The inverse limit of GIT quotients of Grassmannians by the maximal torus

There are finitely many GIT quotients of

The effect of differences in methodology among some recent applications of shearing quotients.

A shearing quotient (SQ) is a way of quantitatively representing the Phase I shearing edges on a molar tooth. Ordinary or phylogenetic least squares regression is fit to data on log molar length (independent variable) and log sum of measured shearing crests (dependent variable). The derived linear equation is used to generate an 'expected' shearing crest length from molar length of included individuals or taxa. Following conversion of all variables to real space, the expected value is subtracted from the observed value for each individual or taxon. The result is then divided by the expected value and multiplied by 100. SQs have long been the metric of choice for assessing dietary adaptations in fossil primates. Not all studies using SQ have used the same tooth position or crests, nor have all computed regression equations using the same approach. Here we focus on re-analyzing the data of one recent study to investigate the magnitude of effects of variation in 1) shearing crest inclusion, and 2) details of the regression setup. We assess the significance of these effects by the degree to which they improve or degrade the association between computed SQs and diet categories. Though altering regression parameters for SQ calculation has a visible effect on plots, numerous iterations of statistical analyses vary surprisingly little in the success of the resulting variables for assigning taxa to dietary preference. This is promising for the comparability of patterns (if not casewise values) in SQ between studies. We suggest that differences in apparent dietary fidelity of recent studies are attributable principally to tooth position examined.

The maximal C*-algebra of quotients as an operator bimodule

We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra A as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product of A with itself labelled by the essential closed right ideals of A into A. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.

Maximal C*-algebras of quotients and injective envelopes of C*-algebras

A new C*-enlargement of a C*-algebra A nested between the local multiplier algebra of A and its injective envelope is introduced. Various aspects of this maximal C*-algebra of quotients are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra such that its second iterated local multiplier algebra is strictly larger than its local multiplier algebra.

Quotients, exactness, and nuclearity in the operator system category

We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP.

Quotients d'une variété algébrique par un groupe algébrique linéairement réductif et ses sous-groupes maximaux unipotents

La construction d'un quotient, en topologie, est relativement simple; si $G$ est un groupe topologique agissant sur un espace topologique $X$, on peut considérer l'application naturelle de $X$ dans $X/G$, l'espace d'orbites muni de la topologie quotient. En géométrie algébrique, malheureusement, il n'est généralement pas possible de munir l'espace d'orbites d'une structure de variété. Dans le cas de l'action d'un groupe linéairement réductif $G$ sur une variété projective $X$, la théorie géométrique des invariants nous permet toutefois de construire un morphisme de variété d'un ouvert $U$ de $X$ vers une variété projective $X//U$, se rapprochant autant que possible d'une application quotient, au sens topologique du terme. Considérons par exemple $X\subseteq P^{n}$, une $k$-variété projective sur laquelle agit un groupe linéairement réductif $G$ et supposons que cette action soit induite par une action linéaire de $G$ sur $A^{n+1}$. Soit $\widehat{X}\subseteq A^{n+1}$, le cône affine au dessus de $\X$. Par un théorème de la théorie classique des invariants, il existe alors des invariants homogènes $f_{1},...,f_{r}\in C[\widehat{X}]^{G}$ tels que $$C[\widehat{X}]^{G}= C[f_{1},...,f_{r}].$$ On appellera le nilcone, que l'on notera $N$, la sous-variété de $\X$ définie par le locus des invariants $f_{1},...,f_{r}$. Soit $Proj(C[\widehat{X}]^{G})$, le spectre projectif de l'anneau des invariants. L'application rationnelle $$\pi:X\dashrightarrow Proj(C[f_{1},...,f_{r}])$$ induite par l'inclusion de $C[\widehat{X}]^{G}$ dans $C[\widehat{X}]$ est alors surjective, constante sur les orbites et sépare les orbites autant qu'il est possible de le faire; plus précisément, chaque fibre contient exactement une orbite fermée. Pour obtenir une application régulière satisfaisant les mêmes propriétés, il est nécessaire de jeter les points du nilcone. On obtient alors l'application quotient $$\pi:X\backslash N\rightarrow Proj(C[f_{1},...,f_{r}]).$$ Le critère de Hilbert-Mumford, dû à Hilbert et repris par Mumford près d'un demi-siècle plus tard, permet de décrire $N$ sans connaître les $f_{1},...,f_{r}$. Ce critère est d'autant plus utile que les générateurs de l'anneau des invariants ne sont connus que dans certains cas particuliers. Malgré les applications concrètes de ce théorème en géométrie algébrique classique, les démonstrations que l'on en trouve dans la littérature sont généralement données dans le cadre peu accessible des schémas. L'objectif de ce mémoire sera, entre autres, de donner une démonstration de ce critère en utilisant autant que possible les outils de la géométrie algébrique classique et de l'algèbre commutative. La version que nous démontrerons est un peu plus générale que la version originale de Hilbert \cite{hilbert} et se retrouve, par exemple, dans \cite{kempf}. Notre preuve est valide sur $C$ mais pourrait être généralisée à un corps $k$ de caractéristique nulle, pas nécessairement algébriquement clos. Dans la seconde partie de ce mémoire, nous étudierons la relation entre la construction précédente et celle obtenue en incluant les covariants en plus des invariants. Nous démontrerons dans ce cas un critère analogue au critère de Hilbert-Mumford (Théorème 6.3.2). C'est un théorème de Brion pour lequel nous donnerons une version un peu plus générale. Cette version, de même qu'une preuve simplifiée d'un théorème de Grosshans (Théorème 6.1.7), sont les éléments de ce mémoire que l'on ne retrouve pas dans la littérature.

