Metric time is the measure of time interval using the metric system , which defines the second as the base unit of time, and multiple and submultiple units formed with SI prefix metric prefixes , such as kiloseconds ... scales, which may be based upon the metric definition of the second. Other units of time, the minute , hour , and day , are accepted for use with the SI modern metric system , but are not part of it. History When the metric system was introduced in France in 1795, it included units for length, area ... of day had been introduced in France two years earlier, but was set aside at the same time the metric system was inaugurated, and did not follow the metric pattern of a base unit and prefixed units. James ... and magnetic metric units, following the recommendation of Carl Friedrich Gauss in 1832. The ephemeris ... metric system, or International System of Units SI , at the 10th Conf rence G n rale des Poids et ... see also Decimal time Numerous proposals have been made for alternative base units of metric time. On March 28, 1794, the president of the commission which developed the metric system , Joseph Louis ..., meck, chi, chron, moment, etc., and multiple and submultiple units formed with metric prefixes ... it was a decimal multiple, not a sexagesimal one . Adopting a metric time system would suggest that a new base unit not named a second would be more consistent with the base 10 structure of the metric ... unit of metric time, but the proposal did not gain acceptance and was eventually abandoned. ref ... Louis Galison ref Alternative meaning Metric time is sometimes used to mean decimal time . Metric ... upon the metric base unit of time, the second. Some proposals for alternative units of metric time .... Other proposals called metric time refer only to decimal time, and therefore are not truly metric. France French decimal time is sometimes called metric time because it was introduced around the same time as the metric system and both were decimal, but it was not part of the decree creating the original ... more details
dablink This article is about the measurement of performance. For metric units, see Metric system and International System of Units . For disambiguous use, see Metric disambiguation A metric is a standard unit of measure , such as meter or mile for length, or gram or ton for weight, or more generally, part of a system of parameters, or systems of measurement , or a set of ways of for quantitatively and periodically measuring, assessing, controlling or selecting a person, process, event, or institution, along with the procedures to carry out measurements and the procedures for the interpretation of the assessment in the light of previous or comparable assessments. Metrics are usually specialized by the subject area, in which case they are valid only within a certain domain and cannot be directly benchmarked or interpreted outside it. This factor severely limits the applicability of metrics, for instance in comparing performance across domains. The prestige attached to them may be said to relate to a quantifiability fallacy , the erroneous belief that if a conclusion is reached by quantitative measurement, it must be vindicated, irrespective of what parameters or purpose the investigation is supposed to have. In business, they are sometimes referred to as key performance indicators , such as overall equipment effectiveness , or key risk indicators . In the field of Facilities Management, a key metric is the Facility Condition Index , or FCI . For a measure to be a metric it has to satisfy four properties 1 non negativity, 2 reflexivity, 3 symmetry, and 4 triangular inequality ref name AG S. Theodoridis and K. Koutroumbas, Pattern Recognition , Fourth Edition, Academic Press, 2009, p. 602. ref references See also Indicator Measure mathematics Measurement Metric mathematics Metrics networking Software metric Category Metrics de Metrik ... more details
The Vaidya metric describes exterior gravitational field due to a radiating star. The metric was proposed by P. C. Vaidya in 1943. It is a non static generalization of the Schwarzschild metric . math ds 2 left 1 frac 2M u r right du 2 2dudr r 2 left d theta 2 sin 2 theta d varphi 2 right math where M u is the mass parameter. References T. Padmanabhan 2010 . http books.google.com books?id BSfe2MjbQ3gC Gravitation Foundations and Frontiers . Cambridge University Press. ISBN 0521882230. pp. 313 314. Category Exact solutions in general relativity Category Gravitation Category Astrophysics ... more details
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 metric space . 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 ... by the others. A metric is called an ultrametric space ultrametric if it satisfies the following ... 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 ... Main Metric space 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 ... x y 1 p n x y math is a metric defining the same topology . One can replace math frac 1 2 n math by any absolute convergence summable sequence math a n math of strictly positive number s. Graph metric , a metric defined in terms of distances in a certain graph. The Hamming distance in coding theory. The Fubini Study metric on complex projective space . Equivalence of metrics For a given set X , two ..., if math d math is a metric, then math min d, 1 math and math d over 1 d math are metrics equivalent to math d. math See also Metric space Notions of metric space equivalence notions of metric ... words, every norm determines a metric, and some metrics determine a norm. Given a normed vector space math X, cdot math we can define a metric on X by math d x,y x y math . The metric d is said to be induced ... more details
