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

generating binary trees using rotations a new algorithm that for the first time exploits the rotational geometry of binary trees is developed in order to allow for the lexicographic generation of computer representations of these trees in average time o 1 per tree rotation codewords for these trees in average time o 1 per tree are also generated it is shown how these codewords relate to lattice paths and using this relationship that n n 1 n 2 is the average number of rotations needed to generate a binary tree on n nodes finally a necessary and sufficient condition that a codeword represent a full binary tree each node has 0 or 2 sons on n 2 m 1 nodes is given and how to contract this codeword to obtain the codeword for the binary tree on m nodes for which this full tree is the extended binary tree is shown

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

snake in the box codes for rank modulation motivated by the rank modulation scheme with applications to flash memory we consider gray codes capable of detecting a single error also known as snake in the box codes we study two error metrics kendall s metric which applies to charge constrained errors and the l metric which is useful in the case of limited magnitude errors in both cases we construct snake in the box codes with rate asymptotically tending to 1 we also provide efficient successor calculation functions as well as ranking and unranking functions finally we also study bounds on the parameters of such codes

generalized gray codes for local rank modulation we consider the local rank modulation scheme in which a sliding window going over a sequence of real valued variables induces a sequence of permutations local rank modulation is a generalization of the rank modulation scheme which has been recently suggested as a way of storing information in flash memory we study gray codes for the local rank modulation scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation unlike the limited scope of previous works we consider code constructions for the entire range of parameters including the code length sliding window size and overlap between adjacent windows we show that the presented codes have asymptotically optimal rate we also provide efficient encoding decoding and next state algorithms

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

generation of spatial orders and space filling curves space filling curves have been found useful for many applications in diverse fields a space filling curve is a path in a 2 r 2 r raster domain which visits each location exactly once in mathematical terms space filling curves linearize a 2d integer space bijectively mapping the space to the integer line an algorithm is presented which generates a large number of space filling curves spatial orders functions are derived such that the code of each location can be calculated from its coordinates and conversely a location code can be decoded to yield the coordinates the algorithm first generates generate 4 4 spatial orders they subsequently may be scaled up to any desired domain of size 2 r 2 r the underlying theory of the algorithm the processes for scaling up encoding and decoding are described in detail the curves are generated as a set of incongruent curves followed if required by the sets of associated congruent curves a number of space filling curves are illustrated

systematic error correcting codes for permutations and multi permutations multi permutations and in particular permutations appear in various applications in an information theory new applications such as rank modulation for flash memories have suggested the need to consider error correcting codes for multi permutations in this paper we study systematic error correcting codes for multi permutations in general and for permutations in particular for a given number of information symbols k and for any integer t we present a construction of k r k systematic t error correcting codes for permutations of length k r where the number of redundancy symbols r is relatively small in particular for a given t and for sufficiently large k we obtain r t 1 while a lower bound on the number of redundancy symbols is shown to be t the same construction is also applied to obtain related systematic error correcting codes for any types of multi permutations

support vector machines and the multiple hypothesis test problem two enhancements are proposed to the application and theory of support vector machines the first is a method of multicategory classification based on the binary classification version of the support vector machine svm the method which is called the m ary svm represents each category in binary format and to each bit of that representation is assigned a conventional svm this approach requires only log sub 2 k svms where k is the number of classes we give an example of classification on an octaphase shift keying 8 psk pattern space to illustrate the main concepts the second enhancement is that of adding equality constraints to the conventional binary classification svm this allows pinning the classification boundary to points that are known a priori to lie on the boundary applications of this method often arise in problems having some type of symmetry we present one such example where the m ary svm is used to classify symbols of a cdma two user multiuser detection pattern space

a laplace transorm potential theoretic method for acoustic propagation in subsonic flows this paper introduces a competitive computational approach for determining time dependent far field sound generated by subsonic flows around lifting airfoils the procedure assumes the linearity of the sound field away from a bounded region surrounding the airfoil it is assumed that the sound pressure on the boundary of this enclosed region referred to as the kirchhoff surface is specified possibly by another procedure such as solving the full euler equations away from the kirchhoff surface the euler equations are linearized about a uniform mean flow it is well known that linearized euler equations can be uncoupled into a scalar convective wave equation however due to the anisotropy present in the convective wave equation it is difficult to compute solutions in this context direct numerical simulation of the convective wave equation requires proper numerical descriptions of far field boundary conditions which is a non trivial task moreover if accurate far field conditions can be formulated the computational cost of direct simulation can be prohibitive even in a modest computational domain in this paper we present an alternate solution procedure first the problem is transformed via the laplace transform with appropriate initial conditions into a reduced wave equation the convective term in the reduced wave equation is removed using a dependent variable transformation then we use gothert s rule to obtain a helmholtz like equation with complex wave number which is subsequently solved using double layer potential theory finally upon application of numerical inverse laplace transform techniques far field acoustic pressure is obtained as a function of space and time

