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Fuchsian Group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving 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) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which 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. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazarus Fuchs
Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-German mathematician who contributed important research in the field of linear differential equations. He was born in Mosina, Moschin in the Grand Duchy of Posen (modern-day Mosina, Poland) and died in Berlin, German Empire, Germany. He was buried in Schöneberg in the Alter St.-Matthäus-Kirchhof, St. Matthew's Cemetery. His grave in section H is preserved and listed as a grave of honour of the State of Berlin. Contribution He is the eponym of Fuchsian groups and functions, and the Picard–Fuchs equation. A singularity (mathematics), singular point ''a'' of a linear differential equation :y''+p(x)y'+q(x)y=0 is called Fuchsian if ''p'' and ''q'' are meromorphic function, meromorphic around the point ''a'', and have poles of orders at most 1 and 2, respectively. According to a Fuchs's theorem, theorem of Fuchs, this condition is necessary and sufficient for the regular singular point, regularity of the singular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Properly Discontinuously
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the Pole (complex analysis), poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and only if its identity is isolated. A subgroup ''H'' of a topological group ''G'' is a discrete subgroup if ''H'' is discrete when endowed with the subspace topology from ''G''. In other words there is a neighbourhood of the identity in ''G'' containing no other element of ''H''. 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. Any group can be endowed with the discrete topology, making it a discrete topological group. 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 catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accumulation Point
In mathematics, a limit point, accumulation point, or cluster point of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood of x contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence (x_n)_ in a topological space X is a point x such that, for every neighbourhood V of x, there are infinitely many natural numbers n such that x_n \in V. This definition of a cluster or accumulation point of a sequence generalizes to Net (mathematics), nets and Filter (set theory), filters. The similarly named notion of a (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the Convergent sequence, sequence converges to (respectively, the Convergent filter, filter conver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically, a Möbius transformation can be obtained by first applying the inverse stereographic projection from the plane to the unit sphere, moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group (mathematics), group called the Möbius group, which is the projective linear group . Together with its subgroups, it has numerous applications in mathematics and physics. Möbius geometry, Möbius geometries and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Fractional Transformations
In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional transformation is a ''transformation'' that is represented by a ''fraction'' whose numerator and denominator are ''linear''. In the most basic setting, , and are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field. The invertibility condition is then . Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When are integers (or, more generally, belong to an integral domain), is supposed to be a rational number (or to belong to the field of fractions of the integral domain. In this case, the invertibility condition is that must be a unit of the domain (that is or in the case of int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action of a group G on a set (mathematics), set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformation (function), transformations form a group (mathematics), group under function composition; for example, the rotation (mathematics), rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a mathematical structure, structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
