on the uniqueness of euclidean distance matrix completions the case of points in general position the euclidean distance matrix completion problem edmcp is the problem of determining whether or not a given partial matrix can be completed into a euclidean distance matrix edm in this paper we investigate the necessary and sufficient conditions for the uniqueness of a given edm completion in the case where this edm completion is generated by points in general position we also show that the problem of checking the validity of these conditions can be formulated as a semidefinite programming problem

non parametric analysis of covariance for confirmatory randomized clinical trials to evaluate dose response relationships in confirmatory randomized clinical trials that are designed to compare multiple doses of a test treatment with a control group and with one another there are often statistical issues regarding compound hypotheses and multiple comparisons which need to be considered in most cases the analysis plan needs a clear specification for the proposed order for conducting statistical tests or for managing the overall significance level which statistical methods will be used and whether adjustment for covariates will be performed there are several benefits of specifying non parametric analysis of covariance ancova for performing the primary confirmatory analyses only minimal assumptions are needed beyond randomization in the study design whereas regression model based methods have assumptions about model fit for which departures may require modifications that are incompatible with a fully prespecified analysis plan non parametric methods provide traditionally expected results of ancova namely a typically small adjustment to the estimate for a treatment comparison so as to account for random imbalance of covariates between treatment groups and variance reduction for this estimate when covariates are strongly correlated with the response of interest the application of non parametric ancova is illustrated for two randomized clinical trials the first has a 3 x 4 factorial response surface design for the comparison of 12 treatments that is combinations of three doses of one drug and four doses of a second drug for change in blood pressure and the second example addresses the comparison of three doses of test treatment and placebo for time to disease progression this clinical trial has comparisons among treatments made for a dichotomous criterion wilcoxon rank scores and averages of cumulative survival rates in each example the non parametric covariance method provides variance reduction relative to its unadjusted counterpart

large scale molecular optimization from distance matrices by a d c optimization approach a so called dca method based on a d c difference of convex functions optimization approach algorithm for solving large scale distance geometry problems is developed different formulations of equivalent d c programs in the l 1 approach are stated via the lagrangian duality without gap relative to d c programming and new nonstandard nonsmooth reformulations in the l infty approach resp the l 1 l infty approach are introduced substantial subdifferential calculations permit us to compute sequences of iterations in the dca quite simply the computations actually require matrix vector products and only one cholesky factorization resp with an additional solution of a convex program in the l 1 approach resp the l 1 l infty approach and allow the exploitation of sparsity in the large scale setting two techniques respectively using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix and the spanning trees procedure are investigated in order to compute a good starting point for the dca finally many numerical simulations of the molecular optimization problems with up to 12567 variables are reported which prove the practical usefulness of the nonstandard nonsmooth reformulations the globality of found solutions and the robustness and efficiency of our algorithms

further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

the kirchhoff indices and the matching numbers of unicyclic graphs the kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph in this paper we determine the minimum kirchhoff index among the unicyclic graphs with fixed number of vertices and matching number and characterize the extremal graphs

on the removal of skewness by transformation summary we suggest empirical transformations for removing most of the skewness of an asymmetric statistic these transformations may be used as the basis for accurate confidence procedures or hypothesis tests and can be employed in conjunction with either a normal approximation or the bootstrap our approach differs from those of other researchers in that the transformation is monotone and invertible in this paper we suggest elementary but general transformations for eliminating skewness from the distribution of a studentized statistic the transformations are monotone and have simple explicit inversion formulae and so are readily applied to confidence interval problems when the transformations are used in conjunction with the bootstrap they can produce confidence intervals with particularly low levels of coverage error that confidence intervals for symmetric statistics have lower coverage error than their counterparts in the case of asymmetry may be deduced from edgeworth expansions the first term in an expansion of size n 1 2 where n is the sample size describes the error in the usual normal approximation and is due entirely to skewness if it can be eliminated for example by transformation then the normal approxima tion will be in error by only o n 1 thus our method involves transforming one statistic into another where the distribution is virtually symmetric applying the normal approximation or the bootstrap to the new statistic and then regaining the asymmetry of the original problem by inverse transformation at the same time retain ing the high coverage accuracy conferred by applying confidence interval procedures to a symmetric statistic

consistency of spectral clustering consistency is a key property of all statistical procedures analyzing randomly sampled data surprisingly despite decades of work little is known about consistency of most clustering algorithms in this paper we investigate consistency of the popular family of spectral clustering algorithms which clusters the data with the help of eigenvectors of graph laplacian matrices we develop new methods to establish that for increasing sample size those eigenvectors converge to the eigenvectors of certain limit operators as a result we can prove that one of the two major classes of spectral clustering normalized clustering converges under very general conditions while the other unnormalized clustering is only consistent under strong additional assumptions which are not always satisfied in real data we conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering

characterizing generic global rigidity a d dimensional framework is a graph and a map from its vertices to ed such a framework is globally rigid if it is the only framework in ed with the same graph and edge lengths up to rigid motions for which underlying graphs is a generic framework globally rigid we answer this question by proving a conjecture by connelly that his sucient condition is also necessary the condition comes from considering the geometry of the length squared mapping essentially the graph is generically locally rigid i the rank of is maximal and it is generically globally rigid i the rank of the gauss map on the image of is maximal we also show that this condition is eciently checkable with a randomized algorithm

