and 25.011. Millennium Problems In mathematics, the Riemannhypothesis , proposed by harvs txt first ... hypothesis for curves over finite fields . The Riemannhypothesis implies results about the distribution ... harv Bombieri 2000 . The Riemannhypothesis is part of Hilbert s eighth problem Problem 8 ..., Pierre Deligne proved an analogue of the RiemannHypothesis for zeta functions of varieties defined ... popular books on the Riemannhypothesis, such as harvtxt Derbyshire 2003 , harvtxt Rockmore ..., as none of the factors have zeros. The Riemannhypothesis discusses zeros outside the region of convergence ... Riemann s statement of the Riemannhypothesis, from harv Riemann 1859 . He was discussing a version ... hypothesis. Consequences of the Riemannhypothesis The practical uses of the Riemannhypothesis include many propositions which are known to be true under the Riemannhypothesis, and some which can be shown to be equivalent to the Riemannhypothesis. Distribution of prime numbers Riemann ... sup harv Ingham 1932 . CITEREFvon Koch1901 Von Koch 1901 proved that the Riemannhypothesis is equivalent ... s result, due to harvtxt Schoenfeld 1976 , says that the Riemannhypothesis is equivalent to math ... functions The Riemannhypothesis implies strong bounds on the growth of many other arithmetic ... to the Riemannhypothesis. From this we can also conclude that if the Mertens function ... positive is equivalent to the Riemannhypothesis harv Titchmarsh 1986 . For the meaning of these symbols ... n , so the Riemannhypothesis can also be stated as a condition on the growth of these determinants. The Riemannhypothesis puts a rather tight bound on the growth of M , since harvtxt Odlyzko te Riele 1985 disproved the slightly stronger Mertens conjecture math M x le sqrt x. math The Riemannhypothesis ... n , math for all n 5040 if and only if the Riemannhypothesis is true, where is the Euler Mascheroni constant . Another example was found by harvtxt Franel Landau 1924 showing that the Riemannhypothesis ... more details
In mathematics , the grand Riemannhypothesis is a generalisation of the Riemannhypothesis and Generalized Riemannhypothesis . It states that the nontrivial zeros of all automorphic function automorphic L function L functions lie on the critical line 1 2 it with t a real number and i the imaginary unit. The modified grand Riemannhypothesis is the assertion that the nontrivial zeros of all automorphic L functions lie on the critical line or the real line . Notes It is widely believed that all global L functions are automorphic L functions. The Siegel zero , conjectured not to exist, is a possible real zero of a Dirichlet L series , rather near s 1. L functions of Maass cusp forms can have trivial zeros which are off the real line. mathanalysis stub Category Zeta and L functions Category Conjectures ... more details
The Riemannhypothesis is one of the most important conjecture s in mathematics . It is a statement about the zeros of the Riemann zeta function . Various geometrical and arithmetical objects can be described by so called global L function s , which are formally similar to the Riemann zeta function. One ... of the Riemannhypothesis. Many mathematicians believe these generalizations of the Riemannhypothesis ... s in which case they are called Dirichlet L series Dirichlet L function s . When the Riemannhypothesis is formulated for Dedekind zeta functions, it is known as the extended Riemannhypothesis ERH and when it is formulated for Dirichlet L functions, it is known as the generalized Riemannhypothesis ... the label generalized Riemannhypothesis to cover the extension of the Riemannhypothesis to all global L functions, not just the special case of Dirichlet L functions. Generalized Riemannhypothesis GRH The generalized Riemannhypothesis for Dirichlet L functions was probably formulated for the first time by Piltz in 1884. Like the original Riemannhypothesis, it has far reaching consequences about the distribution of prime number s. The formal statement of the hypothesis follows. A Dirichlet ... the ordinary Riemannhypothesis. Consequences of GRH Dirichlet s theorem on arithmetic progressions ... Riemannhypothesis is true, then for every coprime a and d and for every 0 math pi x,a,d ... Riemannhypothesis. Assuming the truth of the GRH, the estimate of the character sum in the Character ... , q being the modulus of the character. Extended Riemannhypothesis ERH Suppose K is a number field ... s 0 if the real part of s is between 0 and 1, then it is in fact 1 2. The ordinary Riemannhypothesis ... Springer id R r081940 title Riemannhypothesis, generalized reflist Category Zeta and L functions Category ... can be extended to a meromorphic function defined on the whole complex plane. The generalized Riemannhypothesis asserts that for every Dirichlet character and every complex number s with L , s 0 ... more details
