Search results for "Numbers"

From the fourteenth century to Cabrì: convuleted constructions of star polygons

2014

8-parameter solutions of fifth order to the Johnson equation

2019

We give different representations of the solutions of the Johnson equation with parameters. First, an expression in terms of Fredholm determinants is given; we give also a representation of the solutions written as a quotient of wronskians of order 2N. These solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polyno-mials of degree 2N (N +1) in x, t and 4N (N +1) in y depending on 2N −2 parameters. Here, we explicitly construct the expressions of the rational solutions of order 5 depending on 8 real parameters and we study the patterns of their modulus in the plane (x, y) and their …

From first to fourth order rational solutions to the Boussinesq equation

2020

Rational solutions to the Boussinesq equation are constructed as a quotient of two polynomials in x and t. For each positive integer N , the numerator is a polynomial of degree N (N + 1) − 2 in x and t, while the denominator is a polynomial of degree N (N + 1) in x and t. So we obtain a hierarchy of rational solutions depending on an integer N called the order of the solution. We construct explicit expressions of these rational solutions for N = 1 to 4.

Shape identification in inverse medium scattering problems with a single far-field pattern

2016

Consider time-harmonic acoustic scattering from a bounded penetrable obstacle $D\subset {\mathbb R}^N$ embedded in a homogeneous background medium. The index of refraction characterizing the material inside $D$ is supposed to be Holder continuous near the corners. If $D\subset {\mathbb R}^2$ is a convex polygon, we prove that its shape and location can be uniquely determined by the far-field pattern incited by a single incident wave at a fixed frequency. In dimensions $N \geq 3$, the uniqueness applies to penetrable scatterers of rectangular type with additional assumptions on the smoothness of the contrast. Our arguments are motivated by recent studies on the absence of nonscattering waven…

Discussing Mathematical Learning and Mathematical Praxeologies from a Subject Scientific Perspective

2018

International audience; This programmatic contribution discusses the link between concepts from Anthropological Theory of Didactics (ATD) and the “subject-scientific point of view” according to Holzkamp (1985, 1993). The main common concern of ATD and the subject-scientific approach is to conceptualize and analyse “objects” like “institutionalized mathematical knowledge” and “university” not as conditions that cause reactions but essentially as meanings in the sense of generalized societal reified action possibilities. The link of both approaches is illustrated by the issue of “real numbers” in the transition from school to university: Hypotheses are derived for further actual-empirical res…

Electronic shell structures in bare and protected metal nanoclusters

2016

This short review discusses the concept of the electronic shell structure in the context of metal nanoclusters. Electronic shell structure is a natural consequence of quantization of fermionic states in a quantum confinement, where the symmetry of the confining potential creates energetically close-lying sets of states that reflect the symmetry of the potential. It was introduced in cluster physics in early 1980s and initially influenced greatly by the related model of nuclear shell structure from 1950’s. Three application areas are discussed consisting of free gas phase clusters, clusters supported by insulating oxides or oxide thin films, and clusters that are synthesized by wet chemistry…

First and second order rational solutions to the Johnson equation and rogue waves

2018

Rational solutions to the Johnson equation are constructed as a quotient of two polynomials in x, y and t depending on several real parameters. We obtain an infinite hierarchy of rational solutions written in terms of polynomials of degrees 2N (N + 1) in x, and t, 4N (N + 1) in y, depending on 2N − 2 real parameters for each positive integer N. We construct explicit expressions of the solutions in the cases N = 1 and N = 2 which are given in the following. We study the evolution of the solutions by constructing the patterns of their modulus in the (x, y) plane, and this for different values of parameters.

Sustainable Physical Activity Programs for Young Elderly : A Fuzzy Analytic Hierarchy Process Approach

2020

Physical activity (PA) programs are useful to help young elderly stay in good shape for their senior years. These programs should be sustainable, as this would keep the users active for months and years. A PA program should build on activities that users find meaningful and/or best suited for their history of sports and exercise as well as their present physical capacity. The challenge is to make the best selection from a (long) list of possible activities. We worked out a method to help young elderly to build a sustainable PA program from a set of activities that experts have identified as contributing to health and fitness among young elderly. The method builds on the Analytical Hierarchy…