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G-Reflectors: Analogues of Householder Transformations in Scalar Product Spaces
No full textWe characterize the analogues of Householder transformations in matrix groups associated with scalar products, and precisely delimit their mapping capabilities: given a matrix group Image and vectors x, y, necessary and sufficient conditions are derived for the existence of a Householder-like analogue Image such that Gx=y. When G exists, we show how it can be constructed from x and y. Examples of matrix groups to which these results apply include the symplectic and pseudo-unitary groups
Behavioral and physical masculinization are related to genotype in girls with congenital adrenal hyperplasia
No full textRoyal Manchester Children’s Hospital (C.M.H., J.A.J., D.A.P., P.E.C.), Manchester M27 4HA, Regional Molecular Genetics Laboratory (M.C.), St. Mary’s Hospital, Manchester M13 0JH, and Department of Mathematics (P.F.), University of Manchester, Manchester M13 9PL, United Kingdom; and Department of Psychiatry (H.F.L.M.-B., C.D.), Columbia University, New York, New York 10027
Address all correspondence and requests for reprints to: Catherine M. Hall, Endocrine Department, Royal Manchester Children’s Hospital, Manchester M27 4HA, United Kingdom. E-mail: [email protected].
Girls with congenital adrenal hyperplasia (CAH) exhibit behavioral masculinization. There is controversy about the roles of pre- and postnatal androgens, social factors, and chronic illness in its etiology.
To assess the effect of chronic illness, we compared behavioral masculinity in 24 CAH girls and 25 diabetic girls aged 3–12 yr from Manchester using two sensitive questionnaires, and an overall masculinity score M (high = masculine) was derived.
To assess the contributions of pre- and postnatal androgens, the CAH subjects were categorized into genotype groups (G) according to the reported severity of loss of CYP21 function: G1 (n = 10, null mutations), G2 (n = 9, intron 2G), G3 (n = 3, I172N), and G4 (n = 2, unknown loss of function). In CAH girls, relationships between G, Prader degree of genital masculinization at birth, bone age advance, and M were assessed.
CAH girls were less feminine and more masculine than diabetic girls (P < 0.001), who were not significantly different from U.S. controls. Among the CAH girls, those in G1 and 2 were more genitally masculinized than those in G3 and 4 (P < 0.009) and had higher M (P < 0.025). M was negatively correlated with advanced bone age (r = -0.5; P = 0.02).
CAH girls, but not diabetic girls, demonstrated behavioral masculinization. Both physical and behavioral masculinization were related to each other and to genotype, indicating that behavioral masculinization is a consequence of prenatal androgen exposure
On the spatial development of a dusty wall jet
Get PDFWe consider the flow of an incompressible particle-laden fluid through the application of the so-called �dusty-gas� equations, which treat the fluid/particle suspension as two continua. The two phases are described by their individual field equations and interact through a Stokes-drag mechanism. The particular flow we consider is of boundary-layer type, corresponding to the downstream development of a Glauert-type jet adjacent to a horizontal boundary (the inclusion of the particulate phase requires the flow to be non-self-similar). We solve the governing boundary-layer equations through a numerical spatial marching technique in the three distinct cases of (i) weak gravitational influence, (ii) a jet �above� a wall under the action of gravity and (iii) a jet �below� a wall under the action of gravity.
The qualitative and quantitative features of the three cases are quite different and are presented in detail. Of particular interest is the development of a stagnation point in the particle velocity field at a critical downstream location in case (i), the development of fluid/particle flow reversal in case (ii) and the development of �shock� solutions and particle-free regions in case (iii). Asymptotic descriptions are given of the critical phenomena, which support the numerical results. It is found that inclusion of a Saffman force has no substantial effect on either the location or structure of the stagnation-point region
Golden gaskets: variations on the Sierpinski sieve
No full textWe consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ in (0, 1).
As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2 < λ < 2/3 , i.e. when there are ‘overlaps’ in S_λ as well as
‘holes’. In this introductory paper we show that despite the overlaps (i.e. the breaking down of the open set condition (OSC)), the attractor can still be a totally self-similar fractal, although this happens only for a very special family
of algebraic λ (so-called multinacci numbers). We evaluate the ausdorff dimension of S_λ for these special values by showing that S_λ is essentially the attractor for an infinite IFS that does satisfy the OSC. We also show that the set of points in the attractor with a unique ‘address’ is self-similar and compute its dimension.
