To determine what is lost or gained by using fewer age classes in a Leslie matrix model, I develop a novel method to reduce a Leslie matrix model to a …
To determine what is lost or gained by using fewer age classes in a Leslie matrix model, I develop a novel method to reduce a Leslie matrix model to a smaller one. An aggregated (“reduced”) Leslie matrix model inherits important properties of the original model. To illustrate the influence of aggregation on asymptotic and transient dynamics, I apply the aggregator to 10 Leslie matrices for populations drawn from a diverse set of species.
We give a very short proof of the Theorem: Suppose that $f(x) = a_1x + a_2x^2 + \cdots$ is a formal power series with coefficients in an integral domain, and …
We give a very short proof of the Theorem: Suppose that $f(x) = a_1x + a_2x^2 + \cdots$ is a formal power series with coefficients in an integral domain, and $a_1$ is a primitive $n^{\mathrm {th}}$ root of unity $(n \in {\Bbb N})$. If the $n^{\mathrm {th}}$ iterate $f^{(n)}(x) \equiv f\big (f(\cdots f\big (f(x)\big )\cdots \big )$ satisfies $f^{(n)}(x) = x + b_mx^m + b_{m + 1}x^{m + 1} + \cdots$, with $b_m \neq 0$ and $m > 1$, then $m \equiv 1 \pmod {n}$.
Abstract Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design …
Abstract Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper, we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these association schemes are nonsymmetric.
The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when …
The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when used as a uniform point-wise discretization of an ambient space. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance arising from an Euclidean embedding, including the unitary group as a special case. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: 1) the normalized volume of a metric ball and 2) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.
The kissing number of is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin [Mittelmann and Vallentin 10 [Mittelmann and Vallentin …
The kissing number of is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin [Mittelmann and Vallentin 10 [Mittelmann and Vallentin 10] H. D. Mittelmann and F. Vallentin. "High-accuracy Semidefinite Programming Bounds for Kissing Numbers." Exp. Math. 19 (2010), 175–179.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], based on the semidefinite programming bound of Bachoc and Vallentin [Bachoc and Vallentin 08 [Bachoc and Vallentin 08] C. Bachoc and F. Vallentin. "New Upper Bounds for Kissing Numbers from Semidefinite Programming." J. Am. Math. Soc. 21 (2008), 909–924.[Crossref], [Web of Science ®] , [Google Scholar]], computed the best known upper bounds for the kissing number for several values of n ⩽ 23. In this article, we exploit the symmetry present in the semidefinite programming bound to provide improved upper bounds for n = 9, …, 23.
We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal …
We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.