. ISBN 1 58488 347 2. J Taylor, An Introduction to Measure and Probability , Springer, 1996. ISBN 0 387 94830 9. planetmath id 4002 title singularmeasure Category Integral calculus Category Measures measure ...In mathematics , two positive or signed measure signed or complex measure complex measures &mu and &nu defined on a measurable space &Omega , &Sigma are called singular if there exist two disjoint sets A and B in &Sigma whose set union union is &Omega such that &mu is zero on all measurable subsets of B while &nu is zero on all measurable subsets of A . This is denoted by math mu perp nu. math A refined form of Lebesgue s decomposition theorem decomposes a singularmeasure into a singular continuous measure and a discrete measure . See below for examples. Examples on R sup n sup As a particular case, a measure defined on the Euclidean space R sup n sup is called singular , if it is singular in respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singularmeasure. Example. A discrete measure . The Heaviside step function on the real line , math H x stackrel mathrm def left begin matrix 0, & x 0 1, & x geq 0 end matrix right. math has the Dirac delta function Dirac delta distribution math delta 0 math as its distributional derivative. This is a measure on the real line, a point mass at 0. However, the Dirac measure math delta 0 math is not absolutely continuous with respect to Lebesgue measure math lambda math , nor is math lambda math absolutely continuous with respect to math delta 0 math math lambda 0 0 math but math delta 0 0 1 math if math U math is any open set not containing 0, then math lambda U 0 math but math delta 0 U 0 math . Example. A singular continuous measure. The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous , and indeed its absolutely continuous part is zero it is singular continuous. See also Lebesgue s decomposition theorem Absolutely continuous Singular ... more details
A grammatical number denoting a unit quantity as opposed to the plural and other forms wiktionarypar singular singularity Singular may also refer to Gravitational singularity , a location where the quantities that are used to measure the gravitational field become infinite Technological singularity , a hypothesis that technological progress will reach such an infinite or extremely high value at a point in the near future A group of boar s In mathematics Mathematical singularity SINGULAR , an open source Computer Algebra System CAS Singular matrix , a matrix that is not invertible Singular measure , a measure or probability distribution whose support has zero Lebesgue measure. More generally, one may speak of an object singular with respect to a measure , meaning that replaces the Lebesgue measure above An infinite cardinal number is called singular if it is not a regular cardinal Possible misspelling Singulair , Merck trademark for Montelukast, a common prescription drug used in maintenance treatment of asthma and to relieve symptoms of seasonal allergies. Cingular Wireless , a major mobile network operator in North America disambig es Singular desambiguaci n fr Singulier la Singularis nl Singulier th ... more details
Other uses Singular disambiguation notability date September 2011 SINGULAR typeset span style font variant small caps Singular span is a computer algebra system for polynomial computations with special emphasis on the needs of commutative algebra , algebraic geometry , and singularity theory . S small INGULAR small is free software released under the GNU General Public License . Problems in non commutative algebra can be tackled with the S small INGULAR small offspring PLURAL. S small INGULAR small was designed by Gert Martin Greuel, Gerhard Pfister and Hans Sch nemann. See also Portal Free software Comparison of computer algebra systems External links http www.singular.uni kl.de SINGULAR website http www.singular.uni kl.de plural PLURAL website http www math.science.unitn.it costanti singular Interface from GAP to Singular http www.mathematik.uni kl.de greuel en Gert Martin Greuel s website http www.mathematik.uni kl.de pfister en Gerhard Pfister s website http www.mathematik.uni kl.de hannes de index.html Hans Sch nemann s website Computer algebra systems DEFAULTSORT Singular Category Free computer algebra systems Category Linux computer algebra system software free software stub de Singular Computeralgebrasystem es SINGULAR ja SINGULAR vi SINGULAR ... more details