integral equation procedures for eddy current problems abstract two methods are presented for the solution of a class of two dimensional eddy current problems the specific problem considered is that of a wire carrying a periodic current parallel to a uniform conducting cylinder both methods yield integral equations over the boundary of the conductor the first method is exact while the second is an asymptotic procedure valid for a certain parameter range and reflecting the skin effect numerical implementation procedures are given for both methods and some numerical results are presented

infinity norm permutation covering codes from cyclic groups we study covering codes of permutations with the ell infty metric we provide a general code construction which combines short building block codes into a single long code we focus on cyclic transitive groups as building blocks determining their exact covering radius and showing a linear time algorithm for finding a covering codeword when used in the general construction we show that the resulting covering code asymptotically out performs the best known code while maintaining linear time decoding we also bound the covering radius of relabeled cyclic transitive groups under conjugation showing that the covering radius is quite robust while relabeling cannot reduce the covering radius by much the downside is that we prove the covering radius cannot be increased by more than 1 when using relabeling

systematic error correcting codes for rank modulation the rank modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories in this paper we explore n k d systematic error correcting codes for rank modulation such codes have length n k information symbols and minimum distance d systematic codes have the benefits of enabling efficient information retrieval in conjunction with memory scrubbing schemes we study systematic codes for rank modulation under kendall s t metric as well as under the l metric in kendall s t metric we present k 2 k 3 systematic codes for correcting a single error which have optimal rates unless systematic perfect codes exist we also study the design of multierror correcting codes and provide a construction of k t 1 k 2t 1 systematic codes for large enough k we use nonconstructive arguments to show that for rank modulation systematic codes achieve the same capacity as general error correcting codes finally in the l metric we construct two n k d systematic multierror correcting codes the first for the case of d 0 1 and the second for d n in the latter case the codes have the same asymptotic rate as the best codes currently known in this metric

a loopless algorithm for generating the permutations of a multiset many combinatorial structures can be constructed from simpler components for example a permutation can be constructed from cycles or a motzkin word from a dyck word and a combination in this paper we present a constructor for combinatorial structures called shuffle on trajectories defined previously in a non combinatorial context and we show how this constructor enables us to obtain a new loopless generating algorithm for multiset permutations from similar results for simpler objects

a survey of full length nonlinear shift register cycle algorithms shift registers have been used to generate sequences of 0 s and 1 s for over thirty years a wide variety of applications has been made of these sequences principally communications have made use of the sequences generated one particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences these sequences are periodic and of length 2 n and all 2 n different binary n tuples appear exactly one time in a periodic portion of the sequence in this paper we discuss various algorithms which have been suggested for generating these sequences

long paths and cycles in hypercubes with faulty vertices a fault free path in the n dimensional hypercube q n with f faulty vertices is said to be long if it has length at least 2 n 2f 2 similarly a fault free cycle in q n is long if it has length at least 2 n 2f if all faulty vertices are from the same bipartite class of q n such length is the best possible we show that for every set of at most 2n 4 faulty vertices in q n and every two fault free vertices u and v satisfying a simple necessary condition on neighbors of u and v there exists a long fault free path between u and v this number of faulty vertices is tight and improves the previously known results furthermore we show for every set of at most n 2 10 n 2 1 faulty vertices in q n where n 15 that q n has a long fault free cycle this is a first quadratic bound which is known to be asymptotically optimal

maximum matchings and minimum dominating sets in apollonian networks and extended tower of hanoi graphs abstract the apollonian networks display the remarkable power law and small world properties as observed in most realistic networked systems their dual graphs are extended tower of hanoi graphs which are obtained from the tower of hanoi graphs by adding a special vertex linked to all its three extreme vertices in this paper we study analytically maximum matchings and minimum dominating sets in apollonian networks and their dual graphs both of which have found vast applications in various fields e g structural controllability of complex networks for both networks we determine their matching number domination number the number of maximum matchings as well as the number of minimum dominating sets

