Equal Entries in Totally Positive Matrices

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\textrm{-by-}n$ matrix is $\Theta(n^{4/3})$ (resp. $\Theta(n^{3/2}$)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and $TP$ completability are also presented. We also examine the number and positionings of equal $2\textrm{-by-}2$ minors in a $2\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\textrm{-by-}2$ minors and outerplanar graphs.Comment: 15 page

Genetic Dissection of a QTL Affecting Bone Geometry.

Parameters of bone geometry such as width, length, and cross-sectional area are major determinants of bone strength. Although these traits are highly heritable, few genes influencing bone geometry have been identified. Here, we dissect a major quantitative trait locus (QTL) influencing femur size. This QTL was originally identified in an F2 cross between the C57BL/6J-hg/hg (HG) and CAST/EiJ strains and was referred to as femur length in high growth mice 2 (Feml2). Feml2 was located on chromosome (Chr.) 9 at ∼20 cM. Here, we show that the HG.CAST-(D9Mit249-D9Mit133)/Ucd congenic strain captures Feml2 In an F2 congenic cross, we fine-mapped the location of Feml2 to an ∼6 Mbp region extending from 57.3 to 63.3 Mbp on Chr. 9. We have identified candidates by mining the complete genome sequence of CAST/EiJ and through allele-specific expression (ASE) analysis of growth plates in C57BL/6J × CAST/EiJ F1 hybrids. Interestingly, we also find that the refined location of Feml2 overlaps a cluster of six independent genome-wide associations for human height. This work provides the foundation for the identification of novel genes affecting bone geometry

Interest arbitration, outcomes, and the incentive to bargain : the role of risk preferences

This is a preliminary draf

Eigenvalue Pairing in the Response Matrix for a Class of Network Models with Circular Symmetry

We consider the response matrices in certain weighted networks that display a circular symmetry. It had been observed empirically that these exhibit several paired (multiplicity two) eigenvalues. Here, this pairing is explained analytically for a version of the model more general than the original. The exact number of necessarily paired eigenvalues is given in terms of the structure of the model, and the special structure of the eigenvectors is also described. Examples are provided

The relation between the diagonal entries and the eigenvalues of a symmetric matrix, based upon the sign pattern of its off-diagonal entries

It is known that majorization is a complete description of the relationships between the eigenvalues and diagonal entries of real symmetric matrices. However, for large subclasses of such matrices, the diagonal entries impose much greater restrictions on the eigenvalues. Motivated by previous results about Laplacian eigenvalues, we study here the additional restrictions that come from the off-diagonal sign-pattern classes of real symmetric matrices. Each class imposes additional restrictions. Several results are given for the all nonpositive and all nonnegative classes and for the third class that appears when n = 4. Complete description of the possible relationships are given in low dimensions. (C) 2012 Elsevier Inc. All rights reserved

Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues

Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of any undirected graph with $n$ vertices or more has at least $k$ nonpositive eigenvalues. We show that $NPO(k)$ is well-defined and prove that the values of $NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we prove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$ is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest degree, which generalizes some prior results.Comment: 23 pages, 12 figure

Genome-wide isolation of growth and obesity QTL using mouse speed congenic strains

BACKGROUND: High growth (hg) modifier and background independent quantitative trait loci (QTL) affecting growth, adiposity and carcass composition were previously identified on mouse chromosomes (MMU) 1, 2, 5, 8, 9, 11 and 17. To confirm and further characterize each QTL, two panels of speed congenic strains were developed by introgressing CAST/EiJ (CAST) QTL alleles onto either mutant C57Bl/6J-hg/hg (HG) or wild type C57Bl/6J (B6) genetic backgrounds. RESULTS: The first speed congenic panel was developed by introgressing four overlapping donor regions spanning MMU2 in its entirety onto both HG and B6 backgrounds, for a total of eight strains. Phenotypic characterization of the MMU2 panel confirmed the segregation of multiple growth and obesity QTL and strongly suggested that a subset of these loci modify the effects of the hg deletion. The second panel consisted of individual donor regions on an HG background for each QTL on MMU1, 5, 8, 9, 11 and 17. Of the six developed strains, five were successfully characterized and displayed significant differences in growth and/or obesity as compared to controls. All five displayed phenotypes similar to those originally attributed to each QTL, however, novel phenotypes were unmasked in several of the strains including sex-specific effects. CONCLUSION: The speed congenic strains developed herein constitute an invaluable genomic resource and provide the foundation to identify the specific nature of genetic variation influencing growth and obesity

Serious limitations of the QTL/Microarray approach for QTL gene discovery

<p>Abstract</p> <p>Background</p> <p>It has been proposed that the use of gene expression microarrays in nonrecombinant parental or congenic strains can accelerate the process of isolating individual genes underlying quantitative trait loci (QTL). However, the effectiveness of this approach has not been assessed.</p> <p>Results</p> <p>Thirty-seven studies that have implemented the QTL/microarray approach in rodents were reviewed. About 30% of studies showed enrichment for QTL candidates, mostly in comparisons<b/> between congenic and background strains. Three studies led to the identification of an underlying <it>QTL </it>gene. To complement the literature results, a microarray experiment was performed using three mouse congenic strains isolating the effects of at least 25 biometric QTL. Results show that genes in the congenic donor regions were preferentially selected. However, within donor regions, the distribution of differentially expressed genes was homogeneous once gene density was accounted for. Genes within identical-by-descent (IBD) regions were less likely to be differentially expressed in chromosome 2, but not in chromosomes 11 and 17. Furthermore, expression of <it>QTL </it>regulated in <it>cis </it>(<it>cis </it>eQTL) showed higher expression in the background genotype, which was partially explained by the presence of single nucleotide polymorphisms (SNP).</p> <p>Conclusions</p> <p>The literature shows limited successes from the QTL/microarray approach to identify <it>QTL </it>genes. Our own results from microarray profiling of three congenic strains revealed a strong tendency to select <it>cis-</it>eQTL over <it>trans-</it>eQTL. IBD regions had little effect on rate of differential expression, and we provide several reasons why IBD should not be used to discard eQTL candidates. In addition, mismatch probes produced false <it>cis-</it>eQTL that could not be completely removed with the current strains genotypes and low probe density microarrays. The reviewed studies did not account for lack of coverage from the platforms used and therefore removed genes that were not tested. Together, our results explain the tendency to report QTL candidates as differentially expressed and indicate that the utility of the QTL/microarray as currently implemented is limited. Alternatives are proposed that make use of microarray data from multiple experiments to overcome the outlined limitations.</p