n math space with the Euclidean distance , are completespacecompletemetric spaces. The rational number s with the same distance also form a metricspace, but are not complete. The positive real numbers with distance function math d x,y vert log y x vert math is a completemetricspace. Any normed vector space is a metricspace by defining math d x,y lVert y x rVert math , see also Metric 28mathematics 29 Relation of norms and metrics relation of norms and metrics . If such a space is Completemetricspacecomplete , we call it a Banach space . Examples The Norm mathematics Taxicab norm or Manhattan ... metric or supremum metric, and If math M math is complete, then this function space is complete as well ... in math M math has its limit in math A math . Types of metric spaces Complete spaces A metricspace math M math is said to be Completemetricspacecomplete if every Cauchy sequence converges in math ... value metric math d x,y vert x y vert math , are not complete. Every metricspace has a unique ..., a space is complete iff it is closed in any containing metricspace. Every completemetricspace ... space is itself compact. A metricspace is compact iff it is complete and totally bounded. This is known ... non empty compact subsets of M into a metricspace. One can show that K M is complete if M is complete ...In mathematics , a metricspace is a Set mathematics set where a notion of distance called a metric mathematics metric between elements of the set is defined. The metricspace which most closely corresponds ... geometries such as those used in the theory of general relativity . A metricspace also induces ... abstract topological space s. History Expand section Reasons for generalizing the Euclidean metric, first ... 22 1906 1 74. Definition A metricspace is an ordered pair math M,d math where math M math is a set ... writes math M math for a metricspace if it is clear from the context what metric is used. Examples of metric spaces Finite Metricspace redirects here Ignoring mathematical details, for any system of roads ... more details
up of rational numbers only. If however, math X, d math is a convex metricspace, and, in addition, it is Completemetricspacecomplete , one can prove that for any two points math x ne y math in math ... metricspacecomplete convex metricspace. Yet, if math x math and math y math are two points ...Image Convex metric illustration2.png right thumb An illustration of a convex metricspace. In mathematics , convex metric spaces are, intuitively, metricspace s with the property any segment joining two points in that space has other points in it besides the endpoints. Formally, consider a metricspace ... becomes an equality. A convex metricspace is a metricspace X , d such that, for any two distinct points x and y in X , there exists a third point z in X lying between x and y . Metric convexity does not imply convexity in the usual sense for subsets of Euclidean space see the example of the rational .... Image Circle as convex metric space.png right thumb A circle as a convex metricspace. Any convex set in a Euclidean space is a convex metricspace with the induced Euclidean norm. For closed set s the Contraposition ... distance is a convex metricspace, then it is a convex set this is a particular case of a more general statement to be discussed below . A circle is a convex metricspace, if the distance between ... Let math X, d math be a metricspace which is not necessarily convex . A subset math S math of math ... and math y math is between math x math and math y. math As such, if a metricspace math X, d math admits metric segments between any two distinct points in the space, then it is a convex metricspace. The Contraposition converse is not true, in general. The rational number s form a convex metricspace ... with a closed disc removed . Examples Euclidean spaces, that is, the usual three dimensional space and its analogues for other dimensions, are convex metric spaces. Given any two distinct points math x math and math y math in such a space, the set of all points math z math satisfying the above triangle ... more details
In metric geometry , an injective metricspace , or equivalently a hyperconvex metricspace , is a metricspace with certain properties generalizing those of the real line and of Chebyshev distance L sub sub distances in higher dimensional vector space s. 