italic title prefixed 8 singular they Singular they is the use of wiktionary they they or its inflection ... plural. Though singular they is widespread in everyday English and has a long history of usage, debate ... to either a singular or plural entity. This usage is also known as generic they . For example ... in general notwithstanding the fact that anyone is strictly grammatically singular . In some ... singular or ambiguous, singular they remains morphology linguistics morphologically and syntax syntactically ... Sharp as an example. In neither case is singular they unambiguously a semantic or morpho syntactic singular. What it actually Agreement linguistics agrees with is the plurality implicit in the indeterminacy ... . ref In this kind of analysis, singular they in English is typically an example of a semantically ... Since these make the quantification explicit. ref where the preference many languages show for singular pronouns probably gives rise to the singular in singular they . ref For, specifically, donkey ... as a whole, they are most often conceived of as singular, and a singular pronoun is used. England ... forms are sometimes used. ref Either the plural or the singular may be acceptable for a true bound ... that singular forms are, nonetheless, more natural in distributive constructions is inadvertently ... singular interpretations of they in what are actually translations of plurals. ref http www.crossmyt.com ... LanguageLog http itre.cis.upenn.edu myl languagelog archives 003572.html Singular they God said ... languages because it forms distributives with pronouns and marks for singular and plural. Such languages demonstrate a preference for singular pronouns but attest plurals in a substantial minority ... one person, requires the pronoun to be in the singular. Similar to this, but with even less justification ... exclusively singular nor exclusively plural, they are typically indeterminate in number. ref name ... Y, constant, it and non referring x, variable, they Pinker argues that usage of singular they in English ... more details
wiktionary measure tocright Measure may refer to Legal List of Church of England Measures Measure of the Church of England is a law passed by the General Synod and the UK Parliament equivalent of an Act Measure of the National Assembly for Wales , a law specific to Wales passed by the Welsh Assembly between 2007 and 2011 Measure of the Northern Ireland Assembly 1973 Mathematics and science Measure probability , a mathematical construct that models a real world experiment Measure mathematics , a way to assign non negative real numbers to subsets Measure physics , a way to integrate over all possible histories of a system in quantum field theory Units Measurement , the process of establishing the magnitude of some attribute of an object relative to some unit of measurement Bar music or measure, a unit of time in Western music representing a regular grouping of beats Coal measure is a seam of coal Measure termination , a way to describe termination of processes, algorithms, or functions Measure bartending or jigger, a bartending tool used to measure liquor Other Measure album , a 2000 album from Matt Pond PA Measure data warehouse , a property on which calculations can be made using precomputed aggregates Measure typography , in typography refers to the width of a block or column of text from the left margin to the right margin Countermeasure , a system designed to prevent weapons from acquiring and or destroying a target disambig cv de Ma es Medida fr Mesure it Misura he pt Medida hu M rt k ... more details
upright thumb Facsimile of the title page of Measure for Measure from the First Folio , published in 1623 Measure for Measure is a play by William Shakespeare , believed to have been written in 1603 ... Lodowick. Isabella , a novice nun. Claudio , Isabella s brother. Angelo Measure for Measure Angelo ... Image William Hunt Claudio and Isabella Shakespeare Measure for Measure.jpg thumb right upright Claudio ... judge, Angelo Measure for Measure Angelo . Claudio, a young nobleman, is betrothed unofficially married ... and Mariana publicly petition him, and he hears their claims against Angelo Measure for Measure ... Bawcutt N. W. Bawcutt ed. , Measure for Measure Oxford, 1991 , p. 17 ref Shakespeare was familiar ... way you judge others, you will be judged, and with the measure you use, it will be measured to you. Date, text and authorship Measure for Measure is believed to have been written in 1603 or 1604 ... of Measure that survives today is not in its original form, but rather the product of a revision after ... also Shakespeare s Mediterranean Measure for Measure , in Shakespeare and the Mediterranean The Selected ... recorded performance of Measure for Measure took place on St. Stephen s night, 26 December 1604. During the English Restoration Restoration , Measure was one of many Shakespearean plays adapted ... reading of both the characters and the overall message of the play. ref S. Nagarajan, Measure for Measure, New York, Penguin, 1998 pp. 181 183. ref Notable more recent productions of Measure for Measure ... Wagner with the libretto written by the composer based on Measure for Measure The musical Desperate ..., Round Heads and Pointed Heads , was originally written as an adaptation of Measure for Measure ... different parts of the body contains an underwater recital of Measure for Measure Mr.Beavis, in Aldous ..., to reading the last scene of Measure for Measure References reflist External links wikisource wikiquote http ise.uvic.ca Library SLT plays measuresubj.html Measure for Measure http www.shakespeare literature.com ... more details
