scholarly journals Geometric Integrability of Some Generalizations of the Camassa-Holm Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Ognyan Christov
Keyword(s):  
Point Of View ◽  
Holm Equation ◽  
Pseudospherical Surfaces ◽  
Geometric Point ◽  
Geometric Integrability

We study the Camassa-Holm (CH) equation and recently introducedμCH equation from the geometric point of view. We show that Kupershmidt deformations of these equations describe pseudospherical surfaces and hence are geometrically integrable.

scholarly journals Laws of the iterated logarithm on covering graphs with groups of polynomial volume growth

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ryuya Namba
Keyword(s):  
Random Walks ◽  
Quadratic Forms ◽  
Volume Growth ◽  
Moderate Deviation ◽  
Point Of View ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Geometric Point ◽  
Rate Functions ◽  
The Given

AbstractModerate deviation principles (MDPs) for random walks on covering graphs with groups of polynomial volume growth are discussed in a geometric point of view. They deal with any intermediate spatial scalings between those of laws of large numbers and those of central limit theorems. The corresponding rate functions are given by quadratic forms determined by the Albanese metric associated with the given random walks. We apply MDPs to establish laws of the iterated logarithm on the covering graphs by characterizing the set of all limit points of the normalized random walks.

Dynamics of Multibody Systems: Generalized Mass Metric in Riemannian Velocity Space and Recursive Momentum Formulation

2007 ◽  
Author(s):  
Qiang Zhao ◽  
Hong Tao Wu
Keyword(s):  
Point Of View ◽  
Velocity Space ◽  
Motion Equations ◽  
Closed Loop Systems ◽  
Dynamics Of Multibody Systems ◽  
Operator Factorization ◽  
Geometric Point ◽  
Mass Tensor ◽  
Generalized Momentum ◽  
Serial Chains

This paper describes two aspects of multibody system (MBS) dynamics on a generalized mass metric in Riemannian velocity space and recursive momentum formulation. Firstly, we present a detailed expression of the Riemannian metric and operator factorization of a generalized mass tensor for the dynamics of general-topology rigid MBS. The derived expression allows a clearly understanding the components of the generalized mass tensor, which also constitute a metric of the Riemannian velocity space. It is being the fact that there does exist a common metric in Lagrange and recursive Newton-Euler dynamic equation, we can determine, from the Riemannian geometric point of view, that there is the equivalent relationship between the two approaches to a given MBS. Next, from the generalized momentum definition in the derivation of the Riemannian velocity metrics, recursive momentum equations of MBS dynamics are developed for progressively more complex systems: serial chains, topological trees, and closed-loop systems. Through the principle of impulse and momentum, a new method is proposed for reorienting and locating the MBS form a given initial orientation and location to desired final ones without needing to solve the motion equations.

NONLINEARITY AND LEAST SQUARES

CISM journal ◽  
1988 ◽  
Vol 42 (4) ◽  
pp. 321-330 ◽  
Author(s):  
P.J.G. Teunissen ◽  
E.H. Knickmeyer
Keyword(s):  
Least Squares ◽  
Covariance Structure ◽  
Point Of View ◽  
Helmert Transformation ◽  
Know How ◽  
Least Squares Estimators ◽  
Functional Relations ◽  
Geometric Point ◽  
Rotational Invariant ◽  
Almost All

Since almost all functional relations in our geodetic models are nonlinear, it is important, especially from a statistical inference point of view, to know how nonlinearity manifests itself at the various stages of an adjustment. In this paper particular attention is given to the effect of nonlinearity on the first two moments of least squares estimators. Expressions for the moments of least squares estimators of parameters, residuals and functions derived from parameters, are given. The measures of nonlinearity are discussed both from a statistical and differential geometric point of view. Finally, our results are applied to the 2D symmetric Helmert transformation with a rotational invariant covariance structure.

scholarly journals A teaching proposal for the study of Eigenvectors and Eigenvalues

2017 ◽  
Vol 7 (1) ◽  
pp. 100 ◽  
Author(s):  
María José Beltrán Meneu ◽  
Marina Murillo Arcila ◽  
Enrique Jordá Mora
Keyword(s):  
Rigid Body ◽  
Mathematical Thinking ◽  
Point Of View ◽  
Moment Of Inertia ◽  
Geometric Point ◽  
The Moment ◽  
Eigenvectors And Eigenvalues

In this work, we present a teaching proposal which emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. These concepts are introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software. The proposal was motivated after observing students’ difficulties when treating eigenvectors and eigenvalues from a geometric point of view. It was designed following a particular sequence of activities with the schema: exploration, introduction of concepts, structuring of knowledge and application, and considering the three worlds of mathematical thinking provided by Tall: embodied, symbolic and formal.

scholarly journals Improvements of Rackwitz–Fiessler Method for Correlated Structural Reliability Analysis

2019 ◽  
Vol 17 (06) ◽  
pp. 1950077 ◽  
Author(s):  
Sheng-Tong Zhou ◽  
Qian Xiao ◽  
Jian-Min Zhou ◽  
Hong-Guang Li
Keyword(s):  
Structural Reliability ◽  
Pearson Correlation ◽  
Computational Cost ◽  
Transformation Process ◽  
Point Of View ◽  
Gaussian Copula ◽  
Normal Space ◽  
Copula Theory ◽  
Geometric Point ◽  
Nataf Transformation

Rackwitz–Fiessler (RF) method is well accepted as an efficient way to solve the uncorrelated non-Normal reliability problems by transforming original non-Normal variables into equivalent Normal variables based on the equivalent Normal conditions. However, this traditional RF method is often abandoned when correlated reliability problems are involved, because the point-by-point implementation property of equivalent Normal conditions makes the RF method hard to clearly describe the correlations of transformed variables. To this end, some improvements on the traditional RF method are presented from the isoprobabilistic transformation and copula theory viewpoints. First of all, the forward transformation process of RF method from the original space to the standard Normal space is interpreted as the isoprobabilistic transformation from the geometric point of view. This viewpoint makes us reasonably describe the stochastic dependence of transformed variables same as that in Nataf transformation (NATAF). Thus, a corresponding enhanced RF (EnRF) method is proposed to deal with the correlated reliability problems described by Pearson linear correlation. Further, we uncover the implicit Gaussian copula hypothesis of RF method according to the invariant theorem of copula and the strictly increasing isoprobabilistic transformation. Meanwhile, based on the copula-only rank correlations such as the Spearman and Kendall correlations, two improved RF (IRF) methods are introduced to overcome the potential pitfalls of Pearson correlation in EnRF. Later, taking NATAF as a reference, the computational cost and efficiency of above three proposed RF methods are also discussed in Hasofer–Lind reliability algorithm. Finally, four illustrative structure reliability examples are demonstrated to validate the availability and advantages of the new proposed RF methods.

Fourier-Mukai Transforms in Algebraic Geometry

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Algebraic Geometry ◽  
Smooth Projective Variety ◽  
Derived Category ◽  
Point Of View ◽  
Basic Knowledge ◽  
Research Directions ◽  
Coherent Sheaves ◽  
Geometric Point ◽  
Singular Cohomology

This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.

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