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 isometry isometric embeddings of the space into larger spaces. However ... types of definitions are equivalent. Hyperconvexity A metricspace is said to be hyperconvex if it is convex metric convex and its closed Ball mathematics balls have the binary Helly family Helly property ... sub j sub d p sub i sub , p sub j sub , then there is a point q of the metricspace that is within distance r sub i sub of each p sub i sub . Injectivity A retract metric geometry retraction of a metricspace X is a function &fnof mapping X to a subspace of itself, such that for all x , &fnof &fnof ... . A retract of a space X is a subspace of X that is an image of a retraction. A metricspace X ... sub , but not in higher dimensions The tight span of a metricspace Any real tree Aim X &ndash see Metricspace aimed at its subspace Due to the equivalence between hyperconvexity and injectivity, these spaces are all also injective. Properties In an injective space, the radius of the circumradius minimum ... of Jung s theorem . Every injective space is a completespace Aronszajn and Panitchpakdi 1956 , and every metric map or, equivalently, short map nonexpansive mapping, or short map on a bounded injective space has a Fixed point theorem fixed point Sine 1979 Soardi 1979 . A metricspace is injective if and only if it is an injective object in the category mathematics category of category of metric spaces ... , that subspace Z is a retract of Y . Examples Examples of hyperconvex metric spaces include The real line Any vector space R sup d sup with the Lp space L sub sub distance taxicab geometry Manhattan ... more details
Unreferenced date December 2009 In mathematics , a dilation is a function math f math from a metricspace into itself that satisfies the identity math d f x ,f y rd x,y , math for all points math x, y math where math d x, y math is the distance from math x math to math y math and math r math is some positive real number . In Euclidean space , such a dilation is a similarity geometry similarity of the space. Dilations change the size but not the shape of an object or figure. Every dilation of a Euclidean space that is not a Congruence geometry congruence has a unique fixed point that is called the center of dilation. Some congruences have fixed points and others do not. See also homothety Dilation operator theory DEFAULTSORT Dilation MetricSpace Category Metric geometry ... more details
A probabilistic metricspace is a generalization of metric spaces where the distance is no longer valued in positive real number s, but instead is valued in distribution functions. Let D be the set of all probability distribution function s F such that F 0 0 F is a nondecreasing, left continuous mapping from the real numbers R into 0, 1 such that sup F x 1 where the supremum is taken over all x in R . The ordered pair S , d is said to be a probabilistic metricspace if S is a nonempty set and d S S D In the following, d p , q is denoted by d sub p , q sub for every p , q S S and is a distribution function d sub p , q sub x . The distance distribution function satisfies the following conditions d sub u , v sub x 1 for all x 0 u v u , v S . d sub u , v sub x d sub v , u sub x for all x and for every u , v S . d sub u , v sub x 1 and d sub v , w sub y 1 d sub u , w sub x y 1 for u , v , w S and x , y R . See also Statistical distance refimprove date April 2012 References Chi Woo. probabilistic metricspace version 9 . PlanetMath.org. Freely available at http planetmath.org ProbabilisticMetricSpace.html. DEFAULTSORT Probabilistic MetricSpace Category Theory of probability distributions Category Metric geometry mathanalysis stub probability stub ... more details
cosmology cTopic Expanding universe The metric expansion of space is the increase of distance with time ... tensor metric seen on the left. Over time , the space that makes up the universe is expanding ... upon the properties of the space being discussed, the appropriate metric is mathematically established ... and what is its basis? date March 2012 Hubble s law Technically, the metric expansion of space ... These scientists however did not include Hubble himself. While the metric expansion of space reading ... the metric itself changed exponential growth exponentially , causing space to change from smaller ... math Omega m math . Measuring distance in a metricspace Main comoving coordinates In expanding .... These workings have led to models in which the metric expansion of space is a likely feature ..., metric expansion of space is considered by cosmologists to be an observed feature on the basis that although ... that rely on space expanding through a change in metric. Interestingly, it was not until the discovery ... that the Universe is Expanding Use dmy dates date May 2011 DEFAULTSORT Metric Expansion Of Space ... outward into preexisting space. The universe is not expanding into anything outside of itself ... apart from each other. Metric expansion is a key feature of Big Bang cosmology and is modeled mathematically with the Friedmann Lema tre Robertson Walker metric FLRW metric . This model is valid ... than the speed of light with respect to each other, there is no such theoretical constraint when space ... of space generated near the beginning of the Universe might still be arriving at distant locations hence ... . http arxiv.org abs astro ph 0310808 astro ph 0310808 ref Interpretations of the metric expansion of space are an ongoing subject of debate. ref name Whiting cite journal title The Expansion of Space Free Particle Motion and the Cosmological Redshift arxiv astro ph 0404095 year 2004 journal ... cite journal title Expanding Space The Root of Conceptual Problems of the Cosmological Physics author ... more details
