An Alternative Approach to Lie Groups and Geometric Structures

Author(s):  
Ercüment H. Ortaçgil

This book is about the foundations of geometric symmetry, namely, Lie groups and differential geometry. Although this is a classical subject about which hundreds of books have been written, this book takes a new and innovative approach. The main idea is to replace the Maurer–Cartan form with absolute parallelism and its curvature. Unlike the classical approach, where the model is fixed beforehand by the Maurer–Cartan form, this new approach is model-free, and also revisits the foundational concepts of differential geometry, such as covariant differentiation, from a different perspective.

2013 ◽  
Vol 4 (1) ◽  
pp. 1-17 ◽  
Author(s):  
R. Amala ◽  
R. Vishnu Vardhan

In recent years the ROC curve analysis has got its attention in almost all diversified fields. Basing on the data pattern and its distribution various forms of ROC models have been derived. In this paper, the authors have assumed that the data of two populations (healthy and diseased) follows normal distribution, it is one of the most commonly used forms under parametric approach. The present paper focuses on providing an alternative approach for the tradeoff plot of ROC curve and the computation of AUC using a special function of sigmoid shape called Error function. It is assumed that the test scores of particular biomarker are normally distributed. The entire work has been carried out for providing a new approach for the construction of Binormal ROC curve, which makes use of Error function which can be called as ErROC curve. The summary measure AUC of the resulting ErROC curve has been estimated and defined as ErAUC. The authors have also focused on deriving the expression for obtaining the optimal cut-off point. The new ErROC curve model will provide the true positive rate value at each and every point of false positive rate unlike conventional Binormal ROC model.


2013 ◽  
Vol 2013 (1) ◽  
pp. 1-4
Author(s):  
James K. Dirstein ◽  
Pavol Ihring ◽  
Stano Hroncek

2020 ◽  
Vol 35 (5) ◽  
Author(s):  
Claudio Tennie ◽  
Elisa Bandini ◽  
Carel P. van Schaik ◽  
Lydia M. Hopper

Abstract The zone of latent solutions (ZLS) hypothesis provides an alternative approach to explaining cultural patterns in primates and many other animals. According to the ZLS hypothesis, non-human great ape (henceforth: ape) cultures consist largely or solely of latent solutions. The current competing (and predominant) hypothesis for ape culture argues instead that at least some of their behavioural or artefact forms are copied through specific social learning mechanisms (“copying social learning hypothesis”) and that their forms may depend on copying (copying-dependent forms). In contrast, the ape ZLS hypothesis does not require these forms to be copied. Instead, it suggests that several (non-form-copying) social learning mechanisms help determine the frequency (but typically not the form) of these behaviours and artefacts within connected individuals. The ZLS hypothesis thus suggests that increases and stabilisations of a particular behaviour’s or artefact’s frequency can derive from socially-mediated (cued) form reinnovations. Therefore, and while genes and ecology play important roles as well, according to the ape ZLS hypothesis, apes typically acquire the forms of their behaviours and artefacts individually, but are usually socially induced to do so (provided sufficient opportunity, necessity, motivation and timing). The ZLS approach is often criticized—perhaps also because it challenges the current null hypothesis, which instead assumes a requirement of form-copying social learning mechanisms to explain many ape behavioural (and/or artefact) forms. However, as the ZLS hypothesis is a new approach, with less accumulated literature compared to the current null hypothesis, some confusion is to be expected. Here, we clarify the ZLS approach—also in relation to other competing hypotheses—and address misconceptions and objections. We believe that these clarifications will provide researchers with a coherent theoretical approach and an experimental methodology to examine the necessity of form-copying variants of social learning in apes, humans and other species.


2014 ◽  
Vol 6 ◽  
pp. 869580
Author(s):  
Baozhen Lei ◽  
Harald Löwe ◽  
Xunwei Wang

The present paper provides a first step to a new approach to the theory of gearing, which uses modern differential geometry in order to ensure a strict and coordinate-independent formulation. Here, we are mainly concerned with a basic equation, namely, the equation of meshing, of two rotating surfaces in mesh. Since we are able to solve this equation by the time parameter, we derive parameterizations of the mating pinion from a bevel gear as well as a parameterization for gears produced by special machine tools.


2006 ◽  
Vol 16 (04) ◽  
pp. 887-910 ◽  
Author(s):  
JEAN-MARC GINOUX ◽  
BRUNO ROSSETTO

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables (velocity, acceleration and over-acceleration or jerk). The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra–Gause are proposed.


Fluids ◽  
2018 ◽  
Vol 3 (3) ◽  
pp. 60
Author(s):  
Layachi Hadji

This article deals with the stability problem that arises in the modeling of the geological sequestration of carbon dioxide. It provides a more detailed description of the alternative approach to tackling the stability problem put forth by Vo and Hadji (Physics of Fluids, 2017, 29, 127101) and Wanstall and Hadji (Journal of Engineering Mathematics, 2018, 108, 53–71), and it extends two-dimensional analysis to the three-dimensional case. This new approach, which is based on a step-function base profile, is contrasted with the usual time-evolving base state. While both provide only estimates for the instability threshold values, the step-function base profile approach has one great advantage in the sense that the problem at hand can be viewed as a stationary Rayleigh–Bénard problem, the model of which is physically sound and the stability of which is not only well-defined but can be analyzed by a variety of existing analytical methods using only paper and pencil.


2017 ◽  
Vol 470 (1) ◽  
pp. 576-591 ◽  
Author(s):  
Viktor Radović ◽  
Bojan Novaković ◽  
Valerio Carruba ◽  
Dušan Marčeta

Abstract Asteroid families are a valuable source of information to many asteroid-related researches, assuming a reliable list of their members could be obtained. However, as the number of known asteroids increases fast it becomes more and more difficult to obtain a robust list of members of an asteroid family. Here, we are proposing a new approach to deal with the problem, based on the well-known hierarchical clustering method. An additional step in the whole procedure is introduced in order to reduce a so-called chaining effect. The main idea is to prevent chaining through an already identified interloper. We show that in this way a number of potential interlopers among family members is significantly reduced. Moreover, we developed an automatic online-based portal to apply this procedure, i.e. to generate a list of family members as well as a list of potential interlopers. The Asteroid Families Portal is freely available to all interested researchers.


Sign in / Sign up

Export Citation Format

Share Document