above. Asymptotic forms for the other types of Besselfunction follow straightforwardly from the above ... This section is linked fromBesselfunction For integer order n , J sub n sub is often defined via ... delta function . This property is used to construct an arbitrary functionfrom a series of Bessel ... the order of the Besselfunction the most common and important cases are for an integer or half ... of BesselfunctionBessel s equation arises when finding separable solutions to Laplace s equation ... x are defined by its Besselfunction Properties properties below. It is possible to define the function ... function to non integer values. The graphs of Bessel functions look roughly like oscillating ... in terms of J sub n 1 sub x by the identities Besselfunction Properties below . Image Bessel Functions 1st Kind, n 0,1,2 .svg thumb 300px right Plot of Besselfunction of the first kind, J sub ... linearly independent solution is then found to be the Besselfunction of the second kind, as discussed below. Bessel s integrals Another definition of the Besselfunction, for integer values of math ... mathrm i , n tau x sin tau , mathrm d tau. math This was the approach that Bessel used, and from this definition ... of the Bessel Clifford function . Relation to Laguerre polynomials In terms of the Laguerre polynomials math L k math and arbitrarily chosen parameter math t math , the Besselfunction can be expressed ... 0,1,2 .svg thumb 300px right Plot of Besselfunction of the second kind, Y sub &alpha sub x , for integer ... functions J are entire function s of x . If x is held fixed, then the Bessel functions are entire functions ... if x math . The Besselfunction of the second kind then can be thought to naturally appear as the imaginary ... growing and exponential decay decaying functions, respectively. Like the ordinary Besselfunction J ... frac 4x 2 3 right sqrt 1 frac x 2 3 right dx math The modified Besselfunction of the second kind has also been called by the now rare names Basset function after Alfred Barnard Basset Modified Bessel ... more details
we can see from this that the Besselfunction is not entire. Similarly, the modified Besselfunction ...In mathematical analysis , the Bessel Clifford function , named after Friedrich Bessel and William Kingdon Clifford , is an entire function of two complex variable s that can be used to provide an alternative development of the theory of Besselfunction s. If math pi x frac 1 Pi x frac 1 Gamma x 1 math is the entire function defined by means of the reciprocal Gamma function , then the Bessel Clifford function is defined by the series math mathcal C n z sum k 0 infty pi k n frac z k k math The ratio ... for all regions with bounded z , and hence the Bessel Clifford function is an entire function of the two complex variables n and z . Differential equation of the Bessel Clifford function It follows from the above series on differentiating with respect to x that math mathcal C n x math ... the hypergeometric function being normalized so that its value at z 0 is one. Relation to Bessel functions The Besselfunction of the first kind can be defined in terms of the Bessel Clifford function ... function as math mathcal C n z z n 2 I n 2 sqrt z math but from this starting point we would then need ... C n x , math which defines the recurrence relationship for the Bessel Clifford function. This is equivalent ... function of the second kind The Bessel Clifford differential equation math xy n 1 y y qquad math has ... Hence just as the Besselfunction and modified Besselfunction of the first kind can both be expressed ... pages 346 349 . Citation first A. George last Greenhill title The Bessel Clifford function, and its ... . DEFAULTSORT Bessel Clifford Function Category Complex analysis Category Special hypergeometric ... xy n 1 y y. qquad math This equation is of generalized hypergeometric type, and in fact the Bessel Clifford function is up to a scaling factor a generalized hypergeometric series Pochhammer&ndash Barnes hypergeometric function we have math mathcal C n z pi n 0F 1 n 1 z . math Unless n is a negative ... more details
Wikify date September 2011 In mathematics, the Bessel Maitland function , or Wright generalized Besselfunction , is a generalization of the Besselfunction , introduced by harvs txt authorlink Edward Maitland Wright first Edward Maitland last Wright year 1934 . The word Maitland in the name of the function seems to be the result of confusing Edward Maitland Wright s middle and last names. It is given by math J mu, nu z sum k ge 0 frac z k Gamma k mu nu 1 k . math References Citation last1 Wright first1 E. M. title The asymptotic expansion of the generalized Besselfunction. doi 10.1112 plms s2 38.1.257 jfm 60.0306.02 year 1934 Category Special functions ... more details
