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 paper deals with the filtering problem for a class of discrete time stochastic volatility models in which the disturbances have rational probability density functions. This includes the Cauchy distributions and Student t-distributions with odd number of degrees of freedom. Using state space realizations to represent the rational probability density functions we are able to solve the filter...

In this work we propose to design a minimum phase FIR digital filter transfer function from a given linear phase FIR transfer function which has identical amplitude. We are concentrating on very high degree polynomials for which factorisation procedures for root extraction are unreliable. We also assume that the polynomials have roots on the unit circle. The approach taken involves the use of t...

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...