The Role of Kemeny's Constant in Properties of Markov Chains

In a finite state irreducible Markov chain with stationary probabilities \pi_i and mean first passage times m_(ij) (mean recurrence time when i = j) it was first shown by Kemeny and Snell (1960) that \sum_j \pi_j m_(ij) is a constant K, not depending on i. This constant has since become known as Kemeny's constant. A variety of techniques for finding expressions and various bounds for K are derived. The main interpretation focuses on its role as the expected time to mixing in a Markov chain. Various applications are considered including perturbation results, mixing on directed graphs and its relation to the Kirchhoff index of regular graphs.Comment: 13 page

On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal

"Nested" cryptic diversity in a widespread marine ecosystem engineer: a challenge for detecting biological invasions

<p>Abstract</p> <p>Background</p> <p>Ecosystem engineers facilitate habitat formation and enhance biodiversity, but when they become invasive, they present a critical threat to native communities because they can drastically alter the receiving habitat. Management of such species thus needs to be a priority, but the poorly resolved taxonomy of many ecosystem engineers represents a major obstacle to correctly identifying them as being either native or introduced. We address this dilemma by studying the sea squirt <it>Pyura stolonifera</it>, an important ecosystem engineer that dominates coastal communities particularly in the southern hemisphere. Using DNA sequence data from four independently evolving loci, we aimed to determine levels of cryptic diversity, the invasive or native status of each regional population, and the most appropriate sampling design for identifying the geographic ranges of each evolutionary unit.</p> <p>Results</p> <p>Extensive sampling in Africa, Australasia and South America revealed the existence of "nested" levels of cryptic diversity, in which at least five distinct species can be further subdivided into smaller-scale genetic lineages. The ranges of several evolutionary units are limited by well-documented biogeographic disjunctions. Evidence for both cryptic native diversity and the existence of invasive populations allows us to considerably refine our view of the native versus introduced status of the evolutionary units within <it>Pyura stolonifera </it>in the different coastal communities they dominate.</p> <p>Conclusions</p> <p>This study illustrates the degree of taxonomic complexity that can exist within widespread species for which there is little taxonomic expertise, and it highlights the challenges involved in distinguishing between indigenous and introduced populations. The fact that multiple genetic lineages can be native to a single geographic region indicates that it is imperative to obtain samples from as many different habitat types and biotic zones as possible when attempting to identify the source region of a putative invader. "Nested" cryptic diversity, and the difficulties in correctly identifying invasive species that arise from it, represent a major challenge for managing biodiversity.</p

One step ATRP initiator immobilization on surfaces leading to gradient-grafted polymer brushes

Published: April 30, 2014A method is described that allows potentially any surface to be functionalized covalently with atom transfer radical polymerization (ATRP) initiators derived from ethyl-2-bromoisobutyrl bromide in a single step. In addition, the initiator surface density was variable and tunable such that the thickness of polymer chain grafted from the surface varied greatly on the surfaces providing examples, across the surface of a substrate, of increased chain stretching due to the entropic nature of crowded polymer chains leading toward polymer brushes. An initiator gradient of increasing surface density was deposited by plasma copolymerization of an ATRP initiator (ethyl 2-bromoisobutyrate) and a non-ATRP reactive diluent molecule (ethanol). The deposited plasma polymer retained its chemical ability to surface-initiate polymerization reactions as exemplified by N,N'-dimethyl acrylamide and poly(ethylene glycol) methyl ether methacrylate polymerizations, illustrating linear and bottle-brush-like chains, respectively. A large variation in graft thickness was observed from the low to high chain-density side suggesting that chains were forced to stretch away from the surface interface--a consequence of entropic effects resulting from increased surface crowding. The tert-butyl bromide group of ethyl 2-bromoisobutyrate is a commonly used initiator in ATRP, so a method for covalent linkage to any substrate in a single step desirably simplifies the multistep surface activation procedures currently used.Bryan R. Coad, Katie E. Styan, and Laurence Meaghe

Some properties of matrix partial orderings

AbstractThe matrix partial orderings considered are: (1) the star ordering and (2) the minus ordering or rank subtractivity, both in the set of m × n complex matrices, and (3) the Löwner ordering, in the set of m × m matrices. The problems discussed are: (1) inheriting certain properties under a given ordering, (2) preserving an ordering under some matrix multiplications, (3) relationships between an ordering among direct (or Kronecker) and Hadamard products and the corresponding orderings between the factors involved, (4) orderings between generalized inverses of a given matrix, and (5) preserving or reversing a given ordering under generalized inversions. Several generalizations of results known in the literature and a number of new results are derived

Some inequalities involving statistical expressions (L. V. Foster)

Solutions by J. M. Borwein, G. P. H. Styan and H. Wolkowicz to problem 81-1O, by L. V. Foster (University of Pittsburgh at Johnstown)

Computation of a test statistic in data quality control

When processing observational data, statistical testing is an essential instrument for rendering harmless incidental anomalies and disturbances in the measurements. A commonly used test statistic based on the general linear model is the generalized likelihood ratio test statistic. The standard formula given in the literature for this test statistic is not defined if the noise covariance matrix is singular, and is not suitable for computation if any of the matrices involved are ill-conditioned. Based on Paige’s generalized linear least squares method [Comm. Statist. B—Simulation Comput., 7 (1978), pp. 437–453], a numerically stable approach is proposed for the computation of the test statistic, as well as for the estimates of the parameter vectors, and reliable representations of the error covariance matrices for these estimates are presented. This approach allows the noise covariance matrix to be singular and can be applied directly to the linear model with linear equality constraints.Remote SensingAerospace Engineerin