The standard choice for mutating an individual of an evolutionary algorithm with con tinuous variables is the normal distribution however other distributions especially some versions of the multivariate Cauchy distribution have recently gained increased popularity in practical applications Here the extent to which Cauchy mutation distributions may a ect the local convergence behavior of evoluti...

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...

We give a solution of the blow-up problem for equation u = e, with data close to constants, in any number of space dimensions: there exists a blow-up surface, near which the solution has logarithmic behavior; its smoothness is estimated in terms of the smoothness of the data. More precisely, we prove that for any solution of u = e with Cauchy data on t = 1 close to (ln 2; 2) in H(R) H (R), s is...

In this paper we show that sufficiently smooth coefficients of the elliptic operator ∇x · σ(x)∇x − c(x′), x ∈ R, x′ ∈ R can be uniquely determined from the Cauchy data given on a strip in the plane {x3 = 0}. This is an extension of Tikhonov’s formulation of the one-dimensional inverse problem of electric prospecting to two dimensions. In this formulation, the number of variables in the Cauchy d...

Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero self-distance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over...

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. The new approach evolves flow map using gradient-augmented level set (GALSM). Since can be decomposed into submaps (each over finite time interval), error controlled by choosing remapping times appropriately. This leads to numerical scheme that has ...

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