multi arm trials with multiple primary endpoints and missing values we present an extension of multiple contrast tests for multiple endpoints to the case of missing values the endpoints are assumed to be normally distributed and correlated and to have equal covariance matrices for the different treatments different multivariate t distributions will be applied differing in endpoint specific degrees of freedom in contrast to competing methods the familywise error type i is maintained in the strong sense in an admissible range and the problem of different marginal errors type i is avoided the information of all observations is exploited thereby enabling a gain in power compared with a complete case analysis

optimal data collection for informative rankings expose well connected graphs given a graph where vertices represent alternatives and arcs represent pairwise comparison data the statistical ranking problem is to find a potential function defined on the vertices such that the gradient of the potential function agrees with the pairwise comparisons our goal in this paper is to develop a method for collecting data for which the least squares estimator for the ranking problem has maximal fisher information our approach based on experimental design is to view data collection as a bi level optimization problem where the inner problem is the ranking problem and the outer problem is to identify data which maximizes the informativeness of the ranking under certain assumptions the data collection problem decouples reducing to a problem of finding multigraphs with large algebraic connectivity this reduction of the data collection problem to graph theoretic questions is one of the primary contributions of this work as an application we study the yahoo movie user rating data set and demonstrate that the addition of a small number of well chosen pairwise comparisons can significantly increase the fisher informativeness of the ranking as another application we study the 2011 12 ncaa football schedule and propose schedules with the same number of games which are significantly more informative using spectral clustering methods to identify highly connected communities within the division we argue that the ncaa could improve its notoriously poor rankings by simply scheduling more out of conference games

some characterizations of symmetric inverse m matrices abstract we apply the previously proved equivalence of four models to present new characterizations of matrices whose inverse is a weakly diagonally dominant symmetric m matrix it is shown that the case of strictly ultrametric matrices of martinez michon and san martin is included as a corollary

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

parametrization of approximate algebraic curves by lines it is well known that irreducible algebraic plane curves having a singularity of maximum multiplicity are rational and can be parametrized by lines in this paper given a tolerance e 0 and an e irreducible algebraic plane curve l of degree d having an e singularity of multiplicity d 1 we provide an algorithm that computes a proper parametrization of a rational curve that is exactly parametrizable by lines furthermore the error analysis shows that under certain initial conditions that ensures that points are projectively well defined the output curve lies within the offset region of l at distance at most 2 2e1 2d exp 2

a new approach for computing the asymptotes of a parametric curve abstract in this paper we summarize two algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined the approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers of the same authors from these results we derive a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic curve parametrically defined in n dimensional space

communicability angle and the spatial efficiency of networks we introduce the concept of communicability angle between a pair of nodes in a graph we provide strong analytical and empirical evidence that the average communicability angle for a given network accounts for its spatial efficiency on the basis of the communications among the nodes in a network we determine characteristics of the spatial efficiency of more than a hundred real world networks that represent complex systems arising in a diverse set of scenarios in particular we find that the communicability angle correlates very well with the experimentally measured value of the relative packing efficiency of proteins that are represented as residue networks we finally show how we can modulate the spatial efficiency of a network by tuning the weights of the edges of the networks this allows us to predict the effects of external stresses on the spatial efficiency of a network as well as to design strategies to improve important parameters in real world systems

constructing multiple test procedures for partially ordered hypothesis sets a popular method to control multiplicity in confirmatory clinical trials is to use a so called hierarchical or fixed sequence test procedure this requires that the null hypotheses are ordered a priori for example in order of clinical importance the procedure tests the hypotheses in this order using level tests until one is not rejected it then stops so that no subsequent hypotheses are eligible for rejection this procedure strongly controls the familywise error rate fwe that is to say the probability that any true hypotheses are rejected this paper describes a simple generalization of this approach in which the null hypotheses are partially ordered it is convenient to display the partial ordering in a directed acyclic graph dag we consider sequentially rejective procedures based on the partial ordering in which a hypothesis is tested only when all preceding hypotheses have been tested and rejected in general such procedures do not control the fwe but it is shown that when certain intersection hypotheses are added strong control of the fwe is obtained the purpose of the method is to construct inference strategies for confirmatory clinical trials that better reflect the trial objectives copyright 2007 john wiley sons ltd

approximation of discrete sequential boundaries on obtient des limites conservatives critiques pour des bornes sequentielles discretes lorsque le nombre d instants de decision et de tailles de groupes n est pas fixe d avance

graph realizations associated with minimizing the maximum eigenvalue of the laplacian in analogy to the absolute algebraic connectivity of fiedler we study the problem of minimizing the maximum eigenvalue of the laplacian of a graph by redistributing the edge weights via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one we prove three main results for a slightly generalized form of this embedding problem first given a set of vertices partitioning the graph into several or just one part the barycenter of each part is embedded on the same side of the affine hull of the set as the origin second there is an optimal realization of dimension at most the tree width of the graph plus one and this bound is best possible in general finally bipartite graphs possess a one dimensional optimal embedding

constructing confidence intervals for cost effectiveness ratios an evaluation of parametric and non parametric techniques using monte carlo simulation the statistic of interest in most health economic evaluations is the incremental cost effectiveness ratio since the variance of a ratio estimator is intractable the health economics literature has suggested a number of alternative approaches to estimating confidence intervals for the cost effectiveness ratio in this paper monte carlo simulation techniques are employed to address the question of which of the proposed methods is most appropriate by repeatedly sampling from a known distribution and applying the different methods of confidence interval estimation it is possible to calculate the coverage properties of each method to see if these correspond to the chosen confidence level as the results of a single monte carlo experiment would be valid only for that particular set of circumstances a series of experiments was conducted in order to examine the performance of the different methods under a variety of conditions relating to the sample size the coefficient of variation of the numerator and denominator of the ratio and the covariance between costs and effects in the underlying data response surface analysis was used to analyse the results and substantial differences between the different methods of confidence interval estimation were identified the methods both parametric and non parametric which assume a normal sampling distribution performed poorly as did the approach based on simply combining the separate intervals on costs and effects the choice of method for confidence interval estimation can lead to large differences in the estimated confidence limits for cost effectiveness ratios the importance of such differences is an empirical question and will depend to a large extent on the role of hypothesis testing in economic appraisal however where it is suspected that the sampling distribution is skewed normal approximation methods produce particularly poor results and should be avoided

