Generation of interface for an Allen-Cahn equation with nonlinear diffusion

In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property

Effects of Synaptic and Myelin Plasticity on Learning in a Network of Kuramoto Phase Oscillators

Models of learning typically focus on synaptic plasticity. However, learning is the result of both synaptic and myelin plasticity. Specifically, synaptic changes often co-occur and interact with myelin changes, leading to complex dynamic interactions between these processes. Here, we investigate the implications of these interactions for the coupling behavior of a system of Kuramoto oscillators. To that end, we construct a fully connected, one-dimensional ring network of phase oscillators whose coupling strength (reflecting synaptic strength) as well as conduction velocity (reflecting myelination) are each regulated by a Hebbian learning rule. We evaluate the behavior of the system in terms of structural (pairwise connection strength and conduction velocity) and functional connectivity (local and global synchronization behavior). We find that for conditions in which a system limited to synaptic plasticity develops two distinct clusters both structurally and functionally, additional adaptive myelination allows for functional communication across these structural clusters. Hence, dynamic conduction velocity permits the functional integration of structurally segregated clusters. Our results confirm that network states following learning may be different when myelin plasticity is considered in addition to synaptic plasticity, pointing towards the relevance of integrating both factors in computational models of learning.Comment: 39 pages, 15 figures This work is submitted in Chaos: An Interdisciplinary Journal of Nonlinear Scienc

Fluid Flows of Mixed Regimes in Porous Media

In porous media, there are three known regimes of fluid flows, namely, pre-Darcy, Darcy and post-Darcy. Because of their different natures, these are usually treated separately in literature. To study complex flows when all three regimes may be present in different portions of a same domain, we use a single equation of motion to unify them. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure gradient is either small or large. We estimate the pressure and its gradient for all time in terms of initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on initial and boundary data, and the structural stability for the equation are established.Comment: 33 page

Existence of Ricci flows of incomplete surfaces

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction

On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis

In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function

Self-similar extinction for a diffusive Hamilton-Jacobi equation with critical absorption

International audienceThe behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption ∂_t u − ∆_p u + |∇u|^{p−1} = 0 in (0, ∞) × R^N , and fast diffusion 2N/(N + 1) < p < 2. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as |x| → ∞, it is shown that the corresponding solution u to the above equation approaches a uniquely determined separate variable solution of the form U (t, x) = (T_e − t)^{1/(2−p)} f_* (|x|), (t, x) ∈ (0, T_e) × R^N , as t → T_e , where T_e denotes the finite extinction time of u. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile f_* is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional

Improved catalytic activity of ruthenium–arene complexes in the reduction of NAD+

A series of neutral Ru-II half-sandwich complexes of the type [(eta(6)-arene)Ru(N,N')Cl] where the arene is para-cymene (p-cym), hexamethylbenzene (hmb), biphenyl (bip), or benzene (bn) and N,N' is N-(2-aminoethyl) -4-(trifluoromethyl)benzenesulfonamide (TfEn), N-(2-aminoethyl)-4-toluenesulfonamide (TsEn), or N-(2-aminoethyl)-methylenesulfonamide (MsEn) were synthesized and characterized. X-ray crystal structures of [(p-cym)Ru(MsEn)Cl] (1), [(hmb)Ru(TsEn)Cl] (5), [(hmb)Ru(TfEn)Cl] (6), [(bip)Ru(MsEn)Cl] (7), and [(bip)Ru(TsEn)Cl] (8) have been determined. The complexes can regioselectively catalyze the transfer hydrogenation of NAD(+) to give 1,4-NADH in the presence of formate. The turnover frequencies (TOF) when the arene is varied decrease in the order bn > bip > p-cym > hmb for complexes with the same N,N' chelating ligand. The TOF decreased with variation in the N,N' chelating ligand in the order TfEn > TsEn > MsEn for a given arene. [(bn)Ru(TfEn)Cl] (12) was the most active, with a TOP of 10.4 h(-1). The effects of NAD(+) and formate concentration on the reaction rates were determined for [(p-cym)Ru(TsEn)Cl] (2). Isotope studies implicated the formation of [(arene)Ru(N,N')(H)] as the rate-limiting step. The coordination of formate and subsequent CO2 elimination to generate the hydride were modeled computationally by density functional theory (DFT). CO2 elimination occurs via a two-step process with the coordinated formate first twisting to present its hydrogen toward the metal center. The computed barriers for CO2 release for arene = benzene follow the order MsEn > TsEn > TfEn, and for the Ms En system the barrier followed bn < hmb, both consistent with the observed rates. The effect of methanol on transfer hydrogenation rates in aqueous solution was investigated. A study of pH dependence of the reaction in D2O gave the optimum pH* as 7.2 with a TOF of 1.58 h(-1) for 2. The series of compounds reported here show an improvement in the catalytic activity by an order of magnitude compared to the ethylenediamine analogues

