merge Ornstein Uhlenbeck process discuss Talk Gauss Markov process Merger proposal date March 2012 Unreferenced date December 2009 Distinguish2 the Gauss Markov theorem of mathematical statistics Gauss Markov stochastic processes named after Carl Friedrich Gauss and Andrey Markov are stochastic process es that satisfy the requirements for both Gaussian process es and Markov process es. The stationary Gauss Markov process is a very special case because it is unique, except for some trivial exceptions. Every Gauss Markov process X t possesses the three following properties If h t is a non zero scalar function of t , then Z t h t X t is also a Gauss Markov process If f t is a non decreasing scalar function of t , then Z t X f t is also a Gauss Markov process There exists a non zero scalar function h t and a non decreasing scalar function f t such that X t h t W f t , where W t is the standard Wiener process . Property 3 means that every Gauss Markov process can be synthesized from the standard Wiener process SWP . Properties of the Stationary Gauss Markov Processes A stationary Gauss Markov process with variance math textbf E X 2 t sigma 2 math and time constant math beta 1 math has the following properties. Exponential autocorrelation math textbf R x tau sigma 2 e beta tau . , math A power spectral density PSD function that has the same shape as the Cauchy distribution math textbf S x j omega frac 2 sigma 2 beta omega 2 beta 2 . , math Note that the Cauchy distribution and this spectrum differ by scale factors. The above yields the following spectral factorization math textbf S x s frac 2 sigma 2 beta s 2 beta 2 frac sqrt 2 beta , sigma s beta cdot frac sqrt 2 beta , sigma s beta . math which is important in Wiener filtering and other areas. There are also some trivial exceptions to all of the above. DEFAULTSORT Gauss Markov Process Category Markov processes probability stub he ... more details
Noref date August 2011 The generalized Gauss Newton method is a generalization of the least squares method originally described by Carl Friedrich Gauss and of Newton s method due to Isaac Newton to the case of constrained nonlinear least squares problems. Math stub Category Numerical analysis ar ... more details
Image Gauss alpha 4.9 beta 0.58 cobweb.png thumb right 300px Cobweb plot of the Gauss map for math alpha 4.90 math and math beta 0.58 math . This shows an 8 cycle. In mathematics , the Gauss map also known as Gaussian map ref Chaos and nonlinear dynamics an introduction for scientists and engineers, by Robert C. Hilborn, 2nd Ed., Oxford, Univ. Press, New York, 2004. ref or mouse map , is a nonlinear iterated map of the Real number reals into a real interval given by the Gaussian function math x n 1 exp alpha x 2 n beta, , math where and are real parameters. Named after Carl Friedrich Gauss Johann Carl Friedrich Gauss , the function maps the bell shaped Gaussian function similar to the logistic map . Properties In the parameter space math x n math can be chaotic. The map is also called the mouse map because its bifurcation diagram resembles a mouse see Figures . Image Gauss Orbit Map alpha 4.9.png thumb 300px Bifurcation diagram of the Gauss map with math alpha 4.90 math and math beta math in the range &minus 1 to 1. This graph resembles a mouse. Image Gauss Orbit Map alpha 6.2.png thumb 300px Bifurcation diagram of the Gauss map with math alpha 6.20 math and math beta math in the range &minus 1 to 1. References references references DEFAULTSORT Gauss Iterated Map Category Chaotic maps ... more details
Unreferenced date December 2009 In cartography , the term Gauss Kr ger , named after Carl Friedrich Gauss and Johann Heinrich Louis Kr ger , is used in three slightly different ways. Often, it is just a synonym for the transverse Mercator map map projection projection . Another synonym is Gauss conformal projection . Sometimes, the term is used for a particular computational method for transverse Mercator that is, how to convert between latitude longitude and projected coordinates. There is no simple closed formula to do so when the earth is modelled as an ellipsoid. But the Gauss Kr ger method gives the same results as other methods, at least if you are sufficiently near the central meridian less than 10 degrees of longitude, say. Farther away, some methods become inaccurate. The term is also used for a particular set of transverse Mercator projections used in narrow zones in Europe and South America, at least in Germany, Turkey, Austria, Slovenia, Finland and Argentina. This Gauss Kr ger system is similar to the universal transverse Mercator system, but the central meridians of the Gauss Kr ger zones are only 3 apart, as opposed to 6 in UTM. As a consequence, the scale variation within a Gauss Kr ger zone is about 1 4 of what it is in a UTM zone. More information on this system can be found on the German Wikipedia. DEFAULTSORT Gauss Kruger Coordinate System Category Geographic coordinate systems Cartography stub de Gau Kr ger Koordinatensystem ja pl Odwzorowanie Gaussa Kr gera sq Sistemi koordinativ Gau Kr ger sr sv Gauss Kr ger uk zh ... more details
