Dictionary
(statistics) an arrangement of values of a variable showing their observed or theoretical frequency of occurrence the spatial or geographic property of being scattered about over a range, area, or volume "worldwide in distribution" "the distribution of nerve fibers" "in complementary distribution" the act of distributing or spreading or apportioning the commercial activity of transporting and selling goods from a producer to a consumer
|
Wikipedia
This page deals with generalized functions in mathematical analysis. It is not about probability distributions.''In mathematical analysis, distributions (also known as generalized functions) are objects which generalize function (mathematics)functions and probability distributions. They extend the concept of derivative to all continuous functioncontinuous functions and beyond and are used to formulate generalized solutions of partial differential equations. They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions are distributions, such as the Dirac delta distribution."Generalized functions" were introduced by SobolevSergei Sobolev in 1935. They were independently discovered in late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions.Sometimes, people talk of a "probability theoryprobability distribution" when they just mean "probability measure (mathematics)measure", especially if it is obtained by taking the product of the Lebesgue measure by a positive, real-valued measurable function of integral equal to 1.
Basic idea - The basic idea is as follows. If ''f'' : R → R is an integrationintegrable function, and φ : R → R is a smooth (that is, infinitely derivativedifferentiable) function with compact support (that is, it is identically zero except on some bounded set), then ∫''f''φd''x'' is a real number which linear operatorlinearly and Continuous functioncontinuously depends on φ. One can therefore think of the function ''f'' as a continuous linear functional (mathematics)functional on the space which consists of all the "test functions" φ. Similarly, if ''P'' is a probability distribution on the reals and φ is a test function, then ∫φd''P'' is a real number that continuously and linearly depends on φ: probability distributions can thus also be viewed as continuous linear functionals on the space of test functions. This notion of "continuous linear functional on the space of test functions" is therefore used as the definition of a distribution.Such distributions may be multiplied with real numbers and can be added together, so they form a real vector space. In general it is not possible to define a multiplication for distributions, but distributions may be multiplied with infinitely differentiable functions.To define the derivative of a distribution, we first consider the case of a differentiable and integrable function ''f'' : R → R. If is a test function, then we have :using integration by parts (note that φ is zero outside of a bounded set and that therefore no boundary values have to be taken into account). This suggests that if ''S'' is a ''distribution'', we should define its derivative S' as the linear functional which sends the test function φ to -''S''(φ'). It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.The Dirac delta (so-called Dirac delta function) is the distribution which sends the test function φ to φ(0). It is the derivative of the Heaviside step function ''H''(''x'') = 0 if ''x'' < 0 and ''H''(''x'') = 1 if ''x'' ≥ 0. The derivative of the Dirac delta is the distribution which sends the test function φ to -φ'(0). This latter distribution is our first example of a distribution which is neither a function nor a probability distribution.An alternate definition is the limit of a sequence of functions. For instance the delta function is given bywhere !δa
Formal definition - In the sequel, real-valued distributions on an open setopen subset ''U'' of R''n'' will be formally defined. (With minor modifications, one can also define complex-valued distributions, and one can replace R''n'' by any smooth manifold.) First, the space D(''U'') of test functions on ''U'' needs to be explained. A function φ : ''U'' → R is said to have ''compact support'' if there exists a compact spacecompact subset ''K'' of ''U'' such that φ(''x'') = 0 for all ''x'' in ''U'' \ ''K''. The elements of D(''U'') are the infinitely often differentiable functions φ : ''U'' → R with compact support. This is a real vector space. We turn it into a topological vector space by requiring that a sequence (or net (mathematics)net) !(φ''k'')? converges to 0 if and only if there exists a compact subset ''K'' of ''U'' such that all !φ''k'' 0 and natural number ''d'' ≥ 0 there exists a natural number ''k''0 such that for all ''k'' ≥ ''k''0 the absolute value of all ''d''-th derivatives of !φ''k''complete topological vector space (in fact, a so-called LF-space).The dual space of the topological vector space D(''U''), consisting of all continuous linear functionals ''S'' : D(''U'') → R, is the space of all distributions on ''U''; it is a vector space and is denoted by D'(''U'').The function ''f'' : ''U'' → R is called ''locally integrable'' if it is Lebesgue integrationLebesgue integrable over every compact subset ''K'' of ''U''. This is a large class of functions which includes all continuous functions. The topology on D(''U'') is defined in such a fashion that any locally integrable function ''f'' yields a continuous linear functional on D(''U'') whose value on the test function φ is given by the Lebesgue integral !∫''U''''k'')? converges towards 0 if and only if !''S''''k''strong (operator) topology. This is the case if and only if !''S''''k''converges uniformly to 0 on all bounded subsets of D(''U''). (A subset of ''E'' of D(''U'') is bounded if there exists a compact subset ''K'' of ''U'' and numbers !''d''''n''''n''R is an infinitely often differentiable function and ''S'' is a distribution on ''U'', we define the product ''S''ψ by (''S''ψ)(φ) = ''S''(ψφ) for all test functions φ. The ordinary product rule of calculus remains valid.