In mathematics and computer science , a string metric also known as a string similarity metric or string distance function is a metric mathematics metric that measure similarity or dissimilarity distance between two string computer science text strings for approximate string matching or comparison and in fuzzy string searching . For example the strings Sam and Samuel can be considered to be similar. A string metric provides a number indicating an algorithm specific indication of similarity. The most widely known string metric is a rudimentary one called the Levenshtein distance Levenshtein Distance also known as Edit Distance . It operates between two input strings, returning a score equivalent to the number of substitutions and deletions needed in order to transform one input string into another. Simplistic string metrics such as Levenshtein distance have expanded to include phonetic, token parser token , grammatical and character based methods of statistical comparisons. A widespread example of a string metric is DNA sequence analysis and RNA analysis, which are performed by optimised string metrics to identify matching sequences. String metrics are used heavily in information integration and are currently used in areas including fraud detection , fingerprint analysis , plagiarism detection , ontology merging , DNA analysis , RNA analysis, image analysis , evidence based machine learning, database data deduplication , data mining , Web interfaces, e.g. Ajax programming Ajax style suggestions as you type, data integration , and semantic knowledge integration . List of string ... Soundex distance metric Matching coefficient Dice s coefficient Jaccard similarity or Jaccard coefficient ... Jensen Shannon divergence Harmonic mean Skew divergence Confusion probability Tau metric , an approximation of the Kullback Leibler divergence Fellegi and Sunters metric SFS TFIDF or TF IDF Maximal ... http www.dcs.shef.ac.uk sam stringmetrics.html ukkonen DEFAULTSORT String Metric Category String ... more details
, pp. 1203 1225 ref See also Acoustic metric Apophysis software Complete metric 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
cleanup date January 2012 refimprove date January 2012 Infobox musical artist name Metric image Metric ... br Post punk revival years active 1998&ndash present label Metric Music International br Last Gang ... past members Metric is a Canadian indie rock ref name autogenerated1 and New Wave music New Wave band ... , New York City and Los Angeles . Metric consists of vocalist Emily Haines who also plays the synthesizer ... network.nationalpost.com np blogs theampersand archive 2009 03 12 250672.aspx title Metric moves up ... Awards . Metric won as well in 2010 Juno Award for Group of the Year Group of the Year . History ... title Metric Music Podcast work Access All Areas date 7 September 2009 ref and AAA Shaw said ... to eat. In 1999, Shaw and Haines decided to call their partnership Metric , after a synthesizer beat ... in this period and offered to bring Metric to the UK for a possible record deal. Eager ... acts as Erasure , New Order , and the Pet Shop Boys , Metric worked on a new batch of New Wave ... in the years ahead with their musical mix. Although Metric already had a semi mainstream appeal, they felt ... Metric continued to work on their debut album in the first few months of 2001. Now dubbed Grow Up ... . Image Jimmy Shaw of Metric.jpg thumb right upright Jimmy Shaw of MetricMetric had also gained ... a decade and they met Haines and Shaw at the Brooklyn loft and at local performances. Metric at the time ... show at Black Betty s in Williamsburg. Metric performed in New York in early Spring and late Summer ..., eventually giving the music away for free on the internet. Metric received their first major ... 2003 2004 In 2003, Metric released their first official album Old World Underground, Where Are You .... The album was produced by Michael Andrews Metric was featured as themselves in the 2004 independent ... also features songs by Brian Eno , Daniel Lanois , Emily Haines , Metric and Tricky . Live It Out and Grow Up and Blow Away 2005 2007 Two years later, Metric released their second studio album, Live ... more details
