In this contribution we deal with the development, theoretical examination and numerical examples of a method of lines approximation for the Cauchy problem for elliptic partial differential equations. We restrict ourselves to the Laplace equation. A more general elliptic equation containing a diffusion coefficient will be considered in a forthcoming paper. Our main results are the regularizatio...

An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded `-convergences on the abelian `-group in which A appears as the unit interval. Thus the theory of `-convergence and Cauchy structures transfers to MV-algebras. We outline the general theory, and then apply...

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.

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Togethe...

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

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...