Spelling suggestions: "subject:"2problems inn eometry"" "subject:"2problems inn angeometry""
1 |
Topological methods for strong local minimizers and extremals of multiple integrals in the calculus of variationsShahrokhi-Dehkordi, Mohammad Sadegh January 2011 (has links)
Let Ω ⊂ Rn be a bounded Lipschitz domain and consider the energy functional F[u, Ω] := ∫ Ω F(∇u(x)) dx, over the space Ap(Ω) := {u ∈ W 1,p(Ω, Rn): u|∂Ω = x, det ∇u> 0 a.e. in Ω}, where the integrand F : Mn×n → R is quasiconvex, sufficiently regular and satisfies a p-coercivity and p-growth for some exponent p ∈ [1, ∞[. A motivation for the study of above energy functional comes from nonlinear elasticity where F represents the elastic energy of a homogeneous hyperelastic material and Ap(Ω) represents the space of orientation preserving deformations of Ω fixing the boundary pointwise. The aim of this thesis is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of F and the relation it bares to the domain topology. Our work, building upon previous works of others, explicitly and quantitatively confirms the significant role of domain topology, and provides explicit and new examples as well as methods for constructing such maps. Our approach for constructing strong local minimizers is topological in nature and is based on defining suitable homotopy classes in Ap(Ω) (for p ≥ n), whereby minimizing F on each class results in, modulo technicalities, a strong local minimizer. Here we work on a prototypical example of a topologically non-trivial domain, namely, a generalised annulus, Ω= {x ∈ Rn : a< |x| <b}, with 0 <a<b< ∞. Then the associated homotopy classes of Ap(Ω) are infinitely many when n =2 and two when n ≥ 3. In contrast, for constructing explicitly and directly solutions to the system of Euler-Lagrange equations associated to F we introduce a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group SO(n). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions, modulo isometries, amongst such maps whereas in odd dimensions this number reduces to one. Even more surprising is the fact that in odd dimensions the functional F admits strong local minimizers yet no solution of the Euler-Lagrange equations can be in the form of a generalised twist. Thus the strong local minimizers here do not have the symmetry one intuitively expects!.
|
2 |
Aplicações do método de indução matemática à geometria / Applications of the mathematical induction method to geometryVELOZO NETO, Raimundo do Nascimento 01 June 2017 (has links)
Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-09-12T20:44:07Z
No. of bitstreams: 1
RaimundoVelozoNeto.pdf: 872870 bytes, checksum: ccaffc749ed9ed23b543712ba5273285 (MD5) / Made available in DSpace on 2017-09-12T20:44:07Z (GMT). No. of bitstreams: 1
RaimundoVelozoNeto.pdf: 872870 bytes, checksum: ccaffc749ed9ed23b543712ba5273285 (MD5)
Previous issue date: 2017-06-01 / This work deals with the Method of Mathematical Induction, in particular, its use with
a view to the solution of geometric problems. It initially some considerations are made
about the expression "inductive reasoning" whose it meaning, as appropriately must be
explained in the text, that differs from that of "mathematical induction". We prove the
proposition that guarantees the use of the method based on its foundation, namely the
axiom of mathematical induction (one of the postulates that characterize the natural
numbers). It exhibited some examples of its use of Algebra and the Theory of Numbers.
And then, some applications of the method of mathematical induction to the problems
of Geometry are explored to obtain a geometric measure in terms of another(s), either
for the demonstration of a proposition that insinuates itself true, or for the stages of
construction of a figure given / Este trabalho trata do Método de Indução Matemática, em especial, de seu uso com vistas
à solução de problemas geométricos. Inicialmente, são feitas algumas considerações acerca
da expressão "raciocínio indutivo", cujo sentido, conforme apropriadamente explicado no
texto, difere do de "indução matemática". É provada a proposição que garante o uso do
método com base em seu fundamento, a saber, o axioma de indução matemática (um
dos postulados que caracterizam os números naturais) e exibidos alguns exemplos de
sua utilização em Álgebra e Teoria dos Números. Em seguida, são exploradas algumas
aplicações do método de indução matemática à problemas de Geometria, seja para a
obtenção de uma medida geométrica em termos de outra(s), para a demonstração de uma
proposição que se insinua verdadeira, ou para a exibição das etapas de construção de uma
dada figura.
|
3 |
Middle School Mathematics Teachers' / Problems In Teaching Transformational Geometry And Their Suggestions For The Solution Of These ProblemsIlaslan, Serap 01 March 2013 (has links) (PDF)
The purpose of this study was to reveal and define the problems middle school mathematics teachers experienced in applying transformational geometry and the solutions they proposed to overcome these problems. A total of six elementary mathematics teachers (grades 5-8) in Ankara participated in the study. The data were collected by means of one-to-one interviews with the participants. The findings indicated that the participants&rsquo / problems divided into three parts. These problems were problems arising from teachers, problems arising from students and problems arising from resources. The participants expressed challenges in teaching due to lack of materials, textbooks, and visualization ability of teachers, classroom size, and time. According to the findings, rotation was the most problematic issue. The participants claimed insufficient technological materials were the reason of this problem. Participants did not feel confidence enough to implement transformational geometry especially in rotation since they lacked adequate training and support. The participants claimed that the Ministry&rsquo / s support should be increased, concrete and technological materials should be sufficient in number, and the duration of transformational geometry lesson should be increased.
|
4 |
Non-euclidean geometry and its possible role in the secondary school mathematics syllabusFish, Washiela 01 1900 (has links)
There are numerous problems associated with the teaching of Euclidean geometry at
secondary schools today. Students do not see the necessity of proving results which
have been obtained intuitively. They do not comprehend that the validity of a
deduction is independent of the 'truth' of the initial assumptions. They do not realise
that they cannot reason from diagrams, because these may be misleading or inaccurate.
Most importantly, they do not understand that Euclidean geometry is a particular
interpretation of physical space and that there are alternative, equally valid
interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean
geometry at school level. It is imperative to identify those students who have
the pre-requisite knowledge and skills. A number of interesting teaching strategies,
such as debates, discussions, investigations, and oral and written presentations, can be
used to introduce and develop the content matter. / Mathematics Education / M. Sc. (Mathematics)
|
5 |
Geometry of universal torsors / Geometrie universeller TorsoreDerenthal, Ulrich 13 October 2006 (has links)
No description available.
|
6 |
Non-euclidean geometry and its possible role in the secondary school mathematics syllabusFish, Washiela 01 1900 (has links)
There are numerous problems associated with the teaching of Euclidean geometry at
secondary schools today. Students do not see the necessity of proving results which
have been obtained intuitively. They do not comprehend that the validity of a
deduction is independent of the 'truth' of the initial assumptions. They do not realise
that they cannot reason from diagrams, because these may be misleading or inaccurate.
Most importantly, they do not understand that Euclidean geometry is a particular
interpretation of physical space and that there are alternative, equally valid
interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean
geometry at school level. It is imperative to identify those students who have
the pre-requisite knowledge and skills. A number of interesting teaching strategies,
such as debates, discussions, investigations, and oral and written presentations, can be
used to introduce and develop the content matter. / Mathematics Education / M. Sc. (Mathematics)
|
Page generated in 0.0458 seconds