In mathematics, the original Kobayashi metric is a pseudometric space pseudometric or pseudodistance on complex manifold s introduced by harvs txt authorlink Shoshichi Kobayashi last Kobayashi year 1967 . It can be viewed as the dual mathematics dual of the Carath odory metric , and has been extended to complex analytic space s and almost complex manifold s. On Teichm ller space the Kobayashi metric coincides with the Teichm ller metric on the unit ball, it coincides with the Bergman metric . An analogous pseudodistance was constructed for flat affine and projective structures in harvs txt authorlink Shoshichi Kobayashi last Kobayashi year 1977 and then generalized to normal projective connection s. Essentially the same construction has been applied to normal, pseudo Riemannian conformal connection s and, more recently, to general regular parabolic geometries. Definition If X is a complex manifold, the Kobayashi pseudometric d may be characterized as the largest pseudometric on X such that math d f x ,f y le rho x,y math , for all holomorphic maps f from the unit disk D to X where math rho x,y math denotes distance in the Poincar metric on D . References Cite journal last Kobayashi first Shoshichi title Intrinsic Metrics on Complex Manifolds journal Bull. Amer. Math. Soc. year 1967 volume 73 pages 347 349 url http www.ams.org journals bull 1967 73 03 S0002 9904 1967 11745 2 S0002 9904 1967 11745 2.pdf Citation last1 Kobayashi first1 Shoshichi title Hyperbolic manifolds and holomorphic mappings url http books.google.com books?id rleQdMhML6kC publisher Marcel Dekker Inc. location New York series Pure and Applied Mathematics isbn 978 0 8247 1380 5 mr 0277770 year 1970 volume 2 cite journal last Shoshichi first Kobayashi title Intrinsic distances associated with flat affine or projective structures journal J. Fac. Sci. Univ. Tokyo year 1977 volume 24 pages 129 135 mr 445016 Category Complex manifolds ... more details
General relativity cTopic Exact solutions in general relativity Solutions The Kasner metric is an Exact solutions in general relativity exact solution to Einstein s theory of general relativity . It describes an anisotropic universe without matter i.e., it is a vacuum solution . It can be written in any spacetime dimension math D 3 math and has strong connections with the study of gravitational Chaos theory chaos . The Metric and Kasner Conditions The Metric mathematics metric in math D 3 math spacetime dimensions is math text d s 2 text d t 2 sum j 1 D 1 t 2p j text d x j 2 math , and contains math D 1 math constants math p j math , called the Kasner exponents. The metric describes a spacetime whose equal time slices are spatially flat, however space is expanding or contracting at different rates in different directions, depending on the values of the math p j math . Test particles in this metric whose comoving coordinate differs by math Delta x j math are separated by a physical distance math t p j Delta x j math . The Kasner metric is an exact solution to Einstein s equations in vacuum when the Kasner exponents satisfy the following Kasner conditions, math sum j 1 D 1 p j 1, math math sum j 1 D 1 p j 2 1. math The first condition defines a plane geometry plane , the Kasner plane, and the second describes a sphere , the Kasner sphere. The solutions choices of math p j math satisfying the two conditions therefore lie on the sphere where the two intersect sometimes confusingly ... lie on a math D 3 math dimensional sphere math S D 3 math . Features of the Kasner Metric There are several ... Isotropic metric expansion of space expansion or contraction of space is not allowed. If the spatial ... be satisfied, for math sum j 1 D 1 p j 2 frac 1 D 1 ne 1. math The FRW metric employed in cosmology ... is actually contracting. The Kasner metric is a solution to the vacuum Einstein equations, and so ... in general relativity Category Metric tensors fr M trique de Kasner it Metrica di Kasner ... more details
In game theory , the Helly metric is used to assess the distance between two Strategy game theory strategies . It is named for Eduard Helly . Consider a game math Gamma left langle mathfrak X , mathfrak Y ,H right rangle math , between player I and II. Here, math mathfrak X math and math mathfrak Y math are the sets of pure strategy pure strategies for players I and II respectively and math H H cdot, cdot math is the payoff function. in other words, if player I plays math x in mathfrak X math and player II plays math y in mathfrak Y math , then player I pays math H x,y math to player II . The Helly metric math rho x 1,x 2 math is defined as math rho x 1,x 2 sup y in mathfrak Y left H x 1,y H x 2,y right . math The metric so defined is symmetric, reflexive, and satisfies the triangle inequality . The Helly metric measures distances between strategies, not in terms of the differences between the strategies themselves, but in terms of the consequences of the strategies. Two strategies are distant if their payoffs are different. Note that math rho x 1,x 2 0 math does not imply math x 1 x 2 math but it does imply that the consequences of math x 1 math and math x 2 math are identical and indeed this induces an equivalence relation . If one stipulates that math rho x 1,x 2 0 math implies math x 1 x 2 math then the topology so induced is called the natural topology . The metric on the space of player II s strategies is analogous math rho y 1,y 2 sup x in mathfrak X left H x,y 1 H x,y 2 right . math Note that math Gamma math thus defines two Helly metrics one for each player. Conditional ... epsilon math net in the space math X math with metric math rho math if for any math x in X math there exists math x epsilon in X epsilon math with math rho x,x epsilon epsilon math . A metric space ... math net in math P math . A game that is conditionally compact in the Helly metric has an math ... compact in their Helly metric . References N. N. Vorob ev 1977. Game theory lectures for economists ... more details
