Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions

The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_\lambda^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young lattice with certain tableaux on $k+1$ cores. This algorithm reveals that the $k$-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis \{h_\la\}_{\la\in\CY^k} to \{s_\la^{(k)}\}_{\la\in\CY^k} may now be obtained by counting appropriate classes of tableaux on $k+1$-cores. This suggests that the conjecturally positive $k$-Schur expansion coefficients for Macdonald polynomials (reducing to $q,t$-Kostka polynomials for large $k$) could be described by a $q,t$-statistic on these tableaux, or equivalently on reduced words for affine permutations.Comment: 30 pages, 1 figur

Explicit formulas for the generalized Hermite polynomials in superspace

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version of hep-th/0305038 which has been splitted in two articles. In this revised version, the introduction has been rewritten and a new appendix has been added. To appear in JP

Influence of a local change of depth on the behavior of bouncing oil drops

The work of Couder \textit{et al} (see also Bush \textit{et al}) inspired consideration of the impact of a submerged obstacle, providing a local change of depth, on the behavior of oil drops in the bouncing regime. In the linked videos, we recreate some of their results for a drop bouncing on a uniform depth bath of the same liquid undergoing vertical oscillations just below the conditions for Faraday instability, and show a range of new behaviors associated with change of depth. This article accompanies a fluid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.Comment: High and low resolutions videos included as ancillary file

Alien Registration- Lapointe, Charles J. (Jackman, Somerset County)

https://digitalmaine.com/alien_docs/7039/thumbnail.jp

Missing the Forest for the Trees: The Seventh Circuit’s Refinement of Bloom \u27s Private Gain Test for Honest Services Fraud in \u3cem\u3eUnited States v. Thompson\u3c/em\u3e

Public officials, as well as public employees, owe the public the fiduciary duty of providing their honest services. The mail fraud statute, 18 U.S.C. § 1341, proscribes using the mails to carry out “any scheme or artifice to defraud.” It further defines, in § 1346, “any scheme or artifice to defraud” to include “a scheme or artifice to deprive another of the intangible right of honest services.” In passing these statutes, Congress never defined the terms “scheme or artifice,” “intangible right” or “honest services.” Because of this failure, the appellate courts have adopted limiting principles to prevent minor breaches of fiduciary duties from becoming federal crimes. The Seventh Circuit adopted the limiting principle “misuse of office for private gain” in United States v. Bloom. Recently, the court refined the Bloom standard in United States v. Thompson holding that “neither an increase in salary for doing what one\u27s superiors deem a good job, nor an addition to one\u27s peace of mind, is a ‘private benefit.’” This holding conflicts with previous case law from the District Court for the Northern District of Illinois, concluding that one’s job or the prospect of future employment can be a private gain under Bloom. Because the Seventh Circuit appears to have distinguished Thompson from the cases in the Northern District of Illinois rather than overruling them, the court may have inadvertently created a loophole through which public employees can engage in “subtle schemes” and not be punished under § 1346. This Note concludes that the Seventh Circuit could have distinguished Thompson from the Northern District cases on the grounds that a “scheme,” as defined in its normal usage, existed is those cases, but did not exist in Thompson. The Note further argues that had the court made a threshold determination of whether or not a “scheme” existed in Thompson, it would not have had to reach the private gain issue, thereby preventing the problems raised after Thompson