In probability , a singular distribution is a probability distribution concentrated on a Null set set of Lebesgue measure zero , where the probability of each point in that set is zero. These distributions are sometimes called singular continuous distributions . Such distributions are not absolutely continuous with respect to Lebesgue measure . A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function , since the Lebesgue integral of any such function would be zero. An example is the Cantor distribution . See also Singularmeasure Lebesgue s decomposition theorem External links http eom.springer.de S s085530.htm Springer Encyclopaedia of Mathematics ProbDistributions DEFAULTSORT Singular Distribution Category Probability distributions Category Types of probability distributions Statistics stub probability stub it Distribuzione singolare ... more details
citation last Halmos first P.R. title Measure theory year 1950 publisher v. Nostrand citation last ... year 1928 publisher Gauthier Villars DEFAULTSORT Singular Function Category Fractal curves Category ... more details
Unreferenced date December 2009 Orphan date December 2009 In English language English grammar , a false singular occurs when a Grammatical number singular noun ending in a s or z sound is understood as a plural from which a new singular is constructed. The false singular is a form of back formation . Some false singulars become standard English. For example, pea was originally a false singular from pease pl. peasen . The old word remains in the phrase pease porridge . The non standard historical forms Chinee and Portuguee are also false singulars, from Chinese and Portuguese . DEFAULTSORT False Singular Category Grammar ... more details
In mathematics , in particular functional analysis , the singular values , or s numbers of a compact operator nowrap T &thinsp X Y acting between Hilbert space s X and Y , are the square roots of the eigenvalue s of the nonnegative self adjoint operator nowrap T T &thinsp X X where T denotes the adjoint operator adjoint of T . The singular values are nonnegative real number s, usually listed in decreasing order s sub 1 sub T , s sub 2 sup T , . If T is self adjoint operator self adjoint , then the largest singular value s sub 1 sub T is equal to the operator norm of T see Courant minimax principle . File Singular value decomposition.gif thumb right 280px Visualisation of a singular value decomposition SVD of a 2 dimensional, real en Shear mapping shearing matrix M . First, we see the unit disc in blue together with the two standard basis canonical unit vectors . We then see the action of M ... axes and a second rotation U . &Sigma is a diagonal matrix containing in its diagonal the singular ... geometric interpretation for the singular values Consider the image by T of the N sphere unit sphere this is an ellipsoid , and its semi axes are the singular values of T the figure ... U math and so the singular values are simply the absolute values of the eigenvalues . Most normed ... Fan k norm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm is the p th root of the sum of the p th powers of the singular values. Note ... matrix with the singular values lying on the diagonal. This is the singular value decomposition . History This concept was introduced by Erhard Schmidt in 1907. Schmidt called singular values eigenvalues at that time. The name singular value was first quoted by Smithies in 1937. In 1957, Allakhverdiev ... to operators in Banach space . See also Singular value decomposition Category Operator theory Category Singular value decomposition ar ... more details