In mathematics , the Hilbert metric , also known as the Hilbert projective metric , is an explicitly defined metric mathematics distance function on a bounded convex set convex subset of the n dimensional Euclidean space R sup n sup . It was introduced by harvs txt authorlink David Hilbert first David last Hilbert year 1895 as a generalization of the Cayley metric Cayley s formula for the distance in the Klein model Cayley Klein model of hyperbolic geometry , where the convex set is the n dimensional open unit ball . Hilbert s metric has been applied to Perron Frobenius theorem Perron Frobenius theory and to constructing Gromov hyperbolic space s. Definition Let &Omega be a convex set convex open set open domain in a Euclidean space that does not contain a line. Given two distinct points A and B of &Omega , let X and Y be the points at which the straight line AB intersects the boundary of &Omega , where the order of the points is X , A , B , Y . Then the Hilbert distance d A , B is the logarithm ... 0 and defines a metric mathematics metric on &Omega . If one of the points A and B lies on the boundary ... the rescaling of v and w by positive constants and so descends to a metric on the space of rays ... sup then the induced metric on the projectivization of K is often called simply Hilbert s projective metric . This cone corresponds to a domain &Omega which is a regular simplex of dimension n &minus 1. Motivation and applications Hilbert introduced his metric in order to construct an axiomatic metric geometry in which there exist triangles ABC whose vertices A , B , C are not Line ... , BC , AC do not meet the interior of one of the sides of &Omega . Garrett Birkhoff used Hilbert s metric ... and sufficient conditions for a bounded convex domain in R sup n sup , endowed with its Hilbert metric ... of Jentzsch s theorem , Trans. Amer. Math. Soc. 85 1957 , 219 227 P.J. Bushell, Hilbert s Metric ... year 1895 journal Mathematische Annalen issn 0025 5831 volume 46 pages 91 96 Category 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 metric space . With this definition of a derivative, one can generalize Rademacher s theorem Rademarcher s theorem to metric space 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 metric space 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 space space 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
In group theory , a word metric on a group mathematics group math G math is a way to measure distance between any two elements of math G math . As the name suggests, the word metric is a metric mathematics metric on math G math , assigning to any two elements math g math , math h math of math G math ... set for the group. The word metric on G is very closely related to the Cayley graph of G the word metric measures the length of the shortest path in the Cayley graph between two elements of G. A Generating set of a group generating set for math G math must first be chosen before a word metric ... word metrics. While this seems at first to be a weakness in the concept of the word metric, it can be exploited ... 3 most efficiently is 1 1 1, a word of length 3. The distance between 0 and 3 in the word metric ... metric is equal to m n , because the shortest word representing the difference m n has length equal ... metric between 1,2 and 2,4 is therefore equal to 5. In general, given two elements math v i,j ... w math in the word metric is equal to math i k j l math . Definition Let G be a group, let S be a generating .... Given two elements g,h in G, the distance d g,h in the word metric with respect to S is defined ... g bar w h math . The word metric on G satisfies the axioms for a metric mathematics metric , and it is not hard to prove this. The proof of the symmetry axiom d g,h d h,g for a metric uses the assumption that the generating set S is closed under inverse. Variations The word metric has an equivalent ... set S. When each edge of the Cayley graph is assigned a metric of length 1, the distance between ... g to the vertex h. The word metric on G can also be defined without assuming that the generating ... metric with respect to S to be the word metric with respect to the symmetrization of S. Example ... , the distance between a and b in the word metric equals 2 Suppose that F is the free group on the two ... math g in G math to kg. This action is an isometry of the word metric. The proof is simple the distance ... more details
