and Applied Analysis 3 where we assume that the spatial variable x : x, y, z varies in the cubic domain Ω : { x, y, z : −1 ≤ x, y, z ≤ 1}, andΨ x, μ, θ : Ψ x, y, z, μ, θ is the angular flux in the direction defined by μ ∈ −1, 1 and θ ∈ 0, 2π . σt and σs denote the total and the differential cross section, respectively, σs μ′, φ′ → μ, φ describes the scattering from an assumed pre-collision angu...

I. PROBLEM FORMULATION The Furuta pendulum consists of an arm rotating in the horizontal plane and a pendulum attached to the end of the arm moving freely in the vertical plane, see Fig. 1. The equations of motion of the Furuta pendulum, derived from the standard Euler-Lagrange dynamics, are1 (α + β sin θ)φ̈ + γ cos θθ̈ + 2βθ̇φ̇ sin θ cos θ − γθ̇ sin θ = τφ, γ cos θφ̈ + βθ̈ − βφ̇ sin θ cos θ − δ sin θ ...

We show that the Fourier transform on the Jacobian of a curve interchanges " δ functions " on the curve and the theta divisor. The Torelli theorem is an immediate consequence. 1. Statement of the theorem. 1.1. We live over an algebraically closed base field k. Let J be an abelian variety equipped with a principal polarization θ : J ∼ → J • = Pic 0 (J), so we have the corresponding Fourier trans...

Assuming that the observed gamma-ray burst (GRB) rate as a function of redshift is proportional to a corrected star formation rate, we derive the empirical distribution of the viewing angles of long BATSE GRBs, P (θ), and the distribution of these bursts in the plane of θ against redshift, P (θ, z), by using a tight correlation between collimation-corrected gamma-ray energy (Eγ) and the peak en...

Strong convexity can be characterized in a number of ways. The following facts provide some conditions that are equivalent to Definition 1. For the 2-norm, Fact 2 means that the minimum eigenvalue of the Hessian is lower-bounded by κ. Recall that˜µ(θ) and˜µ(θ) are the gradients of˜Φ(θ) and˜Φ(θ), respectively. Since the conjugate function, ˜ Φ * , is assumed to be κ-strongly convex, we have via ...

In the framework of started in Ref.[1] construction procedure of the general superfield quantization method for gauge theories in Lagrangian formalism the rules for Hamiltonian formulation of general superfield theory of fields (GSTF) are introduced and are on the whole considered. Mathematical means developed in [1] for Lagrangian formulation of GSTF are extended to use in Hamiltonian one. Ham...

This article surveys the θ-intermediate dimensions that were introduced recently which provide a parameterised continuum of run from Hausdorff dimension when θ = 0 to box-counting 1. We bring together diverse properties intermediate we illustrate by examples.

3.1 Maximum likelihood estimates — in exponential families. Let (X, B) be a measurable space and {P θ , θ ∈ Θ} a measurable family of laws on (X, B), dominated by a σ-finite measure v. Let f (θ, x) be a jointly measurable version of the density (dP θ /dv)(x) by Theorem 1.3.3. For each x ∈ X, a maximum likelihood estimate (MLE) of θ is any θ ˆ = θ ˆ (x) such that f (ˆ θ, x) = sup{f (φ, x) : φ ∈ ...