In algebraic topology , a branch of mathematics , singular homology refers to the study of a certain ... . Intuitively spoken, singular homology counts, for each dimension n , the n dimensional holes of a space. Singular homology is a particular example of a homology theory , which has now grown to be a rather ... to understand, being built on fairly concrete constructions. In brief, singular homology is constructed ... sum s, called singular chains . The boundary topology boundary operation on a simplex induces a singular chain complex . The singular homology is then the homology mathematics homology of the chain ... can be applied to all topological spaces, and so singular homology can be expressed in terms of category ... to the category of graded abelian group s. These ideas are developed in greater detail below. Singular simplices A simplex singular n simplex is a continuous mapping math sigma n math from the standard ... n to X math . This mapping need not be injective , and there can be non equivalent singular simplices ... n sigma n Delta n math , is defined to be the formal sum of the singular n &minus 1 simplices represented ... from math p 0 math to math p 1 math is the formal sum or formal difference math p 1 p 0 math . Singular chain complex The usual construction of singular homology proceeds by defining formal sums of simplices .... Consider first the set of all possible singular n simplices math sigma n Delta n math on a topological ... of math C n X math are called singular n chains they are formal sums of singular simplices with integer ... partial math is readily extended to act on singular n chains. The extension, called the boundary operator ..., together with the math C n math , form a chain complex of abelian groups, called the singular ... of singular n cycles . The image of the boundary operator is math B n X operatorname im partial n 1 math , and is called the group of singular n boundaries . It can also be shown that math partial ... H 0 X mathbb Z math . A proof for the homotopy invariance of singular homology groups can be sketched ... more details
In optimal control , problems of singular control are problems that are difficult to solve because a straightforward application of Pontryagin s minimum principle fails to yield a complete solution. Only a few such problems have been solved, such as Merton s portfolio problem in financial economics or trajectory optimization in aeronautics. A more technical explanation follows. The most common difficulty in applying Pontryagin s principle arises when the Hamiltonian depends linearly on the control math u math , i.e., is of the form math H u phi x, lambda,t u cdots math and the control is restricted to being between an upper and a lower bound math a le u t le b math . To minimize math H u math , we need to make math u math as big or as small as possible, depending on the sign of math phi x, lambda,t math , specifically math u t begin cases b, & phi x, lambda,t 0 ?, & phi x, lambda,t 0 a, & phi x, lambda,t 0. end cases math If math phi math is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a bang bang control that switches from math b math to math a math at times when math phi math switches from negative to positive. The case when math phi math remains at zero for a finite length of time math t 1 le t le t 2 math is called the singular control case. Between math t 1 math and math t 2 math the maximization of the Hamiltonian with respect to u gives us no useful information and the solution in that time interval is going to have to be found from other considerations. One approach would be to repeatedly differentiate math partial H partial u math with respect to time until the control u again explicitly appears ... called singular arc will be optimal if it satisfies the Kelley condition math 1 k frac partial partial ... bang singular control refers to a control that has a bang bang portion as well as a singular portion. References reflist DEFAULTSORT Singular Control Category Control theory ... more details
A singular solution y sub s sub x of an ordinary differential equation ordinary differential equation is a solution that is Mathematical singularity singular or one for which the Initial value problem initial value problem also called the Cauchy problem by some authors fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full real line. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be Mathematical singularity singular functions . In some cases, the term singular solution is used to mean a solution at which there is a failure of uniqueness to the initial value problem at every point on the curve. A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By tangent we ... y sub c sub is a solution in a family of solutions parameterized by c . This means that the singular solution is the envelope mathematics envelope of the family of solutions. Usually, singular solutions ... check what happens if the term is equal to zero, and whether it leads to a singular solution. The Picard ... to rule out the existence of singular solutions. Other theorems, such as the Peano existence theorem ... being unique, which can allow for the existence of singular solutions. A divergent solution Consider ... the Picard Lindel f theorem Picard Lindel f theorem . Thus, the solutions above are all singular solutions ... . Thus, the solution math y s math is a singular solution in the stronger sense that uniqueness fails at every value of x . However, it is not a Mathematical singularity singular function since it and all ... are singular, in the sense that the second derivative fails to exist, at math x c 1 math ... x 2 1 4 cdot x 2. , math Now we shall check when these solutions are singular solutions. If two solutions ... Singular solutions in http eom.springer.de default.htm Encyclopaedia of Mathematics . Springer ... more details
