scholarly journals GENERIC UNLABELED GLOBAL RIGIDITY

2019 ◽  
Vol 7 ◽  
Author(s):  
STEVEN J. GORTLER ◽  
LOUIS THERAN ◽  
DYLAN P. THURSTON
Keyword(s):  
Euclidean Distance ◽  
Real Numbers ◽  
Global Rigidity ◽  
Point Pair ◽  
Generic Uniqueness ◽  
Combinatorial Data

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^{d}$ for some $n$ and some $d\geqslant 2$ . Each pair of points has a Euclidean distance in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair distances corresponding to the edges of $G$ . In this paper, we study the question of when a generic $\mathbf{p}$ in $d$ dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair distances together with knowledge of $d$ and $n$ . In this setting the distances are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point pair gave rise to which distance, nor is data about $G$ given. We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair distances (together with $d$ and $n$ ) if and only if it is determined by the labeled distances.

Sets and Numbers

2021 ◽  
pp. 3-27
Author(s):  
James Davidson
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Euclidean Distance ◽  
Final Section ◽  
Real Analysis ◽  
Real Numbers ◽  
The Real ◽  
Basic Ideas ◽  
Distance Sets

This chapter covers set theory. The topics include set algebra, relations, orderings and mappings, countability and sequences, real numbers, sequences and limits, and set classes including monotone classes, rings, fields, and sigma fields. The final section introduces the basic ideas of real analysis including Euclidean distance, sets of the real line, coverings, and compactness.

A Non-Euclidean Distance

1971 ◽  
Vol 64 (7) ◽  
pp. 595-600
Author(s):  
Stanley R. Clemens
Keyword(s):  
Distance Function ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Plane Geometry ◽  
Synthetic Approach ◽  
Real Numbers ◽  
Angle Measure ◽  
The Real ◽  
Measure Function

There are basically two approaches to classical Euclidean plane geometry—the synthetic approach and the metric approach. The older of the two is the synthetic approach followed by Eucliding later by Hilbert. In the Eucliding treatment, one begin by assuming as undefined the relations of betweenness, congruence of segments, and congruence of angles. The metric treatments, initiated by G. D. Birkhoff in the 1930s, assumes the existence of the real numbers (or a set of postulates that guarantees the existence of the real numbers) and the existence of a distance function d and an angle-measure function m.

Pythagorean fuzzy TOPSIS method based on 2-tuple probability weight

2021 ◽  
pp. 1-14
Author(s):  
Tingting Xu ◽  
Hui Zhang ◽  
Boquan Li
Keyword(s):  
Euclidean Distance ◽  
Fuzzy Topsis ◽  
Ideal Solution ◽  
Topsis Method ◽  
Real Numbers ◽  
Fuzzy Environment ◽  
Order Preference ◽  
Real Value ◽  
Probability Weight ◽  
Definition Of

In this paper, the concept of 2-tuple probability weight is presented, and on this basis, the technique for order preference by similarity to ideal solution (TOPSIS) method in Pythagorean fuzzy environment is given. First, the definition of 2-tuple probability weight is put forward, and two examples are provided to illustrate that 2-tuple probability weight can effectively prevent the loss of information. Second, the notion of real-value 2-tuple is defined for any two real numbers, and some basic operations, operation properties, and sorting functions are introduced. Finally, a 2-tuple probability weight Euclidean distance is provided, a new Pythagorean fuzzy TOPSIS method is further proposed, and the flexibility and effectiveness of the proposed methods are illustrated by an example and two comparative analyses.

The Earth Mover’s Distance as a Metric for the Space of Inorganic Compositions

2020 ◽  
Author(s):  
Cameron Hargreaves ◽  
Matthew Dyer ◽  
Michael Gaultois ◽  
Vitaliy Kurlin ◽  
Matthew J Rosseinsky
Keyword(s):  
Euclidean Distance ◽  
Nearest Neighbor ◽  
Nearest Neighbor Search ◽  
Inorganic Crystal Structure Database ◽  
Earth Mover’S Distance ◽  
Chemical Similarity ◽  
Earth Mover's Distance ◽  
Neighbor Search ◽  
The Earth ◽  
Binary Compounds

It is a core problem in any field to reliably tell how close two objects are to being the same, and once this relation has been established we can use this information to precisely quantify potential relationships, both analytically and with machine learning (ML). For inorganic solids, the chemical composition is a fundamental descriptor, which can be represented by assigning the ratio of each element in the material to a vector. These vectors are a convenient mathematical data structure for measuring similarity, but unfortunately, the standard metric (the Euclidean distance) gives little to no variance in the resultant distances between chemically dissimilar compositions. We present the Earth Mover’s Distance (EMD) for inorganic compositions, a well-defined metric which enables the measure of chemical similarity in an explainable fashion. We compute the EMD between two compositions from the ratio of each of the elements and the absolute distance between the elements on the modified Pettifor scale. This simple metric shows clear strength at distinguishing compounds and is efficient to compute in practice. The resultant distances have greater alignment with chemical understanding than the Euclidean distance, which is demonstrated on the binary compositions of the Inorganic Crystal Structure Database (ICSD). The EMD is a reliable numeric measure of chemical similarity that can be incorporated into automated workflows for a range of ML techniques. We have found that with no supervision the use of this metric gives a distinct partitioning of binary compounds into clear trends and families of chemical property, with future applications for nearest neighbor search queries in chemical database retrieval systems and supervised ML techniques.