In linear algebra , Gauss Jordan elimination is an algorithm for getting matrix mathematics matrices ... echelon form. Gauss Jordan elimination goes a step further by placing zeros above and below each pivot ... form, and Gauss Jordan elimination is guaranteed to find it. It is named after Carl Friedrich Gauss ... year. Jordan and Clasen probably discovered Gauss Jordan elimination independently. ref Citation last1 Althoen first1 Steven C. last2 McLaughlin first2 Renate title Gauss Jordan reduction a brief history ... theory complexity theory shows Gauss Jordan elimination has time complexity of order math O n 3 math .... Gaussian elimination shares Gauss Jordan s time complexity order of math O n 3 math , however despite sharing the same order, Gauss Jordan elimination requires approximately 50 more computation steps ... Publishing Company, 1995, Chapter 10 ref . This brings down the fundamental question, why should Gauss Jordan elimination method be used over Gaussian elimination. Gauss Jordan elimination method is used ... speed is the main criteria. It has been shown that even though Gauss Jordan elimination method requires more computation steps than Gaussian elimination, in a multiple processor environment, Gauss Jordan ... cost of the Gauss Jordan elimination method ref G. A. Darmohray and E. D. Brooks, Gaussian Techniques ... . Application to finding inverses If Gauss Jordan elimination is applied on a square matrix , it can ... External links wikibooks 1 Linear Algebra 2 Linear Algebra Gauss Jordan Reduction 3 Gauss Jordan elimination ... for Gauss Jordan elimination in Octave http elonen.iki.fi code misc notes python gaussj index.html Algorithm for Gauss Jordan elimination in Python http lipe.advant.com.br unicenp gauss jordan.php An online tool solve nxm linear systems using Gauss Jordan elimination source code and mobile version ... for Gauss Jordan elimination in Basic http math.fullerton.edu mathews n2003 GaussianJordanMod.html Module for Gauss Jordan Elimination http vivaldi.ucsd.edu 8080 kcheng ece155 hwsoln Gaussian ... more details
dablink Gauss redirects here. For other persons or things named Gauss, see Gauss disambiguation . For many things named after this person, see List of topics named after Carl Friedrich Gauss . Infobox scientist name Carl Friedrich Gauss image Carl Friedrich Gauss.jpg caption Carl Friedrich Gauss 1777 ... known for List of topics named after Carl Friedrich Gauss See full list author abbrev bot author abbrev ... Gauss IPAc en icon a s lang de Gau audio De carlfriedrichgauss.ogg listen , lang la Carolus Fridericus Gauss 30 April 1777 spaced ndash 23 February 1855 was a Germans German mathematician and physical ... , Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one ..., 1927 . http www.mathsong.com cfgauss Dunnington 1927 The Sesquicentennial of the Birth of Gauss dead ..., Wolfgang Sartorius von 1856, repr. 1965 . Gauss zum Ged chtniss. S ndig Reprint Verlag H. R ... of Gauss in Braunschweig.jpg left thumb Statue of Gauss at his birthplace, Braunschweig Carl Friedrich Gauss was born on 30 April 1777 in Braunschweig , in the duchy of Braunschweig Wolfenb ttel, now ... www.math.wichita.edu history men gauss.html title Carl Friedrich Gauss last first date work publisher ... Feast of the Ascension , which itself occurs 40 days after Easter . Gauss would later solve this puzzle ... gauss.htm title Gauss Birthday Problem last first date work publisher ref He was christened and Confirmation ... url http www.gausschildren.org genwiki index.php?title Letter WORTHINGTON, Helen to Carl F. Gauss 1911 07 26 title Letter WORTHINGTON, Helen to Carl F. Gauss 1911 07 26 publisher Susan D. Chambless date 2000 03 11 accessdate 2011 09 14 ref Gauss was a child prodigy . There are many anecdotes ... number theory as a discipline and has shaped the field to the present day. Gauss s intellectual abilities ... of G ttingen University of G ttingen from 1795 to 1798. While in university, Gauss independently ... Greeks , and the discovery ultimately led Gauss to choose mathematics instead of philology as a career ... more details
The Gauss Research Laboratory, Inc. is the corporation in charge of managing the Puerto Rico s Top Level Domain . It is responsible for providing a stable and secure management of the .pr domain. This organization has been recognized and approved by the international oversight agencies, ICANN and ARIN . External links http www.grl.pr Gauss Research Laboratory, Inc. http www.nic.pr Puerto Rico Network Information Center http www.iana.org domains root db pr.html IANA .pr whois information Category Domain name system Category Internet in the United States ... more details