The relation between Wechsler Performance Quotients and quotients on the Snijders-Oomen Non-Verbal Intelligence Scale

This paper examines the relationship between results of the Wechsler-Bellevue Performance Test of Intelligence and the Snijders-Oomen Non-Verbal Intelligence Scale (SONS) as given to hearing-impaired students at Central Institute for the Deaf.

On the cohomology of certain quotients of the spectrum

The aim of this note is to present a new, elementary proof of a result of Baas and Madsen on the mod p cohomology of certain quotients of the spectrum BP.

Quotients of Koszul Algebras and 2-D-Determined Algebras

Vatne [13] and Green and Marcos [9] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.

A tabular series of decimal quotients for all the proper vulgar fractions [microform], of which, when in their lowest terms, neither the numerator nor the denominator is greater than 1000.

Master microform held by: UnM.

Some results on orbifold quotients and related objects

The quotient of a finite-dimensional Euclidean space by a finite linear group inherits different structures from the initial space, e.g. a topology, a metric and a piecewise linear structure. The question when such a quotient is a manifold leads to the study of finite groups generated by reflections and rotations, i.e. by orthogonal transformations whose fixed point subspace has codimension one or two. We classify such groups and thereby complete earlier results by M. A. Mikhaîlova from the 70s and 80s. Moreover, we show that a finite group is generated by reflections and) rotations if and only if the corresponding quotient is a Lipschitz-, or equivalently, a piecewise linear manifold (with boundary). For the proof of this statement we show in addition that each piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly. This solves a challenge by Thurston and confirms a conjecture by Kwasik and Lee. In the topological category a counterexample to the above mentioned characterization is given by the binary icosahedral group. We show that this is the only counterexample up to products. In particular, we answer the question by Davis of when the underlying space of an orbifold is a topological manifold. As a corollary of our results we generalize a fixed point theorem by Steinberg on unitary reflection groups to finite groups generated by reflections and rotations. As an application thereof we answer a question by Petrunin on quotients of spheres.

Swept frequency biompedance analysis for the determination of body water compartments

