Convergence with respect to a valuation is discussed as convergence of a Cauchy sequence. Cauchy sequences of polynomials are defined. They are used to formalize Hensel’s lemma.

This document presents the mechanised proofs of two popular theorems attributed to Augustin Louis Cauchy Cauchy’s Mean Theorem and the Cauchy-Schwarz Inequality.

This paper instigates the multivariate generalization of widely used Cauchy inequality 1 + x ≤ ex where can be any non-negative real number. The results this study solution for Cauchy’s problem particular Ordinary Differential Equation (ODE). is also related to complete monotone function and divided differences theory. proof based on empty product convention notion Beppo Levi theorem Monotone c...

We develop a canonical Wick rotation-rescaling theory in 3-dimensional gravity. This includes (a) A simultaneous classification: this shows how generic maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface, as well as complex projective structures on arbitrary surfaces, are all different materializations of “more fundamental” encoding stru...

Let (M,g,Ω) be the conformal rescaling of a 3+1 dimensional vacuum spacetime (M̃, g̃), with g = Ωg̃, where Ω is a smooth function on M vanishing on the boundary ∂M. If M has smooth null boundary I , with connected components of topology S2×R, it follows that the Weyl tensor of (M,g) vanishes on I . From this follows peeling properties for (M̃, g̃). This picture of isolated systems in general relativ...

Recognizing human facial expression and emotion by computer is an interesting and challenging problem. In this paper we propose a method for recognizing emotions through facial expressions displayed in video sequences. We introduce the Cauchy Naive Bayes classifier which uses the Cauchy distribution as the model distribution and we provide a framework for choosing the best model distribution as...

We formulate an elementary condition on an involutive quantaloidQ under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Q-enriched categories. For such quantaloids, which we call Cauchy-bilateral quantaloids, it follows that the Cauchy completion of any symmetric Q-enriched category is again symmetric. Examples include Lawver...

A completion of a Cauchy space is obtained without the T2 restriction on the space. This completion enjoys the universal property as well. The class of all Cauchy spaces with a special class of morphisms called s-maps form a subcategory CHY′ of CHY. A completion functor is defined for this subcategory. The completion subcategory of CHY′ turns out to be a bireflective subcategory of CHY′. This t...

Recognizing human facial expression and emotion by computer is an interesting and challenging problem. In this paper we propose a method for recognizing emotions through facial expressions displayed in video sequences. We introduce the Cauchy Naive Bayes classifier which uses the Cauchy distribution as the model distribution and we provide a framework for choosing the best model distribution as...

In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results. AMS Classific...