Minimal Rankings and the A-Rank Number of a Path

Given a graph G, a function f:V(G)→ {1,2,…,k} is a k-ranking of G if f(u)=f(v) implies every u-v path contains a vertex w such that f(w)\u3ef(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψr(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine ψr(Pn), a problem posed by Laskar and Pillone in 2000 (Refer to PDF file for exact formulas)

Minimal k-rankings and the a-rank number of a path

Given a graph G, a function f: V(G) -\u3e {1, 2, ..., k} is a k-ranking of G if f(u) = f(v) implies every u - v path contains a vertex w such that f(w) \u3e f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The a-rank number of G, denoted u,(G) equals the largest k such that G has a minimal k-ranking. We establish new results involving minimal rankings of paths and in particular we determine u(Pn), a problem suggested by Laskar and Pillone in 2000. We show u(Pn) = [log2 (n + 1)] + [log2(n + 1 - (2^([log2n]-1))] (Refer to PDF file for exact formulas)

Iron Behaving Badly: Inappropriate Iron Chelation as a Major Contributor to the Aetiology of Vascular and Other Progressive Inflammatory and Degenerative Diseases

The production of peroxide and superoxide is an inevitable consequence of aerobic metabolism, and while these particular "reactive oxygen species" (ROSs) can exhibit a number of biological effects, they are not of themselves excessively reactive and thus they are not especially damaging at physiological concentrations. However, their reactions with poorly liganded iron species can lead to the catalytic production of the very reactive and dangerous hydroxyl radical, which is exceptionally damaging, and a major cause of chronic inflammation. We review the considerable and wide-ranging evidence for the involvement of this combination of (su)peroxide and poorly liganded iron in a large number of physiological and indeed pathological processes and inflammatory disorders, especially those involving the progressive degradation of cellular and organismal performance. These diseases share a great many similarities and thus might be considered to have a common cause (i.e. iron-catalysed free radical and especially hydroxyl radical generation). The studies reviewed include those focused on a series of cardiovascular, metabolic and neurological diseases, where iron can be found at the sites of plaques and lesions, as well as studies showing the significance of iron to aging and longevity. The effective chelation of iron by natural or synthetic ligands is thus of major physiological (and potentially therapeutic) importance. As systems properties, we need to recognise that physiological observables have multiple molecular causes, and studying them in isolation leads to inconsistent patterns of apparent causality when it is the simultaneous combination of multiple factors that is responsible. This explains, for instance, the decidedly mixed effects of antioxidants that have been observed, etc...Comment: 159 pages, including 9 Figs and 2184 reference

Outer Space For 2-Dimensional Raags And Fixed Point Sets Of Finite Subgroups

In [CCV07], Charney, Crisp, and Vogtmann construct an outer space for a 2dimensional right-angled Artin group A[GAMMA] . It is a contractible space on which a finite index subgroup Out0 (A[GAMMA] ) of Out(A[GAMMA] ) acts properly. We construct a different outer space S (A[GAMMA] ) for A[GAMMA] and show that non-empty fixed point sets of finite subgroups of Out0 (A[GAMMA] ) are contractible in this space. While Culler's realization theorem ([Cul84]) implies that fixed point sets of finite subgroups of Out(Fn ) are always non-empty in the Culler-Vogtmann outer space, there is no direct counterpart to this result in the case of right-angled Artin groups and S (A[GAMMA] ). We present some methods for constructing elements in fixed point sets of finite subgroups and examine cases where such methods are applicable

In Vitro Bioavailability Study of an Antiviral Compound Enisamium Iodide

An investigation into the biopharmaceutics classification and a study of the in vitro bioavailability (permeability and solubility) of the antiviral compound enisamium iodide (4-(benzylcarbamoyl)-1-methylpyridinium iodide) were carried out. The solubility of enisamium iodide was determined in four different buffers. Apparent intestinal permeability (Papp) of enisamium iodide was assessed using human colon carcinoma (Caco-2) cells at three concentrations. The solubility of enisamium iodide in four buffer solutions from pH 1.2 to 7.5 is about 60 mg/mL at 25 °C, and ranges from 130 to 150 mg/mL at 37 °C, depending on the pH. Based on these results, enisamium iodide can be classified as highly soluble. Enisamium iodide demonstrated low permeability in Caco-2 experiments in all tested concentrations of 10–100 μM with permeability coefficients between 0.2 × 10−6 cm s−1 and 0.3 × 10−6 cm s−1. These results indicate that enisamium iodide belongs to class III of the Biopharmaceutics Classification System (BCS) due to its high solubility and low permeability. The bioavailability of enisamium iodide needs to be confirmed in animal and human studies