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- World Scientific Publ Co Pte Ltd
- Universidade Estadual Paulista
- Biblioteca Digital da Unicamp
- University of Rzeszów
- MIT - Massachusetts Institute of Technology
- Elsevier
- Springer Berlin Heidelberg; Grupo de Investigaci??n An??lisis Funcional y Aplicaciones; Escuela de Ciencias y Humanidades
- Universidade Cornell
- JUEE Press
- Mais Publicadores...

## Fluctuating commutative geometry

Fonte: World Scientific Publ Co Pte Ltd
Publicador: World Scientific Publ Co Pte Ltd

Tipo: Artigo de Revista Científica
Formato: 2517-2524

ENG

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We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number n of points. The spectral principle of Connes and Chamseddine is used to define dynamics. We show that this simple model has two phases. The expectation value , the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension delta is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, < 2. We also address another discrete model defined on a fixed d = 1 dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.

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## Combining genetic algorithm and simulated annealing: A molecular geometry optimization study

Fonte: Universidade Estadual Paulista
Publicador: Universidade Estadual Paulista

Tipo: Artigo de Revista Científica
Formato: 29-39

ENG

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We introduce a new hybrid approach to determine the ground state geometry of molecular systems. Firstly, we compared the ability of genetic algorithm (GA) and simulated annealing (SA) to find the lowest energy geometry of silicon clusters with six and 10 atoms. This comparison showed that GA exhibits fast initial convergence, but its performance deteriorates as it approaches the desired global extreme. Interestingly, SA showed a complementary convergence pattern, in addition to high accuracy. Our new procedure combines selected features from GA and SA to achieve weak dependence on initial parameters, parallel search strategy, fast convergence and high accuracy. This hybrid algorithm outperforms GA and SA by one order of magnitude for small silicon clusters (Si6 and Si10). Next, we applied the hybrid method to study the geometry of a 20-atom silicon cluster. It was able to find an original geometry, apparently lower in energy than those previously described in literature. In principle, our procedure can be applied successfully to any molecular system. © 1998 Elsevier Science B.V.

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## Um estudo das cônicas na perspectiva da geometria projetiva; A study of the conic in the perspective of projective geometry

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 02/02/2012
PT

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Esta dissertação tem por objetivo apresentar o estudo das Cônicas e suas propriedades, mediante a perspectiva da Geometria Projetiva, bem como propor o software livre GeoGebra como uma alternativa para visualização dos Teoremas de Geometria Projetiva e das propriedades das Cônicas. O trabalho inicia-se com uma introdução histórica do desenvolvimento da projeção, na arte, e da Geometria Projetiva. Em seguida é apresentada a base teórica para o estudo das Cônicas e suas propriedades, o que é tratado em seguida. Por fim, são apresentadas algumas construções, que podem ser executadas no software livre, de geometria dinâmica, GeoGebra.; This dissertation has for objective to present the study of the Conical , and its properties, through the perspective of Projective Geometry, moreover to present free software GeoGebra as an alternative for visualization of the Theorems of Projective Geometry and the properties of the Conical. The work starts with a historical introduction about the development of the projection, in the art, and of Projective Geometry. Next is presented the theoretical basis for the study of conic sections and their properties, which is treated soon after. To finish, some constructions are presented, that can be executed in software of dynamic geometry GeoGebra.

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## Geometry learning: The role of tasks, working models, and dynamic geometry software

Fonte: University of Rzeszów
Publicador: University of Rzeszów

Tipo: Parte de Livro

Publicado em //2008
ENG

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We present several learning experiences that illustrate how three aspects of the geometric competence, constructing and analyzing properties of figures, identifying patterns and investigating and geometric problem solving, were developed by pupils that participated in the implementation of an innovative geometry teaching unit in grade 8. The topics addressed were dealing with properties of two dimensional figures, Pythagoras theorem, loci, translations and similarity of triangles. The development of the geometric competence was clearly supported by the dynamic geometry environment but unfolded in different ways, depending on the way how pupils reacted to the different types of tasks.; Apresentamos várias experiências de aprendizagem que ilustram como três aspectos da competência geométrica, construir e analisar propriedades de figuras, identificar regularidades e investigar e resolver foram desenvolvidos por alunos que participaram de uma experiência de ensino inovadora no campo da Geometria no 8.º ano de escolaridade. Os temas tratados incluem o trabalho com figuras bidimensionais, teorema de Pitágoras, lugares geométricos, translações e semelhança de triângulos. O desenvolvimento da competência geométrica foi claramente apoiado pelo ambiente de geometria dinâmica mas processou-se de formas diferentes...

