In mathematics , Dedekind sums , named after Richard Dedekind , are certain sums of products of a sawtooth function , and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function . They have subsequently been much studied in number theory , and have occurred in some problems of topology . Dedekind sums obey a large number of relationships on themselves this article lists only a tiny fraction of these. Definition Define the sawtooth function math left left right right mathbb R rightarrow mathbb R math as math x begin cases x lfloor x rfloor 1 2, & mbox if x in mathbb R setminus mathbb Z 0,& mbox if x in mathbb Z . end cases math We then let D Z sup 3 sup &rarr R be defined by math D a,b c sum n bmod c left Bigg frac an c Bigg right left left frac bn c right right , math the terms on the right being the Dedekind sums . For the case a 1, one often writes s b , c D 1, b c . Simple formulae Note that D is symmetric in a and b , and hence math D a,b c D b,a c , math and that, by the oddness of , D &minus a , b ... of math sum n bmod c left left frac n x c right right left left x right right , qquad forall x in mathbb ..., we may write s b , c as math s b,c frac 1 c sum omega frac 1 1 omega b 1 omega frac 1 4 frac 1 4c , math where the sum extends over the c th roots of unity other than 1, i.e. over all math ... s b,c frac 1 4c sum n 1 c 1 cot left frac pi n c right cot left frac pi nb c right . math Reciprocity ... of the Dedekind eta function is the following. Let q 3, 5, 7 or 13 and let n 24 q &minus ... law for Dedekind sums ref H. Rademacher, Generalization of the Reciprocity Formula for Dedekind ... math.sfsu.edu beck papers dedekind.slides.pdf Dedekind sums a discrete geometric viewpoint , 2005 or earlier Hans Rademacher and Emil Grosswald , Dedekind Sums , Carus Math. Monographs, 1972. ISBN 0883850168. Category Number theory Category Modular forms fr Somme de Dedekind zh ... more details
Dedekind is the name of People Brendon Dedekind born 1976 , South African swimmer Friedrich Dedekind 1524 1598 , German humanist, theologian, and bookseller Richard Dedekind 1831 1916 , German mathematician Other 19293 Dedekind , asteroid named after Richard Dedekind surname de Dedekind ru ... more details
In number theory , Dedekind function can refer to any of three functions, all introduced by Richard DedekindDedekind eta function Dedekind psi function Dedekind zeta function disambig de Dedekindsche Funktion ... more details
domain Dedekind eta function Dedekind infinite set Dedekind number DedekindsumDedekind zeta function ...for the 16th century humanist Friedrich Dedekind Infobox Scientist name PAGENAME box width image Dedekind.jpeg image size 180px caption Richard Dedekind, c. 1850 birth date birth date October 6, 1831 October ... Dedekind October 6, 1831 &ndash February 12, 1916 was a German people German mathematician who ... TXT 00000255.txt date March 2011 Dedekind was the youngest of four children of Julius Levin Ulrich Dedekind. As an adult, he never employed the names Julius Wilhelm. He was born, lived most of his ..., obtaining a solid grounding in mathematics. In 1850, he entered the University of G ttingen . Dedekind ... mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate ... Dedekind later wrote. At that time, the University of Berlin , not G ttingen , was the leading center for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where .... Dedekind returned to G ttingen to teach as a Privatdozent , giving courses on probability and geometry ... Richard Dedekind In 1858, he began teaching at the ETH Z rich Polytechnic in Z rich today ETH Z rich ... in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching ... married, instead living with his unmarried sister Julia. Dedekind was elected to the Academies of Berlin ... Polytechnic , Dedekind came up with the notion now called a Dedekind cut German language German Schnitt ... locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet Stetigkeit und irrationale Zahlen Continuity and irrational ... , Dedekind met Georg Cantor Cantor . Thus began an enduring relationship of mutual respect, and Dedekind became one of the very first mathematicians to admire Cantor s work on infinite sets ... two sets, Dedekind said that the two sets were similar. He invoked similarity to give the first precise ... more details