Riemann function may refer to one of the several function mathematics functions named after the mathematician Bernhard Riemann , including Riemann zeta function Thomae s function Riemann theta function . dab fr Fonction de Riemann ... more details
Riemann is the surname of a number of notable people Bernhard Riemann , mathematician 1826&ndash 1866 Christel Riemann Hanewinckel , German politician born 1947 Fritz Riemann psychologist , German psychoanalyst 1902&ndash 1979 Hugo Riemann , German musicologist 1849&ndash 1919 Johannes Riemann , German actor 1888&ndash 1959 Katja Riemann born 1963 Manuel Riemann , German soccer player born 1988 Paula Riemann , German actress born 1993 Solomon Riemann , Jewish traveller died c. 1873 Ziska Riemann , German scriptwriter born 1973 See also List of topics named after Bernhard RiemannRiemann crater , a lunar crater surname Riemann, Rieman Riehmann , Riehman Rihmann , Rihman , etc. DEFAULTSORT Riemann Category German language surnames surname stub Germany hist stub de Riemann es Riemann desambiguaci n fr Riemann id Riemann it Riemann ja pt Riemann ro Riemann dezambiguizare ru sl Riemann ... more details
Infobox actor bgcolour name Katja Riemann image Katja riemann 20070607.jpg image size caption Katja Riemann at the w en Deutscher Evangelischer Kirchentag 2007 German Protestant Church Day 2007 birth name Katja Hannchen Leni Riemann birth date Birth date and age 1963 11 1 birth place Weyhe Kirchweyhe, Germany death date death place occupation actress spouse website http www.katja riemann.de Katja Hannchen Leni Riemann born 1 November 1963 in Weyhe Kirchweyhe, Germany is a German actress. Life and work Born as the daughter of two teachers, Katja Riemann spent her childhood in Weyhe , near Bremen . After high school she went to Hamburg to study music and theater. She is the mother of actress Paula Riemann . Awards 1993 Bayerischer Filmpreis Bavarian Film Award , Best Actress 1995 Bavarian Film Award, Best Actress 1997 Bavarian Film Award, Best Film Score http www.bayern.de Anlage19170 PreistraegerdesBayerischenFilmpreises Pierrot.pdf References Katharina Blum Katja Riemann. Mit Charme und Power . Heyne, M nchen 1998. ISBN 3 453 14056 7 External links Commons category imdb name id 0726257 name Katja Riemann http www.katja riemann.de Katja Riemann Official site http katja von garnier.de deutsch katja riemann.htm Katja Riemann Fansite http film.virtual history.com person.php?personid 267 Photographs of Katja Riemann Persondata Metadata see Wikipedia Persondata . NAME Riemann, Katja ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 1 November 1963 PLACE OF BIRTH Weyhe Kirchweyhe, Germany DATE OF DEATH PLACE OF DEATH DEFAULTSORT Riemann, Katja Category German actors Category 1963 births Category Living people Category Best Actress German Film Award winners Category Recipients of the Cross of the Order of Merit of the Federal Republic of Germany germany actor stub de Katja Riemann eo Katja Riemann no Katja Riemann ro Katja Riemann fi Katja Riemann ... more details
Image PaulaRiemann.jpg thumb Paula Riemann Paula Riemann born 3 August 1993 is a Germany German actress. Both of her parents, Katja Riemann and Peter Sattmann , have become well known for their acting roles. She herself has performed in the film Die Wilden H hner 2006 , as well as its sequel Die wilden H hner und die Liebe 2007 , based on the Wild Chicks books by Cornelia Funke . Riemann was also seen in the Joseph Vilsmaier Vilsmaier movie Bergkristall 2004 , in which she acted together with her mother. She has been awarded the Undine Award in 2006. External links imdb name id 1743114 name Paula Riemann Persondata Metadata see Wikipedia Persondata . NAME Riemann, Paula ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 3 August 1993 PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Riemann, Paula Category German child actors Category 1993 births Category Living people Germany actor stub de Paula Riemann es Paula Riemann fr Paula Riemann fi Paula Riemann ... more details
Orphan date February 2009 Solomon Reimann died ca. 1873 was a European Jewish traveler. An account of his travels, Mas ot Shelomoh , based on Riemann s own notes, was written by Wolf Schur and published in 1884. External links http www.jewishencyclopedia.com view.jsp?artid 285&letter R Solomon Riemann article in the Jewish Encyclopedia DEFAULTSORT Riemann, Solomon Category Jewish explorers Category 1870s deaths Category Year of birth missing euro writer stub Jewish hist stub es Solomon Riemann ... more details
