Given three n-tuples {λi}i=1, {μi}i=1, {νi}i=1 of complex numbers, we introduce the problem of when there exists a pair of normal matrices A and B such that σ(A) = {λi}i=1, σ(B) = {μi}i=1, and σ(A+B) = {νi}i=1, where σ(·) denote the spectrum. In the case when λk = 0, k = 2, . . . , n, we provide necessary and sufficient conditions for the existence of A and B. In addition, we show that the solu...

For a given semantics, two nonmonotonic theories Π1 and Π2 can be said to be equivalent if they have the same intended models and strongly (resp., uniformly) equivalent if for any Σ, Π1∪Σ and Π2∪Σ are equivalent, where Σ is a set of sentences (resp., literals). In the general case, no restrictions are placed on the language (signature) of Σ. Relativised notions of strong and uniform equivalence...

Within the framework of a geomechanical model that describes deformation rock mass during subhorizontal coal-bed extraction, mechanism for formation and implementation sudden outburst is substantiated: as working face gets close to weak zone – host joint, areas tensile stresses arise, which creates prerequisites space spalling coal loss with methane. The inverse problem determining conditions a...

avila and molina [1] introduced the notion of generalized weak structures which naturally generalize minimal structures, generalized topologies and weak structures and the structures α(g),π(g),σ(g) and β(g). this work is a further investigation of generalized weak structures due to avila and molina. further we introduce the structures ro(g) and rc(g) and study the properties of the structur...

Given a sampleX1, . . . , Xn of n real-valued observations, the Shapiro– Wilk test (Shapiro and Wilk, 1965) is a test of the composite hypothesis that the data are i.i.d. (independent and identically distributed) and normal, i.e. N(μ, σ) for some unknown real μ and some σ > 0. This test of a parametric hypothesis relates to nonparametrics in that a lot of statistical methods (such as t-tests an...

Bresar in 1993 proved that each biderivation on a noncommutative prime ring is a multiple of a commutatot. A result of it is a characterization of commuting additive mappings, because each commuting additive map give rise to a biderivation. Then in 1995, he investigated biderivations, generalized biderivations and sigma-biderivations on a prime ring and generalized the results of derivations fo...

Let us consider a problem in which random samples are available from two normal distributions. The problem is to determine whether the means of the two distributions are equal. Specifically, we assume that the random variables X1, · · · , Xm form a random sample of size m from a normal distribution for which both the mean μ1 and the variance σ 2 1 are unknown; and that the variables Y1, · · · ,...

A vector-valued random variable X = X 1 · · · X n T is said to have a multivariate normal (or Gaussian) distribution with mean µ ∈ R n and covariance matrix Σ ∈ S n ++ 1 if its probability density function 2 is given by p(x; µ, Σ) = 1 (2π) n/2 |Σ| 1/2 exp − 1 2 (x − µ) T Σ −1 (x − µ). We write this as X ∼ N (µ, Σ). In these notes, we describe multivariate Gaussians and some of their basic prope...