Bioelectrical impedance analysis, (BIA), is a method of body composition analysis first investigated in 1962 which has recently received much attention by a number of research groups. The reasons for this recent interest are its advantages, (viz: inexpensive, non-invasive and portable) and also the increasing interest in the diagnostic value of body composition analysis. The concept utilised by BIA to predict body water volumes is the proportional relationship for a simple cylindrical conductor, (volume oc length2/resistance), which allows the volume to be predicted from the measured resistance and length. Most of the research to date has measured the body's resistance to the passage of a 50· kHz AC current to predict total body water, (TBW). Several research groups have investigated the application of AC currents at lower frequencies, (eg 5 kHz), to predict extracellular water, (ECW). However all research to date using BIA to predict body water volumes has used the impedance measured at a discrete frequency or frequencies. This thesis investigates the variation of impedance and phase of biological systems over a range of frequencies and describes the development of a swept frequency bioimpedance meter which measures impedance and phase at 496 frequencies ranging from 4 kHz to 1 MHz. The impedance of any biological system varies with the frequency of the applied current. The graph of reactance vs resistance yields a circular arc with the resistance decreasing with increasing frequency and reactance increasing from zero to a maximum then decreasing to zero. Computer programs were written to analyse the measured impedance spectrum and determine the impedance, Zc, at the characteristic frequency, (the frequency at which the reactance is a maximum). The fitted locus of the measured data was extrapolated to determine the resistance, Ro, at zero frequency; a value that cannot be measured directly using surface electrodes. The explanation of the theoretical basis for selecting these impedance values (Zc and Ro), to predict TBW and ECW is presented. Studies were conducted on a group of normal healthy animals, (n=42), in which TBW and ECW were determined by the gold standard of isotope dilution. The prediction quotients L2/Zc and L2/Ro, (L=length), yielded standard errors of 4.2% and 3.2% respectively, and were found to be significantly better than previously reported, empirically determined prediction quotients derived from measurements at a single frequency. The prediction equations established in this group of normal healthy animals were applied to a group of animals with abnormally low fluid levels, (n=20), and also to a group with an abnormal balance of extra-cellular to intracellular fluids, (n=20). In both cases the equations using L2/Zc and L2/Ro accurately and precisely predicted TBW and ECW. This demonstrated that the technique developed using multiple frequency bioelectrical impedance analysis, (MFBIA), can accurately predict both TBW and ECW in both normal and abnormal animals, (with standard errors of the estimate of 6% and 3% for TBW and ECW respectively). Isotope dilution techniques were used to determine TBW and ECW in a group of 60 healthy human subjects, (male. and female, aged between 18 and 45). Whole body impedance measurements were recorded on each subject using the MFBIA technique and the correlations between body water volumes, (TBW and ECW), and heighe/impedance, (for all measured frequencies), were compared. The prediction quotients H2/Zc and H2/Ro, (H=height), again yielded the highest correlation with TBW and ECW respectively with corresponding standard errors of 5.2% and 10%. The values of the correlation coefficients obtained in this study were very similar to those recently reported by others. It was also observed that in healthy human subjects the impedance measured at virtually any frequency yielded correlations not significantly different from those obtained from the MFBIA quotients. This phenomenon has been reported by other research groups and emphasises the need to validate the technique by investigating its application in one or more groups with abnormalities in fluid levels. The clinical application of MFBIA was trialled and its capability of detecting lymphoedema, (an excess of extracellular fluid), was investigated. The MFBIA technique was demonstrated to be significantly more sensitive, (P<.05), in detecting lymphoedema than the current technique of circumferential measurements. MFBIA was also shown to provide valuable information describing the changes in the quantity of muscle mass of the patient during the course of the treatment. The determination of body composition, (viz TBW and ECW), by MFBIA has been shown to be a significant improvement on previous bioelectrical impedance techniques. The merit of the MFBIA technique is evidenced in its accurate, precise and valid application in animal groups with a wide variation in body fluid volumes and balances. The multiple frequency bioelectrical impedance analysis technique developed in this study provides accurate and precise estimates of body composition, (viz TBW and ECW), regardless of the individual's state of health.

Genetics of brain fiber architecture and intellectual performance

The study is the first to analyze genetic and environmental factors that affect brain fiber architecture and its genetic linkage with cognitive function. We assessed white matter integrity voxelwise using diffusion tensor imaging at high magnetic field (4 Tesla), in 92 identical and fraternal twins. White matter integrity, quantified using fractional anisotropy (FA), was used to fit structural equation models (SEM) at each point in the brain, generating three-dimensional maps of heritability. We visualized the anatomical profile of correlations between white matter integrity and full-scale, verbal, and performance intelligence quotients (FIQ, VIQ, and PIQ). White matter integrity (FA) was under strong genetic control and was highly heritable in bilateral frontal (a 2 = 0.55, p = 0.04, left; a 2 = 0.74, p = 0.006, right), bilateral parietal (a 2 = 0.85, p < 0.001, left; a 2 = 0.84, p < 0.001, right), and left occipital (a 2 = 0.76, p = 0.003) lobes, and was correlated with FIQ and PIQ in the cingulum, optic radiations, superior fronto- occipital fasciculus, internal capsule, callosal isthmus, and the corona radiata (p = 0.04 for FIQ and p = 0.01 for PIQ, corrected for multiple comparisons). In a cross-trait mapping approach, common genetic factors mediated the correlation between IQ and white matter integrity, suggesting a common physiological mechanism for both, and common genetic determination. These genetic brain maps reveal heritable aspects of white matter integrity and should expedite the discovery of single-nucleotide polymorphisms affecting fiber connectivity and cognition.

The heat kernel on AdS(3) and its applications

We derive the heat kernel for arbitrary tensor fields on S-3 and (Euclidean) AdS(3) using a group theoretic approach. We use these results to also obtain the heat kernel on certain quotients of these spaces. In particular, we give a simple, explicit expression for the one loop determinant for a field of arbitrary spin s in thermal AdS(3). We apply this to the calculation of the one loop partition function of N = 1 supergravity on AdS(3). We find that the answer factorizes into left- and right-moving super Virasoro characters built on the SL(2, C) invariant vacuum, as argued by Maloney and Witten on general grounds.