A theorem on the cores of partitions

If s and t are relatively prime positive integers we show that the s-core of a t-core partition is again a t-core partitionComment: 9 page

Residue symbols and Jantzen-Seitz partitions

Jantzen-Seitz partitions are those $p$-regular partitions of~$n$ which label $p$-modular irreducible representations of the symmetric group $S_n$ which remain irreducible when restricted to $S_{n-1}$; they have recently also been found to be important for certain exactly solvable models in statistical mechanics. In this article we study their combinatorial properties via a detailed analysis of their residue symbols; in particular the $p$-cores of Jantzen-Seitz partitions are determined

Submatrices of character tables and basic sets

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on restricted sets of conjugacy classes, like beautiful combinatorial determinant formulae for submatrices of the character table and Cartan matrices with respect to basic sets; we observe that similar phenomena occur for the transition matrices between power sum symmetric functions to bounded partitions and the $k$-Schur functions introduced by Lapointe and Morse. Arithmetic properties of the numbers occurring in this context are studied via generating functions.Comment: 18 pages; examples added, typos removed, some further minor changes, references update

Restriction of Odd Degree Characters of Sn\mathfrak{S}_n

Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sem. Lothar. Combin. 75 (2015), Art. B75g, 13 pages] and recently developed in [Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., J. Algebra 478 (2017), 271-282]

“Tax Simplification”—Grave Threat to the Charitable Contribution Deduction: The Problem and a Proposed Solution

The present National Administration has continued to support proposed legislative changes aimed at substantially reducing the number of income tax returns in which deductions are itemized. The author contends that these tax simplification proposals are incompatible with the preservation of the charitable contribution deduction and would undermine the position of voluntary charitable organizations by reducing the incentives for giving. He proposes a solution to this dilemma by promoting the charitable contribution deduction, with certain limitations, to the position of a deduction from gross income, rather than a deduction from adjusted gross income

Volume Analysis of the Proximal Tibial Metaphysis

Purpose The Vitrea 2 imaging software (Vital Images Inc, Minnetonka, MN) was used for the volume analysis of the proximal tibial metaphysis. Materials and Methods Eighteen computed tomography scans of the proximal tibia were processed through the software, and 3-dimensional imaging of the proximal tibia was reconstructed. Results The volume and area of the proximal tibia that were generated resulted in a mean area of 127 cm2 and a mean volume of 77.2 cm2. Conclusion This study supports the use of the proximal tibial metaphysis as a source of low to moderate volume of autologous bone. When compared with the accepted average volume of 25 cm2, the computed results showed that there could be up to 3 times the amount of bone available in the proximal tibial metaphysis. The reported volume of bone harvested from previous studies was based on need and not the total amount available; subsequently, the results showed the possibility of a larger resource of bone, which provides the surgeon with the volumetrical limits of the proximal tibia metaphysis

Transient and steady-state shear banding in a lamellar phase as studied by Rheo-NMR

Flow fields and shear-induced structures in the lamellar (L-alpha) phase of the system triethylene glycol mono n-decyl ether (C10E3)/water were investigated by NMR velocimetry, diffusometry, and H-2 NMR spectroscopy. The transformation from multilamellar vesicles (MLVs) to aligned planar lamellae is accompanied by a transient gradient shear banding. A high-shear-rate band of aligned lamellae forms next to the moving inner wall of the cylindrical Couette shear cell while a low-shear-rate band of the initial MLV structure remains close to the outer stationary wall. The band of layers grows at the expense of the band of MLVs until the transformation is completed. This process scales with the applied strain. Wall slip is a characteristic of the MLV state, while aligned layers show no deviation from Newtonian flow. The homogeneous nature of the opposite transformation from well aligned layers to MLVs via an intermediate structure resembling undulated multilamellar cylinders is confirmed. The strain dependence of this transformation appears to be independent of temperature. The shear diagram, which represents the shear-induced structures as a function of temperature and shear rate, contains a transition region between stable layers and stable MLVs. The steady-state structures in the transition region show a continuous change from layer-like at high temperature to MLV-like at lower temperature. These structures are homogeneous on a length scale above a few micrometers