Formal matched asymptotics for degenerate Ricci flow neckpinches

Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on $S^m$, for all $m\geq 3$. In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit

Complete Embedded Self-Translating Surfaces under Mean Curvature Flow

We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.Comment: 42 pages, 8 figures. v2: typos correcte

An automated and versatile ultra-low temperature SQUID magnetometer

We present the design and construction of a SQUID-based magnetometer for operation down to temperatures T = 10 mK, while retaining the compatibility with the sample holders typically used in commercial SQUID magnetometers. The system is based on a dc-SQUID coupled to a second-order gradiometer. The sample is placed inside the plastic mixing chamber of a dilution refrigerator and is thermalized directly by the 3He flow. The movement though the pickup coils is obtained by lifting the whole dilution refrigerator insert. A home-developed software provides full automation and an easy user interface.Comment: RevTex, 10 pages, 10 eps figures. High-resolution figures available upon reques

Effect of Shear on Performance and Microbial Ecology of Continuously Stirred Anaerobic Digesters Treating Animal Manure

We Determined the Effect of Different Mixing Intensities on the Performance, Methanogenic Population Dynamics, and Juxtaposition of Syntrophic Microbes in Anaerobic Digesters Treating Cow Manure from a Dairy Farm. Computer Automated Radioactive Particle Tracking in Conjunction with Computational Fluid Dynamics Was Performed to Quantify the Shear Levels Locally. Four Continuously Stirred Anaerobic Digesters Were Operated at Different Mixing Intensities of 1,500, 500, 250, and 50 Revolutions Per Min (RPM) over a 260-Day Period at a Temperature of 34 ± 1°C. Animal Manure at a Volatile Solids (VS) Concentration of 50 G/L Was Fed into the Digesters Daily at Five Different Organic Loading Rates between 0.6 and 3.5 G vs./L Day. the Different Mixing Intensities Had No Effect on the Biogas Production Rates and Yields at Steady-State Conditions. a Methane Yield of 0.241 ± 0.007 L CH 4/g vs. Fed Was Obtained by Pooling the Data of All Four Digesters during Steady-State Periods. However, Digester Performance Was Affected Negatively by Mixing Intensity during Startup of the Digesters, with Lower Biogas Production Rates and Higher Volatile Fatty Acids Concentrations Observed for the 1,500-RPM Digester. Despite Similar Methane Production Yields and Rates, the Acetoclastic Methanogenic Populations Were Different for the High- and Low-Intensity Mixed Digesters with Methanosarcina Spp. and Methanosaeta Concilii as the Predominant Methanogens, Respectively. for All Four Digesters, Epifluorescence Microscopy Revealed Decreasing Microbial Floc Sizes Beginning at Week 4 and Continuing through Week 26 after Which No Microbial Flocs Remained. This Decrease in Size, and Subsequent Loss of Microbial Flocs Did Not, However, Produce Any Long-Term Upsets in Digester Performance. © 2007 Wiley Periodicals, Inc

Some results on blow up for semilinear parabolic problems

The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)>0 in R, where f(u)=up, p>1, or f(u)=eu. A complete description of the types of blow-up patterns and of the corresponding blow-up final-time profiles is given. In the rescaled variables, both are governed by the structure of the Hermite polynomials H2m(y). The H2-behavior is shown to be stable and generic. The existence of H4-behavior is proved. A nontrivial blow-up pattern with a blow-up set of nonzero measure is constructed. Similar results for the absorption equation ut=uxx−up, 0<p<1, are discussed

On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow

In this article, we study the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older regular surfaces of revolution embedded in R^3 and satisfying periodic boundary conditions. We also give conditions for global existence of solutions and prove that solutions are real analytic in time and space. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability and bifurcation behavior, with the radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana

Dirichlet sigma models and mean curvature flow

The mean curvature flow describes the parabolic deformation of embedded branes in Riemannian geometry driven by their extrinsic mean curvature vector, which is typically associated to surface tension forces. It is the gradient flow of the area functional, and, as such, it is naturally identified with the boundary renormalization group equation of Dirichlet sigma models away from conformality, to lowest order in perturbation theory. D-branes appear as fixed points of this flow having conformally invariant boundary conditions. Simple running solutions include the paper-clip and the hair-pin (or grim-reaper) models on the plane, as well as scaling solutions associated to rational (p, q) closed curves and the decay of two intersecting lines. Stability analysis is performed in several cases while searching for transitions among different brane configurations. The combination of Ricci with the mean curvature flow is examined in detail together with several explicit examples of deforming curves on curved backgrounds. Some general aspects of the mean curvature flow in higher dimensional ambient spaces are also discussed and obtain consistent truncations to lower dimensional systems. Selected physical applications are mentioned in the text, including tachyon condensation in open string theory and the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations

Critical behavior of collapsing surfaces

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the existence of a critical initial surface that develops a degenerate neckpinch. The limiting flow of the Type II singularity is accurately modeled by the rotationally symmetric translating soliton.Comment: 23 pages, 10 figure

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology