Model calculation of orientational effect of deformed aerogel on the order parameter of superfluid 3He

Theory of Rainer and Vuorio of small objects in superfluid ^3He is applied for calculation of the average orientational effect of a deformed aerogel on the order parameter of 3He. The minimum deformation which stabilizes the ordered state is evaluated both for specular and diffusive scattering of quasiparticles by the threads of aerogel.Comment: Contribution to QFS 2007, 6 pages, 1 figur

Comment on "Order parameter of A-like 3He phase in aerogel"

We argue that the inhomogeneous A-phase in aerogel is energetically more preferable than the "robust" phase suggested by I. A. Fomin, JETP Lett. 77, 240 (2003); cond-mat/0302117 and cond-mat/0401639.Comment: 2 page

Geometric optics of whispering gallery modes

Quasiclassical approach and geometric optics allow to describe rather accurately whispering gallery modes in convex axisymmetric bodies. Using this approach we obtain practical formulas for the calculation of eigenfrequencies and radiative Q-factors in dielectrical spheroid and compare them with the known solutions for the particular cases and with numerical calculations. We show how geometrical interpretation allows expansion of the method on arbitrary shaped axisymmetric bodies.Comment: 12 pages, 6 figures, Photonics West 2006 conferenc

Bidimensionality of Geometric Intersection Graphs

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes

On semiring complexity of Schur polynomials

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial {s_\lambda(x_1,\dots,x_k)} labeled by a partition {\lambda=(\lambda_1\ge\lambda_2\ge\cdots)} is bounded by {O(\log(\lambda_1))} provided the number of variables $k$ is fixed