Infobox planet minorplanet yes width 25em bgcolour FFFFC0 apsis name Riemann symbol image caption discovery yes discovery ref discoverer Lyudmila Zhuravleva L. V. Zhuravleva discovery site Nauchnyj discovered October 2, 1978 designations yes mp name 4167 alt names 1978 TQ7 named after Bernhard Riemann mp category orbit ref epoch May 14, 2008 aphelion 2.8176406 perihelion 2.3488720 semimajor eccentricity 0.0907321 period 1516.5253237 avg speed inclination 15.00403 asc node 160.74403 mean anomaly 153.47461 arg peri 113.28193 satellites physical characteristics yes dimensions mass density surface grav escape velocity sidereal day axial tilt pole ecliptic lat pole ecliptic lon albedo temperatures temp name1 mean temp 1 max temp 1 temp name2 max temp 2 spectral type abs magnitude 11.8 4167 Riemann 1978 TQ7 is a Asteroid belt main belt asteroid discovered on October 2, 1978 by Lyudmila Zhuravleva L. V. Zhuravleva at Nauchnyj . References Reflist External links http ssd.jpl.nasa.gov sbdb.cgi?sstr 4167 Riemann JPL Small Body Database Browser on 4167 Riemann Minor planets navigator 4166 Pontryagin 4168 Millan Small Solar System bodies DEFAULTSORT Riemann Category Main Belt asteroids Category Astronomical objects discovered in 1978 beltasteroid stub fa it 4167 Riemann hu 4167 Riemann pl 4167 Riemann pt 4167 Riemann sk 4167 Riemann sr 4167 Riemann ... more details
distinguish2 Bernhard Riemann , the mathematician File Hugo Riemann.jpg thumb Hugo Riemann Hamburg, 1889 Karl Wilhelm Julius Hugo Riemann July 18, 1849 July 10, 1919 was a Germany German music theory music theorist . Biography Riemann was born at Grossmehlra, near Sondershausen . He was educated in theory by Frankenberger, studied the piano with Barthel and Ratzenberger, studied law, and finally philosophy and history at Berlin and T bingen. After going through the Franco German war he decided to devote his life to music, and studied accordingly at the Leipzig Conservatory . He then went to Bielefeld for some years as a teacher and conductor, but in 1878 returned to Leipzig as Privatdozent at the University. As a much desired appointment at the Conservatory did not materialize, Riemann went to Bromberg in 1880, but 1881 90 he was a teacher of piano and theory at Hamburg Conservatory. After a short time at the Sondershausen Conservatory, he held a post in the conservatory at Wiesbaden 1890 95 , but eventually returned to Leipzig University as lecturer in 1895. In 1901, he was appointed professor. Writings In addition to his work as a teacher, lecturer and composer of pedagogical pieces, Riemann had a worldwide reputation as a writer on musical subjects. His best known works are Musik ... harmony References Alexander Rehding Hugo Riemann and the birth of modern musical thought . Cambridge Cambridge University Press, 2003. ISBN 0 521 82073 1 Cite NIE Riemann, Hugo year 1905 External Links IMSLP id Riemann, Hugo Etude Persondata Metadata see Wikipedia Persondata . NAME Riemann, Hugo ... 10, 1919 PLACE OF DEATH DEFAULTSORT Riemann, Hugo Category German musicologists Category German music ... of Leipzig faculty de Hugo Riemann es Hugo Riemann fr Hugo Riemann it Hugo Riemann nl Hugo Riemann no Hugo Riemann pl Hugo Riemann pt Hugo Riemann ru , fi Hugo Riemann sv Hugo Riemann ... more details
for the Riemann surface of a subring of a field Zariski Riemann space Image Riemann sqrt.jpg thumb right Riemann surface for the function &fnof z &radic z . The two horizontal axes represent ... in complex analysis , a Riemann surface , first studied by and named after Bernhard Riemann , is a one dimensional complex manifold . Riemann surfaces can be thought of as deformed versions ... of sheets glued together. The main point of Riemann surfaces is that holomorphic function s may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying ... and other algebraic function s, or the natural logarithm logarithm . Every Riemann surface is a two ... real manifold can be turned into a Riemann surface usually in several inequivalent ways if and only ... strip , Klein bottle and projective plane do not. Geometrical facts about Riemann surfaces ... curves, manifolds or varieties. The Riemann Roch theorem is a prime example of this influence. Definitions There are several equivalent definitions of a Riemann surface. A Riemann surface X is a complex ... of the complex plane to the Riemann surface is called a chart . Additionally, the transition map s between two overlapping charts are required to be Holomorphic function holomorphic . A Riemann surface ... Riemann signifies that X is endowed with an additional structure which allows angle measurement ... geometry Examples Image Riemann sphere1.jpg thumb left 150px The Riemann sphere. The complex plane C is perhaps the most basic Riemann surface. The map f z z the identity map defines a chart ... this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its ... X with a distinct, incompatible Riemann structure. In an analogous fashion, every open subset of the complex plane can be viewed as a Riemann surface in a natural way. More generally, every open subset of a Riemann surface is a Riemann surface. Let S C and let f z z where z is in S and g z 1 z where ... more details
