Jensen's functional equation on the symmetric group Sn\bold{S_n}

Two natural extensions of Jensen's functional equation on the real line are the equations $f(xy)+f(xy^{-1}) = 2f(x)$ and $f(xy)+f(y^{-1}x) = 2f(x)$, where $f$ is a map from a multiplicative group $G$ into an abelian additive group $H$. In a series of papers \cite{Ng1}, \cite{Ng2}, \cite{Ng3}, C. T. Ng has solved these functional equations for the case where $G$ is a free group and the linear group $GL_n(R)$, R=\z,\r, a quadratically closed field or a finite field. He has also mentioned, without detailed proof, in the above papers and in \cite{Ng4} that when $G$ is the symmetric group $S_n$ the group of all solutions of these functional equations coincides with the group of all homomorphisms from $(S_n,\cdot)$ to $(H,+)$. The aim of this paper is to give an elementary and direct proof of this fact.Comment: 8 pages, Abstract changed, the proof of Proposition 2.1 and Lemma 2.4 changed (minor), one reference added, final version, to be published in Aequationes Mathematicae (2011

An O(n3 [square root of] log n) algorithm for the optimal stable marriage problem

We give an O(n^3 √logn) time algorithm for the optimal stable marriage problem. This algorithm finds a stable marriage that minimizes an objective function defined over all stable marriages in a given problem instance.Irving, Leather, and Gusfield have previously provided a solution to this problem that runs in O(n^4) time [ILG87]. In addition, Feder has claimed that an O(n^3 log n) time algorithm exists [F89]. Our result is an asymptotic improvement over both cases.As part of our solution, we solve a special blue-red matching problem, and illustrate a technique for simulating Hopcroft and Karp's maximum-matching algorithm [HK73] on the transitive closure of a graph

Academic reflections between Polynesian tattooing and reflective practice

In Polynesian culture stories which may be generations old are told via tattoo art: the Tahitian word ‘tatu’ or ‘ta-tu’ means to strike something and links directly to the ancient art of tattooing to preserve an ancestral lineage and/or record a particular event or story that has been handed down from generation to generation via the same method (Villequette, 1998). Some scholars such as Gell (1993), and Schrader (2000) and Jones (2000) in Schildkrout (2004), write of tattoos being associated with “subsidiary selves, spirits, ancestors, rulers and victims” that are resident within the tattooed individual, while some write of ethnographic work being inscribed on bodies (Sparkes, 2000, p. 21 and Schildkrout, 2004, p. 322). Auto-ethnographic study (the study of ourselves) is a relatively new field and is often associated with qualitative analysis; as such it has stimulated the author to introduce the term ‘internal’ reflection. I believe that this may describe a ‘personal’ or ‘internal’ reflection that is transmitted to the outside world in the form of a tattoo. Drawing on the work of Sparkes, an auto-ethnography is a narrative of self, although this research offers tattoos as a viable alternative to narrative and suggests that auto-ethnographic tattoos are not only commonplace but that they can also be very real transcripts of the narrative equivalent. Further, this research shows that different cultures reflect in different ways and that the tattoo is a popular and essential method of ethnographic captur

Chronic Wounds: The Persistent Infection Problem

Chronic wounds heal poorly and can have a huge impact on a sufferer’s life. They are caused by a number of factors, one of which is the presence of persistent infections. Many standard treatments are unsuccessful at destroying these infections as the bacteria form a biofilm. Biofilms encase the bacteria, preventing immune cells from destroying them. There are multiple bacterial species within a biofilm, sometimes with antibiotics resistance, and which species are present changes over time. The changing, multi-species nature of biofilms can make finding an effective antibiotic treatment difficult. Also, bacteria in biofilms genetically differ from planktonic bacteria, and are often less susceptible to antibiotics. Additionally, biofilms are thought to reduce the access of antibiotics to the bacteria within. These reasons are discussed in further detail in this review, along with some of the reasons why bacteria can prevent wound closure