Infobox book See Wikipedia WikiProject Novels or Wikipedia WikiProject Books name The Complete Book of Outer Space title orig translator image Image Complete book of outer space.jpg image caption Dust jacket from the first book publication author edited by Jeffrey Logan illustrator Frank R. Paul et al. cover artist Chesley Bonestell country United States language English language English series subject Space exploration publisher Gnome Press release date 1953 english release date media type Print Hardcover Hardback pages 144 pp isbn NA oclc 6824625 preceded by followed by The Complete Book of Outer Space is a 1953 in literature 1953 collection of essays about space exploration edited by Jeffrey Logan. It first appeared as a magazine, published by Maco Magazine Corp. The first book publication was by Gnome Press in 1953 in an edition of 3,000 copies. Contents Preface, by Kenneth MacLeish A Preview of the Future Introduction , by Jeffrey Logan Development of the Space Ship , by Willy Ley Station in Space , by Wernher von Braun Space Medicine , by Heinz Haber Space Suits , by Donald H. Menzel The High Altitude Program , by Robert P. Haviland History of the Rocket Engine , by James H. Wyld Legal Aspects of Space Travel , by Oscar Schachter Exploitation of the Moon , by Hugo Gernsback Life Beyond the Earth , by Willy Ley Interstellar Flight , by Leslie R. Shepard The Spaceship in Science Fiction , by Jeffrey Logan Plea for a Coordinated Space Program , by Wernher von Braun The Flying Saucer Myth , by Jeffrey Logan The Panel of Experts Chart of the Moon Voyage Chart of the Voyage to Mars Timetables and Weights A Space Travel Dictionary References Cite book last Chalker first Jack L. authorlink Jack L. Chalker coauthors Mark Owings title The Science Fantasy Publishers A Bibliographic History, 1923 1998 location Westminster, MD and Baltimore publisher Mirage Press, Ltd. pages 299 date 1998 DEFAULTSORT Complete Book Of Outer Space, The Category 1953 books Category Spaceflight ... more details
Unreferenced stub auto yes date December 2009 In quantum chemistry , a complete active space is a type of classification of molecular orbitals . Spatial orbitals are classified as belonging to three classes core , always hold two electrons active , partially occupied orbitals virtual , always hold zero electrons This classification allows to develop a set of Slater determinant s for the description of the wavefunction as a linear combination of these determinants. Based on the freedom left for the occupation in the active orbitals, a certain number of electrons are allowed to populate all the active orbitals in appropriate combinations, developing a finite size space of determinants. The resulting wavefunction is of Multireference configuration interaction multireference nature, and is blessed by additional properties if compared to other selection schemes. The active classification can theoretically be extended to all the molecular orbitals, to obtain a full CI treatment. In practice, this choice is limited, due to the high computational cost needed to optimize a large CAS wavefunction on medium and large molecular systems. A Complete Active Space wavefunction is used to obtain a first approximation of the so called static correlation , which represents the contribution needed to describe bond dissociation processes correctly. This requires a wavefunction that includes a set of electronic configurations with high and very similar importance. Dynamic correlation , representing the contribution to the energy brought by the instantaneous interaction between electrons, is normally small and can be recovered with good accuracy by means of perturbative evaluations, such as CASPT2 and NEVPT . See also Multi configurational self consistent field Complete Active Space SCF CASSCF ... Active Space SCF RASSCF DEFAULTSORT Complete Active Space Category Quantum chemistry Chem stub it Complete active space ja ... more details
In mathematics , a metricspace aimed at its subspace is a category theory categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope , or tight span , which are basic injective objects of the category of metricspace s. Following harv Holszty ski 1966 , a notion of a metricspace Y aimed at its subspace X is defined. Informally ... , which in a sense of canonical isometric embedding s contains any other space aimed at an isometric image of X . And in the special case of an arbitrary compact metricspace X every bounded subspace of an arbitrary metricspace Y aimed at X is totally bounded i.e. its metric completion is compact . Definitions Let math Y, d math be a metricspace. Let math X math be a subset of math Y math , so that math X,d X 2 math the set math X math with the metric from math Y math restricted to math X math is a metric subspace of math Y,d math . Then Definition . Space math Y math aims at math X math ... Let math text Met X math be the space of all real valued metric map s non contractive of math X math ... a generalisation of the Kuratowski Wojdys awski embedding of bounded metric spaces math X math into math C X math , where we