maximin d optimal designs for longitudinal mixed effects models in this article the optimal selection and allocation of time points in repeated measures experiments is considered d optimal cohort designs are computed numerically for the first and second degree polynomial models with random intercept random slope and first order autoregressive serial correlations because the optimal designs are locally optimal it is proposed to use a maximin criterion it is shown that for a large class of symmetric designs the smallest relative efficiency over the model parameter space is substantial

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

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

embedding fault free hamiltonian paths with prescribed linear forests into faulty ternary n cubes abstract the k ary n cube is an important underlying topology for large scale multiprocessor systems a linear forest in a graph is a subgraph each component of which is a path in this paper we investigate the problem of embedding hamiltonian paths passing through a prescribed linear forest in ternary n cubes with faulty edges given a faulty edge set f with at most 2 n 3 edges and a linear forest l with at most 2 n 3 f edges for two distinct vertices in the ternary n cube we show that the ternary n cube admits a fault free hamiltonian path between u and v passing through l if and only if none of the paths in l has u or v as internal vertices or both of them as end vertices

hyper hamiltonian laceability on edge fault star graph the star graph possess many nice topological properties edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes in this paper we show that the n dimensional star graph is n 3 edge fault tolerant hamiltonian laceable n 3 edge fault tolerant strongly hamiltonian laceable and n 4 edge fault tolerant hyper hamiltonian laceable all these results are optimal in a sense described in this paper

the t k diagnosability of the bc graphs processor fault diagnosis takes an important role in fault tolerant computing on multiprocessor systems there are two classical diagnosis strategies the precise strategy and the pessimistic strategy both of which are based on the well known pmc diagnostic model nevertheless the degree of diagnosability of the system is limited under these two strategies a better method called the t k diagnosis strategy is proposed by somani and peleg in which the identified fault set is allowed to contain at most k fault free processors using this diagnosis strategy the degree of diagnosability of the hypercube increases greatly as the number of the fault free processors in the fault set increases we study the t k diagnosability of so called bc graphs that include hypercubes crossed cubes mobius cubes and twisted cubes etc we show that any n dimensional bc graph is t n k k diagnosable when n spl ges 4 and 0 spl les k spl les n where t n k k 1 n 1 2 k 1 k 2 1 therefore the crossed cube the mobius cube and the twisted cube all have the same t k diagnosability as the hypercube as a result the algorithms developed for diagnosis on the hypercube may also be used to diagnose multiprocessor systems whose network topologies are based on bc graphs

lee distance and topological properties of k ary n cubes in this paper we consider various topological properties of a k ary n cube q sub n sup k using lee distance we feel that lee distance is a natural metric for defining and studying a q sub n sup k after defining a q sub n sup k graph using lee distance we show how to find all disjoint paths between any two nodes given a sequence of radix k numbers a function mapping the sequence to a gray code sequence is presented and this function is used to generate a hamiltonian cycle embedding the graph of a mesh and the graph of a binary hypercube into the graph of a q sub n sup k is considered using a k ary gray code we show the embedding of a k n sub 1 spl times k n sub 2 spl times spl times k n sub m dimensional mesh into a q sub n sup k where n spl sigma sub i l sup m n sub i then using a single digit 4 ary reflective gray code we demonstrate embedding a q sub n into q sub n 2 sup 4 we look at how lee distance may be applied to the problem of resource placement in a q sub n sup k by using a lee distance error correcting code although the results in this paper are only preliminary lee distance error correcting codes have not been applied previously to this problem finally we consider how lee distance can be applied to message routing and single node broadcasting in a q sub n sup k in this section we present two single node broadcasting algorithms that are optimal when single port and multi port i o is used