In abstract algebra , a Dedekind domain or Dedekind ring , named after Richard Dedekind , is an integral ... three other characterizations of Dedekind domains which are sometimes taken as the definition ..., so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains ... consequence of the definition is that every principal ideal domain PID is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain UFD iff it is a PID. The prehistory of Dedekind ... zeta n math is a Dedekind domain. In fact Kummer worked not with ideals but with ideal numbers , and the modern definition of an ideal was given by Dedekind. By the 20th century, algebraists and number ... of being a Dedekind domain is quite robust. For instance the ring of ordinary integers is a PID ... a Dedekind domain. Another illustration of the delicate robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property Properties of commutative rings local property a Noetherian domain math R math is Dedekind iff for every maximal ideal math M math of math R math the localization of a ring localization math R M math is a Dedekind ring. But a local ring local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring DVR , so the same local characterization cannot hold for PIDs rather, one may say that the concept of a Dedekind ... dimension one i.e., every nonzero prime ideal is maximal . Thus a Dedekind domain is a domain ... DD4 . Some examples of Dedekind domains All principal ideal domain s and therefore all discrete valuation ring s are Dedekind domains. The ring math R mathcal O K math of algebraic integers in a number ... , so by DD4 R is a Dedekind domain. As above, this includes all the examples considered by Kummer and Dedekind and was the motivating case for the general definition, and these remain among the most ... more details
1111 1111.svg tautology desc bottom left imagemap In mathematics , the Dedekind numbers are a rapidly growing integer sequence sequence of integers named after Richard Dedekind , who defined them in 1897. The Dedekind number M n counts the number of Monotone boolean function monotonic Boolean functions ..., finding a useful closed form expression for M n remains known as Dedekind s problem . Computing ... to switch from false to true and not from true to false. The Dedekind number M n is the number ... subsets of Boolean variables that can force the function value to be true. Therefore, the Dedekind ... the top and bottom lattice elements and subtract two from the Dedekind numbers. ref Thus, the Dedekind ... Church 1965 harvtxt Zaguia 1993 . ref The Dedekind numbers also count the number of abstract simplicial ... only the empty set. Values The exact values of the Dedekind numbers are known for 0 n 8 2, 3, 6, 20 ... Dedekind number M 5 7581 disproved a conjecture by Garrett Birkhoff that M n is always divisible ... formula harvtxt Kisielewicz 1988 proved the following formula for the Dedekind numbers math M n sum k 1 2 2 n prod j 1 2 n 1 prod i 0 j 1 left 1 b i k b k k prod m 0 log 2 i 1 b m i b m i b m j right ... number of terms in the summation. Asymptotics The logarithm of the Dedekind numbers can be estimated ... last Dedekind first Richard author link Richard Dedekind contribution ber Zerlegungen von Zahlen durch ... approach to Dedekind s problem volume 130 year 2002 . citation last Kisielewicz first Andrzej doi ... 139 144 title A solution of Dedekind s problem on the number of isotone Boolean functions volume ... G. id MR 0382107 journal Transactions of the American Mathematical Society pages 373 390 title On Dedekind ... 1129608 issue 1 journal Order journal Order pages 5 6 title A computation of the eighth Dedekind number ... logic Category Lattice theory Category Set families es N mero de Dedekind ... more details
SUM can refer to The State University of Management Soccer United Marketing Society for the Establishment of Useful Manufactures StartUp Manager Software User s Manual ,as from DOD STD 2 167A, and MIL STD 498 Sveriges unga muslimer , the Young Muslims of Sweden disambig Long comment to avoid being listed on short pages ... more details
