A Widder’s Type Theorem for the Heat Equation with Nonlocal Diffusion

The main goal of this work is to prove that every non-negative strong solution u(x, t) to the problem $$u_t + (-\Delta)^{\alpha/2}{u} = 0 \,\, {\rm for} (x, t) \in {\mathbb{R}^n} \times (0, T ), \, 0 < \alpha < 2,$$ can be written as $$u(x, t) = \int_{\mathbb{R}^n} P_t (x - y)u(y, 0) dy,$$ where $$P_t (x) = \frac{1}{t^{n/ \alpha}}P \left(\frac{x}{t^{1/ \alpha}}\right),$$ and $$P(x) := \int_{\mathbb{R}^n} e^{i x\cdot\xi-|\xi |^\alpha} d\xi.$$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by Widder in [15] to the nonlocal diffusion framework.     

Gradient bounds for anisotropic partial differential equations

We consider solutions in the whole of the space of a partial differential equation driven by the anisotropic Laplacian. We prove a pointwise energy bound and we derive from that some rigidity results.     

Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media

The Ginzburg-Landau-Allen-Cahn equation is a variational model for phase coexistence and for other physical problems. It contains a term given by a kinetic part of elliptic type plus a double-well potential. We assume that the functional depends on the space variables in a periodic way.We show that given a plane with rational normal, there are minimal solutions, satisfying the following properties. These solutions are asymptotic to the pure phases and are separated by an interface. The convergence to the pure phases is exponentially fast. The interface lies at a finite distance M from the chosen plane, where M is a universal constant. Furthermore, these solutions satisfy some monotonicity properties with respect to integer translations (namely, integer translations are always comparable to the function).We then show that all the interfaces of the global periodic minimizers satisfy similar monotonicity and plane-like properties.We also consider the case of possibly irrationally oriented planes. We show that either there is a one parameter family of minimizers whose graphs provide a field of extremals or there are at least two solutions, one which is a minimizer and another one which is not. These solutions also have interfaces bounded by a universal constant, they enjoy monotonicity properties with respect to integer translations and the nonminimal solutions are trapped inside a gap of the lamination induced by the minimizers.     

Sum Rules for the spin surface response of metal clusters within the time dependent local spin density approximation

We study surface spin collective modes of simple metallic clusters using a sum rule approach. We derive analytical expressions for the energy and cubic energy-weighted moments of the time-dependent local-spin-density approximation (TDLSDA) strength function. With these two moments we obtain a mean excitation energy of multipole spin modes and study its dispersion with cluster size up to very large sizes. The crucial roles played by the surface diffuseness and by correlation and kinetic energy effects on the resonance energies are stressed.     

Influence of the draw ratio on the tensile and fracture behavior of NMMO regenerated silk fibers

The tensile properties and fracture surfaces of N‐methylmorpholine‐N‐oxide (NMMO) regenerated silk fibroin fibers produced with a range of draw ratios has been characterized and related to their microstructure with data obtained from Raman spectroscopy and birefringence measurements. The spinning process allows control of two different draw ratios, coagulation, and postspinning, and it has been found that the microstructure and the properties of the fibers can be modified by the proper combination of both draw ratios. NMMO regenerated silk fibroin fibers subjected to postspinning drawing yield tensile properties comparable to other regenerated fibers and strain at breaking comparable to natural Bombyx mori silk fibers. Tensile strength; however, is still significantly lower than that of natural fibers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2568–2579, 2007     

Thiol‐disulfide redox equilibria of glutathione metaboloma compounds investigated by tandem mass spectrometry

The thiol group of cysteine plays a pivotal role in structural and functional biology. We use mass spectrometry to study glutathione‐related homo‐ and heterodimeric disulfides, aiming at understanding the factors affecting the redox potentials of different disulfide/thiol pairs. Several electrospray ionization (ESI)‐protonated disulfides of cysteamine, cysteine, penicillamine, N‐acetylcysteine, N‐acetylpenicillamine, γGluCySH, HSCyGly, and glutathione were analyzed on a triple quadrupole instrument to measure their energy‐resolved tandem mass spectra. Fission of the disulfide bond yields RSH*H⁺ and RS⁺ ions. The logarithm of the intensity ratio of the RS⁺/RSH*H⁺ fragments in homodimeric disulfides is proportional to the normal reduction potential of their RSSR/RSH pairs determined by nuclear magnetic resonance (NMR) in solution, the more reducing ones yielding the higher ratios. Also in some R₁S‐SR₂ disulfides, the ratio of the intensities of the RSH + H⁺ and RS⁺ ions of each participating thiol shows a linear relationship with the Nernst equation potential difference of the corresponding redox pairs. This behavior allows us to measure the redox potentials of some disulfide/thiol pairs by using different thiol‐reducing probes of known oxidoreductive potential as reference. To assist understanding of the fission mechanism of the disulfide bond, the fragments tentatively identified as ‘sulfenium’ were themselves fragmented; accurate mass measurement of the resulting second‐generation fragments demonstrated a loss of thioformaldehyde, thus supporting the assigned structure of this elusive intermediate of the oxidative stress pathway. Understanding this fragmentation process allows us to employ this technique with larger molecules to measure by mass spectrometry the micro‐redox properties of different disulfide bonds in peptides with catalytic and signaling biological activity. Copyright © 2008 John Wiley & Sons, Ltd.     

Inference and Learning in a Latent Variable Model for Beta Distributed Interval Data

Latent Variable Models (LVMs) are well established tools to accomplish a range of different data processing tasks. Applications exploit the ability of LVMs to identify latent data structure in order to improve data (e.g., through denoising) or to estimate the relation between latent causes and measurements in medical data. In the latter case, LVMs in the form of noisy-OR Bayes nets represent the standard approach to relate binary latents (which represent diseases) to binary observables (which represent symptoms). Bayes nets with binary representation for symptoms may be perceived as a coarse approximation, however. In practice, real disease symptoms can range from absent over mild and intermediate to very severe. Therefore, using diseases/symptoms relations as motivation, we here ask how standard noisy-OR Bayes nets can be generalized to incorporate continuous observables, e.g., variables that model symptom severity in an interval from healthy to pathological. This transition from binary to interval data poses a number of challenges including a transition from a Bernoulli to a Beta distribution to model symptom statistics. While noisy-OR-like approaches are constrained to model how causes determine the observables’ mean values, the use of Beta distributions additionally provides (and also requires) that the causes determine the observables’ variances. To meet the challenges emerging when generalizing from Bernoulli to Beta distributed observables, we investigate a novel LVM that uses a maximum non-linearity to model how the latents determine means and variances of the observables. Given the model and the goal of likelihood maximization, we then leverage recent theoretical results to derive an Expectation Maximization (EM) algorithm for the suggested LVM. We further show how variational EM can be used to efficiently scale the approach to large networks. Experimental results finally illustrate the efficacy of the proposed model using both synthetic and real data sets. Importantly, we show that the model produces reliable results in estimating causes using proofs of concepts and first tests based on real medical data and on images.     

Asymptotic Resonance, Interaction of Modes and Subharmonic Bifurcation

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam β-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.