In numerical analysis , Gauss Jacobi quadrature is a method of numerical quadrature based on Gaussian quadrature . Gauss Jacobi quadrature can be used to approximate integrals of the form math int 1 1 f x 1 x alpha 1 x beta , mathrm d x math where is a smooth function on 1, 1 and , 1. The interval 1, 1 can be replaced by any other interval by a linear transformation. Thus, Gauss Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss Legendre quadrature is a special case of Gauss Jacobi quadrature with 0. Similarly, Chebyshev Gauss quadrature arises when one takes . More generally, the special case turns Jacobi polynomials into Gegenbauer polynomials , in which case the technique is sometimes called Gauss Gegenbauer quadrature . Gauss Jacobi quadrature uses x 1 x sup sup 1 x sup sup as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials . Thus, the Gauss Jacobi quadrature rule on n points has the form math int 1 1 f x 1 x alpha 1 x beta , mathrm d x approx lambda 1 f x 1 lambda 2 f x 2 cdots lambda n f x n , math where x sub 1 sub , , x sub n sub are the roots of the Jacobi polynomial of degree n . The weights sub 1 sub , , sub n sub are given by the formula math lambda i frac 2n alpha beta 2 n alpha beta 1 frac Gamma n alpha 1 Gamma n beta 1 Gamma n alpha beta 1 n 1 frac 2 alpha beta P n x i P n 1 x i , math where denotes the Gamma function and P sub n sub the Jacobi polynomial of degree n . References Citation last1 Rabinowitz first1 Philip author1 link Philip Rabinowitz mathematician title A First Course in Numerical Analysis publisher Dover Publications location New York edition 2nd isbn 978 0 486 41454 6 year 2001 chapter 4.8 1 Gauss Jacobi quadrature ... rule free software Matlab, C , and Fortran to evaluate integrals by Gauss Jacobi quadrature rules. http ... Matlab, C , and Fortran for Gauss Gegenbauer quadrature Category Numerical integration quadrature ... more details
In number theory , quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss ... given by a quadratic character for a general character, one obtains a more general Gauss sum . These objects are named after Carl Friedrich Gauss , who studied them extensively and applied ... biquadratic reciprocity laws. Definition Let p be an odd prime number and a an integer. Then the Gauss ... by p , an alternative expression for the Gauss sum with the same value is math G a, chi sum n 1 p 1 ... of the Legendre symbol defines the Gauss sum G &chi . Properties The value of the Gauss sum is an algebraic integer in the p th cyclotomic field Q &zeta sub p sub . The evaluation of the Gauss sum can be reduced to the case a 1 math g a p left frac a p right g 1 p . math The exact value of the Gauss sum, computed by Gauss, is given by the formula math g 1 p sum n 0 p 1 e 2 pi in 2 p begin cases ... left frac 1 p right p math was easy to prove and led to one of Gauss s proofs of quadratic reciprocity . However, the determination of the sign of the Gauss sum turned out to be considerably more difficult Gauss could only establish it after several years work. Later, Dirichlet , Kronecker , Issai Schur Schur and other mathematicians found different proofs. Generalized quadratic Gauss sums Let a , b , c be natural numbers . The generalized Gauss sum G a , b , c is defined by math G a,b,c sum n 0 ... Gauss sum is the sum math G a,c G a,0,c math . Properties The Gauss sum G a , b , c depends only on the residue class of a , b modulo c . Gauss sums are multiplicative function multiplicative , i.e. ... of quadratic Gauss sums one may always assume gcd a , c 1 . Let a , b and c be integers with math ac ... more general Gauss sums math sum n 0 c 1 e pi i a n 2 bn c c a 1 2 e pi i ac b 2 4ac sum n 0 a 1 e ... end cases math for every odd integer m . The values of Gauss sums with b 0 and gcd a , c 1 are explicitly ... Gau . For b 0 the Gauss sums can easily be computed by completing the square in most cases ... more details
In mathematics , the generalized Gauss Bonnet theorem also called Shiing Shen Chern Chern Gauss Pierre Ossian Bonnet Bonnet theorem presents the Euler characteristic of a closed even dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of the Gauss Bonnet theorem named after Carl Friedrich Gauss and Pierre Ossian Bonnet to higher dimensions. Let M be a compact space compact 2 n dimensional Riemannian manifold without boundary, and let math Omega math be the curvature form of the Levi Civita connection . This means that math Omega math is an math mathfrak s mathfrak o 2n math valued 2 form on M . So math Omega math can be regarded as a skew symmetric 2 n 2 n matrix whose entries are 2 forms, so it is a matrix over the commutative ring math bigwedge hbox even T M math . One may therefore take the Pfaffian of math Omega math , math mbox Pf Omega math , which turns out to be a 2 n form. The generalized Gauss Bonnet theorem states that math int M mbox Pf Omega 2 pi n chi M math where math chi M math denotes the Euler characteristic of M . Example dimension 4 In dimension math n 4 math , for a compact oriented manifold ... R math is the scalar curvature . Further generalizations As with the two dimensional Gauss Bonnet Theorem, there are generalizations when M is a manifold manifold with boundary . The Gauss Bonnet Theorem can be seen as a special instance in the theory of characteristic classes . The Gauss ... the metric, and so is an invariant of smooth structure. An extremely far reaching generalization of the Gauss ... index , which can be expressed in terms of characteristic classes. The 2 dimensional Gauss Bonnet Theorem ... index is defined in terms of the Gauss Bonnet integrand. See also Chern Weil homomorphism ... first time that Chern Gauss Bonnet was proven without assuming the manifold to be a hypersurface ... in this paper of Chern. Category Theorems in differential geometry es Teorema de Gauss Bonnet generalizado ... more details
