependent transformation the purpose of this rapid communication is to provide a few clarifying remarks about the decoupling operation it is shown that for homogeneous systems the time dependent configuration space decoupling transformation is real linear and time invariant when cast in state space in addition the configuration space transformation generates a diagonalizing structure preserving transformation in non homogeneous systems both the configuration and associated state transformations are nonlinear and depend continuously on the excitation an example is given of a linear system that can be decoupled in configuration but not in state space

co ordinate transformations for second order systems part i general transformations abstract when the dynamics of any general second order system are cast in a state space format the initial choice of the state vector usually comprises one partition representing system displacements and another representing system velocities co ordinate transformations can be defined which result in more general definitions of the state vector this paper discusses the general case of co ordinate transformations of state space representations for second order systems it identifies one extremely important subset of such coordinate transformations namely the set of structure preserving transformations for second order systems and it highlights the importance of these it shows that one particular structure preserving transformation results in a new system characterized by real diagonal matrices and presents a forceful case that this structure preserving transformation should be considered to be the fundamental definition for the characteristic behaviour of general second order systems in preference to the eigenvalue eigenvector solutions conventionally accepted

the decoupling of damped linear systems in free or forced vibration abstract the purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non oscillatory free vibration or in forced vibration based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations the concept of damped modes of imaginary vibration is introduced by phase synchronization of these real and physically excitable modes a time varying transformation is constructed to decouple non oscillatory free vibration when time drifts caused by viscous damping and by external excitation are both accounted for a time varying decoupling transformation for forced vibration is derived the decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped this paper constitutes the second and final part of a solution to the classical decoupling problem together with an earlier paper a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system

kantorovich problems and conditional measures depending on a parameter abstract we study measurable dependence of measures on a parameter in the following two classical problems constructing conditional measures and the kantorovich optimal transportation for parametric families of measures and mappings we prove the existence of conditional measures measurably depending on the parameter a particular emphasis is made on the borel measurability which cannot be always achieved our second main result gives sufficient conditions for the borel measurability of optimal transports and transportation costs with respect to a parameter in the case where marginal measures and cost functions depend on a parameter as a corollary we obtain the borel measurability with respect to the parameter for disintegrations of optimal plans finally we show that the skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter

on the security of pseudorandomized information theoretically secure schemes in this paper we discuss a naive method of randomness reduction for cryptographic schemes which replaces the required perfect randomness with output distribution of a computationally secure pseudorandom generator prg we propose novel ideas and techniques for evaluating the indistinguishability between the random and pseudorandom cases even against an adversary with computationally unbounded attack algorithm hence the prg based randomness reduction can be effective even for information theoretically secure cryptographic schemes especially when the amount of information received by the adversary is small in comparison to a preceding result of dubrov and ishai stoc 2006 our result removes the requirement of generalized notion of nb prgs and is effective for more general kinds of protocols we give some numerical examples to show the effectiveness of our result in practical situations and we also propose a further idea for improving the effect of the prg based randomness reduction

decoding frequency permutation arrays under chebyshev distance a frequency permutation array fpa of length n m and distance d is a set of permutations on a multiset over m symbols where each symbol appears exactly times and the distance between any two elements in the array is at least d fpa generalizes the notion of permutation array in this paper under the chebyshev distance we first prove lower and upper bounds on the size of fpa then we give several constructions of fpas and some of them come with efficient encoding and decoding capabilities moreover we show one of our designs is locally decodable i e we can decode a message bit by reading at most 1 symbols which has an interesting application to private information retrieval

on the complexity of checking semantic equivalences between pushdown processes and finite state processes simulation preorder equivalence and bisimulation equivalence are the most commonly used equivalences in concurrency theory their standard definitions are often called strong simulation bisimulation while weak simulation bisimulation abstracts from internal t actions we study the computational complexity of checking these strong and weak semantic preorders equivalences between pushdown processes and finite state processes we present a complete picture of the computational complexity of these problems and also study fixed parameter tractability in two important input parameters x the size of the finite control of the pushdown process and y the size of the finite state process all simulation problems are generally exptime complete and only become polynomial if both parameters x and y are fixed weak bisimulation equivalence is pspace complete but becomes polynomial if and only if parameter x is fixed strong bisimulation equivalence is pspace complete but becomes polynomial if either parameter x or y is fixed