local rank modulation for flash memories local rank modulation lrm scheme was suggested for representing the information in flash memories in order to overcome the drawbacks of rank modulation for 0 with s dividing n an s t n lrm scheme is a lrm scheme where the n cells are locally viewed cyclically through a sliding window of size t resulting in a sequence of small permutations which requires less comparisons and less distinct values the gap between two such windows equals to s in this paper encoding decoding and asymptotic enumeration of the 1 t n lrm scheme are studied

snake in the box codes for rank modulation under kendall s tau metric for a gray code in the scheme of rank modulation for flash memories the codewords are permutations and two consecutive codewords are obtained using a push to the top operation we consider the snake in the box code under kendall s tau metric which is a gray code capable of detecting one kendall s tau error we answer two open problems posed by horovitz and etzion first we prove the validity of a construction given by them resulting in a snake of size m 2n 1 2n 1 2 2n 1 second we come up with a different construction aiming at a larger snake of size m 2n 1 2n 1 2 2n 3 the construction is applied successfully to s 7

a fractal set from the binary reflected gray code the permutation associated with the decimal expression of the binary reflected gray code with n bits is considered its cycle structure is studied considered as a set of points its self similarity is pointed out as a fractal it is shown to be the attractor of an ifs for large values of n the set is examined from the point of view of time series analysis

codes in permutations and error correction for rank modulation codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors we study basic coding theoretic problems for such codes representing them as subsets of the set of permutations of n elements equipped with the kendall tau distance we derive several lower and upper bounds on the size of codes these bounds enable us to establish the exact scaling of the size of optimal codes for large values of n we also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors

bounds for permutation rate distortion we study the rate distortion relationship in the set of permutations endowed with the kendall tau metric and the chebyshev metric the ell infty metric this paper is motivated by the application of permutation rate distortion to the average case and worst case distortion analysis of algorithms for ranking with incomplete information and approximate sorting algorithms for the kendall tau metric we provide bounds for various distortion regimes while for the chebyshev metric we present bounds that are valid for all distortions and are especially accurate for small distortions in addition for the chebyshev metric we provide a construction for covering codes

finite element analysis of a scattering problem a finite element method for the solution of a scattering problem for the reduced wave equation is formulated and analyzed the method involves a reformulation of the problem on a bounded domain with a nonlocal boundary condition the space of trial functions includes piecewise polynomial functions and functions arising from spherical harmonics

solution procedures for three dimensional eddy current problems abstract the problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles a common approximation here is that in which the metal is assumed to have infinite conductivity the resulting problem called the perfect conductor problem involves solving maxwell s equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface in the interface problem different sets of maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface solution procedures for this problem are given there is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity both are based on a new integral equation procedure for the perfect conductor problem the asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors

algorithms for the generation of full length shift register sequences two algorithms are presented for the generation of full length shift register cycles also referred to as de bruijn sequences the first algorithm generates 2 k cdot g n k full cycles of length 2 n using 3n k cdot g n k bits of storage where k is a free parameter in the range 1 leq k leq 2 n 4 2 and g n k is of the order of n 2 log k the second algorithm generates about 2 n 2 4 full cycles of length 2 n using about n 2 2 bits of storage in both algorithms the time required to produce the next bit from the last n bits is close to n a possible application to the construction of stream ciphers is indicated

snake in the box codes for rank modulation under kendall s tau metric in s_ 2n 2 snake in the box codes under kendall s tau metric are studied in the rank modulation scheme for flash memories where codewords are a subset of permutations in s n with minimal kendall s tau distance two and two cyclically consecutive codewords are connected via a push to the top operation studies so far restrict the push to the top operations only on odd indices resulting in a snake consisting of permutations with the same parity and thus the minimal distance constraint is easily satisfied asymptotically optimal snake codes have been constructed this way in s 2n 1 as for s 2n 2 this framework keeps the last element fixed and thus a snake in s 2n 2 is equivalent to a snake in s 2n 1 which is rather trivial if one wants to do better then it is inevitable to have some push to the top operations on even indices resulting in a combination of odd and even permutations in the snake which increases the difficulty to guarantee the minimal kendall s tau distance constraint thus horovitz and etzion pose the open problem to prove or disprove that the size of the largest snake in s 2n 2 is not larger than the size of the largest snake in s 2n 1 a first step toward this problem is a negative answer by wang and fu who construct a snake in s 2n 2 with exactly one more permutation than an optimal snake in s 2n 1 in this paper we give an explicit construction of a snake in s 2n 2 with size asymptotically approaching 1 4 s 2n 2