mathcal Z R R neq mathcal Z R R math . This article will develop several notions in terms of the singular submodule and singular ideals, including the definition of singular module , nonsingular module ... a singular module if math mathcal Z M M , math . M is called a nonsingular module if math mathcal Z M 0 , math . R is called right nonsingular if math mathcal Z R R 0 , math . Using the left singular ... R , math , and so right singular ring is not usually defined the same way as singular modules are. Some authors have used singular ring to mean has a nonzero singular ideal , however this usage is not consistent with the usage of the adjectives for modules. Properties Some general properties of the singular ... N N cap mathcal Z M , math . The properties singular and nonsingular are Morita equivalence Morita invariant properties . The singular ideals of a ring contain central nilpotent elements of the ring. Consequently the singular ideal of a commutative ring contains the nilradical of a ring nilradical of the ring ... not necessarily hold for the singular submodule. However if R is a right nonsingular ring, then math ... N is singular. If M is a free module , or if R is right nonsingular, then the converse is true. A semisimple ... more details
Infobox musical artist name Singular image caption Singular origin Bangkok , Thailand genre Pop music Pop , Acoustic music acoustic , jazz years active 2010 present label Sony Music Entertainment operating Thailand associated acts Depapepe , Scrubb , Armchair , Stamp band Stamp current members Sin, Tossaporn Achawanuntakul br Nut, Chotiwut Boonyasit website URL http www.singularband.com background group or band Singular is a band from Thailand . The group includes Sin born on 1 October , a singer songwriter who has performed with the Bangkok Opera , and Nut born on 12 August , the band s guitarist who won a gold medal at the national guitar awards . Their musical character is likened to the Metro Acoustic style. History In May 2010, Singular released its first ever single 24.7 Twenty Four Seven , which reached number one on top radio charts all over Thailand. The music video for 24.7 also reached 590,000 views after three months on Youtube. Singular s second single Bao Bao Tender catapulted the band to greater heights with over four million views on Youtube. Within two weeks of its release, Bao Bao also reached number one on top radio charts all over the Thailand and the phrase lyric for Bao Bao Singular became the sixth most searched phrase on Google Zeitgeist 20 26 September 2010 . The song won a number of national music and media awards, including Song of the Year and Best New Acts for Thailand. Singular also worked on an international single with the Japanese guitar duo, Depapepe . Their joint single release is called Some Other Day , which was written by Sin, while incorporating an existing instrumental piece by Depapepe, Kitto Mata Itsuka . Discography Albums 2011 The White Room Decoration Peak Chart Position 1 ONE Try Tender GAME Days More 24.7 Twenty four ... links http www.singularband.com Official website http www.facebook.com thesingularband Singular ... Singular Press conference http play.you2play.com music 8619 you2play.com http seedspace.dice.in.th event ... more details
In mathematics , more precisely in perturbation theory , a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. This is in contrast to Perturbation theory regular perturbation problems, for which an approximation can be obtained by simply setting the small parameter to zero. More precisely, the solution cannot be uniformly approximated by an asymptotic expansion math varphi x approx sum n 0 N delta n varepsilon psi n x , math as math varepsilon to 0 math . Here math varepsilon math is the small parameter of the problem and math delta n varepsilon math are a sequence of functions of math varepsilon math of increasing order, such as math delta n varepsilon varepsilon n math . This is in contrast to perturbation .... Several classes of singular perturbations are outlined below. Methods of analysis A perturbed ... problem cannot be approximated in this way As seen in the examples below, a singular perturbation generally ... is decreased. Singular perturbation theory is a rich and ongoing area of exploration for mathematicians ... averaging . For books on