for the 19th century mathematician Richard Dedekind Friedrich Dedekind 1524 February 27, 1598 was a Germany German Humanism humanist , theologian , and bookseller . Born in Neustadt am R benberge , he was educated at the University of Marburg universities of Marburg 1543 and University of Wittenberg Wittenberg , where he studied theology . At Wittenberg, his talents were recognized by Philipp Melanchthon . As magister , he became in 1575 a minister of religion minister and inspector of churches in L neburg . He wrote Play theatre plays and in later years became involved in mediating theological disputes. He died on February 27, 1598 at L neburg. Dedekind s Grobianus Dedekind was the author of Grobianus et Grobiana sive, de morum simplicitate, libri tres Cologne , 1558 . This work had first been published in 1549 as Grobianus , but it appeared with additions known as Grobiana in 1554. A poem in Latin elegiac Verse poetry verse , it was first published in two books in 1549, and revised form and enlarged to three books in 1552. Dedekind s work had an immense popularity across Continental Europe . The work describes the fictional Saint Grobian as a counselor who teaches men on how to avoid bad manners, gluttony , and drunkenness . Dedekind s work appeared in England in 1605 as The Schoole of Slovenrie Or, Cato turnd wrong side outward , published by one R.F. . The Schoole was imagined ... tres en icon http leehrsn.stormloader.com dek intro.html Gull s Hornbook BBKL d dedekind f band 20 autor Eberhard Doll artikel Dedekind, Friedrich spalten 373 379 http www.uni mannheim.de mateo camena dede1 te01.html Facsimile des Grobianus de icon http www.bautz.de bbkl d dedekind f.shtml Biographischer Artikel zu Friedrich Dedekind im BBKL DEFAULTSORT Dedekind, Friedrich Category 1524 births Category ... Category German writers Category German theologians Category German booksellers de Friedrich Dedekind ru , sv Friedrich Dedekind ... more details
Infobox planet minorplanet yes width 25em bgcolour FFFFC0 apsis name Dedekind symbol image caption discovery yes discovery ref discoverer P. G. Comba discovery site Prescott Observatory Prescott discovered July 18, 1996 designations yes mp name 19293 alt names 1996 OF named after Richard Dedekind mp category orbit ref epoch May 14, 2008 aphelion 2.5207631 perihelion 2.0175985 semimajor eccentricity 0.1108692 period 1248.5389060 avg speed inclination 6.92112 asc node 105.95226 mean anomaly 90.21510 arg peri 287.75821 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 16.1 19293 Dedekind 1996 OF is a Asteroid belt main belt asteroid discovered on July 18, 1996 by P. G. Comba at Prescott Observatory Prescott . References Reflist External links http ssd.jpl.nasa.gov sbdb.cgi?sstr 19293 Dedekind JPL Small Body Database Browser on 19293 Dedekind MinorPlanets Navigator 19292 1996 NG5 19294 Weymouth MinorPlanets Footer DEFAULTSORT Dedekind Category Main Belt asteroids Category Asteroids named for people Category Discoveries by Paul G. Comba Category Astronomical objects discovered in 1996 beltasteroid stub fa it 19293 Dedekind pl 19293 Dedekind pt 19293 Dedekind ... more details
Refimprove date March 2011 Image Dedekind cut sqrt 2.svg thumb right 350px Dedekind used his cut to construct the irrational number irrational , real number s. In mathematics , a Dedekind cut , named after Richard Dedekind , is a partition of a set partition of the rational number s into two non empty ... the cut. The cut itself is in neither set. More generally, a Dedekind cut is a partition of a totally .... See also completeness order theory . The Dedekind cut resolves the contradiction between the continuous ... rational or irrational number .... Richard Dedekind, Continuity and Irrational Numbers , Section IV Dedekind used the ambiguous word cut Schnitt in the geometric sense. That is, it is an intersection ... number at every point on the continuum. Handling Dedekind cuts It is more symmetrical to use the A , B notation for Dedekind cuts, but each of A and B does determine the other. It can be a simplification ... any downward closed set A without greatest element a Dedekind cut . If the ordered set S is complete, then, for every Dedekind cut A , B of S , the set B must have a minimal element b , hence we ... purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself ... number . Ordering Dedekind cuts Regard one Dedekind cut A , B as less than another Dedekind ... created from set relations. The set of all Dedekind cuts is itself a linearly ordered set of sets . Moreover, the set of Dedekind cuts has the supremum least upper bound property, i.e., every nonempty subset of it that has any upper bound has a least upper bound. Thus, constructing the set of Dedekind ... by Dedekind cuts A typical Dedekind cut of the rational number s is given by math A a in mathbb ... number 2 in Dedekind s construction. To truly establish this, one must show that this really ... . Additional structure on the cuts See Construction of the real numbers Generalization Dedekind ... it generates. Dedekind MacNeille completion MacNeille completion , Macneille completion redirect ... more details