File Riemann sum convergence.png right thumb 300px Four of the Riemann summation Methods methods for approximating ... In mathematics , a Riemann sum is a method for approximating the total area underneath a curve on a graph ... was named after German mathematician Bernhard Riemann . Definition Let f D &rarr R be a function ... 0 sub x sub 1 sub x sub 2 sub ... x sub n sub b . The Riemann sum of f over I with partition P is defined ... a left Riemann sum . If y sub i sub x sub i sub , then S is called a right Riemann sum . If y sub i sub x sub i sub x sub i 1 sub 2, then S is called a middle Riemann sum . The average of the left and right Riemann sum is the trapezoidal sum . If it is given that math S sum i 1 n v i ... to be an upper Riemann sum . Similarly, if v sub i sub is the infimum of f over x sub i &minus 1 sub , x sub i sub , then S is a lower Riemann sum . Any Riemann sum on a given partition that is, for any ... and the upper Riemann sums. A function is defined to be Riemann integral Riemann integrable if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. This fact can also ... Riemann sum methods of x sup 3 sup over 0,2 using 4 subdivisions width 200 image1 LeftRiemann2.svg ... The four methods of Riemann summation are usually best approached with partitions of equal size ... &minus 1 Q , b . Left sum For the left Riemann sum, approximating the function by its value at the left ... . , math The left Riemann sum amounts to an overestimation if f is monotonically decreasing on this interval ... f a Q f a 2Q cdots f b right . , math The right Riemann sum amounts to an overestimation if f is monotonically ... be procedurally computed using Riemann s method. The interval from 0 to 2 is firstly divided into n subintervals, each of which is given a width of math frac 2 n math these are the widths of the Riemann rectangles. Because the right Riemann sum is to be used, the sequence of x coordinates for the boxes ... the n th right Riemann sum will be math S frac 2 n times left frac 2 n right 2 cdots frac 2 n times ... more details
In the branch of mathematics known as real analysis , the Riemann integral , created by Bernhard Riemann ... mathematics interval . While the Riemann integral is unsuitable for many theoretical purposes ..., the Riemann integral can also be readily evaluated by using the fundamental theorem of calculus or approximately by numerical integration . Some of the technical deficiencies in Riemann integration ... the area by math int a b f x ,dx. math The basic idea of the Riemann integral is to use very simple ... of Riemann sums. The numbers in the upper right are the areas of the grey rectangles. They converge ... is greater or equal to another if the former is a refinement of the latter. Riemann sums Choose a real valued function math f math which is defined on the interval math a,b math . The Riemann sum ... represents the area of a rectangle with height math f t i math and width math x i 1 x i math . The Riemann sum is the signed area under all the rectangles. Riemann integral Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be integrable or more specifically Riemann integrable . The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough ... say that the Riemann integral of equals s if the following condition holds For all &epsilon ... it is very difficult to work with. So we will make an alternate definition of the Riemann integral ... new definition says that the Riemann integral of equals s if the following condition holds For all ... 1 y i s right varepsilon. math Both of these mean that eventually, the Riemann sum of with respect ... be trapped, we say that the Riemann sums converge to s . These definitions are actually a special ... that satisfies the condition. Choose any tagged partition whose mesh is less than . Its Riemann sum is within of s , and any refinement of this partition will also have mesh less than , so the Riemann ... more details