upper bounds on matching families in bbz_ pq n matching families are one of the major ingredients in the construction of locally decodable codes ldcs and the best known constructions of ldcs with a constant number of queries are based on matching families the determination of the largest size of any matching family in zmn where zm is the ring of integers modulo m is an interesting problem in this paper we show an upper bound of o pq 0 625n 0 125 for the size of any matching family in zpqn where p and q are two distinct primes our bound is valid when n is a constant p and p q 1 our result improves an upper bound of dvir and coworkers

the transformation of second order linear systems into independent equations the class of second order linear dynamical systems is considered a method and algorithm are presented to transform any system with n degrees of freedom into n independent second order equations the conversion utilizes a real invertible but nonlinear mapping and is applicable to practically every linear system two examples from earthquake engineering are provided to indicate the utility of this approach

an optimal method for the solution of the constrained eigenvalue response problem for large structures comprising rigidly connected substructures dynamic substructuring or modal synthesis is established as a powerful means by which the computational effort required to perform dynamic analyses of large structures comprising many degrees of freedom can be reduced for the case where the substructures are rigidly connected a method of synthesis is proposed which gives very substantial improvements in efficiency over existing methods both in terms of the number of floating point operations which must take place and the on board computer memory required the proposed method makes use of a stable co ordinate transformation which reduces the mass matrix for the constrained composite system to the identity matrix and makes the stiffness matrix banded the bandwidth is exactly equal to twice the total number of constraints plus 1 this is usually very small in comparison with the total number of degrees of freedom in the constrained composite system the proposed method is ideal for combining a number of substructures into a single new substructure in an efficient way without necessarily disposing of any degrees of freedom approximate formulae are presented which give the ratios of the number of operations required for established methods compared with the number needed for the proposed method these formulae determine when it is appropriate to use the method proposed here

eigensolutions of non proportionally damped systems based on continuous damping sensitivity abstract the viscous damping model has been widely used to represent dissipative forces in structures under mechanical vibrations in multiple degree of freedom systems such behavior is mathematically modeled by a damping matrix which in general presents non proportionality that is it does not become diagonal in the modal space of the undamped problem eigensolutions of non proportional systems are usually estimated assuming that the modal damping matrix is diagonally dominant neglecting the off diagonal terms or in the general case using the state space approach in this paper a new closed form expression for the complex eigenvalues of non proportionally damped system is proposed the approach is derived assuming small damping and involves not only the diagonal terms of the modal damping matrix but also the off diagonal terms which appear under higher order the validity of the proposed approach is illustrated through a numerical example

a synthetic approach to markov kernels conditional independence and theorems on sufficient statistics abstract we develop markov categories as a framework for synthetic probability and statistics following work of golubtsov as well as cho and jacobs this means that we treat the following concepts in purely abstract categorical terms conditioning and disintegration various versions of conditional independence and its standard properties conditional products almost surely sufficient statistics versions of theorems on sufficient statistics due to fisher neyman basu and bahadur besides the conceptual clarity offered by our categorical setup its main advantage is that it provides a uniform treatment of various types of probability theory including discrete probability theory measure theoretic probability with general measurable spaces gaussian probability stochastic processes of either of these kinds and many others

2 server pir with subpolynomial communication a 2 server private information retrieval pir scheme allows a user to retrieve the ith bit of an n bit database replicated among two noncommunicating servers while not revealing any information about i to either server in this work we construct a 2 server pir scheme with total communication cost no log log n log n this improves over current 2 server protocols which all require n1 3 communication our construction circumvents the n1 3 barrier of razborov and yekhanin 2007 which holds for the restricted model of bilinear group based schemes covering all previous 2 server schemes the improvement comes from reducing the number of servers in existing protocols based on matching vector codes from 3 or 4 servers to 2 this is achieved by viewing these protocols in an algebraic way using polynomial interpolation and extending them using partial derivatives

frequency dependent operations for 2nd order systems modal expansion and modal correlation abstract in this paper two frequency dependent operations for viscously damped systems are presented the first operation comprises a new frequency dependent modal expansion while the second operation generalises a recently proposed modal correlation approach for these systems such that particular frequencies are targeted the two suggested methods are connected by a common method for specifying frequency in a generalised sense and they are each illustrated using numerical examples

realization of mechanical systems from second order models congruent coordinate transformations are used to convert second order models to a form in which the mass damping and stiffness matrices can be interpreted as a passive mechanical system for those systems which can be constructed from interconnected mass stiffness and damping elements it is shown that the input output preserving transformations can be parametrized by an orthogonal matrix whose dimension corresponds to the number of internal masses those masses at which an input is not applied nor an output measured only a subset of these transformations results in mechanically realizable models for models with a small number of internal masses complete discrete mapping of the transformation space is possible permitting enumeration of all mechanically realizable models sharing the original model s input output behavior when the number of internal masses is large a nonlinear search of transformation space can be employed to identify mechanically realizable models applications include scale model

