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 ... 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 it Somma di Dedekind nl Dedekind som zh ... more details
Dedekind is the name of People Brendon Dedekind born 1976 , South African swimmer Constantin Christian Dedekind 1628 1715 , German poet, dramatist and composer Friedrich Dedekind 1524 1598 , German humanist, theologian, and bookseller Henning Dedekind 1562 1626 , German composer, grandfather of Constantin Christian Dedekind Richard Dedekind 1831 1916 , German mathematician Other 19293 Dedekind , asteroid named after Richard Dedekind surname de Dedekind ru sv Dedekind ... more details
set Dedekind number DedekindsumDedekind zeta function Ideal ring theory Ideal number Vorlesungen ...for the 16th century humanist Friedrich Dedekind Infobox scientist name Richard Dedekind image Dedekind.jpeg image size 180px caption Richard Dedekind, c. 1870 birth date birth date October 6, 1831 October ... Dedekind October 6, 1831 &ndash February 12, 1916 was a German people German mathematician who did ... of the real number s. Life Dedekind s father was Julius Levin Ulrich Dedekind, an administrator at TU Braunschweig Collegium Carolinum in Braunschweig city Braunschweig . Dedekind had three older ... Carolinum in 1848 before moving to the University of G ttingen in 1850. There, Dedekind studied number ... level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis ... evident in Dedekind s subsequent publications. 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 ... the habilitation in 1854. Dedekind returned to G ttingen to teach as a Privatdozent , giving courses ... stamp from 1981, commemorating Richard Dedekind In 1858, he began teaching at the ETH Z rich Polytechnic ... Institute of Technology in 1862, Dedekind returned to his native Braunschweig, where he spent the rest ... to publish. He never married, instead living with his unmarried sister Julia. Dedekind was elected ... time at the ETH Z rich Polytechnic , Dedekind came up with the notion now called a Dedekind cut ... number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet Stetigkeit und irrationale Zahlen ..., while on holiday in Interlaken , Dedekind met Georg Cantor Cantor . Thus began an enduring relationship of mutual respect, and Dedekind became one of the very first mathematicians to admire ... correspondence between two sets, Dedekind said that the two sets were similar. He invoked similarity ... 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
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 ... domain with Krull dimension one i.e., every nonzero prime ideal is maximal . Thus a Dedekind ... easiest to verify 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 ... domain is a field , 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 ... more details
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 j k prod ... 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 ... expression as a summation , ref harvtxt Kisielewicz 1988 . ref are known. However Dedekind s problem ... from false to true and not from true to false. The Dedekind number M n is the number of different monotonic ... variables that can force the function value to be true. Therefore, the Dedekind number M n ... lattice elements and subtract two from the Dedekind numbers. ref Thus, the Dedekind numbers count ... Zaguia 1993 . ref The Dedekind numbers also count the number of abstract simplicial complex es .... Values The exact values of the Dedekind numbers are known for 0 n 8 2, 3, 6, 20, 168, 7581, 7828354 ... must also be even. ref harvtxt Yamamoto 1953 . ref The calculation of the fifth Dedekind number M 5 ... The logarithm of the Dedekind numbers can be estimated accurately via the bounds math n choose ... . As cited by harvtxt Wiedemann 1991 . citation last Dedekind first Richard author link Richard Dedekind ... Entropy, independent sets and antichains a new approach to Dedekind s problem volume 130 year 2002 ... die Reine und Angewandte Mathematik pages 139 144 title A solution of Dedekind s problem on the number ... Society pages 373 390 title On Dedekind s problem the number of isotone Boolean functions. II ... of the eighth Dedekind number volume 8 year 1991 . citation last Yamamoto first Koichi mr 0070608 ... Dedekind ... more details
Notability Astro date February 2012 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 uk 19293 vi 19293 Dedekind yo 19293 Dedekind ... 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 Friedrich Dedekind was found in Hannover 2008 by Eberhard Doll, hinden for more than 400 years ... 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 Persondata Metadata see Wikipedia Persondata . NAME Dedekind, Friedrich ALTERNATIVE ... PLACE OF DEATH DEFAULTSORT Dedekind, Friedrich Category 1524 births Category 1598 deaths Category People ... de Friedrich Dedekind hu Friedrich Dedekind ru , sv Friedrich 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