Unreferenced date December 2009 Clarence Edward Gauss January 12, 1887 1960 was an United States American diplomat . Gauss was born in Washington, D.C. , as the son of Herman Gauss and Emile J. Eisenman Gauss. He married Rebecca Louise Barker in 1917. He was a Republican Party United States Republican and a Protestant . Diplomatic career Foreign Service Officer for the United States Foreign Service U.S. Vice Consul in Shanghai , 1912 15 U.S. Consul in Shanghai, 1916 Amoy, 1916 20 Tsinan, 1920 23 U.S. Consul General in Mukden, 1923 24 Tsinan, 1924 26 Shanghai, 1926 27, 1935 38 Tientsin, 1927 31 Paris, 1935 U.S. Minister to Australia, 1940 41 was the United States ambassador to the Republic of China before and during the Second World War . He resigned from the post in November 1944, and was replaced by Patrick Hurley . Chronology U.S. Vice Consul Shanghai , 1912 15 U.S. Consul Shanghai, 1916 Amoy , 1916 20 Tsinan , 1920 23 U.S. Consul General Mukden , 1923 24 Tsinan, 1924 26 Shanghai, 1926 27, 1935 38 Tientsin , 1927 31 Paris , 1935 S start S dip Succession box before Nelson T. Johnson title US Ambassador to China years 1941&ndash 1944 after Patrick Hurley Succession box title List of United States ambassadors to Australia U.S. Ambassador to Australia before first incumbent after Nelson T. Johnson years 1940 1941 S end US Ambassadors to the PRC Persondata Metadata see Wikipedia Persondata . NAME Gauss, Clarence E. ALTERNATIVE NAMES SHORT DESCRIPTION American diplomat DATE OF BIRTH January 12, 1887 PLACE OF BIRTH DATE OF DEATH 1960 PLACE OF DEATH DEFAULTSORT Gauss, Clarence E. Category 1887 births Category 1960 deaths Category Ambassadors of the United States to Australia Category Ambassadors of the United States to China ... more details
In general relativity , Gauss Bonnet gravity , also referred to as Einstein Gauss Bonnet gravity , ref Citation last Lovelock first David title The Einstein tensor and its generalizations journal J. Math. Phys. volume 12 issue 3 pages 498 year 1971 url http link.aip.org link JMAPAQ v12 i3 p498 s1 ref is a modification of the Einstein Hilbert action to include the generalized Gauss Bonnet theorem Gauss Bonnet term named after Carl Friedrich Gauss and Pierre Ossian Bonnet math G R 2 4R mu nu R mu nu R mu nu rho sigma R mu nu rho sigma math math int d Dx sqrt g , G math This term is only nontrivial in 4 1D or greater, and as such, only applies to extra dimensional models. In 3 1D and lower, it reduces to a topological divergence theorem surface term . This follows from the Gauss Bonnet theorem on a 4D manifold math frac 1 8 pi 2 int d 4x sqrt g , G Chi M math . Despite being quadratic in the Riemann tensor and Ricci tensor , terms containing more than 2 partial derivatives of the metric tensor metric cancel out, making the Euler Lagrange equations partial differential equation Equations of second order second order quasilinear partial differential equations in the metric. Consequently, there are no additional dynamical degrees of freedom, as in say f R gravity . More generally, we may consider math int d Dx sqrt g , f left G right math term for some function f . Nonlinearities in f render this coupling nontrivial even in 3 1D. However, fourth order terms reappear with the nonlinearities. See also Einstein Hilbert action f R gravity Lovelock gravity References reflist Theories of gravitation Category Theories of gravitation relativity stub ... more details
In numerical analysis and scientific computing , the Gauss Legendre methods are a family of numerical methods for ordinary differential equations . Gauss Legendre methods are implicit Runge Kutta methods . More specifically, they are collocation method s based on the points of Gauss Legendre quadrature . The Gauss Legendre method based on s points has order 2 s . ref harvnb Iserles 1996 p 47 ref All Gauss Legendre methods are A stability A stable . ref harvnb Iserles 1996 p 63 ref The Gauss Legendre method of order two is the implicit midpoint rule . Its Butcher tableau is cellpadding 3px cellspacing 0px style text align center style border right 1px solid border bottom 1px solid 1 2 style border bottom 1px solid 1 2 style border right 1px solid 1 The Gauss Legendre method of order four has Butcher tableau cellpadding 3px cellspacing 0px style text