Compact support and convolution - We say that a distribution ''S'' has ''compact support'' if there is a compact subset ''K'' of ''U'' such that for every test function φ whose support is completely outside of ''K'', we have ''S''(φ) = 0. Alternatively, one may define distributions with compact support as continuous linear functionals on the space !C∞∞''k''''k''R''n'' and one of them has compact support, then one can define a new distribution, the ''convolution'' ''S''*''T'' of ''S'' and ''T'', as follows: if φ is a test function in D(R''n'') and ''x'', ''y'' elements of R''n'', write !φ''x''''x''< ;/sub>)? and (''S''*''T'')(φ) = ''S''(ψ).This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense::d/d''x'' (''S'' * ''T'') = (d/d''x'' ''S'') * ''T'' = ''S'' * (d/d''x'' ''T'').
Tempered distributions and Fourier transform - By using a larger space of test functions, one can define the ''tempered distributions'', a subspace of D'(R''n''). These distributions are useful if one studies the Fourier transform in generality: all tempered distributions have a Fourier transform, but not all distributions have one.The space of test functions employed here, the so-called Schwartz-space, is the space of all infinitely differentiable rapidly decreasing functions, where φ : R''n'' → R is called ''rapidly decreasing'' if any derivative of φ, multiplied with any power of ''x'', converges towards 0 for ''x'' → ∞. These functions form a complete topological vector space with a suitably defined family of seminorms. More precisely, let:for α, β multi-indices of size ''n''. Then φ is rapidly-decreasing if all the values:The family of seminorms ''p''α, β defines a locally convex topology on the Schwartz-space. It is metrizable and complete.The derivative of a tempered distribution is again a tempered distribution.Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions can be viewed as tempered distributions. To study the Fourier transform, it is best to consider ''complex''-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform ''F'' yields then an automorphism of Schwartz-space, and we can define the Fourier transform of the tempered distribution ''S'' by (''FS'')(φ) = ''S''(''F''φ) for every test function φ. ''FS'' is thus again a tempered distribution. The Fourier transform is a continuous, linear, bijective operator from the space of tempered distributions to itself. This operation is compatible with differentiation in the sense that F'' (d/d''x'' ''S'') = ''ix'' ''FS'' and also with convolution: if ''S'' is a tempered distribution and ψ is a ''slowly increasing'' infinitely differentiable function on R''n'' (meaning that all derivatives of ψ grow at most as fast as polynomials), then ''S''ψ is again a tempered distribution and F''(''S''ψ) = ''FS'' * ''F''ψ.
Using holomorphic functions as test functions - The success of the theory led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular by Mikio Sato, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example Path integral formulation Feynman integrals.
