Phase Segregation Dynamics in Particle Systems with Long Range Interactions I: Macroscopic Limits

We present and discuss the derivation of a nonlinear non-local integro-differential equation for the macroscopic time evolution of the conserved order parameter of a binary alloy undergoing phase segregation. Our model is a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, i.e. a (Poisson) nearest-neighbor exchange process, reversible with respect to the Gibbs measure for a Hamiltonian which includes both short range (local) and long range (nonlocal) interactions. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (part II), we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.Comment: amstex with macros (included in the file), tex twice, 20 page

Point Interaction in two and three dimensional Riemannian Manifolds

We present a non-perturbative renormalization of the bound state problem of n bosons interacting with finitely many Dirac delta interactions on two and three dimensional Riemannian manifolds using the heat kernel. We formulate the problem in terms of a new operator called the principal or characteristic operator. In order to investigate the problem in more detail, we then restrict the problem to one particle sector. The lower bound of the ground state energy is found for general class of manifolds, e.g., for compact and Cartan-Hadamard manifolds. The estimate of the bound state energies in the tunneling regime is calculated by perturbation theory. Non-degeneracy and uniqueness of the ground state is proven by Perron-Frobenius theorem. Moreover, the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics. Renormalization procedure does not lead to any radical change in these cases. Finally, renormalization group equations are derived and the beta-function is exactly calculated. This work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions, developed by S. G. Rajeev.Comment: 43 page

A spinorial energy functional: critical points and gradient flow

On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, {\phi}) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor {\phi}. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.Comment: Small changes, final versio

AdS spacetimes from wrapped D3-branes

We derive a geometrical characterisation of a large class of AdS_3 and AdS_2 supersymmetric spacetimes in IIB supergravity with non-vanishing five-form flux using G-structures. These are obtained as special cases of a class of supersymmetric spacetimes with an $\mathbb{R}^{1,1}$ or $\mathbb{R}$ (time) factor that are associated with D3-branes wrapping calibrated 2- or 3- cycles, respectively, in manifolds with SU(2), SU(3), SU(4) and G_2 holonomy. We show how two explicit AdS solutions, previously constructed in gauged supergravity, satisfy our more general G-structure conditions. For each explicit solution we also derive a special holonomy metric which, although singular, has an appropriate calibrated cycle. After analytic continuation, some of the classes of AdS spacetimes give rise to known classes of BPS bubble solutions with $\mathbb{R}\times SO(4)\times SO(4)$, $\mathbb{R}\times SO(4)\times U(1)$, and $\mathbb{R}\times SO(4)$ symmetry. These have 1/2, 1/4 and 1/8 supersymmetry, respectively. We present a new class of 1/8 BPS geometries with $\mathbb{R}\times SU(2)$ symmetry, obtained by analytic continuation of the class of AdS spacetimes associated with D3-branes wrapped on associative three-cycles.Comment: 1+30 pages; v2, references added; v3, typos corrected, reference adde

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II

Let $L=\Delta-\nabla\varphi\cdot\nabla$ be a symmetric diffusion operator with an invariant measure $d\mu=e^{-\varphi}dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla \varphi|\leq\theta$ and $\infty$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator $L$ on this kind manifold, and thereby generalize a Cheng's result on the Laplacian case (Math. Z., 143 (1975) 289-297).Comment: Final version. The original proof of Theorem 2.1 using Li-Yau gradient estimate method has been moved to the appendix. The new proof is simple and direc

The UK Crop Microbiome Cryobank: a utility and model for supporting Phytobiomes research

Plant microbiomes are the microbial communities essential to the functioning of the phytobiome—the system that consist of plants, their environment, and their associated communities of organisms. A healthy, functional phytobiome is critical to crop health, improved yields and quality food. However, crop microbiomes are relatively under-researched, and this is associated with a fundamental need to underpin phytobiome research through the provision of a supporting infrastructure. The UK Crop Microbiome Cryobank (UKCMC) project is developing a unique, integrated and open-access resource to enable the development of solutions to improve soil and crop health. Six economically important crops (Barley, Fava Bean, Oats, Oil Seed Rape, Sugar Beet and Wheat) are targeted, and the methods as well as data outputs will underpin research activity both in the UK and internationally. This manuscript describes the approaches being taken, from characterisation, cryopreservation and analysis of the crop microbiome through to potential applications. We believe that the model research framework proposed is transferable to different crop and soil systems, acting not only as a mechanism to conserve biodiversity, but as a potential facilitator of sustainable agriculture systems

Expert consensus guidelines for the genetic diagnosis of Alport syndrome

Recent expert guidelines recommend genetic testing for the diagnosis of Alport syndrome. Here, we describe current best practice and likely future developments. In individuals with suspected Alport syndrome, all three COL4A5, COL4A3 and COL4A4 genes should be examined for pathogenic variants, probably by high throughput-targeted next generation sequencing (NGS) technologies, with a customised panel for simultaneous testing of the three Alport genes. These techniques identify up to 95% of pathogenic COL4A variants. Where causative pathogenic variants cannot be demonstrated, the DNA should be examined for deletions or insertions by re-examining the NGS sequencing data or with multiplex ligation-dependent probe amplification (MLPA). These techniques identify a further 5% of variants, and the remaining few changes include deep intronic splicing variants or cases of somatic mosaicism. Where no pathogenic variants are found, the basis for the clinical diagnosis should be reviewed. Genes in which mutations produce similar clinical features to Alport syndrome (resulting in focal and segmental glomerulosclerosis, complement pathway disorders, MYH9-related disorders, etc.) should be examined. NGS approaches have identified novel combinations of pathogenic variants in Alport syndrome. Two variants, with one in COL4A3 and another in COL4A4, produce a more severe phenotype than an uncomplicated heterozygous change. NGS may also identify further coincidental pathogenic variants in genes for podocyte-expressed proteins that also modify the phenotype. Our understanding of the genetics of Alport syndrome is evolving rapidly, and both genetic and non-genetic factors are likely to contribute to the observed phenotypic variability

A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of K\"ahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an $n$-dimensional cone a rescaling of the canonical potential is an $n$-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde