The initial-boundary problem for the heat conduction equation inside a bounded domain is considered. It supposed that on boundary of this exchange takes place according to Newton’s law. control parameter equal magnitude output hot air and defined givenmpart boundary. Then we determined dependence T(θ) parameters temperature process when θ close critical value. Рассмотрена начально-краевая задач...

We consider the probabilistic approach to the problems treated in [7]. We focus on the diffusion models generated by L θ,a u(x) := θ 2 x 2a u + (θ 2 a x 2a−1 + θ 1 x a) u , θ = (θ 1 , θ 2) T ∈ R × (0, +∞), when a = 1 2 or a = 1 and face the problem of finding optimal (in the asymptotic sense) estimators of the unknown parameter vector θ.

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...

We study the threshold θ ≥ 2 contact process on a homogeneous tree Tb of degree κ = b + 1, with infection parameter λ ≥ 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point λMF c (κ, θ) and for λ ≥ λ MF c (κ, θ) it survives iff p ≥ p MF c (κ, θ, λ), where this critical density satisfies 0 < pMF c (κ, θ, λ...

In this work, we studied the transformations involving coherent (θ”/S”), semicoherent (θ’/S’), and incoherent (θ/S) precipitates (θ = Al 2 Cu, S CuMg) formed during heat treatment of an annealed nanostructured powder 2024 produced by cryogenic milling. These form over a wide temperature range (300–500 °C) depending on chemical composition. They influence grain growth sintering may facilitate fo...

We begin with three technical lemmas. Lemma 2 Let y ̸ = 0 and 0 < λ 1 ≤ ∥X T y∥ ∞. We have ⟨ y λ 1 − θ * 1 , θ * 1 ⟩ ≥ 0. Proof Since the Euclidean projection of y λ1 onto {θ : ∥X T θ∥ ∞ ≤ 1} is θ * 1 , it follows from Lemma 1 that ⟨θ * 1 − y λ 1 , θ − θ * 1 ⟩ ≥ 0, ∀θ : ∥X T θ∥ ∞ ≤ 1. (46) As 0 ∈ {θ : ∥X T θ∥ ∞ ≤ 1}, we have Eq. (45). Lemma 3 Let y ̸ = 0 and 0 < λ 1 ≤ ∥X T y∥ ∞. If θ * 1 parallel...

θ 1 + · · · + θ k = 1. Show that θ 1 x 1 + · · · + θ k x k ∈ C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k. Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x 1 , x 2 , x 3 ∈ C, and θ 1 + θ 2 + θ 3 = 1 w...

In many problems, before any observations from f(x|θ) are available, the experimenter or statistician will be able to summarize his previous information and knowledge about where in Θ the value of θ is likely to lie by constructing a probability distribution for θ on the set Θ. In other words, before any experimental data have been collected or observed, the experimenter’s past experience and k...

In the previous class we already mentioned that many of the most powerful probabilistic models contain hidden variables. We will denote these variables with y. It is usually also the case that these models are most easily written in terms of their joint density, p(d,y,θ) = p(d|y,θ) p(y|θ) p(θ) (1) Remember also that the objective function we want to maximize is the log-likelihood (possibly incl...