Problem of multiplication - The main problem of the theory of distributions (and hyperfunctions) is that it is a purely linear theory, in the sense that the product of two distributions cannot consistently be defined (in general), as has been proved by Laurent Schwartz in the 1950's.Thus, nonlinear problems cannot be posed and thus not solved in distribution theory.In the context of quantum field theory, the non-respect of this fact is one of the sources of the "divergencies". Although in the context of the latter theory, Epstein and Glaser developed a mathematically rigorous (but extremely technical) theory, this does not solve the problem. Many other interesting theories are non linear, like for example Navier-Stokes equations of fluid dynamics.In view of this, several theories of algebra (ring theory)algebras of generalized functions have been developed, among which Colombeau algebraColombeau's (simplified) algebra is maybe the most popular in use today.
See also - generalized functionColombeau algebra
References - M. J. Lighthill (1958). ''Introduction to Fourier Analysis and Generalized Functions''. Cambridge University Press. ISBN 0-521-09128-4 (defines distributions as limits of sequences of functions under integrals) L. Schwartz (1954), ''Sur l'impossibilité de la multiplications des distributions'', C.R.Acad.Sci. Paris 239, pp 847-848.Category:Functional analysisde:Distribution (Mathematik)fr:Distribution (analyse !mathématique)nl:Distributieja :超関数
|
|
Websites
Galileo Kenya : The leading Global Travel Distribution in Kenya
Provides Global Distribution System(Galileo) and Internet Connectivity and travel information to the travel agents,hotels,car rentals , airlines and tour operators in kenya.
http://www.galileo-kenya.com/
Radio Airplay Data and Distribution
ArtistMonitor™ can tell you whenever and wherever your music is played across a network of 2,500 college, non-commercial and commercial radio stations. In partnership with RadioWave Airplay Monitor, artists can also distribute and track their releases across 4,800 Internet radio stations.
http://www.artistmonitor.com/
record label for punk hardcore expirimental hiphop and electronical music
dutch label, booking agency, disto, and online store.
http://www.hecticrecs.com/
www.courage.ca
Canada's leading Window film distributor
http://www.courage.ca/
WOFEX - World's Food Expo
WOFEX is the biggest international food show in the Philippines.
http://www.wofex.com/
Steve Whitney Ministries
A Non Profit Organization. Food Distribution Center
http://www.stevewhitneyministries.org/
Baron Paper Company
Precision Sheeting, Trimming, Winding, and Converting. Quality Plant Sleeves and many innovative packaging products for Potted Plants and Cut Flowers.
http://www.baronpaper.com/
CAD Drafting & Design
Computer Aided Drafting & Design Large Format Digital Printing & Scanning
http://www.uniquedrafting.com/
SCADA for Electric Utilities
iPower SCADA provides a safe and intuitive environment for electricity network operations. Intelligent configuration reduces implementation costs while producing systems that are consistent and easy to maintain. An open, client-server architecture with redundancy, complete a safe and robust SCADA system.
http://www.i-scada.com/
Air Charter Broker and Distribution Cosultants
Gate 4 provides any support needed to lease or charter aircrafts both passengers and cargo. We also support distribution of touristic prodacts as car rentals, hotels, and tour operators.
http://www.gate4.it/
Snapdata International Group
Specialists in the publication of industry overviews covering over 25 industries in nearly 40 countries on a global, regional and country basis.
http://www.snapdata.com/
Business consulting in Poland
Consulting services for foreign companies intending to do business in Poland and central Europe
http://www.integralis.pl/
Ketka's illustrations
illustrations, sketches, posters
http://www.ketka.ru/
Textile Enzymes
Manufactures of Textile Cellulase Enzymes
http://www.megapacific.com/
Learn Languages Worldwide : e-learning and coaching
Languages online provides training services combining e-learning and distant coaching by native speakers.
http://www.comefica.com/
http://www.diabetesduringpregnancy.org/
Thsi website describes an educational and clinical tool for women who have been diagnosed with gestational diabetes to manage this condition during pregnancy.