lunar crater data latitude 39.5 N or S N longitude 87.2 E or W E diameter 110 km depth Unknown colong 274 eponym Bernhard Riemann G. F. Bernhard Riemann Riemann pronounced REE mahn is a Moon lunar impact crater crater that is located near the northeastern limb of the Moon , and can just be observed edge on when libration effects bring it into sight. It lies to the east northeast of the large walled plain Gauss crater Gauss . To the southeast, beyond sight on the Far side Moon far side , is the crater Vestine crater Vestine . This is a heavily battered and eroded formation that is only a remnant of its former self. The outer rim has been worn away in many places, and now forms an irregular series of ridges in a rough circle. The rim is overlain along the south southwestern rim by Beals crater Beals , and several smaller craters lie along the western and southeast rim. The most intact portion of the outer wall is along the eastern edge. The interior floor is a mixture of level terrain mixed with rough ground where impacts have stirred up the surface. It is generally less rough in the eastern half, especially near the center. A small, bowl shaped crater lies on the floor in the southeastern part of the interior, and the faint remnants of several other lesser craters can be observed in the surface. 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 Riemann. class wikitable width 25 style background eeeeee Riemann width 25 style background eeeeee Latitude width 25 style background eeeeee Longitude width 25 style background eeeeee Diameter align center B align center 41.6 N align center 85.2 E align center 24 km align center J align center 37.4 N align center 90.2 E align center 39 km The following craters have been renamed by the International Astronomical Union IAU . Riemann A &mdash See Beals crater . References Lunar crater references Category Impact craters on the Mo ... more details
In mathematics , a Riemann form in the theory of abelian varieties and modular forms , is the following data A Lattice group Lattices in complex space lattice in a complex vector space C sup g sup . An bilinear form alternating bilinear form from to the integer s satisfying the following two conditions ol li the real linear extension sub R sub C sup g sup C sup g sup R of satisfies sub R sub iv , iw sub R sub v , w for all v , w in C sup g sup C sup g sup li the associated hermitian form H v , w sub R sub iv , w i sub R sub v , w is positive definite . ol Note the hermitian form written here is linear in the first variable, in opposition to the standard definition of this encyclopedia, but in accord with the standard use in this specific subject . Riemann forms are important because of the following The alternatization of the Chern class of any factor of automorphy is a Riemann form. Conversely, given any Riemann form, we can construct a factor of automorphy such that the alternatization of its Chern class is the given Riemann form. References Citation last Milne first James title Abelian Varieties year 1998 url http www.jmilne.org math CourseNotes av.html accessdate 2008 01 15 Citation last Hindry first Marc last2 Silverman first2 Joseph H. title Diophantine Geometry, An Introduction publisher location New York series Graduate Texts in Mathematics isbn 0 387 98981 1 id MathSciNet id 1745599 year 2000 volume 201 Citation last Mumford first David author link David Mumford title Abelian Varieties publisher Oxford University Press location London series Tata Institute of Fundamental Research Studies in Mathematics id MathSciNet id 0282985 year 1970 volume 5 Springer title Abelian function id A a010220 Springer title Theta function id T t092600 DEFAULTSORT Riemann Form Category Abelian varieties ... more details
About the German chess master the German psychologist and astrologer of the same name Fritz Riemann psychologist Fritz Riemann 2 January 1859, Weistritz, near widnica Schweidnitz 25 November 1932, Erfurt was a German chess master. Born in Silesia then Prussia , he was a chess pupil of Adolf Anderssen in Breslau. In 1876, he won a match against Arnold Schottl nder 5 0 there. In 1879, he took 5th in Leipzig 1st DSB Congress , Berthold Englisch won , and took 2nd in Wesselburen. ref http www.anders.thulin.name SUBJECTS CHESS CTCIndex.pdf Name Index to Jeremy Gaige s Chess Tournament Crosstables , An Electronic Edition, Anders Thulin, Malm , 2004 09 01 ref In 1880, he took 2nd, behind Louis Paulsen , in Braunschweig 13th WSB Congress , and drew a match with Emil Schallopp 2 2 2 in Berlin. In 1881, he tied for 13 14th in Berlin 2nd DSB Congress , Joseph Henry Blackburne won . In 1883, he tied for 6 7th in Nuremberg 3rd DSB Congress , Szymon Winawer won . In 1885, he tied for 8 9th in Hamburg 4th DSB Congress , Isidor Gunsberg won , and drew a match with Ernst Flechsig 5 5 0 in Breslau. ref http members.shaw.ca edo2 players p386.html Edo Historical Chess Ratings ref In 1888, he shared 1st with Curt von Bardeleben in Leipzig. ref http xoomer.alice.it cserica scacchi storiascacchi tornei pagine itornei1880 99.htm I tornei dal 1880 al 1899 Bot generated title ref He wrote a book Riemann, Fritz Schach Erinnerungen des j ngsten Anderssen Sch lers. Mit vielen Diagrammen im Text und einem Bildnis des Verfassers. de Gruyter, Berlin und Leipzig 1925. References references External links chessgames player id 10376 Persondata Metadata see Wikipedia Persondata . NAME Riemann, Fritz ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 2 January 1859 PLACE OF BIRTH DATE OF DEATH 25 November 1932 PLACE OF DEATH DEFAULTSORT Riemann, Fritz Category 1859 births Category 1932 deaths Category German chess players Category People from the Province of Silesia ca Fritz Riemann de Fritz Riemann Schachspieler ... more details