In mathematical physics , a metric mathematics metric describes the arrangement of relative distances within a surface or volume, usually measured by signals passing through the region essentially describing the intrinsic geometry of the region. An acoustic metric will describe the signal carrying properties characteristic of a given particulate medium in acoustics , or in fluid dynamics . Other descriptive names such as sonic metric are also sometimes used, interchangeably. A simple fluid example For simplicity, we will assume that the underlying background geometry is Euclidean , and that this space is filled with an isotropic inviscid fluid at zero temperature e.g. a superfluid . This fluid is described by a density field and a velocity field math vec v math . The speed of sound at any given point depends upon the compressibility which in turn depends upon the density at that point. This can be specified by the speed of sound field c. Now, the combination of both isotropy and Galilean covariance tells us that the permissible velocities of the sound waves at a given point x, math ... that sound is like light moving though a spacetime described by an effective metric tensor called the acoustic metric . The acoustic metric math mathbf g g 00 dt otimes dt 2g 0i dx i otimes dt g ij .... Acoustic horizons Main Sonic black holes An acoustic metric can give rise to acoustic horizons also known as sonic horizons , analogous to the event horizons in the spacetime metric of general relativity. However, unlike the spacetime metric, in which the invariant speed is the absolute upper limit on the propagation of all causal effects, the invariant speed in an acoustic metric is not the upper ... metric. It is possible for certain physical effects to propagate back across an acoustic ..., since a generic analog model depends upon BOTH the acoustic metric AND the underlying background ... geometry. See also Analog models of gravity Hawking radiation gravastar acoustics metric mathematics ... more details
In music a metric modulation is a change modulation music modulation from one time signature tempo meter music meter to another, wherein a note value from the first is made equivalent to a note value in the second, like a pivot . The term was invented to describe the practice of Elliott Carter , who prefers to call it temporal modulation. Citation needed date April 2010 For metric modulation to exist 3 things have to occur There has to be an exact relationship between two different tempi A common pulse music pulse must exist between these two tempi The name and function of the pulse changes. The following formula illustrates how to determine the tempo before or after a metric modulation, or, alternatively, how many of the associated note values will be in each measure before or after the modulation math frac new tempo old tempo frac number of pivot note values in new measure number of pivot note values in old measure math Winold 1975, 230 31 Thus if the two half notes in 4 4 time at a tempo of quarter note 84 are made equivalent with three half notes at a new tempo, that tempo will be math begin align qquad frac x 84 & frac 3 2 , 84 cdot frac x 84 & frac 3 2 cdot 84 , qquad x & frac 3 cdot 84 2 , qquad x & 126 end align math Winold 1975, p.230, example taken from Carter s Eight Etudes and a Fantasy for woodwind quartet 1950 , Fantasy, mm. 16 17. A tempo or metric modulation causes ... uses of tempo modulations, such as tempo networks and beat subdivision spaces. Score notation Metric modulations are generally notated as note value note value . br For example, br Image 4 5 Metric Modulation.JPG ... 5. Further reading Arlin, Mary I. 2000 . Metric Mutation and Modulation The Nineteenth Century ... share Metric Modulation 4 over 3 Conor Guilfoyle . http www.youtube.com watch?v RFVdZ9YV15U&list UU8W1Z7Iurx8aQdS NZEHjjA&index 4&feature plcp Metric modulation 3 over 2 Conor Guilfoyle . DEFAULTSORT Metric Modulation Category Rhythm Category Musical techniques ... more details
In mathematics , the Poincar metric , named after Henri Poincar , is the metric tensor describing a two dimensional surface of constant negative curvature . It is the natural metric commonly used in a variety ... forms are reviewed below. Overview of metrics on Riemann surfaces A metric on the complex plane may ... form . The determinant of the metric is equal to math lambda 4 math , so the square root of the determinant ... math Phi z, overline z math is said to be the potential of the metric if math 4 frac partial partial ... curvature of the metric is given by math K Delta log lambda. , math This curvature is one half ... are invariant under isometries. Thus, for example, let S be a Riemann surface with metric math lambda 2 z, overline z , dz , d overline z math and T be a Riemann surface with metric math mu ... frac partial partial overline z w z 0. math Metric and volume element on the Poincar plane The Poincar metric tensor in the Poincar half plane model is given on the upper half plane H as math ds 2 frac dx 2 dy 2 y 2 frac dz , d overline z y 2 math where we write math dz dx i ,dy. math This metric ... math dz d overline z frac dz ,d overline z cz d 4 math thus making it clear that the metric tensor ... The metric is given by math rho z 1,z 2 2 tanh 1 frac z 1 z 2 z 1 overline z 2 math math rho z ... math . Another interesting form of the metric can be given in terms of the cross ratio . Given any four ... z 1 z 2 z 3 z 4 z 2 z 3 z 4 z 1 . math Contradict inline article Cross ratio date 2011 07 11 Then the metric ... math and math z 2 math . The geodesics for this metric tensor are circular arcs perpendicular to the real ..., and 0 to the bottom of the disk. Metric and volume element on the Poincar disk The Poincar metric ... by math d mu frac dx ,dy 1 x 2 y 2 2 frac dx ,dy 1 z 2 2 . math The Poincar metric is given ... The geodesics for this metric tensor are circular arcs whose endpoints are orthogonal to the boundary .... The Poincar metric on the upper half plane induces a metric on the q disk math ds 2 frac 4 ... more details
