Powerful sets: a generalisation of binary matroids

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a \textit{powerful set} if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. Our motivation is that, to each powerful set, there is an associated nonnegative-integer-valued rank function (by a construction of Farr), although it does not in general satisfy all the matroid rank axioms. In this paper we investigate the combinatorial properties of powerful sets. We prove fundamental results on special elements (loops, coloops, frames, near-frames, and stars), their associated types of single-element extensions, various ways of combining powerful sets to get new ones, and constructions of nonlinear powerful sets. We show that every powerful set is determined by its clutter of minimal nonzero members. Finally, we show that the number of powerful sets is doubly exponential, and hence that almost all powerful sets are nonlinear.Comment: 19 pages. This work was presented at the 40th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (40ACCMCC), University of Newcastle, Australia, Dec. 201

Mixed Statistics on 01-Fillings of Moon Polyominoes

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let \M be a moon polyomino with $n$ rows and $m$ columns. Consider all the 01-fillings of \M in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of \M. More precisely, let $S \subseteq \{1,2,..., n\}$ and $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $\alpha(S; M)$ (resp. $\beta(S; M)$) is the sum of the number of northeast chains whose top (resp. bottom) cell is in $\mathcal{R}(S)$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $\mathcal{R}(S)$. Similarly, we define the left-mixed and right-mixed statistics $\gamma(T; M)$ and $\delta(T; M)$, where $T$ is a subset of the column index set $\{1,2,..., m\}$. Let $\lambda(A; M)$ be any of these four statistics $\alpha(S; M)$, $\beta(S; M)$, $\gamma(T; M)$ and $\delta(T; M)$, we show that the joint distribution of the pair $(\lambda(A; M), \lambda(\bar A; M))$ is symmetric and independent of the subsets $S, T$. In particular, the pair of statistics $(\lambda(A;M), \lambda(\bar A; M))$ is equidistributed with (\se(M),\ne(M)), where \se(M) and $\ne(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.Comment: 20 pages, 6 figure

Underwater Endoscopic Mucosal Resection of Large Duodenal Adenomas (Video)

AbstractBackground and aimsEndoscopic mucosal resection (EMR) is a well-established method for the removal of neoplastic polyps throughout the GI tract. EMR typically involves insufflation of the lumen using air or CO2, followed by submucosal lifting of a polyp to minimize the risk of deep tissue injury and perforation, followed by hot-snare polypectomy. Underwater endoscopic mucosal resection (UEMR) is a new technique that uses water to enable lesion visualization in a lumen that is not distended by air or CO2, followed by piecemeal hot-snare resection of large mucosally-based neoplasms. UEMR does not require submucosal injection to create a fluid cushion. Very few published examples of UMER in the duodenum exist. This video case series describes the use of UEMR for the resection of several large duodenal adenomas.ProcedureUnderwater endoscopic mucosal resection was utilized for the removal of several large duodenal adenomas.ResultsThree duodenal lesions ranging from 1.8cm to 5cm were successfully resected by UEMR. The mean time for resection was 18min. There were no adverse events.ConclusionsUEMR is an efficacious technique for the resection of large mucosally-based neoplasms of the duodenum

Monodisperse core/shell and other complex structured nanocrystals and methods of preparing the same

Provides new compositions containing nearly monodisperse colloidal core/shell semiconductor nanocrystals with high photoluminescence quantum yields (PL QY), as well as other complex structured semiconductor nanocrystals. This invention also provides new synthetic methods for preparing these nanocrystals, and new devices comprising these compositions. In addition to core/shell semiconductor nanocrystals, this patent also provides complex semiconductor nanostructures, quantum shells, quantum wells, doped nanocrystals, and other multiple-shelled semiconductor nanocrystals