On residuals of finite groups

A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in \cite{DHKL}, we establish the following generalisation: if $\mathfrak{X}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups $G$ with trivial $\mathfrak{X}$-radical, \begin{equation} \left\lvert G^{\overline{\mathfrak{X}}}\right\rvert \,>\, \vert G\vert^\gamma, \end{equation} where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When $\mathfrak{X} = \mathfrak{N}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J.\,G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations $\mathfrak{X}$ of full characteristic such that $\mathfrak{S} \subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of \cite[Thm.~C]{DHKL}.Comment: 19 page

The structure of separated flow regions occurring near the leading edge of airfoils including transition

All the time and effort was directed toward acquiring, reducing, and analyzing more hot-wire anemometer data. Some static pressure distribution data were also acquired to support the analysis of the velocity profile data. Laser Doppler Velocimetry data were not acquired due to equipment problems. Included were seven combinations of chord Reynolds number, angle of attack, and acoustic forcing using the NACA 663-018 airfoil. This research has as its objective the detailed documentation of the structure and behavior of the transitional separation bubble and the redeveloping boundary layer after reattachment over an airfoil at low Reynolds numbers. The intent is to further the understanding of the complex flow phenomena so that analytic methods for predicting their formulation and development can be improved. These analytic techniques have applications in the design and performance prediction of airfoils operating in the low Reynolds number flight regime

An Evolutionary Interpretation of Teleostean Forebrain Anatomy

During the past few years, our investigations of the forebrain in the zebrafish (a teleost fish) have shown that its molecular anatomy and expression patterns of genes involved in the regulation of neuronal transmitter phenotypes, such as gamma-aminobutyric acid- (GABA-)ergic neurons, are very similar to those seen in mammalian model organisms such as mouse and rat. For example, we have been able to identify previously undiscovered homologies, such as subpallial regions in the zebrafish that are homologous to the medial and lateral ganglionic eminences in mammals, as well as regions homologous to the larval eminentia thalami and its adult derivative, the bed nucleus of the stria medullaris. Furthermore, in what we term the partial eversion model of the telencephalon in teleosts, we propose homologies to all four mammalian pallial areas and conclude that the posterior zone of the dorsal telencephalic area in teleosts is homologous to the piriform cortex and is formed by a migratory stream of cells originating in a dorsomedial zone of the pallium (the primordial medial zone of area dorsalis telencephali). In this review we critically discuss and justify these findings in the context of forebrain evolution in fishes. Copyright (C) 2009 S. Karger AG, Base

Electron-capture supernovae as sources of 60Fe

We investigate the nucleosynthesis of the radionuclide 60Fe in electron-capture supernovae (ECSNe). The nucleosynthetic results are based on a self-consistent, two-dimensional simulation of an ECSN as well as models in which the densities are systematically increased by some factors (low-entropy models). 60Fe is found to be appreciably made in neutron-rich ejecta during the nuclear quasi-equilibrium phase with greater amounts being produced in the lower-entropy models. Our results, combining them with the yields of core-collapse supernovae (CCSNe) in the literature, suggest that ECSNe account for at least 4-30% of live 60Fe in the Milky Way. ECSNe co-produce neutron-rich isotopes, 48Ca, 50Ti, 54Cr, some light trans-iron elements, and possibly weak r-process elements including some radionuclides such as 93Zr, 99Tc, and 107Pd, whose association with 60Fe might have been imprinted in primitive meteorites or in the deep ocean crust on the Earth.Comment: 6 pages, 2 figures, accepted for publication in ApJ