Proof. To see L2-convergence, we will prove that (Sn)n>1 is Cauchy in L 2. It is enough to show that ||Sn − Sm||2 < ε for all n,m > N(ε). Suppose n > m, then we obtain ||Sn − Sm||2 = E(Sn − Sm) = E(Xm+1 +Xm+2 + · · ·+Xn) = Var(Xm+1 +Xm+2 + · · ·+Xn) = Var(Xm+1) + Var(Xm+2) + · · ·+ Var(Xn). For any ε > 0, there exists N = N(ε) with ∑∞ i=N Var(Xi) < ε 2, thus we have ||Sn−Sm||2 < ε for all n,m >...

In this paper, we establish a new version of Ekeland’s variational principle in a new setting of cone pseudo-quasimetric spaces. In constrast to metric spaces, we do not require that each forward Cauchy sequence is forward convergent and that each forward convergent sequence has the unique forward limit. The motivation of this paper comes from applications in Behavioral Sciences since a cone ps...

1. The Set Q 1 2. Addition and multiplication of rational numbers 4 2.1. Definitions and properties. 4 2.2. Comments 7 2.3. Connections with Z. 9 2.4. Better notation. 10 2.5. Solving the equations Ea,b and Ma,b. 12 3. Ordering the rational numbers 13 4. Sequences and limits in Q 18 5. Non-convergent Cauchy sequences of rationals 20 5.1. An irrational sequence of rationals 20 5.2. An irrational...

A family of generalized Cauchy distributions, T-Cauchy{Y} family, is proposed using the T-R{Y} framework. The family of distributions is generated using the quantile functions of uniform, exponential, log-logistic, logistic, extreme value, and Fréchet distributions. Several general properties of the T-Cauchy{Y} family are studied in detail including moments, mean deviations and Shannon’s entrop...

Observations of the multi-TeV spectra of the nearby BL objects Mkn 421 and Mkn 501 exhibit the high energy cutoffs predicted to be the result of intergalactic annihilation interactions, primarily with infrared photons having a flux level as determined by various astronomical observations. After correction for this absorption effect, the derived intrinsic spectra of these multi-TeV sources can b...

We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal com...

In this article, we generalize, improve, unify and enrich some results for Jaggi-W-contraction-type mappings in the framework of b-metric-like spaces. Our supplement numerous methods existing literature, created new approach to prove that a Picard sequence is Cauchy space. Among other things, Wardowski’s theorem, but now by using only property (W1). proofs article are much shorter than ones rec...

We introduce the notion of entropy solutions (e.s.) to a conservation law with an arbitrary jump continuous flux vector and prove existence largest smallest e.s. Cauchy problem. The monotonicity stability properties these are also established. In case periodic initial function, we derive uniqueness Generally, property can be violated, which is confirmed by example. Finally, that in single space...

In this paper is introduced a new type of generalization of metric spaces called $S_b$ metric space. For this new kind of spaces it has been proved a common fixed point theorem for four mappings which satisfy generalized contractive condition. We also present example to confirm our theorem.

In this paper we define concepts of statistical convergence and statistical Cauchy on probabilistic normed spaces. Then we give a useful characterization for statistically convergent sequences. Furthermore, we display an example such that our method of convergence is stronger than the usual convergence on probabilistic normed spaces. We also introduce statistical limit points, statistical clust...