Let α be an algebraic integer whose all conjugates lie in a sector |arg z| ≤ θ with 0 ≤ θ < 90◦. Using the method of explicit auxiliary functions, we compute the greatest lower bound v(θ) of the absolute trace of α, for θ belonging to seven subintervals of [0, 90◦). The polynomials involved in the auxiliary functions are found by Wu’s algorithm.

Likelihood principle concerns foundations of statistical inference and it is often invoked in arguments about correct statistical reasoning. Let f(x|θ) be a conditional distribution for X given the unknown parameter θ. For the observed data, X = x, the function `(θ) = f(x|θ), considered as a function of θ, is called the likelihood function. The name likelihood implies that, given x, the value o...

Likelihood principle concerns foundations of statistical inference and it is often invoked in arguments about correct statistical reasoning. Let f(x|θ) be a conditional distribution for X given the unknown parameter θ. For the observed data, X = x, the function `(θ) = f(x|θ), considered as a function of θ, is called the likelihood function. The name likelihood implies that, given x, the value o...

For a positive integer n and d ∈ [0, 1], let G(n, d) be the set of graphs on n vertices with minimum degree at least d. Let G be constructed by taking H ∈ G(n, d) and adding a set R of random edges, R ⊆ [n] 2 \ E(H), m = |R|. We write G = H + R. The term whp means that a property occurs with high probability. That is, the property occurs with probability approaching 1 as n → ∞. A graph is said ...

C Maximum likelihood estimation C.1 Building the likelihoood function Let Θ = (Θ1; Θ2) = (σ , σ η, σ u, π; β1, λ1, β2, λ2, φ) be the vector of parameters. Let X = (x1,s,t, x2,s,t, r1,s,t, r2,s,t) be the vector of observables, with r2 = p2− p1 and r1 = p1. Let 1(x) be an indicator variable equal to 1 if the event x has occured, and 0 otherwise. Let g(x) be the standard normal PDF and G(x) the st...

In Harding (1996; see also Harding, 1998, 1999), a method was given to construct an orthomodular poset Fact X from the direct product decompositions of a set. As this forms the basis of our study we briefly review the pertinent facts. For a set X let Eq X denote the set of equivalence relations on X, use ◦ for relational product, 1 for the least equivalence relation on X, and ∇ for the largest....

Let H be a family of connected graphs. A graph G is said to be H-free if G does not contain any members of H as an induced subgraph. Let F(H) be the family of connected H-free graphs. In this context, the members of H are called forbidden subgraphs. In this paper, we focus on two pairs of forbidden subgraphs containing a common graph, and compare the classes of graphs satisfying each of the two...

This paper aims to investigate properties of involution palindrome languages. Let θ be either a morphic or antimorphic involution. we show that if θ is an antimorphic involution, then the set of all θ-palindrome words is a proper subset of all skew θ-palindrome words. We give a characterization for that a word is skew θ-palindrome. That is, A word is skew θ-palindrome if and only if it is a pro...

1 Frame of the first-order asymptotic decision theory Consider a sequence of statistical experiments ET = (X T ,AT , {P T θ }θ∈Θ) (T ∈ R+). Let θ̂T : X T → Θ be a sequence of estimators of the unknown parameter θ. A basic property θ̂T should have is the consistency : θ̂T →PT θ θ (T → ∞) for every θ ∈ Θ. The analyst should not use any estimator without checking this property. For example, if one us...

Let Q be an infinite subset of Z, let Ψ : Z→ [0,∞), and let θ ∈ R. Define E(Q,Ψ, θ) = {x ∈ R : ‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q}. We prove a lower bound on the Fourier dimension of E(Q,Ψ, θ). This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimens...