The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance

For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenetic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Under the uniform model, the limiting distribution is the Airy distribution. The cornerstone of this study is the fact that the probabilistic models for phylogenetic trees are strongly related to the random permutation and the Catalan models for binary search trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000547 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Creation and Growth of Components in a Random Hypergraph Process

Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to 1 as $\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that $\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also show that the expected number of vertices that ever belong to an $\ell$-component is approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate consequence, it follows that with high probability the largest $\ell$-component during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$. Our results give insight about the size of giant components inside the phase transition of random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend

Fully Analyzing an Algebraic Polya Urn Model

This paper introduces and analyzes a particular class of Polya urns: balls are of two colors, can only be added (the urns are said to be additive) and at every step the same constant number of balls is added, thus only the color compositions varies (the urns are said to be balanced). These properties make this class of urns ideally suited for analysis from an "analytic combinatorics" point-of-view, following in the footsteps of Flajolet-Dumas-Puyhaubert, 2006. Through an algebraic generating function to which we apply a multiple coalescing saddle-point method, we are able to give precise asymptotic results for the probability distribution of the composition of the urn, as well as local limit law and large deviation bounds.Comment: LATIN 2012, Arequipa : Peru (2012

An Interesting Class of Operators with unusual Schatten-von Neumann behavior

We consider the class of integral operators Q_\f on $L^2(\R_+)$ of the form (Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy. We discuss necessary and sufficient conditions on $\phi$ to insure that $Q_{\phi}$ is bounded, compact, or in the Schatten-von Neumann class \bS_p, $1<p<\infty$. We also give necessary and sufficient conditions for $Q_{\phi}$ to be a finite rank operator. However, there is a kind of cut-off at $p=1$, and for membership in \bS_{p}, $0<p\leq1$, the situation is more complicated. Although we give various necessary conditions and sufficient conditions relating to Q_{\phi}\in\bS_{p} in that range, we do not have necessary and sufficient conditions. In the most important case $p=1$, we have a necessary condition and a sufficient condition, using $L^1$ and $L^2$ modulus of continuity, respectively, with a rather small gap in between. A second cut-off occurs at $p=1/2$: if \f is sufficiently smooth and decays reasonably fast, then \qf belongs to the weak Schatten-von Neumann class \wS{1/2}, but never to \bS_{1/2} unless \f=0. We also obtain results for related families of operators acting on $L^2(\R)$ and $\ell^2(\Z)$. We further study operations acting on bounded linear operators on $L^{2}(\R^{+})$ related to the class of operators Q_\f. In particular we study Schur multipliers given by functions of the form $\phi(\max\{x,y\})$ and we study properties of the averaging projection (Hilbert-Schmidt projection) onto the operators of the form Q_\f.Comment: 87 page

Intrinsic peculiarities of real material realizations of a spin-1/2 kagome lattice

Spin-1/2 magnets with kagome geometry, being for years a generic object of theoretical investigations, have few real material realizations. Recently, a DFT-based microscopic model for two such materials, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, was presented [O. Janson, J. Richter and H. Rosner, arXiv:0806.1592]. Here, we focus on the intrinsic properties of real spin-1/2 kagome materials having influence on the magnetic ground state and the low-temperature excitations. We find that the values of exchange integrals are strongly dependent on O--H distance inside the hydroxyl groups, present in most spin-1/2 kagome compounds up to date. Besides the original kagome model, considering only the nearest neighbour exchange, we emphasize the crucial role of the exchange along the diagonals of the kagome lattice.Comment: 4 pages, 4 figures. A paper for the proceedings of the HFM 2008 conferenc

Monotone graph limits and quasimonotone graphs

The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a `quasi-monotonicity' property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.Comment: 38 page

Environmental effects on progesterone profile measures of dairy cow fertility

Environmental effects on fertility measures early in lactation, such as the interval from calving to first luteal activity (CLA), proportion of samples with luteal activity during the first 60 days after calving (PLA) and interval to first ovulatory oestrus (OOE) were studied. In addition, traditional measurements of fertility, such as pregnancy to first insemination, number of inseminations per service period and interval from first to last insemination were studied as well as associations between the early and late measurements. Data were collected from an experimental herd during 15 years and included 1106 post-partum periods from 191 Swedish Holsteins and 325 Swedish Red and White dairy cows. Individual milk progesterone samples were taken twice a week until cyclicity and thereafter less frequently. First parity cows had 14.8 and 18.1 days longer CLA (LS-means difference) than second parity cows and older cows, respectively. Moreover, CLA was 10.5 days longer for cows that calved during the winter season compared with the summer season and 7.5 days longer for cows in tie-stalls than cows in loose-housing system. Cows treated for mastitis and lameness had 8.4 and 18.0 days longer CLA, respectively, compared with healthy cows. OOE was affected in the same way as CLA by the different environmental factors. PLA was a good indicator of CLA, and there was a high correlation (−0.69) between these two measurements. Treatment for lameness had a significant influence on all late fertility measurements, whereas housing was significant only for pregnancy to first insemination. All fertility traits were unfavourably associated with increased milk production. Regression of late fertility measurements on early fertility measurements had only a minor association with conception at first AI and interval from first to last AI for cows with conventional calving intervals, i.e. a 22 days later, CLA increased the interval from first to last insemination by 3.4 days. Early measurements had repeatabilities of 0.14–0.16, indicating a higher influence by the cow itself compared with late measurements, which had repeatabilities of 0.09–0.10. Our study shows that early fertility measurements have a possibility to be used in breeding for better fertility. To improve the early fertility of the cow, there are a number of important factors that have to be taken into account

Random trees with superexponential branching weights

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case when $w_n$ grows faster than exponentially with $n$. In this case the measures on trees of finite size $N$ converge weakly as $N$ tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form $w_n=((n-1)!)^\alpha$ with $\alpha >0$ we obtain more refined results about the approach to the infinite volume limit.Comment: 19 page