An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen--Cahn equation

This paper studies traveling fronts to the Allen–Cahn equation in RN for N ≥ 3. Let (N −2)-dimensional smooth surfaces be the boundaries of compact sets in RN−1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation

Existence and Uniqueness of Solutions to a Nonlocal Equation with Monostable Nonlinearity

Let $J \in C(\mathbb{R})$, $J\ge 0$, \int_{\tiny\mathbb{R}} J = 1 and consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$, where $f$ is a KPP-type nonlinearity, periodic in $x$. We show that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, $J$ is symmetric, then the nontrivial solution is unique

Global exponential convergence to variational traveling waves in cylinders

We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for "pushed" fronts.Comment: 23 page

Large-time Behavior of Solutions to the Inflow Problem of Full Compressible Navier-Stokes Equations

Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations is investigated on the half line $R^+ =(0,+\infty)$. The wave structure which contains four waves: the transonic(or degenerate) boundary layer solution, 1-rarefaction wave, viscous 2-contact wave and 3-rarefaction wave to the inflow problem is described and the asymptotic stability of the superposition of the above four wave patterns to the inflow problem of full compressible Navier-Stokes equations is proven under some smallness conditions. The proof is given by the elementary energy analysis based on the underlying wave structure. The main points in the proof are the degeneracies of the transonic boundary layer solution and the wave interactions in the superposition wave.Comment: 27 page

Ballistic-Electron-Emission Microscopy at Epitaxial Metal/Semiconductor Interfaces(STM-BEEM interfaces)

The invention of ballistic-electron-emission microscopy (BEEM) has made it possible to study hot electron transport across interfaces with a spatial resolution unparalleled before. In order to exploit the limits of the method we have applied BEEM experiments carried out in UHV and at 77 K to epitaxial CoSi_2 films on silicon. CoSi_2/Si may be considered as a model system for the metal/semiconductor interface, because its atomic structure can be rather well controlled experimentally and has been well characterized by transmission electron microscopy. This overview contains a discussion of the various processes leading to contrast in BEEM images for CoSi_2/Si interfaces. The BEEM current may be affected by (a) the atomic surface structure or surface defects, both of which can change the tunneling distribution, (b) inelastic and elastic scattering processes within the metal films and (c) interface scattering or variations of the Schottky barrier height, resulting from interfacial defects. Scattering processes will be shown to be dominant in the case of CoSi_2/Si(111) interfaces, since the Schottky barrier height is not measurably affected by interfacial dislocations and other defects. Here the ultimate resolution limits of the BEEM technique have been reached, in the sense that individual point defects can be resolved. The CoSi_2/Si(100) interface represents a more complicated case, where extended defects lead to significant barrier lowering, whereas interface scattering is obscured by the strong modification of the tunneling distribution by surface reconstructions

Endomorphisms of quantized Weyl algebras

Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity

A Generalization of Mathieu Subspaces to Modules of Associative Algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $\mathcal A$, which was introduced recently in [Z4] and [Z6], to $\mathcal A$-modules $\mathcal M$. The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets $\sigma(N)$ and $\tau(N)$ of stable elements and quasi-stable elements, respectively, for all $R$-subspaces $N$ of $\mathcal A$-modules $\mathcal M$, where $R$ is the base ring of $\mathcal A$. We then prove some general properties of the sets $\sigma(N)$ and $\tau(N)$. Furthermore, examples from certain modules of the quasi-stable algebras [Z6], matrix algebras over fields and polynomial algebras are also studied.Comment: A new case has been added; some mistakes and misprints have been corrected. Latex, 31 page

Metallic liquid hydrogen and likely Al2O3 metallic glass

Dynamic compression has been used to synthesize liquid metallic hydrogen at 140 GPa (1.4 million bar) and experimental data and theory predict Al2O3 might be a metallic glass at ~300 GPa. The mechanism of metallization in both cases is probably a Mott-like transition. The strength of sapphire causes shock dissipation to be split differently in the strong solid and soft fluid. Once the 4.5-eV H-H and Al-O bonds are broken at sufficiently high pressures in liquid H2 and in sapphire (single-crystal Al2O3), electrons are delocalized, which leads to formation of energy bands in fluid H and probably in amorphous Al2O3. The high strength of sapphire causes shock dissipation to be absorbed primarily in entropy up to ~400 GPa, which also causes the 300-K isotherm and Hugoniot to be virtually coincident in this pressure range. Above ~400 GPa shock dissipation must go primarily into temperature, which is observed experimentally as a rapid increase in shock pressure above ~400 GPa. The metallization of glassy Al2O3, if verified, is expected to be general in strong oxide insulators. Implications for Super Earths are discussed.Comment: 8 pages, 5 figures, 14th Liquid and Amorphous Metals Conference, Rome 201