Search results for "Calculus"

Conditional measures and their applications to fuzzy sets

1991

Abstract Given a ⊥-decomposable measure with respect to a continuous t-conorm, as introduced by the author in an earlier paper (see Section 1), we can construct ⊥-conditional measures as implications. These fulfil a ‘generalized product law’ replacing the product in the classical law by any other strict t-norm ⊥ and turn out to be decomposable with respect to an operation ⊥ V depending on ⊥, ⊥ and the condition set V (Section 2). More general, conditional measures are introduced axiomatically and are shown to be ⊥-conditional measures with respect to some ⊥-decomposable measure (Section 3). ‘Bayesian-like’ models are given which are alternatives to that presented by the author in a recent p…

Vectors and Vector Fields

2012

The purpose of this book is to explain in a rigorous way Stokes’s theorem and to facilitate the student’s use of this theorem in applications. Neither of these aims can be achieved without first agreeing on the notation and necessary background concepts of vector calculus, and therein lies the motivation for our introductory chapter.

Vectors, Tensors, Manifolds and Special Relativity

2015

Assuming that the reader is familiar with the notion of vectors, within a few pages, with a few examples, the reader will get to be familiar with the generic picture of tensors. With the specific notions given in this chapter, the reader will be able to understand more advanced tensor courses with no further effort. The transition between tensor algebra and tensor calculus is done naturally with a very familiar example. The notion of manifold and a few basic key aspects on Special Relativity are also presented.

Volume growth and parabolicity

2001

On the enhancement of diffusion by chaos, escape rates and stochastic instability

1999

We consider stochastic perturbations of expanding maps of the interval where the noise can project the trajectory outside the interval. We estimate the escape rate as a function of the amplitude of the noise and compare it with the purely diffusive case. This is done under a technical hypothesis which corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map. We also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure.

Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams

2000

Present and future high-precision tests of the Standard Model and beyond for the fundamental constituents and interactions in Nature are demanding complex perturbative calculations involving multi-leg and multi-loop Feynman diagrams. Currently, large effort is devoted to the search for closed expressions of loop integrals, written whenever possible in terms of known - often hypergeometric-type - functions. In this work, the scalar three-point function is re-evaluated by means of generalized hypergeometric functions of two variables. Finally, use is made of the connection between such Appell functions and dilogarithms coming from a previous investigation, to recover well-known results.

Indefinite integrals of products of special functions

2016

ABSTRACTA method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals o…

Fixed point theorems for multivalued maps via new auxiliary function

2016

We introduce a contractive condition involving new auxiliary function and prove a fixed point theorem for closed multivalued maps on complete metric spaces. An example and an application to integral equation are given in support of our findings.

On the number of solutions of a Duffing equation

1991

The exact number of solutions of a Duffing equation with small forcing term and homogeneous Neumann boundary conditions is given. Several bifurcation diagrams are shown.

Recent progress in electrical impedance tomography

2003

We consider the inverse problem of finding cavities within some body from electrostatic measurements on the boundary. By a cavity we understand any object with a different electrical conductivity from the background material of the body. We survey two algorithms for solving this inverse problem, namely the factorization method and a MUSIC-type algorithm. In particular, we present a number of numerical results to highlight the potential and the limitations of these two methods.