fault tolerant hamiltonian laceability of balanced hypercubes the balanced hypercube as a new variant of hypercube has many desirable properties such as strong connectivity high regularity and symmetry the particular property of the balanced hypercube is that each processor has a backup processor sharing the same neighborhood a hamiltonian bipartite graph g v 0 v 1 e is said to be hamiltonian laceable if there is a hamiltonian path between any two vertices x v 0 and y v 1 it has been proved that the balanced hypercube bh n is hamiltonian laceable for all n 1 in this paper we have proved that after at most 2 n 2 faulty edges occur bh n remains hamiltonian laceable for all n 2 this result is optimal with respect to the number of faulty edges can be tolerated in bh n

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

nonlinear boundary value problems for first order impulsive functional differential equations this paper is concerned with solutions of nonlinear boundary value problems for a class of first order functional differential equations we introduce new concept of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid obviously we extend previous results

condition for perfect dimensionality recovery by variational bayesian pca having shown its good performance in many applications variational bayesian vb learning is known to be one of the best tractable approximations to bayesian learning however its performance was not well understood theoretically in this paper we clarify the behavior of vb learning in probabilistic pca or fully observed matrix factorization more specifically we establish a necessary and sufficient condition for perfect dimensionality or rank recovery in the large scale limit when the matrix size goes to infinity our result theoretically guarantees the performance of vb pca at the same time it also reveals the conservative nature of vb learning it offers a low false positive rate at the expense of low sensitivity by contrasting with an alternative dimensionality selection method we characterize vb learning in pca in our analysis we obtain bounds of the noise variance estimator and a new and simple analytic form solution for the other parameters which themselves are useful for implementation of vb pca

antiperiodic boundary value problem for first order impulsive ordinary differential equations this paper discusses the antiperiodic boundary value problem for first order impulsive ordinary differential equations we establish several existence results by using the leray schauder alternative the lower and upper solution method and the monotone iterative technique

periodic boundary value problem for first order impulsive ordinary differential equations this paper investigates the existence of minimal and maximal solutions of the periodic boundary value problem for first order impulsive differential equations by establishing two comparison results and using the method of upper and lower solutions and the monotone iterative technique

displacement operator based decompositions of matrices using circulants or other group matrices abstract we show how an arbitrary square matrix can be expressed as sums of products of circulant and upper or lower triangular toeplitz matrices and as sums of products of matrices derived from finite groups group matrices and matrices which are close to group matrices the results obtained are interesting from several points of view they lead to different methods for computing linear transforms using ffts or fast convolution algorithms to faster methods for solving toeplitz systems to potential methods for mapping linear transforms to parallel computer architectures with interconnection scheme given by the group graph of a finite group and possibly to matrix theoretic methods for expressing relationship between finite groups

regular quantum graphs we introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries the method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way incoming and outgoing channels at vertex scattering processes are connected symmetry properties of the quantum graph as well as its spectral statistics depend on the particular choice of permutation matrices also called connectivity matrices and can now be controlled easily the method may find applications in the study of quantum random walks and may also prove to be useful in analysing universality in spectral statistics

first order impulsive ordinary differential equations with anti periodic and nonlinear boundary conditions the monotone iterative technique is a powerful method that have been used to approximate the extremal solutions of several problems 1 2 on the other hand impulsive differential equations are a basic tool to study some problems of biology medicine engineering and physics 3 in this paper we deal with the following nonlinear boundary value problem for first order differential equation with impulses at fixed points

the spectra of manhattan street networks the multidimensional manhattan street networks constitute a family of digraphs with many interesting properties such as vertex symmetry in fact they are cayley digraphs easy routing hamiltonicity and modular structure from the known structural properties of these digraphs we determine their spectra which always contain the spectra of hypercubes in particular in the standard two dimensional case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity

