Search results for "calculus"

Determination of molecular weights and Stokes' radii of non-denatured proteins by polyacrylamide gradient gel electrophoresis. 1. An equation relatin…

1982

Untreated and processed gel plates of polyacrylamide (PAA) gradient gels were cut into strips perpendicularly to their length, and the wet and dry matter of the sections was determined. In untreated gels the apparent dry matter, as well as the relative dry matter, are a linear function of the gel length. In processed gels, however, only the apparent gel concentration increases linearly with the gel length, whereas the relative dry matter increases linearly with the square root of the gel length. The %T (content in polyacrylamide) was calculated from the apparent dry matter. The gel gradients used were found to be linear with respect to %T. Six different calibration proteins were run and the…

Fractional viscoelastic characterization of laminated glass beams under time-varying loading

2021

Abstract Laminated glass is a composite made of elastic glass layers sandwiching thin viscoelastic polymeric interlayers. There are several types of polymers, traditionally modelled as linear viscoelastic materials using a Prony’s series of units in the Maxwell-Wiechert arrangement. We show that one single element with fractional viscoelastic properties (two constitutive parameters that depend on environmental temperature), is sufficient to provide an accurate description of the polymer response under arbitrary time-varying actions. This is a great advantage over the classical viscoelastic characterization, which requires at least 10–15 terms in the Prony’s series, each one characterized by…

1986

The variation of the Huggins coefficient KH with the relative molecular mass M of the polymers was measured for solutions of polystyrene and of polyisobutylene and found to be most pronounced in the case of thermodynamically good solvents but vanishing at the theta-temperature, where the individual curves kH (T; M) intersect with each other. The experimental results are interpreted as a consequence of the rheological inequality of inter- and intra-molecular contacts between polymer segments. A model is presented according to which kH should be a linear function of M−(a−0,5), where a is the exponent of the intrinsic viscosity-relative molecular mass relationship (Kuhn-Mark-Houwink). The eval…

Stoïlow’s theorem revisited

2020

Stoilow's theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps z -> z(k) and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested in continuous, open and discrete maps. (C) 2019 Elsevier GmbH. All rights reserved. Peer reviewed

Convex analysis and dual problems

2018

Tässä tutkielmassa tarkastellaan valittujen variaatiolaskennan ongelmien ja näiden duaaliongelmien välisiä suhteita. Tutkielmassa esitetään aiheen yleinen teoria ja annetaan esimerkkejä sovelluksista.

Propuesta de unidad didáctica del concepto de integral y del cálculo de primitivas

2010

En este Trabajo Fin de Máster se va a proceder a desarrollar una propuesta de unidad didáctica sobre el concepto de integral y el cálculo de primitivas. Ambos temas, en orden contrario, fueron de los que me encargué de desarrollar en mis prácticas. Como trataré más adelante, la forma tradicional de introducir estos temas a los alumnos es comenzar con el cálculo de primitivas o antiderivadas para después, una vez que los alumnos tienen suficiente destreza en el cálculo de las mismas, continuar con el teorema fundamental del cálculo y la regla de Barrow, es decir, las integrales definidas, convirtiéndose dichos conceptos en transparentes para los alumnos y sin llegar a ser asimilados correcta…

An empirical study of the understanding of formal propositions about sequences, with a focus on infinite limits

2018

International audience; In this paper, we analyze the answers of one group of high-school students and two groups of first-year University students to a questionnaire designed to test their level of recognition and understanding of the formal definition of the concept of infinite limit. Although this empirical study is ancillary to a larger project centred on didactic engineering, its analysis sheds light on the key issue of the logical prerequisites for the learning of the fundamental concepts of analysis. It also provides a new tool to investigate students' concept-image of limits, and assess the impact of teaching contexts and teaching paths.

New degrees of freedom for differential forms on cubical meshes

2022

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Significantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices.

Are all denumerable sets of numbers order-isomorphic?

2018

International audience; In this paper we study cognitive conflicts on the issue of number sets being dense, ordered and denumerable. We first provide historical-epistemological background related to these notions. Then we consider the cognitive conflicts under the lenses of concept image and concept definition, which we use to analyse empirical data collected in order to understand better the didactical and cognitive issues at stake.

Approximate Taylor methods for ODEs

2017

Abstract A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the ODEs, and not their derivatives, are needed, just as in classical Runge–Kutta schemes. Compared to Runge–Kutta methods, the number of function evaluations to achieve a given order is higher, however with the present procedure it is much easier to produce arbitrary high-order schemes, which may be important in some applications. In many cases the new approach leads to an asymptotically lower computational cost when compared to the Taylor expansio…