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Simplicial Manifold
In physics, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of Regge calculus. These objects combine attributes of a simplex with those of a manifold. There is no standard usage of this term in mathematics, and so the concept can refer to a triangulation in topology, or a piecewise linear manifold, or one of several different functors from either the category of sets or the category of simplicial sets to the category of manifolds. A manifold made out of simplices A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold. This is essentially the concept of a triangulation in topology. This can mean simply that a neighborhood of each vertex (i.e. the set of simplices that contain that point as a vertex) is homeomorphic to a ''n''-dimensional ball. A simplicial object built from manifolds A simplicial manifold is also a simplicial object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Manifold
In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure. Formal definition A topological space ''X'' is called locally Euclidean if there is a non-negative integer ''n'' such that every point in ''X'' has a neighborhood which is homeomorphic to real ''n''-space R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groupoid
In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations :s,t : \operatorname \to \operatorname are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by Charles Ehresmann under the name ''differentiable groupoids''. Definition and basic concepts A Lie groupoid consists of * two smooth m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nerve (category Theory)
In category theory, a discipline within mathematics, the nerve ''N''(''C'') of a small category ''C'' is a simplicial set constructed from the objects and morphisms of ''C''. The geometric realization of this simplicial set is a topological space, called the classifying space of the category ''C''. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory. Motivation The nerve of a category is often used to construct topological versions of moduli spaces. If ''X'' is an object of ''C'', its moduli space should somehow encode all objects isomorphic to ''X'' and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Space
In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ..... See in particulap. 8 References Homotopy theory Topological spaces {{math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Object
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Simplicial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set. Definitions Neighbourhood of a point If X is a topological space and p is a point in X, then a neighbourhood of p is a subset V of X that includes an open set U containing p, p \in U \subseteq V \subseteq X. This is equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need not be an open subset of X. When V is open (resp. closed, compact, etc.) in X, it is called an (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors require neighbourhoods to be open, so it is important to note their conventions. A set that is a neighbourhood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri PoincarĂ©), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
