Continuous Symmetry Measures. VI. The Relations Between Polyhedral Point-Group/Subgroup Symmetries

We extend our analysis of the symmetry content of the classical polyhedra [1] to the analysis of the degree of polyhedral subgroup symmetries. The quantitative levels of the hierarchical polyhedral symmetries series of O_h, D_4h and D_2h of hexacoordinated structures, as well as the relations between them, serve as an example. A distinction is made between two types of measures: quantitative evaluation of the degree of symmetry, and quantitative evaluation of the degree of content of a reference shape.     

A Constitutive Model for Swelling Pressure and Volumetric Behavior of Highly-Hydrated Connective Tissue

The fluid pressure within water-filled connective tissues such as cartilage, intervertebral disk and cornea facilitates a vital part of their functionality. Cartilage and intervertebral disk must resist compressive loading, and even the cornea uses fluid pressure loading to form its precise refractive geometry. The fluid pressure is composed of hydrostatic and swelling pressure components, with the latter deriving primarily from osmotic forces associated with ion concentrations. The tissues, like electrolyte gels in general, have a strong tendency to absorb water and swell. The goal of this work is to formulate an in vivo tissue theory from first principles and to arrive at the simplest possible model which captures the essential features of tissue charge effects and swelling behavior. The Gibbs free energy of the tissue, including elastic, hydrostatic, and electrostatic components, is characterized and equilibrium thermodynamics is employed to find explicit constitutive equations for the tissue osmotic pressure and osmotic compressibility over a unit cell. It is shown that the osmotic compressibility essentially defines the tissue macroscopic pressure-volume relationship at equilibrium. To illustrate the theory in detail, the human cornea is taken as a model tissue, and it is shown how the nanometer features of the glycosaminoglycan charge distribution can be modeled within the proposed theory. The cornea model is further extended to include the effects of the pump-leak hydration control mechanism based on active ion and passive water transport across the corneal endothelium. The model has been implemented in a standard finite element code and is shown to be capable of reproducing fundamental in vitro swelling experiments, including massive swelling, and typical in vivo swelling observed in disease states such as Fuch’s dystophy.     

Approximate Bogomol'nyi-Prasad-Sommerfield states

We consider dimensionally reduced three-dimensional supersymmetric Yang-Mills-Chern-Simons theory. Although the N=1 supersymmetry of this theory does not allow local massive Bogomol'nyi-Prasad-Sommerfield (BPS) states, we find approximate BPS states which have nonzero masses that are almost independent of the Yang-Mills coupling constant and which are a reflection of the massless BPS states of the underlying N=1 super-Yang-Mills theory. The masses of these states at large Yang-Mills coupling are exactly at the n-particle continuum thresholds. This leads to a relation between their masses at zero and large Yang-Mills coupling.     

Simulation of Space Shuttle neutron measurements with FLUKA

FLUKA is an integrated particle transport code that has enhanced multigroup low-energy neutron transport capability similar to the well-known MORSE transport code. Gammas are produced in groups but many important individual lines are specifically included, and subsequently transported by the main FLUKA routines which use a modified version of EGS4 for electromagnetic (EM) transport. Recoil protons are also transported by the primary FLUKA transport simulation. The neutron cross-section libraries employed within FLUKA were supplied by Giancarlo Panini (ENEA, Italy) based upon the most recent data from JEF-1, JEF-2.2, ENDF/B-VI, JENDL-3, etc. More than 60 different materials are included in the FLUKA databases with temperature ranges including down to cryogenic temperatures. This code has been used extensively to model the neutron environments near high-energy physics experiment shielding. A simulation of the Space Shuttle based upon a spherical aluminum equivalent shielding distribution has been performed with reasonable results. There are good prospects for extending this calculation to a more realistic 3-D geometrical representation of the Shuttle including an accurate representation of its composition, which is an essential ingredient for the improvement of the predictions. A proposed project to develop a combined analysis and simulation package based upon FLUKA and the analysis infrastructure provided by the ROOT software is under active consideration. The code to be developed for this project will be of direct application to the problem of simulating the neutron environment in space, including the albedo effects.     

A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions

Let , , be a dimensional slab. Denote points by , where and . Denoting the boundary of the slab by , let

where is an ordered sequence of intervals on the right half line (that is, b_{n}$">). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let . Let and denote respectively the cone of bounded, positive harmonic functions in and the cone of positive harmonic functions in which satisfy the Dirichlet boundary condition on and the Neumann boundary condition on .

Letting , the main result of this paper, under a modest assumption on the sequence , may be summarized as follows when :

1. If , then and are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if with 2$">.

2. If and , then and is one-dimensional. In particular, this occurs if .

3. If , then and the set of minimal elements generating is isomorphic to (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if with .

When , as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for is as above.

     

A Comment on the Light-Cone Vacuum in 1+1 Dimensional Super-Yang–Mills Theory

The discrete light-cone quantization (DLCQ) of a supersymmetric gauge theory in 1+1 dimensions is discussed, with particular attention given to the inclusion of the gauge zero mode. Interestingly, the notorious zero-mode problem is now tractable because of special supersymmetric cancellations. In particular, we show that anomalous zero-mode contributions to the currents are absent, in contrast to what is observed in the nonsupersymmetric case. An analysis of the vacuum structure is provided by deriving the effective quantum mechanical Hamiltonian of the gauge zero mode. It is shown that the inclusion of the zero modes of the adjoint scalars and fermions is crucial for probing the phase properties of the vacua. We find that the ground-state energy is zero and thus consistent with unbroken supersymmetry and conclude that the light-cone Fock vacuum is unchanged with or without the presence of matter fields.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.