gray code permutation algorithm for high dimensional data encryption abstract we present a novel permutation algorithm for fast encryption of a large amount of data such as 3d images and real time videos the proposed p box algorithm takes advantage of gray code properties and allows fast encryption with high information diffusion the algorithm is optimized for integer q bit operations q 8 16 32 allowing a direct implementation in almost any hardware platform while avoiding rounding errors of floating point operations by combining the p box with chaotic s box based on the logistic map we design a complete highly secure and fast cryptosystem

a boundary element method for a two dimensional interface problem in electromagnetics this paper presents galerkin approximations for solutions of two dimensional interface problems by solving corresponding boundary integral equations these are obtained by simple layer potential operators only due to the strong ellipticity of the integral equations the galerkin procedure converges with optimal order smoothness of the given data implies high convergence rates for the layers

trimming and gluing gray codes abstract we consider the algorithmic problem of generating each subset of n 1 2 n whose size is in some interval k l 0 k l n exactly once cyclically by repeatedly adding or removing a single element or by exchanging a single element for k 0 and l n this is the classical problem of generating all 2 n subsets of n by element additions removals and for k l this is the classical problem of generating all n k subsets of n by element exchanges we prove the existence of such cyclic minimum change enumerations for a large range of values n k and l improving upon and generalizing several previous results for all these existential results we provide optimal algorithms to compute the corresponding gray codes in constant o 1 time per generated set and o n space rephrased in terms of graph theory our results establish the existence of almost hamilton cycles in the subgraph of the n dimensional cube q n induced by all levels k l we reduce all remaining open cases to a generalized version of the middle levels conjecture which asserts that the subgraph of q 2 k 1 induced by all levels k c k 1 c c 0 1 k has a hamilton cycle we also prove an approximate version of this generalized conjecture showing that this graph has a cycle that visits a 1 o 1 fraction of all vertices

compression in the space of permutations we investigate lossy compression source coding of data in the form of permutations this problem has direct applications in the storage of ordinal data or rankings and in the analysis of sorting algorithms we analyze the rate distortion characteristic for the permutation space under the uniform distribution and the minimum achievable rate of compression that allows a bounded distortion after recovery our analysis is with respect to different practical and useful distortion measures including kendall tau distance spearman s footrule chebyshev distance and inversion ell 1 distance we establish equivalence of source code designs under certain distortions and show simple explicit code designs that incur low encoding decoding complexities and are asymptotically optimal finally we show that for the mallows model a popular nonuniform ranking model on the permutation space both the entropy and the maximum distortion at zero rate are much lower than the uniform counterparts which motivates the future design of efficient compression schemes for this model

simulating size constrained galton watson trees we discuss various methods for generating random galton watson trees conditional on their sizes being equal to n a linear expected time algorithm is proposed

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

correcting charge constrained errors in the rank modulation scheme we investigate error correcting codes for a the rank modulation scheme with an application to flash memory devices in this scheme a set of n cells stores information in the permutation induced by the different charge levels of the individual cells the resulting scheme eliminates the need for discrete cell levels overcomes overshoot errors when programming cells a serious problem that reduces the writing speed and mitigates the problem of asymmetric errors in this paper we study the properties of error correcting codes for charge constrained errors in the rank modulation scheme in this error model the number of errors corresponds to the minimal number of adjacent transpositions required to change a given stored permutation to another erroneous one a distance measure known as kendall s distance we show bounds on the size of such codes and use metric embedding techniques to give constructions which translate a wealth of knowledge of codes in the lee metric to codes over permutations in kendall s metric specifically the one error correcting codes we construct are at least half the ball packing upper bound

base change and theta correspondences for supercuspidal representations of sl 2 abstract let f be a p adic field with p odd quadratic base change and theta lifting are shown to be compatible for supercuspidal representations of sl 2 f the argument involves the theory of types and the lattice model of the weil representation