align center style border right 1px solid math tfrac12 tfrac16 sqrt3 math math tfrac14 math math tfrac14 tfrac16 sqrt3 math style border right 1px solid border bottom 1px solid math tfrac12 tfrac16 sqrt3 math style border bottom 1px solid math tfrac14 tfrac16 sqrt3 math style border bottom 1px solid math tfrac14 math style border right 1px solid math tfrac12 math math tfrac12 math The Gauss Legendre method of order six has Butcher tableau cellpadding 3px cellspacing 0px style text align center style border right 1px solid math tfrac12 tfrac1 10 sqrt 15 math math tfrac5 36 math math tfrac29 tfrac1 15 sqrt 15 math math tfrac5 36 tfrac1 30 sqrt 15 math style border right 1px solid math tfrac12 math math tfrac5 36 tfrac1 24 sqrt 15 math math tfrac29 math math tfrac5 36 tfrac1 24 sqrt 15 math style border right 1px solid border bottom 1px solid math tfrac12 tfrac1 10 sqrt 15 math style border bottom 1px solid math tfrac5 36 tfrac1 30 sqrt 15 math style border bottom 1px solid math tfrac29 tfrac1 15 sqrt 15 math style border bottom ... tfrac5 18 math The computational cost of higher order Gauss Legendre methods is usually too high ... more details
File Gauss Bonnet theorem.svg thumb 300px An example of complex region where Gauss Bonnet theorem can apply. Shows the sign of geodesic curvature. The Gauss Bonnet theorem or Gauss Bonnet formula in differential geometry is an important statement about surface s which connects their geometry in the sense of curvature to their topology in the sense of the Euler characteristic . It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848. Statement of the theorem Suppose math M math is a Compact ... 1 the Gauss Bonnet formula does not work. It holds true however for the compact closed unit ... specify a triangle on M formed by three geodesic s. Then we can apply Gauss Bonnet to the surface T formed ... cases of Gauss Bonnet. Triangles In spherical trigonometry and hyperbolic trigonometry , the area ... case of Gauss Bonnet, where the curvature is concentrated at discrete points the vertices . Thinking of curvature as a Measure mathematics measure , rather than as a function, Descartes theorem is Gauss Bonnet where the curvature is a discrete measure , and Gauss Bonnet for measures generalizes both Gauss Bonnet for smooth manifolds and Descartes theorem. Combinatorial analog There are several combinatorial analogs of the Gauss Bonnet theorem. We state the following one. Let math M math be a finite ... Generalizations of the Gauss Bonnet theorem to n dimensional Riemannian manifolds were ... see generalized Gauss Bonnet theorem and Chern Weil homomorphism . The Riemann Roch theorem can also be seen as a generalization of Gauss Bonnet. An extremely far reaching generalization of all the above ... mathworld.wolfram.com Gauss BonnetFormula.html Gauss Bonnet Theorem at Wolfram Mathworld Category ... Bonnet es Teorema de Gauss Bonnet fr Formule de Gauss Bonnet ko it Teorema di Gauss Bonnet nl Stelling van Gauss Bonnet ru sv Gauss Bonnets sats uk zh ... more details
as the Gauss Kuzmin distribution . This follows in part because the Gauss map acts as a truncating ...... OEIS A038517 and its absolute value is known as the Gauss Kuzmin Wirsing constant . Analytic ... formed, and thus very numerically tractable. Note that each entry is a finite rational zeta series . The Gauss ... 8 See section 15 . K. I. Babenko, On a Problem of Gauss , Soviet Mathematical Doklady 19 136 140 1978 MR 57 12436 K. I. Babenko and S. P. Jur ev, On the Discretization of a Problem of Gauss , Soviet Mathematical Doklady 19 731 735 1978 . MR 81h 65015 A. Durner, On a Theorem of Gauss Kuzmin L vy. Arch. Math. 58 , 251 256, 1992 . MR 93c 11056 A. J. MacLeod, High Accuracy Numerical Values of the Gauss ... of Gauss Kuzmin L vy and a Frobenius Type Theorem for Function Spaces. Acta Arith. 24 , 507 ... computation of the Gauss Kuzmin Wirsing constant 2003 Contains a very extensive collection of references ... gauss kuzmin.ps On the Gauss Kuzmin Wirsing Constant 1995 . Linas Vepstas http www.linas.org math gkw.pdf The Bernoulli Operator, the Gauss Kuzmin Wirsing Operator, and the Riemann Zeta 2004 PDF External links MathWorld urlname Gauss Kuzmin WirsingConstant title Gauss Kuzmin Wirsing Constant SloanesRef ... Gauss Kuzmin Wirsing it Costante di Gauss Kuzmin Wirsing ... more details
action principle . See also Appell s equation of motion References Gauss CF. 1829 Crelle s Journal f. Math., 4 , 232. Gauss CF. Werke , 5 , 23. Hertz H. 1896 Principles of Mechanics , in Miscellaneous Papers , vol. III, Macmillan. External links http eom.springer.de g g043500.htm Gauss principle ... more details
