bipolar fuzzy graphs in this paper we introduce the notion of bipolar fuzzy graphs describe various methods of their construction discuss the concept of isomorphisms of these graphs and investigate some of their important properties we then introduce the notion of strong bipolar fuzzy graphs and study some of their properties we also discuss some propositions of self complementary and self weak complementary strong bipolar fuzzy graphs

on the k tuple domination of generalized de brujin and kautz digraphs a vertex u in a digraph g v a is said to dominate itself and vertices v such that u v a for a positive integer k a k tuple dominating set of g is a subset d of vertices such that every vertex in g is dominated by at least k vertices in d the k tuple domination number of g is the minimum cardinality of a k tuple dominating set of g this paper deals with the k tuple domination problem on generalized de bruijn and kautz digraphs we establish bounds on the k tuple domination number for the generalized de bruijn and kautz digraphs and we obtain some conditions for the k tuple domination number attaining the bounds

the heat equation with rough boundary conditions and holomorphic functional calculus abstract in this paper we consider the laplace operator with dirichlet boundary conditions on a smooth domain we prove that it has a bounded h calculus on weighted l p spaces for power weights which fall outside the classical class of a p weights furthermore we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity in particular we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data

edge fault tolerant panconnectivity and edge pancyclicity of the complete graph the complete graphs are an important class of graphs and are also fundamental interconnection networks recently fu investigated their edge fault tolerant hamiltonicity and ho et al investigated their edge fault tolerant hamiltonian connectivity in this paper we improve the result of fu and point out that the proof of the result of ho et al fails then we consider the edge fault tolerant panconnectivity of the complete graphs and obtain the following result let f be any set of at most 2n 10 faulty edges in the complete graph k n with n vertices such that every vertex of the graph g k n f is incident with at least three edges and g k 8 e k 4 k 1 0 e k 5 then g is nearly panconnected i e for any two vertices u and v there exists a path connecting u and v in g of any length from 3 to n 1 as a corollary every edge in the graph g lies on a cycle of any length from 4 to n moreover the number 2n 10 of faulty edges tolerated is sharp

embedding hamiltonian cycles in alternating group graphs under conditional fault model in this paper assuming that each node is incident with two or more fault free links we show that an n dimensional alternating group graph can tolerate up to 4n 13 link faults where n 4 while retaining a fault free hamiltonian cycle the proof is computer assisted the result is optimal with respect to the number of link faults tolerated previously without the assumption at most 2n 6 link faults can be tolerated for the same problem and the same graph

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

the sharp weighted bound for general calderon zygmund operators for a general calderon zygmund operator t on r n it is shown that ktfkl2 w c t sup q a q w q w 1 a k fkl2 w for all muckenhoupt weights w 2 a2 this optimal estimate was known as the a2 conjecture a recent result of perez treil volberg reduced the problem to a testing condition on indicator functions which is verified in this paper the proof consists of the following elements i a variant of the nazarov treil volberg method of random dyadic systems with just one random system and completely without bad parts ii a resulting representation of a general calderon zygmund operator as an average of dyadic shifts and iii improvements of the lacey petermichl reguera estimates for these dyadic shifts which allow summing up the series in the obtained representation

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

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

the sharp bound for the hilbert transform on weighted lebesgue spaces in terms of the classical ap characteristic we show that the norm of the hilbert transform as an operator in the weighted space l p r for 2 p oo is bounded by a constant multiple of the first power of the classical ap characteristic of this result is sharp we also prove a bilinear imbedding theorem with simple conditions

differential hopf algebra structures on the universal enveloping algebra of a lie algebra we discuss a method to construct a de rham complex differential algebra of poincar e birkhoff witt type on the universal enveloping algebra of a lie algebra g we determine the cases in which this gives rise to a differential hopf algebra that naturally extends the hopf algebra structure of u g the construction of such differential structures is interpreted in terms of colour lie superalgebras