Confusing date December 2006 The signature of a metric tensor or more generally a symmetric bilinear form , thought of as a quadratic form is the number of positive, negative and zero eigenvalue s of the metric. That is, the corresponding real symmetric matrix is diagonalisation diagonalised , and the diagonal entries of each sign counted. If the matrix mathematics matrix of the metric tensor is nowrap n n , then the number of positive, negative and zero eigenvalues p , q and r may take values from 0 to n with nowrap 1 p q r n . The signature may be denoted by a pair of integers nowrap p , q implying nowrap 1 r 0 , a triple nowrap p , q , r , or as an explicit list such as nowrap , , , or nowrap , , , , in this case nowrap 1, 3 resp. nowrap 3, 1 . ref Rowland, Todd. Matrix Signature. From MathWorld A Wolfram Web Resource, created by Eric W. Weisstein. http mathworld.wolfram.com MatrixSignature.html ... if r is nonzero. A Riemannian metric is a metric with a definite bilinear form positive definite signature. A Lorentz metric Lorentzian metric is one with signature nowrap p , 1 , or sometimes nowrap 1, q . There is another definition of signature of a nondegenerate metric tensor given by a single .... More generally, the metric signature nowrap p , q , r of A is a group of three natural numbers defined ... metric is positive definite on the space like subspace, and negative definite on the time like subspace. In the specific case of the Minkowski metric , whose metric has coordinates math ds 2 dx 2 dy 2 dz 2 c 2 dt 2 math , the metric signature is nowrap 3, 1, 0 , since it is positive definite ... in analogy to the Minkowski metric discussed above. The more general signatures are often referred .... Signature change If a metric is regular everywhere then the signature of the metric is constant ... of the metric may change at these surfaces. ref cite journal last1 Dray first1 Tevian last2 ... Sign convention Notes reflist Category Differential geometry Category Metric tensors 2 de Signatur ... more details
In differential geometry , the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold . It is so called because it is derived from the Bergman kernel , both of which are named for Stefan Bergman . Definition Let math G subset mathbb C n math be a domain and let math K z,w math be the Bergman kernel on G . We define a Hermitian metric on the tangent bundle math T z mathbb C n math by math g ij z frac partial 2 partial z i , partial bar z j log K z,z , math for math z in G math . Then the length of a tangent vector math xi in T z mathbb C n math is given by math left vert xi right vert B,z sqrt sum i,j 1 n g ij z xi i bar xi j . math This metric is called the Bergman metric on G . The length of a piecewise smooth function C sup 1 sup curve math gamma colon 0,1 to mathbb C n math is then computed as math ell gamma int 0 1 left vert frac partial gamma partial t t right vert B, gamma t dt . math The distance math d G p,q math of two points math p,q in G math is then defined as math d G p,q inf ell gamma mid text all piecewise C 1 text curves gamma text such that gamma 0 p text and gamma 1 q . math The distance d sub G sub is called the Bergman distance . The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain. More importantly, the distance d sub G sub is invariant under biholomorphic mappings of G to another domain math G math . That is if f is a biholomorphism of G and math G math , then math d G p,q d G f p ,f q math . References Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992. PlanetMath attribution id 6803 title Bergman metric Category Complex manifolds differential geometry stub ... more details
Infobox Television episode Title Trust Metric Series Numb3rs Image Caption Season 4 Episode 1 Airdate ... Gary Wilmes as Michael Kirkland Episode list Prev The Janus List Next Hollywood Homicide Trust Metric ... by series writer Ken Sanzel, Trust Metric is set five weeks after the events in The Janus List . In Trust Metric s story, a Federal Bureau of Investigation FBI team attempts to find a pair of espionage ... as a guest star. Trust Metric first aired in the United States on September 28, 2007. Critics gave ... into Lancer. At the FBI office, Charlie uses a trust metric to determine Colby s credibility. The rest ... to find Colby and Dwayne and find Lancer s SUV at the port. Charlie s trust metric reveals that they could ... begin discussing the revision of Charlie s paper on friendship math. Production Filming Trust Metric ... Trust Metric . ref name Trust Metric Pre Production cite video people Tony Scott executive producer ... Numb3rs Trust Metric Pre Production medium DVD Numb3rs Season 4 publisher CBS Studios, Inc. ref Scott wanted to reinvent the show s effects. ref name Trust Metric Pre Production Specifically ... Trust Metric Tony Touch cite video people Ken Sanzel executive producer writer , Nicolas Falacci co ... Trust Metric Tony s Touch medium DVD Numb3rs Season 4 publisher CBS Studios, Inc. ref In order to maintain ... Metric Tony Touch To give the title shot a new look, Skip Chaisson changed the color shot at the end ... slo mo and freeze frame to increase the intensity of the storytelling. ref name Trust Metric Post ... editor date 2008 title Crunching Numb3rs Trust Metric Post Production medium DVD Numb3rs Season 4 publisher CBS Studios, Inc. ref He changed several sets for texture. ref name Trust Metric Tony ... seasons. ref name Trust Metric Tony Touch He commented and made suggestions about the actors hair. ref ... September 28, 2007 publisher Zap2It.com ref Trust Metric took nine days to film, three days on set and six days on location. ref name Trust Metric Pre Production Numb3rs location managers John Armstrong ... more details
Orphan date April 2011 The Bures metric defines the infinitesimal distance between density matrix operators defining quantum states, according to the following formula math d rho, rho d rho 2 frac 1 2 tr d rho G , math where math G math is implicitly given by math rho G G rho d rho , math Some of the applications of the Bures metric include that given a target error, it allows the calculation of the minimum number of measurements to distinguish two different states ref distinct and the use of the volume element as a candidate for the Jeffreys prior probability density ref priorprob for mixed quantum states. Bures Distance The Bures distance is the finite version of the infinitesimal square distance described above and is given by math D B rho 1, rho 2 2 2 1 sqrt F rho 1, rho 2 , math where ... The Bures metric can be seen as the quantum equivalent of the Fisher information metric and can be rewritten ... mu d theta nu math where the quantum Fisher metric tensor components is identified as math J mu nu tr rho frac L mu L nu L nu L mu 2 . math The definition of the SLD implies that the quantum Fisher metric is 4 times the Bures metric. In other words, given that math g mu nu math are components of the Bures metric tensor, one has math J mu nu 4 g mu nu math As it happens with the classical Fisher information metric, the quantum Fisher metric can be used to find the Cram r Rao bound of the covariance . Explicit Formulas The actual computation of the Bures metric is not evident from the definition ... form of the Bures metric, valid for 2x2 and 3x3 systems, respectively math d rho, rho d ... 2 2 x 3 2 le 1 math . The components of the Bures metric in this parametrization can be calculated as math ... metric References Uhlmann, A., http www.ma.utexas.edu mp arc c 92 92 44.ps.gz The Metric of Bures ... Mixed States , Seminar Sophus Lie, 73 , 1993. Dittmann J., Explicit formulae for the Bures metric ..., M., Computation of Uhlmann s parallel transport for density matrices and the Bures metric on three ... more details
In mathematics, the concept of a generalised metric is a generalisation of that of a metric space metric , in which the distance is not a real number but taken from an arbitrary ordered field . In general, when we define metric space 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 metric space 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 Normality. The matter of wonder is that, even without choice, general metrics are monotonically normal ... Category Metric geometry ... more details
ROUGE , or Recall Oriented Understudy for Gisting Evaluation , is a set of metrics and a software package used for evaluating automatic summarization and machine translation software in natural language processing . The metrics compare an automatically produced summary or translation against a reference or a set of references human produced summary or translation. Various forms of the metric are based on n grams ROUGE N least common substrings ROUGE L and ROUGE W and skip bigrams and unigrams ROUGE S and ROUGE SU . See also BLEU F1 Score F Measure METEOR NIST metric Word error rate Word Error Rate WER Noun Phrase Chunking References http www.law.kuleuven.ac.be icri conferences Lin.pdf Slides of talk by Chin Yew Lin Lin, Chin Yew. 2004. ROUGE a Package for Automatic Evaluation of Summaries. In Proceedings of the Workshop on Text Summarization Branches Out WAS 2004 , Barcelona, Spain, July 25 26, 2004. External links http berouge.com ROUGE web site Category Machine translation ... more details
In mathematics , the Carath odory metric is a metric mathematics metric defined on the open set open unit ball of a complex number complex Banach space that has many similar properties to the Poincar metric of hyperbolic geometry . It is named after the Greece Greek mathematician Constantin Carath odory . Definition Let X , be a complex Banach space and let B be the open unit ball in X . Let denote the open unit disc in the complex plane C , thought of as the Poincar disc model for 2 dimensional real 1 dimensional complex hyperbolic geometry. Let the Poincar metric on be given by math rho a, b tanh 1 frac a b 1 bar a b math thus fixing the curvature to be &minus 4 . Then the Carath odory metric d on B is defined by math d x, y sup rho f x , f y f B to Delta mbox is holomorphic . math What it means for a