In mathematics, a Jackson q Besselfunction or basic Besselfunction is one of the three q analog s of the Besselfunction introduced by harvs txt authorlink F. H. Jackson last Jackson year1 1903 year2 1903b year3 1905 year4 1905b . The third Jackson q Besselfunction is the same as the Hahn Exton q Besselfunction . Definition The three Jackson q Bessel functions are given in terms of the Pochhammer symbol and the basic hypergeometric function by math J nu 1 x q frac q nu 1 q infty q q infty x 2 nu 2 phi 1 0,0 q nu 1 q, x 2 4 math math J nu 2 x q frac q nu 1 q infty q q infty x 2 nu 0 phi 1 q nu 1 q, x 2q nu 1 4 math math J nu 3 x q frac q nu 1 q infty q q infty x 2 nu 1 phi 1 0 q nu 1 q,qx 2 4 math References Citation last1 Ismail first1 Mourad E. H. title The zeros of basic Bessel functions, the functions J sub ax sub x , and associated orthogonal polynomials doi 10.1016 0022 247X 82 90248 7 mr 649849 year 1982 journal Journal of Mathematical Analysis and Applications issn 0022 247X volume 86 issue 1 pages 1 19 Citation last1 Jackson first1 F. H. title On generalized functions of Legendre and Bessel year 1903 journal Trans. Roy. Soc. Edinburgh volume 41 pages 1 28 Citation last1 Jackson first1 F. H. title Theorems relating to a generalization of the Bessel functions year 1903 journal Trans. Roy. Soc. Edinburgh volume 41 pages 105 118 Citation last1 Jackson first1 F. H. title Theorems relating to a generalization of Bessel s function. jfm 36.0513.02 year 1904 journal Edinb. Roy. Soc. Trans. volume 41 pages 399 408 Citation last1 Jackson first1 F. H. title The Application of Basic Numbers to Bessel s and Legendre s Functions doi 10.1112 plms s2 2.1.192 year 1905 journal Proc. London Math. Soc. volume 2 issue 1 pages 192 220 Citation last1 Jackson first1 F. H. title The Application of Basic Numbers to Bessel s and Legendre s Functions Second paper doi 10.1112 plms s2 3.1.1 year 1905 journal Proc. London Math. Soc. volume 3 issue 1 pages 1 23 Category Special functions ... more details
In mathematics, the Hahn Exton q Besselfunction or the third Jackson q Besselfunction is a q analog q analog of the Besselfunction , introduced by harvs txt last Hahn authorlink Wolfgang Hahn year 1953 in a special case and by harvs txt last Exton authorlink Harold Exton year 1983 in general. The Hahn Exton q Besselfunction is given by math J nu 3 x q frac x nu q nu 1 q infty q q infty sum k ge 0 frac 1 kq k k 1 2 x 2k q nu 1 q k q q k math References Citation last1 Exton first1 Harold title q hypergeometric functions and applications url http books.google.com books?id 3kHvAAAAMAAJ publisher Ellis Horwood Ltd. location Chichester series Ellis Horwood Series Mathematics and its Applications isbn 978 0 85312 491 7 mr MR708496 year 1983 Citation last1 Hahn first1 Wolfgang title Die mechanische Deutung einer geometrischen Differenzengleichung language German doi 10.1002 zamm.19530330811 zbl 0051.15502 year 1953 journal Zeitschrift f r Angewandte Mathematik und Mechanik issn 0044 2267 volume 33 pages 270 272 citation first M. E. H. last Ismail title Some Properties of Jacksons Third q BesselFunction series preprint year 2003 Citation last1 Swarttouw first1 Ren F. title An addition theorem and some product formulas for the Hahn Exton q Bessel functions doi 10.4153 CJM 1992 052 6 mr 1178574 year 1992 journal Canadian Journal of Mathematics issn 0008 414X volume 44 issue 4 pages 867 879 Category Special functions Category Q analogs ... more details
Bessel may refer to Bessel beam Bessel ellipsoid Besselfunction in mathematics Bessel s inequality in mathematics Bessel filter , a linear filter often used in audio crossover systems Bessel crater , a small lunar crater Bessel transform, also known as Fourier Bessel transform or Hankel transform Window functionBessel window Bessel window Besselian date, see Epoch astronomy Besselian years MV Bessel , a German merchant ship in service 1928 45, latterly for the Kriegsmarine Bessel is the name of Friedrich Bessel 1784 1846 , German mathematician, astronomer, and systematizer of the Bessel functions Bessel Kok born 1941 , Dutch businessman and chess organiser See also Bessell disambiguation Category Surnames Category Given names de Bessel Begriffskl rung fr Bessel nl Bessel ja sk Bessel sr ... more details
their name from body parts legs of a table are meant, regular plural must be used. Latin , though ...Refimprove date January 2012 Wiktionary plural In linguistics , plurality or a plural is a concept of quantity ... to the plural morpheme the value more than one yet that is a mistake. Consider the form of the noun ... value of the so called plural is, then, other than one . Typically applied to noun s, a plural ... such as stress or implicit markers context. In the English language , singular and plural are the only .... A plural is commonly abbreviated pl. in dictionaries. In part of speech tagging it has other notation ... main English plural In English, the plural is usually formed with the addition of s e.g., one ... plural forms require more noticeable changes in word structure. Most words ending in a y preceded ... . A small class of words have identical singular and plural forms e.g., one sheep , two sheep one aircraft , two aircraft . See English plural Irregular plurals for more examples of irregular pluralisation ... . In languages with dual, trial, or paucal numbers, plural refers to numbers higher than those. However, numbers besides singular, plural, and to a lesser extent dual are extremely rare. Languages with measure ... grammatical number at all, though they are likely to have plural personal pronoun s. Some languages like Mele Fila language Mele Fila distinguish between a plural and a greater plural. A greater plural ... between the paucal, the plural, and the greater plural is often relative to the type ..., paucal, greater paucal, and plural. Such languages as these have the most complex grammatical number in the world. An interesting difference from Romance Germanic languages is found in some Slavic ..., Polish language Polish has singular and plural, and a special form paucal for numbers where the last digit is 2, 3 or 4, excluding endings of 12, 13 and 14 . Russian uses plural form of words ending ... and genitive plural form of words for all other numbers. In addition, Slovene language Slovene ... more details
