In mathematics , a set A is Dedekindinfinite if some proper subset B of A is equinumerous to A . Explicitly ..., most mathematician s simply assumed that a set is infinite iff if and only if it is Dedekindinfinite. However, this equivalence cannot be proved with the axiomatic set theory axioms of Zermelo ... set countably infinite subset. Every Dedekindinfiniteset A also satisfies the following condition ... that A is Dedekindinfinite. For example, if B is an infinite but Dedekind finite set, and A is the set ... Dedekindinfiniteset satisfies the following equivalent conditions There exists a surjective map from A onto a countably infiniteset. The powerset of A is Dedekindinfinite Sets satisfying these properties ... sets are infinite. ZF also shows that every well ordered infiniteset is Dedekindinfinite. Relation to the axiom of choice Since every infinite, well ordered set is Dedekindinfinite, and since ... the general AC implies that every infiniteset is Dedekindinfinite. 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 infiniteset is Dedekindinfinite ..., there exists a model of ZF in which every infiniteset is Dedekindinfinite, yet the CC fails ... entertain the thought that there might be a distinction between the notions of infiniteset and Dedekind ... is Dedekind finite if it is not Dedekindinfinite. A vaguely related notion is that of a Dedekind finite ... definition of infiniteset This definition of infiniteset should be compared with the usual definition a set A is finite setinfinite when it cannot be put in bijection with a finite ordinal number ordinal , namely a set of the form 0,1,2,..., n &minus 1 for some natural number n an infiniteset is one ... than the axiom of countable choice CC . See the references below. Dedekindinfinite sets in ZF The following ... to be equivalent without using the AC. A is Dedekindinfinite . There is a function mathematics ... more details
Refimprove date September 2011 In set theory , an infiniteset is a Set mathematics set that is not a finite set . Infinite sets may be countable set countable or uncountable set uncountable . Some examples are the set of all integer s, ..., 1, 0, 1, 2, ... , is a countably infiniteset and the set of all real number s is an uncountable set uncountably infiniteset . Properties The set of natural numbers whose existence is postulated by the axiom of infinity is infinite. It is the only set which is directly required by the axioms to be infinite. The existence of any other infiniteset can be proved in Zermelo Fraenkel set theory ZFC only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The powerset of an infiniteset is infinite. Any superset of an infiniteset is infinite. If an infiniteset is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infiniteset is infinite. The Cartesian product of an infiniteset and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite if the axiom of choice holds, then it is infinite. If an infiniteset is a well ordered set , then it must have a nonempty subset which has no greatest element. In ZF, a set is infinite if and only if the powerset of its powerset is a Dedekindinfiniteset , having a proper subset equinumerous to itself. If the axiom of choice is also true, infinite sets are precisely the Dedekindinfinite sets. If an infiniteset ... in infiniteset theory Category Cardinal numbers bg cs Nekone n mno ina de ... 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 sv Dedekind ... more details
to give the first precise definition of an Dedekindinfinitesetinfiniteset a set is infinite ... in Dedekind s work. Dedekind defined an ring ideal ideal as a subset of a set of numbers, composed ... his definition of an infiniteset . He also proposed an axiom atic foundation for the natural numbers ... Peano Peano , citing Dedekind, formulated an equivalent but simpler Peano axioms set of axioms , now the standard ones. See also Dedekind cut Dedekind domain Dedekind eta function DedekindinfinitesetDedekind number Dedekind sum Dedekind 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 ... more details
, the integral closure of a Dedekind domain in an infinite algebraic extension is a Pr fer domain ...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 ... 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
Set families es N mero de Dedekind ... 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 ... element set, the number of elements in a free distributive lattice with n generators, or the number ... 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 ... of sets, none of which is contained in any other set. If V is a set of n Boolean variables, an antichain ... set of inputs if some subset of the true inputs to f belongs to A and false otherwise. Conversely ... variables that can force the function value to be true. Therefore, the Dedekind number M n equals the number of different antichains of subsets of an n element set. ref harvtxt Kahn 2002 . ref ... given by Birkhoff s representation theorem from the partially ordered set of subsets of the n variables with set inclusion as the partial order. This construction produces the free distributive lattice ... 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 on n elements, families of sets with the property that any subset of a set in the family also belongs ... of the two element set x , y The function f x , y false that ignores its input values and always returns false corresponds to the empty set empty antichain . The logical conjunction f x , y x y corresponds to the antichain x , y containing the single set x , y ... corresponds to the antichain x containing the single set x The function f x , y y that ignores ... the single set y The logical disjunction f x , y x y corresponds to the antichain ... 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 ... 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 ... relations. The set of all Dedekind cuts is itself a linearly ordered set of sets . Moreover, the set ... any upper bound has a least upper bound. Thus, constructing the set of Dedekind cuts serves ... sets Main Dedekind MacNeille completion More generally, if S is a partially ordered set , a completion ... is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals ... 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 ordered set into two non empty parts A and B , such that A is closed downwards meaning that for all ... element. See also completeness order theory . In particular, as discussed below, Dedekind cuts ... element of the B set . In other words, the number line where every real number is defined 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 ... of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent ... 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 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