stochastic stability of continuous time consensus protocols a unified approach to studying convergence and stochastic stability of continuous time consensus protocols cps is presented in this work our method applies to networks with directed information flow both cooperative and noncooperative interactions networks under weak stochastic forcing and those whose topology and strength of connections may vary in time the graph theoretic interpretation of the analytical results is emphasized we show how the spectral properties such as algebraic connectivity and total effective resistance as well as the geometric properties such as the dimension and the structure of the cycle subspace of the underlying graph shape stability of the corresponding cps in addition we explore certain implications of spectral graph theory to cp design in particular we point out that expanders sparse highly connected graphs generate cps whose performance remains uniformly high when the size of the network grows unboundedly similarly we highlight the benefits of using random

on graphs with algebraic connectivity equal to minimum edge density abstract given an undirected graph g v e and a nonempty subset x v the edge density of x is given by x v ex x v x where ex is the set of all edges with one end in x and the other end in v x it is known that the algebraic connectivity of g denoted by a g satisfies a g minx v x in this paper we study the graphs g for which equality holds in the above inequality

resistance distances in extended or contracted networks abstract in this paper we introduce new effective resistances on a given network associated with a positive value and two weights on the vertex set and investigate under which conditions they determine a distance we prove that this property is closely related with superharmonicity moreover we analyze the behavior of these distances under the usual network transformations specially the so called star mesh transformation we also compute the effective resistance for an extended network that is the network obtained from the former one by joining a new vertex and then study the effect of the contraction of this new network that is we apply a star mesh transformation with center in the joined vertex

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

modified sequentially rejective multiple test procedures abstract suppose that n hypotheses h 1 h 2 h n with associated test statistics t 1 t 2 t n are to be tested by a procedure with experimentwise significance level the probability of rejecting one or more true hypotheses smaller than or equal to some specified value a commonly used procedure satisfying this condition is the bonferroni b procedure which consists of rejecting h i for any i iff the associated test statistic t i is significant at the level n holm 1979 introduced a modified bonferroni procedure with greater power than the b procedure under holm s sequentially rejective bonferroni srb procedure if any hypothesis is rejected at the level n the denominator of for the next test is n 1 and the criterion continues to be modified in a stagewise manner with the denominator of reduced by 1 each time a hypothesis is rejected so that tests can be conducted at successively higher significance levels holm proved that the experimentwise significance level

on closed test procedures for dose response analysis this paper concerns the testing of a dose response effect in medical studies we consider two situations the first is when the drug effect on a parameter of interest is likely to increase or decrease with increasing doses we propose a procedure based on the closure principle of peritz combined with an application of a closed testing procedure of marcus et al to a test proposed by turkey et al the second situation is the analysis of a drug effect where one might observe a reversal at higher doses for this situation we propose simultaneous examination of contrasts among the doses at each of several stages of the testing scheme we calculate critical values for this procedure incorporating the correlation structure among the contrasts both procedures strongly control the family wise error rate at the pre determined alpha level they also provide information about the shape of the dose response lacking in other commonly used procedures for testing against an ordered alternative hypothesis we illustrate the procedures on two datasets

the communicability distance in graphs let g be a simple connected graph with adjacency matrix a the communicabilitygpq between two nodes p and q of the graph is defined as the pq entry of g exp a we prove here that p q gpp gqq 2gpq 1 2 is a euclidean distance and give expressions for it in paths cycles stars and complete graphs with n nodes the sum of all communicabilitydistances in a graph is introduced as a new graph invariant g we compare this index with the wiener and kirchhoff indices of graphs and conjecture about the graphs with maximum and minimum values of this index

an sdp based divide and conquer algorithm for large scale noisy anchor free graph realization we propose the disco algorithm for graph realization in mathbb r d given sparse and noisy short range intervertex distances as inputs our divide and conquer algorithm works as follows when a group has a sufficiently small number of vertices the basis step is to form a graph realization by solving a semidefinite program the recursive step is to break a large group of vertices into two smaller groups with overlapping vertices these two groups are solved recursively and the subconfigurations are stitched together using the overlapping atoms to form a configuration for the larger group at intermediate stages the configurations are improved by gradient descent refinement the algorithm is applied to the problem of determining protein moleculer structure tests are performed on molecules taken from the protein data bank database for each molecule given 20 30 of the inter atom distances less than 6a that are corrupted by a high level of noise disco is able to reliably and efficiently reconstruct the conformation of large molecules in particular given 30 of distances with 20 multiplicative noise a 13000 atom conformation problem is solved within an hour with a root mean square deviation of 1 6a