http://www.diabetesduringpregnancy.org/
Intelligent Home Technologies
Provides home automation system design and installation in Maryland including home theater, audio/video distribution, Beam central vacuum, telephone systems, and networks
http://www.intelligenthometechnologies.com/
Home Inspections services in GTA,Mississauga,Oakville,Milton,Georgetown,
BUYING OR SELLING A HOUSE? Buying a home is for most people one of the biggest decisions you will make in your life. Fortunately, you don’t have to face it alone. You already have real estate professional and now is the time to include professional home inspector. HOME PROOF INSPECTION is for your security, peace of mind, and smoother closing, whether you’re buying or selling a home. A certified home inspector will be at your home for 2 to 3 hours, and you’re invited to follow the inspector step by step analysis of your home. Our reports are handed to you immediately at the end of your inspection.
http://www.homeproofinspections.com/
Conectivity Products
Satellite, Cable, Video Cameras, Security, and Audio Video Distribution products.
http://www.swidendist.com/
Paknovus UAB
Plastic packaging Plastic articles Panel houses
http://www.paknovus.lt/
Cinema, Movies
Official movies websites
http://www.tfmdistribution.fr/
MRI Executive Solutions
Specialists in Talent Acquistion Our business is people. Meeting the needs of clients ranging from private, regional providers to globally recognized public corporations. And we have assisted top talent in career transitions from executive leadership at the C-level to VPs, Directors and Managers in a variety of functional areas.
http://www.mriexecutivesolutions.com/
fosters mjb shipping.com Compare carriers and manage freight shipments online
freight,shipping,trucking,freight services,ltl,freight shipping,truck load,freight quote,air shipping,freight rate,shipping services,international freight,shipping a freight carrier and freight services, book freight shipments, and manage your freight transportation. Freightquote.com provides services for shipping customers, freight carriers, freight forwarders, cargo agents, and e-commerce marketplaces like eBay,oversized shipment>
http://www.mjbshipping.com/
3Lune Records
3Lune Records, founded in 1997 in Rome, is a Records Label and Distributor of Instrumental & Contemporary music, Jazz, Traditional & World-music, Early & Classical music. Catalogue online.
http://www.trelunerecords.it/
Tritium Tri-phazer Audio Electronics
High End Cables, Air Bearing Turntables, Triphazers, audio modifications, Tube Electronic Parts.
http://www.triphazer.com/
Commercial Real Estate & Investments in Florida at UNIWEB Commercial Inc.
Insider tips on real estate, available local properties, company details, what every buyer should know.
http://www.uniwebcommercial.com/
GM-Software
Software Solutions
http://www.gm-software.de/
Heminge & Condell
Counselors to leading businesses, not-for-profits, and governments, world-wide
http://www.hemingeandcondell.com/
Simonton Cove Weather
Amature weather station 1/4 mile from Simonton Cove South Portland Maine. Local weather with links to tides and marine forecast and a Flash weather portal with 5 second update.
http://www.simontoncoveweather.com/
Quality Time Graphics Inc.
Superior quality digital, offset and web printers. Full prepress, printing, finishing and distribution capabilities. Within budgets, on time, reliable and top quality.
http://www.qualitytimegraphics.com/
HELL BENT FOR RECORDS
Mailorder - Heavy Metal Vinyl & CDs / Underground Distribution
http://www.hbfr.de/
LOGISTICS DEVELOPMENT SERVICES
Supply Chain-Freight Management consulting firm
http://www.logisticsds.com/
Forcom Forwarding & Transport
Forcom offers you all transport services incl. customs clearance and handling services at rotterdam
http://www.forcom.nl/
Hamrick Software
Produces VueScan, a utility for obtaining high-quality images from most scanners, for Windows, Mac OS, and Linux. Also produces VuePrint, a JPEG and GIF image viewer for Windows. [Shareware]
http://www.hamrick.com/
Red Hat, Inc.
Develops and provides open source software products and services. (Nasdaq: RHAT).
http://www.redhat.com/
Debian GNU/Linux
Official site. One of the most important distributions, uses only Free Software as defined by FSF. Descriptions (Social Contract, partners, donations), news, sources, packages, documents, support, developer corner.
http://www.debian.org/
Festival Distribution Inc.
Distributing independent roots music to retailers across Canada.
http://www.festival.bc.ca/
|
|