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 ... definite cut there corresponds a definite rational or irrational number .... Richard Dedekind, Continuity and Irrational Numbers , Section IV More generally, a Dedekind cut is a partition of a totally ... element. See also completeness order theory . In particular, as discussed below, Dedekind cuts ... as a Dedekind cut of rationals is a Complete metric space complete linear continuum continuum without any further gaps. Dedekind used the German word Schnitt cut in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to cut the other. Dedekind s construction of the number ... defines a Dedekind cut on the other. Representations 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, in terms ... 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 must have that A is the interval ... of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent ... . Ordering of cuts Regard one Dedekind cut A , B as less than another Dedekind cut C , D if A is a proper ... 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 cuts serves ... of the real numbers See also Construction of the real numbers Construction by Dedekind cuts A typical Dedekind cut of the rational number s is given by math A a in mathbb Q a 2 2 lor a le 0 , math ... more details
Infobox sportsperson birth date February 14, 1976 birth place medaltemplates 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 2000 FINA World Swimming Championships 25 m Men s 50 m breaststroke 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 Brendon Dedekind born February 14, 1976 in Pietermaritzburg , KwaZulu Natal . He won an international championship gold medal 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 Swimmer 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 SouthAfrica swimming bio stub it Brendon Dedekind no Brendon Debekind fi Brendon Dedekind ... more details
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 ...hatnote 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 , let math q e 2 pi rm i tau , math , and define the eta function by, math eta tau e frac pi rm i tau 12 prod n 1 infty 1 q n . math The notation math q equiv e 2 pi rm i tau , math is now standard in number theory , though many older books use q for the nome mathematics nome math e pi rm i tau , math . Note that, math Delta 2 pi 12 eta 24 tau math where math Delta math is the modular discriminant . The presence of 24 number 24 can be understood by connection with other occurrences, such as in the 24 dimensional Leech lattice . The eta function is holomorphic on the upper half plane but cannot be continued analytically beyond it. File Q Eulero.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 ... values of the arguments math eta z sum n 1 infty chi n exp tfrac 1 12 pi i n 2 z , math where math chi ... phi q sum n infty infty 1 n q 3n 2 n 2 . math Because the eta function is easy to compute numerically ... ca Funci eta de Dedekind de Dedekindsche Funktion es Funci n eta de Dedekind fr Fonction ta de Dedekind nl Dedekind functie ja pl Funkcja modularna Dedekinda fi Dedekindin eetafunktio ... 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 ... 1 2. The Dedekind zeta function is named for Richard Dedekind who introduced them in his ... 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 ... ring O sub K sub I . This sum converges absolutely for all complex numbers s with real ... zeta function. Euler product The Dedekind zeta function of K has an Euler product which is a product ... 1 1 N K mathbf Q P s , text for Re s 1. math This is the expression in analytic terms of the Dedekind ... group and class group of K . The Dedekind zeta function satisfies a functional equation relating its ... function, the values of the Dedekind zeta function at integers encode at least conjecturally important ... to other L functions For the case in which K is an abelian extension of Q , its Dedekind zeta function ... group G , its Dedekind zeta function is the Artin L function Artin L function of the regular representation ... equivalent if they have the same Dedekind zeta function. harvs txt last1 Bosma first1 Wieb ... different class numbers, so the Dedekind zeta function of a number field does not determine its class ... functions Category Algebraic number theory ar ca Funci zeta de Dedekind de Dedekindsche Zeta Funktion es Funci n zeta de Dedekind fr Fonction z ta de Dedekind nl Dedekind zeta functie ja 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 This is also a consequence of the fact that we can write as a Dirichlet convolution of math psi n epsilon 2 math where math epsilon 2 math is the Indicator function characteristic function of the squares. Higher Orders The generalization to higher orders via ratios of Jordan s totient function Jordan s totient is math psi k n frac J 2k n J k n math with Dirichlet series math sum n ge 1 frac psi k n n s frac zeta s zeta s k zeta 2s math . It is also the Dirichlet convolution of a power and the square of the Mobius function , math psi k n n k mu 2 n math . If math epsilon 2 1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0 ldots math is the Indicator function characteristic function of the squares, another Dirichlet convolution leads to the generalized Divisor function &sigma function , math epsilon 2 n psi k n sigma k n math . References Goro Shimura , Introduction to the Arithmetic Theory of Automorphic Functions , Princeton, 1971 page ... Functions year 2010 External links MathWorld title Dedekind Function urlname DedekindFunction Category ... 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 mathdab ... 