here consider arbitrary metric spaces bounded or unbounded . It is clear that the space ... d f,g sup x in X f x g x infty math for every math f, g in text Aim X math is a metric on math ... X to Y math be an isometric embedding. Then there exists a natural metric map math j colon Y to operatorname ... x in X , math and math y in Y , math . Theorem The space Y above is aimed at subspace X if and only ... that every space aimed at X can be isometrically mapped into Aim X , with some additional essential categorical requirements satisfied. The space Aim X is injective metricspace injective hyperconvex in the sense of Aronszajn Panitchpakdi given a metricspace M, which contains Aim X as a metric subspace, there is a canonical and explicit metric retraction of M onto Aim X harv Holszty ski 1966 ... more details
Wiktionary metricMetric s may refer to the metric system of measurement International System of Units , or Syst me International SI , the modern form of the metric system Metric ton , a measurement of mass equal to 1,000 kg an analytical measurement intended to quantify the state of a system. For example population density is one metric which may be used to describe a city. Metric mathematics , an abstraction of the notion of distance in a metricspaceMetric tensor , in mathematics, a symmetric rank 2 tensor, used to measure length and angle Metric band , a Canadian indie rock band Metrics networking , set of properties of a communication path Font Metrics Font metrics , a group of properties describing a font Reuse metrics , a quantitative indicator of an attribute for software reuse and reusability Router metrics , used by a router to make routing decisions Software metric s, a measure of some property of a piece of software or its specifications Performance metric s, a measure of an organization s activities and performance. METRIC a computer model Mapping EvapoTranspiration at high Resolution with Internalized Calibration that uses Landsat satellite data to compute and map evapotranspiration ET . See also Meter disambiguation Units of measurement Metric expansion of space disambiguation cs Metrika da Metrik de Metrik el es M trica desambiguaci n eo Metriko fr M trique pl Metryka ru sk Metrika sr sv Metrik uk ... more details
Other uses Metric disambiguation MetricMETRIC is a computer model Mapping EvapoTranspiration at high Resolution with Internalized Calibration that uses Landsat satellite data to compute and map evapotranspiration ET developed by Richard Allen et.al. at the University of Idaho . ref Idaho Department of Water Resources Mapping Evapotranspiration http www.idwr.idaho.gov GeographicInfo METRIC et.htm ref Climate Change is a world wide problem that affects every country on earth. Understanding the processes and factors that control or affect ecosystem response to climate change is essential to mitigate impacts. Studying the regional to global radiation balance, hydrologic energy, water and carbon dioxide fluxes is important to understand effects of climate change on ecosystem health and evolution, including impacts of invasive species such as cheatgrass bromus tectorum in the western US. The exchanges of water and carbon at the plant atmosphere interface are coupled through the active control of leaf and needle stomata on the exchange of gas into or out of leaves with the atmosphere and by effects of direct evaporation from soil. Given the importance of carbon uptake and vapor outputs, the ability to quantify the uptake is important, particularly in large areas with significant capability to assimilate carbon. In the upper Snake River system of Idaho, the most common ecosystems, besides farms, are alpine forest, sage brush, invasive cheatgrass and bunch grass. A primary goal of this research is to contribute toward a better understanding and methods of quantifying the magnitude, timing, distribution and coupling of carbon, energy and water fluxes in these three dominant natural ecosystems, including effects of burning, and improve the accuracy of modeling sensible heat flux H ..., hydrosphere, atmosphere, and biosphere. METRIC helps us estimate surface energy flux s on a large ... that why METRIC is so valuable because it can produce ET maps that have values across the entire landscape ... more details
wiktionarypar complete compl te To be complete is to be in the state of requiring nothing else to be added. Complete may also refer to Complete Lila McCann album Complete Lila McCann album Complete News from Babel album Complete News from Babel album Complete complexity , in mathematics Completemetricspace , in mathematics Complete , a song by Kutless from To Know That You re Alive Complete , a ballad song from Girls Generation s 2007 debut album Girls Generation 2007 album Girls Generation See also Completely disambiguation Completeness disambig ... more details