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## 13.472J Computational Geometry, Spring 2003; Computational Geometry

Fonte: MIT - Massachusetts Institute of Technology
Publicador: MIT - Massachusetts Institute of Technology

EN-US

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#surface modeling#b-splines#deformable surfaces#generalized cylinders#offsets#filleting surfaces#Non-linear solvers and intersection problems#Solid modeling#boundary representation#non-manifold and mixed-dimension boundary representation models#octrees

Topics in surface modeling: b-splines, non-uniform rational b-splines, physically based deformable surfaces, sweeps and generalized cylinders, offsets, blending and filleting surfaces. Non-linear solvers and intersection problems. Solid modeling: constructive solid geometry, boundary representation, non-manifold and mixed-dimension boundary representation models, octrees. Robustness of geometric computations. Interval methods. Finite and boundary element discretization methods for continuum mechanics problems. Scientific visualization. Variational geometry. Tolerances. Inspection methods. Feature representation and recognition. Shape interrogation for design, analysis, and manufacturing. Involves analytical and programming assignments.

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## Automatic Deduction in (Dynamic) Geometry: Loci Computation

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/article; acceptedVersion

ENG

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A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail.

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## An algebraic taxonomy for locus computation in dynamic geometry

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/article; submittedVersion

ENG

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The automatic determination of geometric loci is an important issue in Dynamic Geometry. In Dynamic Geometry systems, it is often the case that locus determination is purely graphical, producing an output that is not robust enough and not reusable by the given software. Parts of the true locus may be missing, and extraneous objects can be appended to it as side products of the locus determination process. In this paper, we propose a new method for the computation, in dynamic geometry, of a locus defined by algebraic conditions. It provides an analytic, exact description of the sought locus, making possible a subsequent precise manipulation of this object by the system. Moreover, a complete taxonomy, cataloging the potentially different kinds of geometric objects arising from the locus computation procedure, is introduced, allowing to easily discriminate these objects as either extraneous or as pertaining to the sought locus. Our technique takes profit of the recently developed GröbnerCover algorithm. The taxonomy introduced can be generalized to higher dimensions, but we focus on 2-dimensional loci for classical reasons. The proposed method is illustrated through a web-based application prototype, showing that it has reached enough maturity as to be considered a practical option to be included in the next generation of dynamic geometry environments.

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## Álgebra geométrica conforme e geometria de distâncias; Conformal geometric algebra and distance geometry

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 05/08/2015
PT

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#Geometria de distâncias#Álgebra geométrica#Geometria conforme#Distance geometry#Geometric algebra#Conformal geometry#Proteins - Conformation

Neste trabalho apresentaremos um método alternativo à resolução de Problemas de Geometria de Distâncias, em que suas instâncias podem ser discretizadas. Essencialmente, estes problemas consistem em determinar um conjunto de pontos num dado espaço geométrico, no qual algumas distâncias entre dois desses pontos são conhecidas. A abordagem considerada difere da tradicional por tratar os objetos geométricos em um espaço projetivo de uma imersão conformal, do espaço euclidiano em um espaço de Minkowski (Álgebra Geométrica Conforme). Seu modelo de geometria proporciona uma interpretação natural dos objetos relevantes ao problema. Além do estudo teórico apresentado, propomos uma aplicação do método que resolve a principal classe de problemas com esta característica, conhecida na literatura por Discretizable Molecular Distance Geometry Problem - DMDGP, referente ao cálculo de estrutura tridimensional de proteínas (conformação de proteínas.; In this work we present an alternative method to solve distances geometry problems in that their instances can be discretized. Essentially, these problems consists in determining a set of points in a given geometric space in which certain distances between two of these points are known. The considered approach differs from the traditional because it deals with geometric objects in a projective space of a conformal immersion of Euclidean space into a Minkowski space (Conformal Geometric Algebra). This geometric model provides a natural interpretation of the relevant objects to the problem. In addition...