counting techniques to label constant weight gray codes with links to minimal generating sets of semigroups let be a partition of the positive integer n a partition of the set xn 1 2 n is said to be of type if the sizes of its classes form the partition of n given 1 r n an r element subset a of xn and a partition of xn are said to be orthogonal if every class of meets a in exactly one element let gn r be the graph whose vertices are the r element subsets of xn with two sets being adjacent if they intersect in r 1 elements the graph gn r is hamiltonian hamiltonian cycles of gn r are early examples of error correcting codes where they came to be known as constant weight gray codes a hamiltonian cycle a1 a2 anr in gn r is said to be orthogonally labeled if there exists a list of distinct partitions 1 2 nr of type such that i is orthogonal to both ai and ai 1 where i 1 2 nr taken modulo nr for all but a finite class of partition types we present counting arguments to prove that any hamiltonian cycle in gn r can be orthogonally labeled the remaining cases with the exception of cases which are equivalent to the celebrated middle levels conjecture are treated via constructive arguments in a sequel a semigroup of transformations of xn is sn normal if it is closed under conjugation by the permutations of xn the combinatorics results here and in the sequel lead to a determination of the rank and idempotent rank of all sn normal semigroups thereby broadly generalizing the known result that the rank and idempotent rank of k n r is s n r a stirling number of the second kind

perfect snake in the box codes for rank modulation for odd n the alternating group on n elements is generated by the permutations that jump an element from an odd position to position 1 we prove hamiltonicity of the associated directed cayley graph for all odd n neq 5 a result of rankin implies that the graph is not hamiltonian for n 5 this solves a problem arising in rank modulation schemes for flash memory our result disproves a conjecture of horovitz and etzion and proves another conjecture of yehezkeally and schwartz

generalized gray codes with prescribed ends abstract an n bit gray code is a sequence of all n bit vectors such that consecutive vectors differ in a single bit it is well known that given 0 1 n an n bit gray code between and exists iff the hamming distance d of and is odd we generalize this classical result to k pairwise disjoint pairs i i 0 1 n if d i i is odd for all i and k n then the set of all n bit vectors can be partitioned into k sequences such that the i th sequence leads from i to i and consecutive vectors differ in a single bit this holds for every n 1 with one exception in the case when n k 1 4 our result is optimal in the sense that for every n 2 there are n pairwise disjoint pairs i i 0 1 n with d i i odd for which such sequences do not exist

a heuristic approach for constructing symmetric gray codes interest in the symmetric gray sg code is high because of its importance in the design and testing of many types of electrical circuits and information systems the conventional methods for constructing such a code are complicated a heuristic approach is described in this paper examples of constructing the sg code of order 5 6 and 7 are included

constant weight gray codes for local rank modulation we consider the local rank modulation lrm scheme in which a sliding window going over a sequence of real valued variables induces a sequence of permutations lrm is a generalization of the rank modulation scheme which has been recently suggested as a way of storing information in flash memory we study constant weight gray codes for the lrm scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation we present a practical construction of codes with asymptotically optimal rate and weight asymptotically half the length thus having an asymptotically optimal charge difference between adjacent cells next we turn to examine the existence of optimal codes by specifically studying codes of weight 2 and 3 in the former case we upper bound the code efficiency proving that there are no such asymptotically optimal cyclic codes in contrast for the latter case we construct codes which are asymptotically optimal we conclude by providing necessary conditions for the existence of cyclic and cyclic optimal gray codes

a survey of combinatorial gray codes the term combinatorial gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified small way this notion generalizes the classical binary reflected gray code scheme for listing n bit binary numbers so that successive numbers differ in exactly one bit position as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families including permutations and combinations the area of combinatorial gray codes was popularized by herbert wilf in his invited address at the siam conference on discrete mathematics in 1988 and his subsequent siam monograph combinatorial algorithms an update 1989 in which he posed some open problems and variations on the theme this resulted in much recent activity in the area and most of the problems posed by wilf are now solved in this paper we survey the area of combinatorial gray codes describe recent results variations and trends and highlight some open problems

numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via lagrange multipliers this paper deals with the finite element solution of the eddy current problem in a bounded conducting domain crossed by an electric current and subject to boundary conditions appropriate from a physical point of view two different cases are considered depending on the boundary data input current density flux or input current intensities the analysis of the former is an intermediate step for the latter which is more realistic in actual applications weak formulations in terms of the magnetic field are studied the boundary conditions being imposed by means of appropriate lagrange multipliers the resulting mixed formulations are analyzed depending on the regularity of the boundary data finite element methods are introduced in each case and error estimates are proved finally some numerical results to assess the effectiveness of the methods are reported

mixed radix gray codes in lee metric gray codes where two consecutive codewords differ in exactly one position by plusmn1 are given in a single radix code all dimensions have the same base say kappa whereas in a mixed radix code the base in one dimension can be different from the base in another dimension constructions of new classes of mixed radix gray codes are presented it is shown how these codes can be used as a basis for constructing edge disjoint hamiltonian cycles in mixed radix toroidal networks when the number of dimensions n 2r for some r ges 0 efficient algorithms for the generation of these codes are then shown