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 ring . A ring is said to be a Dedekind finite ring if ab 1 implies ba 1 for any two ring elements ..., most mathematician s simply assumed that a set is infinite iff if and only if it is Dedekind ... than the axiom of countable choice CC . See the references below. Dedekind infinite sets in ZF The following ... to be equivalent without using the AC. A is Dedekind infinite . There is a function mathematics ... set countably infinite subset. Every Dedekind infinite set A also satisfies the following condition ... 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 ... 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 equivalent conditions There exists a surjective map from A onto a countably infinite set. The powerset 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 the axiom of choice Since every infinite, well ordered set is Dedekind infinite, and since ... the general AC implies that every infinite set is Dedekind infinite. However, the equivalence of the two ... an infinite, Dedekind finite set. By the above, such a set cannot be well ordered in this model. If we assume the CC AC sub sub , then it follows that every infinite set is Dedekind infinite ..., there exists a model of ZF in which every infinite set is Dedekind infinite, yet the CC fails. History The term is named after the German mathematician Richard Dedekind , who first explicitly ... 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
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
File Constantin Christian Dedekind.jpg thumb Constantin Christian Dedekind Constantin Christian Dedekind 2 April 1628 1715 was a German poet, dramatist, librettist, composer and bass singer of the Baroque era . Biography Dedekind was born in Reinsdorf, Thuringia into a musical family, the son of musician Stefan Dedekind 1595 1636 and the grandson of composer Henning Dedekind 1562 1626 . He was educated at Quedlinburg Abbey . From about 1647 he lived in Dresden . Early recognition of his poetic talent came in 1652 when Johann Rist , in his role of Count Palatine Imperial Imperial Count Palatine , awarded him the Dichterkrone equivalent to making him Poet Laureate . A few years later Dedekind became a member of the Elbschwanenorden Elbe Swan order , Rist s poetical society. He also pursued ..., though not a very successful one. Dedekind s wide circle of friends in Dresden included composers .... 193 194 ref and Georg Neumark , as well as Dedekind himself. ref Buelow 2004 . p. 247 ref He also ... Rede bind und Dicht Kunst 1679 . He and Dedekind exchanged correspondence for many years. In 1680 the outbreak of plague disease plague caused Dedekind to flee Dresden for Meissen . In his last .... Dedekind died in Dresden and was buried on 2 September 1715. Works Die Aelbianische Musen Lust 1657 ... pages 371 373 author Aikin, Judith P. title Constantin Christian Dedekind Neue geistliche Schauspiele ... Companion to Baroque Music chapter Dedekind, Constantin Christian last Sadie first Julie Anne year ... links ChoralWiki IckingArchive idx Dedekind name Constantin Christian Dedekind Persondata Metadata see Wikipedia Persondata . NAME Dedekind, Constantin Christian ALTERNATIVE NAMES SHORT DESCRIPTION ... 1715 PLACE OF DEATH Dresden DEFAULTSORT Dedekind, Constantin Christian Category 1628 births ... from Dresden Germany poet stub Germany composer stub ca Constantin Christian Dedekind de Constantin Christian Dedekind ... more details
Sum of permutations may refer to Direct sum of permutations Skew sum of permutations disambig Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... 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
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
In mathematics , statistics and elsewhere, sums of squares occur in a number of contexts Statistics For partitioning of variance, see Partition of sums of squares 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 and algebraic geometry For representing a polynomial as the sum of squares of polynomials , see Polynomial SOS . For computational optimization , see Sum of squares optimization . 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. Euclidean geometry and other inner product spaces 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 mathdab ... 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 ar eo Aksiomo de Cantor Dedekind nl Axioma van Cantor Dedekind ... 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 of the n ... 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 mu left frac ... mid n end cases math and this leads to the formula math c q n sum d , mid , q,n mu left frac q d ... that c sub q sub n is always an integer. Compare it with the formula math phi q sum d , mid ,q mu ... sums... , Ramanujan, Papers , p. 371 ref The equivalence of it and Ramanujan s sum is due ... 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 Let m sub 1 sub , m sub 2 sub 0, m lcm m sub 1 sub ... orthogonality property math frac 1 m sum k 1 m c m 1 k c m 2 k begin cases phi m , & text if m 1 m ... sum stackrel d mid n gcd d,k 1 d frac mu tfrac n d phi d frac mu n c n k phi n , math known as the Richard ... ref T th, external links, eq. 8. ref math sum stackrel 1 le k le n gcd k,n 1 c n k a mu n c n a , math due to Cohen. Table class wikitable style text align center cellpadding 2 Ramanujan Sum c sub ... mathematics convergent infinite series of the form 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 where the a sub q sub are complex ... more details
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