So far in this course we have dealt entirely with the evolution of characters that are controlled by simple Mendelian inheritance at a single locus. There are notes on the course website about gametic disequilibrium and how allele frequencies change at two loci simultaneously, but we didn’t discuss them. In every example we’ve considered we’ve imagined that we could understand something about evolution by examining the evolution of a single gene. That’s the domain of classical population genetics. For the next few weeks we’re going to be exploring a field that’s actually older than classical population genetics, although the approach we’ll be taking to it involves the use of population genetic machinery. If you know a little about the history of evolutionary biology, you may know that after the rediscovery of Mendel’s work in 1900 there was a heated debate between the “biometricians” (e.g., Galton and Pearson) and the “Mendelians” (e.g., de Vries, Correns, Bateson, and Morgan). Biometricians asserted that the really important variation in evolution didn’t follow Mendelian rules. Height, weight, skin color, and similar traits seemed to

Human biochemical genetics

It has been firmly established that the rapid uptake of Ca2+ by mitochondria from a wide range of sources is mediated by a uniporter which permits transport of the ion down its electrochemical gradient. Several mechanisms of Ca2+ efflux from mitochondria have also been extensively discussed in the literature. Energized mitochondria must expend a significant amount of energy to transport Ca2+ against its electrochemical gradient from the matrix space to the external space. Two separate mechanisms have been found to mediate this outward transport: a Ca2+/nNa+ exchanger and a Na(+)-independent efflux mechanism. These efflux mechanisms are considered from the perspective of available energy. In addition, a reversible Ca2(+)-induced increase in inner membrane permeability can also occur. The induction of this permeability transition is characterized by swelling of the mitochondria, leakiness to small ions such as K+, Mg2+, and Ca2+, and loss of the mitochondrial membrane potential. It has been suggested that the permeability transition and its reversal may also function as a mitochondrial Ca2+ efflux mechanism under some conditions. The characteristics of each of these mechanisms are discussed, as well as their possible physiological functions.

Mechanisms by which mitochondria transport calcium.

Magnesium. An update on physiological, clinical and analytical aspects.

0. Introduction. An n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n. In this paper we investigate certain group actions on manifolds, which have a simple convex polytope as orbit space. Let P\" denote such a simple polytope. We have two situations in mind. (1) The group is Z, M\" is n-dimensional and (2) The group is T\", M is 2n-dimensional and M2\"/T\" P\". Up to an automorphism of the group, the action is required to be locally isomorphic to the standard representation of Z. on in the second case. In the first case, we call M a \"small cover\" of P\"; in the second, it is a \"toric manifold\" over P. First examples are provided by the natural actions of Z. and T\" on RP\" and CP\", respectively. In both cases the orbit space is an n-simplex. Associated to a small cover of P\", there is a homomorphism 2\" Z’ Z, where m is the number of codimension-one faces of P\". The homomorphism 2 specifies an isotropy subgroup for each codimension-one face. We call it a \"characteristic function\" of the small cover. Similarly, the characteristic function ofa toric manifold over pn is a map Z --) 7/\". A basic result is that small covers and toric manifolds over P\" are classified by their characteristic functions (see Propositions 1.7 and 1.8). The algebraic topology of these manifolds is very beautiful. The calculation of their homology and cohomology groups is closely related to some well-known constructions in commutative algebra and the combinatorial theory of convex polytopes. We discuss some of these constructions below. Let f denote the number of/-faces of P\" and let h denote the coefficient of \"in f(t 1). Then (fo, f,) is called the f-vector and (ho, h,) the h-vector of P\". The f-vector and the h-vector obviously determine one another. The Upper Bound Theorem, due to McMullen, asserts that the inequality h < (,-,{-x), holds for all n-dimensional convex polytopes with m faces of codimension one. In 1971 McMullen conjectured simple combinatorial conditions on a sequence (h0, h,) of integers necessary and sufficient for it to be the h-vector of a simple convex polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley (see [Bronsted] for more details and references). Research on

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Convex polytopes, Coxeter orbifolds and torus actions

Contents: Matrix Eigenvalue Methods.- Double Bracket Isospectral Flows.- Singular Value Decomposition.- Linear Programming.- Approximation and Control.- Balanced Matrix Factorizations.- Invariant Theory and System Balancing.- Balancing via Gradient Flows.- Sensitivity Optimization.- Linear Algebra.- Dynamical Systems.- Global Analysis.