doctoral students Heinrich Scherk known for Besselfunction s br Stellar parallax prizes Gold Medal of the Royal Astronomical Society 1829 & 1841 religion footnotes Friedrich Wilhelm Bessel 22 July 1784 &ndash 17 March 1846 was a Germany German mathematician , astronomer , and systematizer of the Besselfunction s which were discovered by Daniel Bernoulli . He was a contemporary of Carl Friedrich Gauss Carl Gauss , also a mathematician and physicist . The asteroid 1552 Bessel was named in his honour. Life and work Bessel was born in Minden in Minden Ravensberg , the son of a civil servant. At the age of 14 Bessel was apprenticed to the import export concern Kulenkamp . He soon became the company ... won him the Lalande Prize from the French Academy of Sciences in 1811. Bessel was able to pin down the position of over 50,000 stars during his time at K nigsberg . With this work under his belt, Bessel ... systems, Bessel developed what are now known as Besselfunction s. Critical for the solution of certain ...Infobox scientist name Friedrich Bessel image Friedrich Wilhelm Bessel 1839 painting .jpg image size 225px caption Christian Albrecht Jensen C. A. Jensen , Friedrich Wilhelm Bessel , 1839 Ny Carlsberg ... longitude . Bessel came to the attention of a major figure of German astronomy at the time, Heinrich ... two years Bessel had left Kulenkamp and become an assistant at Lilienthal, Lower Saxony Lilienthal Observatory ... of 26, Bessel was appointed director of the K nigsberg Observatory by King Frederick William III of Prussia ... parallax accurately. In 1838 Bessel won the race, announcing that 61 Cygni had a parallax of 0.314 ... ly away. If the currently accepted figure of 11.4 ly see 61 Cygni and discussion page is correct, Bessel .... As well as helping determine the parallax of 61 Cygni, Bessel s precise measurements allowed him ... led to the discovery of Sirius B . Despite lacking a university education, Bessel was a major figure ... is named Bessel crater Bessel after him. Bessel s work in 1840 contributed in some degree to the discovery ... more details
File Bessel beam.svg thumb Diagram of Axicon and resulting Bessel Beam File Bessek beam intensity.svg thumb Cross section of the Bessel beam and graph of intensity File Bessel beam reform.svg thumb Bessel beam re forming central bright area after obstruction A Bessel beam is a field of electromagnetic, acoustic or even gravitational radiation whose amplitude is described by a Besselfunction of the first ... 3539 3547 ref The mathematics mathematical function which describes a Bessel beam is a solution of Bessel ... pmid 12226659 issue 6903 bibcode 2002Natur.419..145G ref ref D. McGloin, K. Dholakia, Bessel beams diffraction in a new light, Contemporary Physics 46 2005 15 28 ref A true Bessel beam is non diffractive ... spot. Bessel beams are also self healing , meaning that the beam can be partially obstructed at one point, but will re form at a point further down the beam axis . As with a plane wave a true Bessel ... because they exhibit little or no diffraction over a limited distance. Approximations to Bessel beams are made in practice by focusing a Gaussian beam with an axicon lens to generate a Bessel Gauss beam. The properties of Bessel beams ref F. O. Fahrbach, P. Simon, A. Rohrbach, Microscopy with self ... of an electromagnetic zero order Bessel beam by a dielectric sphere. Optics Letters 36 2011 766 768 ref make them extremely useful for Optical tweezers optical tweezing , as a narrow Bessel beam will maintain ... manipulation with acoustical tweezers may be feasible with a Bessel beam that scatters ref P. L. Marston, Scattering of a Bessel beam by a sphere, J. Acoust. Soc. Am. 121 2007 753 758 ref ref G. T. Silva, Off axis scattering of an ultrasound bessel beam by a sphere. IEEE Trans. Ultrason. Ferroelectr ... order Bessel vortex beam by a rigid sphere, Wave Motion 48 2011 392 400 ref and produces a radiation pressure radiation force resulting from the exchange of acoustic momentum between the wave field ... and quasi standing zero order Bessel beam tweezers, Annals of Physics 323 2008 1604 1620 ref ... more details
Bessel polynomial from which the filter gets its name and math omega 0 math is a frequency chosen ... order Bessel filter, illustrating flat unit delay in the passband The transfer function for a third ... also Butterworth filter Comb filter Chebyshev filter Elliptic filter Besselfunction Group delay ... filter3 hide In electronics and signal processing , a Bessel filter is a type of linear filter with a maximally flat group delay maximally linear phase response . Bessel filters are often used in audio crossover systems. Analog Bessel filters are characterized by almost constant group delay across ... s name is a reference to Friedrich Bessel , a German mathematician 1784 1846 , who developed the mathematical theory on which the filter is based. The filters are also called Bessel Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design. ref ... function Image Bessel4 GainDelay.png right thumb A plot of the gain and group delay for a fourth order low pass Bessel filter. Note that the transition from the pass band to the stop band is much slower than for other filters, but the group delay is practically constant in the passband. The Bessel filter maximizes the flatness of the group delay curve at zero frequency. A Bessel low pass filter is characterized by its transfer function ref name bianchi cite book title Electronic filter ...