fault tolerant edge bipancyclicity of faulty hypercubes under the conditional fault model it is well known that the n dimensional hypercube qn is one of the most versatile and efficient interconnection network architecture yet discovered for building massively parallel or distributed systems let f be the faulty set of qn and let fv fe be the numbers of faulty vertices and faulty edges in f respectively an edge e x y is said to be free if e x y are not in f and a cycle is said to be fault free if there is no faulty vertex or faulty edge on the cycle in this paper we prove that each free edge x y in qn for n 3 lies on a fault free cycle of any even length from 6 to 2 n 2 f v if f v f e 2 n 5 f e n 2 and both x and y are incident to at least two free edges this result confirms a conjecture reported in the literature

hamiltonian path embedding and pancyclicity on the mobius cube with faulty nodes and faulty edges a graph g v e is said to be pancyclic if it contains fault free cycles of all lengths from 4 to v in g let f sub v and f sub e be the sets of faulty nodes and faulty edges of an n dimensional mobius cube mq sub n respectively and let f f sub v spl cup f sub e a faulty graph is pancyclic if it contains fault free cycles of all lengths from 4 to v f sub v in this paper we show that mq sub n f contains a fault free hamiltonian path when f spl les n 1 and n spl ges 1 we also show that mq sub n f is pancyclic when f spl les n 2 and n spl ges 2 since mq sub n is regular of degree n both results are optimal in the worst case

two node disjoint paths in balanced hypercubes the balanced hypercube bhn proposed by wu and huang is a variation of the hypercube it has been proved that the balanced hypercube is a node transitive and bipartite graph assume that the nodes are divided into two bipartite node sets x and y u and x are two different nodes in x and v and y are two different nodes in y in this paper we prove that there exist two node disjoint paths p x y and r u v in bhn and v p x y v r u v v bhn where n 1 the hamiltonian laceability of bhn which was proved by xu et al is also obtained from the corollary of our result

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

embedding long cycles in faulty k ary 2 cubes abstract the class of k ary n cubes represents the most commonly used interconnection topology for distributed memory parallel systems given an even k 4 let v 1 v 2 be the bipartition of the k ary 2 cube f v 1 f v 2 be the numbers of faulty vertices in v 1 and v 2 respectively and f e be the number of faulty edges in this paper we prove that there exists a cycle of length k 2 2max f v 1 f v 2 in the k ary 2 cube with 0 f v 1 f v 2 f e 2 this result is optimal with respect to the number of faults tolerated

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

the triangular pyramid routing and topological properties in this paper a new topology for multicomputer interconnection networks based on triangular mesh is proposed the new network referred to as the triangular pyramid or tripy for short has l levels of triangular mesh we study some basic important properties of the proposed network as well as introduce a routing algorithm for the tripy network based on the routing of triangular meshes we prove that this form of pyramidal network is hamiltonian hamiltonian connected and pancyclic we also prove that the proposed network is 6 colorable and conduct a brief comparison of the tripy and its traditional pyramid counterpart our results show that the proposed network has higher scalability connectivity and total network bandwidth while preserving the important properties of the traditional pyramid network

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

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

conditional edge fault tolerant edge bipancyclicity of hypercubes the hypercube has been one of the most popular interconnection networks for parallel computer communication systems in this paper we assume that each node is incident with at least two fault free links under this assumption we show that every fault free edge lies on a fault free cycle of every even length from 6 to 2 n inclusive even if it has up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

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

fault free hamiltonian cycles in crossed cubes with conditional link faults the crossed cube which is a variation of the hypercube possesses some properties superior to the hypercube in this paper assuming that each node is incident with at least two fault free links we show that an n dimensional crossed cube contains a fault free hamiltonian cycle even if there are up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated we also verify that the assumption is practically meaningful by evaluating its occurrence probability which is very close to 1

conditional edge fault tolerant hamiltonicity of dual cubes the dual cube is an interconnection network for linking a large number of nodes with a low node degree it uses low dimensional hypercubes as building blocks and keeps the main desired properties of the hypercube a dual cube dc n has n 1 links per node where n is the degree of a cluster n cube and one more link is used for connecting to a node in another cluster in this paper assuming each node is incident with at least two fault free links we show a dual cube dc n contains a fault free hamiltonian cycle even if it has up to 2n 3 link faults the result is optimal with respect to the number of tolerant edge faults