total decoupling of general quadratic pencils part i theory abstract the notion of quadratic pencils 2 m c k where m c and k are n n real matrices with or without some additional properties such as symmetry or positive definiteness plays critical roles in many important applications it has been long desirable yet with very limited success to reduce a complicated high degree of freedom system to some simpler low degree of freedom subsystems recently garvey friswell and prells proposed a promising approach by which under some mild assumptions a general quadratic pencils can be converted by real valued isospectral transformations into a totally decoupled system this approach if numerically feasible would reduce the original n degree of freedom second order system to n totally independent single degree of freedom second order subsystems such a claim would be a striking breakthrough in the common knowledge that generally no three matrices m c and k can be diagonalized simultaneously this paper intends to serve three purposes to clarify some of the ambiguities in the original proposition to simplify some of the computational details and most importantly to complete the theory of existence by matrix polynomial factorization tactics

eigenvalue and eigenvector derivatives of second order systems using structure preserving equivalences abstract a structure preserving equivalence in the sense intended here is a mapping between the stiffness damping and mass matrices describing some initial second order system and the corresponding three matrices of another second order system having identical spectrum most second order systems can be diagonalised through a mapping of this sort the mapping provides a new approach to the evaluation and the understanding of eigenvalue and eigenvector derivatives in place of pairs of eigenvalues we think of real scalar stiffness damping and mass quantities representing decoupled single degree of freedom systems in place of pairs of eigenvectors we think of individual columns of the matrices involved in the mapping this approach resolves the completely artificial phenomenon that the eigenvalue and eigenvector derivatives become undefined at instants when modification of say a damping parameter causes a pair of complex eigenvalues to turn into a pair of real eigenvalues or vice versa it also has the advantage of being applicable to cases where any one or more of the system matrices are singular

general isospectral flows for linear dynamic systems the matrix a a0 a1 2a2 kak lal with matrix coefficients a0 a1 a2 al cm n defines a linear dynamic system of dimension m n when m n and when det a 0 for some values of the eigenvalues of this system are well defined a one parameter trajectory of such a system a0 a1 a2 al is an isospectral flow if the eigenvalues and the dimensions of the associated eigenspaces are the same for all parameter values ir this paper presents the most general form for isospectral flows of linear dynamic systems of orders l 2 3 4 and the forms for isospectral flows for even higher order systems are evident from the patterns emerging based on the definition of a class of coordinate transformations called structure preserving transformations the concept of isospectrality and the associated flows is seen to extend to cases where m n

on the labeling problem of permutation group codes under the infinity metric we consider codes over permutations under the infinity norm given such a code we show that a simple relabeling operation which produces an isomorphic code may drastically change the minimal distance of the code thus we may choose a code structure for efficient encoding procedures and then optimize the code s minimal distance via relabeling to establish that the relabeling problem is hard and is of interest we formally define it and show that all codes may be relabeled to get a minimal distance at most 2 on the other hand the decision problem of whether a code may be relabeled to distance 2 or more is shown to be np complete and calculating the best achievable minimal distance after relabeling is proved to be hard to approximate up to a factor of 2 we then consider general bounds on the relabeling problem we specifically construct the optimal relabeling for transitive cyclic groups we conclude with the main result a general probabilistic bound which we then use to show both the agl p group and the dihedral group on p elements may be relabeled to a minimal distance of p o p in p

the complexity of counting quantifiers on equality languages an equality language is a relational structure with infinite domain whose relations are first order definable in equality we classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers in doing this we find ourselves in various worlds in which dichotomies or trichotomies subsist

uniform random sampling not recommended for large graph size estimation abstract the norm of data size estimation is to use uniform random samples whenever possible there have been tremendous efforts in obtaining uniform random samples using methods such as metropolis hasting random walk or importance sampling 2 this paper shows that on the contrary to the common practice uniform random sampling should be avoided when pps probability proportional to size sampling is available for large data to develop intuition of the sampling process we discuss the sampling and estimation problem in the context of graph the size is the number of nodes in the graph uniform random sampling corresponds to uniform random node rn sampling and pps sampling is approximated by random edge re sampling in this setting we show that for large graphs re sampling outperforms rn sampling with a ratio proportional to the normalized graph degree variance this result is particularly important in the era of big data when data are typically large and scale free 3 resulting in large degree variance we derive the result by giving the variances of rn and re estimators each step of the derivation is supported and demonstrated by simulation studies assuming power law distributions then we use 18 real world networks to verify the result furthermore we show that the performance of random walk rw sampling is data dependent and can be significantly worse than rn and re more specifically rw can estimate online social networks but not web graphs due to the difference of the graph conductance