kirchhoff indexes of a network in this work we define the effective resistance between any pair of vertices with respect to a value 0 and a weight on the vertex set this allows us to consider a generalization of the kirchhoff index of a finite network it turns out that is the lowest eigenvalue of a suitable semi definite positive schrodinger operator and is the associated eigenfunction we obtain the relation between the effective resistance and hence between the kirchhoff index with respect to and and the eigenvalues of the associated schrodinger operator however our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network from these expressions we derive a full generalization of foster s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk finally we compute the effective resistances and the generalized kirchhoff index with respect to a and for some families of networks with symmetries specifically for weighted wagon wheels and circular ladders

on minimizing the spectral width of graph laplacians and associated graph realizations extremal eigenvalues and eigenvectors of the laplace matrix of a graph form the core of many bounds on graph parameters and graph optimization problems for example the value of a uniform sparsest cut is bounded from below and above by the second smallest and the largest eigenvalue of the weighted laplacian divided by the number of nodes of the graph minimizing the difference between maximum and second smallest eigenvalue over edge weighted laplacians of a graph reduces the size of this interval we study this problem of minimizing the spectral width for its own sake with the goal of advancing the understanding of connections between structural properties of the graph and the corresponding eigenvectors and eigenvalues building on previous work where these eigenvalues were investigated separately we show that a corresponding dual problem allows us to view eigenvectors to optimized eigenvalues as graph realizations in euclidean space whose structure is tightly linked to the separator structure of the g

the molecule problem exploiting structure in global optimization the molecule problem is that of determining the relative locations of a set of objects in euclidean space relying only upon a sparse set of pairwise distance measurements this np hard problem has applications in the determination of molecular conformation the molecule problem can be naturally expressed as a continuous global optimization problem but it also has a rich combinatorial structure this paper investigates how that structure can be exploited to simplify the optimization problem in particular we present a novel divide and conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones since the cost of the optimization can grow exponentially with problem size this approach holds the promise of a substantial improvement in performance our algorithmic development relies upon some recently published results in graph theory we describe an implementation of this algorithm and report some results of its performance on a sample molecule

a stagewise rejective multiple test procedure based on a modified bonferroni test summary simes 1986 has proposed a modified bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses in contrast to the classical bonferroni procedure it is not obvious how statements about individual hypotheses are to be made for this procedure in the present paper a multiple test procedure allowing statements on individual hypotheses is proposed it is based on the principle of closed test procedures marcus peritz gabriel 1976 and controls the multiple level a

a note on multiple testing procedures in dose finding it is shown that in some recently proposed multiple test procedures for dose finding tamhane hochberg and dunnett 1996 biometrics 52 21 37 strong control of the familywise error is only guaranteed if order relations are assumed to hold among the means procedures that can be applied safely without assuming any restrictions on the means can be based on the classical many one comparisons of doses with the zero dose

asymptotes of space curves in blasco and perez diaz 2014 see 3 a method for computing generalized asymptotes of a real algebraic plane curve implicitly defined is presented generalized asymptotes are curves that describe the status of a branch at points with sufficiently large coordinates and thus it is an important tool to analyze the behavior at infinity of an algebraic curve this motivates that in this paper we analyze and compute the generalized asymptotes of a real algebraic space curve which could be parametrically or implicitly defined we present an algorithm that is based on the computation of the infinity branches this concept was already introduced for plane curves in blasco and perez diaz 2014 1 in particular we show that the computation of infinity branches in the space can be reduced to the computation of infinity branches in the plane and thus the methods in blasco and perez diaz 2014 see 1 can be applied

multiple test procedures in clinical dose finding studies abstract test procedures in clinical dose finding studies are considered when it can be assumed that increasing dose levels lead to a larger or at least equal efficacy if one is interested in various comparisons within the set of dose levels one arrives at a problem of multiple testing against ordered alternatives some multiple test procedures controlling the multiple level of significance are compared in a simulation study it follows that a simple procedure that only uses comparisons between neighboring dose levels can be recommended the degree of approximation of this bonferroni type procedure is investigated under normal assumptions

extended maxt tests of one sided hypotheses in many statistical applications of one sided tests of multiple hypotheses researchers are often concerned not only with global tests of the intersection of individual hypotheses but also with multiple tests of individual hypotheses for example in clinical trial studies researchers often need to find out the efficacy of a treatment as well as the significance of each outcome measurement endpoint of the treatment this article proposes maxt type tests aiming at improving the global power of existing maxt tests our extended maxt tests are constructed by adding an extra component to the maximand set of existing maxt tests the added component is a weighted sum of other components some power properties relating to choices of weight are studied our simulation study shows that the proposed tests can considerably improve the global power of existing maxt tests and can also outperform many other global tests under some alternatives and or some nonnormal distributions furthermore it is shown that such g

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

embedded in the shadow of the separator eigenvectors to the second smallest eigenvalue of the laplace matrix of a graph also known as fiedler vectors are the basic ingredient in spectral graph partitioning heuristics maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by fiedler who proved tight connections of this value to the connectivity of the graph our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity by exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the n nodes of the graph in n space so that their barycenter is the origin the distance between adjacent nodes is bounded by one and the nodes are spread as much as possible the sum of the squared norms is maximized for connected graphs we prove that for any separator in the graph at least one of the two separated node sets is embedded in the shadow with the origin being the light source of the convex hull of the separator furthermore we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one

characterizing the finiteness of the hausdorff distance between two algebraic curves in this paper we present a characterization for the hausdorff distance between two given algebraic curves in the n dimensional space parametrically or implicitly defined to be finite the characterization is related with the asymptotic behavior of the two curves and it can be easily checked more precisely the hausdorff distance between two curves c and c is finite if and only if for each infinity branch of c there exists an infinity branch of c such that the terms with positive exponent in the corresponding series are the same and reciprocally