Probability distribution name Gauss Kuzmin type mass pdf image cdf image parameters none support math k in 1,2, ldots math pdf math log 2 left 1 frac 1 k 1 2 right math cdf math 1 log 2 left frac k 2 k 1 right math mean math infty math median math 2 , math mode math 1 , math variance math infty math skewness not defined kurtosis not defined entropy 3.432527514776... ref cite journal last1 Blachman first1 N. year 1984 title The continued fraction as an information source Corresp. url journal IEEE Transactions onInformation Theory volume 30 issue 4 pages 671 674 doi 10.1109 TIT.1984.1056924 ref ref name KornerupMatula cite journal last Kornerup first P. first2 D. last2 Matula title LCF A lexicographic binary representation of the rationals journal Journal of Universal Computer Science month July ... Fractions Gauss Kuzmin Entropy year 2008 url http linas.org math entropy.pdf ref can show 3.432527514..., but via nontrivial math mgf char In mathematics , the Gauss Kuzmin distribution is a discrete ... in 0, 1 . ref MathWorld title Gauss Kuzmin Distribution urlname Gauss KuzminDistribution ref The distribution is named after Carl Friedrich Gauss , who derived it around 1800, ref cite book last Gauss first C.F. url http gdz.sub.uni goettingen.de dms load img ?PPN PPN236018647 pages 552 ... in 1929. ref cite journal last Kuzmin first R.O. title On a problem of Gauss journal DAN ... of Gauss journal Atti del Congresso Internazionale dei Matematici, Bologna year 1932 volume 6 pages ..., Eduard Wirsing showed ref cite journal last Wirsing first E. title On the theorem of Gauss Kusmin ... pages pp. 507 528 ref that, for &lambda 0.30366... the Gauss Kuzmin Wirsing constant , the limit math ... cite journal last Babenko first K.I. title On a problem of Gauss journal Soviet Math. Dokl. year 1978 volume 19 pages pp. 136 140 ref See also Khinchin s constant Gauss Kuzmin Wirsing operator References ... fr Loi de Gauss Kuzmin ru ... more details
File Gauss Hermite quadrature weights.svg thumb Weights versus x sub i sub for four choices of n In numerical analysis , Gauss Hermite quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind math int infty infty e x 2 f x ,dx. math In this case math int infty infty e x 2 f x ,dx approx sum i 1 n w i f x i math where n is the number of sample points to use for the approximation. The x sub i sub are the roots of the physicists Hermite polynomial H sub n sub x i 1,2,..., n and the associated weights w sub i sub are given by ref name AS Abramowitz and Stegun Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions , 10th printing with corrections 1972 , Dover, ISBN 978 0 486 61272 0. Equation 25.4.46. ref math w i frac 2 n 1 n sqrt pi n 2 H n 1 x i 2 . math References references Further reading http dlmf.nist.gov 3.5 P6 3.5. Quadrature Gauss Hermite Formula at the Digital Library of Mathematical Functions For tables of Gauss Hermite abscissae and weights up to order n 32 see http www.efunda.com math num integration findgausshermite.cfm. External links http people.sc.fsu.edu jburkardt cpp src gen hermite rule gen hermite rule.html Generalized Gauss Hermite quadrature , free software in C , Fortran, and Matlab Category Numerical integration quadrature ... more details
In numerical mathematics , the Gauss Kronrod quadrature formula is a method for numerical integration calculating approximate values of integral s . Gauss Kronrod quadrature is a variant of Gaussian quadrature , in which the evaluation points are chosen so that an accurate approximation can be computed by re using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order the latter called an embedded rule . The difference between these two approximations is used to estimate the calculational error of the integration. These formulas are named after Alexander Kronrod , who invented them in the 1960s, and Carl Friedrich Gauss . Gauss Kronrod quadrature is used in the QUADPACK library, the GNU Scientific Library , the NAG Numerical Libraries and R programming language R . ref http stat.ethz.ch R manual R patched library stats html integrate.html http stat.ethz.ch R manual R patched library stats html integrate.html ref Description The problem in numerical integration is to approximate definite integrals ... is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous ... must be evaluated at every point. Gauss Kronrod formulas are extensions of the Gauss quadrature ... order estimate. The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error. Example A popular example combines a 7 point Gauss rule with a 15 point Kronrod rule Harv Kahaner Moler Nash 1989 loc § 5.5 . Because the Gauss points ... Gauss nodes Weights 0.94910 79123 42759 &lowast 0.12948 49661 68870 0.74153 11855 99394 &lowast 0.27970 ... based on Gauss Kronrod rules. SLATEC at Netlib is a large public domain library for numerical computing ... http www.advanpix.com 2011 11 07 gauss kronrod quadrature nodes weights Arbitrary precision Gauss ... more details
