geometric invariant
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2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Yuchin Sun

AbstractGiven a closed manifold of dimension at least three, with non-trivial homotopy group $$\pi _3(M)$$ π 3 ( M ) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bounded by one, with sum of their energies realizing a geometric invariant width.


2021 ◽  
Vol 15 (2) ◽  
pp. 77
Author(s):  
Agus Probo Sutejo ◽  
Haerul Ahmadi ◽  
Tasih Mulyono

The examination of defects in radiographic films necessitates specialized knowledge, as indicated by an expert radiographer (AR) degree, yet the subjectivity of AR in identifying defects is problematic. To overcome this subjectivity, an automatic welding defect identification is needed. This is executed by using Matlab to create artificial neural networks, which is beneficial for users with the graphical user interface (GUI) feature. One of the breakthroughs in the figure extraction into seven feature vector values is the geometric invariant moment theory. This prevents translation, rotation, and scaling from changing the figure's characteristics. Therefore, a welding defect identification system with a geometric invariant moment was created in the digital radiographic film figure to overcome the reading error by AR. The identification system obtained an accuracy rating of 89.9%.


2021 ◽  
Vol 67 (3) ◽  
pp. 301-330
Author(s):  
Alexander H.W. Schmitt

Author(s):  
Jérémie Brieussel ◽  
Thibault Godin ◽  
Bijan Mohammadi

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bing He ◽  
Jun Liu ◽  
Tengfei Yang ◽  
Bin Xiao ◽  
Yanguo Peng

AbstractInspired by quaternion algebra and the idea of fractional-order transformation, we propose a new set of quaternion fractional-order generalized Laguerre orthogonal moments (QFr-GLMs) based on fractional-order generalized Laguerre polynomials. Firstly, the proposed QFr-GLMs are directly constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and fractional-order transformations, which extract only the global features from color images, our proposed QFr-GLMs can extract both the global and local color features. This paper also derives a new set of invariant color-image descriptors by QFr-GLMs, enabling geometric-invariant pattern recognition in color images. Finally, the performances of our proposed QFr-GLMs and moment invariants were evaluated in simulation experiments of correlated color images. Both theoretical analysis and experimental results demonstrate the value of the proposed QFr-GLMs and their geometric invariants in the representation and recognition of color images.


2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Nicolas Tholozan ◽  
Jérémy Toulisse

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.


2021 ◽  
Vol 53 ◽  
Author(s):  
Mohamd Saleem Lone

In this paper, we investigate the geometric invariant properties of a normal curve on a smooth immersed surface under conformal transformation. We obtain an invariant-sufficient condition for the conformal image of a normal curve. We also find the deviations of normal and tangential components of the normal curve under the same motion.


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