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

on embedding cycles into faulty twisted cubes the twisted cube tq n is an alternative to the popular hypercube network recently some interesting properties of tq n were investigated in this paper we study the pancycle problem on faulty twisted cubes let f e and f v be the numbers of faulty edges and faulty vertices in tq n respectively we show that with f e f v 3

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

the 2 path bipanconnectivity of hypercubes in this paper we introduce the concept of the k path bi panconnectivity of bipartite graphs it is a generalization of the bi panconnectivity and of the paired many to many k disjoint path cover the 2 path bipanconnectivity with only one exception of the n cube q n n 4 is proved precisely the following result is obtained in an n cube with n 4 given any four vertices u 1 v 1 u 2 v 2 such that two of them are in one partite set and the another two are in the another partite set let s t 5 if c u 1u 2v 1v 2 is a cycle of length 4 and s d u 1 v 1 1 and t d u 2 v 2 1 otherwise where d u v denotes the distance between two vertices u and v and let i and j be any two integers such that both i s 0 and j t 0 are even with i j 2 n then there exist two vertex disjoint u 1 v 1 path p and u 2 v 2 path r with v p i and v r j as consequences many properties of hypercubes such as bipanconnectivity bipanpositionable bipanconnectivity 18 bipancycle connectivity 12 two internally disjoint paths with two given lengths and the 2 disjoint path cover with a path of a given length 21 follow from our result

_ estimates via extrapolation of carleson measures and applications to divergence form elliptic operators we revisit the extrapolation method for carleson measures introduced in lm to prove a1 estimates for certain caloric measures and we present a purely real variable version of the method suitable for establishing a1 estimates to illustrate the use of this technique we then reprove a well known result of fkp the results of lm and of hl are examples of carleson a1 extrapolation in which a given non negative measure is shown to belong to a1 or weak a1 using properties of some controlling carleson measure the results of ahlt and of ahmtt involve carleson carleson extrapolation in which a non negative measure in the half space r n 1 is shown to be a carleson measure using properties of another controlling

fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs the arrangement graph a sub n k is a generalization of the star graph there are some results concerning fault hamiltonicity and fault hamiltonian connectivity of the arrangement graph however these results are restricted in some particular cases and thus are less completed we improve these results and obtain a stronger and simpler statement let n k spl ges 2 and f spl sube v a sub n k spl cup e a sub n k we prove that a sub n k f is hamiltonian if f spl les k n k 2 and a sub n k f is hamiltonian connected if f spl les k n k 3 these results are optimal

entropy conditions in two weight inequalities for singular integral operators abstract a new type of bumping of the muckenhoupt a 2 condition on weights is introduced it is based on bumping the entropy integral of the weights in particular one gets assuming mild regularity conditions on the corresponding young functions the bump conjecture proved in 20 25 as a corollary of entropy bumping but our entropy bumps cannot be reduced to the bumping with orlicz norms in the solution of bump conjecture they are effectively smaller henceforth we get somewhat stronger results than the one that solves the bump conjecture in 20 25 new results concerning one sided bumping conjecture are obtained all the results hold in the general non homogeneous situation

on reliability of the folded hypercubes in this paper we explore the 2 extra connectivity and 2 extra edge connectivity of the folded hypercube fq n we show that k 2 fq n 3n 2 for n 8 and l 2 fq n 3n 1 for n 5 that is for n 8 resp n 5 at least 3n 2 vertices resp 3n 1 edges of fq n are removed to get a disconnected graph that contains no isolated vertices resp edges when the folded hypercube is used to model the topological structure of a large scale parallel processing system these results can provide more accurate measurements for reliability and fault tolerance of the system