hatnote This article is about Gauss s law concerning the gravitational field. For analogous laws concerning different fields, see Gauss s law electricity or Gauss s law for magnetism . For Gauss s theorem, a mathematical theorem relevant to all of these laws, see Divergence theorem . In physics , Gauss s law for gravity , also known as Gauss s flux theorem for gravity , is a law of physics which is essentially equivalent to Newton s law of universal gravitation . Although Gauss s law for gravity is physically equivalent to Newton s law, there are many situations where Gauss s law for gravity offers a more convenient and simple way to do a calculation than Newton s law. The form of Gauss s law for gravity is mathematically similar to Gauss s law for electrostatics , one of Maxwell s equations . Gauss s law for gravity has the same mathematical relation to Newton s law that Gauss s law for electricity ..., analogous to how magnetic flux is a surface integral of the magnetic field. Gauss s law for gravity ... . Integral form The integral form of Gauss s law for gravity states Equation box 1 indent equation ... negative or zero , and never positive. This can be contrasted with Gauss s law for electricity, where ... or negative, while mass can only be positive. Differential form The differential form of Gauss s law .... Relation to the integral form The two forms of Gauss s law for gravity are mathematically equivalent ... M int V rho dV math we can apply the divergence theorem to the integral form of Gauss s law for gravity ... at math nabla cdot mathbf g 4 pi G rho math which is the differential form of Gauss s law for gravity ... computation. Relation to Newton s law Deriving Gauss s law from Newton s law Gauss s law ... in the box below. It is mathematically identical to the proof of Gauss s law in electrostatics ... nabla cdot mathbf g mathbf r 4 pi G rho mathbf r math which is the differential form of Gauss s law for gravity, as desired. Deriving Newton s law from Gauss s law and irrotationality It is impossible ... more details
Unreferenced date April 2008 In complex analysis , a branch of mathematics, the Gauss Lucas theorem gives a geometry geometrical relation between the root of a function root s of a polynomial P and the roots of its derivative P nowiki nowiki . The set of roots of a real or complex polynomial is a set of point geometry points in the complex plane . The theorem states that the roots of P nowiki nowiki all lie within the convex hull of the roots of P , that is the smallest convex polygon containing the roots of P . When P has a single root then this convex hull is a single point and when the roots lie on a line geometry line then the convex hull is a line segment segment of this line. The Gauss Lucas theorem, named after Carl Friedrich Gauss and F lix Lucas is similar in spirit to Rolle s theorem . Formal statement If P is a nonconstant polynomial with complex coefficients, all root of a function zeros of P nowiki nowiki belong to the convex hull of the set of zeros of P . Special cases It is easy to see that if P x ax sup 2 sup bx c is a second degree polynomial , the zero of P nowiki nowiki x 2 ax b is the average of the roots of P . In that case, the convex hull is the line segment with the two roots as endpoints and it is clear that the average of the roots is the middle point of the segment. In addition, if a polynomial of degree n of real number real coefficients has n distinct real zeros math x 1 x 2 cdots x n , math , we see, using Rolle s theorem , that the zeros of the derivative polynomial are in the interval math x 1,x n , math which is the convex hull of the set ... s theorem Sturm s theorem Properties of polynomial roots Gauss s lemma polynomial Polynomial function ... Lucas Gauss Theorem by Bruce Torrence, the Wolfram Demonstrations Project . DEFAULTSORT Gauss Lucas ... analysis de Satz von Gau Lucas es Teorema de Gauss Lucas fr Th or me de Gauss Lucas ko he nl Stelling van Gauss Lucas ru ... more details
In numerical analysis Chebyshev Gauss quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind math int 1 1 frac f x sqrt 1 x 2 ,dx math and math int 1 1 sqrt 1 x 2 g x ,dx. math In the first case math int 1 1 frac f x sqrt 1 x 2 ,dx approx sum i 1 n w i f x i math where math x i cos left frac 2i 1 2n pi right math and the weight math w i frac pi n . math ref name AS1 Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions , 10th printing with corrections 1972 , Dover, ISBN 978 0 486 61272 0. Equation 25.4.38. ref In the second case math int 1 1 sqrt 1 x 2 g x ,dx approx sum i 1 n w i g x i math where math x i cos left frac i n 1 pi right math and the weight math w i frac pi n 1 sin 2 left frac i n 1 pi right . , math ref name AS2 Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions , 10th printing with corrections 1972 , Dover, ISBN 978 0 486 61272 0. Equation 25.4.40. ref See also Chebyshev nodes References references External links http mathworld.wolfram.com Chebyshev GaussQuadrature.html Chebyshev Gauss Quadrature from Wolfram MathWorld http people.sc.fsu.edu jburkardt cpp src chebyshev1 rule chebyshev1 rule.html Gauss Chebyshev type 1 quadrature and http people.sc.fsu.edu jburkardt cpp src chebyshev2 rule chebyshev2 rule.html Gauss Chebyshev type 2 quadrature , free software in C , Fortran, and Matlab. Category Numerical integration quadrature ... more details