constructions of rank modulation codes rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines modeling rank modulation involves construction of packings of the space of permutations equipped with the kendall tau distance as our main set of results we present several general constructions of codes in permutations that cover a broad range of code parameters in particular we show a number of ways in which conventional error correcting codes can be modified to correct errors in the kendall space our constructions are nonasymptotic and afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the hamming metric as an example from binary bose chaudhuri hocquenghem codes we obtain codes correcting t kendall errors in n memory cells that support the order of n log2n t messages for any constant t 1 2 we give many examples of rank modulation codes with specific parameters turning to asymptotic analysis we construct families of rank modulation codes that correct a number of errors that grows with n at varying rates from n to n2 one of our constructions gives rise to a family of rank modulation codes for which the tradeoff between the number of messages and the number of correctable kendall errors approaches the optimal scaling rate

limited magnitude error correcting gray codes for rank modulation we construct error correcting codes over permutations under the infinity metric which are also gray codes in the context of rank modulation i e are generated as simple circuits in the rotator graph these errors model limited magnitude or spike errors for which only single error detecting gray codes are currently known surprisingly the error correcting codes we construct achieve a better asymptotic rate than that of presently known constructions not having the gray property and exceed the gilbert varshamov bound additionally we present efficient ranking and unranking procedures as well as a decoding procedure that runs in linear time finally we also apply our methods to solve an outstanding issue with error detecting rank modulation gray codes also known in this context as snake in the box codes under a different metric the kendall tau metric in the group of permutations over an even number of elements s 2n where we provide asymptotically optimal codes

deflating quadratic matrix polynomials with structure preserving transformations given a pair of distinct e s l 1 l 2 of an nbyn quadratic matrix polynomial q l with nonsingular leading coefficient and their corresponding ev s we show how to transform q l into a quadratic of the form twobytwoa qd l 0 0 q l having the same e s as q l with qd l an n 1 times n 1 quadratic matrix py and q l a scalar quadratic py with roots l 1 and l 2 this block diagonalization cannot be achieved by a similarity transformation applied directly to q l unless the ev s corresponding to l 1 and l 2 are parallel we identify conditions under which we can construct a family of 2n times 2n elementary similarity transformations that a are rank two modifications of the identity matrix b act on linearizations of q l c preserve the block structure of a large class of block symmetric linearizations of q l thereby defining new quadratic matrix polynomials q 1 l that have the same e s as q l d yield quadratics q 1 l with the property that their ev s associated with l 1 and l 2 are parallel and hence can subsequently be deflated by a similarity applied directly to q 1 l this is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action

computing the ball size of frequency permutations under chebyshev distance abstract let s n be the set of all permutations over the multiset 1 1 ts m m where n m a frequency permutation array fpa of minimum distance d is a subset of s n in which every two elements have distance at least d fpas have many applications related to error correcting codes in coding theory the gilbert varshamov bound and the sphere packing bound are derived from the size of balls of certain radii we propose two efficient algorithms that compute the ball size of frequency permutations under chebyshev distance here it is equivalent to computing the permanent of a special type of matrix which generalizes the toepliz matrix in some sense both methods extend previous known results the first one runs in o 2 d d 2 376 log n time and o 2 d d 2 space the second one runs in o 2 d d d n time and o 2 d d space for small constants and d both are efficient in time and use constant storage space

spectral decomposition of real symmetric quadratic λ matrices and its applications spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions the canonical form often facilitates discussions which otherwise would be complicated and involved spectral decomposition is of fundamental importance in many applications the well known glr theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees but its development is based on complex numbers this paper modifies the glr theory for the special application to real symmetric quadratic matrix polynomials q a mx2 cx k m nonsingular subject to the specific restriction that all matrices in the representation be realvalued it is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which in turn can be constructed from the spectral data applications to a variety of challenging inverse problems are discussed

