Covariance structure in the skull of Catarrhini: a case of pattern stasis and magnitude evolution

The study of the genetic variance/covariance matrix (G-matrix) is a recent and fruitful approach in evolutionary biology, providing a window of investigating for the evolution of complex characters. Although G-matrix studies were originally conducted for microevolutionary timescales, they could be extrapolated to macroevolution as long as the G-matrix remains relatively constant, or proportional, along the period of interest. A promising approach to investigating the constancy of G-matrices is to compare their phenotypic counterparts (P-matrices) in a large group of related species; if significant similarity is found among several taxa, it is very likely that the underlying G-matrices are also equivalent. Here we study the similarity of covariance and correlation structure in a broad sample of Old World monkeys and apes (Catarrhini). We made phylogenetically structured comparisons of correlation and covariance matrices derived from 39 skull traits, ranging from between species to the superfamily level. We also compared the overall magnitude of integration between skull traits (r(2)) for all Catarrhim genera. Our results show that P-matrices were not strictly constant among catarrhines, but the amount of divergence observed among taxa was generally low. There was significant and positive correlation between the amount of divergence in correlation and covariance patterns among the 30 genera and their phylogenetic distances derived from a recently proposed phylogenetic hypothesis. Our data demonstrate that the P-matrices remained relatively similar along the evolutionary history of catarrhines, and comparisons with the G-matrix available for a New World monkey genus (Saguinus) suggests that the same holds for all anthropoids. The magnitude of integration, in contrast, varied considerably among genera, indicating that evolution of the magnitude, rather than the pattern of inter-trait correlations, might have played an important role in the diversification of the catarrhine skull. (C) 2009 Elsevier Ltd. All rights reserved.

The Evolution of Modularity in the Mammalian Skull II: Evolutionary Consequences

Changes in patterns and magnitudes of integration may influence the ability of a species to respond to selection. Consequently, modularity has often been linked to the concept of evolvability, but their relationship has rarely been tested empirically. One possible explanation is the lack of analytical tools to compare patterns and magnitudes of integration among diverse groups that explicitly relate these aspects to the quantitative genetics framework. We apply such framework here using the multivariate response to selection equation to simulate the evolutionary behavior of several mammalian orders in terms of their flexibility, evolvability and constraints in the skull. We interpreted these simulation results in light of the integration patterns and magnitudes of the same mammalian groups, described in a companion paper. We found that larger magnitudes of integration were associated with a blur of the modules in the skull and to larger portions of the total variation explained by size variation, which in turn can exert a strong evolutionary constraint, thus decreasing the evolutionary flexibility. Conversely, lower overall magnitudes of integration were associated with distinct modules in the skull, to smaller fraction of the total variation associated with size and, consequently, to weaker constraints and more evolutionary flexibility. Flexibility and constraints are, therefore, two sides of the same coin and we found them to be quite variable among mammals. Neither the overall magnitude of morphological integration, the modularity itself, nor its consequences in terms of constraints and flexibility, were associated with absolute size of the organisms, but were strongly associated with the proportion of the total variation in skull morphology captured by size. Therefore, the history of the mammalian skull is marked by a trade-off between modularity and evolvability. Our data provide evidence that, despite the stasis in integration patterns, the plasticity in the magnitude of integration in the skull had important consequences in terms of evolutionary flexibility of the mammalian lineages.

MODULARITY, NOISE, AND NATURAL SELECTION

Most biological systems are formed by component parts that are to some degree interrelated. Groups of parts that are more associated among themselves and are relatively autonomous from others are called modules. One of the consequences of modularity is that biological systems usually present an unequal distribution of the genetic variation among traits. Estimating the covariance matrix that describes these systems is a difficult problem due to a number of factors such as poor sample sizes and measurement errors. We show that this problem will be exacerbated whenever matrix inversion is required, as in directional selection reconstruction analysis. We explore the consequences of varying degrees of modularity and signal-to-noise ratio on selection reconstruction. We then present and test the efficiency of available methods for controlling noise in matrix estimates. In our simulations, controlling matrices for noise vastly improves the reconstruction of selection gradients. We also perform an analysis of selection gradients reconstruction over a New World Monkeys skull database to illustrate the impact of noise on such analyses. Noise-controlled estimates render far more plausible interpretations that are in full agreement with previous results.