MedalTableTop Replace this image male.svg 150px MedalSport Men s Swimming sport swimming MedalCountry RSA MedalCompetition FINA World Championships Short Course World Championships SC MedalSilver 2000 FINA Short Course World Championships 2000 Athens 50 m freestyle MedalSilver 2000 FINA Short Course World Championships 2000 Athens 50 m breaststroke MedalCompetition Pan Pacific Swimming Championships Pan Pacific Championships MedalGold 1999 Pan Pacific Swimming Championships 1999 Sydney 50 m freestyle MedalCompetition Universiade MedalBronze Swimming at the 1997 Summer Universiade 1997 Catania 50 m freestyle MedalBottom Brendon Dedekind born February 14, 1976 in Pietermaritzburg , KwaZulu Natal is the first swimmer from South Africa to win an international gold medal. He did so in the 50 m freestyle at the 1999 Pan Pacific Swimming Championships . Nickname d Skinny Man , he competed in two consecutive Summer Olympics for his native country, starting in 1996, when he was a finalist in the 50 m freestyle. References http www.sports reference.com olympics athletes de brendon dedekind 1.html sports reference Footer Pan Pacific Champions 50m Freestyle Men Persondata Metadata see Wikipedia Persondata . NAME Dedekind, Brendon ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH February 14, 1976 PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Dedekind, Brendon Category 1976 births Category Living people Category Freestyle swimmers Category Swimmers at the 1996 Summer Olympics Category Swimmers at the 2000 Summer Olympics Category Olympic swimmers of South Africa Category South African swimmers Dedekind SouthAfrica swimming bio stub it Brendon Dedekind no Brendon Debekind fi Brendon Dedekind ... more details
In mathematics , the Dedekind zeta function of an algebraic number field K , generally denoted sub K sub s , is a generalization of the Riemann zeta function &mdash which is obtained by specializing to the case where K is the rational number s Q . In particular, it can be defined as a Dirichlet series , it has an Euler product expansion, it satisfies a functional equation L function functional equation , it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s 1, and its values encode arithmetic data of K . The extended Riemann hypothesis states that if sub K sub s 0 and 0 Re s 1, then Re s 1 2. The Dedekind zeta function is named for Richard Dedekind who introduced them in his supplement to Johann Peter Gustav Lejeune Dirichlet P.G.L. Dirichlet s Vorlesungen ber Zahlentheorie ... number field K . Its Dedekind zeta function is first defined for complex numbers s with real part Re s 1 by the Dirichlet series math zeta K s sum I subseteq mathcal O K frac 1 N K mathbf ... O sub K sub I . This sum converges absolutely for all complex numbers s with real part ... function. Euler product The Dedekind zeta function of K has an Euler product which is a product over ... Q P s , text for Re s 1. math This is the expression in analytic terms of the Dedekind domain uniqueness ... group of K . The Dedekind zeta function satisfies a functional equation relating its values at s and 1 ..., the values of the Dedekind zeta function at integers encode at least conjecturally important arithmetic ... L functions For the case in which K is an abelian extension of Q , its Dedekind zeta function can ... , its Dedekind zeta function is the Artin L function Artin L function of the regular representation ... zeta de Dedekind fr Fonction z ta de Dedekind ja nl Dedekind zeta functie pt Fun o zeta de Dedekind ... more details