on the locality of codeword symbols consider a linear n k d q code c we say that the ith coordinate of c has locality r if the value at this coordinate can be recovered from accessing some other r coordinates of c data storage applications require codes with small redundancy low locality for information coordinates large distance and low locality for parity coordinates in this paper we carry out an in depth study of the relations between these parameters we establish a tight bound for the redundancy n k in terms of the message length the distance and the locality of information coordinates we refer to codes attaining the bound as optimal we prove some structure theorems about optimal codes which are particularly strong for small distances this gives a fairly complete picture of the tradeoffs between codewords length worst case distance and locality of information symbols we then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes using our structure theorem we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings we prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance

the decoupling of second order linear systems with a singular mass matrix abstract it was demonstrated in earlier work that a nondefective linear dynamical system with an invertible mass matrix in free or forced motion may be decoupled in the configuration space by a real and isospectral transformation we extend this work by developing a procedure for decoupling a linear dynamical system with a singular mass matrix in the configuration space transforming the original differential algebraic system into decoupled sets of real independent first and second order differential equations numerical examples are provided to illustrate the application of the decoupling procedure

co ordinate transformations for second order systems part ii elementary structure preserving transformations abstract it has been shown in a previous paper that there is a real valued transformation from the generaln degree of freedom second order system to a second order system characterized by diagonal matrices an immediate extension of this fact is that for any second order system there is a set of real valued transformations thestructure preserving transformations which transform this system to a different second order system having identical characteristic behaviour there are several possible reasons why it may be very useful to achieve a particular structure in the transformed system it is obvious that a diagonal structure is extremely useful and a method has been devised for determining the diagonalizing transformation from the solution of the usual complex eigenvalue eigenvector problem this paper begins by outlining the usefulness of some other structures then it defines a class of elementary structure preserving co ordinate transformations that transform from one n degree of freedom second order system to another the termelementary is applied because any one of these transformations is the minimum rank modification of the identity transformation the changes occurring in the system matrices as a result of the application of one such elementary transformation transpire to be very simple in form they are low rank and they can be computed very efficiently this paper provides the fundamental tools to enable the design of structure preserving co ordinate transformations which transform a second order system originally characterized by three general matrices in stages into a mathematically similar second order system characterized by three diagonal matrices the procedure by which the individual elementary transformations are obtained is still under development and it is not discussed in this paper however an illustration is given of a five degree of freedom self adjoint system being transformed into tridiagonal form

towards 3 query locally decodable codes of subexponential length a q query locally decodable code ldc encodes an n bit message x as an n bit codeword c x such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword c x even after some constant fraction of codeword bits has been corrupted we give new constructions of three query ldcs of vastly shorter length than that of previous constructions specifically given any mersenne prime p e 2t 1 we design three query ldcs of length n e exp o n1 t for every n based on the largest known mersenne prime this translates to a length of less than exp o n10 7 compared to exp o n1 2 in the previous constructions it has often been conjectured that there are infinitely many mersenne primes under this conjecture our constructions yield three query locally decodable codes of length n e exp no 1 log log n for infinitely many n we also obtain analogous improvements for private information retrieval pir schemes we give 3 server pir schemes with communication complexity of o n10 7 to access an n bit database compared to the previous best scheme with complexity o n1 5 25 assuming again that there are infinitely many mersenne primes we get 3 server pir schemes of communication complexity no 1 log logn for infinitely many n previous families of ldcs and pir schemes were based on the properties of low degree multivariate polynomials over finite fields our constructions are completely different and are obtained by constructing a large number of vectors in a small dimensional vector space whose inner products are restricted to lie in an algebraically nice set

bounds on the size of permutation codes with the kendall tau metric the rank modulation scheme has been proposed for efficient writing and storing data in nonvolatile memory storage error correction in the rank modulation scheme is done by considering permutation codes in this paper we consider codes in the set of all permutations on n elements s n using the kendall tau metric the main goal of this paper is to derive new bounds on the size of such codes for this purpose we also consider perfect codes diameter perfect codes and the size of optimal anticodes in the kendall tau metric structures which have their own considerable interest we prove that there are no perfect single error correcting codes in s n where n 4 is a prime or 4 leq n leq 10 we present lower bounds on the size of optimal anticodes with odd diameter as a consequence we obtain a new upper bound on the size of codes in s n with even minimum kendall tau distance we present larger single error correcting codes than the known ones in s 5 and s 7