is defined included Finsler manifold s and sub Riemannian manifold s. Any completemetricspacecomplete and convex metricspace is a length metricspace harv Khamsi Kirk 2001 loc Theorem 2.16 , a result ...In the mathematics mathematical study of metric spaces , one can consider the arclength of paths in the space ... to that distance. The distance between two points of a metricspace relative to the intrinsic metric is defined as the infimum of the length of all paths from one point to the other. A metricspace is a length metricspace if the intrinsic metric agrees with the original metric of the space. Definitions Let math M, d , math be a metricspace . We define a new metric math d I , math on math M , math , known as the induced intrinsic metric , as follows math d I x,y , math is the infimum of the lengths ... M, d , math is a length space or a path metricspace and the metric math d , math is intrinsic . We say that the metric math d , math has approximate midpoints if for any math varepsilon 0 math ... 2 varepsilon math . Examples Euclidean space R sup n sup with the ordinary Euclidean metric is a path metricspace. R sup n sup 0 is as well. The unit circle S sup 1 sup with the metric inherited from the Euclidean metric of R sup 2 sup the chordal metric is not a path metricspace. The induced intrinsic metric on S sup 1 sup measures distances as angle s in radian s, and the resulting length metricspace is called the Riemannian circle . In two dimensions, the chordal metric on the sphere is not intrinsic ... manifold can be turned into a path metricspace by defining the distance of two points as the infimum ... by d . The space M , d sub l sub is always a path metricspace with the caveat, as mentioned above, that d sub l sub can be infinite . The metric of a length space has approximate midpoints. Conversely, every completespacecompletemetricspace with approximate midpoints is a length space. The Hopf Rinow theorem states that if a length space math M,d math is completespacecomplete and locally ... more details
Germanic origin . Definition Let M , d be a metricspace for which every probability measure on M is a Radon measure a so called Radon space . For p 1, let P sub p sub M denote the collection of all probability measures on M with Moment mathematics Moments in metric spaces finite ...In mathematics , the Wasserstein or Vasershtein metric is a metric mathematics distance function defined between probability measure probability distribution s on a given metricspace M . Intuitively, if each distribution is viewed as a unit amount of dirt piled on M , the metric is the minimum cost ... times the distance it has to be moved. Because of this analogy, the metric is known in computer science ... sub p sub notation. The Wasserstein metric may be equivalently defined by math W p mu, nu p inf ... and respectively. Applications The Wasserstein metric is a natural way to compare the probability ... uniform perturbations random or deterministic . In computer science, for example, the metric W sub ... images . Properties Metric structure It can be shown that W sub p sub satisfies all the axiom s of a metric mathematics metric on P sub p sub M . Furthermore, convergence with respect to W sub p sub ... Radon metric math rho mu, nu sup left left. int M f x , mathrm d mu nu x right mbox continuous f M to 1, 1 right . math If the metric d is bounded by some constant C , then math 2 W 1 mu, nu leq C rho mu, nu , math and so convergence in the Radon metric also known as strong convergence implies convergence in the Wasserstein metric, but not vice versa. Separability and completeness For any p 1, the metricspace P sub p sub M , W sub p sub is Separable space separable , and is Completespacecomplete if M , d is separable and complete. See also L vy metric L vy Prokhorov metric Transportation theory References cite book author Ambrosio, L., Gigli, N. & Savar , G. title Gradient Flows in Metric Spaces and in the Space of Probability Measures publisher ETH Z rich, Birkh user Verlag location ... more details
In mathematics , the term metric dimension has various meanings. The Metric dimension graph theory metric dimension of an undirected graph G is the minimum number of vertices in a subset S of G such that all other vertices are uniquely determined by their distances to the vertices in S . The Minkowski Bouligand dimension also called the metric dimension is a way of determining the dimension of a fractal set in a Euclidean space by counting the number of fixed size boxes needed to cover the set as a function of the box size. The equilateral dimension of a metricspace also called the metric dimension is the maximum number of points at equal distances from each other. The Hausdorff dimension is an Extended real number line extended non negative real number associated with any metricspace that generalizes the notion of the dimension of a real vector space. mathdab ... more details
, pp. 1203 1225 ref See also Acoustic metric Apophysis software Completemetric Fractal compression Fractal image compression Image differencing Metric tensor Multifractal system Sources and notes ... of the Hutchinson Metric Between Digitized Images Category Metric geometry Category Topology ... more details