&dq Bessel filter polynomial&lr &as brr 3&ei gyeWSvTbIpmwNPyaqNcH v onepage&q Bessel 20filter 20polynomial ... 0 math . Bessel polynomials File Bessel 3rd order poles.svg right thumb The roots of the third order Bessel polynomial are the pole zero plot poles of filter transfer function in the s plane s plane , here plotted as crosses. The transfer function of the Bessel filter is a rational function whose denominator is a reverse Bessel polynomial , such as the following math n 1 quad s 1 math math n 2 quad ... n 5 quad s 5 15s 4 105s 3 420s 2 945s 945 math The reverse Bessel polynomials are given by ref name ... more details
In mathematics , a Bessel process , named after Friedrich Bessel , Why is it named after him? is a type of stochastic process . The Bessel process of order n is the real number real valued process X given by math X t W t , math where · denotes the Norm mathematics Euclidean norm Euclidean norm in R sup n sup and W is an n dimensional Wiener process Brownian motion started from the origin. The Bessel process of order n is the solution to the stochastic differential equation math dX t dZ t frac n 1 2 frac dt X t math where Z is a 1 dimensional Wiener process Brownian motion . Note that this SDE makes sense for any real parameter math n math although the drift term is singular at zero . Since W was assumed to have started from the origin the initial condition is X sub 0 sub 0. For n &ge 2, the n dimensional Wiener process is Markov chain Recurrence transient from its starting point almost surely with probability one , X sub t sub > 0 for all t > 0. It is, however, neighbourhood recurrent for n 2, meaning that with probability 1, for any r 0, there are arbitrarily large t with X sub t sub r on the other hand, it is truly transient for n 2, meaning that X sub t sub &ge r for all t sufficiently large. A notation for the Bessel process of order n started at zero is BES sub 0 sub n . References cite book author ksendal, Bernt title Stochastic Differential Equations An Introduction with Applications publisher Springer, Berlin year 2003 isbn 3 540 04758 1 Williams D. 1979 Diffusions, Markov Processes and Martingales, Volume 1 Foundations. Wiley. ISBN 0 471 99705 6. Category Stochastic processes probability stub ... more details
are the same as the first but in reverse order. For example, the third degree Bessel polynomial is math y 3 x 15x 3 15x 2 6x 1 , math while the third degree reverse Bessel polynomial is math theta 3 x x 3 6x 2 15x 15 , math The reverse Bessel polynomial is used in the design of Bessel filter Bessel electronic filters . Properties Definition in terms of Bessel functions The Bessel polynomial may also be defined using Besselfunction s from which the polynomial draws its name. math y n x ... frac 2 pi x ,e 1 x K n frac 1 2 1 x math where math K n x math is a modified Besselfunction of the second ... as a hypergeometric function The Bessel polynomial may also be defined as a confluent hypergeometric function Dita, 2006 math y n x , 2F 0 n,n 1 x 2 left frac 2 x right n U left n, 2n, frac 2 x right left frac 2 x right n 1 U left n 1,2n 2, frac 2 x right . math The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial math theta n x frac n 2 n ,L n 2n 1 2x math from which it follows that it may also be defined as a hypergeometric function math theta n x frac 2n n 2 n , , 1F 1 n 2n 2x math where math 2n n math is the Pochhammer symbol rising factorial . Generating function The Bessel polynomials have the generating function math sum n 0 sqrt frac 2 pi x n frac 1 2 e x K n frac 1 2 x frac t n n e x 1 sqrt 1 2t . math Recursion The Bessel polynomial may also be defined ...More footnotes date September 2009 In mathematics , the Bessel polynomials are an orthogonal polynomials ..., is sometimes known as the reverse Bessel polynomials See Grosswald 1978, Berg 2000 . math theta ... 2 theta n 2 x , math Differential equation The Bessel polynomial obeys the following differential equation ... of the Bessel polynomials have been suggested in literature Krall, Fink , as following math y n ... n left frac 1 x alpha, beta right . math For the weighting function math rho x alpha, beta , 1F 1 left ... math a curve surrounding the 0 point. They specialize to the Bessel polynomials for math alpha beta ... more details
of a barrel or cask is a back formation from staves , which is its plural. See the Plural to singular ... n plurals The plural of a few nouns can also be formed from the singular by adding n or en , stemming ... plurals from Latin and Greek See also Plural form of words ending in us English has borrowed ... . Since the word comes from Latin processus , whose plural in the Latin declension Fourth declension ... testes IPA t sti z Axes IPA ksi z , the plural of axis , is pronounced differently from axes IPA ... declension the plural has a long to differentiate it from the singular short style margin left ... from Inuktitut retain traditional plurals see also English plural Plurals of names of peoples Plurals ..., for example, was originally pluralfrom agendum something to be acted on but is nowadays used ... and roll, impressionism, surrealism. This includes those that look plural but function as grammatically ... case. Both words