Suppose that at any stage of a statistical experiment a control variable X that affects the distribution of the observed data Y can be used. The distribution of Y depends on some unknown parameter θ, and we consider the classical problem of testing a simple hypothesis H0 : θ = θ0 against a simple alternative H1 : θ = θ1 allowing the data to be controlled by X, in the following sequential contex...

The purpose of this contribution is to report a new approach, recently proposed by us in [1], to simulate numerically systems with a θ-vacuum term. To fix the ideas let us consider the partition function ZV (θ) of any θ-vacuum like model in a finite space-time lattice volume V . This action can be written, up to a normalization constant, as the discrete Fourier transform of the probability dist...

Let M(A, θ) be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet B ⊂ A such that the right factor of a bisection M(A, θ) = M(B, θB).T be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of M(A, θ) and associated bases of LK(A, θ).

Consider the single-agent decision problem, where the agent chooses the optimal action a ∈ A to maximize her expected utility. The utility depends on the state of nature θ ∈ Θ, and for the remainder of this lecture, we assume that Θ = {θ1, ..., θN} is a finite set with N elements. Assume also that the (state dependent) utility function u(a, θ) is continuous in a for all θ and A is compact. Let ...

The situation is somewhat different when the aligned rectangles are replaced by similar copies of a given convex polygon. More precisely, suppose that 0 is a distribution of N points in the unit square U ̄ [0, 1]#, treated as a torus. Suppose that AXU is a closed convex polygon of diameter less than 1 and with centre of gravity at the origin O. For every real number r satisfying 0% r% 1 and ever...

We first establish the following Lemma which relates to the results in [1]. Lemma 1. Consider random variables X ∈ R and Y ∈ R. Let f Y |X be the Radon-Nikodym derivative of the probability measure P θ Y |X with respect to arbitrary measures QY provided that P θ Y |X QY . θ ∈ R is a parameter. f θ Y is the Radon-Nikodym derivative of probability measure P θ Y with respect to QY provided that P ...

Abstract. We describe a universal transition mechanism characterizing the passage to an annealed behavior and to a regime where the fluctuations about this behavior are Gaussian, for the long time asymptotics of the empirical average of the expected value of the number of random walks which branch and annihilate on Z, with stationary random rates. The random walks are independent, continuous ti...

Let G be a real form of a complex reductive group. Suppose that we are given involutions σ and θ of G. Let H = G denote the fixed group of σ and let K = G denote the fixed group of θ. We are interested in calculating the double coset space H\G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our res...

Suppose a right triangle has edges A,B,C with corresponding lengths a, b, c. Suppose C is the hypotenuse. Let θ be the angle of the triangle formed by the edges A and C. From the definitions of trigonometric functions that we used in high school, cos θ = a/c and sin θ = b/c. Also, from an identity that we learned in high school, (cos θ) + (sin θ) = 1, so (a/c) + (b/c) = 1, i.e. a + b = c. Howev...

Let βi, i = 1, . . . , k, denote parameters whose values must be determined by deformation measurements. The results of measurements are given by a realization y of an n-dimensional random vector Y . The class of distribution functions connected with Y is F = {F (·, β, θ) : β ∈ R, θ ∈ θ}, where β = (β1, . . . , βk) (′ denotes the transposition of a matrix) ∈ R (k-dimensional Euclidean space), θ...