|
Geodesic Bicombing
In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space. The study of metric spaces with distinguished geodesics traces back to the work of the mathematician Herbert Busemann. The convention to call a collection of paths of a metric space bicombing is due to William Thurston. By imposing a weak global non-positive curvature condition on a geodesic bicombing several results from the theory of CAT(0) spaces and Banach space theory may be recovered in a more general setting. Definition Let (X,d) be a metric space. A map \sigma\colon X\times X\times ,1to X is a ''geodesic bicombing'' if for all points x,y\in X the map \sigma_(\cdot):=\sigma(x,y,\cdot) is a unit speed metric geodesic from x to y, that is, \sigma_(0)=x, \sigma_(1)=y and d(\sigma_(s), \sigma_(t))=\vert s-t\vert d(x,y) for all real numbers s,t\in ,1/math>. Different classes of geodesic bicombings A geodesic bicombing \sigma\colon X\times X\times ,1to X is: * ''reversible'' if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Geometry
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun '' geodesic'' and the adjective '' geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Busemann
Herbert Busemann (12 May 1905 – 3 February 1994) was a German-American mathematician specializing in convex and differential geometry. He is the author of Busemann's theorem in Euclidean geometry and geometric tomography. He was a member of the Royal Danish Academy and a winner of the Lobachevsky Medal (1985), the first American mathematician to receive it. He was also a Fulbright scholar in New Zealand in 1952. Biography Herbert Busemann was born in Berlin to a well-to-do family. His father, Alfred Busemann, was a director of Krupp, where Busemann also worked for several years. He studied at University of Munich, Paris, and Rome. He defended his dissertation in University of Göttingen in 1931, where his advisor was Richard Courant. He remained in Göttingen as an assistant until 1933, when he escaped Nazi Germany to Copenhagen (he had a Jewish grandfather). He worked at the University of Copenhagen until 1936, when he left to the United States. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C. to Margaret Thurston (), a seamstress, and Paul Thurston, an aeronautical engineer. William Thurston suffered from congenital strabismus as a child, causing issues with depth perception. His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones. He received his bachelor's degree from New College in 1967 as part of its inaugural class. For his undergraduate thesis, he developed an intuitionist foundation for topology. Following this, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-positive Curvature
In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature extends to the category of geodesic metric spaces, where one can use comparison triangles to quantify the curvature of a space; in this context, non-positively curved spaces are known as (locally) CAT(0) spaces. Riemann Surfaces If S is a closed, orientable Riemann surface then it follows from the Uniformization theorem that S may be endowed with a complete Riemannian metric with constant Gaussian curvature of either 0, 1 or -1. As a result of the Gauss–Bonnet theorem one can determine that the surfaces which have a Riemannian metric of constant curvature 0 -1 i.e. Riemann surfaces with a complete, Riemannian metric of non-positive constant curvature, are exac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CAT(k) Space
In mathematics, a \mathbf(k) space, where k is a real number, is a specific type of metric space. Intuitively, triangles in a \operatorname(k) space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a \operatorname(k) space, the curvature is bounded from above by k. A notable special case is k=0; complete \operatorname(0) spaces are known as " Hadamard spaces" after the French mathematician Jacques Hadamard. Originally, Aleksandrov called these spaces “\mathfrak_k domain”. The terminology \operatorname(k) was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications). Definitions For a real number k, let M_k denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature k. Denote by D_k the diameter of M_k, which is \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometriae Dedicata
''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the Netherlands.. It is published by Springer Netherlands. The Editors-in-Chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The highest-ranking editor of a publication may also be titled editor, managing ... are John R. Parker and Jean-Marc Schlenker.Journal website References External links Springer site Mathematics journals Springer Science+Business Media academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Metric Space
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher- dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of that these two different types of definitions are equivalent. Hyperconvexity A metric space X is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is: #Any two points x and y can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. X is a path space). #If F is any family of closed balls _r(p) = \ such that each pair of balls in F meets, then there exists a point x common to all the balls in F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wasserstein Distance
In mathematics, the Wasserstein distance or Kantorovich– Rubinstein metric is a distance function defined between probability distributions on a given metric space M. It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on ''M'', the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of earth that needs to be moved times the mean distance it has to be moved. This problem was first formalised by Gaspard Monge in 1781. Because of this analogy, the metric is known in computer science as the earth mover's distance. The name "Wasserstein distance" was coined by R. L. Dobrushin in 1970, after learning of it in the work of Leonid Vaseršteĭn on Markov processes describing large systems of automata (Russian, 1969). However the metric was first defined by Leonid Kantorovich in ''The Mathematical Method of Production Planning and Organization'' (Russian original 1939) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultralimit
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and uses an ultrafilter to avoid the process of repeatedly passing to subsequences to ensure convergence. An ultralimit is a generalization of the notion of Gromov–Hausdorff convergence of metric spaces. Ultrafilters An ultrafilter ''ω'' on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and which, given any subset ''X'' of , contains either ''X'' or . An ultrafilter ''ω'' on is ''non-principal'' if it contains no finite set. Limit of a sequence of points with respect to an ultrafilter Let ''ω'' be a non-principal ultrafilter on \mathbb N . If (x_n)_ is a sequence of points in a metric space (''X'',''d'') a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
