Curse of dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces (often with hundreds or thousands of dimensions) that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The term curse of dimensionality was coined by Richard E. Bellman when considering problems in dynamic optimization.[1][2]
Eigenvalues and eigenvectors
If the vector space V is finite-dimensional, then the linear transformation T can be represented as a square matrix A, and the vector v by a column vector, rendering the above mapping as a matrix multiplication on the left hand side and a scaling of the column vector on the right hand side in the equation There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space to itself, given any basis of the vector space. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations.[1][2] Geometrically an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction that is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed.[3]
Manifold
The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features.
Wave function
Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. The classical process (A–B) is represented as the motion of a particle along a trajectory. The quantum process (C–H) has no such trajectory.
Vector space
Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are best thought of as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.
Five-dimensional space
Physics[edit] Much of the early work on five dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity and electromagnetism. German mathematician Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which used the fifth dimension to unify gravity with electromagnetic force. Although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century.[1]
Oskar Klein
Oskar Benjamin Klein (Swedish: [klajn]; 15 September 1894 – 5 February 1977) was a Swedish theoretical physicist.[1] Biography[edit] Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Gottlieb Klein from Humenné in Slovakia and Antonie (Toni) Levy. He became a student of Svante Arrhenius at the Nobel Institute at a young age and was on the way to Jean-Baptiste Perrin in France when World War I broke out and he was drafted into the military. From 1917, he worked a few years with Niels Bohr in the University of Copenhagen and received his doctoral degree at the University College of Stockholm (now Stockholm University) in 1921.
BKS theory
The Bohr-Kramers-Slater (BKS) theory[1][2][3] was perhaps the final attempt at understanding the interaction of matter and electromagnetic radiation on the basis of the so-called old quantum theory, in which quantum phenomena are treated by imposing quantum restrictions on classically describable behaviour. It was advanced in 1924, and sticks to a classical wave description of the electromagnetic field. It was perhaps more a research program than a full physical theory, the ideas that are developed not being worked out in a quantitative way. [4]:236 One aspect, the idea of modelling atomic behaviour under incident electromagnetic radiation using "virtual oscillators" at the absorption and emission frequencies, rather than the (different) apparent frequencies of the Bohr orbits, significantly led Born, Heisenberg and Kramers to explore mathematics that strongly inspired the subsequent development of matrix mechanics, the first form of modern quantum mechanics. Origins[edit] Jump up ^ N.