the decoupling of defective linear dynamical systems in free motion abstract it was demonstrated in two earlier papers that there exists a real linear time varying transformation that decouples any non defective linear dynamical system in free vibration in the configuration space as an extension of this work the present paper represents the first systematic effort to decouple defective systems it is shown that the decoupling of defective systems is a rather delicate procedure that depends on the multiplicities of the system eigenvalues while any defective system can be decoupled with the eigenvalues kept invariant the geometric multiplicities of these eigenvalues may not be preserved several numerical examples are provided to illustrate the theoretical developments

duality for borel measurable cost functions we consider the monge kantorovich transport problem in an abstract measure theoretic setting our main result states that duality holds if c x x y 0 is an arbitrary borel measurable cost function on the product of polish spaces x y in the course of the proof we show how to relate a non optimal transport plan to the optimal transport costs via a subsidy function and how to identify the dual optimizer we also provide some examples showing the limitations of the duality relations

computing pure bayesian nash equilibria in games with finite actions and continuous types we extend the well known fictitious play fp algorithm to compute pure strategy bayesian nash equilibria in private value games of incomplete information with finite actions and continuous types g facts we prove that if the frequency distribution of actions fictitious play beliefs converges then there exists a pure strategy equilibrium strategy that is consistent with it we furthermore develop an algorithm to convert the converged distribution of actions into an equilibrium strategy for a wide class of games where utility functions are linear in type this algorithm can also be used to compute pure e nash equilibria when distributions are not fully converged we then apply our algorithm to find equilibria in an important and previously unsolved game simultaneous sealed bid second price auctions where various types of items e g substitutes or complements are sold finally we provide an analytical characterisation of equilibria in games with linear utilities specifically we show how equilibria can be found by solving a system of polynomial equations for a special case of simultaneous auctions we also solve the equations confirming the results obtained numerically

inverse problems for damped vibrating systems abstract linear damped vibrating systems are defined by three real definite matrices m 0 d 0 and k 0 the mass damping and stiffness matrices respectively it is assumed that all eigenvalues of the system are simple and nonreal so that the eigenvectors columns of a matrix x c c n n are also complex it is shown that when properly defined the eigenvectors have a special structure consistent with x c x r i i where x r r n n x r is nonsingular and is orthogonal by taking advantage of this structure solutions of the inverse problem are obtained i e given complete information on the eigenvalues and eigenvectors it is shown how m d and k can be found three points of view are developed and compared namely using spectral theory structure preserving similarities and factorisation theory

error correcting data structures we study data structures in the presence of adversarial noise we want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object even if the data structure has been corrupted by a constant fraction of errors we measure the efficiency of a data structure in terms of its length the number of bits in its representation and query answering time measured by the number of bit probes to the possibly corrupted representation the main issue is the trade off between these two this new model is the common generalization of static data structures and locally decodable error correcting codes ldcs we prove a number of upper and lower bounds on various natural error correcting data structure problems in particular we show that the optimal length of t probe error correcting data structures for the membership problem where we want to store subsets of size s from a universe of size n such that membership queries can be answered effic

kernel and fast algorithm for dense triplet inconsistency we study the parameterized complexity of inferring supertrees from sets of rooted triplets an important problem in phylogenetics for a set l of labels and a dense set t of triplets distinctly leaf labeled by 3 subsets of l we seek a tree distinctly leaf labeled by l and containing all but at most k triplets from t as homeomorphic subtree our results are the first polynomial kernel for this problem with o k 2 labels and a subexponential fixed parameter algorithm running in time o l 4 2 o k 1 3 l o g k

a note on regular conditional probabilities in doob s sense it is pointed out that an example due to j pachl shows the nonexistence of regular conditional probabilities in doob s sense when the underlying space is a perfect probability space this answers a question of sazonov

correcting limited magnitude errors in the rank modulation scheme we study error correcting codes for permutations under the infinity norm motivated by a novel storage scheme for flash memories called rank modulation in this scheme a set of n flash cells are combined to create a single virtual multilevel cell information is stored in the permutation induced by the cell charge levels spike errors which are characterized by a limited magnitude change in cell charge levels correspond to a low distance change under the infinity norm we define codes protecting against spike errors called limited magnitude rank modulation codes lmrm codes and present several constructions for these codes some resulting in optimal codes these codes admit simple recursive and sometimes direct encoding and decoding procedures we also provide lower and upper bounds on the maximal size of lmrm codes both in the general case and in the case where the codes form a subgroup of the symmetric group in the asymptotic analysis the codes we construct outperform the gilbert varshamov like bound estimate

divisors of mersenne numbers we add to the heuristic and empirical evidence for a conjecture of gillies about the distribution of the prime divisors of mersenne numbers we list some large prime divisors of mersenne numbers mp in the range 17000 p 105

