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Protorus
In mathematics, a protorus is a compact connected topological abelian group. Equivalently, it is a projective limit of tori (products of a finite number of copies of the circle group), or the Pontryagin dual of a discrete torsion-free abelian group. Some examples of protori are given by solenoid groups. See also * Duocylinder - Cartesian product of two disks * Proprism In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher. The term was coined by John Horton Conway for ''product prism''. The dimension of the ... References * Topological groups {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is '' locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Abelian Group
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or ''inductive limit'', and a ''limit'' becomes a colimit. Formal definition Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) be a directed poset (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a family of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f_ is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. The circle group forms a subgroup of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure \theta: \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Dual
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a -adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solenoid Group
:''This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.'' In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms :f_i: S_ \to S_i \quad \forall i \ge 0 where each S_i is a circle and ''f''''i'' is the map that uniformly wraps the circle S_ for n_ times (n_ \geq 2) around the circle S_i. This construction can be carried out geometrically in the three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of a compact topological group. Solenoids were first introduced by Vietoris for the n_i = 2 case, and by van Dantzig the n_i = n case, where n\geq 2 is fixed. Such a solenoid arises as a one-dimensional expanding attractor, or Smale–Williams attractor, and forms an important example in the theory of hyperbolic dynamical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duocylinder
The duocylinder, also called the double cylinder or the bidisc, is a geometric object embedded in 4- dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii ''r''1 and ''r''2: :D = \left\ It is analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment. But unlike the cylinder, both hypersurfaces (of a regular duocylinder) are congruent. Its dual is a duospindle, constructed from two circles, one at the XY plane and the other in the ZW plane. Geometry Bounding 3-manifolds The duocylinder is bounded by two mutually perpendicular 3-manifolds with torus-like surfaces, respectively described by the formulae: :x^2 + y^2 = r_1^2, z^2 + w^2 \leq r_2^2 and :z^2 + w^2 = r_2^2, x^2 + y^2 \leq r_1^2 The duocylinder is so called because these two bounding 3-manifolds may be thought of as 3-dimensional cylinders 'bent around' in 4-dimensional space such that they form closed loops in the XY and ZW ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''- tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Examples A deck of cards A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