designing optimal multiplex networks for certain laplacian spectral properties we discuss the design of interlayer edges in a multiplex network under a limited budget with the goal of improving its overall performance we analyze the following three problems separately first we maximize the smallest nonzero eigenvalue also known as the algebraic connectivity second we minimize the largest eigenvalue also known as the spectral radius and finally we minimize the spectral width maximizing the algebraic connectivity requires identical weights on the interlayer edges for budgets less than a threshold value however for larger budgets the optimal weights are generally nonuniform the dual formulation transforms the problem into a graph realization embedding problem that allows us to give a fuller picture namely before the threshold budget the optimal realization is one dimensional with nodes in the same layer embedded to a single point while beyond the threshold the optimal embeddings generally unfold into spaces with dimension bounded by the multiplicity of the algebraic connectivity finally for extremely large budgets the embeddings again revert to lower dimensions minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector before a threshold the total budget is distributed among interlayer edges corresponding to the nodal lines of this eigenvector and the optimal largest eigenvalue of the laplacian remains constant for larger budgets the weight distribution tends to be almost uniform in the dual picture the optimal graph embedding is one dimensional and nonhomogeneous at first with the nodes corresponding to the layer with the largest spectral radius distributed on a line according to its eigenvector while the other layer is embedded at the origin beyond this threshold the optimal embedding expands to be multidimensional and for larger values of the budget the two layers fill the embedding space finally we show how these two problems are connected to minimizing the spectral width

conditions for unique graph realizations the graph realization problem is that of computing the relative locations of a set of vertices placed in euclidean space relying only upon some set of inter vertex distance measurements this paper is concerned with the closely related problem of determining whether or not a graph has a unique realization both these problems are np hard but the proofs rely upon special combinations of edge lengths if one assumes the vertex locations are unrelated then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths this paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension efficient sequential and nc algorithms are presented for each condition although these algorithms have very different flavors in different dimensions

multiple comparisons by rerandomization tests abstract classical methods of simultaneous inference cannot be used with rerandomization tests without risking excessive type i errors this is illustrated by examples that show some published methods to be invalid for multiple comparisons however the use of stepwise procedures is shown to remain valid with rerandomization tests in particular this article advocates an adaptation of begun and gabriel s 1981 technique and provides the algorithm for using it

a geometric confidence ellipse approach to the estimation of the ratio of two variables one is often interested in the ratio of two variables for example in genetics assessing drug effectiveness and in health economics in this paper we derive an explicit geometric solution to the general problem of identifying the two tangents from an arbitrary external point to an ellipse this solution permits numerical integration of a bivariate normal distribution over a wedge shaped region bounded by the tangents which yields an evaluation of the tangent slopes as confidence limits on the ratio of the component variables after suitable adjustment of the confidence coverage of the ellipse these confidence limits are shown to be equivalent to those from fieller s method however the geometric approach allows additional interpretation of the data through identification of the points of tangency the ellipse itself and expressions for the coverage probability of the confidence interval numerical evaluations using the theoretical expressions for the geometric confidence intervals but ignoring sample variation in the underlying parameters suggested that they perform well overall and are slightly conservative simulations that do take account of sample variation in the underlying parameters again suggested that the intervals perform well overall although here they are slightly anti conservative coverage probabilities for the confidence intervals were only weakly dependent on the distance and correlation of the ellipse but there were asymmetries in the failure rates of the upper and lower confidence limits in some configurations the probability of no real solution existing was also evaluated these ideas are illustrated by a practical example

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

graph weight allocation to meet laplacian spectral constraints we adjust the node and edge weightings of graphs using convex optimization to impose bounds on their laplacian spectra first we derive necessary and sufficient conditions that characterize the feasibility of spectral bounds given positive node and edge weightings synthesizing these conditions leads naturally to algorithms that exploit convexity to achieve several eigenvalue bounds simultaneously the algorithms we propose apply to many graph design problems as well as multi agent systems control finally we suggest efficient ways to accommodate larger graphs and show that dual formulations lead to substantial improvement in the size of graphs that can be addressed

bounds on the laplacian spectral radius of graphs the spectral radius of a graph is the largest eigenvalue of adjacency matrix of the graph and its laplacian spectral radius is the largest eigenvalue of the laplacian matrix which is the difference of the diagonal matrix of vertex degrees and the adjacency matrix some sharp bounds are obtained for the laplacian spectral radii of connected graphs as consequences some sharp upper bounds of the nordhaus gaddum type are also obtained for the sum of laplacian spectral radii of a connected graph and its connected complement

distance geometry algorithms in molecular modelling of polymer and composite systems abstract in this paper microscopic atomistically detailed models of polymer and composite systems are studied in the context of distance geometry algorithms the development of such algorithms in the general framework of molecular modelling is closely associated with problems in computational optimisation and is pursued here as a key to a better understanding of properties of polymer and composite systems at the macroscopic level applications of two distance geometry algorithms to topological optimisation of alkane chains and bulk decane structures are discussed with results of numerical simulations

euclidean distance geometry and applications euclidean distance geometry is the study of euclidean geometry based on the concept of distance this is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in euclidean space realizing those given distances we survey the theory of euclidean distance geometry and its most important applications with special emphasis on molecular conformation problems

