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

une dichotomie de hopf pour les flots geodesiques associes aux groupes discrets d isometries des arbres let x be a complete locally compact metric tree and f a group of isometries of x acting properly on this space the space of bi infinite geodesics in x constitutes a space gx on which f acts properly let q be the quotient of gx by this action the geodesic flow associated to f is the flow on q which is the quotient of the geodesic flow on gx defined by the time shift on geodesics to any f conformal measure on the boundary ox there is an associated measure m on q which is invariant by the geodesic flow we prove the following results the geodesic flow on q m is either conservative or dissipative if it is conservative then it is ergodic if it is dissipative then it is not ergodic unless it is measurably conjugate to the action of r on itself by conjugation we prove also a dichotomy in terms of the conical limit set a c ox of f the flow on q m is conservative if and only if u a 4ox and it is dissipative if and only if u a 0 the results are analogous to results of e hopf and d sullivan in the case of riemannian manifolds of constant negative curvature

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

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

topological mixing in 1 spaces if x is a proper cat 1 space and a non elementary discrete group of isometries acting properly discontinuously on x it is shown that the geodesic flow on the quotient space y x is topologically mixing provided that the generalized busemann function has zeros on the boundary x and the non wandering set of the flow equals the whole quotient space of geodesics gy gx r the latter being redundant when y is compact applications include the proof of topological mixing for a compact negatively curved polyhedra b compact quotients of proper geodesically complete cat 1 spaces by a one ended group of isometries and c finite n dimensional ideal polyhedra

the lorentzian splitting theorem without the completeness assumption on demontre le theoreme de decomposition lorentzien pour des espace temps globalement hyperboliques obeissant a la condition d energie forte sans hypothese de completude geodesique du genre temps

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

holomorphic representation of constant mean curvature surfaces in minkowski space consequences of non compactness in loop group methods abstract we give an infinite dimensional generalized weierstrass representation for spacelike constant mean curvature cmc surfaces in minkowski 3 space r 2 1 the formulation is analogous to that given by dorfmeister pedit and wu for cmc surfaces in euclidean space replacing the group su 2 with su 1 1 the non compactness of the latter group however means that the iwasawa decomposition of the loop group used to construct the surfaces is not global we prove that it is defined on an open dense subset after doubling the size of the real form su 1 1 and prove several results concerning the behavior of the surface as the boundary of this open set is encountered we then use the generalized weierstrass representation to create and classify new examples of spacelike cmc surfaces in r 2 1 in particular we classify surfaces of revolution and surfaces with screw motion symmetry as well as studying another class of surfaces for which the metric is rotationally invariant

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

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

maximal hypersurfaces in stationary asymptotically flat spacetimes existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike near infinity the first class consists of strongly causal asymptotically flat spacetimes which contain no black hole or white hole but may contain ergoregions where the killing orbits fail to be timelike the second class of spacetimes possess a black hole and a white hole with the black and white hole horizons intersecting in a compact 2 surfaces

a strong maximum principle for weak solutions of quasi linear elliptic equations with applications to lorentzian and riemannian geometry the strong maximum principle is proved to hold for weak in the sense of support functions sub and supersolutions to a class of quasi linear elliptic equations that includes the mean curvature equation for c0 space like hypersurfaces in a lorentzian manifold as one application a lorentzian warped product splitting theorem is given 1998 john wiley sons inc

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

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

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

polynomial quadratic differentials on the complex plane and light like polygons in the einstein universe abstract we construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light like polygons in the 2 dimensional einstein universe as an application we find a class of minimal lagrangian maps between ideal polygons in the hyperbolic plane

splitting theorems for spatially closed space times a lorentzian splitting theorem is obtained for spatially closed spacetimes the proof employs and extends some recent results of bartnik and gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space times a splitting theorem for spatially closedtime periodic space times which generalizes a result first considered by avez is derived as a corollary

surfaces of prescribed mean curvature vector in semi riemannian manifolds abstract solving a second order quasilinear initial value problem we prove existence and uniqueness of both space like and time like immersions of prescribed mean curvature vector in general semi riemannian manifolds such surfaces arise for example as critical points of generalized area functionals as well as self similar solutions of the mean curvature flow

on maximal surfaces in asymptotically flat space times existenc of maximal and almost maximal hypersurfaces in asymptotically flat space times is established under boundary conditions weaker than those considered previously we show in particular that every vacuum evolution of asymptotically flat data for the einstein equations can be foliated by slices maximal outside a spatially compact set and that every strictly stationary asymptotically flat space time can be foliated by maximal hypersurfaces amongst other uniqueness results we show that maximal hypersurfaces can be used to partially fix an asymptotic poincare group

ergodic properties of boundary actions and the nielsen schreier theory we study the basic ergodic properties ergodicity and conservativity of the action of a subgroup h of a free group f on the boundary pt f with respect to the uniform measure our approach is geometrical and combinatorial and it is based on choosing a system of nielsen schreier generators in h associated with a geodesic spanning tree in the schreier graph x h bs f we give several mod 0 equivalent descriptions of the hopf decomposition of the boundary into the conservative and the dissipative parts further we relate conservativity and dissipativity of the action with the growth of the schreier graph x and of the subgroup h equiv cogrowth of x respectively on the other hand our approach sheds a new light on entirely algebraic properties of subgroups of a free group we also construct numerous examples illustrating the connections between various relevant notions

cat 1 spaces divergence groups and their commensurators a cat 1 space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane b briha gr the cat 1 property is one among many possible generalizations to singular spaces of the notion of negative curvature important examples of cat 1 spaces include riemannian manifolds of sectional curvature k 1 and their convex subsets b g s metric trees and piecewise hyperbolic cell complexes mou da hag be 1 be 2 b br in this paper we establish certain superrigidity results for isometric actions of a group on a cat 1 space in the following two settings a the group is a subgroup of a locally compact group g with comg where g is a sufficiently large discrete subgroup and comg g g g 1 g and share a subgroup of finite index is the commensurator of in g b the group is an irreducible lattice in g n 1g k where each g is a semisimple algebraic group defined over a local field k the issues addressed in this paper are motivated on one hand by earlier work of g a margulis ma dealing with the linear representation theory of where in case a g is a semisimple group and g a lattice and on the other hand by the results of lubotzky mozes and zimmer l m z concerning isometric actions of on trees where comg g is the group of automorphisms of a regular tree and g is a lattice our approach to establishing superrigidity results is based on ergodic theoretic methods developed by margulis ma zi 3 a c b in this context the following notion of boundary of a locally compact group will be useful let b be a standard borel space on which acts by borel automorphisms preserving a finite measure class

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

regularity of lorentzian busemann functions a general theory of regularity for lorentzian busemann functions in future timelike geodesically complete spacetimes is presented this treatment simplifies and extends the local regularity developed by eschenburg galloway and newman to prove the lorentzian splitting theorem criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of bartnik

spacelike hypersurfaces with prescribed boundary values and mean curvature we consider the boundary value problem for the mean curvature operator in minkowski space and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions our main results hold for non constant mean curvature and make no assumptions about the smoothness of the boundary or boundary data

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

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

gluing initial data sets for general relativity we establish an optimal gluing construction for general relativistic initial data sets the construction is optimal in two distinct ways first it applies to generic initial data sets and the required generically satisfied hypotheses are geometrically and physically natural second the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place using this construction we establish the existence of cosmological maximal globally hyperbolic vacuum space times with no constant mean curvature spacelike cauchy surfaces