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## Geometria da informação : métrica de Fisher; Information geometry : Fisher's metric

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 23/08/2013
PT

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#Geometria da informação#Matriz de informação de Fisher#Distância de Rao#Geometria diferencial#Estatística matemática#Information geometry#Fisher information metric#Rao distance#Differential geometry#Mathematical statistics

A Geometria da Informação é uma área da matemática que utiliza ferramentas geométricas no estudo de modelos estatísticos. Em 1945, Rao introduziu uma métrica Riemanniana no espaço das distribuições de probabilidade usando a matriz de informação, dada por Ronald Fisher em 1921. Com a métrica associada a essa matriz, define-se uma distância entre duas distribuições de probabilidade (distância de Rao), geodésicas, curvaturas e outras propriedades do espaço. Desde então muitos autores veem estudando esse assunto, que está naturalmente ligado a diversas aplicações como, por exemplo, inferência estatística, processos estocásticos, teoria da informação e distorção de imagens. Neste trabalho damos uma breve introdução à geometria diferencial e Riemanniana e fazemos uma coletânea de alguns resultados obtidos na área de Geometria da Informação. Mostramos a distância de Rao entre algumas distribuições de probabilidade e damos uma atenção especial ao estudo da distância no espaço formado por distribuições Normais Multivariadas. Neste espaço, como ainda não é conhecida uma fórmula fechada para a distância e nem para a curva geodésica, damos ênfase ao cálculo de limitantes para a distância de Rao. Conseguimos melhorar...

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## A Riemannian geometry in the q-Exponential Banach manifold induced by q-Divergences

Fonte: Springer Berlin Heidelberg; Grupo de Investigaci??n An??lisis Funcional y Aplicaciones; Escuela de Ciencias y Humanidades
Publicador: Springer Berlin Heidelberg; Grupo de Investigaci??n An??lisis Funcional y Aplicaciones; Escuela de Ciencias y Humanidades

Tipo: article; Art??culo; publishedVersion

ENG

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#Espacios de Orlicz#ALGORITMOS#ESPACIOS DE BANACH#ESPACIOS VECTORIALES#TEOREMA DE BANACH#GEOMETR??A DE RIEMANN#GEOMETR??A DIFERENCIAL#ESPACIOS M??TRICOS#MATEM??TICAS#INTELIGENCIA ARTIFICIAL#PROCESAMIENTO DE IM??GENES

For the family of non-parametric q-exponential statistical models, in a former paper, written by the same authors, a differentiable Banach manifold modelled on Lebesgue spaces of real random variables has been built -- In this paper, the geometry induced on this manifold is characterized by q-divergence functionals -- This geometry turns out to be a generalization of the geometry given by Fisher information metric and Levi-Civita connections -- Moreover, the classical Amari??s ??-connections appears as special case of the q???connections (q) -- The main result is the expected one, namely the zero curvature of the manifold

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## Future Directions of Research in Geometry: A Summary of the Panel Discussion at the 2007 Midwest Geometry Conference

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2007

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#Mathematics - Differential Geometry#General Relativity and Quantum Cosmology#High Energy Physics - Theory#Mathematical Physics

The 2007 Midwest Geometry Conference included a panel discussion devoted to
open problems and the general direction of future research in fields related to
the main themes of the conference. This paper summarizes the comments made
during the panel discussion.; Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA/

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## Harmonic Maps and Hypersymplectic Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We study the hypersymplectic geometry of the moduli space of solutions to
Hitchin's harmonic map equations on a $G$-bundle. This is the split-signature
analogue of Hitchin's Higgs bundle moduli space. Due to the lack of
definiteness, this moduli space is globally not well-behaved. However, we are
able to construct a smooth open set consisting of solutions with small Higgs
field, on which we can investigate the hypersymplectic geometry. Finally, we
reinterpret our results in terms of the Riemannian geometry of the moduli space
of $G$-connections.; Comment: revised version, 22 pages, to appear in J. Geom. Phys