come from a Latin word meaning kind , but they do not form a singular plural ..., people can also be a singular noun with its own plural for example, We are many persons, from many ... singular or plural . This article discusses the variety of ways in which English plurals are formed ... provided in this article are for Received Pronunciation and General American . Regular plurals The plural ... in a sibilant sound IPA s , IPA z , IPA , IPA , IPA t or IPA d the plural is formed by adding ... other than a sibilant IPA p , IPA t , IPA k , IPA f or IPA the plural is formed by adding IPA ... plural adds IPA z , represented orthography orthographically by s style margin left 2em text align left border collapse separate border spacing 10px 0px Plural form is plataPOO boy boys IPA b z girl ... ref http www2.gsu.edu wwwesl egw crump.htm English Irregular Plural Nouns ref ref http hopelive.hope.ac.uk ... spacing 10px 0px P&O Ferries from ferry Other exceptions include lay by s and stand by s . Words ..., but money can also form its plural regularly. ref http dictionary.reference.com browse money Entry ... more details
lunar crater data image Image Bessel crater Apollo 15.jpg 200px caption Lunar crater Besselfrom Apollo 15 . NASA photo. latitude 21.8 N or S N longitude 17.9 E or W E diameter 16 km depth 1.7 km colong 342 eponym Friedrich Wilhelm Bessel Friedrich W. BesselBessel is a small moon lunar Impact crater crater that is located in the southern half of the Mare Serenitatis . Despite its small size, this is the largest crater to lie entirely within the Lunar mare mare . It lies to the north northeast of the crater Menelaus crater Menelaus . This crater is circular and bowl shaped with a rim that has a higher albedo than the floor or the surrounding mare. The outer rim is not significantly worn, and there are no features of note on the interior, apart from some slumping of material from the inner walls to the floor. Bessel is not of sufficient size to have developed the wiktionary terrace terrace structures of larger craters. It was named after Friedrich Wilhelm Bessel . A large Ray system ray , most likely from Tycho crater Tycho , crosses the mare from north to south, passing Bessel s western side. Satellite craters By convention these features are identified on lunar maps by placing the letter on the side of the crater mid point that is closest to Bessel. class wikitable width 25 style background eeeeee Bessel width 25 style background eeeeee Latitude width 25 style background eeeeee Longitude width 25 style background eeeeee Diameter align center D align center 27.3 N align center 19.9 E align center 5 km align center F align center 21.2 N align center 13.8 E align center 1 km align ... Astronomical Union IAU . Bessel A &mdash See Sarabhai crater . Bessel E &mdash See Bobillier crater ... title Radar Strip Showing Crater Bessel date November 23, 2009 publisher NASA url http www.nasa.gov ... craters on the Moon da Bessel m nekrater de Bessel Mondkrater fa fr Bessel crat re ru sk Bessel kr ter ... more details
Bessel Kok born Hilversum , 13 December 1941 is a Dutch business man and chess organiser living in Prague . He has served in top management positions in telecommunications companies in Belgium Belgacom and in the Czech Republic . He was also President of the Belgian based Banking communications company Society for Worldwide Interbank Financial Telecommunication SWIFT and responsible for SWIFT s sponsorship of several major International chess events. Kok was the Chairman of World Chess Grandmaster Association from 1985 until 1991, and helped to establish the 2002 Prague Agreement concerning the World Chess Championship . Kok is also active in the art s he was the producer of the movie film Nicholas Winton, the power of Good about British businessman Nicholas Winton who rescued children from the Nazi s. The film won an Emmy Award in 2002. Kok ran a campaign for presidency of FIDE the World Chess Federation FIDE against the incumbent Kirsan Ilyumzhinov in 2006. The election was held in June, 2006 in Torino , Italy during the World Chess Olympiad and Kok lost 54 96 to Ilyumzhinov. External links http www.fide.com news.asp?id 960 Two Tickets Announced for FIDE Elections http www.chessville.com Editorials Interviews 20Questions Kok.htm Chessville Interviews 20 Questions with Bessel Kok Persondata Metadata see Wikipedia Persondata . NAME Kok, Bessel ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 13 December 1941 PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Kok, Bessel Category 1941 births Category Living people Category Dutch chess players Category Dutch businesspeople Category People from Hilversum Category Chess officials Category Belgacom Group Netherlands business bio stub netherlands chess bio stub ca Bessel Kok de Bessel Kok fr Bessel Kok nl Bessel Kok fi Bessel Kok ... more details
