Michael P.B. Gallaugher

A matrix variate skew-t distribution

Although there is ample work in the literature dealing with skewness in the multivariate setting, there is a relative paucity of work in the matrix variate paradigm. Such work is, for example, useful for modelling three-way data. A matrix variate skew-t distribution is derived based on a mean-variance matrix normal mixture. An expectation-conditional maximization algorithm is developed for parameter estimation. Simulated data are used for illustration. Copyright

Finite mixtures of skewed matrix variate distributions

Abstract Clustering is the process of finding underlying group structures in data. Although mixture model-based clustering is firmly established in the multivariate case, there is a relative paucity of work on matrix variate distributions and none for clustering with mixtures of skewed matrix variate distributions. Four finite mixtures of skewed matrix variate distributions are considered. Parameter estimation is carried out using an expectation-conditional maximization algorithm, and both simulated and real data are used for illustration.

A matrix variate skew-t distribution: A matrix variate skew-t distribution

Although there is ample work in the literature dealing with skewness in the multivariate setting, there is a relative paucity of work in the matrix variate paradigm. Such work is, for example, useful for modelling three-way data. A matrix variate skew-t distribution is derived based on a mean-variance matrix normal mixture. An expectation-conditional maximization algorithm is developed for parameter estimation. Simulated data are used for illustration. Copyright

Non-Kähler expanding Ricci solitons, Einstein metrics, and exotic cone structures