function on a Banach space to be holomorphic is defined in the article on Infinite dimensional holomorphy . Properties For any point x in B , math d 0, x rho 0, x . math d can also be given by the following formula, which Carath odory attributed to Erhard Schmidt math d x, y sup left left. 2 tanh 1 left frac f x f y 2 right right f B to Delta mbox is holomorphic right math For all a and b in B , math a b leq 2 tanh frac d a, b 2 , qquad qquad 1 math with equality if and only if either a b or there exists a bounded linear functional &isin X sup &lowast sup such that 1, a b 0 and math rho ell a , ell b d a, b . math Moreover, any satisfying these three conditions has a &minus b a &minus b . Also, there is equality in 1 if a b and a &minus b a b . One way to do this is to take b &minus a . If there exists a unit vector u in X that is not an extreme point of the closed unit ball in X , then there exist ... Cambridge Univ. Press location Cambridge year 2003 Category Hyperbolic geometry Category Metric geometry ... more details
about the metric system in general information about specific versions of the system, such as the International ... See introduction Introduction to the metric system Image FourMetricInstruments.JPG thumb 280px The metric ... that have metric calibrations are shown. Three of the objects, a tape measure calibrated in centimetre ... use by tradesmen . The metric system is an international decimal ised systems of measurement ... have been refined and the metric system extended to incorporate many more units. Although a number of variants of the metric system emerged in the late nineteenth and early twentieth centuries ... industrialized country that does not use the metric system as its official system of measurement, although the metric system has been officially sanctioned for use there since 1866. Although the United Kingdom committed to officially adopting the metric system for many measurement applications ... of these prototypes can be retired by 2014. From its beginning, the main feature of the metric system .... The uncoordinated use of the metric system by different scientific and engineering disciplines, particularly ... standard internationally recognised standard metric system. Features Although the metric system ... internationally recognised characters. The metric system was, in the words of the French philosopher ... of prefixes that have now been defined in SI. ref SI Brochure 3.1 SI prefixes p 103 ref The metric .... However, these overtures failed and the custody of the metric system remained in the hands of the French ... Decimal multiples The metric system is decimal, except where the non SI units for time and plane angle ... MacTutor Decimal prefixes are a characteristic of the metric system conversions between units of different ... to overcome this problem while retaining the other benefits of the metric system by replacing all ... of Great Britain accessdate 23 April 2012 ref Prefixes common metric prefixes Main metric prefix A common ... version of the metric system the base units could be derived from a specified length the metre and the weight ... more details
In the mathematics mathematical field of differential geometry , a metric tensor is a type of function ... way as an inner product, a metric tensor is used to define the length of and angle between tangent ... with a metric tensor is known as a Riemannian manifold . By integral integration , the metric tensor ... of length, a Riemannian manifold is a metric space , meaning that it has a metric mathematics ..., the metric tensor itself is the derivative of the distance function taken in a suitable manner . Thus the metric tensor gives the infinitesimal distance on the manifold. While the notion of a metric .... The metric tensor is an example of a tensor field , meaning that relative to a local coordinate system on the manifold, a metric tensor takes on the form of a symmetric matrix whose entries transform ... a metric tensor is a covariant symmetric tensor . From the coordinate independent point of view, a metric tensor is defined to be a nondegenerate form nondegenerate symmetric bilinear form on each ... led Gauss to introduce the predecessor of the modern notion of the metric tensor. Arclength ... law EquationNote 3 is known as the metric tensor of the surface. Invariance of arclength ... product of two tangent vectors Another interpretation of the metric tensor, also considered by Gauss .... In contemporary terms, the metric tensor allows one to compute the dot product of tangent vectors ... , consisting of all tangent vectors to the manifold at the point p . A metric at p is a function ... , the metric tensor was allowed to be non symmetric however, the antisymmetric part of such a tensor ... sub p sub X sub p sub , Y sub p sub 0. A metric tensor g on M assigns to each point p of M a metric g sub p sub in the tangent space at p in a way that varies smooth function smoothly with p ... of the metric Hatnote This section assumes some familiarity with coordinate vector s. The components of the metric in any basis of a vector space basis of vector field s, or frame bundle frame , f ... more details
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