Unreferenced date February 2012 The Bessel ellipsoid or Bessel 1841 is an important reference ellipsoid of geodesy . It is currently used by several countries for their national geodetic surveys, in Europe and on other continents, but will be replaced in the next decades by modern ellipsoids of satellite geodesy . The Bessel ellipsoid was derived 1841 by Friedrich Wilhelm Bessel , based on several meridian arc s and other data of continental geodetic network s of Europe , Russia and the British Survey of India . It is based on 10 meridional arcs and 38 precise measurements of the astro geographic latitude and longitude see also Astro geodetic astro geodesy . The dimensions of the ellipsoid axes were defined by logarithm s in keeping with former calculation methods. The Bessel and GPS ellipsoids The Bessel ellipsoid fits especially well to the geoid curvature of Europe and Eurasia . Therefore it is optimal for National survey networks in these regions, despite of the fact that its axes are about 700 m shorter than that of the mean Earth ellipsoid derived by satellites. Below the are the two axes a , b and the flattening a &minus b a . For comparison, the data ... and the Global Positioning System GPS system. Bessel ellipsoid 1841 defined by log a and a 6,377,397.155 ... data published by Bessel 1841 were then the best and most modern data mapping the figure of the Earth ... of network data. Every reference ellipsoid deviates from the worldwide data e.g. of satellite geodesy in the same way as the pioneering work of Bessel. In 1950 about 50 of the European triangulation networks and about 20 of other continent s networks were based on the Bessel ellipsoid. In the following ... of about 1940. As of 2010 the Bessel ellipsoid is the geodetic system for Germany , for Austria ... coordinate system UTM coordinates Category Geodesy de Bessel Ellipsoid fr Ellipso de de Bessel nl Ellipso de van Bessel ru sv Bessels ellipsoid ... more details
Unreferenced stub auto yes date December 2009 1552 Bessel alternate designation 1938 DE is a asteroid belt Main Belt asteroid . It was discovered on February 24, 1938 by Yrj V is l at Turku , Finland . It was named in honor of the German astronomer Friedrich Wilhelm Bessel . Minor planets navigator 1551 Argelander 1553 Bauersfelda Small Solar System bodies DEFAULTSORT Bessel Category Main Belt asteroids Category Asteroids named for people Category Discoveries by Yrj V is l Category Astronomical objects discovered in 1938 Beltasteroid stub de 1552 Bessel el 1552 eo 1552 Bessel eu 1552 Bessel fa it 1552 Bessel la 1552 Bessel hu 1552 Bessel ja no 1552 Bessel nn 1552 Bessel pl 1552 Bessel pt 1552 Bessel sk 1552 Bessel sr 1552 sv 1552 Bessel tl 1552 Bessel uk 1552 vi 1552 Bessel yo 1552 Bessel ... more details
In mathematics , especially functional analysis , Bessel s inequality is a statement about the coefficients of an element math x math in a Hilbert space with respect to an orthonormal sequence . Let math H math be a Hilbert space, and suppose that math e 1, e 2, ... math is an orthonormal sequence in math H math . Then, for any math x math in math H math one has math sum k 1 infty left vert left langle x,e k right rangle right vert 2 le left Vert x right Vert 2 math where , denotes the inner product space inner product in the Hilbert space math H math . If we define the infinite sum math x sum k 1 infty left langle x,e k right rangle e k, math consisting of infinite sum of vector resolute math x math in direction math e k math , Bessel s inequality mathematics inequality tells us that this series mathematics series Limit of a sequence converges . One can think of it that there exists math x in H math which can be described in terms of potential basis math e 1, e 2, ... math . For a complete orthonormal sequence that is, for an orthonormal sequence which is a Orthonormal basis basis , we have Parseval s identity , which replaces the inequality with an equality and consequently math x math with math x math . Bessel s inequality follows from the identity math 0 le left x sum k 1 n langle x, e k rangle e k right 2 x 2 2 sum k 1 n langle x, e k rangle 2 sum k 1 n langle x, e k rangle 2 x 2 sum k 1 n langle x, e k rangle 2, math which holds for any natural n . See also Cauchy Schwarz inequality External links http mathworld.wolfram.com BesselsInequality.html Bessel s Inequality the article on Bessel s Inequality on MathWorld. PlanetMath attribution title Bessel inequality id 3089 Category Hilbert space Category Inequalities ca Desigualtat de Bessel de Besselsche Ungleichung es Desigualdad de Bessel eo Neegala o de Bessel fr In galit de Bessel it Disuguaglianza di Bessel kk hu Bessel egyenl tlens g ro Inegalitatea lui Bessel ru fi Besselin ... more details
In mathematics , the Bessel potential is a potential theory potential named after Friedrich Wilhelm Bessel similar to the Riesz potential but with better decay properties at infinity. If s is a complex number with positive real part then the Bessel potential of order s is the operator math I Delta s 2 math where is the Laplace operator and the fractional calculus fractional power is defined using Fourier transforms. See also Riesz potential Fractional integration Sobolev space Fractional Schr dinger equation References eom id B b110420 title Bessel potential operator first R. last Duduchava Citation last1 Grafakos first1 Loukas title Modern Fourier analysis publisher Springer Verlag location Berlin, New York edition 2nd series Graduate Texts in Mathematics isbn 978 0 387 09433 5 doi 10.1007 978 0 387 09434 2 mr 2463316 year 2009 volume 250 eom id B b120170 title Bessel potential space first L.I. last Hedberg eom id B b015870 first E.D. last Solomentsev citation first Elias last Stein authorlink Elias Stein title Singular integrals and differentiability properties of functions publisher Princeton University Press location Princeton, NJ year 1970 isbn 0 691 08079 8 Category Fractional calculus Category Partial differential equations Category Potential theory Category Singular integrals ... more details