In number theory , the Dedekind psi function is the multiplicative function on the positive integers defined by math psi n n prod p n left 1 frac 1 p right , math where the product is taken over all primes p dividing n by convention, 1 is the empty product and so has value 1 . The function was introduced by Richard Dedekind in connection with modular function s. The value of n for the first few integers n is 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24 ... OEIS id A001615 . n is greater than n for all n greater than 1, and is even for all n greater than 2. If n is a square free number then n divisor function n . The function can also be defined by setting p sup n sup p 1 p sup n 1 sup for powers of any prime p , and then extending the definition to all integers by multiplicitivity. This also leads to a proof of the generating function in terms of the Riemann zeta function , which is math sum frac psi n n s frac zeta s zeta s 1 zeta 2s . math References Goro Shimura , Introduction to the Arithmetic Theory of Automorphic Functions , Princeton, 1971 page 25, equation 1 External links MathWorld title Dedekind Function urlname DedekindFunction Category Multiplicative functions bg de Dedekindsche Psi Funktion ... more details
,d e rm i pi frac a d 12c s d,c frac 1 4 quad c 0 . math Here math s h,k , math is the Dedekindsum math s h,k sum n 1 k 1 frac n k left frac hn k left lfloor frac hn k right rfloor frac 1 2 right . math ...dablink For the Dirichlet series see Dirichlet eta function . Image Dedekind Eta.jpg right thumb 500px Dedekind function in the complex plane The Dedekind eta function , named after Richard Dedekind , is a function defined on the upper half plane of complex number s, where the imaginary part is positive. For any such complex number math tau , math , we define math q e 2 pi rm i tau , math , and define the eta function by math eta tau q frac 1 24 prod n 1 infty 1 q n . math The notation math q equiv e 2 rm i pi tau , math is now standard in number theory , though many older books use q for the nome mathematics nome math q equiv e pi rm i tau , math . The presence of 24 number 24 can be understood by connection with other occurrences, as in the modular discriminant and the Leech lattice . The eta function is holomorphic on the upper half plane but cannot be continued analytically beyond it. Image Q euler.jpeg thumb right Modulus of Euler phi on the unit disc, colored so that black 0, red 4 Image Discriminant real part.jpeg thumb right The real part of the modular discriminant as a function of q . The eta function satisfies the functional equation s ref cite journal author Siegel, C.L. title A Simple Proof of math eta 1 tau eta tau sqrt tau rm i , math journal Mathematika year 1954 volume 1 page 4 doi 10.1112 S0025579300000462 ref math eta tau 1 e frac pi rm i 12 eta tau , , math math eta tau 1 sqrt rm i tau eta tau . , math More generally, suppose math a, b, c, d , math are integers ... for special values of the arguments math eta z sum n 1 infty chi n exp tfrac 1 12 pi i n 2 z , math ... identity math phi q sum n infty infty 1 n q 3n 2 n 2 . math Because the eta function is easy to compute ... Funktion es Funci n eta de Dedekind fr Fonction ta de Dedekind ja pl Funkcja ... more details
In mathematics , a set A is Dedekind infinite if some proper subset B of A is equinumerous to A . Explicitly ... is Dedekind finite if it is not Dedekind infinite. A vaguely related notion is that of a Dedekind finite ... iff if and only if it is Dedekind infinite. However, this equivalence cannot be proved with the axiomatic ... below. Dedekind infinite sets in ZF The following conditions are equivalent in ZF . In particular, note that all these conditions can be proved to be equivalent without using the AC. A is Dedekind ... of all natural number s. A has a countable set countably infinite subset. Every Dedekind infinite .... This is sometimes written as A is dually Dedekind infinite . It is not provable in ZF without the AC that dual Dedekind infinity implies that A is Dedekind infinite. For example, if B is an infinite but Dedekind finite set, and A is the set of finite one to one sequences from B , then drop the last Element mathematics element is a surjective but not injective function from A to A , yet A is Dedekind finite. It can be proved in ZF that every dually Dedekind infinite set satisfies the following ... of A is Dedekind infinite Sets satisfying these properties are sometimes called weakly Dedekind infinite . It can be shown in ZF that weakly Dedekind infinite sets are infinite. ZF also shows that every well ordered infinite set is Dedekind infinite. Relation to AC and AC Since