principal component analysis eigenvalue spectra from data with symmetry breaking structure principal component analysis pca is a ubiquitous method of multivariate statistics that focuses on the eigenvalues ensuremath lambda and eigenvectors of the sample covariance matrix of a data set we consider p n dimensional data vectors mathit ensuremath xi drawn from a distribution with covariance matrix mathbf c we use the replica method to evaluate the expected eigenvalue distribution ensuremath rho ensuremath lambda as stackrel ensuremath rightarrow n ensuremath infty with p ensuremath alpha n for some fixed ensuremath alpha in contrast to existing studies we consider the case where mathbf c contains a number of symmetry breaking directions so that the sample data set contains some definite structure explicitly we set mathbf c ensuremath sigma 2 mathbf i ensuremath sigma 2 ensuremath sum m 1 s a m mathbf b m mathbf b m t with a m g0 ensuremath forall m we find that the bulk of the eigenvalues are distributed as for the case when the elements of mathit ensuremath xi are independent and identically distributed with increasing ensuremath alpha a series of phase transitions are observed at ensuremath alpha a m ensuremath 2 m 1 2 dots s each time a single ensuremath delta function ensuremath delta ensuremath lambda ensuremath ensuremath lambda u a m separates from the upper edge of the bulk distribution where ensuremath lambda u a ensuremath sigma 2 1 a 1 ensuremath alpha a ensuremath 1 we confirm the results of the replica analysis by studying the stieltjes transform of ensuremath rho ensuremath lambda this suggests that the results obtained from the replica analysis are universal irrespective of the distribution from which mathit ensuremath xi is drawn provided the fourth moment of each element of mathit ensuremath xi exists

doubly linked ring networks we consider networks of processors where each processor either has one in link and one out link or two in links and two out links we study three properties of such networks 1 diameter 2 connectivity and 3 the ring property we propose a class of networks which seem to achieve the optimum as far as these three properties are concerned

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

asymptotic power of sphericity tests for high dimensional data this paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity we establish the convergence under the null hypothesis and the alternative of the log ratio of the joint densities of the sample covariance eigenvalues to a gaussian process indexed by the norm of the perturbation when the perturbation norm is larger than the phase transition threshold studied in baik et al 2005 the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain when the norm is below the threshold the process is non degenerate so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous using the asymptotic theory of statistical experiments we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region in particular we show that the asymptotic power of the tracy widom type tests is trivial whereas that of the eigenvalue based likelihood ratio test is strictly larger than the size and close to the power envelope

periodic boundary value problems for first order impulsive dynamic equations on time scales this paper deals with the periodic boundary value problem for a class of first order impulsive dynamic equations on time scales we introduce a new notion of lower and upper solutions which extends the classical lower and upper solutions the existence of extremal solutions is presented by virtue of the method of lower and upper solutions coupled with monotone iterative technique an example is also included to illustrate the main results

periodic boundary value problem for first order impulsive delay differential equations in this paper by means of the method of upper and lower solutions and the monotone iterative technique the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive delay differential equations is considered

large graphs with given degree and diameter part i the following problem arises in the study of interconnection networks find graphs of given diameter and degree having the maximum number of vertices in this correspondence we give some constructions of graphs proving in particular that lim inf n d d 2 d where n d is the maximum number of vertices of a graph with degree a and diameter d

periodic boundary value problem for the second order impulsive functional differential equations this paper considers the existence of extreme solutions of the periodic boundary value problem for second order functional differential equations by using the method of lower and upper solutions coupled with monotone iterative technique

line digraph iterations and the d k digraph problem this paper studies the behavior of the diameter and the average distance between vertices of the line digraph of a given digraph the results obtained are then applied to the so called d k digraph problem that is to maximize the number of vertices in a digraph of maximum out degree d and diameter k by line digraph iterations it is possible to construct digraphs with a number of vertices larger than d2 l d2times the nonattainable moore bound in particular this solves the d k digraph problem for k 2 also the line digraph technique provides us with a simple local routing algorithm for the corresponding networks

first order impulsive ordinary differential equations with advanced arguments this paper deals with impulsive advanced ordinary differential equations with boundary conditions we investigate the existence of solutions and quasisolutions for advanced impulsive differential equations to obtain such results we apply schauder s fixed point theorem corresponding results are also formulated for differential inequalities

statistical mechanics of learning multiple orthogonal signals asymptotic theory and fluctuation effects the learning of signal directions in high dimensional data through orthogonal decomposition or principal component analysis pca has many important applications in physics and engineering disciplines e g wireless communication information theory and econophysics the accuracy of the orthogonal decomposition can be studied using mean field theory previous analysis of data produced from a model with a single signal direction has predicted a retarded learning phase transition below which learning is not possible i e if the signal is too weak or the data set is too small then it is impossible to learn anything about the signal direction or magnitude in this contribution we show that the result can be generalized to the case where there are multiple signal directions each nondegenerate signal is associated with a retarded learning transition however fluctuations around the mean field solution lead to large finite size effects unless the signal strengths are very well separated we evaluate the one loop contribution to the mean field theory which shows that signal directions are indistinguishable from one another if their corresponding population eigenvalues are separated by o n tau with exponent tau 1 3 where n is the data dimension numerical simulations are consistent with the analysis and show that finite size effects can persist even for very large data sets

nonhomogeneous boundary value problem for first order impulsive differential equations with delay the authors employ the method of upper and lower solution coupled with the monotone iterative technique to obtain results of existence and uniqueness for a nonhomogeneous boundary value problem of impulsive differential equations with delay