Citations missing article date January 2008 Vasily Vasil yevich Bessel lang ru April 25 OS April 13 , 1843 1842? St Petersburg March 1, OS February 16 1907, Zurich was a Russian music publisher. Bessel graduated from the Saint Petersburg Conservatory in 1865 studying violin with Henryk Wieniawski and viola with Hieronymus Weickmann . He co founded the thriving publishing firm V. Bessel and Co. since 1869 and a print shop since 1871 , which published works by prominent Russian composers, notably Pyotr Tchaikovsky , Anton Rubinstein , Alexander Dargomyzhsky and the members of the New Russian Musical School Modest Musorgsky , Nikolai Rimsky Korsakov Nikolay Rimsky Korsakov , C sar Cui , Mily Balakirev , and Alexander Borodin . He was the journal s publisher and also the sole editor of a weekly St Petersburg magazine Muzykal ny listok The Musical Leaf from September 3, 1872 to June 5, 1877, that appeared on Sundays during the nine month Russian musical season, from September October to May June. The journal s purpose was to offer an in depth view of the many aspects of Russian and foreign musical life. Later he published the journal Muzykal noye obozrenie The musical revue 1885 1888 . He also wrote a book Notnoe delo The Note Matter , published in 1901. His brother N. V. Bessel was a co owner of the firm V. Bessel and Co. External links http www.ripm.org journal info.php?ABB MUL Muzykal ny listok Persondata Metadata see Wikipedia Persondata . NAME Bessel, Vasily ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 1843 PLACE OF BIRTH DATE OF DEATH 1907 PLACE OF DEATH DEFAULTSORT Bessel, Vasily Category Russian editors Category 1843 births Category 1907 deaths ru , ... more details
logic , plural quantification is the theory that an individual Variable mathematics variable x may take on plural , as well as singular values. As well as substituting individual objects such as Alice ... an objective existence distinct from the individual objects that fall under them, but is neither ... ideas for a semantics of plurals. Plural quantification Standard first order logic has difficulties ... be systematically interpreted in terms of plural quantification, and that, therefore, 2nd order monadic ... account of plural terms that allows for both distributive and non distributive satisfaction ... are predicates of sets of individuals or of mereological sums . Several writers have suggested that plural ... compete against each other e.g. as teams in an online game , while Nicolas 2008 has argued that plural ..., that plural expressions in ordinary language are manifestly and obviously free of existential commitment ... logical difference, since in the first case the plural is distributive, in the second, it is collective ... University Press. Burgess, J.P., From Frege to Friedman A Dream Come True? , 2004, E Pluribus Unum Plural Logic and Set Theory, Philosophia Mathematica 12 3 193&ndash 221. Cameron, J. R., 1999, Plural Reference, Ratio . Cite journal doi 10.1023 A 1015190829525 last1 Cocchiarella first1 Nino ... issue 3 pages 186 97 citation last McKay first Thomas J. author link Thomas J. McKay title Plural ... 9033 2 last1 Nicolas first1 David year 2008 title Mass nouns and plural logic url http d.a.nicolas.free.fr Nicolas Mass nouns and plural logic Revised 2.pdf journal Linguistics and Philosophy volume ... , 1982, Plural Reference and Set Theory, in Barry Smith , ed., Parts and Moments Studies in Logic and Formal ... last1 Uzquiano first1 Gabriel year 2003 title Plural Quantification and Classes url journal Philosophia ... sep entry plural quant Plural quantification ystein Linnebo Web pages of some people important in the field ... David Nicholas. Category Quantification es Cuantificaci n plural ... more details
In statistics , Bessel s correction , named after Friedrich Bessel , is the use of n 1 instead ... estimation theory estimating the population variance and standard deviation from a sample when the population ... 0 19 954145 4 entry for Variance data ref the factor n n 1 is itself called Bessel s correction . A subtle point is that, while the sample variance using Bessel s correction is an unbiased ... estimate of the population standard deviation because the square root is a concave function , the bias ..., such as the normal see unbiased estimation of standard deviation for details. One can understand Bessel ... from the population math 2051, quad 2053, quad 2055, quad 2050, quad 2051 , math One may compute ... terms, we are seeking the sum of squared distances from the population mean, but end up calculating the sum of squared differences from the sample mean, which, as will be seen, is the number that minimizes ... the deviation from an individual to the sample mean, and math b math representing the deviation from the sample mean to the population mean. Note that we ve simply decomposed the actual deviation from ... to the squares of deviations from the population mean math begin align , underbrace 2053 2050 begin smallmatrix text Deviation from text the population text mean end smallmatrix , 2 & , overbrace , underbrace 2053 2052 begin smallmatrix text Deviation from text the sample mean end smallmatrix , text ... from the sample average must be zero. When the middle column has vanished, we then observe ... from the sample mean The sum of all of the entries in the remaining two columns a sup 2 sup and b sup 2 sup is the sum of squares of the deviations from the population mean, because of the way ... The sum of squares of the deviations from the population mean will be bigger than the sum of squares of the deviations from the sample mean except when the population mean is the same as the sample mean, in which case the two are equal . That is why the sum of squares of the deviations from the sample ... more details