every infinite, well ordered set is Dedekind infinite, and since the AC is equivalent to the well ordering theorem ... set is Dedekind infinite. However, the equivalence of the two definitions is much weaker than the full ... with no denumerable subset. Hence, in this model, there exists an infinite, Dedekind finite set. By the above ... that every infinite set is Dedekind infinite. However, the equivalence of these two definitions ... every infinite set is Dedekind infinite, yet the CC fails. History The term is named after the German mathematician Richard Dedekind , who first explicitly introduced the definition. It is notable that this definition ... more details
In mathematics , in particular the study of abstract algebra , a Dedekind Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domain s. Definition Let R be an integral domain and g R Z sub 0 sub be a function from R to the non negative Integer rational integers . Denote by 0 sub R sub the additive identity of R . The function g is called a Dedekind Hasse norm on R if the following three conditions are satisfied g 0 sub R sub 0, if a 0 sub R sub then g a 0, for any nonzero elements a and b in R either b divides a in R , or there exist elements x and y in R such that 0 g xa &minus yb g b . The third condition is a slight generalisation of condition EF1 of Euclidean functions, as defined on the Euclidean domain article. If the value of x can always be taken as 1 then g will in fact be a Euclidean function and R will hence be a Euclidean domain. Integral and principal ideal domains The notion of a Dedekind Hasse norm was developed independently by Richard Dedekind and, later, by Helmut Hasse . They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a principal ideal domain . To wit, they proved that an integral domain R is a principal ideal domain if and only if R has a Dedekind Hasse norm. Example Let F be a Field mathematics field and consider the polynomial ring F X . The function g on this domain that maps a nonzero polynomial p to 2 sup deg p sup , where deg p is the degree of p , and maps the zero polynomial to zero, is a Dedekind Hasse norm on F X . The first two conditions are satisfied simply by the definition of g , while the third condition can be proved using polynomial long division . References R. Sivaramakrishnan, Certain number theoretic episodes in algebra , CRC Press , 2006. External links planetmath reference id 3188 title Dedekind Hasse valuation ... more details
In mathematical logic , the phrase Cantor Dedekind axiom has been used to describe the thesis that the real number s are order isomorphic to the linear continuum of geometry . In other words the axiom states that there is a one to one correspondence between real numbers and points on a line. This axiom is the cornerstone of analytic geometry . The Cartesian coordinate system developed by Ren Descartes explicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor . This is sometimes referred to as the real number line blend ref cite book author George Lakoff and Rafael E. N ez title Where Mathematics Comes From How the embodied mind brings mathematics into being publisher Basic Books year 2000 isbn 0 465 03770 4 ref A consequence of this axiom is that Alfred Tarski Alfred Tarski s proof of the decidability logic decidability of the ordered real field could be seen as an algorithm to solve any problem in Euclidean geometry . Notes reflist References Erlich, P.. 1994 . General introduction . Real Numbers, Generalizations of the Reals, and Theories of Continua , vi xxxii. Edited by P. Erlich, Kluwer Academic Publishers, Dordrecht Category Real numbers Category Mathematical axioms mathlogic stub eo Aksiomo de Cantor Dedekind nl Axioma van Cantor Dedekind ... more details
Wiktionary sumSum or summation is the process or result of addition . Sum may also refer to Sum category theory , a mathematical term Sum book Sum book , a 2009 collection of short stories by David Eagleman Sum Unix , a program for generating checksums Sum country subdivision , an administrative division in Mongolia and nearby regions the local term for one of the districts of Mongolia Som currency , the unit of currency used in Turkic speaking countries of Central Asia the IATA airport code for the Sumter Airport in Sumter County, South Carolina, USA the ISO 639 3 code for the Sumo language As an acronym, SUM may refer to Senter for utvikling og milj Centre for Development and the Environment , part of the University of Oslo SUM interbank network Soccer United Marketing The State University of Management , a Russian university See also 3SUM , a term from computational complexity theory Cogito ergo sum Direct sum , in mathematics Sum 41 , a Canadian punk band Sum certain , a legal term Disambig de Sum fr Sum it SUM nl Sum no Sum andre betydninger pl Sum uk ... more details
