Let f and g be two continuous functions on the unit sphere Sn−1 in Rn , n ≥ 3, and let their restrictions to any one-dimensional great circle E coincide after some rotation φE of this circle: f (φE (θ)) = g(θ)∀θ ∈ E . We prove that in this case f (θ) = g(θ) or f (θ) = g(−θ) for all θ ∈ Sn−1. This answers the question posed by Richard Gardner and Vladimir Golubyatnikov. Published by Elsevier Inc.

Let (X,m, H) be a hereditary m-space. A subset of X is said to θ-H-compact relative if for every cover U by m(θ)-open sets X, there exists finite U0 such that \ ∪ ∈ H. We obtain several properties these sets. Also, we define and investigate two kinds strong forms “θ-H-compact X”.

3.9 A likelihood ratio test for nested composite hypotheses: Wilks's theorem. Let Θ be a d-dimensional parameter space, specifically, an open set in R d. Let H 0 be a k-dimensional subset of Θ, in a sense to be made more precise below, for some k < d. For example, H 0 could be the intersection with Θ of a k-dimensional flat hyperplane. Let {P θ , θ ∈ Θ} be an equivalent family of laws on a samp...

Let A be a self-adjoint operator on a Hilbert space H. Assume that the spectrum of A consists of two disjoint components σ0 and σ1 such that the convex hull of the set σ0 does not intersect the set σ1 . Let V be a bounded self-adjoint operator on H offdiagonal with respect to the orthogonal decomposition H = H0 ⊕H1 where H0 and H1 are the spectral subspaces of A associated with the spectral set...

Theorem 1. Let p(x,θ) be a strictly positive, continuous probability distribution over continuous or discrete states xn and continuous parameters θ = (θ1, . . . , θM ), twice continuously differentiable with respect to θ. Let b(θ) be a strictly positive, twice continuously differentiable function. Then the set of stochastic differential equations (S1) leaves the distribution p∗(θ) invariant. Fu...

A large number of experiments now appear to confirm the existence of the exotic, strangeness +1 θ baryon. The I = 3/2 exotic Ξ3/2 signal observed by NA49 remains to be confirmed (it is not seen by HERA-B, and its compatibility with earlier high statistics Ξ production experiments has also been questioned). Recently, H1 reported evidence for an anti-charmed exotic, though this state was not seen...

Let κ be a regular cardinal. Consider the Baire numbers of the spaces (2)κ for various θ ≥ κ. Let l be the number of such different Baire numbers. Models of set theory with l = 1 or l = 2 are known and it is also known that l is finite. We show here that if κ > ω, then l could be any given finite number. The Baire number of a topological space with no isolated points is the minimal cardinality ...

The following result is proved: Let R be a 2-torsion free semiprime ring, and let T : R → R be an additive mapping, related to a surjective homomorphism θ : R → R, such that 2T (x2) = T (x)θ(x) + θ(x)T (x) for all x ∈ R. Then T is both a left and a right θ-centralizer.

• Let X be a short-hand notation for the vector [X0, X1, X2, . . . , Xq]. Under the Bayesian paradigm, x0 is no longer viewed as a point value but as a a random variable. Under both, the frequentist and Bayesian paradigms, X1, X2, . . . , Xq are all random variables defined by the Markov process Xt (see eq. 12). • Let pY|θ(y|θ) be the sampling density function of the observations given θ. • Let...

Let cov[Y ] denote the covariance matrix of a multivariate random variable Y , i. Proposition 7 (Cramér-Rao inequality). Suppose that X is a multivariate random variable with distribution p(x; θ), for parameter θ ∈ Θ n , and let T(X) = (T 1 (X),. .. , T r (X)) be any unbiased estimator of ψ(θ) = (ψ 1