For non-multinacci values of λ we show that if λ is close to 2/3 , then S_λ has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question
Injective objects in triangulated categories
Get PDFWe extend ideas and results of Benson and Krause on pure-injectives in triangulated categories. Given a generating set of compact objects in a compactly generated triangulated category T we define notions of monomorphism, exactness and injectivity relative to this set. We show that the injectives correspond to injective objects in a localisation of the functor category Mod T c where Tc denotes the subcategory of compact objects of T. The paper begins by setting up the required localisation theory.
Benson and Krause [BK] showed that injective modules over the Tate cohomology ring of a finite group algebra kG, where k is a field of characteristic p and G is a p-group, correspond to certain pure-injective objects in the (compactly generated, triangulated) stable module category of kG. We generalise this to arbitrary compactly generated triangulated categories, replacing the trivial module k by any compact object and the Tate cohomology ring by the graded endomorphism ring of that object. We obtain the strongest results in the case that this graded endomorphism ring is coherent
Numerical solution of the Navier-Stokes equations for the flow in a cylinder cascade
Get PDFA numerical study of the steady, two-dimensional incompressible flow past a cascade of circular cylinders is presented. The Navier–Stokes equations are written in terms of the streamfunction and vorticity and solved using a novel numerical technique based on using the Chebychev collocation method in one direction and high-order finite differences in the other direction. A direct solver combined with Newton–Raphson linearization is used to solve the discrete equations. Steady flow solutions have been obtained for large Reynolds numbers, far higher than those obtained previously, and for varying gap widths between the cylinders. Three distinct types of solutions, dependent on the gap width, have been found. Comparisons with theoretical predictions for various flow quantities show good agreement, especially for the narrow gap width case. However, existing theories are unable to explain the solution properties which exist for intermediate gap widths
When every finitely generated flat module is projective
No full textWe investigate the class of rings over which every finitely generated flat right module is projective
Accretion disc dynamos opened up by external magnetic fields
Get PDFWe show that a relatively weak poloidal external magnetic field, imposed in the outer parts of a Keplerian accretion disc, can open magnetic lines of a much stronger magnetic field generated by the dynamo in the inner disc. The resulting magnetic configuration can be favourable for launching a centrifugally driven wind. Even for a relatively weak external magnetic field (with energy density in excess of about only 10% of the thermal energy density at the outer disc radius) the geometry of the poloidal field in the disc neigbourhood is almost independent of the dynamo action, and is determined by the external field. The radial profile of the poloidal magnetic field on the disc surface is similar to that in the self-similar solution of Blandford & Payne (\cite{BP82}). We conclude that poloidal fields resulting from dynamo action can be important for launching a magneto-centrifugally driven outflow in accretion discs that occur in weakly magnetized environments, i.e. where an external field is too weak to be important. However, even a relatively weak external field can be wound up by the differential rotation to give a much stronger azimuthal field, and so modify dynamo action in the disc
The Hinfinity-norm calculation for large sparse systems.
Get PDFIn this paper, we describe an algorithm for estimating the H∞-norm of a large linear time
invariant dynamical system described by a discrete time state-space model. The algorithm is
designed to be efficient for state-space models defined by {A,B,C,D} where A is a large sparse
matrix of order n which is assumed very large relative to the input and output dimensions of
the system
THE DEVELOPMENT OF A NOVEL METHODOLOGY FOR TOMOGRAPHIC PHOTOELASTICITY
Get PDFA description is given of the current development of a novel, non-destructive, experimental method which allows the determination of 3-D stress distributions in birefringent materials. A conventional polariscope is used through which to view the model with the addition of a component positioning device. A minimal set of measurements has been devised which can be easily and efficiently realized in an actual laboratory arrangement, and which produces precisely the sufficient amount of data in order to tomographically reconstruct the full stress tensor within the bulk matter of the medium.
The paper includes an outline of the mathematical procedure, the results of numerical tests of the algorithms and a discussion of further work necessary to realise a truly practical technique