Palatini Variational Principle for an Extended Einstein-Hilbert Action

We consider a Palatini variation on a generalized Einstein-Hilbert action. We find that the Hilbert constraint, that the connection equals the Christoffel symbol, arises only as a special case of this general action, while for particular values of the coefficients of this generalized action, the connection is completely unconstrained. We discuss the relationship between this situation and that usually encountered in the Palatini formulation.Comment: 14 pages, LaTe

Hyperspectral chemical imaging reveals spatially varied degradation of polycarbonate urethane (PCU) biomaterials

Hyperspectral chemical imaging (HCI) is an emerging technique which combines spectroscopy with imaging. Unlike traditional point spectroscopy, which is used in the majority of polymer biomaterial degradation studies, HCI enables the acquisition of spatially localised spectra across the surface of a material in an objective manner. Here, we demonstrate that attenuated total reflectance Fourier transform infra-red (ATR-FTIR) HCI reveals spatial variation in the degradation of implantable polycarbonate urethane (PCU) biomaterials. It is also shown that HCI can detect possible defects in biomaterial formulation or specimen production; these spatially resolved images reveal regional or scattered spatial heterogeneity. Further, we demonstrate a map sampling method, which can be used in time-sensitive scenarios, allowing for the investigation of degradation across a larger component or component area. Unlike imaging, mapping does not produce a contiguous image, yet grants an insight into the spatial heterogeneity of the biomaterial across a larger area. These novel applications of HCI demonstrate its ability to assist in the detection of defective manufacturing components and lead to a deeper understanding of how a biomaterial’s chemical structure changes due to implantation. Statement of Signifance The human body is an aggressive environment for implantable devices and their biomaterial components. Polycarbonate urethane (PCU) biomaterials in particular were investigated in this study. Traditionally one or a few points on the PCU surface are analysed using ATR-FTIR spectroscopy. However the selection of acquisition points is susceptible to operator bias and critical information can be lost. This study utilises hyperspectral chemical imaging (HCI) to demonstrate that the degradation of a biomaterial varies spatially. Further, HCI revealed spatial variations of biomaterials that were not subjected to oxidative degradation leading to the possibility of HCI being used in the assessment of biomaterial formulation and/or component production

Non-Coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation

This paper presents three results on dependent site percolation on the square lattice. First, there exists no positively associated probability measure on {0,1}^{Z^2} with the following properties: a) a single infinite 0cluster exists almost surely, b) at most one infinite 1*cluster exists almost surely, c) some probabilities regarding 1*clusters are bounded away from zero. Second, we show that coexistence of an infinite 1*cluster and an infinite 0cluster is almost surely impossible when the underlying probability measure is ergodic with respect to translations, positively associated, and satisfies the finite energy condition. The third result analyses the typical structure of infinite clusters of both types in the absence of positive association. Namely, under a slightly sharpened finite energy condition, the existence of infinitely many disjoint infinite self-avoiding 1*paths follows from the existence of an infinite 1*cluster. The same holds with respect to 0paths and 0clusters.Comment: 17 pages, 1 figur

A theoretical and empirical investigation of nutritional label use

Due in part to increasing diet-related health problems caused, among others, by obesity, nutritional labelling has been considered important, mainly because it can provide consumers with information that can be used to make informed and healthier food choices. Several studies have focused on the empirical perspective of nutritional label use. None of these studies, however, have focused on developing a theoretical economic model that would adequately describe nutritional label use based on a utility theoretic framework. We attempt to fill this void by developing a simple theoretical model of nutritional label use, incorporating the time a consumer spends reading labels as part of the food choice process. The demand equations of the model are then empirically tested. Results suggest the significant role of several variables that flow directly from the model which, to our knowledge, have not been used in any previous empirical work

Trapping in the random conductance model

We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time $2n$. We show that in the situations when the heat kernel exhibits subdiffusive decay --- which is known to occur in dimensions $d\ge4$ --- the walk gets trapped for a time of order $n$ in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.Comment: 21 pages, version to appear in J. Statist. Phy