simultaneous tridiagonalization of two symmetric matrices we show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations no assumptions are made on the non singularity or definiteness of the two matrices the reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via householder reflectors two algorithms are proposed one using non orthogonal rank one modifications of the identity matrix and the other more costly but more stable using a combination of householder reflectors and non orthogonal rank one modifications of the identity matrix with minimal condition numbers each of these tridiagonalization processes requires o n 3 arithmetic operations and respects the symmetry of the problem we illustrate and compare the two algorithms with some numerical experiments

improved lower bounds on the size of balls over permutations with the infinity metric we study the size or volume of balls in the metric space of permutations s n under the infinity metric we focus on the regime of balls with radius r rho cdot n 1 rho in 0 1 i e a radius that is a constant fraction of the maximum possible distance we provide new lower bounds on the size of such balls these new lower bounds reduce the asymptotic gap to the known upper bounds to at most 0 029 bits per symbol additionally they imply an improved ball packing bound for error correcting codes and an improved upper bound on the size of optimal covering codes

computing pure nash and strong equilibria in bottleneck congestion games bottleneck congestion games properly model the properties of many real world network routing applications they are known to possess strong equilibria a strengthening of nash equilibrium to resilience against coalitional deviations in this paper we study the computational complexity of pure nash and strong equilibria in these games we provide a generic centralized algorithm to compute strong equilibria which has polynomial running time for many interesting classes of games such as e g matroid or single commodity bottleneck congestion games in addition we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics using unilateral improvement dynamics in matroid games pure nash equilibria can be reached efficiently in contrast computing even a single coalitional improvement move in matroid and single commodity games is strongly np hard in addition we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics they continue to hold even for convergence to approximate equilibria

total decoupling of general quadratic pencils part ii structure preserving isospectral flows abstract quadratic pencils 2 m c k where m c and k are n n real matrices with or without some additional properties such as symmetry connectivity bandedness or positive definiteness arise in many important applications recently an existence theory has been established showing that almost all n degree of freedom second order systems can be reduced to n totally independent single degree of freedom second order subsystems by real valued isospectral transformations in contrast to the common knowledge that generally no three matrices can be diagonalized simultaneously by equivalence transformations these isospectral transformations endeavor to maintain a special linearization form called the lancaster structure and do break down m c and k into diagonal matrices simultaneously however these transformations depend on the matrices in a rather complicated way and hence are difficult to construct directly in this paper a second part of a continuing study a closed loop control system that preserves both the lancaster structure and the isospectrality is proposed as a means to achieve the diagonal reduction consequently these transformations are acquired

structure preserving isospectral transformation for total or partial decoupling of self adjoint quadratic pencils abstract quadratic pencils 2m c k where m c and k are n n real matrices arise in a broad range of important applications its spectral properties affect the vibration behavior of the underlying system which often consists of many elements coupled together through an intricate network of inter connectivities it is known that an n degree of freedom system with semi simple eigenvalues can be reduced to without tampering with the innate vibration properties n mutually independent single degree of freedom subsystems referred to as total decoupling this paper revisits the problem with the additional constraint that the masses should stay invariant throughout the reduction process rescaling the masses if necessary m is assumed to be the identity matrix isospectral flows are derived to either totally or partially decouple c and k to independent units of modules indeed the same framework can be tailored to handle any kinds of desired structure two new results are obtained first the global convergence is guaranteed by using the ojasiewicz gradient inequality second bounds on errors due to numerical integration and floating point arithmetic calculation are derived which can be used for assessing the quality of the transformation numerical experiments on four distinct scenarios are given to demonstrate the capabilities of the framework of handling the decoupling problem

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

fast mixed searching and related problems on graphs abstract in this paper we introduce the fast mixed search model which is a combination of the fast search model and the mixed search model given a graph g in which a fugitive hides on vertices or along edges the fast mixed search problem is to find the minimum number of searchers to capture the fugitive in the fast mixed search model we establish a relation between the fast mixed search problem and the induced path cover problem we also consider relations between the fast mixed search problem and other graph search problems we prove that fast mixed searching from leaves is np complete and it remains np complete for planar graphs with maximum degree 3 we also prove that fast mixed searching between leaves is np complete and it remains np complete for graphs with maximum degree 4 we present linear time algorithms for computing the fast mixed search number and optimal search strategies of some classes of graphs including trees cacti and proper interval graphs