Equivariant Birch-Swinnerton-Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over Q and let K/Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h(1)(E x(Q) K)(1) viewed as amotive over Q with coefficients in Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S-3-extension of Q.

A Conjecture Concerning Transformation Of A Supercooled Hard-Sphere Liquid To A Metastable Disordered Solid

It is conjectured that the hard sphere system has several distinct solid phases, all but one of which are metastable. The bifurcation theory analysis of freezing is extended to the description of the transition between a supercooled liquid and a disordered solid by defining a restricted phase space for the disordered solid. This approach leads to the prediction of a first order transition between a supercooled hard sphere fluid and a disordered metastable hard sphere solid. The results of the calculation are in qualitative agreement with the results of Woodcock's molecular dynamics computer simulations. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

Curves on threefolds and a conjecture of Griffiths-Harris

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in . We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.

Proof of a conjecture of Frankl and Furedi

We give a simple linear algebraic proof of the following conjecture of Frankl and Furedi [7, 9, 13]. (Frankl-Furedi Conjecture) if F is a hypergraph on X = {1, 2, 3,..., n} such that 1 less than or equal to /E boolean AND F/ less than or equal to k For All E, F is an element of F, E not equal F, then /F/ less than or equal to (i=0)Sigma(k) ((i) (n-1)). We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace of R-/F/. Finally, the desired bound on /F/ is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray-Chaudhuri-Wilson theorem. (C) 1997 Academic Press.

The Herzog-Vasconcelos conjecture for affine semigroup rings

Let S be a simplicial affine semigroup such that its semigroup ring A = k[S] is Buchsbaum. We prove for such A the Herzog-Vasconcelos conjecture: If the A-module Der(k)A of k-linear derivations of A has finite projective dimension then it is free and hence A is a polynomial ring by the well known graded case of the Zariski-Lipman conjecture.

OOCs, Partial Relative Difference Families and a Conjecture of Golomb

The cyclic difference sets constructed by Singer are also examples of perfect distinct difference sets (DDS). The Bose construction of distinct difference sets, leads to a relative difference set. In this paper we introduce the concept of partial relative DDS and prove that an optical orthogonal code (OOC) construction due to Moreno et. al., is a partial relative DDS. We generalize the concept of ideal matrices previously introduced by Kumar and relate it to the concepts of this paper. Another variation of ideal matrices is introduced in this paper: Welch ideal matrices of dimension n by (n - 1). We prove that Welch ideal matrices exist only for n prime. Finally, we recast an old conjecture of Golomb on the Welch construction of Costas arrays using the concepts of this paper. This connection suggests that our construction of partial relative difference sets is in a sense, unique

Ontheproof of the Poincare´ conjecture

In 2002, Perelman proved the Poincare conjecture, building on the work of Richard Hamilton on the Ricci flow. In this article, we sketch some of the arguments and attempt to place them in a broader dynamical context.

On the proof of the Poincare´ conjecture

In 2002, Perelman proved the Poincare conjecture, building on the work of Richard Hamilton on the Ricci flow. In this article, we sketch some of the arguments and attempt to place them in a broader dynamical context.

A unified approach to known and unknown cases of Berge's conjecture

Berge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection Ck of k disjoint independent sets, where each dipath P?P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the GreeneKleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the GallaiMilgram Theorem, for k?? (where ?is the number of vertices in the longest dipath in the graph), by the GallaiRoy Theorem, and when the optimal path partition P contains only dipaths P with |P|?k. Recently, it was proved (Eur J Combin (2007)) for k=2. There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=?-1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.

On linear series and a conjecture of D. C. Butler

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H-0 (L) of dimension n+1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V circle times O-C -> L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

Beyond Schur's Conjecture

Schur 4] conjectured that the maximum length N of consecutive quadratic nonresidues modulo a prime p is less than root p if p is large enough. This was proved by Hummel in 2003. In this note, we outline a clear improvement over Hummel's bound for p > 23.