Optimization and Dynamical Systems

These notes are intended as an introduction to the theory of Coxeter groups. They closely follow my talk in the Lectures on Modern Mathematics Series at the Mathematical Sciences Center in Tsinghua University on May 10, 2013. They were prepared from the beamer presentation which I used during my talk. AMS classification numbers. Primary: 20F55, 20F65

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The geometry and topology of Coxeter groups

A Coxeter system (r, V) is a group r (a "Coxeter group") together with a set of generators V such that each element of V has order two and such that all relations in r are consequences of relations of the form (VW)m(v, w) = 1, where v, w E V and m(v, w) denotes the order of vw. The m(v, w)'s (which are positive integers or oo) obviously determine the Coxeter system up to isomorphism. The list of finite Coxeter groups is short and well-known; however, general Coxeter groups are much more flexible. In fact, the set V and the m(v, w)'s can be specified arbitrarily (at least if V is finite) subject only to the conditions: m(v, v) = 1 and m(v, w) = m(w, v) ? 2 if v -# w. Suppose that (r, V) is a Coxeter system, that X is a Hausdorff space and that (Xv)v~v is a locally finite family of closed subspaces indexed by V. (The Xv are called the "panels" of X.(2)) There is a classical method for constructing a transformation group from these data. For each x E X, let V(x) denote the set of v in V such that x E Xv. For each subset S of V, let rs be the subgroup generated by S and let Xs be the "face" of X defined by

Groups Generated by reflections and aspherical manifolds not covered by Euclidean space

Hyperbolization is a process for converting a simplicial complex into a metric space with "nonpositive curvature" in the sense of Gromov Several such processes are described in [19, §34] One of the purposes of this paper is to elaborate this idea of Gromov Another purpose is to use it to construct the three examples described below Our approach to hyperbolization is based on the following construction of Williams [32] Suppose that X is a space and that / : X -> σ is a map onto the standard λz-simplex Suppose, also, that K is an ^-dimensional simplicial complex To these data Williams associates a space XAK, constructed by replacing each n-simplex in the barycentric subdivision of K by a copy of X The pair (X, /) is a "hyperbolized ^-simplex" if X is a nonpositively curved manifold with boundary and / has appropriate properties (It is proved in §4 that hyperbolized simplices exist) If (X, /) is a hyperbolized ^-simplex, then XAK is nonpositively curved; it is called a "hyperbolization of K" In all three examples we begin with a polyhedral homology manifold having a desired feature; a hyperbolization then has the added feature of nonpositive curvature The first example is a closed aspherical fourmanifold which cannot be triangulated Taking the product of this example with a A2-torus, we obtain an aspherical manifold of any dimension > 4 which is not homotopy equivalent to a PL manifold The second example is a closed smooth manifold of dimension n > 5 which carries a topological metric of nonpositive curvature, while its universal cover, though contractible, is not homeomorphic to a Euclidean n-space R As we shall see, such a manifold cannot carry a PL or smooth metric of nonpositive curvature The third example is a further refinement: M carries a topological metric of strict negative curvature and its universal

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Hyperbolization of polyhedra

https://people.math.osu.edu/davis.12/jpaa.pdf

Michael W. Davis

Papers

Convex polytopes, Coxeter orbifolds and torus actions

The geometry and topology of Coxeter groups

Groups Generated by reflections and aspherical manifolds not covered by Euclidean space

Hyperbolization of polyhedra

Right-angled Artin groups are commensurable with right-angled Coxeter groups