Two-point coordinate rings for GK-curves

Giulietti and Korchm\'aros presented new curves with the maximal number of points over a field of size q^6. Garcia, G\"uneri, and Stichtenoth extended the construction to curves that are maximal over fields of size q^2n, for odd n >= 3. The generalized GK-curves have affine equations x^q+x = y^{q+1} and y^{q^2}-y^q = z^r, for r=(q^n+1)/(q+1). We give a new proof for the maximality of the generalized GK-curves and we outline methods to efficiently obtain their two-point coordinate ring.Comment: 16 page

Universal features of cluster numbers in percolation

The number of clusters per site $n(p)$ in percolation at the critical point $p = p_c$ is not itself a universal quantity---it depends upon the lattice and percolation type (site or bond). However, many of its properties, including finite-size corrections, scaling behavior with $p$, and amplitude ratios, show various degrees of universal behavior. Some of these are universal in the sense that the behavior depends upon the shape of the system, but not lattice type. Here, we elucidate the various levels of universality for elements of $n(p)$ both theoretically and by carrying out extensive studies on several two- and three-dimensional systems, by high-order series analysis, Monte-Carlo simulation, and exact enumeration. We find many new results, including precise values for $n(p_c)$ for several systems, a clear demonstration of the singularity in $n''(p)$, and metric scale factors. We make use of the matching polynomial of Sykes and Essam to find exact relations between properties for lattices and matching lattices. We propose a criterion for an absolute metric factor $b$ based upon the singular behavior of the scaling function, rather than a relative definition of the metric that has previously been used.Comment: 8 pages, 3 figure

Contested Meaning of the Nation-state Through Historical Border Narratives a Case Study of the Batang Kanyau Iban, West Kalimantan

Nation as a cultural-psychological phenomenon is best understood in terms of how a sense of nationhood operates in order to construct social identities or a social imagination about the modern nation-state (Anderson 1983). The forging of nationalism as a national identity cannot be seen in isolation of the rise of modernization and industrialization (Gellner 1987). Although the nation appears to be a modern phenomenon, Smith (1991) stresses that every nation preserves its own past historical artefacts, narratives, and symbols for present-day needs. This model needs to be elaborated further as it is insufficient to understand how a sense of nationhood operates among borderlanders of a state. This paper relates the story of Kalimantan\u27s Iban borderlanders who are officially registered as Indonesian subjects but live on the dividing line between two countries. This makes them appear to be ambiguous subjects who are torn between the two different historical timelines of British and Dutch colonial history (as well as postcolonial Malaysian-Indonesian history). They are marginalized in every aspect and are the forgotten subjects in the history of the broader picture of Indonesia\u27s so-called nationalism project. The explanation is twofold. The first explains how identity is constructed as multi-layered historical narratives involving local and national cultures, and second, how transnational borderlanders give meaning to nation as narrative. The primary data for this article were collected in 2002 through a series of interviews in the village of Benua Sadap, an Iban settlement on the Batang Kanyau River, close to the West Kalimantan (Indonesia) and Sarawak (Malaysia) borderline