Total variation denoisingIn signal processing, particularly , total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process (filter). It is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the absolute is high. According to this principle, reducing the total variation of the signal—subject to it being a close match to the original signal—removes unwanted detail whilst preserving important details such as .
Riesz potentialIn mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable. If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by where the constant is given by This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α.
Fejér's theoremIn mathematics, Fejér's theorem, named after Hungarian mathematician Lipót Fejér, states the following: Explicitly, we can write the Fourier series of f as where the nth partial sum of the Fourier series of f may be written as where the Fourier coefficients are Then, we can define with Fn being the nth order Fejér kernel. Then, Fejér's theorem asserts that with uniform convergence.
Matrix unitIn linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ith row and jth column is denoted as . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is A vector unit is a standard unit vector. A single-entry matrix generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1. The set of m by n matrix units is a basis of the space of m by n matrices.
PyramidA pyramid (from πυραμίς pyramís) is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilateral, or of any polygon shape. As such, a pyramid has at least three outer triangular surfaces (at least four faces including the base). The square pyramid, with a square base and four triangular outer surfaces, is a common version.
Banach–Alaoglu theoremIn functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.
Great Pyramid of GizaThe Great Pyramid of Giza is the largest Egyptian pyramid and served as the tomb of pharaoh Khufu, who ruled during the Fourth Dynasty of the Old Kingdom. Built in the early 26th century BC, over a period of about 27 years, the pyramid is the oldest of the Seven Wonders of the Ancient World, and the only wonder that has remained largely intact. It is the most famous monument of the Giza pyramid complex, which is part of the UNESCO World Heritage Site "Memphis and its Necropolis".
Square pyramidIn geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has C_4v symmetry. If all edge lengths are equal, it is an equilateral square pyramid, the Johnson solid J_1. A possibly oblique square pyramid with base length l and perpendicular height h has volume: In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles.
