Search results for "calculu"
showing 10 items of 642 documents
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…
On Variational Measures Related to Some Bases
2000
Abstract We extend, to a certain class of differentiation bases, some results on the variational measure and the δ-variation obtained earlier for the full interval basis. In particular the theorem stating that the variational measure generated by an interval function is σ-finite whenever it is absolutely continuous with respect to the Lebesgue measure is extended to any Busemann–Feller basis.
Discrete exterior calculus for photonic crystal waveguides
2022
The discrete exterior calculus (DEC) is very promising, though not yet widely used, discretization method for photonic crystal (PC) waveguides. It can be seen as a generalization of the finite difference time domain (FDTD) method. The DEC enables efficient time evolution by construction and fits well for nonhomogeneous computational domains and obstacles of curved surfaces. These properties are typically present in applications of PC waveguides that are constructed as periodic structures of inhomogeneities in a computational domain. We present a two-dimensional DEC discretization for PC waveguides and demonstrate it with a selection of numerical experiments typical in the application area. …
First-year engineering students’ use of their mathematics textbook - opportunities and constraints
2012
Published version of an article in the journal: Mathematics Education Research Journal. Also available from the publlisher at: http://dx.doi.org/10.1007/s13394-012-0040-9 The role of the mathematics textbook at tertiary level has received limited exposure in previous research although it is likely that students work individually and that some of this work depends on the use of the textbook. The aim of this study was to investigate the process of approaching the textbook from epistemological, cognitive, and didactical perspectives. The focus was on identifying and discussing the opportunities and constraints in the process. The study was an explorative case study and the participants were fi…
Enseñanza de las ciencias : revista de investigación y experiencias didácticas
1996
Resumen basado en el de la publicación Resumen en inglés La evolución histórica del principio de Le Chatelier está relacionada con su uso en el aula de química. La aparente simplicidad en la cual el principio de Le Chatelier fue formulado por primera vez y el éxito logrado en la implementación de algunos procesos industriales importantes le dio un reconocimiento inicial que aún perdura. Sin embargo, desde el comienzo de este siglo, diferentes autores han señalado el carácter limitado de la regla cualitativa y su formulación vaga y ambigua. Además, la termodinámica dada por el Principio de Le Chatelier en algunas formulaciones cuantitativas limitan su aplicabilidad. Sin embargo, la mayoría d…
The use of integrals in Mechanics of Materials textbooks for engineering students: the case of the first moment of an area
2018
International audience; Research has reported on the difficulties engineering students face in relating the content of their mathematics courses to what is taught in their professional courses. One way to address these difficulties is by better understanding how mathematical notions are used in professional engineering courses. This paper analyses how the notion of first moment of an area – which is defined as an integral – is used in civil engineering courses. Basing our analysis on elements from the anthropological theory of the didactic, we are currently analysing a classic Mechanics of Materials book. Our findings indicate that although first moments are introduced as an integral, the t…
An explicit mechanical interpretation of Eringen non-local elasticity by means of fractional calculus
2009
If the attenuation function of strain is expressed as a power law, the formalism of fractional calculus may be used to handle Eringen non-local elastic model. Aim of the present paper is to provide a mechanical interpretation to this non-local fractional elastic model by showing that it is equivalent to a discrete, point-spring model. A one-dimensional geometry is considered; static, kinematic and constitutive equations as well as the proper boundary conditions are derived and discussed.