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Geometry of Rotation
Published in Kenichi Kanatani, 3D Rotations, 2020
There exist linear transformations that preserve lengths and angles but not the sense. Such a transformation is a composition of a rotation and a reflection (Fig. 2.1(b)). A reflection is a mapping into a position symmetric with respect to some plane passing through the origin. If a rotation is composed with a reflection, a right-handed system {a, b, c} is mapped to a left-handed system {a′, b′, c′}. The volume of the parallelepipeds they define have reversed signs; |a′, b′, c′| = −|a, b, c|. Composition of a rotation and a reflection is sometimes called improper rotation, while the usual rotation is called proper rotation. In this book, rotations always mean proper rotations.
When is deep learning better and when is shallow learning better: qualitative analysis
Published in International Journal of Parallel, Emergent and Distributed Systems, 2022
Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich
A function is invariant with respect to a shift means that , i.e. that the function is periodic with period T. One can easily check that non-constant polynomials cannot be periodic. So, since we are considering polynomials, the only transformation with respect to which a polynomial can be invariant are reflection with respect to a point a. Reflection means that for each original point x, we take a new point which is located at the same distance from a as the original point x, but at the opposite side of the point a. In other words, we have , i.e. equivalently, .
Two-Dimensional Plane Wave Reflection and Transmission in a Layered Highly Anisotropic Media under Initial Stress
Published in Journal of Earthquake Engineering, 2020
Akanksha Srivastava, Amares Chattopadhyay, Pooja Singh, Abhishek Kumar Singh
An earthquake is a sudden movement inside the Earth’s surface causing brittle fractures in the Earth’s crust. This sudden movement generates seismic waves. When a seismic wave comes across the discontinuities between different layers, it produces a reflection and transmission phenomena. In recent years, the subject of wave propagation and their reflection and transmission from a plane boundary interface in solids have been a considerable interest in the field of seismology. The phenomena of reflection and transmission are used extensively in several contexts, such as petroleum exploration, land survey acquisition, marine survey acquisition, rock mechanics, seismic data processing, and seismic attribute analysis. The velocity of seismic wave propagating through Earth’s different layers and interfaces is highly influenced by the material properties of the medium. Generally there are three kinds of body waves traveling through an anisotropic media with different velocities which are referred to as quasi-P (qP), quasi-SV (qSV), and quasi-SH (qSH) waves. It has been discovered that the movement of particles is neither purely longitudinal nor transverse.