A plural society is defined by Fredrik Barth as a society combining Ethnicity ethnic contrasts the Economics economic interdependence of those groups, and the Ecology ecological specialization i.e., use ..., but only in the marketplace in buy ing and sell ing. There is a plural society, with different sections of the community living side by side, within the same Politics political unit. Plural Societies and Democratic Regimes During research about plural societies, Asim Ejaz, Student of M.phil Political ... of Arend Lijphart, democracy in plural societies that it is so much difficult to achieve and stable democratic government in plural society. As Aristotle says about stable governing system that, a state ... impossible to achieve and maintain stable democratic government in plural societies. Consociational .... For the successful democratic regimes in the third world, due to plural societies, Consociational democracy is based, also on normative model. A Plural society is a society, divided by segmental ... due to plural societies within European countries. According to Geberial Almond, Separation of power doctrine is also concerned with political stability. He extends idea of separation of power from ... have created homogeneity among their plural societies, as like idealize British society. But Geberial ... Democracy and the Segments of Plural Society As Arend Lijhpart argue that there is constitutionalization for the segments of plural societies, and its better solution is consociational or semi ... countries in the list of Consociational and Semi consociational Democracy regarding study of plural .... Segments of plural Society can represent their self within that political party and can create ..., even successful in People Republic of China nowadays, where Mangols from Inner Mangolia , Haan and Yugur from Xinjiang , Tibetan from Tibet and several other segments of society are joining ... Plural Society Category Cultural concepts Category Sociological terms ms Masyarakat majmuk nl ... more details
A double plural is a plural form to which an extra suffix has been added, mainly because the original plural suffix had become improductive and therefore irregular. So the form as a whole was no longer seen as a plural this is an instance of morphological leveling . Examples of this can be seen in the history of English language English and Dutch language Dutch . Historically, the general English plural markers were not only s or en but also in certain specific declensions ra ru which is still rather general today in German language German under the form er . The ancient plural of child was cildra cildru , to which an en suffix was later added when the ra ru became unproductive the Dutch plural form kind er en and the corresponding Zeelandic form kind er s are also double plurals which were formed in the same way as the English double plurals, while for example German and Limburgian have historically conservative single plurals such as Kind er . wikt double plural See also Reduplicated plural Category Grammatical number Category Historical linguistics ling morph stub nl Stapelmeervoud ... more details
Infobox Political Party party name Concertaci n Plural colorcode blue party logo leader Cristina Fern ndez de Kirchner foundation 2006 ideology Peronism , Social democracy , Centre left , Democratic socialism , Political radicalism Radicalism and Christian democracy headquarters Buenos Aires , Argentina international colours Blue , White website Plural Consensus lang es Concertaci n Plural is an Kirchnerism Kirchnerist electoral alliance in Argentina acting as a political bloc in the Argentine Senate and elsewhere. The coalition is largely made up of members of the Radical Civic Union , a group of smaller parties and dissident Socialist Party Argentina Socialist Party members who support the governing Peronist s led by President of Argentina President Cristina Fern ndez de Kirchner . Former President N stor Kirchner proposed the coalition as a new centre left force in Argentine politics. http www.clarin.com diario 2006 12 18 um m 01330273.htm The smaller parties originally included the Broad Front , the Party of Victory , the Intransigent Party and the Christian Democratic Party Argentina Christian Democratic Party . The leading former Radical supporting the Kirchners was Julio Cobos , who was elected Vice President. Having been expelled from the Radicals, he set up his own party, which eventually settled on the name ConFe Federal Consensus . Formally, the bloc s two senators are Senator Dora S nchez of Corrientes Province and Pablo Verani of R o Negro Province , both former Radicals. Other leading members of the concertation, often termed Radicales K , are governors Miguel Saiz , Gerardo Zamora and Arturo Colombi and deputies such as Hugo Nelson Prieto . In the Argentine ... de la Concertaci n exists, led by Prieto. The Plural Consensus tag has been widely used in the media ... Coalitions of parties in Argentina Argentina party stub es Concertaci n Plural it Concertaci n Plural ... more details
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