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

nonlinear boundary problem of first order impulsive integro differential equations this paper focuses on first order impulsive integro differential equations with nonlinear boundary the lower and upper solutions combined with monotone iterative techniques are used to obtain the existence of extremal solutions of the problem numerical examples are given to prove its validity

periodic boundary value problems for delay differential equations with impulses existence and approximation of solutions for an impulsive delay differential equation with periodic boundary value conditions are presented we use comparison principles and monotone iterative techniques

representations and routing for cayley graphs computer networks it is shown that all finite cayley graphs can be represented by generalized chordal rings gcr an example borel cayley graph is used to illustrate the generation of gcr representations a sufficient condition is given for the representation of a cayley graph as a chordal ring cr with the integer labeling of gcr representations a straightforward progressive routing algorithm based on table look up is summarized

accuracy of pseudo inverse covariance learning a random matrix theory analysis for many learning problems estimates of the inverse population covariance are required and often obtained by inverting the sample covariance matrix increasingly for modern scientific data sets the number of sample points is less than the number of features and so the sample covariance is not invertible in such circumstances the moore penrose pseudo inverse sample covariance matrix constructed from the eigenvectors corresponding to nonzero sample covariance eigenvalues is often used as an approximation to the inverse population covariance matrix the reconstruction error of the pseudo inverse sample covariance matrix in estimating the true inverse covariance can be quantified via the frobenius norm of the difference between the two the reconstruction error is dominated by the smallest nonzero sample covariance eigenvalues and diverges as the sample size becomes comparable to the number of features for high dimensional data we use random matrix theory techniques and results to study the reconstruction error for a wide class of population covariance matrices we also show how bagging and random subspace methods can result in a reduction in the reconstruction error and can be combined to improve the accuracy of classifiers that utilize the pseudo inverse sample covariance matrix we test our analysis on both simulated and benchmark data sets

monotone iterative technique for impulsive integro differential equations with periodic boundary conditions this paper investigates the existence of minimal and maximal solutions of impulsive integro differential equations with periodic boundary conditions by establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique

boundary value problems for first order functional differential equations with impulsive integral conditions in this paper by using the method of lower and upper solutions coupled with the monotone iterative technique we give conditions for existence of extreme solutions for first order functional differential equations with a new impulsive integral condition some comparison results are also formulated

extension of de bruijn graph and kautz graph abstract we introduce a method of division of vectors to extend the definitions of de bruijn graphs and kautz graphs basic properties of extended de bruijn graphs and extended kautz graphs are given and finally several new results on isomorphic factorization of those graphs are given

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

nonlinear boundary value problem of first order impulsive functional differential equations this paper discusses nonlinear boundary value problem for first order impulsive functional differential equations we establish several existence results by using the lower and upper solutions and monotone iterative techniques two examples are discussed to illustrate the efficiency of the obtained results

integral boundary value problems for first order integro differential equations with impulsive integral conditions this paper is concerned with the integral boundary value problems for first order integro differential equations with impulsive integral conditions sufficient conditions are obtained for the existence of extremal solutions

some new results about the d k graph problem the d k graph problem which is a stiu open extremal problem in graph theory has received very much attention from many authors due to its theoretic interest and also due to its possible applications in communication network design the problem consists in maximizing the number of nodes n of an undirected regular graph d k of degree d and diameter k in this paper after a survey of the known results we present two new families of graphs and two methods of generating graphs given some existing ones leading to further substantial improvements of some of the results gathered by storwick 21 and recently improved by arden and lee 3 and also by imase and itoh 11

