scholarly journals Approximate Triangulations of Grassmann Manifolds

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 172
Author(s):  
Kevin P. Knudson
Keyword(s):  
Euclidean Space ◽  
Persistent Homology ◽  
Simplicial Complexes ◽  
Grassmann Manifolds ◽  
The Family

We define the notion of an approximate triangulation for a manifold M embedded in Euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in M and use persistent homology to find a complex in the family whose homology agrees with that of M. Our key examples are various Grassmann manifolds G k ( R n ) .

Holomorphic Mappings of the Hyperbolic Space into the Complex Euclidean Space and the Bloch Theorem

1975 ◽  
Vol 27 (2) ◽  
pp. 446-458 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Hyperbolic Space ◽  
Holomorphic Mapping ◽  
Continuous Functions ◽  
Holomorphic Mappings ◽  
Euclidean Metric ◽  
The Family ◽  
Complex Euclidean Space ◽  
Bounded Holomorphic

This paper is to study various properties of holomorphic mappings defined on the unit ball B in the complex euclidean space Cn with ranges in the space Cm. Furnishing B with the standard invariant Kähler metric and Cm with the ordinary euclidean metric, we define, for each holomorphic mapping f : B → Cm, a pair of non-negative continuous functions qf and Qf on B ; see § 2 for the definition.Let (Ω), Ω > 0, be the family of holomorphic mappings f : B → Cn such that Qf(z) ≦ Ω for all z ∈ B. (Ω) contains the family (M) of bounded holomorphic mappings as a proper subfamily for a suitable M > 0.

scholarly journals Estimating Multidimensional Persistent Homology Through a Finite Sampling

2015 ◽  
Vol 25 (03) ◽  
pp. 187-205 ◽  
Author(s):  
Niccolò Cavazza ◽  
Massimo Ferri ◽  
Claudia Landi
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Persistent Homology ◽  
Betti Numbers ◽  
Finite Sample ◽  
Combinatorial Description ◽  
Persistent Betti Numbers ◽  
Finite Sampling ◽  
The One ◽  
Union Of Balls

An exact computation of the persistent Betti numbers of a submanifold [Formula: see text] of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of [Formula: see text] is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of [Formula: see text] from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.

scholarly journals On Constant Metric Dimension of Some Generalized Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xuewu Zuo ◽  
Abid Ali ◽  
Gohar Ali ◽  
Muhammad Kamran Siddiqui ◽  
Muhammad Tariq Rahim ◽  
...  
Keyword(s):  
Image Processing ◽  
Euclidean Space ◽  
Global Positioning System ◽  
Metric Space ◽  
Network Theory ◽  
Convex Polytopes ◽  
Metric Dimension ◽  
Positioning System ◽  
The Family ◽  
Global Positioning

Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) to the arbitrary metric space. A family ℱ = G n of connected graphs with n ≥ 3 is a family of constant metric dimension if dim G = k (some constant) for all graphs in the family. Family ℱ has bounded metric dimension if dim G n ≤ M , for all graphs in ℱ . Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.

ASYMPTOTICS FOR THE SPECTRAL AND WALK DIMENSION AS FRACTALS APPROACH EUCLIDEAN SPACE

Fractals ◽  
2002 ◽  
Vol 10 (04) ◽  
pp. 403-412 ◽  
Author(s):  
B. M. HAMBLY ◽  
T. KUMAGAI
Keyword(s):  
Euclidean Space ◽  
Sierpinski Gasket ◽  
Two Dimensions ◽  
Spectral Dimension ◽  
Sierpiński Gasket ◽  
The Family ◽  
Physics Literature ◽  
Dynamic Dimension ◽  
Sierpinski Carpets ◽  
Made In

We discuss the behavior of the dynamic dimension exponents for families of fractals based on the Sierpinski gasket and carpet. As the length scale factor for the family tends to infinity, the lattice approximations to the fractals look more like the tetrahedral or cubic lattice in Euclidean space and the fractal dimension converges to that of the embedding space. However, in the Sierpinski gasket case, the spectral dimension converges to two for all dimensions. In two dimensions, we prove a conjecture made in the physics literature concerning the rate of convergence. On the other hand, for natural families of Sierpinski carpets, the spectral dimension converges to the dimension of the embedding Euclidean space. In general, we demonstrate that for both cases of finitely and infinitely ramified fractals, a variety of asymptotic values for the spectral dimension can be achieved.

scholarly journals Determination of features based on the formation of persistent spectra of eigenvalues of Laplace matrices

2020 ◽  
pp. 49-64
Author(s):  
S. V. Lejhter ◽  
S. N. Chukanov
Keyword(s):  
Persistent Homology ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Wasserstein Distance ◽  
Computer Hardware ◽  
Euler Characteristics ◽  
Topological Features ◽  
Laplace Matrix ◽  
Definition Of

An algorithm for determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes has been developed in the paper. The spectrum of eigenvalues of the Laplace matrix is used as features in the data structure for image analysis. Similarly to the method of persistent homology, the filtering of embedded simplicial complexes is formed, approximating the image of the object, but the topological features at each stage of filtration is the spectrum of eigenvalues of the Laplace matrix of simplicial complexes. The spectrum of eigenvalues of the Laplace matrix allows to determine the Betti numbers and Euler characteristics of the image. Based on the method of finding the spectrum of eigenvalues of the Laplace matrix, an algorithm is formed that allows you to obtain topological features of images of objects and quantitative estimates of the results of image comparison. Software has been developed that implements this algorithm on computer hardware. The method of determining the spectrum of eigenvalues of the Laplace matrix has the following advantages: the method does not require a bijective correspondence between the elements of the structures of objects; the method is invariant with respect to the Euclidean transformations of the forms of objects. Determining the spectrum of eigenvalues of the Laplace matrix for simplicial complexes allows you to expand the number of features for machine learning, which allows you to increase the diversity of information obtained by the methods of computational topology, while maintaining topological invariants. When comparing the shapes of objects, a modified Wasserstein distance can be constructed based on the eigenvalues of the Laplace matrix of the compared shapes. Using the definition of the spectrum of eigenvalues of the Laplace matrix to compare the shapes of objects can improve the accuracy of determining the distance between images.

scholarly journals Family of Enneper Minimal Surfaces

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 281
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Minimal Surface ◽  
Algebraic Equation ◽  
Minimal Surfaces ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
The Family ◽  
Free Representation ◽  
Positive Integers

We consider a family of higher degree Enneper minimal surface E m for positive integers m in the three-dimensional Euclidean space E 3 . We compute algebraic equation, degree and integral free representation of Enneper minimal surface for m = 1 , 2 , 3 . Finally, we give some results and relations for the family E m .

NONLINEAR STATISTICS OF HUMAN SPEECH DATA

2009 ◽  
Vol 19 (07) ◽  
pp. 2307-2319 ◽  
Author(s):  
KENNETH A. BROWN ◽  
KEVIN P. KNUDSON
Keyword(s):  
Time Delay ◽  
Betti Number ◽  
Persistent Homology ◽  
Point Clouds ◽  
Simplicial Complexes ◽  
Data Sets ◽  
Human Speech ◽  
Speech Data ◽  
Nonlinear Statistics

We study the structure of point clouds obtained as time delay embeddings of human speech signals by approximating the data sets with certain simplicial complexes and analyzing their persistent homology. Results for several different sounds are presented in embedding dimensions 3 and 4. The first Betti number allows a coarse classification of sounds into three groups: vowels, nasals and noise.

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