S-convex,monotone, and positive interpolation with rational bicubic splines of C2-continuity

In this paper we deal with shape preserving interpolation of data sets given on rectangular grids. The aim is to show that there exist spline interpolants of the continuity classC ² which areS-convex, monotone, or positive if the data sets have these properties. This is done by using particular rational bicubic splines defined on the grids introduced by the data. Interpolants of the desired type can be constructed by a simple search procedure.     

Positivity of cubic polynomials on intervals and positive spline interpolation

A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubicC ¹-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization.     

Projective Extensions of Fields

Every field K admits proper projective extensions, that is,Galois extensions where the Galois group is a non-trivial projectivegroup, unless K is separably closed or K is a pythagorean formallyreal field without cyclic extensions of odd degree. As a consequence,it turns out that almost all absolute Galois groups decomposeas proper semidirect products. We show that each local field has a unique maximal projectiveextension, and that the same holds for each global field ofpositive characteristic. In characteristic 0, we prove thatLeopoldt's conjecture for all totally real number fields isequivalent to the statement that, for all totally real numberfields, all projective extensions are cyclotomic. So the realizabilityof any non-procyclic projective group as Galois group over Qproduces counterexamples to the Leopoldt conjecture.     

Existence of weak solutions to doubly degenerate diffusion equations

We prove existence of weak solutions to doubly degenerate diffusion equations

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

A hypergeometric function approach to the persistence problem of single sine-Gordon breathers

It is shown that for an interesting class of perturbation functions, at most one of the continuum of sine-Gordon breathers can persist for the perturbed equation. This question is much more subtle than the question of persistence of large portions of the family, because analytic continuation arguments in the amplitude parameter are no longer available. Instead, an asymptotic analysis of the obstructions to persistence for large Fourier orders is made, and it is connected to the asymptotic behaviour of the Taylor coefficients of the perturbation function by means of an inverse Laplace transform and an integral transform whose kernel involves hypergeometric functions in a way that is degenerate in that asymptotic analysis involves a splitting monkey saddle. Only first order perturbation theory enters into the argument. The reasoning can in principle be carried over to other perturbation functions than the ones considered here.

     

Student-Invented Numeration Systems: Pattern-Analysis and Mathematical Understanding

Thirteen fifth graders were given an assignment to invent their own numeration systems, following a unit on bases and a look at early events in the history of numbers. The task presented options that required the students to make decisions (such as whether to use a base, which base to use, design of symbols, etc.), and build a rationale for the elements of their system. Analyses of patterns embedded in their invented systems provided an assessment of student understanding of numeration. The progression of more and less complex thinking related to the student's choice of a base other than 10, consistency of logic throughout the system in words and symbols, rationale for change, and perception of real life examples that would change if the system was adopted. The invention task is presented as another way to make connections.     

Numerical study on the fluid-structure interaction in a model aquatic canopy flow

This work presents numerical simulations and selected results of the flow over aquatic canopies, consisting of artificial flexible rectangular blades, arranged in a well-defined order. The results obtained with three different Reynolds and Cauchy numbers are compared with experimental data achieving good agreement. The considered range of Cauchy numbers represents three different types of canopies ranging from rigid up to highly flexible plants. The transient flow data and blade positions are statistically analyzed to gain deeper understanding of the complex physical processes for this kind of fluid structure interaction. For example, the correlation of role of large scale motion of the flexible blades in conjunction with coherent vortex structures of the flow is addressed. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability criteria for positive linear discrete-time systems

Positivity - We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have...     

Convex interval interpolation with cubic splines,II

The problem of convex interval interpolation with cubicC ¹-splines has an infinite number of solutions, if it is solvable at all. For selecting one of the solutions a regularized mean curvature is minimized. The arising finite dimensional constrained program is solved numerically by means of a dualization approach.Dedicated to Professor Julius Albrecht on the occasion of his 65th birthday.