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

quantitative boundedness of littlewood paley functions on weighted lebesgue spaces in the schrodinger setting abstract let l v be the schrodinger operator on r n with n 3 where v is a non negative potential which belongs to certain reverse holder class r h q r n with q n 2 in this article the authors obtain the quantitative weighted boundedness of littlewood paley functions g l s l and g l associated to l on weighted lebesgue spaces l p w where w belongs to the class of muckenhoupt a p weights adapted to l

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

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

a_p weights and quantitative estimates in the schrodinger setting suppose l delta v is a schrodinger operator on mathbb r n with a potential v belonging to certain reverse holder class rh sigma with sigma ge n 2 the aim of this paper is to study the a p weights associated to l denoted by a p l which is a larger class than the classical muckenhoupt a p weights we first prove the quantitative a p l bound for the maximal function and the maximal heat semigroup associated to l then we further provide the quantitative a p q l bound for the fractional integral operator associated to l we point out that all these quantitative bounds are known before in terms of the classical a p q constant however since a p q subset a p q l the a p q l constants are smaller than a p q constant hence our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to l next we prove two weight inequalities for the fractional integral operator these have been unknown up to this point finally we also have a study on the exp log link between a p l and bmo l the bmo space associated with l and show that for w in a p l log w is in bmo l and that the reverse is not true in general

fault tolerant embedding of cycles of various lengths in k ary n cubes the k ary n cube has been one of the most popular interconnection networks for distributed memory parallel systems in this paper we study the problem of embedding cycles of various lengths into faulty k ary n cubes it is proved that a faulty k ary n cube with f v faulty vertices and f e faulty edges admits a fault free cycle of every even length from 4 to k n 2f v if k 4 is even and f v f e 4 is even and f v f e 2n 3 the results are optimal with respect to the number of vertex and edge faults tolerated

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

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

the bipancycle connectivity of the hypercube paths and cycles are popular interconnection networks due to their simplicity and low degrees and therefore an efficient way of embedding them into an interconnection network is of particular importance many path embedding aspects and cycle embedding aspects such as hamiltonicity hamiltonian connectivity panconnectivity bipanconnectivity pancyclicity vertex pancyclicity and edge pancyclicity have been proposed and extensively studied in fact these discussions are meaningful for interconnection networks of resource allocated systems or heterogeneous computing systems in this paper we propose a new cycle embedding aspect called bipancycle connectivity that combines the concepts of vertex pancyclicity and bipanconnectivity a bipartite graph is bipancycle connected if an arbitrary pair of vertices x y is contained by common cycles called bipanconnected cycles that include a cycle of every even length ranging from minc x y to n where minc x y denotes the length of the minimum cycle that contains x and y and n is the number of vertices in this paper we introduce a new graph called the cycle of ladders we show that the hypercube is bipancycle connected by presenting algorithms to embed the cycle of ladders into the hypercube

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

classes of weights related to schrodinger operators abstract in this work we obtain boundedness on weighted lebesgue spaces on r d of the semi group maximal function riesz transforms fractional integrals and g function associated to the schrodinger operator v where v satisfies a reverse holder inequality with exponent greater than d 2 we consider new classes of weights that locally behave as muckenhoupt s weights and actually include them the notion of locality is defined by means of the critical radius function of the potential v given in shen 1995 8

a kind of conditional vertex connectivity of cayley graphs generated by 2 trees let g be a graph then t v g is called an r k vertex cut if g t is disconnected and each vertex in v g t has at least k neighbors in g t the size of a smallest r k vertex cut is the r k vertex connectivity of g and is denoted by k k g in this paper we determine the numbers k 1 and k 2 for cayley graphs generated by 2 trees including the popular alternating group graphs