singular perturbation in ODE and PDE s, see for example Holmes, Introduction ... . Springer, 1999. ISBN 978 0387989310 ref Examples of singular perturbative problems Each ... is regular instead of singular, will fail. Some show how the problem may be solved by more sophisticated singular methods. Vanishing coefficients in ordinary differential equations Differential equations ... one of them separately. Through a singular perturbation technique, we can make these two subsystems .... Methods and Applications of Singular Perturbations Boundary Layers and Multiple Timescale Dynamics ... math 1 x 2 math with roots at math x pm 1 math . Singular perturbation analysis suggests that the cubic ... math decreases, while the third becomes arbitrarily large. References references DEFAULTSORT Singular Perturbation Category Differential equations Category Nonlinear control id Usikan singular ... more details
Orphan date December 2010 Unreferenced date December 2006 A Singular Conspiracy 1974 by Barry Perowne is a fictional treatment of the unaccounted period in Edgar Allan Poe s life from January to May 1844. Poe visits Paris under an assumed name, having used travel papers belonging to a stranger he has seen wounded in a bar room brawl in New York. He has the intention of parlaying his West Point training into a position in the Lafayette Circle , a group of volunteer soldiers about to aid Poland against its Russia n occupiers. This inveigles Poe with a young Charles Baudelaire in a plot to expose Baudelaires stepfather who rejects Poe s application to blackmail, to free up Baudelaires patrimony. See also portal Novels Edgar Allan Poe in popular culture DEFAULTSORT Singular Conspiracy Category 1974 novels Category Historical novels Category Novels by Barry Perowne Category 1844 in fiction Category Novels about writers Hist novel stub ... more details
In mathematics , singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator ... R sup n sup is Mathematical singularity singular along the diagonal x y . Specifically ... n sup . The Hilbert transform main Hilbert transform The archetypal singular integral operator is the Hilbert ... Harmonic Analysis work pages publisher Princeton University Press year 1993 url accessdate ref Singular integrals of convolution type A singular integral of convolution type is an operator T defined by convolution ... publisher Pearson Education, Inc. place New Jersey year 2004 url accessdate ref Singular integrals ... whenever y y leq frac 1 2 max x y , x y math Singular integrals of non convolution type T is said to be a singular integral operator of non convolution type associated to the Calder n&ndash Zygmund ... Zygmund operators A singular integral of non convolution type T associated to a Calder n&ndash ... with 1 p . The T b theorem The T b theorem provides sufficient conditions for a singular integral operator to be a Calder n&ndash Zygmund operator, that is for a singular integral ... b . The T b theorem states that a singular integral operator T associated to a Calder n&ndash Zygmund ... of certain singular integrals issue 1 mr 0052553 zbl 0047.10201 year 1952 journal Acta Mathematica ... title On singular integrals mr 0084633 zbl 0072.11501 year 1956 journal American Journal of Mathematics ... 0 521 42001 6 Citation last Mikhlin first Solomon G. author link Solomon Mikhlin title Singular integral ... link Solomon Mikhlin title Multidimensional singular integrals and integral equations place Oxford ... Pr ssdorf first2 Siegfried title Singular Integral Operators place Berlin &ndash Heidelberg &ndash ... first Elias last Stein authorlink Elias Stein title Singular integrals and differentiability properties ... Singular integrals and differentiability properties of functions year 1970 pages XIV 287 ... more details
There is no really adequate definition of singular term . Here are some definitions proposed by different writers A term that tells us which individual is being talked about. John Stuart Mill , Arthur Prior , P. F. Strawson ref Strawson 1950, Prior 1976, Mill 1908 ref A term that is grammatically singular, i.e. a proper name proprium nomen , a demonstrative pronoun pronomen demonstrativum or a demonstrative pronoun with a common name cum termino communi . William of Ockham ref Ockham, loc. cit. ref A term that is inherently about the object to which it applies or Reference refers . Gottlob Frege ref Frege 1892 ref A term that is true in the same sense of only one object. Peter of Spain ref Hispanus 1947 ref References reflist Works cited Frege, G. 1892 On Sense and Reference , originally published as ber Sinn und Bedeutung in Zeitschrift f r Philosophie und philosophische Kritik , vol. 100, pp. 25 50. Transl. Geach & Black 56 78. Mill, J.S., A System of Logic , London 1908 8th edition . Petrus Hispanus, Summulae Logicales , ed. I. M. Bochenski Turin, 1947 also quoted in Prior 1976 Prior, A.N. The Doctrine of Propositions & Terms London 1976 Strawson, P.F. On Referring , Mind 1950 pp. 320 44. William of Ockham, Sum of Logic Summa logicae Paris 1448, Bologna 1498, Venice 1508, Oxford 1675 Category Grammar Category Names Category Concepts in logic Category Philosophy of language ur ... more details