proof of the middle levels conjecture define the middle layer graph as the graph whose vertex set consists of all bitstrings of length 2n 1 that have exactly n or n 1 entries equal to 1 with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit the middle levels conjecture asserts that this graph has a hamilton cycle for every n geq 1 this conjecture originated probably with havel buck and wiedemann but has also been attributed to dejter erd h o s trotter and various others and despite considerable efforts it resisted all attacks during the last 30 years in this paper we prove the middle levels conjecture in fact we construct 2 2 omega n different hamilton cycles in the middle layer graph which is best possible

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

counting sequences a gray sequence is a counter sequence where the hamming distance between successive states is one a construction for any fixed odd distance between successive states is given based on error correcting codes such sequences could be used in testing and fault diagnosis when sequencing through all possible input combinations sequences with the maximum possible difference between successive states alternately n and n 1 are given a characterization of such maximal change sequences is given the most nearly uniform even difference sequences are described the distribution of bit changes or transition counts for individual variables for a counting sequence is considered

supercuspidal representations and the theta correspondence ii sl 2 and the anisotropic o 3 a parametrization is given of the local theta correspondence attached to the reductive dual pair sl 2 f o f where f is a nonarchimedean local field of odd residual characteristic and 0 is the orthogonal group of a ternary quadratic form which is anisotropic over f the parametrization is in terms of inducing data various lattice models of the oscillator representation are used

reordering columns for smaller indexes column oriented indexes such as projection or bitmap indexes are compressed by run length encoding to reduce storage and increase speed sorting the tables improves compression on realistic data sets permuting the columns in the right order before sorting can reduce the number of runs by a factor of two or more unfortunately determining the best column order is np hard for many cases we prove that the number of runs in table columns is minimized if we sort columns by increasing cardinality experimentally sorting based on hilbert space filling curves is poor at minimizing the number of runs

on the n t antipodal gray codes an n bit gray code is a circular listing of all 2 nn bit binary strings in which consecutive strings differ at exactly one bit for n t 2 n 1 an n t antipodal gray code is a gray code in which the complement of any string appears t steps away from the string clockwise or counterclockwise killian and savage proved that an n n antipodal gray code exists when n is a power of 2 or n 3 and does not exist for n 6 or odd n 3 motivated by these results we prove that for odd n 3 an n t antipodal gray code exists if and only if t 2 n 1 1 for even n we establish two recursive constructions for n t codes from smaller n t consequently various n t antipodal gray codes are found for even n s examples are for t 2 n 1 2 k with k odd and 1 k n 3 when n 4 for t 2 n k when n 2k with 1 k 3 for t n when n 2 k 2 an alternative proof for killian and savage s result etc

dynamical decoupling schemes derived from hamilton cycles we address the problem of decoupling the interactions in a spin network governed by a pair interaction hamiltonian combinatorial schemes for decoupling and for manipulating the couplings of hamiltonians have been developed which use selective pulses in this paper we consider an additional requirement on these pulse sequences as few different control operations as possible should be used this requirement is motivated by the fact that to find an optimal implementation of each individual selective pulse will be expensive since it requires to solve a pulse shaping problem hence it is desirable to use as few different selective pulses as possible as a first result we show that for d dimensional systems where d 2 the ability to implement only two control operations is sufficient to turn off the time evolution next we address the case of a bipartite system with local control and show that four different control operations are sufficient finally turning to networks consisting of several d dimensio

an axiomatic approach to constructing distances for rank comparison and aggregation we propose a new family of distance measures on rankings derived through an axiomatic approach that consider the nonuniform relevance of the top and bottom of ordered lists and similarities between candidates the proposed distance functions include specialized weighted versions of the kendall distance and the cayley distance and are suitable for comparing rankings in a number of applications including information retrieval and rank aggregation in addition to proposing the distance measures and providing the theoretical underpinnings for their applications we also analyze algorithmic and computational aspects of weighted distance based rank aggregation we present an aggregation method based on approximating weighted distance measures by a generalized version of spearman s footrule distance as well as a markov chain method inspired by pagerank where transition probabilities of the markov chain reflect the chosen weighted distances