dcc linear congruential graphs a new class of interconnection networks let n be an integer and f f sub 1 1 spl les i spl les t for some integer t be a finite set of linear functions we define a linear congruential graph g f n as a graph on the vertex set v 0 1 n 1 in which any x spl isin v is adjacent to f sub i x mod n 1 spl les i spl les t for a linear function g and a subset v sub 1 of v we define a linear congruential graph g f n g v sub 1 as a graph on vertex set v in which any x spl isin v is adjacent to f sub i x mod n 1 spl les i spl les t and any x spl isin v sub 1 is also adjacent to g x mod n these graphs generalize several well known families of graphs e g the de bruijn graphs we give a family of linear functions called dcc linear functions that generate regular highly connected graphs which are of substantially larger order than de bruijn graphs of the same degree and diameter some theoretical and empirical properties of these graphs are given and their structural properties are studied

sandpile groups of generalized de bruijn and kautz graphs and circulant matrices over finite fields we determine the critical groups of the generalized de bruijn graphs db n d and generalized kautz graphs kautz n d thus extending and completing earlier results for the classical de bruijn and kautz graphs moreover for a prime p the critical groups of db n p are shown to be in close correspondence with groups of n n circulant matrices over fp which explains numerical data in 11 and suggests the possibility to construct normal bases in fp n from spanning trees in db n p

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

small diameter symmetric networks from linear groups a report is presented on a collection of constructions of symmetric networks that provide the largest known values for the number of nodes that can be placed in a network of a given degree and diameter some of the constructions are in the range of current potential engineering significance the constructions are cayley graphs of linear groups obtained by experimental computation

a gaussian scenario for unsupervised learning we consider random patterns on the n sphere which are uniformly distributed with the exception of a single symmetry breaking orientation along which they are gaussian distributed the unsupervised recognition of this orientation by different learning rules is studied in the large n limit using the replica method the model is simple enough to be analytically tractable and rich enough to exhibit most of the phenomena observed with other pattern distributions a learning algorithm based on the minimization of a cost function is identified which reaches the upper theoretical limit imposed by the optimal bayes learning scenario an implementation of this algorithm is proposed and tested numerically

interconnection networks based on a generalization of cube connected cycles a generalization of the cube connected cycles of preparata and vuillemin is described which retains the symmetry of these architectures while allowing for constructions of greater density and of arbitrary degree these constructions are of a type known as cayley graphs and their analysis is greatly facilitated by the applicability of methods from abstract algebra

class congruence property and two phase routing of borel cayley graphs dense symmetric graphs are useful interconnection models for multicomputer systems borel cayley graphs the densest degree 4 graphs for a range of diameters are attractive candidates however the group theoretic representation of these graphs makes the development of efficient routing algorithms difficult in earlier reports we showed that all degree 4 borel cayley graphs have generalized chordal ring gcr and chordal ring cr representations we present the class congruence property and use this property to develop the two phase routing algorithm for borel cayley graphs in a special gcr representation the algorithm requires a small space complexity of o p k for n p spl times k nodes although suboptimal the algorithm finds paths with length bounded by 2d where d is the diameter furthermore our computer implementation of the algorithm on networks with 1 081 and 15 657 nodes shows that the average path length is on the order of the diameter the performance of the algorithm is compared with that of existing optimal and suboptimal algorithms

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

automatic pca dimension selection for high dimensional data and small sample sizes bayesian inference from high dimensional data involves the integration over a large number of model parameters accurate evaluation of such high dimensional integrals raises a unique set of issues these issues are illustrated using the exemplar of model selection for principal component analysis pca a bayesian model selection criterion based on a laplace approximation to the model evidence for determining the number of signal principal components present in a data set has previously been show to perform well on various test data sets using simulated data we show that for d dimensional data and small sample sizes n the accuracy of this model selection method is strongly affected by increasing values of d by taking proper account of the contribution to the evidence from the large number of model parameters we show that model selection accuracy is substantially improved the accuracy of the improved model evidence is studied in the asymptotic limit d at fixed ratio n d with 1 in this limit model selection based upon the improved model evidence agrees with a frequentist hypothesis testing approach

the k tuple twin domination in generalized de bruijn and kautz networks given a digraph network g v a a vertex u in g is said to out dominate itself and all vertices v such that the arc u v a similarly u in dominates both itself and all vertices w such that the arc w u a a set d of vertices of g is a k tuple twin dominating set if every vertex of g is out dominated and in dominated by at least k vertices in d respectively the k tuple twin domination problem is to determine a minimum k tuple twin dominating set for a digraph in this paper we investigate the k tuple twin domination problem in generalized de bruijn networks g b n d and generalized kautz g k n d networks when d divides n we provide construction methods for constructing minimum k tuple twin dominating sets in these networks these results generalize previous results given by araki t araki the k tuple twin domination in de bruijn and kautz digraphs discrete mathematics 308 2008 6406 6413 for de bruijn and kautz networks

design to minimize diameter on building block network this paper proposes a simple algorithm for the graph design of small diameter networks for given nodes n and degree d this algorithm can be used to construct a directed graph with diameter logd n which is at most one larger than the lower bound logd n d 1 1 1 its average distance is also close to the lower bound the algorithm can be used to construct a directed graph with smaller diameter compared with any other conventional methods when n is larger

