16 research outputs found
Recognition of plane-to-plane map-germs
We present a complete set of criteria for determining A-types of
plane-to-plane map-germs of corank one with A-codimension <7, which provides a
new insight into the A-classification theory from the viewpoint of recognition
problem. As an application to generic differential geometry, we discuss about
projections of smooth surfaces in 3-space.Comment: 22 page
Contact unimodal map germs from the plane to the plane
In this article, we correct the classification of unimodal map germs from the plane to the plane of Boardman symbol given by Dimca and Gibson. Also, we characterize this classification of unimodal map germs in terms of certain invariants. Moreover, on the basis of this characterization we present an algorithm to compute the type of unimodal map germs of the Boardman symbol without computing the normal form and give its implementation in the computer algebra system Singular [8]
Contact unimodal map germs from the plane to the plane
In this article, we correct the classification of unimodal map germs from the plane to the plane of Boardman symbol given by Dimca and Gibson. Also, we characterize this classification of unimodal map germs in terms of certain invariants. Moreover, on the basis of this characterization we present an algorithm to compute the type of unimodal map germs of the Boardman symbol without computing the normal form and give its implementation in the computer algebra system Singular [8]
C. T. C. Wall's contributions to the topology of manifolds
C. T. C. Wall spent the first half of his career, roughly from 1959 to
1977, working in topology and related areas of algebra. In this period, he
produced more than 90 research papers and two books. Above all, Wall was responsible for major advances in the topology of
manifolds. Our aim in this survey is to give an overview of how his work has
advanced our understanding of classification methods. Wall's approaches to
manifold theory may conveniently be divided into three phases
On kinematic singularities of low dimension.
This thesis is an investigation into the types of singularities that can appear
on trajectories of rigid motions, kinematic singularities, motivated by problems
in mechanical engineering of designing mechanisms. Here we consider rigid motions
of the plane and space with one and two degrees of freedom.
In order to study these singularities weprove a multi-germ transversality result
and also a result about the restrictions on the codimension of the singularity
given by the number of degrees of freedom of the motion. Some of the classifications
of the singularities we are interested in have already been completed
but all the simple singularities of space curves and also most of the multi-germs,
both of the plane and of space, are classified here. We also study the unfoldings
and bifurcation sets of all the kinematic singularities on our lists
Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand
Listed in 2020 Dean's List of Exceptional ThesesCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis
(a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived.
(b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem.
(c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for -series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations.
(d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation.
(e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version