Habitable Zones and UV Habitable Zones around Host Stars

Ultraviolet radiation is a double-edged sword to life. If it is too strong, the terrestrial biological systems will be damaged. And if it is too weak, the synthesis of many biochemical compounds can not go along. We try to obtain the continuous ultraviolet habitable zones, and compare the ultraviolet habitable zones with the habitable zones of host stars. Using the boundary ultraviolet radiation of ultraviolet habitable zone, we calculate the ultraviolet habitable zones of host stars with masses from 0.08 to 4.00 \mo. For the host stars with effective temperatures lower than 4,600 K, the ultraviolet habitable zones are closer than the habitable zones. For the host stars with effective temperatures higher than 7,137 K, the ultraviolet habitable zones are farther than the habitable zones. For hot subdwarf as a host star, the distance of the ultraviolet habitable zone is about ten times more than that of the habitable zone, which is not suitable for life existence.Comment: 5 pages, 3 figure

The mass-to-light ratio of rich star clusters

We point out a strong time-evolution of the mass-to-light conversion factor eta commonly used to estimate masses of unresolved star clusters from observed cluster spectro-photometric measures. We present a series of gas-dynamical models coupled with the Cambridge stellar evolution tracks to compute line-of-sight velocity dispersions and half-light radii weighted by the luminosity. We explore a range of initial conditions, varying in turn the cluster mass and/or density, and the stellar population's IMF. We find that eta, and hence the estimated cluster mass, may increase by factors as large as 3 over time-scales of 50 million years. We apply these results to an hypothetic cluster mass distribution function (d.f.) and show that the d.f. shape may be strongly affected at the low-mass end by this effect. Fitting truncated isothermal (Michie-King) models to the projected light profile leads to over-estimates of the concentration parameter c of delta c ~ 0.3 compared to the same functional fit applied to the projected mass density.Comment: 6 pages, 2 figures, to appear in the proceedings of the "Young massive star clusters", Granada, Spain, September 200

A low-memory algorithm for finding short product representations in finite groups

We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-rho approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random sequence S of length d log_2 n, where n=#G and d >= 2 is a constant, we find that its expected running time is O(sqrt(n) log n) group operations (we give a rigorous proof for d > 4), and it only needs to store O(1) group elements. We consider applications to class groups of imaginary quadratic fields, and to finding isogenies between elliptic curves over a finite field.Comment: 12 page

Constrained Markovian dynamics of random graphs

We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the `mobility' (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph's adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology.Comment: 28 pages, 6 figure

On a problem of colouring the real plane

summary:What is the least number of colours which can be used to colour all points of the real Euclidean plane so that no two points which are unit distance apart have the same colour? This well known problem, open more than 25 years is studied in the paper. Some partial results and open subproblems are presented

Recognizing Outer 1-Planar Graphs in Linear Time

A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1-planar graphs, whose recognition is N P-hard. Our main result is a linear-time algorithm that first tests whether a graph G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors (see Fig. 3), which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p

Picking Planar Edges; or, Drawing a Graph with a Planar Subgraph

Given a graph G and a subset F ⊆ E(G) of its edges, is there a drawing of G in which all edges of F are free of crossings? We show that this question can be solved in polynomial time using a Hanani-Tutte style approach. If we require the drawing of G to be straight-line, but allow up to one crossing along each edge in F, the problem turns out to be as hard as the existential theory of the real numbers