Properties of 3-manifolds for relativists

In canonical quantum gravity certain topological properties of 3-manifolds are of interest. This article gives an account of those properties which have so far received sufficient attention, especially those concerning the diffeomorphism groups of 3-manifolds. We give a summary of these properties and list some old and new results concerning them. The appendix contains a discussion of the group of large diffeomorphisms of the $l$-handle 3-manifold.Comment: 20 pages. Plain-TeX, no figures, 1 Table (A4 format

An Analysis of the Representations of the Mapping Class Group of a Multi-Geon Three-Manifold

It is well known that the inequivalent unitary irreducible representations (UIR's) of the mapping class group $G$ of a 3-manifold give rise to ``theta sectors'' in theories of quantum gravity with fixed spatial topology. In this paper, we study several families of UIR's of $G$ and attempt to understand the physical implications of the resulting quantum sectors. The mapping class group of a three-manifold which is the connected sum of $\R^3$ with a finite number of identical irreducible primes is a semi-direct product group. Following Mackey's theory of induced representations, we provide an analysis of the structure of the general finite dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables one to draw several interesting qualitative conclusions about the geons' behavior in different quantum sectors, without requiring an explicit knowledge of the UIR's corresponding to the individual primes.Comment: 52 pages, harvmac, 2 postscript figures, epsf required. Added an appendix proving the semi-direct product structure of the MCG, corrected an error in the characterization of the slide subgroup, reworded extensively. All our analysis and conclusions remain as befor

State space c-reductions for concurrent systems in rewriting logic

We present c-reductions, a state space reduction technique. The rough idea is to exploit some equivalence relation on states (possibly capturing system regularities) that preserves behavioral properties, and explore the induced quotient system. This is done by means of a canonizer function, which maps each state into a (non necessarily unique) canonical representative of its equivalence class. The approach exploits the expressiveness of rewriting logic and its realization in Maude to enjoy several advantages over similar approaches: exibility and simplicity in the definition of the reductions (supporting not only traditional symmetry reductions, but also name reuse and name abstraction); reasoning support for checking and proving correctness of the reductions; and automatization of the reduction infrastructure via Maude's meta-programming features. The approach has been validated over a set of representative case studies, exhibiting comparable results with respect to other tools

Cyclic Statistics In Three Dimensions

While 2-dimensional quantum systems are known to exhibit non-permutation, braid group statistics, it is widely expected that quantum statistics in 3-dimensions is solely determined by representations of the permutation group. This expectation is false for certain 3-dimensional systems, as was shown by the authors of ref. [1,2,3]. In this work we demonstrate the existence of ``cyclic'', or $Z_n$, {\it non-permutation group} statistics for a system of n > 2 identical, unknotted rings embedded in $R^3$. We make crucial use of a theorem due to Goldsmith in conjunction with the so called Fuchs-Rabinovitch relations for the automorphisms of the free product group on n elements.Comment: 13 pages, 1 figure, LaTex, minor page reformattin

Numerical investigation of the impact behaviour of bioinspired nacre-like aluminium composite plates

Inspired by the hierarchical structure of nacre, an aluminium alloy (AA) 7075 based composite featuring layer waviness and cohesive interface is studied as a low weight impact resistant material. To investigate the mechanical response and the ballistic performance of this laminated structure, a numerical study of the proposed nacre-like composite plates made of 1.1-mm thick AA 7075 tablets bonded with toughened epoxy resin was performed using Abaqus/Explicit. Target thicknesses of 5.4-mm, 7.5-mm and 9.6-mm impacted by a rigid hemi-spherical projectile were simulated. The epoxy material was modelled using a user-defined interface cohesive element with compressive strength enhancement. A significant performance improvement was recorded for the 5.4-mm nacre-like plate (compared to the same thickness bulk plate), which was explained by the hierarchical structure facilitating both localised energy absorption (by deformation of the tablet) and more globalized energy absorption (by inter-layered delamination and friction). For a given projectile, however, the performance improvement of using the proposed composite decreased with increasing laminate thickness, which was attributed to the increased likelihood of ductile failure occurring prior to perforation in thicker bulk plates. For 5.4-mm thick plates impacted at high velocity, the nacre-like plate had a better ballistic performance than that of the plates made of continuous (flat and wavy) layers, which was attributed to the larger area of plastic deformation (observed in the nacre-like plate after impact) due to the tablets arrangement.The Australian Research Council Centre of Excellence for Design in Light Metals (CE0561574); National Natural Science Foundation of China (No. 11232003); The Australian Research Council via project DP1093485

Engineering photocycle dynamics. Crystal structures and kinetics of three photoactive yellow protein hinge-bending mutants

Crystallographic and spectroscopic analyses of three hinge-bending mutants of the photoactive yellow protein are described. Previous studies have identified Gly(47) and Gly(51) as possible hinge points in the structure of the protein, allowing backbone segments around the chromophore to undergo large concerted motions. We have designed, crystallized, and solved the structures of three mutants: G47S, G51S, and G47S/G51S. The protein dynamics of these mutants are significantly affected. Transitions in the photocycle, measured with laser induced transient absorption spectroscopy, show rates up to 6-fold different from the wild type protein and show an additive effect in the double mutant. Compared with the native structure, no significant conformational differences were observed in the structures of the mutant proteins. We conclude that the structural and dynamic integrity of the region around these mutations is of crucial importance to the photocycle and suggest that the hinge-bending properties of Gly(51) may also play a role in PAS domain proteins where it is one of the few conserved residues

Stationary Kolmogorov Solutions of the Smoluchowski Aggregation Equation with a Source Term

In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. We derive a ``locality criterion'', expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels,$K_\zeta(m_1,m_2)=(m_1m_2)^{\zeta/2}$ which includes both gelling and non-gelling cases, reproducing the known solution in the case $\zeta=0$. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.Comment: This article is an expanded version of a talk given at IHP workshop "Dynamics, Growth and Singularities of Continuous Media", Paris July 2003. Updated 01/04/04. Revised version with additional discussion, references added, several typographical errors corrected. Revised version accepted for publication by Phys. Rev.

Quantitative measurement of sliding friction dynamics at mesoscopic scales: The lateral force apparatus

We describe an apparatus designed to quantitatively measure friction dynamics at the mesoscopic scale. This lateral force apparatus, LFA, uses double parallel leaf springs in leaf-spring units as force transducers and two focus error detection optical heads, optical heads, to measure deflections. The design of the leaf-spring units is new. Normal spring constants are in the range of 20–4000 N/m, and lateral spring constants are 7–1000 N/m. The optical heads combine a 10 nm sensitivity with a useful range of about 100 µm. The proven range of normal forces is 400 nN–150 mN. The leaf-spring units transduce friction and normal forces independently. Absolute values of normal and friction forces are calibrated. Typical errors are less than 10%. The calibration is partly in situ, for the sensitivity of the optical heads, and partly ex situ for the normal and lateral spring constants of the leaf-spring units. There is minimal coupling between the deflection measurements in the lateral and normal directions. This coupling is also calibrated in situ. It is typically 1% and can be as low as 0.25%. This means that the displacements of the tip can be measured accurately in the sliding direction and normal to the surface. Together, these characteristics make the LFA, well suited for quantitative study of friction dynamics at mesoscopic scales. Furthermore the design of the leaf-spring unit allows exchange of tips which may be fabricated (e.g., etched) from wire material (d0.4 mm) and can have customized shapes, e.g., polished flat squares. The ability of the LFA to study friction dynamics is briefly illustrated by results of stick-slip measurements on soft polymer surfaces