The Gauss Matuyama Reversal was a geologic event approximately 2.588 million years ago when the Earth Earth s magnetic field underwent Geomagnetic reversal reversal . This event, which separates the Piacenzian from the Gelasian and marks the start of the Quaternary ref name INQUA 16 1 http www.inqua.tcd.ie documents QP 2016 1.pdf Clague, John et al. 2006 Open Letter by INQUA Executive Committee Quaternary Perspective, the INQUA Newsletter International Union for Quaternary Research 16 1 ref , is useful in dating sediments. Biological effects The event is marked by the extinction of calcareous nannofossils Discoaster pentaradiatus and D. surculus , among others. References nowiki See http en.wikipedia.org wiki Wikipedia Footnotes for an explanation of how to generate footnotes using the ref and ref tags, and the template below. nowiki references See also Geomagnetic reversal Category Paleomagnetism Geology stub es L mite magnetoestratigr fico Gauss Matuyama ... more details
In numerical analysis Gauss Laguerre quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind math int 0 infty e x f x ,dx. math In this case math int 0 infty e x f x ,dx approx sum i 1 n w i f x i math where x sub i sub is the i th root of Laguerre polynomial L sub n sub x and the weight w sub i sub is given by ref name AS Abramowitz, M & Stegun, I A, Handbook of Mathematical Functions , 10th printing with corrections 1972 , Dover, ISBN 978 0 486 61272 0. Equation 25.4.45. http www.nrbook.com abramowitz and stegun page 890.htm access online ref math w i frac x i n 1 2 L n 1 x i 2 . math For more general functions To integrate the function math f math math int 0 infty f left x right dx int 0 infty f left x right e x e x dx int 0 infty g left x right e x dx math . Generalized Gauss Laguerre quadrature More generally, one can also consider integrands that have a known math x alpha math power law singularity at x 0, for some real number math alpha 1 math , leading to integrals of the form math int 0 infty x alpha e x f x ,dx. math This allows one to efficiently evaluate such integrals for polynomial or smooth f x even when is not an integer. ref Philip Rabinowitz and George Weiss, Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form math int 0 infty exp x x n f x dx math , Mathematical Tables and Other Aids to Computation , vol. 13, pp. 285 294 1959 . ref Program to Calculate Modified Gauss Laguerre Abscissa And Weights In Mathematica 7 Suppose want to calculate for n 5 n 5 t Array N Root LaguerreL n, x , &, n w t n 1 LaguerreL n 1 , t 2 modW Array Exp t t n 1 LaguerreL n 1 , t 2 &, n t is an Abscissa array. w is the corresponding array of weights and modW is the corresponding array ... 8067 gauss laguerre Matlab routine for Gauss Laguerre quadrature http people.sc.fsu.edu jburkardt m src gen laguerre rule gen laguerre rule.html generalized Gauss Laguerre quadrature , free ... more details
Image Double gauss.png thumb The double Gauss design with optical ray traces 200px right The double Gauss ... right thumb 350px Development of the Double Gauss The double Gauss lens consists of two back to back Gauss lens es a design with a positive Lens optics Types of simple lenses meniscus ... , and has sometimes been made at f 1.0. Extra wide aperture f 1.4 Double Gauss lenses usually have ... elements. Moderate aperture f 2.8 versions can be simplified to five elements. The Double Gauss ..., pp 245 248. ref Double Gauss Planar tweaks were the standard wide aperture, normal and near normal prime lens for sixty years. History The original two element Gauss was a telescope Objective optics ... Gauss as an improvement to the Joseph von Fraunhofer Fraunhofer Fraunhofer telescope objective ... in 1888 by taking two of these lenses and placing them back to back making a double Gauss ref ... photographic results. ref Kingslake, p 118. ref Current double Gauss lenses can be traced ... aberration . It was the original six element symmetric f 4.5 Double Gauss lens. ref Kingslake ... unsuccessful, but its asymmetry is the foundation of the modern Double Gauss, including ... designed in 1927, had a six element asymmetric Double Gauss formula. Post World War II Zeiss Oberkochen, West Germany no longer uses the Biotar name instead lumping any Double Gauss variant under ..., pp 122 123. ref For example, three asymmetric Double Gauss lenses were produced in 1934 for Ihagee ..., West Sussex, UK Hove Collectors Books, 1987. ISBN 0 906447 38 0. pp 25 26. ref Early Double Gauss ... M3 1953, West Germany . During the 1960s and 70s, every optical house had Double Gauss normal ... Gauss normal lenses. Zooms continue to dominate the digital era, but many new prestige low production Double Gauss lenses have appeared. Compare the Canon EF 50mm f 1.2L USM 2007, Japan , ref Anonymous ... of Double Gauss lens designs widths 120px perrow 4 Image DoubleGauss2text.svg Double Gauss lens designs ... more details
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