Large time behavior for a quasilinear diffusion equation with critical gradient absorption

International audienceWe study the large time behavior of non-negative solutions to thenonlinear diffusion equation with critical gradient absorption\partial_t u-\Delta_{p}u+|\nabla u|^{q_*}=0 \quad \hbox{in} \(0,\infty)\times\mathbb{R}^N\ ,for $p\in(2,\infty)$ and $q_*:=p-N/(N+1)$. We show that theasymptotic profile of compactly supported solutions is given by asource-type self-similar solution of the $p$-Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results

Modern optical astronomy: technology and impact of interferometry

The present `state of the art' and the path to future progress in high spatial resolution imaging interferometry is reviewed. The review begins with a treatment of the fundamentals of stellar optical interferometry, the origin, properties, optical effects of turbulence in the Earth's atmosphere, the passive methods that are applied on a single telescope to overcome atmospheric image degradation such as speckle interferometry, and various other techniques. These topics include differential speckle interferometry, speckle spectroscopy and polarimetry, phase diversity, wavefront shearing interferometry, phase-closure methods, dark speckle imaging, as well as the limitations imposed by the detectors on the performance of speckle imaging. A brief account is given of the technological innovation of adaptive-optics (AO) to compensate such atmospheric effects on the image in real time. A major advancement involves the transition from single-aperture to the dilute-aperture interferometry using multiple telescopes. Therefore, the review deals with recent developments involving ground-based, and space-based optical arrays. Emphasis is placed on the problems specific to delay-lines, beam recombination, polarization, dispersion, fringe-tracking, bootstrapping, coherencing and cophasing, and recovery of the visibility functions. The role of AO in enhancing visibilities is also discussed. The applications of interferometry, such as imaging, astrometry, and nulling are described. The mathematical intricacies of the various `post-detection' image-processing techniques are examined critically. The review concludes with a discussion of the astrophysical importance and the perspectives of interferometry.Comment: 65 pages LaTeX file including 23 figures. Reviews of Modern Physics, 2002, to appear in April issu

Synthesis and characterization of p-n junction ternary mixed oxides for photocatalytic coprocessing of CO2 and H2O

In the present paper, we report the synthesis and characterization of both binary (Cu2 O, Fe2 O3, and In2 O3 ) and ternary (Cu2 O-Fe2 O3 and Cu2 O-In2 O3 ) transition metal mixed-oxides that may find application as photocatalysts for solar driven CO2 conversion into energy rich species. Two different preparation techniques (High Energy Milling (HEM) and Co-Precipitation (CP)) are compared and materials properties are studied by means of a variety of characterization and analytical techniques UV-Visible Diffuse Reflectance Spectroscopy (UV-VIS DRS), X-ray Photoelectron Spectroscopy (XPS), X-Ray Diffraction (XRD), Transmission Electron Microscopy (TEM), and Energy Dispersive X-Ray spectrometry (EDX). Appropriate data elaboration methods are used to extract materials bandgap for Cu2 O@Fe2 O3 and Cu2 O@In2 O3 prepared by HEM and CP, and foresee whether the newly prepared semiconductor mixed oxides pairs are useful for application in CO2-H2 O coprocessing. The experimental results show that the synthetic technique influences the photoactivity of the materials that can correctly be foreseen on the basis of bandgap experimentally derived. Of the mixed oxides prepared and described in this work, only Cu2 O@In2 O3 shows positive results in CO2-H2 O photo-co-processing. Preliminary results show that the composition and synthetic methodologies of mixed-oxides, the reactor geometry, the way of dispersing the photocatalyst sample, play a key role in the light driven reaction of CO2 –H2 O. This work is a rare case of full characterization of photo-materials, using UV-Visible DRS, XPS, XRD, TEM, EDX for the surface and bulk analytical characterization. Surface composition may not be the same of the bulk composition and plays a key role in photocatalysts behavior. We show that a full material knowledge is necessary for the correct forecast of their photocatalytic behavior, inferred from experimentally determined bandgaps