A general method is developed by using nonstandard analysis for formulating and proving a theorem about upper Banach density parallel to each theorem about Shnirel'man density or lower asymptotic density. There are many interesting results about Shnirel'man density or lower asymptotic density (see [4, Chapter 1] for example) in additive number theory. There are also a few interesting results ab...

In the classical theory of probability we start by a pair (Ω, ), with Ω set of elementary events and σ-algebra of subsets of Ω, called events. A probability on is defined as a countable additive and positive function p such that p(Ω)=1. If A and B are two events, the conditional probability p(A/B) is defined if and only if p(B) is not null, by considering the quotient p(AB)/p(B). In the theory ...

We develop a test for the linear no cointegration null hypothesis in a threshold vector error correction model. We adopt a sup-Wald type test and derive its null asymptotic distribution. A residual-based bootstrap is proposed, and the first-order consistency of the bootstrap is established. A set of Monte Carlo simulations shows that the bootstrap corrects size distortion of asymptotic distribu...

We consider goodness–of–fit testing for multivariate stable distributions. The proposed test statistics exploit a characterizing property of the characteristic function of these distributions and are consistent under some conditions. The asymptotic distribution is derived under the null hypothesis as well as under local alternatives. Conditions for an asymptotic null distribution free of parame...

Testing a low-dimensional null hypothesis against a high-dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model. Using asymptotic arguments, we show that the distribution of such a test statistic can be approximated by a ratio of quadratic forms in normal variables, for which algorithms are readily availa...

We present an analytic method for calculating the late-time tails of a linear scalar field outside a Kerr black hole. We give the asymptotic behavior at timelike infinity (for fixed r), at future null infinity, and along the event horizon (EH). In all three asymptotic regions we find a power-law decay. We show that the power indices describing the decay of the various modes at fixed r differ fr...

It is shown that the integrability conditions that arise in the Null Surface Formulation (NSF) of general relativity (GR) impose a field equation on the local null surfaces which is equivalent to the vanishing of the Bach tensor. This field equation is written explicitly to second order in a perturbation expansion. The field equation is further simplified if asymptotic flatness is imposed on th...

In this work, we explore the existence of traversable wormhole solutions supported by double gravitational layer thin shells and satisfying null energy condition (NEC) throughout whole spacetime, in a quadratic-linear form generalized hybrid metric-Palatini gravity. We start showing that for particular action, junction conditions on continuity Ricci scalar $R$ Palatini $\mathcal{R}$ theory can ...