Image Stereographic projection in 3D.png thumb right The Riemann sphere can be visualized as the complex ... below . In mathematics , the Riemann sphere or extended complex plane , named after the 19th century mathematician Bernhard Riemann , is the sphere obtained from the complex plane by adding a point ... function on the Riemann sphere, with the Pole complex analysis poles of the rational function ... function whose codomain is the Riemann sphere. In geometry , the Riemann sphere is the prototypical example of a Riemann surface , and is one of the simplest complex manifold s. In projective geometry ... space of all complex line s in math mathbb C 2 math . As with any compact space compact Riemann ... numbers is referred to as the Riemann sphere or extended complex plane . Arithmetic operations Addition ... f z g z h z math can be extended to a continuous function on the Riemann sphere. Specifically, if math ... these definitions, math f math becomes a continuous function from the Riemann sphere to itself ... the Riemann sphere to itself. As a complex manifold As a one dimensional complex manifold, the Riemann ... maps are holomorphic function holomorphic , they define a complex manifold, called the Riemann sphere . Intuitively, the transition maps indicate how to glue two planes together to form the Riemann ... every point in the Riemann sphere has both a math zeta math value and a math xi math value, and the two .... However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well defined ... theorem , a central result in the classification of Riemann surfaces, states that the only simply ... plane , and the Riemann sphere. Of these, the Riemann sphere is the only one that is a Closed manifold ... manifold. As the complex projective line The Riemann sphere can also be defined as the complex ... of the Riemann sphere connects most readily to projective geometry. For example, any line or smooth ... for studying the sphere s automorphism s, later in this article. As a sphere Image Riemann ... more details
A Riemann problem , named after Bernhard Riemann , consists of a conservation law together with piecewise constant data having a single discontinuity . The Riemann problem is very useful for the understanding of hyperbolic partial differential equation s like the Euler equations because all properties, such as shocks and rarefaction waves, appear as Method of characteristics characteristic s in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations fluid dynamics Euler equations . In numerical analysis , Riemann problems appear in a natural way in finite volume method s for the solution of equation of conservation laws due to the discreteness of the grid. For that it is widely used in computational fluid dynamics and in Computational Magnetohydrodynamics MHD simulations. In these fields Riemann problems are calculated using Riemann solver s. The Riemann problem in linearized gas dynamics As a simple example, we investigate the properties of the one dimensional Riemann problem in gas dynamics , which is defined by math begin bmatrix rho u end bmatrix begin bmatrix rho L u L end bmatrix text for x leq 0 qquad text and qquad begin bmatrix rho u end bmatrix begin bmatrix rho R u R end bmatrix text for x 0 math where x 0 separates two different states, together with the linearised gas dynamic equation see gas dynamics for derivation math begin align frac partial rho partial t rho 0 frac partial u partial x & 0 8pt frac partial u partial t frac a 2 rho 0 frac partial rho partial x & 0 end align math we can rewrite the above equation in conservative form math U t A U x 0 math math U begin bmatrix rho u end bmatrix ... 30em cite book first Eleuterio F. last Toro year 1999 title Riemann Solvers and Numerical Methods ... Computational magnetohydrodynamics Riemann solver DEFAULTSORT Riemann Problem Category Hyperbolic partial differential equations Category Fluid dynamics Category Computational fluid dynamics de Riemann ... more details