understanding waveshrink variance and bias estimation summary donoho johnstone s waveshrink procedure has proved valuable for function estimation and nonparametric regression waveshrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise waveshrink has very broad asymptotic near optimality properties and achieves the optimal risk to within a factor of log n in this paper we derive computationally efficient formulae for computing the exact bias variance and l2 risk of waveshrink estimates in finite sample situations we use these formulae to understand the behaviour of waveshrink estimators construct approximate confidence intervals and bias estimates for waveshrink and compute ideal thresholds for a given function we show that hard shrinkage has smaller bias but larger variance than soft shrinkage and that significantly smaller thresholds should be used for soft shrinkage we also compute minimax thresholds for waveshrink estimators and demonstrate that the minimax thresholds can nearly achieve the ideal rank for a range of functions

embedding meshes into crossed cubes crossed cubes are important variants of hypercubes in this paper we consider embeddings of meshes in crossed cubes the major research findings in this paper are 1 for any integer n 1 a 2x2 n 1 mesh can be embedded in the n dimensional crossed cube with dilation 1 and expansion 1 2 for any integer n 4 two node disjoint 4x2 n 3 meshes can be embedded in the n dimensional crossed cube with dilation 1 and expansion 2 the obtained results are optimal in the sense that the dilations of the embeddings are 1 the embedding of the 2x2 n 1 mesh is also optimal in terms of expansion because it has the smallest expansion 1

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

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

hamiltonian paths and cycles with prescribed edges in the 3 ary n cube the k ary n cube has been one of the most popular interconnection networks for massively parallel systems given a set p of at most 2n 2 n 2 prescribed edges and two vertices u and v we show that the 3 ary n cube contains a hamiltonian path between u and v passing through all edges of p if and only if the subgraph induced by p consists of pairwise vertex disjoint paths none of them having u or v as internal vertices or both of them as end vertices as an immediate result the 3 ary n cube contains a hamiltonian cycle passing through a set p of at most 2n 1 prescribed edges if and only if the subgraph induced by p consists of pairwise vertex disjoint paths

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

mapping cycles and trees on wrap around butterfly graphs we give a new algebraic representation for the wrap around butterfly interconnection network this new representation is based on the direct product of groups and finite fields and allows an algebraic expression of the network connectivity the abstract algebraic tools may then be employed to explore the structural properties of the butterfly in this paper we exploit this model to map guest graphs on the butterfly in particular we provide designs of unit dilation mappings of all possible length cycles on butterflies we also map the largest possible binary trees on butterfly networks with a dilation 2 if the network degree is less than 16 3 if it is less than 32 and 4 if it is less than 64 this is a great improvement over previous results

hamiltonian path embeddings in conditional faulty k ary n cubes the class of k ary n cubes represents the most commonly used interconnection topology for distributed memory parallel systems a k ary n cube is bipartite if and only if k is even in this paper we consider the faulty k ary n cube with even k 4 and n 2 such that each vertex of the k ary n cube is incident with at least two healthy edges based on this requirement we prove that the k ary n cube contains a hamiltonian path joining every pair of vertices which are in different parts even if it has up to 4n 5 edge faults and this result is optimal

geodesic pancyclicity of twisted cubes the hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement an n dimensional twisted cube tq n is an important variation of the hypercube q n and preserves many of its desirable properties the problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years the geodesic cycle embedding problem is for any two distinct vertices to find all the possible lengths of cycles that include a shortest path joining them in this paper we prove that tq n is geodesic 2 pancyclic for each odd integer n 3 this result implies that tq n is edge pancyclic for each odd integer n 3 moreover tq nxk 2 is also demonstrated to be geodesic 4 pancyclic

model selection using wavelet decomposition and applications in this paper we discuss how to use wavelet decompositions to select a regression model the methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector we then apply the selection rule to specific problems including testing for pure white noise the power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression

the effect of dropout on the efficiency of d optimal designs of linear mixed models dropout is often encountered in longitudinal data optimal designs will usually not remain optimal in the presence of dropout in this paper we study d optimal designs for linear mixed models where dropout is encountered moreover we estimate the efficiency loss in cases where a d optimal design for complete data is chosen instead of that for data with dropout two types of monotonically decreasing response probability functions are investigated to describe dropout our results show that the location of d optimal design points for the dropout case will shift with respect to that for the complete and uncorrelated data case owing to this shift the information collected at the d optimal design points for the complete data case does not correspond to the smallest variance we show that the size of the displacement of the time points depends on the linear mixed model and that the efficiency loss is moderate