Level Set Method Optimized with the Euclidean Distance Transform

2019 ◽  
Author(s):  
Luis Fernando Segalla ◽  
Alexandre Zabot ◽  
Diogo Nardelli Siebert ◽  
Fabiano Wolf
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Euclidean Distance ◽  
Distance Transform ◽  
Euclidean Distance Transform

On Some I-convergence of Difference Double Sequence Classes of Fuzzy Real Numbers Defined by Modulus Function

2017 ◽  
Vol 8 (12) ◽  
pp. 758-768
Author(s):  
Manmohan Das ◽  
Sanjay Kumar Das
Keyword(s):  
Double Sequence ◽  
Modulus Function ◽  
Real Numbers ◽  
Fuzzy Real Numbers

Improving the Computational Cost for Copied Region Detection in Forensic Images

2016 ◽  
Vol 2 (1) ◽  
pp. 55
Author(s):  
Tu Huynh-Kha ◽  
Thuong Le-Tien ◽  
Synh Ha ◽  
Khoa Huynh-Van
Keyword(s):  
Wavelet Transform ◽  
Euclidean Distance ◽  
Research Work ◽  
Computational Cost ◽  
Correlation Coefficients ◽  
Zernike Moments ◽  
Computational Time ◽  
Discrete Wavelet ◽  
Feature Vectors ◽  
Region Detection

This research work develops a new method to detect the forgery in image by combining the Wavelet transform and modified Zernike Moments (MZMs) in which the features are defined from more pixels than in traditional Zernike Moments. The tested image is firstly converted to grayscale and applied one level Discrete Wavelet Transform (DWT) to reduce the size of image by a half in both sides. The approximation sub-band (LL), which is used for processing, is then divided into overlapping blocks and modified Zernike moments are calculated in each block as feature vectors. More pixels are considered, more sufficient features are extracted. Lexicographical sorting and correlation coefficients computation on feature vectors are next steps to find the similar blocks. The purpose of applying DWT to reduce the dimension of the image before using Zernike moments with updated coefficients is to improve the computational time and increase exactness in detection. Copied or duplicated parts will be detected as traces of copy-move forgery manipulation based on a threshold of correlation coefficients and confirmed exactly from the constraint of Euclidean distance. Comparisons results between proposed method and related ones prove the feasibility and efficiency of the proposed algorithm.

IMPLEMENTASI METODE EUCLIDEAN DISTANCE UNTUK EKSTRAKSI FITUR JARAK PADA CITRA SKELETON

2019 ◽  
Vol 24 (2) ◽  
pp. 134-139
Author(s):  
Miftahul Jannah ◽  
Nurul Humaira
Keyword(s):  
Euclidean Distance

Gait adalah cara atau sikap berjalan kaki seseorang. Tiap orang memiliki cara berjalan yang berbeda, sehingga gerak jalan seseorang sulit untuk disembunyikan ataupun direkayasa. Analisis gait adalah ilmu pengetahuan yang mempelajari tentang kemampuan atau cara bergerak manusia. Dalam bidang kedokteran, analisis gait digunakan untuk menentukan penanganan dan terapi bagi pasien rehabilitasi medik. Dalam penelitian ini digunakan fitur jarak pada citra skeleton. Ekstraksi fitur jarak pada citra skeleton menggunakan metode euclidean distance terbagi dalam beberapa tahapan, dimulai dengan mengambil citra skeleton, konversi citra RGB menjadi citra Biner, proses menemukan titik koordinat dari titik akhir dan titik percabangan, dan ekstraksi fitur pada skeleton. Metode yang digunakan menghasilkan persentase tingkat keberhasilan sebesar 87.84%.

Algorithm of point pattern matching based on quasi Laplacian spectrum and point pair topological characteristics

2013 ◽  
Vol 33 (6) ◽  
pp. 1686-1690
Author(s):  
Guanliang ZHANG ◽  
Huanxin ZOU ◽  
Chunyan LU ◽  
Jian ZHAO
Keyword(s):  
Pattern Matching ◽  
Point Pattern ◽  
Laplacian Spectrum ◽  
Point Pair ◽  
Point Pattern Matching ◽  
Topological Characteristics

Spheric radial basis functions in Mahalanobis measuring for displaying local field features

2019 ◽  
Vol 952 (10) ◽  
pp. 2-9
Author(s):  
Yu.M. Neiman ◽  
L.S. Sugaipova ◽  
V.V. Popadyev
Keyword(s):  
Gravitational Field ◽  
Quadratic Form ◽  
Local Field ◽  
Euclidean Distance ◽  
Basis Functions ◽  
Spherical Functions ◽  
Local Models ◽  
Stationary Field ◽  
Modeling Process ◽  
The Earth

As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.

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