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

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

a kind of conditional connectivity of cayley graphs generated by wheel graphs for a connected graph g v e a subset fv is called an rk vertex cut of g if gf is disconnected and each vertex in vf has at least k neighbors in gf the cardinality of the minimum rk vertex cut is the rk vertex connectivity of g and is denoted by k g the conditional connectivity is a measure to explore the structure of networks beyond the vertex connectivity let sym n be the symmetric group on 1 2 n and t be a set of transpositions of sym n denote by g t the graph with vertex set 1 2 n and edge set ij ij t if g t is a wheel graph then simply denote the cayley graph cay sym n t by wgn in this paper we determine the values of 1 and 2 for cayley graphs generated by wheel graphs and prove that 1 wgn 4n6 and 2 wgn 8n18

a characterization of two weight norm inequalities for fractional and poisson integrals caracterisation de deux inegalites de normes ponderees pour les integrales fractionnaires et de poisson applications aux operateurs differentiels elliptiques degeneres

weighted inequalities for commutators of schrodinger riesz transforms abstract in this work we obtain weighted l p 1 p and weak l log l estimates for the commutator of the riesz transforms associated to a schrodinger operator v where v satisfies some reverse holder inequality the classes of weights as well as the classes of symbols are larger than a p and bmo corresponding to the classical riesz transforms

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

complete path embeddings in crossed cubes crossed cubes are popular variants of hypercubes in this paper we study path embeddings between any two distinct nodes in crossed cubes we prove two important results in the n dimensional crossed cube a for any two nodes all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1 b for any two integers n 2 and l with 1

quantization of differential calculi on universal enveloping algebras a method to construct differential calculi on quantized universal enveloping algebras is discussed these differential calculi are obtained by quantizing calculi on classical enveloping algebras provided with appropriate co poisson structures the procedure is demonstrated by applying it to the standard quantizations of the heisenberg algebra and the algebragl 2

sharp a2 inequality for haar shift operators as a corollary to the main result of the paper we give a new proof of the inequality rm t f l 2 w lesssim w a 2 f l 2 w where t is either the hilbert transform amer j math 129 5 1355 1375 2007 a riesz transform proc amer math soc 136 4 1237 1249 2008 or the beurling operator duke math j 112 2 281 305 2002 the weight w is non negative and the linear growth in the a2 characteristic on the right is sharp prior proofs relied strongly on haar shift operators cr acad sci paris ser i math 330 6 455 460 2000 and bellman function techniques the new proof uses haar shifts and then uses an elegant two weight t 1 theorem of nazarov treil volberg math res lett 15 3 583 597 2008 to immediately identify relevant carleson measure estimates which are in turn verified using an appropriate corona decomposition of the weight w

many to many disjoint paths in faulty hypercubes this paper considers the problem of many to many disjoint paths in the hypercube q n with f v faulty vertices and f e faulty edges and obtains the following result for any integer k with 1 k n 1 any two sets s and t of k fault free vertices in different parts if f v f e n k 1 then there exist k disjoint fault free s t paths in q n which contains at least 2 n 2 f v vertices this result is optimal in the worst case

a two weight weak type inequality for fractional integrals for 1 p q oo 0 a n and w x v x nonnegative weight functions on r 1 we show that the weak type inequality f xw x dx a q if x l v x dx holds for allf 0 if and only if f ta xqw x v x i dx p 1b aw cc for all cubes q in rn here t7 denotes the fractional integral of order a ta f x f i x y i f y dy more generally we can replace ta by any suitable convolution operator with radial kernel decreasing in i x i

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

various cycles embedding in faulty balanced hypercubes consider the balanced hypercube bh n with f e 2 n 3 faulty edges prove that every edge of bh n f e lies on fault free cycles of even lengths from 6 to 22n prove that the lower limit of the length 6 is sharp quite a lot of interconnection networks are served as the underlying topologies of large scale multiprocessor systems the hypercube is one of the most popular interconnection networks in this paper we consider the balanced hypercube which is a variant of the hypercube huang and wu showed that the balanced hypercube has better properties than hypercube with the same number of links and processors let f e be the set of faulty edges in an n dimensional balanced hypercube bh n where n 2 in this paper we consider bh n with f e 2 n 3 faulty edges and prove that every fault free edge lies on a fault free cycle of every even length from 6 to 2 2 n in bh n f e furthermore we prove that the lower limit of the length 6 is sharp by giving a counter example

linear bound for the dyadic paraproduct on weighted lebesgue space l2 w the dyadic paraproduct is bounded in weighted lebesgue spaces lp w if and only if the weight w belongs to the muckenhoupt class apd however the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest l2 w case in this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted lebesgue space l2 w depends linearly on the a2d characteristic of the weight w using bellman function techniques and extrapolate this result to the lp w case

sharp bekolle estimates for the bergman projection we prove sharp estimates for the bergman projection in weighted bergman spaces in terms of the bekolle constant our main tools are a dyadic model dominating the operator and an adaptation of a method of cruz uribe martell and perez c 2013 elsevier inc all rights reserved