pancyclicity and bipancyclicity of conditional faulty folded hypercubes a graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order the pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network s topology can simulate rings of various lengths an n dimensional folded hypercube fq n is an attractive variant of an n dimensional hypercube q n that is obtained by establishing some extra edges between the vertices of q n fq n for any odd n is known to be bipartite in this paper we explore the pancyclicity and bipancyclicity of fq n for any fq n n 2 with at most 2n 3 faulty edges where each vertex is incident to at least two fault free edges we prove that there exists a fault free cycle of every even length from 4 to 2 n and when n 2 is even we prove there also exists a fault free cycle of every odd length from n 1 to 2 n 1 the result is optimal with respect to the number of faulty edges tolerated

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

embedding meshes into twisted cubes the n dimensional twisted cube tn n is a variation of the hypercube in this paper we study embedding of meshes into tn n we prove three major results in this paper 1 for any integer n 1 a 2x2 n 1 mesh can be embedded into tn n with dilation 1 and expansion 1 2 for any integer n 4 an mxk m 3 k 3 mesh cannot be embedded into tn n with dilation 1 3 for any integer n 4 two node disjoint 4x2 n 3 meshes can be embedded into tn n with dilation 2 and expansion 1

wavelet based data reduction techniques for process fault detection this article presents new data reduction methods based on the discrete wavelet transform to handle potentially large and complicated nonstationary data curves the methods minimize objective functions to balance the trade off between data reduction and modeling accuracy theoretic investigations provide the optimality of the methods and the large sample distribution of a closed form estimate of the thresholding parameter an upper bound of errors in signal approximation or estimation is derived based on evaluation studies with popular testing curves and real life datasets the proposed methods demonstrate their competitiveness with the existing engineering data compression and statistical data denoising methods for achieving the data reduction goals further experimentation with a tree based classification procedure for identifying process fault classes illustrates the potential of the data reduction tools extension of the engineering scalogram to the reduced size semiconductor fabrication data leads

some new topological properties of the triangular pyramid networks abstract the triangular pyramid proposed by razavi and sarbazi azad the triangular pyramid routing and topological properties information sciences 180 2010 2328 2339 is a hierarchical structure based on triangular meshes in this paper we show that the triangular pyramid shares some nice symmetry properties of the pyramid we also determine the connectivity of the triangular pyramid and prove that it is 1 fault tolerant vertex pancyclic

embedding graphs onto the supercube in this paper we consider the supercube a new interconnection network derived from the hypercube the supercube introduced by a sen 1989 has the same diameter and connectivity as a hypercube but can be realized for any number of nodes not only powers of 2 we study the supercube s ability to execute parallel programs using graph embedding techniques we show that complete binary trees and bidimensional meshes with a side length power of 2 are spanning subgraphs of the supercube we then prove that the supercube is hamiltonian and when the number of nodes is not a power of 2 it contains all cycles of length greater than 3 as subgraphs

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

a functional data analytic approach to signal discrimination motivated by specific problems involving radar range profiles we suggest techniques for real time discrimination in the context of signal analysis the key to our approach is to regard the signals as curves in the continuum and employ a functional data analytic fda method for dimension reduction based on the fda technique for principal coordinates analysis this has the advantage relative to competing methods such as canonical variates analysis of providing a signal approximation that is best possible in an l2 sense for a given dimension as a result it produces particularly good discrimination we explore the use of both nonparametric and gaussian based discriminators applied to the dimensionreduced data

hamiltonian paths passing through prescribed edges in balanced hypercubes abstract the balanced hypercube was proposed as a novel interconnection network for large scale parallel systems it is known that the balanced hypercube is edge bipancyclic given a set of edges p with p n 1 and two vertices x and y in different bipartite sets we show that there exists a hamiltonian path from x to y passing through p in b h n if and only if p is a linear forest and neither x nor y is an internal vertex of p and x and y are not end vertices of a component of p simultaneously consequently the balanced hypercube contains a hamiltonian cycle passing through a set p of at most n prescribed edges if and only if p is a linear forest moreover our result improves some known results