by the others. A metric is called an ultrametric space ultrametric if it satisfies the following ... Main Metricspace Examples of metric spaces The discrete space discrete metric if x y then d x , y 0. Otherwise, d x , y 1. The Euclidean metric is translation and rotation invariant. The Taxicab geometry taxicab metric is translation invariant. More generally, any metric induced by a norm .... The Fubini Study metric on complex projective space . Equivalence of metrics For a given set X , two ... to math d. math See also Metricspace Notions of metricspace equivalence notions of metric ... words, every norm determines a metric, and some metrics determine a norm. Given a normed vector space ... by the norm math cdot math . Conversely if a metric d on a vector space X satisfies the properties ... metricspace, we get a pseudosemimetric, i.e. a symmetric premetric. Any premetric gives rise to a preclosure ... behaved of the metricspace categories. One can take arbitrary products and coproducts and form quotient ... metric functions. They are defined as inner product s on the tangent space with an appropriate ... is also called the invariant distance . See also Acoustic metricCompletemetric Notes references ...In mathematics , a metric or distance function is a function mathematics function which defines a distance between elements of a Set mathematics set . A set with a metric is called a metricspace . A metric induces a topology on a set but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable . In differential geometry , the word metric is also used to refer to a structure defined only on a differentiable manifold which is more properly termed a metric tensor or Riemannian or pseudo Riemannian metric . Definition A metric ... x , y , z in X , d x , z max d x , y , d y , z A metric d on X is called intrinsic metric intrinsic ... invariance translation invariant metric if d x , y d x a , y a for all x , y and a in X . Notes ... more details
In the mathematics mathematical theory of metricspace s, a metric map is a Function mathematics function between metric spaces that does not increase any distance such functions are always continuous function continuous . These maps are the morphism s in the category of metric spaces , Met Isbell 1964 . They are also called Lipschitz continuity Lipschitz functions with Lipschitz constant 1, nonexpansive maps , nonexpanding maps , weak contractions , or short maps . Specifically, suppose that X and Y are metric spaces and is a function mathematics function from X to Y . Thus we have a metric map when, for any points x and y in X , math d Y f x ,f y leq d X x,y . math Here d sub X sub and d sub Y sub denote the metrics on X and Y respectively. A map between metric spaces is an isometry if and only if 1 it is metric, 2 it is a bijection , and 3 its inverse functions inverse is also metric. The composite function composite of metric maps is also metric. Thus metric spaces and metric maps form a category theory category Category of metric spaces Met Met is a subcategory of the category of metric spaces and Lipschitz functions, and the isomorphism s in Met are the isometries. One can say that is strictly metric if the inequality mathematics inequality is strict for every two different points. Thus a contraction mapping is strictly metric, but not necessarily the other way around. Note that an isometry is never strictly metric, except in the degeneracy mathematics degenerate case of the empty set empty space or a single point space. References cite journal author Isbell, J. R. authorlink John R. Isbell title Six theorems about injective metric spaces journal Comment. Math. Helv. volume 39 year 1964 pages 65 76 url http www.digizeitschriften.de resolveppn GDZPPN002058340 doi 10.1007 BF02566944 Category Metric geometry Category Lipschitz maps Geometry stub es Funci n corta fr Application non expansive it Funzione non espansiva pl Odwzorowanie nierozszerzaj ce pt Fun o ... more details
In mathematics, the concept of a generalised metric is a generalisation of that of a metricspacemetric , in which the distance is not a real number but taken from an arbitrary ordered field . In general, when we define metricspace the distance function is taken to be a real valued function mathematics function . The real numbers form an ordered field which is archimedean property Archimedean and complete ordered field order complete . So, the metric spaces have some nice properties like in a metricspace compactness, sequential compactness and countable compactness are equivalent etc etc. These properties may not, however, hold so easily if the distance function is taken in an arbitrary ordered field, instead of in math scriptstyle mathbb R math . Preliminary definition Let math F, , cdot, math be an arbitrary ordered field, and math M math a nonempty set a function math d M times M to F cup 0 math is called a metric on math M math , iff the following conditions hold math d x,y 0 Leftrightarrow x y math math d x,y d y,x math , commutativity math d x,y d y,z le d x,z math , triangle inequality. It is not difficult to verify that the open balls math B x, delta y in M d x,y delta math form a basis for a suitable topology, the latter called the metric topology on math M math , with the metric in math F math . In view of the fact that math F math in its order topology is monotonically normal , we would expect math M math to be at least Regular space regular . Further properties However, under axiom of choice , every general metric is monotonically normal , for, given math x in G math , where math G math is open, there is an open ball math B x, delta math such that math x in B x, delta subseteq G math . Take math mu x,G B x, delta 2 math . Verify the conditions for Monotone ... . We would show that with respect to this mu operator, the space is monotonically normal. Note that math ... Category Metric geometry ... more details
