Formation of singularities on the surface of a liquid metal in a strong electric field

The nonlinear dynamics of the free surface of an ideal conducting liquid in a strong external electric field is studied. It is establish that the equations of motion for such a liquid can be solved in the approximation in which the surface deviates from a plane by small angles. This makes it possible to show that on an initially smooth surface for almost any initial conditions points with an infinite curvature corresponding to branch points of the root type can form in a finite time.Comment: 14 page

Nonlinear equation for curved stationary flames

A nonlinear equation describing curved stationary flames with arbitrary gas expansion $\theta = \rho_{{\rm fuel}}/\rho_{{\rm burnt}}$, subject to the Landau-Darrieus instability, is obtained in a closed form without an assumption of weak nonlinearity. It is proved that in the scope of the asymptotic expansion for $\theta \to 1,$ the new equation gives the true solution to the problem of stationary flame propagation with the accuracy of the sixth order in $\theta - 1.$ In particular, it reproduces the stationary version of the well-known Sivashinsky equation at the second order corresponding to the approximation of zero vorticity production. At higher orders, the new equation describes influence of the vorticity drift behind the flame front on the front structure. Its asymptotic expansion is carried out explicitly, and the resulting equation is solved analytically at the third order. For arbitrary values of $\theta,$ the highly nonlinear regime of fast flow burning is investigated, for which case a large flame velocity expansion of the nonlinear equation is proposed.Comment: 29 pages 4 figures LaTe

The stability for the Cauchy problem for elliptic equations

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.Comment: 57 pages, review articl

MAP4 Mechanism that Stabilizes Mitochondrial Permeability Transition in Hypoxia: Microtubule Enhancement and DYNLT1 Interaction with VDAC1

Mitochondrial membrane permeability has received considerable attention recently because of its key role in apoptosis and necrosis induced by physiological events such as hypoxia. The manner in which mitochondria interact with other molecules to regulate mitochondrial permeability and cell destiny remains elusive. Previously we verified that hypoxia-induced phosphorylation of microtubule-associated protein 4 (MAP4) could lead to microtubules (MTs) disruption. In this study, we established the hypoxic (1% O2) cell models of rat cardiomyocytes, H9c2 and HeLa cells to further test MAP4 function. We demonstrated that increase in the pool of MAP4 could promote the stabilization of MT networks by increasing the synthesis and polymerization of tubulin in hypoxia. Results showed MAP4 overexpression could enhance cell viability and ATP content under hypoxic conditions. Subsequently we employed a yeast two-hybrid system to tag a protein interacting with mitochondria, dynein light chain Tctex-type 1 (DYNLT1), by hVDAC1 bait. We confirmed that DYNLT1 had protein-protein interactions with voltage-dependent anion channel 1 (VDAC1) using co-immunoprecipitation; and immunofluorescence technique showed that DYNLT1 was closely associated with MTs and VDAC1. Furthermore, DYNLT1 interactions with MAP4 were explored using a knockdown technique. We thus propose two possible mechanisms triggered by MAP4: (1) stabilization of MT networks, (2) DYNLT1 modulation, which is connected with VDAC1, and inhibition of hypoxia-induced mitochondrial permeabilization