Theorem 1. Assume that β ∈ R and that X is a n × k matrix of rank k < n. Let Y1, . . . , Yn are independent normally distributed random variables with mean vector μ = Xβ. Then, the likelihood ratio test of the hypothesis H0 : Aβ = 0 versus H1 : Aβ 6= 0. where A is a q × k matrix has critical region C = {y;F (y) ≥ F0}. F is given by F (y) = ∑n i=1(yi − ̂̂ μi) −∑ni=1(yi − μ̂i) ∑n i=1(yi − μ̂i) n− k q...

The current high level of interest in neutrino properties is well justified. Neutrino properties (such as masses, mixings, magnetic moments etc.) are of interest for a variety of reasons: (i) in their own right as fundamental parameters and (ii) as harbingers of new physics beyond the standard model (if e.g. mi 6= 0, θi 6= 0, μi 6= 0 etc.). I will not review here the kinematic limits on masses ...

Let G be a graph with n vertices and m edges. Let λ1, λ2, . . . , λn be the eigenvalues of the adjacency matrix of G, and let μ1, μ2, . . . , μn be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity E(G) = ∑ni=1 |λi | is the energy of the graph G. We now define and investigate the Laplacian energy as LE(G) = ∑ni=1 |μi − 2m/n|. There is a great deal of analogy between...

Let C(t) : I 7→ R be a simple closed unit-speed C curve in R with normal ~n(t). The curve C generates a distribution Γ which acts on vector fields ~v(x1, x2) : R 2 7→ R by line integration according to Γ(~v) = ∫ ~v(C(t)) · ~n(t)dt. We consider the problem of efficiently approximating such functionals. Suppose we have a vector basis or frame Φ = (~ φμ); then an m-term approximation to Γ can be f...

A leading-power factorization formula for weight functions relating the B̄ → Xsγ photon spectrum to arbitrary partial decay rates in B̄ → Xu lν̄ is derived. These weight functions are independent of the hadronic shape function and allow for the determination of |Vub| in a model-independent way. We calculate the weight function in renormalizationgroup improved perturbation theory to complete next-t...