We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form R × M2 × · · · × Mr, where r ≥ 2 and Mi are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or Sp(1) bundles over quaternionic projective space. Mathematics Subject Classification (2000): 53C25, 53C44 0. Introduction In [BDGW] we constructed complete steady gradient Ricci soliton structures (including Ricci-flat metrics) on manifolds of the form R ×M2 × . . . × Mr where Mi, 2 ≤ i ≤ r, are arbitrary closed Einstein manifolds with positive scalar curvature. We also produced numerical solutions of the steady gradient Ricci soliton equation on certain non-trivial R and R bundles over quaternionic projective spaces. In the current paper we will present the analogous results for the case of expanding solitons on the same underlying manifolds. Recall that a gradient Ricci soliton is a manifold M together with a smooth Riemannian metric g and a smooth function u, called the soliton potential, which give a solution to the equation: (0.1) Ric(g) + Hess(u) + ǫ 2 g = 0 for some constant ǫ. The soliton is then called expanding, steady, or shrinking according to whether ǫ is greater, equal, or less than zero. A gradient Ricci soliton is called complete if the metric g is complete. The completeness of the vector field ∇u follows from that of the metric (cf [Zh]). If the metric of a gradient Ricci soliton is Einstein, then either Hessu = 0 (i.e., ∇u is parallel) or we are in the case of the Gaussian soliton (cf [PW] or [PRS]). At present most examples of non-Kählerian expanding solitons arise from left-invariant metrics on nilpotent and solvable Lie groups (resp. nilsolitons, solvsolitons), as a result of work by J. Lauret [La1], [La3], M. Jablonski [Ja], and many others (cf the survey [La2]). These expanders are however not of gradient type, i.e., they satisfy the more general equation (0.2) Ric(g) + 1 2 LXg + ǫ 2 g = 0 where X is a vector field on M and L denotes Lie differentiation. A large class of complete, non-Einstein, non-Kählerian expanders of gradient type (with dimension ≥ 3) consists of an r-parameter family of solutions to (0.1) on R ×M2 × . . . ×Mr where k > 1 and Mi are positive Einstein manifolds. The special case r = 1 (i.e., no Mi) is due to Bryant [Bry] and the solitons have positive sectional curvature. The r = 2 case is due to Gastel and Kronz [GK], who adapted Böhm’s construction of complete Einstein metrics with negative scalar curvature to the soliton case. The case of arbitrary r was treated in [DW3] via a generalization Date: revised November 21, 2013. M. Wang is partially supported by NSERC Grant No. OPG0009421. 1 2 M. BUZANO, A. S. DANCER, M. GALLAUGHER, AND M. WANG of the dynamical system studied by Bryant. The soliton metrics in this family are all of multiple warped product type. In other words, the manifold is thought of as being foliated by hypersurfaces of the form Sk ×M2× . . .×Mr each equipped with a product metric depending smoothly on a real parameter t. More recently, Schulze and Simon [SS] constructed expanding gradient Ricci solitons with nonnegative curvature operator in arbitrary dimensions by studying the scaling limits of the Ricci flow on complete open Riemannian manifolds with non-negative bounded curvature operator and positive asymptotic volume ratio. As pointed out in [BDGW], the situation of multiple warped products on nonnegative Einstein manifolds is rather special because of the automatic lower bound on the scalar curvature of the hypersurfaces. This leads, in the case where all factors have positive scalar curvature, i.e., k > 1, to definiteness of certain energy functionals occurring in the analysis of the dynamical system arising from (0.1), and hence to coercive estimates on the flow. In the present case, where one factor is a circle, i.e., k = 1, we can pass, as in [BDGW], to a subsystem where coercivity holds, and this is enough for the analysis to proceed. The new solitons obtained, like those of [DW3], have conical asymptotics, and are not of Kähler type (Theorem 2.14). We note that the lowest dimensional solitons we obtain form a 2-parameter family on R × S2. The special case r = 1 was analysed earlier by the physicists Gutperle, Headrick, Minwalla and Schomerus [GHMS]. As in [BDGW] we also obtain a family of solutions to our soliton equations that yield complete Einstein metrics of negative scalar curvature (Theorem 3.1). These are analogous to the metrics discovered by Böhm in [Bo]. Recall that for Böhm’s construction the fact that the hyperbolic cone over the product Einstein metric on the hypersurface acts as an attractor plays an important role in the convergence proof for the Einstein trajectories. When k = 1, however, no product metric on the hypersurface can be Einstein with positive scalar curvature, so the hyperbolic cone construction cannot be exploited directly. It turns out that the analysis of the soliton case already contains most of the analysis required for the Einstein case. The new Einstein metrics we obtain have exponential volume growth. Since the underlying smooth manifolds in the present paper are identical to those in [BDGW], our constructions give rise to pairs of homeomorphic but not diffeomorphic non-Einstein expanding gradient Ricci solitons as well as similar pairs of complete Einstein manifolds with negative scalar curvature. Furthermore, since our expanders and Einstein metrics have asymptotically conical structures, we also obtain pairs whose asymptotic cones are homeomorphic but not diffeomorphic. The details can be found at the end of §3. To make further progress in the search for expanders, we need to consider more complicated hypersurface types where the scalar curvature may not be bounded below. In [BDGW] we carried out numerical investigations of steady solitons where the hypersurfaces are the total spaces of Riemannian submersions for which the hypersurface metric involves two functions, one scaling the base and one the fibre of the submersion. We now look numerically at expanding solitons with such hypersurface types, in particular where the hypersurfaces are S2 or S3 bundles over quaternionic projective space. We produce numerical evidence of complete expanding gradient Ricci soliton structures in these cases. Before undertaking our theoretical and numerical investigations, we first prove some general results about expanding solitons of cohomogeneity one type. Some of the results follow from properties of general expanding gradient Ricci solitons. However, the proofs are much simpler and sometimes the statements are sharper, which is helpful in numerical studies. The results include monotonicity and concavity properties for the soliton potential similar to those proved in [BDGW] in the steady case, as well as an upper bound for the mean curvature of the hypersurfaces. To derive this bound, we need to know that complete non-Einstein expanding gradient Ricci solitons have infinite volume. We include a proof of this fact here (Prop. 1.22) since we were not able to NON-KÄHLER EXPANDING RICCI SOLITONS II 3 find an explicit statement in the literature. Finally we derive an asymptotic lower bound for the gradient of the soliton potential, which is in turn used to exhibit a general Lyapunov function for the cohomogeneity one expander equations. 1. Background on cohomogeneity one expanding solitons We briefly review the formalism [DW1] for Ricci solitons of cohomogeneity one. We work on a manifold M with an open dense set foliated by equidistant diffeomorphic hypersurfaces Pt of real dimension n. The dimension of M , the manifold where we construct the soliton, is therefore n+1. The metric is then of the form ḡ = dt2 + gt where gt is a metric on Pt and t is the arclength coordinate along a geodesic orthogonal to the hypersurfaces. This set-up is more general than the cohomogeneity one ansatz, as it allows us to consider metrics with no symmetry provided that appropriate additional conditions on Pt are satisfied, see the following as well as Remarks 2.18 and 3.18 in [DW1]. We will also suppose that u is a function of t only. We let rt denote the Ricci endomorphism of gt, defined by Ric(gt)(X,Y ) = gt(rt(X), Y ) and viewed as an endomorphism via gt. Also let Lt be the shape operator of the hypersurfaces, defined by the equation ġt = 2gtLt where gt is regarded as an endomorphism with respect to a fixed background metric Q. The Levi-Civita connections of ḡ and gt will be denoted by ∇ and ∇ respectively. The relative volume v(t) is defined by dμgt = v(t) dμQ We assume that the scalar curvature St = tr(rt) and the mean curvature tr(Lt) (with respect to the normal ν = ∂ ∂t) are constant on each hypersurface. These assumptions hold, for example, if M is of cohomogeneity one with respect to an isometric Lie group action. They are satisfied also when M is a multiple warped product over an interval. The gradient Ricci soliton equation now becomes the system − tr(L̇)− tr(L) + ü+ ǫ 2 = 0, (1.1) r − (trL)L− L̇+ u̇L+ ǫ 2 I = 0, (1.2) d(trL) + δ∇L = 0. (1.3) The first two equations represent the components of the equation in the directions normal and tangent to the hypersurfaces P , respectively. The third equation represents the equation in mixed directions—here δ∇L denotes the codifferential for TP -valued 1-forms. In the warped product case the final equation involving the codifferential automatically holds. This is also true for cohomogeneity one metrics that are monotypic, i.e., when there are no repeated real irreducible summands in the isotropy representation of the principal orbits (cf [BB], Prop. 3.18). There is a conservation law (1.4) ü+ (−u̇+ trL) u̇− ǫu = C for some constant C. Using our equations we may rewrite this as (1.5) S + tr(L)− (u̇− trL) − ǫu+ 1