Unreferenced date March 2007 In mathematics , specifically in measure theory , the trivial measure on any measurable space X , is the measure which assigns zero measure to every measurable set A 0 for all A in . Properties of the trivial measure Let denote the trivial measure on some measurable space X , . A measure &nu is the trivial measure &mu if and only if &nu X 0. is an invariant measure and hence a quasi invariant measure for any measurable function f X &rarr X . Suppose that X is a topological space and that &Sigma is the Borel sigma algebra Borel algebra on X . trivially satisfies the condition to be a regular measure . is never a strictly positive measure , regardless of X , , since every measurable set has zero measure. Since X 0, is always a finite measure, and hence a locally finite measure . If X is a Hausdorff space Hausdorff topological space with its Borel algebra, then trivially satisfies the condition to be an tight measure . Hence, is also a Radon measure . In fact, it is the vertex of the cone mathematics pointed cone of all non negative Radon measures on X . If X is an infinity infinite dimension al Banach space with its Borel algebra, then is the only measure on X , that is locally finite and invariant under all translations of X . See the article There is no infinite dimensional Lebesgue measure . If X is n dimensional Euclidean space R sup n sup with its usual algebra and n dimensional Lebesgue measure sup n sup , is a singularmeasure with respect to sup n sup simply decompose R sup n sup as A R sup n sup 0 and B 0 and observe that A sup n sup B 0. Category Measures measure theory nl Triviale maat pl Miara trywialna ... more details
mu n prod i 1 n 1 a cos 3 i 2 pi x ,dx, a 1. math One can construct another singular doubling measure ... a singularmeasure on 0, 1 singular to Lebesgue measure. ref cite journal last Kahane first ... , where d and L depend on M and . Doubling Measures Definition A Measure mathematics measure on a metric space X is said to be doubling if the measure of any ball is approximately the measure ... leq C mu B x,r , math for all x in X and r 0. In this case, we say is C doubling . A measure space that supports a doubling measure is necessarily a doubling metric space, where the doubling ... a doubling measure. ref cite journal last Wu first Jang Mei title Hausdorff dimension and doubling ... ref Examples A simple example of a doubling measure is Lebesgue measure on a Euclidean space. One can, however, have doubling measures on Euclidean space that are Singular measures singular with respect to Lebesgue measure. One example on the real line is the weak limit of the following sequence ... I denote its middle third interval. Fix 0 1 and let be the measure such that ... 15 year 1969 pages 185 192 ref Applications The definition of a doubling measure may seem arbitrary ... of metric spaces with doubling measures. references DEFAULTSORT Doubling Measure Category ... more details
and n dimensional Lebesgue measure sup n sup , then sub x sub is a singularmeasure with respect ...In mathematics , a Dirac measure is a measure mathematics measure sub x sub on a set X with any sigma algebra &sigma algebra of subset s of X defined by math delta x A 1 A x begin cases 0, & x not in A 1, & x in A. end cases math for a given math x in X math and any measurable set measurable set A X . The Dirac measure is a probability measure , and in terms of probability it represents the almost sure outcome x in the sample space X . We can also say that the measure is a single Atom measure theory atom at x however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence . The Dirac measures are the extreme point s of the convex set of probability measures on X . The name is a back ... as a theorem of Lebesgue integration . Properties of the Dirac measure Let sub x sub denote the Dirac measure centred on some fixed point x in some measurable space X , . sub x sub is a probability measure, and hence a finite measure. Suppose that X , T is a topological space and that ... positive measure if and only if the topology T is such that x lies within every non empty open set, e.g. in the case of the trivial topology , X . Since sub x sub is probability measure, it is also a locally finite measure . If X is a Hausdorff space Hausdorff topological space with its Borel algebra, then sub x sub satisfies the condition to be an inner regular measure , since Singleton ... is also a Radon measure . Assuming that the topology T is fine enough that x is closed, which is the case in most applications, the Support measure theory support of sub x sub is x . Otherwise, supp sub x sub is the closure of x in X , T . Furthermore, sub x sub is the only probability measure ... publisher CRC Press See also Discrete measure DEFAULTSORT Dirac Measure Category Measures measure ... more details