In mathematics , the metric derivative is a notion of derivative appropriate to Parametric equation parametrized path topology paths in metricspace s. It generalizes the notion of speed or absolute velocity to spaces which have a notion of distance i.e. metric spaces but not direction such as vector space s . Definition Let math M, d math be a metricspace. Let math E subseteq mathbb R math have a limit point at math t in mathbb R math . Let math gamma E to M math be a path. Then the metric derivative of math gamma math at math t math , denoted math gamma t math , is defined by math gamma t lim s to 0 frac d gamma t s , gamma t s , math if this Limit mathematics limit exists. Properties Recall that absolute continuity AC sup p sup I X is the space of curves I X such that math d left gamma s , gamma t right leq int s t m tau , mathrm d tau mbox for all s, t subseteq I math for some m in the Lp space L sup p sup space L sup p sup I R . For AC sup p sup I X , the metric derivative of exists for Lebesgue measure Lebesgue almost all times in I , and the metric derivative is the smallest m L sup p sup I R such that the above inequality holds. If Euclidean space math mathbb R n math is equipped with its usual Euclidean norm math math , and math dot gamma E to V math is the usual Fr chet derivative with respect to time, then math gamma t dot gamma t , math where math d x, y x y math is the Euclidean metric. References cite book author Ambrosio, L., Gigli, N. & Savar , G. title Gradient Flows in Metric Spaces and in the Space of Probability Measures publisher ETH Z rich, Birkh user Verlag, Basel year 2005 isbn 3 7643 2428 7 Category Differential calculus Category Metric geometry ... more details
In mathematics , the product metric is a definition of metric mathematics metric on the Cartesian product of two metric spaces . As described below, the p product metric of the Cartesian product of n metric spaces is the Lp space p norm of the n vector of the norms of the n subspaces math d p mathbf x 1, dots, mathbf x n d 1 mathbf x 1 , dots, d n mathbf x n p math Definition Let math X, d X math and math Y, d Y math be metric spaces and let math 1 leq p leq infty math . Define the math p math product metric math d p math on math X times Y math by math d p left x 1 , y 1 , x 2 , y 2 right left d X x 1 , x 2 p d Y y 1 , y 2 p right 1 p math for math 1 leq p infty math math d infty left x 1 , y 1 , x 2 , y 2 right max left d X x 1 , x 2 , d Y y 1 , y 2 right . math for math x 1 , x 2 in X math , math y 1 , y 2 in Y math . Choice of norm For Euclidean space s, using the L sub 2 sub norm gives rise to the Euclidean metric in the product space however, any other choice of p will lead to a topologically equivalent metricspace. In the category of metric spaces , the sup norm is used. References citation last1 Deza first1 Michel Marie author1 link Michel Deza last2 Deza first2 Elena page 83 publisher Springer Verlag title Encyclopedia of Distances url http books.google.com books?id LXEezzccwcoC&pg PA83 year 2009 . DEFAULTSORT Product Metric Category Metric geometry ... more details
In mathematics , the L vy metric is a metric mathematics metric on the space of cumulative distribution function s of one dimensional random variable s. It is a special case of the L vy Prokhorov metric , and is named after the France French mathematician Paul Pierre L vy . Definition Let math F, G mathbb R to 0, infty math be two cumulative distribution functions. Define the L vy distance between them to be math L F, G inf varepsilon 0 F x varepsilon varepsilon leq G x leq F x varepsilon varepsilon mathrm ,for ,all , x in mathbb R . math Intuitively, if between the graphs of F and G one inscribes squares with sides parallel to the coordinate axes at points of discontinuity of a graph vertical segments are added , then the side length of the largest such square is equal to L F , G . See also C dl g L vy Prokhorov metric Wasserstein metric References springer author V.M. Zolotarev id l l058310 title L vy metric Category Measure theory Levy metric Category Metric geometry Levy metric Category Probability theory Levy metric ... more details