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

on cardinalities of k abelian equivalence classes two words u and v are k abelian equivalent if for each word x of length at most k x occurs equally many times as a factor in both u and v the notion of k abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words in this paper we study the equivalence classes induced by the k abelian equivalence mainly focusing on the cardinalities of the classes in particular we are interested in the number of singleton k abelian classes i e classes containing only one element we find a connection between the singleton classes and cycle decompositions of the de bruijn graph we show that the number of classes of words of length n containing one single element is of order o n n m k 1 1 where n m l 1 l d l d m l d is the number of necklaces of length l over an m ary alphabet we conjecture that the upper bound is sharp we also remark that for k even and m 2 the lower bound n n m k 1 1 follows from an old conjecture on the existence of gray codes for necklaces of odd length we verify this conjecture for necklaces of length up to 15

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

error correction in flash memories via codes in the ulam metric we consider rank modulation codes for flash memories that allow for handling arbitrary charge drop errors unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements the proposed translocation rank codes account for more general forms of errors that arise in storage systems translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding our results include derivation of the asymptotic capacity of translocation rank codes construction techniques for asymptotically good codes as well as simple decoding methods for one class of constructed codes as part of our exposition we also highlight the close connections between the new code family and permutations with short common subsequences deletion and insertion error correcting codes for permutations and permutation codes in the hamming distance

supercuspidal duality for the two fold cover of sl2 and the split 03 introduction as first introduced the shimura shintani niwa correspondence was a correspondence between homomorphic cusp forms of half integral weight and certain modular forms of integral weight various authors have studied this correspondence in a representation theoretic framework gelbart s seminal work g1 provides some tools for translation and miscellaneous results using whittaker models and l functions gelbart and piatetski shapiro gps1 gps2 obtained a generalized correspondence between automorphic cuspidal representations of gl2 the metaplectic group and automorphic representations of gl2 flicker f used a variation on the selberg trace formula to extend the work of gelbart and piatetski shapiro finally waldspurger w returned to whittaker models andl functions to obtain a correspondence between automorphic cuspidal representations of sl2 the nontrivial two fold cover of sl2 and automorphic representations of pgl2 our more modest results complement the above and are obtained by purely algebraic means we now attempt a brief summary

computing binary combinatorial gray codes via exhaustive search with sat solvers the term binary combinatorial gray code refers to a list of binary words such that the hamming distance between two neighboring words is one and the list satisfies some additional properties that are of interest to a particular application e g circuit testing data compression and computational biology new distance preserving and circuit codes are presented along with a complete list of equivalence classes of the coil in the box codes for codeword length with respect to symmetry transformations of hypercubes a gray ordered code composed of all necklaces of the length is presented improving the known result with length

optimal permutation recovery in permuted monotone matrix model motivated by recent research on quantifying bacterial growth dynamics based on genome assemblies we consider a permuted monotone matrix model y z where the rows represent different samples th

a potential theoretic method for far field sound radiation calculations abstract the far field acoustic radiation due to the interaction of upstream unsteady vortical disturbances with an airfoil in subsonic compressible flow is calculated using potential theory a kirchhoff surface is placed in the near field surrounding the airfoil and the pressure on this surface is calculated from the unsteady flow field obtained using a second order finite difference code the governing equation is reduced to the helmholtz equation in the frequency domain and the solution is written in terms of an integral over the kirchhoff surface involving the free space green s function and an unknown single layer density function the single layer density is then determined from the boundary condition on the kirchhoff surface this method is presented as an alternative to classical kirchhoff methods it has the advantage of being able to accommodate arbitrarily shaped kirchhoff surfaces and is also readily extendable to three dimensional problems numerical results are presented for thin symmetric and loaded airfoils thin airfoil results are compared to the analytical solution and thick airfoil results are checked for numerical convergence and compared to results obtained from a time domain euler solver

