The distribution of height and diameter in random non-plane binary trees

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to admit a limiting theta distribution, both in a central and local sense, as well as obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height

H-integral normal mixed Cayley graphs

A mixed graph is called integral if all the eigenvalues of its Hermitian adjacency matrix are integers. A mixed Cayley graph $Cay(\Gamma, S)$ is called normal if $S$ is the union of some conjugacy classes of a finite group $\Gamma$. In 2014, Godsil and Spiga characterized integral normal Cayley graphs. We give similar characterization for the integrality of a normal mixed Cayley graph $Cay(\Gamma,S)$ in terms of $S$. Xu and Meng (2011) and Li (2013) characterized the set $S\subseteq \mathbb{Z}_n$ for which the eigenvalues $\sum\limits_{k\in S} w_n^{jk}$ of the circulant digraph $Cay(\mathbb{Z}_n, S)$ are Gaussian integers for all $j=1,...,h$. Here the adjacency matrix of $Cay(\mathbb{Z}_n, S)$ is considered to be the $n\times n$ matrix $[a_{ij}]$, where $a_{ij}=1$ if $(i,j)$ is an arc of $Cay(\mathbb{Z}_n, S)$, and $0$ otherwise. Let $\{\chi_1,\ldots,\chi_h\}$ be the set of the irreducible characters of $\Gamma$. We prove that $\frac{1}{\chi_j(1)} \sum\limits_{s \in S} \chi_j(s)$ is a Gaussian integer for all $j=1,...,h$ if and only if the normal mixed Cayley graph $Cay(\Gamma, S)$ is integral. As a corollary to this, we get an alternative and easy proof of the characterization, as obtained by Xu, Meng and Li, of the set $S\subseteq \mathbb{Z}_n$ for which the circulant digraph $Cay(\mathbb{Z}_n, S)$ is Gaussian integral

Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

A cellular topological field theory

We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect to composition of cobordisms

Integral trees of diameter 4

An integral tree is a tree whose adjacency matrix has only integer eigenvalues. While most previous work by other authors has been focused either on the very restricted case of balanced trees or on finding trees with diameter as large as possible, we study integral trees of diameter 4. In particular, we characterize all diameter 4 integral trees of the form T(m1, t1) T(m2, t2). In addition we give elegant parametric descriptions of infinite families of integral trees of the form T(m1, t1) · · · T(mn, tn) for any n > 1. We conjecture that we have found all such trees