Diophantine approximation on Veech surfaces

We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments

New limits on the β+\beta^{+}EC and ECEC processes in 120^{120}Te

New limits on the double beta processes for $^{120}$Te have been obtained using a 400 cm$^3$ HPGe detector and a source consisting of natural Te0$_2$ powder. At a confidence level of 90% the limits are $0.19\times 10^{18}$ y for the $\beta^+$EC$(0\nu + 2\nu)$ transition to the ground state, $0.75\times 10^{18}$ y for the ECEC$(0\nu + 2\nu)$ transition to the first 2$^+$ excited state of $^{120}$Sn (1171.26 keV) and $(0.19-0.6)\times 10^{18}$ y for different ECEC($0\nu$) captures to the ground state of $^{120}$Sn.Comment: 9 pages, 4 figures; v2: minor change

Rational, Replacement, and Local Invariants of a Group Action

The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame method in differential geometry. The generating set of rational invariants appears as the coefficients of a Groebner basis, reduction with respect to which allows to express a rational invariant in terms of the generators. The replacement invariants, introduced in the paper, are tuples of algebraic functions of the rational invariants. Any invariant, whether rational, algebraic or local, can be can be rewritten terms of replacement invariants by a simple substitution.Comment: 37 page

A Jang Equation Approach to the Penrose Inequality

We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominat energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data, and are applied to give a new proof of the general Penrose Inequality for these data sets. When appropriately coupled with an inverse mean curvature flow, analogous existence and regularity results for the associated system of equations in the nonspherical setting would yield a proof of the full Penrose Conjecture. Thus it remains as an important and challenging open problem to determine whether this system does indeed admit the desired solutions.Comment: 31 page

Spontaneous spin textures in dipolar spinor condensates

We have mapped out a detailed phase diagram that shows the ground state structure of a spin-1 condensate with magnetic dipole-dipole interactions. We show that the interplay between the dipolar and the spin-exchange interactions induces a rich variety of quantum phases that exhibit spontaneous magnetic ordering in the form of intricate spin textures.Comment: 4.1 pages, 4 figure