Secants of minuscule and cominuscule minimal orbits

We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian

Two sides tangential filtering decomposition

AbstractIn this paper we study a class of preconditioners that satisfy the so-called left and/or right filtering conditions. For practical applications, we use a multiplicative combination of filtering based preconditioners with the classical ILU(0) preconditioner, which is known to be efficient. Although the left filtering condition has a more sound theoretical motivation than the right one, extensive tests on convection–diffusion equations with heterogeneous and anisotropic diffusion tensors reveal that satisfying left or right filtering conditions lead to comparable results. On the filtering vector, these numerical tests reveal that e=[1,…,1]T is a reasonable choice, which is effective and can avoid the preprocessing needed in other methods to build the filtering vector. Numerical tests show that the composite preconditioners are rather robust and efficient for these problems with strongly varying coefficients

Minimal decomposition of binary forms with respect to tangential projections

Let $C\subset \mathbb{P}^n$ be a rational normal curve and let $\ell_O:\mathbb{P}^{n+1}\dashrightarrow \mathbb{P}^n$ be any tangential projection form a point $O\in T_AC$ where $A\in C$. Hence $X:= \ell_O(C)\subset \mathbb{P}^n$ is a linearly normal cuspidal curve with degree $n+1$. For any $P = \ell_O(B)$, $B\in \mathbb{P}^{n+1}$, the $X$-rank $r_X(P)$ of $P$ is the minimal cardinality of a set $S\subset X$ whose linear span contains $P$. Here we describe $r_X(P)$ in terms of the schemes computing the $C$-rank or the border $C$-rank of $B$.Comment: 7 page

Dynamics of particle-particle collisions in a viscous liquid

When two solid spheres collide in a liquid, the dynamic collision process is slowed by viscous dissipation and the increased pressure in the interparticle gap as compared with dry collisions. This paper investigates liquid-immersed head-on and oblique collisions, which complements previously investigated particle-on-wall immersed collisions. By defining the normal from the line of centers at contact, the experimental findings support the decomposition of an oblique collision into its normal and tangential components of motion. The normal relative particle motion is characterized by an effective coefficient of restitution and a binary Stokes number with a correlation that follows the particle-wall results. The tangential motion is described by a collision model using a normal coefficient of restitution and a friction coefficient that are modified for the liquid effects

Coarse-grained description of thermo-capillary flow

A mesoscopic or coarse-grained approach is presented to study thermo-capillary induced flows. An order parameter representation of a two-phase binary fluid is used in which the interfacial region separating the phases naturally occupies a transition zone of small width. The order parameter satisfies the Cahn-Hilliard equation with advective transport. A modified Navier-Stokes equation that incorporates an explicit coupling to the order parameter field governs fluid flow. It reduces, in the limit of an infinitely thin interface, to the Navier-Stokes equation within the bulk phases and to two interfacial forces: a normal capillary force proportional to the surface tension and the mean curvature of the surface, and a tangential force proportional to the tangential derivative of the surface tension. The method is illustrated in two cases: thermo-capillary migration of drops and phase separation via spinodal decomposition, both in an externally imposed temperature gradient.Comment: To appear in Phys. Fluids. Also at http://www.scri.fsu.edu/~vinals/dj1.p