This article is about the concept in Riemannian geometry . For the typographic concept, see Typeface Font metrics . Merge to metric connection discuss Talk metric connection merge metric compatibility here? date September 2011 In mathematics, given a metric tensor math g ab math , a covariant derivative is said to be compatible with the metric if the following condition is satisfied math nabla c , g ab 0. math Although other covariant derivatives may be supported within the metric, usually one only ever considers the metric compatible one. This is because given two covariant derivatives, math nabla math and math nabla math , there exists a tensor for transforming from one to the other math nabla a x b nabla a x b C ab c x c. math If the space is also torsion tensor torsion free , then the tensor math C ab c math is symmetric in its first two indices. References cite arxiv last Rodrigues first W. A. last2 Fern ndez first2 V. V. last3 Moya first3 A. M. year 2005 title Metric compatible covariant derivatives eprint math 0501561 Citation last Wald first Robert M. title General Relativity book General Relativity publisher University of Chicago Press year 1984 isbn 0 226 87033 2 Category Differential geometry Category Riemannian geometry Relativity stub ... more details
About the data structure the type of metricspace Real tree A metric tree is any tree data structure tree data structure specialized to index data in metricspace s. Metric trees exploit properties of metric spaces such as the triangle inequality to make accesses to the data more efficient. Examples include the M tree , vp tree s, cover tree s, MVP Tree s, and bk tree s. ref name Samet cite book last Samet first Hanan title Foundations of multidimensional and metric data structures year 2006 publisher Morgan Kaufmann isbn 978 0 12 369446 1 url http books.google.dk books?id KrQdmLjTSaQC ref should have a list and summary of metric trees, with links to the main articles. Multidimensional search Most algorithms and data structures for searching a dataset are based on the classical binary search algorithm, and generalizations such as the k d tree or range tree work by interleaving the binary search algorithm over the separate coordinates and treating each spatial coordinate as an independent search constraint. These data structures are well suited for range query problems asking for every point math x,y math that satisfies math mbox min x leq x leq mbox max x math and math mbox min y leq y leq mbox max y math . A limitation of these multidimensional search structures is that they are only defined for searching over objects that can be treated as vectors. They aren t applicable for the more ... to a given query image. Metric data structures If there is no structure to the similarity measure .... The first article on metric trees, as well as the first use of the term metric tree , published ... title Satisfying General Proximity Similarity Queries with Metric Trees journal Information Processing ... data structures, and research on metric tree data structures blossomed in the late 1990s and included ... conference last Brin first Sergey title Near Neighbor Search in Large Metric Spaces booktitle 21st International Conference on Very Large Data Bases VLDB date 1995 ref The first textbook on metric ... more details
Orphan date September 2011 Refimprove date September 2011 In mathematical analysis , a metric differential is a generalization of a derivative for a Lipschitz continuity Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metricspace . With this definition of a derivative, one can generalize Rademacher s theorem Rademarcher s theorem to metricspace valued Lipschitz functions. Discussion Rademacher s theorem states that a Lipschitz map f R sup n sup R sup m sup is differentiable amost everywhere in R sup n sup in other words, for almost every x , f is approximately linear when you look in a small enough neighborhood of x . If f is a function from a Euclidean space R sup n sup that takes values instead in a metricspace X , it doesn t immediately make sense to talk about differentiability since X has no linear structure a priori. Even if you assume that X is a Banach space and ask whether a Fr chet derivative exists almost everywhere, this does not hold. For example, consider the function f 0,1 L sup 1 sup 0,1 , mapping the unit interval into the Lp spacespace of integrable functions , defined by f x sub 0, x sub , this function is Lipschitz and in fact, an isometry since, if 0 ... as you zoom in on almost every point, then such a theorem exists but is stated in terms of the metric properties of f instead of its linear properties. Definition and existence of the metric differential A substitute for a derivative of f R sup n sup X is the metric differential of f ... r rightarrow 0 frac d X f z rx ,f z r math whenever the limit exists here d sub X sub denotes the metric ... Bernd Kirchheim title Rectifiable metric spaces local structure and regularity of the Hausdorff measure ... theorem in terms of metric differentials holds for almost every z in R sup m sup , MD f , ... an isometry from R sup n sup with respect to the seminorm MD f , z into the metric space ... more details
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