A Mixture of Matrix Variate Skew-t Distributions

Although there is ample work in the literature dealing with skewness in the multivariate setting, there is a relative paucity of work in the matrix variate paradigm. Such work is, for example, useful for modelling three-way data. A matrix variate skew-t distribution is derived based on a mean-variance matrix normal mixture. An expectation-conditional maximization algorithm is developed for parameter estimation. Simulated data are used for illustration. Copyright

Factors associated with heterogeneity in microarray gene expression in peripheral blood mononuclear cells from large pedigrees

BackgroundGenome-wide microarray expression is a rich source of functional genomic data. We examined evidence for differences in expression from peripheral blood mononuclear cells between individuals, examined some of factors that may be responsible and provide recommendations for analysis.MethodsA total of 643 individuals from 17 large Mexican American pedigrees had microarray gene expression data generated from peripheral blood mononuclear cells. This data has previously been used to map cis- and trans-expression quantitative trait loci using genome-wide linkage analysis. We estimated both principal components and cell proportions in these data, and tested them for association with clinical factors to provide insight into causes of variation in gene expression between individuals.ResultsWe identified that there were highly significant differences in the second principal component of gene expression between pedigrees, with 3 pedigrees being outliers. The estimated cell proportions identified 1 individual who was a gross outlier, as well as pedigrees that differed from others in their estimated proportions of helper and cytotoxic T cells.ConclusionsThese phenomena could be from either pedigree-specific genetic variation, technical artefacts, or clinical factors. Incorporating factors that influence gene expression into genetic analysis, and exclusion of outliers could improve the power of genetic mapping of expression traits.

Muscle Androgen Receptor Content but Not Systemic Hormones Is Associated With Resistance Training-Induced Skeletal Muscle Hypertrophy in Healthy, Young Men

The factors that underpin heterogeneity in muscle hypertrophy following resistance exercise training (RET) remain largely unknown. We examined circulating hormones, intramuscular hormones, and intramuscular hormone-related variables in resistance-trained men before and after 12 weeks of RET. Backward elimination and principal component regression evaluated the statistical significance of proposed circulating anabolic hormones (e.g., testosterone, free testosterone, dehydroepiandrosterone, dihydrotestosterone, insulin-like growth factor-1, free insulin-like growth factor-1, luteinizing hormone, and growth hormone) and RET-induced changes in muscle mass (n = 49). Immunoblots and immunoassays were used to evaluate intramuscular free testosterone levels, dihydrotestosterone levels, 5α-reductase expression, and androgen receptor content in the highest- (HIR; n = 10) and lowest- (LOR; n = 10) responders to the 12 weeks of RET. No hormone measured before exercise, after exercise, pre-intervention, or post-intervention was consistently significant or consistently selected in the final model for the change in: type 1 cross sectional area (CSA), type 2 CSA, or fat- and bone-free mass (LBM). Principal component analysis did not result in large dimension reduction and principal component regression was no more effective than unadjusted regression analyses. No hormone measured in the blood or muscle was different between HIR and LOR. The steroidogenic enzyme 5α-reductase increased following RET in the HIR (P < 0.01) but not the LOR (P = 0.32). Androgen receptor content was unchanged with RET but was higher at all times in HIR. Unlike intramuscular free testosterone, dihydrotestosterone, or 5α-reductase, there was a linear relationship between androgen receptor content and change in LBM (P < 0.01), type 1 CSA (P < 0.05), and type 2 CSA (P < 0.01) both pre- and post-intervention. These results indicate that intramuscular androgen receptor content, but neither circulating nor intramuscular hormones (or the enzymes regulating their intramuscular production), influence skeletal muscle hypertrophy following RET in previously trained young men.