On trees, tanglegrams, and tangled chains

International audience Tanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared.

On axisymmetric free vibrations of a thin disc-rod structure

A simple asymptotic formula for the principal natural frequency of a thin disc-rod structure with variable material and geometric characteristics of the rod is constructed.

The study of oscillations localized near a layer of density discontinuity in an example of solar model

If at some radius r = rj there is a jump of the density and composition then a simple asymptotic formula can be derived which gives ω, the oscillation frequency for large degree modes (ℓ ≫ 1). Using relations of Gabriel and Scuflaire (1979) and taking into account the fact that the oscillation amplitude at R < rj or R > rj is equal to rℓ or r−ℓ multiplied by some functions only weakly dependent on ℓ, we obtain where g is the gravitational acceleration, and ρi(ρe) are the densities at the lover (upper) sides of the layer in question. The comparison with the exact ω-values considered below shows that the asymptotic values are larger by 13, 8.8, and 6 percents for ℓ equal to 5, 7, and 10, respectively. Because ω should be less than the maximum value of the Brunt - Väisälä frequency, we have for a finite thickness Δ of the jump layer and for finite ℓ that ℓ ≪ 2rj/Δ. We have performed numerical study of the above modes which may be called interior modes. The equilibrium model was similar to that studied by Vandakurov (1984 a,b) but now we assume that a small iron-like core (as in Rouse's (1983) model) is present. We assume the hydrogen content in the core to be large enough for convection to occur. Above the core, there are the inhomogeneous convectively neutral zone (rj ≤ r ≤ rj1), the homogeneous radiative zone (rj1 ≤ r ≤ ru), and the convective envelope (ru ≤ r ≤ R), whose structure have been taken from Spruit's (1974) paper.

Supersonic Bangs

SummaryThe non-linear theory of supersonic bangs, obtained in Part 1 for a body accelerating along a straight path, is extended to include curved paths. The basic theory remains the same. The important parameter, which appears in the theory, is the acceleration component along the ray, the rays being lines drawn from points on the flight path at an angle cos-1(1/M) with the direction of motion. It is found that the only essential effect of the curvature of the path is in the modification of this acceleration component to include a term due to the transverse acceleration. With this modification the main results are formally the same as in Part I.The strength of the bow shock is obtained, and it is found that the effect of the curvature of the path is more pronounced at points on the inside of the curve, and in general it becomes greater as the distance from the body increases. A simple asymptotic formula is obtained which predicts the strength of the shock with an error of less than five per cent, at distances of the order of a hundred body-lengths. Finally, the theory is compared and contrasted with the recent work by Warren.