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

comments on low diameter interconnections for routing in high performance parallel systems with connections and extensions to arc coloring of coset graphs for original paper see r melhem ibid vol 56 p 502 10 2007 recently melhem presented a new class of low diameter interconnection ldi networks ieee trans computers vol 56 no 4 pp 502 510 we note that ldi networks are the same as the previously known generalized de bruijn graphs point out an error in the decomposition of ldi networks into permutations and find that the correct decomposition scheme is an instance of arc coloring for coset graphs hence we pursue a number of general results on arc coloring of coset graphs that can be applied to this particular decomposition problem as well as within many other contexts including complete arc coloring and normality of coset graphs

connectivity of regular directed graphs with small diameters this paper clarifies the relation between the diameter k and the edge or node connectivity ce or c of digraphs the following two inequalities are derived where n is the number of nodes d is the maximum degree and d is the minimum degree

realization of permutations on generalized indra networks abstract the realization of permutations of data on a class of redundant path interconnection networks called generalized indra networks is studied this class of networks is constructed by combining l layers copies of subnetworks in parallel preceded by a distribution stage the subnetwork chosen being a delta network constructed from r r switching elements redundant links are provided on the input side of the network so that failures in the distribution stage could be sustained the problem of realizing an arbitrary permutation on such networks has a one to one correspondence with the problem of performing the same permutation in multiple passes on a delta network this problem is reduced to that of finding a vertex coloring for the conflict graph of the permutation although the general problem of finding optimal colorings for graphs of arbitrary permutations is unsolved classes of permutations with some regularity can be mapped optimally we illustrate this with the bpc class of permutations conflicts in a permutation in the bpc class are described by means of conflict cubes and these cubes are used to obtain an optimal mapping of the permutation on the network using a minimum number of layers several examples are provided to illustrate the underlying concepts of our work

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

periodic boundary value problems for the first order impulsive functional differential equations this paper is concerned with the existence of extreme solutions of the periodic boundary value problems for a class of first order functional differential equations we introduce new concept of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid meanwhile we extend previous results

the gamma network the gamma network is an interconnection network connecting n 2n inputs to n outputs it is a multistage network with n switches per stage each of which is a 3 input 3 output crossbar the stages are linked via power of two and identify connections in such a way that redundant paths exist between the input and output terminals in this network a path from a source to a destination may be represented using one of the redundant forms of the difference between the source and destination numbers the redundancy in paths may thus be studied using the theory of redundant number systems results are obtained on the distribution of paths connecting inputs and outputs and the permuting capabilities of the gamma network frequently used permutations and control mechanisms are discussed briefly we also perform a detailed terminal reliability analysis of the gamma network deriving expressions for the reliability between an input and output terminal

communication aspects of networks based on geometric incidence relations abstract we explore the communication properties of a family of networks based on the incidence relation of the middle subspaces of a projective space network theoretic issues as well as implementation details are discussed the networks are shown to be symmetric and nearly optimal in diameter they support a natural routing scheme with efficient implementation of optimal complexity an extremely high redundancy makes the networks robust a parallel routing algorithm is analysed and is shown to achieve running time o diameter which is also the lower bound

the spectra of some families of digraphs abstract let g be a digraph or a graph when seen as a symmetric digraph with adjacency matrix a having the eigenvalue with associated eigenvector v as it is well known the entries of v can be interpreted as charges in each vertex then the linear transformation v av corresponds to a natural displacement of charges where each vertex sends a copy of its charge to its in neighbors and absorbs a copy of the charges of its out neighbors so the resulting charge distribution is just v in this work we use this approach to derive some old and new results about the spectral characterization of g more precisely we show how to obtain the spectra of some families of di graphs such as the partial line digraphs and the line graphs of regular or semiregular graphs

periodic boundary value problem for the first order impulsive functional differential equations this paper is concerned with the existence of extreme solutions of the periodic boundary value problem for a class of first order impulsive functional differential equations we introduce new concept of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid meanwhile we extend previous results