Sum rule may refer to Sum rule in differentiation Sum rule in integration Rule of sum , a counting principle in combinatorics Sum rule in quantum mechanics in quantum field theory , a sum rule is a property of the sum of the scattering probability over all energies which is independent of the particular dynamical details and can be calculated precisely. It is a special case of a quantum mechanical sum rule, with specific applications. In the theory of current algebra quark currents , the Adler Weissberger sum rule gives the normalization of weak interaction strength in terms of pion scattering amplitudes, the SVZ sum rules Mikhail Shifman Shifman Vainshtein Valentin I. Zakharov Zakharov sum rules in quantum chromodynamics predict some of the low lying meson properties from some universal vacuum parameters and short distance quark gluon interactions, while the finite energy sum rules are phenomenological properties of the hadronic spectral density which were significant for the discovery of string theory . mathdab ... more details
dablink The symbol math oplus math denotes direct sum it is also the astrological and astronomical symbol ... a direct sum of objects already known, giving a new one. This is generally the Cartesian product of the underlying ..., the direct sum is often, but not always, the coproduct in the Category mathematics category in question. In cases where an object is expressed as a direct sum of subobjects, the direct sum can be referred to as an internal direct sum . The direct sum of a family of objects A sub i sub , with i I ... include the direct sum of abelian groups , the direct sum of modules , the direct sum of rings , the direct sum of matrices , and the direct sum topology direct sum of topological spaces . A related concept is that of the direct product , which is sometimes the same as the direct sum, but at other times can be entirely different. span id abgrps span Direct sum of abelian groups The direct sum of abelian groups is a prototypical example of a direct sum. Given two abelian groups A , and B , , their direct sum A B is the same as their direct product of groups direct product , i.e. its ... groups A sub i sub for i I , the direct sum math bigoplus i in I A i math is a proper subgroup ... sum is indeed the coproduct in the category of abelian groups . Direct sum of modules e.g. vector spaces main Direct sum of modules span id reps span Direct sum of representations Group representations The direct sum of group representations generalizes the direct sum of modules direct sum of the underlying ... G modules , the direct sum of the representations is V W with the action of g G given component wise, i.e. g v , w g v , g w . Direct sum of rings main Product of rings Given a finite family of rings ... called the direct sum. Note that in the category of commutative rings , the direct sum is not the coproduct ... I.11 ref span id internal span Internal direct sum An internal direct sum is simply a direct sum ... sum of the x axis x , 0 x R and the y axis 0, y y R , and the sum of x , 0 and 0, y is the internal ... more details
In mathematics , statistics and elsewhere, sums of squares occur in a number of contexts Statistics For partitioning a sum of squares, see Sum of squares statistics For the sum of squared deviations , see Least squares For the sum of squared differences , see Mean squared error For the sum of squared error , see Residual sum of squares For the sum of squares due to lack of fit , see Lack of fit sum of squares For sums of squares relating to model predictions, see Explained sum of squares For sums of squares relating to observations, see Total sum of squares For sums of squared deviations, see Squared deviations For modelling involving sums of squares, see Analysis of variance For modelling involving the multivariate generalisation of sums of squares, see Multivariate