Computational physics A Riemann solver is a numerical method used to solve a Riemann problem . They are heavily used in computational fluid dynamics and computational magnetohydrodynamics . Exact solvers Sergei K. Godunov Godunov is credited with introducing the first exact Riemann solver for the Euler equations ref Citation last Godunov first S. K. title A difference scheme for numerical computation of discontinuous solution of hyperbolic equation journal Math. Sbornik volume 47 pages 271&ndash 306 year 1959 ref , by extending the previous CIR Courant Isaacson Reeves method to non linear systems of hyperbolic conservation laws. Modern solvers are able to simulate relativistic effects and magnetic fields. For the hydrodynamic case latest research results showed the possibility to avoid the iterations to calculate the exact solution for the Euler equations. Approximate solvers As iterative solutions are too costly, especially in Magnetohydrodynamics, some approximations have to be made. The most popular solvers are. Roe solver main Roe solver Philip L. Roe Roe used the linearisation of the Jacobian, which he then solves exactly. ref Citation last Roe first P. L. title Approximate Riemann ... Leer van Leer and Einfeldt solver is an approximate solution to the Riemann problem, which is only ... Speares first3 W. title Restoration of the contact surface in the HLL Riemann solver journal Shock ... but somewhat more diffusive. ref Citation last Quirk first J. J. title A contribution to the great Riemann ... 10.1002 fld.1650180603 postscript . ref Rotated hybrid Riemann solvers These solvers were introduced ... first2 K. title Very simple, carbuncle free, boundary layer resolving, rotated hybrid Riemann solvers ... solver at the same time. They developed robust and accurate Riemann solvers by combining the Roe ... Riemann solvers can be combined into a single Roe type solver the Roe solver with modified wave ... 2 References Citation first Eleuterio F. last Toro year 1999 title Riemann Solvers and Numerical Methods ... more details
In mathematics, the Lindel f hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindel f see harvtxt Lindel f 1908 about the rate of growth of the Riemann zeta function on the critical line that is implied by the Riemannhypothesis . It says that, for any 0, math zeta left frac12 it right ... last Huxley year1 2002 year2 2005 Relation to the Riemannhypothesis harvtxt Backlund 1918 1919 showed that the Lindel f hypothesis is equivalent to the following statement about the zeros of the zeta ... and imaginary part between T and T 1 is o log T as T tends to infinity. The Riemannhypothesis implies that there are no zeros at all in this region and so implies the Lindel f hypothesis ... theorem implies that is convex. The Lindel f hypothesis states 1 2 0, which together ... the Lindel f hypothesis seems only slightly stronger than what has already been proved, but in spite ... The Lindel f hypothesis is equivalent to the statement that math int 0 T zeta 1 2 it 2k ,dt O T 1 ... Ingham , shows that the Lindel f hypothesis implies that, for any 0, math p n 1 ... conjecture for the Riemann zeta function doi 10.1016 j.jnt.2007.05.013 id MathSciNet id 2419176 year ... last1 Conrey first1 J. B. last2 Ghosh first2 A. title A conjecture for the sixth power moment of the Riemann ... Edwards first1 H. M. authorlink Harold Edwards mathematician title Riemann s Zeta Function publisher ... Roger Heath Brown last1 Heath Brown first1 D. R. title The fourth power moment of the Riemann ... A K Peters id MathSciNet id 1956254 year 2002 chapter Integer points, exponential sums and the Riemann zeta function pages 275 290 Citation last1 Huxley first1 M. N. title Exponential sums and the Riemann ... citation first A. E. last Ingham title Mean Value Theorems in the Theory of the Riemann Zeta Function ... Voronin first2 S. M. title The Riemann zeta function publisher Walter de Gruyter & Co. location ... the Riemann zeta function and the hyperbolic Laplacian url http www.numdam.org item?id ASNSP 1995 ... more details
main Statistical hypothesis testing In statistical hypothesis testing , the alternative hypothesis or maintained hypothesis or research hypothesis and the null hypothesis are the two rival hypotheses which are compared by a statistical hypothesis testing statistical hypothesis test . An example might be where water quality in a stream has been observed over many years and a test is made of the null hypothesis that there is no change in quality between the first and second halves of the data against the alternative hypothesis that the quality is poorer in the second half of the record. The concept of an alternative hypothesis in testing was devised by Jerzy Neyman and Egon Pearson , and it is used in the Neyman Pearson lemma . It forms a major component in modern statistical hypothesis testing . However it was not part of Ronald Fisher Ronald Fisher s formulation of statistical hypothesis testing, and he violently opposed its use. ref name Cohen Jacob Cohen statistician Cohen, J. 1990. Things I have learned so far . American Psychologist 45 1304&ndash 1312. ref In Fisher s approach to testing, the central idea is to assess whether the observed dataset could have resulted from chance if the null hypothesis were assumed to hold, notionally without preconceptions about what other model might hold. Modern statistical hypothesis testing accommodates this type of test since the alternative hypothesis can be just the negation of the null hypothesis. References reflist Statistics Category Hypothesis testing Category Statistical inference eo Alternativa hipotezo ko ... more details
