Dirac-Foldy term and the electromagnetic polarizability of the neutron

We reconsider the Dirac-Foldy contribution $\mu^2/m$ to the neutron electric polarizability. Using a Dirac equation approach to neutron-nucleus scattering, we review the definitions of Compton continuum ($\bar{\alpha}$), classical static ($\alpha^n_E$), and Schr\"{o}dinger ($\alpha_{Sch}$) polarizabilities and discuss in some detail their relationship. The latter $\alpha_{Sch}$ is the value of the neutron electric polarizability as obtained from an analysis using the Schr\"{o}dinger equation. We find in particular $\alpha_{Sch} = \bar{\alpha} - \mu^2/m$ , where $\mu$ is the magnitude of the magnetic moment of a neutron of mass $m$. However, we argue that the static polarizability $\alpha^n_E$ is correctly defined in the rest frame of the particle, leading to the conclusion that twice the Dirac-Foldy contribution should be added to $\alpha_{Sch}$ to obtain the static polarizability $\alpha^n_E$.Comment: 11 pages, RevTeX, to appear in Physical Review

On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest

Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

In the sl\_n case, A. Berenstein and A. Zelevinsky studied the Sch\"{u}tzenberger involution in terms of Lusztig's canonical basis, [3]. We generalize their construction and formulas for any semisimple Lie algebra. We use for this the geometric lifting of the canonical basis, on which an analogue of the Sch\"{u}tzenberger involution can be given. As an application, we construct semitoric degenerations of Richardson varieties, following a method of P. Caldero, [6]Comment: 22 pages, 3 figure