Incompatible sets of gradients and metastability

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an $L^1$ neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea

Quasiconvexity at the boundary and the nucleation of austenite

Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut

Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance

Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker-D\"oring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.Comment: 28 page

Lattice-Constrained Parametrizations of Form Factors for Semileptonic and Rare Radiative B Decays

We describe the form factors for semileptonic B to rho l nu and radiative B to K* gamma decays with just two parameters and the two form factors for semileptonic B to pi l nu decays with three parameters. The parametrizations are constrained by lattice results and are consistent with heavy quark symmetry, kinematic constraints and light cone sum rule scaling relations.Comment: 3 pages, latex, 2 eps files, uses epsf.sty and espcrc2.sty, poster presented at Lattice 97, Edinburgh, 22-26 July 199

The Meat Standards Australia Index indicates beef carcass quality

A simple index that reflects the potential eating quality of beef carcasses is very important for producer feedback. The Meat Standards Australia (MSA) Index reflects variation in carcass quality due to factors that are influenced by producers (hot carcass weight, rib fat depth, hump height, marbling and ossification scores along with milk fed veal category, direct or saleyard consignment, hormonal growth promotant status and sex). In addition, processor impacts on meat quality are standardised so that the MSA Index could be compared across time, breed and geographical regions. Hence, the MSA Index was calculated using achilles hung carcasses, aged for 5 days postmortem. Muscle pH can be impacted by production, transport, lairage or processing factors, hence the MSA Index assumes a constant pH of 5.6 and loin temperature of 7 o C for all carcasses. To quantify the cut weight distribution of the 39 MSA cuts in the carcass, 40 Angus steers were sourced from the low (n=13), high (n=15) and myostatin (n=12) muscling selection lines. The left side of each carcass was processed down to the 39 trimmed MSA cuts. There was no difference in MSA cut distribution between the low and high muscling lines (P>0.05), although there were differences with nine cuts from the myostatin line (P<0.05). There was no difference in the MSA Index calculated using actual muscle percentages and using the average from the low and high muscling lines (R 2 =0.99). Different cooking methods impacted via a constant offset between eating quality and carcass input traits (R 2 =1). The MSA Index calculated for the four most commercially important cuts was highly related to the index calculated using all 39 MSA cuts (R 2 =0.98), whilst the accuracy was lower for an index calculated using the striploin (R 2 =0.82). Therefore, the MSA Index was calculated as the sum of the 39 eating quality scores predicted at 5 days ageing, based on their most common cooking method, weighted by the proportions of the individual cut relative to total weight of all cuts. The MSA Index provides producers with a tool to assess the impact of management and genetic changes on the predicted eating quality of the carcass. The MSA Index could also be utilised for benchmarking and to track eating quality trends at farm, supply chain, regional, state or national levels

A stability version of H\"older's inequality

We present a stability version of H\"older's inequality, incorporating an extra term that measures the deviation from equality. Applications are given.Comment: Journal of Mathematical Analysis and Applications, Volume 343, Issue 2, Pages 842-852. This version differs from the published one in that it contains a new reference, and a trivial improvement of Corollary 3.2. fo

Second moment of the pion's distribution amplitude

We present preliminary results from the QCDSF/UKQCD collaborations for the second moment of the pion's distribution amplitude with two flavours of dynamical fermions. We use nonperturbatively determined renormalisation coefficients to convert our results to the MSbar scheme at 5 GeV^2. Employing a linear chiral extrapolation from our large pion masses >550 MeV, we find =0.281(28), leading to a value of a_2=0.236(82) for the second Gegenbauer moment.Comment: 6 pages, 5 figures, 1 table. To appear in: Procs. of the Workshop on Light-Cone QCD and Nonperturbative Hadron Physics 2005 (LC 2005), Cairns, Australia, 200

Heavy-to-light transition form factors and their relations in light-cone QCD sum rules

The improved light-cone QCD sum rules by using chiral current correlator is systematically reviewed and applied to the calculation of all the heavy-to-light form factors, including all the semileptonic and penguin ones. By choosing suitable chiral currents, the light-cone sum rules for all the form factors are greatly simplified and depend mainly on one leading twist distribution amplitude of the light meson. As a result, relations between these form factors arise naturally. At the considered accuracy these relations reproduce the results obtained in the literature. Moreover, since the explicit dependence on the leading twist distribution amplitudes is preserved, these relations may be more useful to simulate the experimental data and extract the information on the distribution amplitude.Comment: 1+16 pages, no figure

B→ϕπB\to \phi \pi and B0→ϕϕB^0 \to \phi\phi in the Standard Model and new bounds on R parity violation

We study the pure penguin decays $B \to \phi\pi$ and $B^0 \to \phi\phi$. Using QCD factorization, we find ${\cal B}(B^\pm \to\phi\pi^{\pm} )=2.0^{+0.3}_{-0.1}\times 10^{-8}$. For the pure penguin annihilation process $B^0 \to \phi\phi$, analyzed here for the first time, ${\cal B}(B^0 \to\phi\phi)=2.1^{+1.6}_{-0.3}\times 10^{-9}$. The smallness of these decays in the Standard Model makes them sensitive probes for new physics. From the upper limit of $B\to \phi\pi$,we find constraints on R parity violating couplings, $| \lambda{''}_{i23}\lambda{''}_{i21}|<6\times10^{-5}$, $| \lambda'_{i23}\lambda'_{i21}|<4\times10^{-4}$ and $| \lambda'_{i32}\lambda'_{i12}|<4\times10^{-4}$ for $i=1,2,3$. Our new bounds on $|\lambda{''}_{i23}\lambda{''}_{i21}|$ are one order of magnitude stronger than before. Within the available upper bounds for $| \lambda{''}_{i23}\lambda{''}_{i21}|$, $|\lambda'_{i23}\lambda'_{i21}|$ and $|\lambda'_{i32}\lambda'_{i12}|$, we find that ${\cal B}(B\to\phi\phi)$ could be enhanced to $10^{-8}\sim 10^{-7}$. Experimental searches for these decays are strongly urged.Comment: 5 pages, 3 figures embede