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

a systematic approach for embedding of hamiltonian cycles through a prescribed edge in locally twisted cubes the locally twisted cube interconnection network has been recognized as an attractive alternative to the hypercube network previously the locally twisted cube has been shown to contain a hamiltonian cycle the main contribution of this paper is to provide the necessary and sufficient conditions for determining a characterization of permutations of link dimensions constructing hamiltonian cycles in a locally twisted cube for those permutations we propose a linear algorithm for finding a hamiltonian cycle through a given edge as a result we obtain a lower bound for the number of hamiltonian cycles through a given edge in an n dimensional locally twisted cube

conditional connectivity of star graph networks under embedding restriction the n dimensional star graph s n is one of the most attractive interconnection networks of large scale multiprocessor systems the k embedding restricted edge connectivity h k s n z k s n of s n is defined to be the cardinality of a minimum subset of edges nodes if any whose deletion disconnects s n and each node of the remaining components lies in an undamaged k dimensional substar s k in this paper we investigate the k embedding restricted edge connectivity of s n and determine the values of h k s n and z k s n for some k s

conditional diagnosability of matching composition networks under the mm model diagnosability is a critical metric for determining the reliability of a multiprocessor system in 2005 lai et al proposed a new measure for the fault diagnosis of a system i e conditional diagnosability in which it is assumed that at least one of the neighbors of an arbitrary node in the system is not faulty in this paper we obtain a sufficient condition for a class of networks called matching composition networks mcns which are conditionally t diagnosable under the mm model then we apply the sufficient condition to show the conditional diagnosability of bijective connection bc networks finally we show that the sufficient condition can be applied to networks other than bc networks

fault free hamiltonian cycles in twisted cubes with conditional link faults the n dimensional twisted cube denoted by tq n a variation of the hypercube possesses some properties superior to the hypercube in this paper assuming that each vertex is incident with at least two fault free links we show that tq n can tolerate up to 2n 5 edge faults while retaining a fault free hamiltonian cycle the result is optimal with respect to the number of edge faults tolerated

linear array and ring embeddings in conditional faulty hypercubes the n dimensional hypercube qn is a graph having 2n vertices labeled from 0 to 2n 1 two vertices are connected by an edge if their binary labels differ in exactly one bit position in this paper we consider the faulty hypercube qn with n 3 that each vertex of qn is incident to at least two nonfaulty edges based on this requirement we prove that qn contains a hamiltonian path joining any two different colored vertices even if it has up to 2n 5 edge faults moreover we show that there exists a path of length 2n 2 between any two the same colored vertices in this faulty qn furthermore we also prove that the faulty qn still contains a cycle of every even length from 4 to 2n inclusive

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc

hamiltonian properties of honeycomb meshes meshes are widely used topologies for networks on chip noc honeycomb meshes have better topological properties than meshes in order to communicate efficiently in a linear or cyclic manner it is benefited that there is a hamiltonian path or hamiltonian cycle in noc in this paper we give a necessary and sufficient condition for the existence of hamiltonian path between any pair of vertices in a honeycomb mesh and for the existence of hamiltonian path in a honeycomb mesh with one faulty vertex besides we give a systematic method to construct a hamiltonian path in honeycomb meshes

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

sharp weighted bounds for fractional integral operators abstract the relationship between the operator norms of fractional integral operators acting on weighted lebesgue spaces and the constant of the weights is investigated sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions as an application improved sobolev inequalities are obtained some of the techniques used include a sharp off diagonal version of the extrapolation theorem of rubio de francia and characterizations of two weight norm inequalities

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