The innateness hypothesis is a linguistic theory of language acquisition which holds that at least some linguistic knowledge exists in humans at birth. ref http dictionary.reference.com browse innateness hypothesis Based on the Random House Dictionary, Random House, Inc. 2009. ref Facts about the complexity of human language systems, the universality of language acquisition, the facility that children demonstrate in acquiring these systems, and the comparative performance of adults in attempting the same task are all commonly invoked in support. The idea that there may be an age by which this learning must be accomplished is known as the critical period hypothesis . Noam Chomsky is responsible for the innateness hypothesis. Hilary Putnam published a critique of the innateness hypothesis entitled The Innateness Hypothesis and Explanatory Models in Linguistics . ref http www.springerlink.com content w476u76126j58330 fulltext.pdf ref References references See also Language acquisition Category Linguistics Category Philosophy of language Category Hypotheses ... more details
For the periodical Null Hypothesis The Journal of Unlikely Science Main Statistical hypothesis testing The practice of science involves formulating and testing hypothesis hypotheses , assertions that are falsifiable using a test of observed data. The null hypothesis typically corresponds to a general or default position. For example, the null hypothesis might be that there is no relationship between ... title null hypothesis definition publisher Businessdictionary.com date accessdate 2010 07 29 ... proven guilty can be interpreted as saying that his or her innocence is the null hypothesis. Other legal systems may exist in which the null hypothesis is that the defendant is guilty. The term was originally ... statistics.berkeley.edu stark SticiGui Text gloss.htm null hypothesis title Glossary publisher Statistics.berkeley.edu ... hypothesis, the alternative hypothesis , which asserts a particular relationship between the phenomena ... negation of the null hypothesis and predicts the results from the experiment if the alternative hypothesis is true. The use of alternative hypotheses was not part of Fisher s formulation, but became standard. Principle Hypothesis testing works by Sampling statistics collecting data and measuring how probability probable the data are, assuming the null hypothesis is true. If the data ... that the null hypothesis is false. If the data do not contradict the null hypothesis, then no conclusion is made. In this case, the null hypothesis could be true or false the data give insufficient ... attack and this drug has no effect on the chances of having a heart attack . The test of the hypothesis ... hypothesis is rejected. Choice of H sub 0 The choice of null hypothesis H sub 0 and consideration ... i.e. that on average it lands heads up 50 of the time . A potential null hypothesis is this coin ... result of 5 tosses is 5 heads. Under this null hypothesis, the data are considered unlikely with a fair coin, the probability of this is 3 . The data refute the null hypothesis the coin is biased ... more details
Orphan date February 2009 otheruses comparator disambiguation The comparator hypothesis is a hypothesis in the field of the psychology of motivation and learning . ref http books.google.com books?hl en&lr &id k6ufhxSYXe8C&oi fnd&pg PA51&dq comparator&ots 0kZ3T3e4dw&sig ObK0QBEZCAeLsxDUjVsSafHIBWg PPA53,M1 ref Created by Ralph Miller, it established that responses are due to a comparison between the direct activation of the outcome and the indirect activation of the outcome. The comparator hypothesis was the first model which successfully accounts for retrospective reevaluation phenomena. However, after the publication of the comparator hypothesis, traditional models like Wagner s SOP and the Rescorla Wagner model were modified to be able to account for retrospective reevaluation phenomena. Today, the comparator hypothesis can successfully account for counteraction phenomena, a topic in which both the traditional models and their reformulation tends to fail. References reflist Category Learning psychology Category Motivation psych stub ... more details
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