kernels for below upper bound parameterizations of the hitting set and directed dominating set problems in the hitting set problem we are given a collection f of subsets of a ground set v and an integer p and asked whether v has a p element subset that intersects each set in f we consider two parameterizations of hitting set below tight upper bounds p m k and p n k in both cases k is the parameter we prove that the first parameterization is fixed parameter tractable but has no polynomial kernel unless conp np poly the second parameterization is w 1 complete but the introduction of an additional parameter the degeneracy of the hypergraph h v f makes the problem not only fixed parameter tractable but also one with a linear kernel here the degeneracy of h v f is the minimum integer d such that for each x v the hypergraph with vertex set v x and edge set containing all edges of f without vertices in x has a vertex of degree at most d in nonblocker directed nonblocker we are given an undirected graph a directed graph g on n vertices and an integer k and asked whether g has a set x of n k vertices such that for each vertex y x there is an edge arc from a vertex in x to y nonblocker can be viewed as a special case of directed nonblocker replace an undirected graph by a symmetric digraph dehne et al proc sofsem 2006 proved that nonblocker has a linear order kernel we obtain a linear order kernel for directed nonblocker

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

green s theorem and isolation in planar graphs we show a simple application of email protected s theorem from multivariable calculus to the isolation problem in planar graphs in particular we give a log space construction of a skew symmetric polynomially bounded edge weight function for directed planar graphs such that the weight of any simple cycle in the graph is non zero with respect to this weight function as a direct consequence of the above weight function we are able to isolate a directed path between two fixed vertices in a directed planar graph we also show that given a bipartite planar graph we can obtain an edge weight function using the above function in log space which isolates a perfect matching in the given graph earlier this was known to be true only for grid graphs which is a proper subclass of planar graphs we also look at the problem of obtaining a straight line embedding of a planar graph in log space although we do not quite achieve this goal we give a piecewise straight line embedding of the given planar graph in log space

modal correlation approaches for general second order systems matching mode pairs and an application to campbell diagrams abstract modal correlation is well developed for undamped and proportionally damped vibrating systems it is less well defined for generally damped linear systems this paper addresses the fundamental problem of comparing two general second order linear systems through modal information it considers precisely the problem of how to achieve matching of modes mode pairs there are several possible motivations for modal correlation of which the most important is probably the model updating application in that application one set of modes derives from a numerical model and the other from measured data this paper focuses mainly on a different application constructing campbell diagrams for rotating machines there are two significant differences here a the two sets of modes being compared at any one time are from the same numerical model but for different spin speeds and b there is generally a strong distinction between the left and right modes of the system without some modal correlation approach the campbell diagram is constructed simply as a set of points on the frequency speed graph with modal correlation the eigenvalue problem can be solved at far fewer speeds and the points can be joined meaningfully a dimensionless n n modal matching array is produced whose entries indicate which pairs of modes from the first system best correlate with any particular pair of modes from the second system the presented work is motivated mainly by the application of developing campbell diagrams for rotating machines by means which are more effective than simply plotting a large set of discrete points wider applications of this paper include model updating procedures where mode pairs must be matched initially to ensure convergence towards the exact system

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

two query pcp with subconstant error we show that the np complete language 3sat has a pcp verifier that makes two queries to a proof of almost linear size and achieves subconstant probability of error eeo 1 the verifier performs only projection tests meaning that the answer to the first query determines at most one accepting answer to the second query the number of bits representing a symbol in the proof depends only on the error e previously by the parallel repetition theorem there were pcp theorems with two query projection tests but only arbitrarily small constant error and polynomial size there were also pcp theorems with subconstant error and almost linear size but a constant number of queries that is larger than 2 as a corollary we obtain a host of new results in particular our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem a partial list includes the following 1 3sat cannot be efficiently approximated to within a factor of 7 8po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 7 8pe for any constant e 0 under polynomial reductions hastad 2 3lin cannot be efficiently approximated to within a factor of 1 2po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 1 2pe for any constant e 0 under polynomial reductions hastad 3 a pcp theorem with amortized query complexity 1 p o 1 and amortized free bit complexity o 1 previously the best known amortized query complexity and free bit complexity were 1pe and e respectively for any constant e 0 samorodnitsky and trevisan one of the new ideas that we use is a new technique for doing the composition step in the classical proof of the pcp theorem without increasing the number of queries to the proof we formalize this as a composition of new objects that we call locally decode reject codes ldrc the notion of ldrc was implicit in several previous works and we make it explicit in this work we believe that the formulation of ldrcs and their construction are of independent interest

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159