ARTICLE | doi:10.20944/preprints201610.0089.v1
Subject: Mathematics & Computer Science, Analysis Keywords: improper integral; explicit expression; unified expression; beta function; Wallis ratio; integral representation; Catalan number
Online: 21 October 2016 (08:29:08 CEST)
In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.
ARTICLE | doi:10.20944/preprints202107.0711.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Improper integral; Mathematical modelling; Mathematical instrumentation.
Online: 31 July 2021 (14:39:43 CEST)
Background: There is little clarity in the application of content related to improper integrals in university students, due to the absence of meaning, which prevents them from making a connection with everyday problem situations. Methods: we designed a mathematical modelling proposal where a specific situation involving the instrumentation, use and application of this type of integrals is experimented and solved with a population of engineering students, who learn to use them. Results: The importance of using mathematical modelling as a didactic-dynamic resource is highlighted because it helps students to reach an understanding of real situations involving improper integrals in different contexts. Conclusions: Despite the numerous errors detected in the students, this strategy made it possible to demonstrate the development of advanced mathematical thinking skills in young people.
ARTICLE | doi:10.20944/preprints202206.0227.v1
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: Chebfun; differential equation; non-linearity; singularity; convergence; Bernstein growth; improper integrals; boundary layer
Online: 16 June 2022 (02:53:41 CEST)
The Chebyshev collocation method (ChC) implemented as Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for toughest problems we have to loosen the default Chebfun tolerance in Newton's solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).
ARTICLE | doi:10.20944/preprints201710.0130.v1
Subject: Materials Science, General Materials Science Keywords: Multiferroicity; LiCuVO4; Spin-driven improper ferroelectricity; Hysteresis in magnetic fields, Multiferroic Hysteresis Loop
Online: 19 October 2017 (17:42:26 CEST)
Multiferroics, showing both ferroelectric and magnetic order, are promising candidates for future electronic devices. Especially, the fundamental understanding of ferroelectric switching is of key relevance for further improvements, which however is rarely reported in literature. On a prime example for a spin-driven multiferroic, LiCuVO4, we present an extensive study of the ferroelectric order and the switching behavior as function of external electric and magnetic fields. From frequency-dependent polarization switching and using the Ishibashi-Orihara theory, we deduce the existence of ferroelectric domains and domain-walls. These have to be related to counterclockwise and clockwise spin-spirals leading to the formation of multiferroic domains. A novel measurement – multiferroic hysteresis loop – is established to analyze the electrical polarization simultaneously as a function of electrical and magnetic fields. This technique allows characterizing the complex coupling between ferroelectric and magnetic order in multiferroic LiCuVO4.