We present improved upper and lower bounds on the spanning ratio of θgraphs with at least six cones. Given a set of points in the plane, a θ-graph partitions the plane around each vertex into m disjoint cones, each having aperture θ = 2π/m, and adds an edge to the ‘closest’ vertex in each cone. We show that for any integer k ≥ 1, θ-graphs with 4k+ 2 cones have a spanning ratio of 1 + 2 sin(θ/2)...

This Article contains typographical errors. In the Results section under subheading 'Simultaneous silence and HOIs of hippocampal neurons' " The parameters {θ i, θ i j} were adjusted to fit these statistics. " should read: " The parameters {θ i, θ ij } were adjusted to fit these statistics. " In the Results section under subheading 'Simultaneous silence is a ubiquitous feature of HOIs' " By exp...

We are given a finite set of n points (guards) G in the plane R 2 and an angle 0 ≤ Θ ≤ 2π. A Θ-cone is a cone with apex angle Θ. We call a Θ-cone empty (with respect to G) if it does not contain any point of G. A point p ∈ R is called Θ-guarded if every Θ-cone with its apex located at p is non-empty. Furthermore, the set of all Θ-guarded points is called the Θ-guarded region, or the Θ-region fo...

This is a short tutorial on the EM algorithm, including modern proofs of monotonicity, and several examples focusing on the use of EM to handle heavy-tailed models (Laplace, Student) and on finite mixture estimation. 1 The Algorithm Consider a general scenario in which we have observed data x, and a set of unknown parameters θ. Let us also assume some prior p(θ) for the parameters, which could ...

The authors point to several problems occurring if DIC = 2D(θ) − D(θ) is applied to missing data models: (i) the focus of the model depends on whether the missing data z are considered as observations or as parameters or integrated out; (ii) the parameter θ may be non-identifiable; (iii) θ can be a poor estimator of θ. They then investigate modifications of DIC mainly based on estimates other t...

We focus on Θ-rich and almost Θ-rich words over a finite alphabet A, where Θ is an involutive antimorphism over A∗. We show that any recurrent almost Θ-rich word u is an image of a recurrent Θ′-rich word under a suitable morphism, where Θ′ is also an involutive antimorphism. Moreover, if the word u is uniformly recurrent, we show that Θ′ can be set to the reversal mapping. We also treat one spe...

For robustness check, in this section we extend our analysis to the continuous type model, which can be regarded as the limiting case of many finite types. As an overview, with a continuum of types, the tax schedule chosen under each regime is characterized by a second-order differential equation with two boundary values. By focusing on the case where the vertical types are distributed uniforml...

Let K denote a field. Let d denote a nonnegative integer and consider a sequence p = (θi, θ ∗ i , i = 0...d;φj , φj , j = 1...d) consisting of scalars taken from K. We call p a parameter array whenever: (PA1) θi 6= θj, θ ∗ i 6= θ ∗ j if i 6= j, (0 ≤ i, j ≤ d); (PA2) φi 6= 0, φi 6= 0 (1 ≤ i ≤ d); (PA3) φi = φ1 ∑i−1 h=0 θh−θd−h θ0−θd + (θ i − θ ∗ 0)(θi−1 − θd) (1 ≤ i ≤ d); (PA4) φi = φ1 ∑i−1 h=0 ...

Traveling wave solutions to a class of dispersive models, ut − utxx + uux = θuuxxx + (1− θ)uxuxx, are investigated in terms of the parameter θ, including two integrable equations, the Camassa-Holm equation, θ = 1/3, and the Degasperis-Procesi equation, θ = 1/4, as special models. It was proved in [H. Liu and Z. Yin, Contemporary Mathematics, 2011, 526, pp273–294] that when 1/2 < θ ≤ 1 smooth so...

We are given a finite set of m points (guards) G in the plane R 2 and an angle 0 ≤ Θ ≤ 2π. A Θ-cone is a cone with apex angle Θ. We call a Θ-cone empty (with respect to G) if it does not contain any point of G. A point p ∈ R is called Θ-guarded if every Θ-cone with apex located at p is non-empty. Moreover, the set of all Θ-guarded points is called Θ-region. We first show how to construct a data...