multiple testing in clinical trials the basic ideas of multiple testing are outlined and the problem of how to control the probability of erroneous decisions is discussed the main emphasis is on the concept of the multiple level of significance controlling the experiment or family error in the strong sense which can be achieved by applying the principle of closed tests various practical situations encountered in multiple testing in clinical trials are considered more than one end point more than two treatments such as comparisons with a single control comparisons using ordered alternatives all pairwise comparisons and contrast methods and more than one trial tests based on global statistics the union intersection principle and other criteria are discussed the application of the multiple test concept in sequential sampling is investigated finally some comments are made on multiple power multiple confidence intervals and directed decisions

evaluation of experiments with adaptive interim analyses summary a general method for statistical testing in experiments with an adaptive interim analysis is proposed the method is based on the observed error probabilities from the disjoint subsamples before and after the interim analysis formally an intersection of individual null hypotheses is tested by combining the twop values into a global test statistic stopping rules for fisher s product criterion in terms of critical limits for the p value in the first subsample are introduced including early stopping in the case of missing effects the control of qualitative treatment stage interactions is considered a generalization to three stages is outlined the loss of power when using the product criterion instead of the optimal classical test on the whole sample is calculated for the test of the mean of a normal distribution depending on increasing proportions of the first subsample in relation to the total sample size an upper bound on the loss of power due to early stopping is derived a general example is presented and rules for assessing the sample size in the second stage of the trial are given the problems of interpretation and precautions to be taken for applications are discussed finally the sources of bias for estimation in such designs are described

on the power of tests for multiple comparison of three normal means abstract this article gives theoretical reasons for the findings of a simulation study by shaffer 1981 two well established procedures for multiple comparison of three normal means are compared an f procedure and an r procedure it turns out that for a large set of parameters the average number of rejected pairwise hypotheses is larger when the f procedure is applied to achieve our result we have to prove a special instance of a conjecture by david lachenbruch and brandis 1972

a comparison of two modified bonferroni procedures summary it is shown that hommel s 1988 modified bonferroni procedure is at least as powerful as hochberg s 1988 procedure and in general more powerful

lra local rigid averaging of stretchable non rigid shapes we present a novel algorithm for generating the mean structure of non rigid stretchable shapes following an alignment process which supports local affine deformations we translate the search of the mean shape into a diagonalization problem where the structure is hidden within the kernel of a matrix this is the first step required in many practical applications where one needs to model bendable and stretchable shapes from multiple observations

multiple tests of cost effectiveness angles cost effectiveness angles are an attractive measure of performance when comparing effects and costs of health care therapies because they have a clear interpretation and are well suited for statistical inference in clinical trials a common setup is the comparison of multiple new therapies with a single control if cost effectiveness angles are calculated for each comparison multiplicity issues should be taken into account when quantifying uncertainty of the point estimates therefore this paper proposes a parametric test for multiple cost effectiveness angles that guarantees strong family wise error rate control the idea is to replace the test of m cost effectiveness angles as a union intersection test of 3m linear hypotheses considering the correlation structure of the individual test statistics for the linear hypotheses leads to a maximum type test for the intersection hypothesis inverting these test decisions then gives simultaneous cis of cost effectiveness angles with the appropriate coverage probabilities

computing the asymptotes for a real plane algebraic curve the purpose of this paper is to present an algorithm for computing all the asymptotes of a real plane algebraic curve by this algorithm all the asymptotes of a real plane algebraic curve may be represented via polynomial real root isolation

the fastest mixing markov process on a graph and a connection to a maximum variance unfolding problem we consider a markov process on a connected graph with edges labeled with transition rates between the adjacent vertices the distribution of the markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue lambda 2 of the laplacian of the weighted graph in this paper we consider the problem of assigning transition rates to the edges so as to maximize lambda 2 subject to a linear constraint on the rates this is the problem of finding the fastest mixing markov process fmmp on the graph we show that the fmmp problem is a convex optimization problem which can in turn be expressed as a semidefinite program and therefore effectively solved numerically we formulate a dual of the fmmp problem and show that it has a natural geometric interpretation as a maximum variance unfolding mvu problem the problem of choosing a set of points to be as far apart as possible measured by their variance while respecting local distance constraints this mvu problem is closely related to a problem recently proposed by weinberger and saul as a method for unfolding high dimensional data that lies on a low dimensional manifold the duality between the fmmp and mvu problems sheds light on both problems and allows us to characterize and in some cases find optimal solutions

nonparametric statistical methods for cost effectiveness analyses two measures often used in a cost effectiveness analysis are the incremental cost effectiveness ratio icer and the net health benefit nhb inferences on these two quantities are often hindered by highly skewed cost data in this article we derive the edgeworth expansions for the studentized t statistics for the two measures and show how they could be used to guide inferences in particular we use the expansions to study the theoretical performance of existing confidence intervals based on normal theory and to derive new confidence intervals for the icer and the nhb we conduct a simulation study to compare our new intervals with several existing methods the methods evaluated include taylor s interval fieller s interval the bootstrap percentile interval and the bootstrap bias corrected acceleration interval we found that our new intervals give good coverage accuracy and are narrower compared to the current recommended intervals

average resistance of toroidal graphs abstract the average effective resistance of a graph is a relevant performance index in many applications including distributed estimation and control of network systems in this paper we study how the average resistance depends on the graph topology and specifically on the dimension of the graph we concentrate on d dimensional toroidal grids and we exploit the connection between resistance and laplacian eigenvalues our analysis provides tight estimates of the average resistance which are key to study its asymptotic behavior when the number of nodes grows to infinity in dimension two the average resistance diverges in this case we are able to capture its rate of growth when the sides of the grid grow at different rates in higher dimensions the average resistance is bounded uniformly in the number of nodes in this case we conjecture that its value is of order 1 d for large d we prove this fact for hypercubes and when the side lengths go to infinity