group action graphs and parallel architectures the authors develop an algebraic framework that exposes the structural kinship among the debruijn shujcfle exchange butterjy and cube connected cycles networks and illustrate algorithmic benefits that ensue from the exposed relationships the framework builds on two alge braically specified genres of graphs a group action graph gag for short is given by a set v of vertices and a set h of permutations of v for each v e v and each r eii there is an arc labeled r from vertex v to vertex v a cayley graph is a gag v h where v is the group gr h generated by h and where each r 6 h acts on each g 6 gr h by right multiplication the graphs gr h h and v h are called associated graphs it is shown that every gag is a quotient graph of its associated cayley graph by applying such general results the authors determine the following the butterfly network a cayley graph and the debruijn network a gag are associated graphs the cube connected cycles network a cayley graph and the shuffle exchange network a gag are associated graphs the order n instance of both the butterfly and the cube connected cycles share the same underlying group but have slightly different generator sets ii by analyzing these algebraic results it is delimited for any cayley graph g and associated gag 7 a family of leveled algorithms which run as efficiently on t as they do on the much larger g further analysis of the results yields new surprisingly efficient simulations by the shuffle oriented networks the shuffle exchange and debruijn networks of like sized butterfly oriented networks the butterfly and cube connected cycles networks an n vertex butterfly oriented network can be simulated by the smallest shuffle oriented network that is big enough to hold it with slowdown o log log n this simulation is exponentially faster than the anticipated logarithmic slowdown the mappings that underlie the simulation can be computed in linear time and they afford one an algorithmic tech nique for translating any program developed for a butterfly oriented architecture into an equivalent program for a shuffle oriented architecture the latter program incurring only the indicated slowdown factor

computing scattering amplitudes for arbitrary cylinders under incident plane waves a numerical method to compute scattering amplitudes for time harmonic waves scattered from infinite cylinders with arbitrary uniform cross section is described a nonlocal boundary condition is used to develop a variational formulation of the scattering problem and the finite element method is applied to determine approximations to the near field scattering amplitudes are then determined by means of an integral representation obtained from green s formula and properties of the nonlocal boundary operator computational results are presented to illustrate the method s application

gray codes and lexicographical combinatorial generation for nonnesting and sparse nonnesting set partitions we present combinatorial gray codes and explicit designs of efficient algorithms for lexicographical combinatorial generation of the sets of nonnesting and sparse nonnesting set partitions of length n

gray coding cubic planar maps the idea of combinatorial gray codes is to list objects in question in such a way that two successive objects differ in some pre specified small way in this paper we utilize description trees to cyclicly gray code three classes of cubic planar maps namely bicubic planar maps 3 connected cubic planar maps and cubic non separable planar maps

constructions of snake in the box codes for rank modulation snake in the box code is a gray code which is capable of detecting a single error gray codes are important in the context of the rank modulation scheme which was suggested recently for representing information in flash memories for a gray code in this scheme the codewords are permutations two consecutive codewords are obtained using the push to the top operation and distance measure is defined on permutations in this paper the kendall s metric is used as the distance measure we present a general method for constructing such gray codes we apply the method recursively to obtain a snake of length m2n 1 2n 1 2n 1 m2n 1 for permutations of s2n 1 from a snake of length m2n 1 for permutations of s2n 1 thus we have lim n m2n 1 s2n 1 0 4338 improving on the previous known ratio of lim n 1 n using the general method we also present a direct construction this direct construction is based on necklaces and it might yield snakes of length 2n 1 2 2n 1 for permutations of s2n 1 the direct construction was applied successfully for s7 and s9 and hence lim n m2n 1 s2n 1 0 4743

generating linear extensions fast one of the most important sets associated with a poset cal p is its set of linear extensions e cal p this paper presents an algorithm to generate all of the linear extensions of a poset in constant amortized time that is in time o e cp where e cp e cp the fastest previously known algorithm for generating the linear extensions of a poset runs in time o n cdot e cp where n is the number of elements of the poset the algorithm presented here is the first constant amortized time algorithm for generating a naturally defined class of combinatorial objects for which the corresponding counting problem is p complete furthermore it is shown that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions the algorithm is practical and can be modified to count linear extensions efficiently and to compute p x y for all pairs x y in time o n 2 e cal p

the structure of single track gray codes single track gray codes are cyclic gray codes with codewords of length n such that all the n tracks which correspond to the n distinct coordinates of the codewords are cyclic shifts of the first track we investigate the structure of such binary codes and show that there is no such code with 2 sup n codewords when n is a power of 2 this implies that the known codes with 2 sup n 2n codewords when n is a power of 2 are optimal this result is then generalized to codes over gf p where p is a prime a subclass of single track gray codes called single track gray codes with k spaced heads is also defined all known systematic constructions for single track gray codes result in codes from this subclass we investigate this class and show it has a strong connection with two classes of sequences the full order words and the full order self dual words we present an iterative construction for binary single track gray codes which are asymptotically optimal if an infinite family of asymptotically optimal seed codes exists this construction is based on an effective way to generate a large set of distinct necklaces and a merging method for cyclic gray codes based on necklaces representatives