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

edge disjoint hamiltonian cycles in k ary n cubes and hypercubes solutions for decomposing a higher dimensional torus to edge disjoint lower dimensional tori in particular edge disjoint hamiltonian cycles are obtained based on the coding theory approach first lee distance gray codes in z sub k sup n are presented and then it is shown how these codes can directly be used to generate edge disjoint hamiltonian cycles in k ary n cubes further some new classes of binary gray codes are designed from these lee distance gray codes and using these new classes of binary gray codes edge disjoint hamiltonian cycles in hypercubes are generated

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

path embeddings in faulty 3 ary n cubes the class of k ary n cubes represents the most commonly used interconnection topology for distributed memory parallel systems in this paper we study the problem of embedding paths of various lengths into faulty 3 ary n cubes and prove that a faulty 3 ary n cube with f 2n 3 faulty vertices admits a path of every length from 2n 1 to 3 n f 1 connecting any two distinct healthy vertices this result is optimal with respect to the number of vertex faults tolerated

panconnectivity and pancyclicity of the 3 ary n cube network under the path restrictions the k ary n cube denoted by q n k is one of the most important interconnection networks for parallel computing in this paper we consider the problem of embedding paths and cycles into 3 ary n cubes under the path restrictions let p be a path in q n 3 we show that when v p 2n 3 there exists a path of any length from n 1 to v q n 3 p 1 between two arbitrary nodes in q n 3 p we also prove that when e p 2n 1 there exists a cycle of any length from e p n to v q n 3 in q n 3 passing through p our results are best possible in some sense

embedding binary trees into crossed cubes the recently introduced interconnection network crossed cube has attracted much attention in the parallel processing area due to its many attractive features like the ordinary hypercube the n dimensional crossed cube is a regular graph with 2 sup n vertices and n2 sup n 1 edges the diameter of the crossed cube is approximately half that of the ordinary hypercube these advantages of the crossed cube motivated the study of how well it can simulate other networks such as the complete binary tree we show that the 2 sup n 1 node complete binary tree can be embedded into the n dimensional crossed cube with dilation 1

wavelet shrinkage and generalized cross validation for image denoising we present a denoising method based on wavelets and generalized cross validation and apply these methods to image denoising we describe the method of modified wavelet reconstruction and show that the related shrinkage parameter vector can be chosen without prior knowledge of the noise variance by using the method of generalized cross validation by doing so we obtain an estimate of the shrinkage parameter vector and hence the image which is very close to the best achievable mean squared error result that given by complete knowledge of the underlying clean image

conditional edge fault hamiltonicity of augmented cubes the augmented cube is a variation of hypercubes it possesses many superior properties in this paper we show that for any n dimensional augmented cube n 3 with faulty edges up to 4n 8 in which each vertex is incident to at least two fault free edges there exists a fault free hamiltonian cycle our result is optimal with respect to the number of faulty edges tolerated

a note on hamiltonian paths and cycles with prescribed edges in the 3 ary n cube in wang et al 2011 the authors proved that the 3 ary n cube admits a hamiltonian path between u and v passing through every edges in a prescribed set p with at most 2 n 2 edges if the subgraph induced by p consists of pairwise vertex disjoint paths none of which has u or v as internal vertices or both of them as end vertices they proved the above results based on some lemmas including lemma 4 7 in the paper however the proof of case 2 4 of lemma 4 7 is incorrect in this note we will give a counterexample and provide a correct proof therefore the main result in the above paper is still correct

fault free hamiltonian cycles passing through a linear forest in ternary n cubes with faulty edges a linear forest in a graph is a subgraph each component of which is a path in this paper we investigate the existence of a hamiltonian cycle passing through a linear forest in a ternary n cube qn3 n 2 with faulty edges let f be a faulty edge set of qn3 and l be a prescribed linear forest in qn3 f if e l 2n 1 and f n e l 2 1 then there is a hamiltonian cycle passing through l in qn3 f