The symmetric shape of some inequalities between two sequences real numbers generates the same in operator theory. In this paper, we study a new refinement Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several bounded linear operators on Hilbert space, where mention Bohr’s Bergström’s operators. We present an type operators, by technique monotony sequence. also prove Aczél spac...

We derive strong laws of large numbers and central limit theorems for Bajraktarević, Gini exponential- (also called Beta-type) logarithmic Cauchy quotient means independent identically distributed (i.i.d.) random variables. The a sequence i.i.d. variables behave asymptotically normal with the usual square root scaling just like geometric given Somewhat surprisingly, multiplicative in rather dif...

By studying Cameron’s operator in terms of determinants, two kinds “integer” sequences incomplete numbers were introduced. One was the sequence restricted numbers, including s-step Fibonacci sequences. Another associated Lamé higher order. classical Trudi’s formula and inverse relation, more expressions able to be obtained. These relations identities can extended those negative integers or rati...

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes t...

n this paper, we apply Picard’s Iteration Method followed by Adomian Decomposition Method to solve a nonlinear Singular Cauchy Problem of Euler- Poisson- Darboux Equation. The solution of the problem is much simplified and shorter to arriving at the solution as compared to the technique applied by Carroll and Showalter (1976)in the solution of Singular Cauchy Problem. 

Classical computations can not capture the essence of infinite computations very well. This paper will focus on a class of infinite computations called convergent infinite computations. A logic for convergent infinite computations is proposed by extending first order theories using Cauchy sequences, which has stronger expressive power than the first order logic. A class of fixed points characte...

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...

This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spac...

Bill Lawvere’s 1973 milestone paper “Metric spaces, generalized logic, and closed categories” helped us to detect categorical structures in previously unexpected surroundings. His revolutionary idea was not only to regard individual metric spaces as categories (enriched over the monoidal-closed category given by the non-negative extended real half-line, with arrows provided by ≥ and tensor by +...

The most basic concept is that of an infinite sequence (of real or complex numbers in these notes). For p ∈ Z, let Np = {k ∈ Z : k ≥ p}. An infinite sequence of (complex) numbers is a function a : Np → C. Usually, for n ∈ Np we write a(n) = an, and denote the sequence by a = {an}n=p. The sequence {an}n=p is said to converge to the limit A ∈ C provided that for each > 0 there is an N ∈ Z such th...