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Parabolic Transform
In geometry and complex analysis, a Möbius transformation
Möbius transformation
of the complex plane is a rational function of the form f ( z ) = a z + b c z + d displaystyle f(z)= frac az+b cz+d of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0. Geometrically, a
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Möbius Transform
In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. Other Möbius inversion formulas are obtained when different locally finite partially ordered sets replace the classic case of the natural numbers ordered by divisibility; for an account of those, see incidence algebra.Contents1 Statement of the formula 2 Series relations 3 Repeated transformations 4 Generalizations 5 On Posets 6 Multiplicative notation 7 Proofs of generalizations 8 Contributions of Weisner, Hall, and Rota 9 See also 10 References 11 External linksStatement of the formula[edit] The classic version states that if g and f are arithmetic functions satisfying g ( n )
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Kleinian Group
In mathematics, a Kleinian group
Kleinian group
is a discrete subgroup of PSL(2, C). The group PSL(2, C) of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic space H3, and as orientation preserving conformal maps of the open unit ball B3 in R3 to itself. Therefore, a Kleinian group
Kleinian group
can be regarded as a discrete subgroup acting on one of these spaces. There are some variations of the definition of a Kleinian group: sometimes Kleinian groups are allowed to be subgroups of PSL(2, C).2 (PSL(2, C) extended by complex conjugations), in other words to have orientation reversing elements, and sometimes they are assumed to be finitely generated, and sometimes they are required to act properly discontinuously on a non-empty open subset of the Riemann sphere
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Physics
Physics
Physics
(from Ancient Greek: φυσική (ἐπιστήμη), translit. physikḗ (epistḗmē), lit. 'knowledge of nature', from φύσις phýsis "nature"[1][2][3]) is the natural science that studies matter[4] and its motion and behavior through space and time and that studies the related entities of energy and force.[5] Physics
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Identity Component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group. Similarly, the identity path component of a topological group G is the path component of G that contains the identity element of the group.Contents1 Properties 2 Component group 3 Examples 4 ReferencesProperties[edit] The identity component G0 of a topological group G is a closed normal subgroup of G. It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological group are continuous maps by definition. Moreover, for any continuous automorphism a of G we havea(G0) = G0.Thus, G0 is a characteristic subgroup of G, so it is normal. The identity component G0 of a topological group G need not be open in G. In fact, we may have G0 = e , in which case G is totally disconnected
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Celestial Sphere
In astronomy and navigation, the celestial sphere is an abstract sphere, with an arbitrarily large radius, that is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer. If centered on the observer, half of the sphere would resemble a hemispherical screen over the observing location. The celestial sphere is a practical tool for spherical astronomy, allowing astronomers to specify the apparent positions of objects in the sky if their distances are unknown or irrelevant.Contents1 Introduction 2 Celestial coordinate systems 3 History 4 Star globe 5 Bodies other than Earth 6 See also 7 Notes 8 References 9 External linksIntroduction[edit]Celestial Sphere, 18th century. Brooklyn Museum.Because astronomical objects are at such remote distances, casual observation of the sky offers no information on their actual distances
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Twistor Theory
Twistor theory
Twistor theory
was proposed by Roger Penrose
Roger Penrose
in 1967[1] as a possible path[2] to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic arena for physics from which space-time itself should emerge
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Simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected[1]) if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other such path while preserving the two endpoints in question
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Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them
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Fundamental Group
In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group. Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. The abelianization of the fundamental group can be identified with the first homology group of the space
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Discrete Subgroup
In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology. Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups
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Fuchsian Group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R)
PSL(2,R)
can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) = PSL(2,R)
PSL(2,R)
(so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group
Kleinian group
(a discrete group of PSL(2,C)) that is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface
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Modular Group
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified
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Hyperbolic Space
In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature. When embedded to a Euclidean space
Euclidean space
(of a higher dimension), every point of a hyperbolic space is a saddle point
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Fractal
Mandelbrot set: Self-similarity
Self-similarity
illustrated by image enlargements. This panel, no magnification.The same fractal as above, magnified 6-fold. Same patterns reappear, making the exact scale being examined difficult to determine.The same fractal as above, magnified 100-fold.The same fractal as above, magnified 2000-fold, where the Mandelbrot set fine detail resembles the detail at low magnification.In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects. Artificially created fractals commonly exhibit similar patterns at increasingly small scales.[1] It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger sponge.[2] Fractals can also be nearly the same at different levels
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Modular Form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that, like a modular form, is invariant with respect to the modular group, but without the condition that f (z) be holomorphic at infinity
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