The Percolation Signature of the Spin Glass Transition

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin-Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering -- both in short-range (EA) and infinite-range (SK) models -- within a two-replica FK representation and also within the different Chayes-Machta-Redner two-replica graphical representation. Based on numerical studies of the $\pm J$ EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of {\it two} percolating clusters of {\it unequal} densities.Comment: 13 pages, 6 figure

Optimal designs for rational function regression

We consider optimal non-sequential designs for a large class of (linear and nonlinear) regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. The proposed method treats D-, E-, A-, and $\Phi_p$-optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well established numerical optimization methods. In contrast to previous optimization-based methods proposed for similar design problems, it also has theoretical guarantee of its algorithmic efficiency; in fact, the running times of all numerical examples considered in the paper are negligible. The stability of the method is demonstrated in an example involving high degree polynomials. After discussing linear models, applications for finding locally optimal designs for nonlinear regression models involving rational functions are presented, then extensions to robust regression designs, and trigonometric regression are shown. As a corollary, an upper bound on the size of the support set of the minimally-supported optimal designs is also found. The method is of considerable practical importance, with the potential for instance to impact design software development. Further study of the optimality conditions of the main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory and additional example

Simulations of denuded-zone formation during growth on surfaces with anisotropic diffusion

We have investigated the formation of denuded zones during epitaxial growth on surfaces exhibiting anisotropic diffusion of adparticles, such as Si(001)-2x1, using Monte Carlo simulations and a continuum model. In both the simulations, which were mainly for low-temperature cases (small critical clusters), and the continuum model, appropriate for high-temperature cases (large critical clusters), it was found that the ratio of denuded-zone widths Wf and Ws in the fast- and slow-diffusion directions scales with the ratio Df/Ds of the diffusion constants in the two directions with a power of 1/2, i.e., Wf/Ws ≈ (Df/Ds)1/2, independent of various conditions including the degree of diffusion anisotropy. This supplies the foundation of a method for extracting the diffusion anisotropy from the denuded zone anisotropy which is experimentally measurable. Further, we find that unequal probabilities of a diffusing particle sticking to different types of step edges [e.g., S A and SB steps on Si(001)] does not affect the relation Wf/Ws ≈ (Df/Ds)1/2 seriously unless the smaller of the two sticking probabilities is less than about 0.1. Finally, we examined the relation between the number of steps and the number of sites visited in anisotropic random walks, finding it is better described by a crossover from one-dimensional to two-dimensional behavior than by scaling behavior with a single exponent. This result has bearing on scaling arguments relating denuded-zone widths to diffusion constants for anisotropic diffusion.open7

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <= m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

Can forest management based on natural disturbances maintain ecological resilience?

Given the increasingly global stresses on forests, many ecologists argue that managers must maintain ecological resilience: the capacity of ecosystems to absorb disturbances without undergoing fundamental change. In this review we ask: Can the emerging paradigm of natural-disturbance-based management (NDBM) maintain ecological resilience in managed forests? Applying resilience theory requires careful articulation of the ecosystem state under consideration, the disturbances and stresses that affect the persistence of possible alternative states, and the spatial and temporal scales of management relevance. Implementing NDBM while maintaining resilience means recognizing that (i) biodiversity is important for long-term ecosystem persistence, (ii) natural disturbances play a critical role as a generator of structural and compositional heterogeneity at multiple scales, and (iii) traditional management tends to produce forests more homogeneous than those disturbed naturally and increases the likelihood of unexpected catastrophic change by constraining variation of key environmental processes. NDBM may maintain resilience if silvicultural strategies retain the structures and processes that perpetuate desired states while reducing those that enhance resilience of undesirable states. Such strategies require an understanding of harvesting impacts on slow ecosystem processes, such as seed-bank or nutrient dynamics, which in the long term can lead to ecological surprises by altering the forest's capacity to reorganize after disturbance