0 math . Moreover, harmonic measure on D is Singularmeasure mutually singular with respect to t dimensional Hausdorff measure for all t 1. Dahlberg s theorem ref cite journal last Dahlberg first Bj rn E. J. title Estimates of harmonic measure journal Arch. Rat. Mech. Anal. volume 65 issue 3 year 1977 ...In mathematics , especially potential theory , harmonic measure is a concept related to the theory of harmonic ... thumb Harmonic measure is the exit distribution of Brownian motion In probability theory , harmonic measure of a bounded domain in Euclidean space math R n math , math n geq .... More generally, harmonic measure of an It diffusion X describes the distribution of X as it hits the boundary of D . In the complex plane , harmonic measure can be used to estimate the absolute ... harmonic measure and the theory of conformal map s. The term harmonic measure was introduced by R ... . The connection between harmonic measure and Brownian motion was first identified by Kakutani ten ... principle H sub f sub x determines a probability measure &omega x , D on &part D by math H f x int partial D f y , mathrm d omega x, D y . math The measure &omega x , D is called the harmonic measure of the domain D with pole at x . Properties For any Borel subset E of &part D , the harmonic measure &omega x , D E is equal to the value at x of the solution to the Dirichlet ... and D , &omega x , D is a probability measure on &part D . If &omega x , D E ... case E is said to be a set of harmonic measure zero . This is a consequence of Harnack s inequality . Since explicit formulas for harmonic measure are not typically available, we are interested in determining conditions which guarantee a set has harmonic measure zero. F. and M. Riesz Theorem ... , then harmonic measure is mutually absolutely continuous with respect to arc length for all math ... R n math is a bounded Lipschitz domain , then harmonic measure and n 1 dimensional Hausdorff measure ... more details
wiktionary unit of measure Unit of measure may refer to Units of measurement for relevance to weights and measures Unit of account for relevance in economics Unit of Measure album Unit of Measure album , a 2000 album by Tony Rice disambiguation ... more details
Image Angelo Measure for Measure.jpg right thumb 125px Angelo seduces Isabella in a Los Angeles production of Measure for Measure . Angelo is a character in Shakespeare Shakespeare s play Measure for Measure . Role in the play Angelo is the deputy to Vincentio, the Duke of Vienna , who begins the play by departing the city under mysterious circumstances and leaves the straight laced Angelo in power. Angelo s first act is to begin the enforcement of an old law that makes fornication out of wedlock punishable by death, but proves himself a hypocrite when Isabella, the sister of Claudio, the first man sentenced under the law, comes to plead for her brother s life. Angelo agrees to commute the sentence only if she will sleep with him. Angelo is ultimately duped by being set up with Mariana, a woman he was once betrothed to, who masquerades as Isabella at the assignation. And after Angelo thinks he has attained the object of desire, he covers his tracks by ordering the execution of Claudio after all. But before the scheme is revealed to him, he admits his angst over his behaviour This deed unshapes me quite, makes me unpregnant br And dull to all proceedings. A deflower d maid br And by an eminent body that enforced br The law against it But that her tender shame br Will not proclaim against her maiden loss, br How might she tongue me Yet reason dares her no br For my authority bears of a credent bulk, br That no particular scandal once can touch br But it confounds the breather. He should have lived, br Save that riotous youth, with dangerous sense, br Might in the times to come have ta en revenge, br By so receiving a dishonour d life br With ransom of such shame. Would yet he had lived br A lack, when once our grace we have forgot, br Nothing goes right we would, and we would not. ref Act 4, Scene 4 ref Analysis The role is similar to Shylock in A Merchant of Venice in that it is the antagonist .... Performances Measure for Measure was rarely performed until Samuel Phelps played Vincentio Shakespeare ... more details
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