characterization of symmetric m matrices as resistive inverses abstract we aim here at characterizing those nonnegative matrices whose inverse is an irreducible stieltjes matrix specifically we prove that any irreducible stieltjes matrix is a resistive inverse to do this we consider the network defined by the off diagonal entries of the matrix and we identify the matrix with a positive definite schrodinger operator whose ground state is determined by the lowest eigenvalue of the matrix and the corresponding positive eigenvector we also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric m matrices

on the uniqueness of euclidean distance matrix completions the euclidean distance matrix completion problem edmcp is the problem of determining whether or not a given partial matrix can be completed into a euclidean distance matrix in this paper we present necessary and sufficient conditions for a given solution of the edmcp to be unique

a symbolic numerical approach to approximate parameterizations of space curves using graphs of critical points a simple algorithm for computing an approximate parameterization of real space algebraic curves using their graphs of critical points is designed and studied in this paper the first step is determining a suitable space graph which contains all critical points of a real algebraic space curve c implicitly defined as the complete intersection of two surfaces the construction of this graph is based on one projection of c in a general position onto an xy plane and on an intentional choice of vertices the second part of the designed method is a computation of a spline curve which replaces the edges of the constructed graph by segments of a chosen free form curve this step is formulated as an optimization problem when the objective function approximates the integral of the squared euclidean distance of the constructed approximate curve to the intersection curve the presented method based on combining symbolic and numerical steps to the approximation problem provides approximate parameterizations of space algebraic curves from a small number of approximating arcs it may serve as a first step to several problems originating in technical practice where approximation curve parameterizations are needed

sample size determination for multiple comparison studies treating confidence interval width as random methods for optimal sample size determination are developed using four popular multiple comparison procedures scheffe s bonferroni s tukey s and dunnett s procedures where random samples of the same size n are to be selected from k 2 normal populations with common variance sigma2 and where primary interest concerns inferences about a family of l linear contrasts among the k population means for a simultaneous coverage probability of 1 alpha the optimal sample size is defined to be the smallest integer value n m such that simultaneously for all l confidence intervals the width of the lth confidence interval will be no greater than tolerance 2deltal l 1 2 l with tolerance probability at least 1 gamma treating the pooled sample variance s2p as a random variable using scheffe s procedure as an illustration comparisons are made to usual sample size methods that incorrectly ignore the stochastic nature of s2p the latter approach can lead to serious underestimation of required sample sizes and hence to unacceptably low values of the actually tolerance probability 1 gamma our approach guarantees a lower bound of 1 alpha gamma for the probability that the l confidence intervals will both cover the parametric functions of interest and also be sufficiently narrow recommendations are provided regarding the choices among the four multiple comparison procedures for sample size determination and inference making

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

some new inequalities for the range distribution with application to the determination of optimum significance levels of multiple range tests abstract a problem arising in multiple range tests is the monotonicity of the defining critical values which ws first treated by lehmann and shaffer 1977 this article proves a general inequality for the distribution function of the range which guarantees the monotonicity of a special set of critical values it also discusses the relation to some other inequalities the importance of these inequalities is pointed out in considering some concepts of admissibility and optimality of multiple range tests

sequential multiple comparisons with the best abstract in this article we give a method of designing and analyzing an experiment for comparing k 2 treatments such that a a treatment is terminated as soon as there is statistical evidence that it is not the best treatment and b at the conclusion of the experiment one can assert at some prespecified confidence level simultaneously that those terminated treatments are not the best and those remaining treatments are respectively within certain small distances of the unknown best treatment the inference given by our method is stronger than existing sequential elimination selection inference

a new graph parameter related to bounded rank positive semidefinite matrix completions the gram dimension mathrm gd g of a graph g is the smallest integer k ge 1 such that any partial real symmetric matrix whose entries are specified on the diagonal and at the off diagonal positions corresponding to edges of g can be completed to a positive semidefinite matrix of rank at most k assuming a positive semidefinite completion exists for any fixed k the class of graphs satisfying mathrm gd g le k is minor closed hence it can be characterized by a finite list of forbidden minors we show that the only minimal forbidden minor is k k 1 for k le 3 and that there are two minimal forbidden minors k 5 and k 2 2 2 for k 4 we also show some close connections to euclidean realizations of graphs and to the graph parameter nu g of van der holst combinatorica 23 4 633 651 2003 in particular our characterization of the graphs with mathrm gd g le 4 implies the forbidden minor characterization of the 3 realizable graphs of belk discret comput geom 37 139 162 2007 and belk and connelly discret comput geom 37 125 137 2007 and of the graphs with nu g le 4 of van der holst combinatorica 23 4 633 651 2003

cuts matrix completions and graph rigidity this paper brings together several topics arising in distinct areas polyhedral combinatorics in particular cut and metric polyhedra matrix theory and semidefinite programming in particular completion problems for positive semidefinite matrices and euclidean distance matrices distance geometry and structural topology in particular graph realization and rigidity problems cuts and metrics provide the unifying theme indeed cuts can be encoded as positive semidefinite matrices this fact underlies the approximative algorithm for max cut of goemans and williamson and both positive semidefinite and euclidean distance matrices yield points of the cut polytope or cone after applying the functions 1 arccos or when fixing the dimension in the euclidean distance matrix completion problem we find the graph realization problem and the related question of unicity of realization which leads to the question of graph rigidity our main objective here is to present in a unified setting a number of results and questions concerning matrix completion graph realization and rigidity problems these problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross fertilization between the various fields involved