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## On the development of differential geometry in Estonia

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/09/2005

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#Mathematics - History and Overview#Mathematics - Differential Geometry#53-03#01A55#01A60#53A07#53A10#53A35#53C07#53D05#81R60

We give a brief review of a research made in the field of differential
geometry in Estonia in the period from the beginning of the 19th century to the
present time. The biographic data of mathematicians who made a valuable
contribution to the development of differential geometry in Estonia in
mentioned period are presented. The material presented in the introduction
covers the period from the beginning of the 19th century to the middle of the
20th century and it can be considered as a brief historical sketch of the
development of differential geometry in Estonia in this period. The next
sections give an idea of the modern trends of development of differential
geometry in Estonia.; Comment: 22 pages, based on the talk presented at the Finnish-Estonian
Colloquium held in Tallinn (Estonia), May 27-29, 2002

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## Affine and projective universal geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/12/2006

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By recasting metrical geometry in a purely algebraic setting, both Euclidean
and non-Euclidean geometries can be studied over a general field with an
arbitrary quadratic form. Both an affine and a projective version of this new
theory are introduced here, and the main formulas extend those of rational
trigonometry in the plane. This gives a unified, computational model of both
spherical and hyperbolic geometries, allows the extension of many results of
Euclidean geometry to the relativistic setting, and provides a new metrical
approach to algebraic geometry.; Comment: 22 pages

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## Contact Geometry of Curves

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/10/2009

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Cartan's method of moving frames is briefly recalled in the context of
immersed curves in the homogeneous space of a Lie group $G$. The contact
geometry of curves in low dimensional equi-affine geometry is then made
explicit. This delivers the complete set of invariant data which solves the
$G$-equivalence problem via a straightforward procedure, and which is, in some
sense a supplement to the equivariant method of Fels and Olver. Next, the
contact geometry of curves in general Riemannian manifolds $(M,g)$ is
described. For the special case in which the isometries of $(M,g)$ act
transitively, it is shown that the contact geometry provides an explicit
algorithmic construction of the differential invariants for curves in $M$. The
inputs required for the construction consist only of the metric $g$ and a
parametrisation of structure group SO(n); the group action is not required and
no integration is involved. To illustrate the algorithm we explicitly construct
complete sets of differential invariants for curves in the Poincare half-space
$H^3$ and in a family of constant curvature 3-metrics. It is conjectured that
similar results are possible in other Cartan geometries.

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## Courant Algebroids in Parabolic Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Let $p$ be a Lie subalgebra of a semisimple Lie algebra $g$ and $(G,P)$ be
the corresponding pair of connected Lie groups. A Cartan geometry of type
$(G,P)$ associates to a smooth manifold $M$ a principal $P$-bundle and a Cartan
connection, and a parabolic geometry is a Cartan geometry where $P$ is
parabolic. We show that if $P$ is parabolic, the adjoint tractor bundle of a
Cartan geometry, which is isomorphic to the Atiyah algebroid of the principal
$P$-bundle, admits the structure of a (pre-)Courant algebroid, and we identify
the topological obstruction to the bracket being a Courant bracket. For
semisimple $G$, the Atiyah algebroid of the principal $P$-bundle associated to
the Cartan geometry of $(G,P)$ admits a pre-Courant algebroid structure if and
only if $P$ is parabolic.; Comment: 24 pages. v3: Added Section 4.3. v4: Modified abstract, added
concluding remarks

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## Generalised Complex Geometry and the Planck Cone

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/11/2004

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Complex geometry and symplectic geometry are mirrors in string theory. The
recently developed generalised complex geometry interpolates between the two of
them. On the other hand, the classical and quantum mechanics of a finite number
of degrees of freedom are respectively described by a symplectic structure and
a complex structure on classical phase space. In this letter we analyse the
role played by generalised complex geometry in the classical and quantum
mechanics of a finite number of degrees of freedom. We identify generalised
complex geometry as an appropriate geometrical setup for dualities. The latter
are interpreted as transformations connecting points in the interior of the
Planck cone with points in the exterior, and viceversa. The Planck cone bears
some resemblance with the relativistic light-cone. However the latter cannot be
traversed by physical particles, while dualities do connect the region outside
the Planck cone with the region inside, and viceversa.; Comment: 9 pages

