Basic Principles of Data Mining

2010 ◽  
pp. 266-289
Author(s):  
Karl-Ernst Erich Biebler
Keyword(s):  
Data Mining ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Topological Order ◽  
Data Types ◽  
Basic Principles ◽  
Mathematical Structures ◽  
Related Data ◽  
Higher Dimensional ◽  
Discriminant Analyses

This chapter gives a summary of data types, mathematical structures, and associated methods of data mining. Topological, order theoretical, algebraic, and probability theoretical mathematical structures are introduced. The n-dimensional Euclidean space, the model used most for data, is defined. It is executed briefly that the treatment of higher dimensional random variables and related data is problematic. Since topological concepts are less well known than statistical concepts, many examples of metrics are given. Related classification concepts are defined and explained. Possibilities of their quality identification are discussed. One example each is given for topological cluster and for topological discriminant analyses.

scholarly journals The mathematical foundations of anelasticity: existence of smooth global intermediate configurations

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200462
Author(s):  
Christian Goodbrake ◽  
Alain Goriely ◽  
Arash Yavari
Keyword(s):  
Euclidean Space ◽  
Deformation Gradient ◽  
Deformation Tensor ◽  
Dimensional Euclidean Space ◽  
Multiplicative Decomposition ◽  
Mathematical Basis ◽  
Isometric Embeddings ◽  
Higher Dimensional ◽  
Mathematical Foundations ◽  
Central Tool

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

The Integral of Spatial Data Mining in the Era of Big Data

2017 ◽  
pp. 90-126
Author(s):  
Gebeyehu Belay Gebremeskel ◽  
Chai Yi ◽  
Zhongshi He
Keyword(s):  
Data Mining ◽  
Data Warehouse ◽  
Spatial Data ◽  
High Volume ◽  
Spatial Data Mining ◽  
Research Field ◽  
Data Sets ◽  
Data Types ◽  
Basic Principles ◽  
Gis Data

Data Mining (DM) is a rapidly expanding field in many disciplines, and it is greatly inspiring to analyze massive data types, which includes geospatial, image and other forms of data sets. Such the fast growths of data characterized as high volume, velocity, variety, variability, value and others that collected and generated from various sources that are too complex and big to capturing, storing, and analyzing and challenging to traditional tools. The SDM is, therefore, the process of searching and discovering valuable information and knowledge in large volumes of spatial data, which draws basic principles from concepts in databases, machine learning, statistics, pattern recognition and 'soft' computing. Using DM techniques enables a more efficient use of the data warehouse. It is thus becoming an emerging research field in Geosciences because of the increasing amount of data, which lead to new promising applications. The integral SDM in which we focused in this chapter is the inference to geospatial and GIS data.

scholarly journals Confidence Analysis of Standard Deviational Ellipse and Its Extension into Higher Dimensional Euclidean Space

PLoS ONE ◽  
2015 ◽  
Vol 10 (3) ◽  
pp. e0118537 ◽  
Author(s):  
Bin Wang ◽  
Wenzhong Shi ◽  
Zelang Miao
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Higher Dimensional ◽  
Standard Deviational Ellipse

Inequalities for random flats meeting a convex body

1985 ◽  
Vol 22 (03) ◽  
pp. 710-716 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Finite Number ◽  
Random Point ◽  
Dimensional Euclidean Space ◽  
Random Direction ◽  
Higher Dimensional

We choose a uniform random point in a given convex bodyKinn-dimensional Euclidean space and through that point the secant ofKwith random direction chosen independently and isotropically. Given the volume ofK, the expectation of the length of the resulting random secant ofKwas conjectured by Enns and Ehlers [5] to be maximal ifKis a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meetingK, and we prove that certain geometric probabilities connected with these again become maximal whenKis a ball.

scholarly journals Higher horospherical limit sets for G-modules over CAT(0)-spaces

2021 ◽  
pp. 1-31
Author(s):  
ROBERT BIERI ◽  
ROSS GEOGHEGAN
Keyword(s):  
Group Theory ◽  
Euclidean Space ◽  
Discrete Group ◽  
Tropical Geometry ◽  
Dimensional Euclidean Space ◽  
Limit Sets ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Suitable Space

Abstract The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

scholarly journals Neohookean deformations of annuli in the higher dimensional Euclidean space

2019 ◽  
Vol 189 ◽  
pp. 111575
Author(s):  
Jian-Feng Zhu ◽  
David Kalaj
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Higher Dimensional

Solutions of equivariance for a polynomial-like iterative equation

2000 ◽  
Vol 130 (5) ◽  
pp. 1153-1163 ◽  
Author(s):  
Weinian Zhang
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Closed Subgroup ◽  
General Linear Group ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Dimensional Euclidean Space ◽  
Continuous Solutions ◽  
Iterative Equation ◽  
Higher Dimensional

Using fixed point theorems we discuss continuous solutions of Γ-equivariance for a polynomial-like iterative equation on the real line, where Γ is a closed subgroup of the general linear group GL(R). Our main results guarantee the existence of solutions with certain kinds of symmetry. We show that, under restrictive hypotheses, similar results can be proved in a higher-dimensional case, where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in N-dimensional Euclidean space.

The Integral of Spatial Data Mining in the Era of Big Data

2019 ◽  
pp. 863-899
Author(s):  
Gebeyehu Belay Gebremeskel ◽  
Chai Yi ◽  
Zhongshi He
Keyword(s):  
Data Mining ◽  
Data Warehouse ◽  
Spatial Data ◽  
High Volume ◽  
Spatial Data Mining ◽  
Research Field ◽  
Data Sets ◽  
Data Types ◽  
Basic Principles ◽  
Gis Data

Data Mining (DM) is a rapidly expanding field in many disciplines, and it is greatly inspiring to analyze massive data types, which includes geospatial, image and other forms of data sets. Such the fast growths of data characterized as high volume, velocity, variety, variability, value and others that collected and generated from various sources that are too complex and big to capturing, storing, and analyzing and challenging to traditional tools. The SDM is, therefore, the process of searching and discovering valuable information and knowledge in large volumes of spatial data, which draws basic principles from concepts in databases, machine learning, statistics, pattern recognition and 'soft' computing. Using DM techniques enables a more efficient use of the data warehouse. It is thus becoming an emerging research field in Geosciences because of the increasing amount of data, which lead to new promising applications. The integral SDM in which we focused in this chapter is the inference to geospatial and GIS data.

scholarly journals Symmetries of Monocoronal Tilings

2015 ◽  
Vol Vol. 17 no.2 (Combinatorics) ◽  
Author(s):  
Dirk Frettlöh ◽  
Alexey Garber
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Dimensional Euclidean Space ◽  
Symmetry Groups ◽  
Translation Group ◽  
One Dimensional ◽  
Higher Dimensional ◽  
International Audience

International audience The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.

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