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In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. The anti-de Sitter space and de Sitter space are named after Willem de Sitter
Willem de Sitter
(1872–1934), professor of astronomy at Leiden University
Leiden University
and director of the Leiden
Leiden
Observatory
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Coordinate Patch
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fibre bundles.Contents1 Charts 2 Formal definition of atlas 3 Transition maps 4 More structure 5 See also 6 References 7 External linksCharts[edit] See also: Manifold
Manifold
§ Charts The definition of an atlas depends on the notion of a chart
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Pseudosphere
In geometry, the term pseudosphere is used to describe various surfaces with constant negative Gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, a tractricoid, or a hyperboloid.Contents1 Theoretical pseudosphere 2 Tractricoid 3 Universal covering space 4 Hyperboloid 5 See also 6 References 7 External linksTheoretical pseudosphere[edit] In its general interpretation, a pseudosphere of radius R is any surface of curvature −1/R2, by analogy with the sphere of radius R, which is a surface of curvature 1/R2
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Saddle Surface
In mathematics, a saddle point or minimax point[1] is a point on the surface of the graph of a function where the slopes (derivatives) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes.[2] An example of a saddle point shown on the right is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form
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Gabriel's Horn
Gabriel's horn (also called Torricelli's trumpet) is a geometric figure which has infinite surface area but finite volume. The name refers to the tradition identifying the Archangel Gabriel
Gabriel
as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite
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Trumpet
BrassWind Brass Aerophone Hornbostel–Sachs classification 423.233 (Valved aerophone sounded by lip movement)Playing rangeWritten range:Related instrumentsFlugelhorn, cornet, cornett, Flumpet, bugle, natural trumpet, bass trumpet, post horn, Roman tuba, buccina, cornu, lituus, shofar, dord, dung chen, sringa, shankha, lur, didgeridoo, Alphorn, Russian horns, serpent, ophicleide, piccolo trumpet, horn, alto horn, baritone horn, pocket trumpetPart of a series onMusical instrumentsWoodwindsPiccolo Flute Oboe Cor anglais Clarinet Saxophone Bassoon Contrabassoon Bagpipes RecorderGarklein in C6 (c‴) Sopranino in F5 (f″) Soprano in C5 (c″) Alto in F4 (f′)
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Mass-energy Equivalence
In physics, mass–energy equivalence states that anything having mass has an equivalent amount of energy and vice versa, with these fundamental quantities directly relating to one another by Albert Einstein's famous formula:[1] E = m c 2 displaystyle E=mc^ 2 This formula states that the equivalent energy (E) can be calculated as the mass (m) multiplied by the speed of light (c = about 7008300000000000000♠3×108 m/s) squared. Similarly, anything having energy exhibits a corresponding mass m given by its energy E divided by the speed of light squared c². Because the speed of light is a very large number in everyday units, the formula implies that even an everyday object at rest with a modest amount of mass has a very large amount of energy intrinsically
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Newton's Law Of Universal Gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation
states that a particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.[note 1] This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning.[1] It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1686. When Newton's book was presented in 1686 to the Royal Society, Robert Hooke
Robert Hooke
made a claim that Newton had obtained the inverse square law from him. In today's language, the law states: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points
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Gravitational Constant
The gravitational constant, also known as the universal gravitational constant, or as Newton's constant, denoted by the letter G, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity
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Non-Euclidean Geometry
In mathematics, non- Euclidean geometry
Euclidean geometry
consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry
Euclidean geometry
lies at the intersection of metric geometry and affine geometry, non- Euclidean geometry
Euclidean geometry
arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines
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Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry
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Black Hole
A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it.[1] The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole.[2][3] The boundary of the region from which no escape is possible is called the event horizon. Although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, no locally detectable features appear to be observed.[4] In many ways a black hole acts like an ideal black body, as it reflects no light.[5][6] Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass
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Isometric Embedding
In mathematics, an embedding (or imbedding[1]) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f : X → Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map f : X → Y is an embedding is often indicated by the use of a "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK);[2] thus: f : X ↪ Y . displaystyle f:Xhookrightarrow Y
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Metric Tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. A metric tensor is called positive definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive definite metric tensor is known as a Riemannian manifold
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Strong Force
In particle physics, the strong interaction is the mechanism responsible for the strong nuclear force (also called the strong force or nuclear strong force), and is one of the four known fundamental interactions, with the others being electromagnetism, the weak interaction, and gravitation. At the range of 10−15 m (1 femtometer), the strong force is approximately 137 times as strong as electromagnetism, a million times as strong as the weak interaction, and 1038 times as strong as gravitation.[1] The strong nuclear force holds most ordinary matter together because it confines quarks into hadron particles such as the proton and neutron. In addition, the strong force binds neutrons and protons to create atomic nuclei
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Quasi-sphere
In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case.Contents1 Notation and terminology 2 Definition 3 Geometric characterizations3.1 Centre and radius squared 3.2 Diameter and radius4 Partitioning 5 See also 6 Notes 7 ReferencesNotation and terminology[edit] This article uses the following notation and terminology:A pseudo-Euclidean vector space, denoted Rs,t, is a real vector space with a nondegenerate quadratic form with signature (s, t)
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