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## Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in Kyoto

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/06/2014

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#Computer Science - Computational Geometry#Computer Science - Computational Complexity#Mathematics - Combinatorics#Mathematics - Geometric Topology#57M50, 57N35, 57Q35, 57N10, 57Q15, 68Q17, 68U05, 52B05,#F.1.3#F.4.1#G.2.1#G.2.2#I.3.5

This workshop about triangulations of manifolds in computational geometry and
topology was held at the 2014 CG-Week in Kyoto, Japan.
It focussed on computational and combinatorial questions regarding
triangulations, with the goal of bringing together researchers working on
various aspects of triangulations and of fostering a closer collaboration
within the computational geometry and topology community.
Triangulations are highly suitable for computations due to their clear
combinatorial structure. As a consequence, they have been successfully employed
in discrete algorithms to solve purely theoretical problems in a broad variety
of mathematical research areas (knot theory, polytope theory, 2- and 3-manifold
topology, geometry, and others). However, due to the large variety of
applications, requirements vary from field to field and thus different types of
triangulations, different tools, and different frameworks are used in different
areas of research. This is why today closely related research areas are
sometimes largely disjoint leaving potential reciprocal benefits unused.
To address these potentials a workshop on Triangulations was held at
Oberwolfach Research Institute in 2012. Since then many new collaborations
between researchers of different mathematical communities have been
established. Regarding the computational geometry community...

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## Lectures on Calabi-Yau and special Lagrangian geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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This paper gives a leisurely introduction to Calabi-Yau manifolds and special
Lagrangian submanifolds from the differential geometric point of view, followed
by a survey of recent results on singularities of special Lagrangian
submanifolds, and their application to the SYZ Conjecture.
It is aimed at graduate students in Geometry, String Theorists, and others
wishing to learn the subject, and is designed to be fairly self-contained. It
is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in
June and July 2001.
We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the
Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry.
`Almost Calabi-Yau m-folds' (a generalization of Calabi-Yau m-folds useful in
special Lagrangian geometry) are explained and the deformation theory and
moduli spaces of compact special Lagrangian submanifolds in (almost) Calabi-Yau
m-folds is described.
In the final part we consider isolated singularities of special Lagrangian
m-folds, focussing mainly on singularities locally modelled on cones, and the
expected behaviour of singularities of compact special Lagrangian m-folds in
generic (almost) Calabi-Yau m-folds. String Theory, Mirror Symmetry and the SYZ
Conjecture are briefly discussed...

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## Verification of the influence of Urban Geometry on the Nocturnal Heat Island Intensity

Fonte: JUEE Press
Publicador: JUEE Press

Tipo: info:eu-repo/semantics/article; info:eu-repo/semantics/publishedVersion;
Formato: application/pdf

Publicado em 08/12/2013
ENG

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#Urban Engineering#Civil Engineering#Architecture and Urbanism#Architecture and Urbanism.#numerical model#urban geometry#heat island#urban canyon.

Nocturnal heat island formation is a prominent phenomenon in urban planning research and thermal comfort. It is characterized by an increased air temperature from the periphery to the center of the cities and is partly caused by the urban geometry. The phenomenon is a result of the influence of the characteristics of urbanization, which alter the energy balance of the cities. Among surveys in this context, the model proposed by Oke (1981) shows its relevance in the area. Using simulations with reduced models, Oke found that the fraction of visible sky negatively correlated with the heat buildup on the surfaces and increased the air temperature. The relationship between the height of the buildings and the width of the path (H/W ratio, height/width) was used to measure the urban geometry in this previous study. Based on that study, this research verified the role of geometry in the formation of an urban heat island in a Brazilian city, aiming to adapt this model to real urban conditions. The methodological procedures rely on the following steps: a study of a theoretical-numerical base model (Oke model, 1981) and adjustment (validation). Thus, the methodology to be employed included urban data collection, which included urban geometry and air temperature...

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