On the Stanley Depth of Squarefree Veronese Ideals

Let $K$ be a field and $S=K[x_1,...,x_n]$. In 1982, Stanley defined what is now called the Stanley depth of an $S$-module $M$, denoted \sdepth(M), and conjectured that \depth(M) \le \sdepth(M) for all finitely generated $S$-modules $M$. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when $M = I / J$ with $J \subset I$ being monomial $S$-ideals. Specifically, their method associates $M$ with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in $S$. In particular, if $I_{n,d}$ is the squarefree Veronese ideal generated by all squarefree monomials of degree $d$, we show that if $1\le d\le n < 5d+4$, then \sdepth(I_{n,d})= \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d, and if $d\geq 1$ and $n\ge 5d+4$, then d+3\le \sdepth(I_{n,d}) \le \floor{\binom{n}{d+1}\Big/\binom{n}{d}}+d.Comment: 10 page

Back to the drawing board : architecture, sculpture, and the digital image

Wissenschaftliches Kolloquium vom 19. bis 22. April 2007 in Weimar an der Bauhaus-Universität zum Thema: ‚Die Realität des Imaginären. Architektur und das digitale Bild

The Reversal Ratio of a Poset

Felsner and Reuter introduced the linear extension diameter of a partially ordered set $\mathbf{P}$, denoted \mbox{led}(\mathbf{P}), as the maximum distance between two linear extensions of $\mathbf{P}$, where distance is defined to be the number of incomparable pairs appearing in opposite orders (reversed) in the linear extensions. In this paper, we introduce the reversal ratio $RR(\mathbf{P})$ of $\mathbf{P}$ as the ratio of the linear extension diameter to the number of (unordered) incomparable pairs. We use probabilistic techniques to provide a family of posets $\mathbf{P}_k$ on at most $k\log k$ elements for which the reversal ratio $RR(\mathbf{P}_k)\leq C/\log k$, where $C$ is a constant. We also examine the questions of bounding the reversal ratio in terms of order dimension and width.Comment: 10 pages, 2 figures; Accepted for publication in ORDE

Asymptotic Enumeration of Labelled Interval Orders

Building on work by Zagier, Bousquet-M\'elou et al., and Khamis, we give an asymptotic formula for the number of labelled interval orders on an $n$-element set.Comment: 6 page

Responsibility accounting (RA) as a tool to enhance cost control. A position paper of retail outlets in Zimbabwe.

This research sought to explore the impact of responsibility accounting as a tool to enhance cost control in businesses. A case study of Zimbabwean retail outlets was used and both theoretical and historical data was collected over half a year period to deduce recommendations from this study. The research observed unwillingness from management in most retail outlets to delegate responsibilities. Management highlighted their greatest fear to delegation being, unskilled subordinates yet the research exposed power hungriness on management’s part. To improve the businesses’ operations, responsibility accounting was recommended, were willingness to delegation of responsibilities was encouraged, a clearly definition of lines of command and decentralisation of decision making to branches was vied for. Key terms: Responsibility accounting, Cost control, Control activity, Segregation of duties, Retail outlets, Zimbabwe.

Electronic structure of the substitutional versus interstitial manganese in GaN

Density-functional studies of the electron states in the dilute magnetic semiconductor GaN:Mn reveal major differences for the case of the Mn impurity at the substitutional site Mn_Ga versus the interstitial site Mn_I. The splitting of the two-fold and the three-fold degenerate Mn(d)states in the gap are reversed between the two cases, which is understood in terms of the symmetry-controlled hybridization with the neighboring atoms. In contrast to Mn_Ga, which acts as a deep acceptor, Mn_I acts as a donor, suggesting the formation of Coulomb-stabilized complexes such as (Mn_Ga Mn_I Mn_Ga), where the acceptor level of Mn_Ga is passivated by the Mn_I donor. Formation of such passivated clusters might be the reason for the observed low carrier-doping efficiency of Mn in GaN. Even though the Mn states are located well inside the gap,the wave functions are spread far away from the impurity center. This is caused by the hybridization with the nitrogen atoms, which acquire small magnetic moments aligned with the Mn moment. Implications of the differences in the electronic structure for the optical properties are discussed