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

exact reconstruction of euclidean distance geometry problem using low rank matrix completion the euclidean distance geometry problem arises in a wide variety of applications from determining molecular conformations in computational chemistry to localization in sensor networks when the distance information is incomplete the problem can be formulated as a nuclear norm minimization problem in this paper this minimization program is recast as a matrix completion problem of a low rank r gram matrix with respect to a suitable basis the well known restricted isometry property cannot be satisfied in this scenario instead a dual basis approach is introduced to theoretically analyze the reconstruction problem if the gram matrix satisfies certain coherence conditions with parameter nu the main result shows that the underlying configuration of n points can be recovered with very high probability from o nr nu log 2 n uniformly random samples computationally simple and fast algorithms are designed to solve the euclidean distance geometry problem numerical tests on different 3 d data and protein molecules validate the effectiveness and efficiency of the proposed algorithms

universal rigidity and edge sparsification for sensor network localization owing to their high accuracy and ease of formulation there has been great interest in applying convex optimization techniques particularly that of semidefinite programming sdp relaxation to tackle the sensor network localization problem in recent years however a drawback of such techniques is that the resulting convex program is often expensive to solve in order to speed up computation various edge sparsification heuristics have been proposed whose aim is to reduce the number of edges in the input graph although these heuristics do reduce the size of the convex program and hence make it faster to solve they are often ad hoc in nature and do not preserve the localization properties of the input as such one often has to face a tradeoff between solution accuracy and computational effort in this paper we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input at the heart of our heuristic is a graph decomposition procedure which allows us to identify certain sparse generically universally rigid subgraphs of the input graph our computational results show that the proposed approach can significantly reduce the computational and memory complexities of sdp based algorithms for solving the sensor network localization problem moreover it compares favorably with existing speedup approaches both in terms of accuracy and solution time

an o log n log log n approximation algorithm for the asymmetric traveling salesman problem we present a randomized o log n log log n approximation algorithm for the asymmetric traveling salesman problem atsp this provides the first asymptotic improvement over the long standing log n approximation bound stemming from the work of frieze et al 1982 frieze am galbiati g maffioki f 1982 on the worst case performance of some algorithms for the asymmetric traveling salesman problem networks 12 1 23 39 the key ingredient of our approach is a new connection between the approximability of the atsp and the notion of so called thin trees to exploit this connection we employ maximum entropy rounding a novel method of randomized rounding of lp relaxations of optimization problems we believe that this method might be of independent interest

algorithms for leader selection in stochastically forced consensus networks we are interested in assigning a pre specified number of nodes as leaders in order to minimize the mean square deviation from consensus in stochastically forced networks this problem arises in several applications including control of vehicular formations and localization in sensor networks for networks with leaders subject to noise we show that the boolean constraints which indicate whether a node is a leader are the only source of nonconvexity by relaxing these constraints to their convex hull we obtain a lower bound on the global optimal value we also use a simple but efficient greedy algorithm to identify leaders and to compute an upper bound for networks with leaders that perfectly follow their desired trajectories we identify an additional source of nonconvexity in the form of a rank constraint removal of the rank constraint and relaxation of the boolean constraints yields a semidefinite program for which we develop a customized algorithm well suited for large networks several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms

power calculations for large multi arm placebo controlled studies of dichotomous outcomes several statistical methods can be used to test for differences in placebo controlled dichotomous multi arm studies these tests include linear and quadratic trends dunnett s test hochberg s test and the overall chi square test we computed the power of these tests under a wide variety of alternatives for two arms versus a control the power was computed exactly for more than two arms the power was calculated from 100000 simulations per alternative against monotonic alternatives the linear trend test tended to be superior against convex alternatives the quadratic trend test tended to be superior against general alternatives dunnett s test hochberg s test and the overall chi square test were superior and had very similar power curves the hochberg test was generally less powerful than dunnett s test but the difference was offset by its increased power to reject multiple hypotheses these results remained nearly constant over a wide range of sample sizes average event rates and average effect sizes

complex networks in the euclidean space of communicability distances we study the properties of complex networks embedded in a euclidean space of communicability distances the communicability distance between two nodes is defined as the difference between the weighted sum of walks self returning to the nodes and the weighted sum of walks going from one node to the other we give some indications that the communicability distance identifies the least crowded routes in networks where simultaneous submission of packages is taking place we define an index q based on communicability and shortest path distances which allows reinterpreting the small world phenomenon as the region of minimum q in the watts strogatz model it also allows the classification and analysis of networks with different efficiency of spatial uses consequently the communicability distance displays unique features for the analysis of complex networks in different scenarios

closure of the newman keuls multiple comparisons procedure abstract closure of the family of hypotheses tested by the newman keuls procedure has been proposed in order to avoid excessive type i errors this proposal is examined in some detail and a practical procedure for carrying it out is described and programmed in fortran the relation of such closure to other multiple comparison procedures is discussed and its relatively high power is pointed out

bounds on graph eigenvalues i abstract we prove three results about the spectral radius g of a graph g a let t r n be the r partite turan graph of order n if g is a k r 1 free graph of order n then g t r n unless g t r n b for most irregular graphs g of order n and size m g 2 m n 1 2 m 2 n c let 0 k l if g is a graph of order n with no k 2 k k 1 and no k 2 l 1 then g min g k l 1 k l 1 2 4 l n 1 2

bounds on graph eigenvalues abstract upper and lower estimates are found for the maximum of the kth eigenvalue of a graph as a function of the number of vertices or edges