analysis of variance Number theory For the sum of squares of consecutive integers, see Square pyramidal number For representing an integer as a sum of squares of integers, see Lagrange s four square theorem Fermat s theorem on sums of two squares says which integers are sums of two squares. A separate article discusses Proofs of Fermat s theorem on sums of two squares Algebra For representing a polynomial as sum of squares of polynomials, see Polynomial SOS . For representing a multivariate polynomial that takes only non negative values over the reals as a sum of squares of rational functions, see Hilbert s seventeenth problem . The Brahmagupta Fibonacci identity says the set of all sums of two squares is closed under multiplication. Geometry The Pythagorean theorem says that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares on the legs Control engineering Sum of squares optimization mathdab ... more details
The topic of power sums is treated at Please don t replace this with anything saying that the term power sums refers to either of these topics. It does not. Actually it means three slightly different things in the four topics below, but only the first topic is one actually referred to as power sum. Or worse, anything saying that power sums plural refer to those topics. Power sum symmetric polynomial Finite sum of equal powers of variables Newton s identities Certain identities involving power sum symmetric polynomials of functions Symmetric function A power sum symmetric function is a formal infinite sum of equal powers of variables Faulhaber s formula Formulas involving sums of equal powers of finitely many successive integers disambig ... more details
s sum , usually denoted c sub q sub n , is a function of two positive integer variables q and n defined by the formula math c q n sum a 1 atop a,q 1 q e 2 pi i tfrac a q n , math where a , q 1 means ... Dedekind Vorlesungen ber Zahlentheorie , 4th ed. ref In addition to the expansions ... sufficiently large odd number is the sum of three primes. ref Nathanson, ch. 8 ref Notation For integers ... sum d , mid ,m f d math means that d goes through all the positive divisors of m , e.g. math sum d , mid ... q primitive q sup th sup roots of unity. Thus, the Ramanujan sum c sub q sub n is the sum ... 12 sub sup 12 sup 1 is the primitive first root of unity. blockquote Therefore, if math eta q n sum k 1 q zeta q kn math is the sum of the n sup th sup powers of all the roots, primitive and imprimitive, math eta q n sum d , mid , q c d n , math and by M bius inversion , math c q n sum d , mid ,q ... q& mbox if q mid n end cases math and this leads to the formula math c q n sum d , mid , q,n mu left ... This shows that c sub q sub n is always an integer. Compare it with the formula math phi q sum d ... trigonometrical sums... , Ramanujan, Papers , p. 371 ref The equivalence of it and Ramanujan s sum ... the absolute value of the sequence c sub 1 sub n , c sub 2 sub n , ... is bounded by n , the sum of the divisors of n . If q 1 math sum n a a q 1 c q n 0. math Table class wikitable style text align center cellpadding 2 Ramanujan Sum c sub s sub n colspan 2 rowspan 2 colspan 30 n width 25 1 ... math f n sum q 1 infty a q c q n math or of the form math f n sum q 1 infty a q c n q math ... theory of prime numbers, namely that the series math sum n 1 infty frac mu n n math converges to 0 ... s sum delta , mid ,q mu left frac q delta right delta 1 s sum n 1 infty frac c q n n s math is a generating ... frac sigma r 1 n n r zeta r sum q 1 infty frac c q n q r math is a generating function for the sequence ... series math frac zeta s zeta r s 1 zeta r sum q 1 infty sum n 1 infty frac c q n q r n s . math ... more details
A lump sum is a single payment of money , as opposed to a series of payments made over time such as an annuity . It could be an agreement where in real estate development a developer or owner pays for the completed project by a general contractor and does not require a detailed breakdown of every cost . See also Lump sum tax Lottery payouts Structured settlement Distortions economics J.G. Wentworth External Links http www.sec.gov investor pubs lump sum payouts.htm SEC guidelines for lump sum payouts Category Money finance stub de Pauschale nl Lumpsum pl Rycza t ru uk ... more details
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