Starting with a σ-finite measure on an algebra, we define a pseudometric and show how measurable sets from the Caratheodory Extension Theorem can be thought of as limit points of Cauchy sequences in the algebra.

In this article, we have proposed a new distribution having three parameters using half Cauchy family of called Half-Cauchy Exponential extension Distribution. The statistical properties as well different characteristics like the hazard rate function (HRF), probability density (PDF), cumulative (CDF), quantile function, skewness, and kurtosis are discussed. estimated by least-square estimation ...

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, (M,gab), with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be int...

The aim of this paper is to establish an extension of Minkowski’s inequality when p, s, t ∈ R−{0}, p > 0 are different such that s, t < 1, s, t = 0 and s−t p−t < 1, to isotonic linear functionals taking into account that the time scale Cauchy delta, Cauchy nabla, α-diamond, multiple Riemann, and multiple Lebesque integrals all are isotonic linear functionals. Several applications of these resul...

This paper continues the study of a kernel family which uses the Cauchy-Stieltjes kernel 1/(1 − θx) in place of the celebrated exponential kernel exp(θx) of the exponential families theory. We extend the theory to cover generating measures with support that is unbounded on one side. We illustrate the need for such an extension by showing that cubic pseudo-variance functions correspond to free-i...

In this paper, we develop Terence Tao’s harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if ∅ 6= A,B ⊆ Z/pZ with p a prime, then |A+ B| > min{p, |A|+ |B| − 1}, where A + B = {a + b : a ∈ A, b ∈ B}. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new form of the uncertainty princip...

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy-Kovalevskaya extension theorem for discre...

An extension of Kato’s stability condition for nonautonomous Cauchy problems is presented. It is proved that in the commutative case this condition and a mild regularity assumption imply wellposedness. If one supposes the Kato-stability, then the solutions are given by an integral formula. By means of examples we show that in general these stability conditions cannot be omitted in our results. ...