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Dendrite (mathematics)
In mathematics, a dendrite is a certain type of topological space that may be characterized either as a locally connected dendroid or equivalently as a locally connected continuum that contains no simple closed curves. Importance Dendrites may be used to model certain types of Julia set. For example, if 0 is pre-periodic, but not periodic, under the function f(z) = z^2 + c, then the Julia set of f is a dendrite: connected, without interior.. References See also *Misiurewicz point *Real tree, a related concept defined using metric spaces instead of topological spaces *Dendroid (topology) and unicoherent space In mathematics, a unicoherent space is a topological space X that is connected and in which the following property holds: For any closed, connected A, B \subset X with X=A \cup B, the intersection A \cap B is connected. For example, any closed i ..., two more general types of tree-like topological space Continuum theory Trees (topology) {{Topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Connected
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dendroid (topology)
In mathematics, a dendroid is a type of topological space, satisfying the properties that it is hereditarily unicoherent (meaning that every subcontinuum of ''X'' is unicoherent), arcwise connected, and forms a continuum. The term dendroid was introduced by Bronisław Knaster lecturing at the University of Wrocław,. although these spaces were studied earlier by Karol Borsuk and others.. proved that dendroids have the fixed-point property: Every continuous function from a dendroid to itself has a fixed point. proved that every dendroid is ''tree-like'', meaning that it has arbitrarily fine open covers whose nerve is a tree. The more general question of whether every tree-like continuum has the fixed-point property, posed by , was solved in the negative by David P. Bellamy, who gave an example of a tree-like continuum without the fixed-point property. In Knaster's original publication on dendroids, in 1961, he posed the problem of characterizing the dendroids which can be embed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum (topology)
In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. Definitions * A continuum that contains more than one point is called nondegenerate. * A subset ''A'' of a continuum ''X'' such that ''A'' itself is a continuum is called a subcontinuum of ''X''. A space homeomorphic to a subcontinuum of the Euclidean plane R2 is called a planar continuum. * A continuum ''X'' is homogeneous if for every two points ''x'' and ''y'' in ''X'', there exists a homeomorphism ''h'': ''X'' → ''X'' such that ''h''(''x'') = ''y''. * A Peano continuum is a continuum that is locally connected at each point. * An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum ''X'' is hereditarily indecomposable if every subcontinuum of ''X'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dendrite Julia
Dendrites (from Greek δένδρον ''déndron'', "tree"), also dendrons, are branched protoplasmic extensions of a nerve cell that propagate the electrochemical stimulation received from other neural cells to the cell body, or soma, of the neuron from which the dendrites project. Electrical stimulation is transmitted onto dendrites by upstream neurons (usually via their axons) via synapses which are located at various points throughout the dendritic tree. Dendrites play a critical role in integrating these synaptic inputs and in determining the extent to which action potentials are produced by the neuron. Dendritic arborization, also known as dendritic branching, is a multi-step biological process by which neurons form new dendritic trees and branches to create new synapses. The morphology of dendrites such as branch density and grouping patterns are highly correlated to the function of the neuron. Malformation of dendrites is also tightly correlated to impaired nervous syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia Set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic". The Julia set of a function is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Formal definition Let f(z) be a non-constant holomorphic function from the Riemann sphere on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Misiurewicz Point
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By analogy, the term ''Misiurewicz point'' is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic. This term makes less sense for maps in greater generality that have more than one free critical point because some critical points might be periodic and others not. These points are named after the Polish-American mathematician Michał Misiurewicz, who was the first to study them. Mathematical notation A parameter c is a Misiurewicz point M_ if it satisfies the equations: :f_c^(z_) = f_c^(z_) and: :f_c^(z_) \neq f_c^(z_) so: :M_ = c : f_c^(z_) = f_c^(z_) where: * z_ is a critical point of f_c, * k and n are positive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Tree
In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the simplest examples of Gromov hyperbolic spaces. Definition and examples Formal definition A metric space X is a real tree if it is a geodesic space where every triangle is a tripod. That is, for every three points x, y, \rho \in X there exists a point c = x \wedge y such that the geodesic segments rho,x rho,y/math> intersect in the segment rho,c/math> and also c \in ,y/math>. This definition is equivalent to X being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin"). Real trees can also be characterised by a topological property. A metric space X is a real tree if for any pair of points x, y \in X all topological embeddings \sigma of the segment ,1/math> into X such that \sigma(0) = x, \, \sigma(1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unicoherent Space
In mathematics, a unicoherent space is a topological space X that is connected and in which the following property holds: For any closed, connected A, B \subset X with X=A \cup B, the intersection A \cap B is connected. For example, any closed interval on the real line is unicoherent, but a circle is not. If a unicoherent space is more strongly hereditarily unicoherent (meaning that every subcontinuum is unicoherent) and arcwise connected, then it is called a dendroid. If in addition it is locally connected then it is called a dendrite. The Phragmen–Brouwer theorem In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if ''X'' is a normal connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong ... states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space. References * External links * Ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Theory
In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. Definitions * A continuum that contains more than one point is called nondegenerate. * A subset ''A'' of a continuum ''X'' such that ''A'' itself is a continuum is called a subcontinuum of ''X''. A space homeomorphic to a subcontinuum of the Euclidean plane R2 is called a planar continuum. * A continuum ''X'' is homogeneous if for every two points ''x'' and ''y'' in ''X'', there exists a homeomorphism ''h'': ''X'' → ''X'' such that ''h''(''x'') = ''y''. * A Peano continuum is a continuum that is locally connected at each point. * An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum ''X'' is hereditarily indecomposable if every subcontinuum of ''X'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
