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Fourier Series: Basic Concepts
Published in Steven G. Krantz, Differential Equations, 2022
Many problems of mathematics and mathematical physics are set on all of Euclidean space—not on an interval. Thus it is appropriate to have analytical tools designed for that setting. The Fourier transform is one of the most important of these devices. In this Anatomy we explore the basic ideas behind the Fourier transform. We shall present the concepts in Euclidean space of any dimension. Throughout, we shall use the standard notation f∈L1 or f∈L1(ℝN) to mean that f is integrable.
Different Machine Learning Models
Published in Neeraj Kumar, Aaisha Makkar, Machine Learning in Cognitive IoT, 2020
In geometry, hyper-plane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these data sets are closed (a set is closed if and only if it coincides with its closure) and at least one of them is compact (compactness is a property that generalizes the notion of a subset containing all its limit points and bounded), then we can say that there is a hyper-plane in between the sets and even two parallel hyper-planes in between them separated by a gap. Consider the Figure 7.2, in which a represents the maximum margin hyper-plane and b represents the support vectors. Observations: The outline appears when you connect points of the set.Convex null cannot overlap in linearly separable classes.The highest hyper-plane margin is always the one that is as far away from both convex hulls as feasible.There is at least one support vector in each class.The set of support vectors here describes the maximum hyper-plane margin for the learning issue in a unique way.
Physiology, Anatomy and Fractal Properties
Published in Dinesh K. Kumar, Sridhar P. Arjunan, Behzad Aliahmad, Fractals, 2017
Dinesh K. Kumar, Sridhar P. Arjunan, Behzad Aliahmad
If we consider the question, What is a dimension in the spatial domain? In Euclidean space, a line is considered to have one dimension, while a rectangle has a dimension of two. This is because there is only one linearly independent direction in a straight line, and two linearly independent directions in a plane. In a line there is only one way to move while the plane has dimension of two because there are two independent directions. Similarly, cube has three dimensions: length, width, and height. An alternative way to view the concept of dimension is for a self-similar object. The dimension N is the exponent of the number of self-similar pieces created with magnification factor N. Thus, when a square is divided in 32 segments, the magnification is three, and the dimension is two.
Human spatial learning strategies in wormhole virtual environments
Published in Spatial Cognition & Computation, 2023
Christopher Widdowson, Ranxiao Frances Wang
Another important theoretical concept needing clarification is the distinction between Euclidean vs. non-Euclidean geometry and metric vs non-metric space (e.g., Montello, 1992). A metric space satisfies the following properties: 1) the distance from A to B is zero if and only if A and B are the same point; 2) the distance between two distinct points is positive (positivity); 3) the distance from A to B is the same as the distance from B to A (symmetry); and 4) the distance from A to B is less than or equal to the distance from A to B via any third point C (triangle inequality). A Euclidean space is a type of metric space that also satisfies the parallel postulate, therefore a space can be Euclidean, non-Euclidean but metric, or non-metric at all. A spatial representation that does not conform to Euclidean geometry can have violations specific to Euclidean metric (e.g., parallel postulate), or violations of general metric principles that are not specific to Euclidean geometry (e.g., symmetry or triangle inequality). Therefore, it is important to distinguish between Euclidean vs non-Euclidean and metric vs non-metric spaces. When the experimental evidence only involves violation of the general metric properties, it is more appropriate to call it “non-metric” than non-Euclidean, and the theoretical distinction should be referred to as metric vs non-metric instead of Euclidean vs non-Euclidean.