Further Paradoxes in Generalized Levy Problems

The classical solution of Levy's plate problem breaks down for a critical plate angle, with both stress and displacement fields becoming unbounded throughout the plate. The same sort of paradoxical response occurs in traditional solutions for other critical angles if the pressure in Levy's original problem is replaced by a constant shear traction. These breakdowns can be rectified by suitably supplementing classical fields. In this note, a further generalization of Levy's problem is considered. This entails clamping the plate edge without applied tractions. Critical angles for which traditional analysis breaks down are identified, and new solutions are developed for these angles. This revised version was published online in July 2006 with corrections to the Cover Date.     

Uniform indentation of the elastic half-space by a rigid rectangular punch

The problem considered is that of a rigid flat-ended punch with rectangular contact area pressed into a linear elastic half-space to a uniform depth. Both the lubricated and adhesive cases are treated. The problem reduces to solving an integral equation (or equations) for the contact stresses. These stresses have a singular nature which is dealt with explicitly by a singularity-incorporating finite-element method. Values for the stiffness of the lubricated punch and the adhesive punch are determined: the effect of adhesion on the stiffness is found to be small, producing an increase of the order of 3%.     

Synthesis and Properties of Lanthanide Ruthenium(III) Oxide Perovskites

An extensive series of new LnRuO₃ perovskites has been synthesized at high pressure. These ruthenium(III)‐based oxides are ruthenium deficient, and high‐pressure samples have compositions close to LnRu_0.9O₃. These phases stabilize ruthenium(III) which is very unusual in oxides. X‐ray and neutron powder diffraction studies show that the materials adopt orthorhombic perovskite superstructures in which the RuO₆ octahedra are tetragonally compressed. These distortions, and the Mott insulator properties of the materials, are driven by strong spin‐orbit coupling.     

Isopeptide crosslinks--their occurrence and importance in protein structure

Isopeptide links refer to bonds between the ?-amino group of lysine and the side-chain carboxyl groups of glutamic or aspartic acid. Covalent crosslinks of this kind, which undoubtedly occur in protein feedstuffs and reduce their nutritional value, also participate In the fibrinogen → fibrin transformation and play a part in stabilizing the structure of keratins. Isopeptide links also arise on heating of both fibrous and globular proteins, the amount formed depending upon the severity of heat treatment. The mode of formation and the significance of these crosslinks require further elucidation.     

Abstracts of interest

Experimental Techniques -     

Delaying satisfiability for random 2SAT

Let (C₁,C′_(*)),(C₂,C′_(*)),…,(C _m,C′_(*)) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4 \begin{align*}\binom{n}{2}\end{align*} clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on‐line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on‐line choice algorithm in the above process which results whp in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)^1/4. This contrasts with the well‐known fact that a random m ‐clause formula constructed without the choice of two clauses at each step is unsatisfiable whp whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as “Achlioptas processes.” This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on‐line setting above, we also consider an off‐line version in which all m clause‐pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off‐line setting, we show that the two‐choice satisfiability threshold for k ‐SAT for any fixed k coincides with the standard satisfiability threshold for random 2k ‐SAT.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Preparation,structural and magnetic characterization of synthetic anti-ferromagnetic (SAF) nanoparticles

Synthetic anti-ferromagnetic nanoparticles (SAFs) are a novel type of magnetic nanoparticle (MNP) fabricated using nanoimprint lithography, direct deposition of multilayer films and retrieval in liquid phase via an ‘etching’ release process. Such physical fabrication techniques enable accurate control of particle shape, size and composition. We systematically varied the processing conditions to produce different configurations of SAF nanoparticles and performed extensive characterization using transmission electron microscopy (TEM) and alternating gradient magnetometry (AGM) to study their corresponding structural and magnetic behavior. A key focus of this paper is the preparation of TEM cross-section specimens of SAF nanoparticles for their structural characterization. This is not a trivial task, but is useful and revealing in terms of structural features. A major finding from our study shows that the introduction of oxygen during deposition of the copper release layer gives significantly improved flatness of the multilayer structure but no significant change in the magnetic properties. Magnetic measurements show that these nanoparticles have nearly zero magnetic remanence, a linear response of magnetization and more than twice the saturation magnetization compared to iron oxide nanoparticles.     

Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most  $d$ , throughout the uniqueness region of the Gibbs measure on the infinite $d$ -regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree  $d$ for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.