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 c &minus D a , b c , D a , b &minus c D a , b c . By the periodicity of D in its first two arguments, the third argument being the length of the period for both, D a , b c D a kc , b lc c , for all integers k , l . If d is a positive integer, then D ad , bd cd dD a , b c , D ad , bd c D a , b c ... 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
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 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
Infobox Single Name Infinite... Cover Artist Beni Arashiro from Album Beni Released October 20, 2004 Format CD Recorded Genre Pop music Pop Length Label Avex Trax Producer Junpei Takada, CMJK, Hirofumi Asamoto, S.Hoffs, B. Steinberg, T. Kelly Last single Harmony Beni Arashiro Harmony br 2004 This single Infinite... br 2004 Next single Here Alone br 2004 Infinite... is Beni Arashiro s second single, which served as the outro theme for TBS s Count Down TV in October 2004. ref ja icon cite web url http www.amazon.co.jp dp B0002L4DPU title Amazon.co.jp Infinite... single Maxi publisher Amazon date 2009 06 27 accessdate 2009 06 27 ref Track listing Infinite... Flower on the earth Eternal flame Infinite... Instrumental Flower on the earth Instrumental Charts Oricon Sales Chart Japan class wikitable width 500px align left Release align left Chart align left Peak Position align left First Week Sales align left Sales Total align left Chart Run align left 20 October 2004 align left Oricon Weekly Singles Chart align left 24 align left 7,201 align left align left 4 weeks References Reflist Category Beni singer songs Category 2004 singles 2000s Japan single stub ja Infinite... more details
File Dedekind Macneille completion.svg thumb 240px The Hasse diagram of a partially ordered set left and its Dedekind MacNeille completion right . In order theory order theoretic mathematics , the Dedekind MacNeille completion of a partially ordered set also called the completion by cuts or normal completion ... set , gives an equivalent definition of the Dedekind MacNeille completion. With the alternative ... s, viewed as a totally ordered set with the usual numerical order, then each element of the Dedekind ... numbers is an example of the Dedekind completion of a totally ordered set , and the Dedekind MacNeille ... a set of elements no two of which are comparable then the Dedekind MacNeille completion of mvar ... the Dedekind MacNeille completion of a finite partially ordered set. Because the Dedekind ... Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cut s used by Dedekind to construct the real number s from the rational number s. Order embeddings and lattice completions A partially ordered set consists of a set of elements together with a binary relation math x y on pairs of elements that is reflexive ... ordered set, a completion of mvar S means a complete lattice mvar L with an order embedding of mvar ... and &minus &infin are a completion in this sense of the rational numbers the set of rational numbers ... numbers it has the upper bound pi . A given partially ordered set may have several different completions. For instance, one completion of any partially ordered set mvar S is the set of its lower set downwardly closed subsets ordered by subset inclusion . mvar S is embedded in this lattice by sending each element mvar x to the lower set that it generates. The result is a distributive lattice ... 2004 isbn 9780470850558 page 10 . ref Among all possible lattice completions, the Dedekind MacNeille ... Definition For each subset mvar A of a partially ordered set mvar S , let math A sup u sup denote ... more details
Confusing date November 2007 The Absolute Infinite is mathematician Georg Cantor s concept of an infinity that transcended the transfinite number s. Cantor equated the Absolute Infinite with God . ref 3.2, cite journal author Ignacio Jan title The role of the absolute infinite in Cantor s conception of set journal Erkenntnis volume 42 issue 3 month May year 1995 pages 375 402 doi 10.1007 BF01129011 ref He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller object Citation needed date November 2007 . Cantor s view Cantor is quoted as saying quote The actual infinite arises in three contexts first ... , where I call it the Absolute Infinite or simply Absolute second when it occurs in the contingent ... Mifflin, 1987 ISBN 0395383153. ref Cantor also mentioned the idea in his letters to Richard Dedekind ... from Cantor to Dedekind, dated July 28, 1899. However, as Ivor Grattan Guinness has discovered, ref http www.digizeitschriften.de home services pdfterms ?ID 516899 The Rediscovery of the Cantor Dedekind ... , of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August ..., then as a well ordered set, a number would correspond to it which would be greater than ... of all ordinal numbers is an inconsistent, absolutely infinite multiplicity. The Burali Forti ... that, for every property that can be logically defined, there exists a set of all objects that have ... generally, as noted by A.W. Moore mathematician A.W. Moore , there can be no end to the process of set mathematics set formation, and thus no such thing as the totality of all sets , or the set hierarchy . Any such totality would itself have to be a set, thus lying somewhere within the cumulative hierarchy hierarchy and thus failing to contain every set. A standard solution to this problem is found in Zermelo set theory Zermelo s set theory , which does not allow the unrestricted formation of sets ... 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
wiktionarypar set TOCright Set may refer to Mathematics and programming Set mathematics , A collection of well defined and distinct objects Set theory Category of sets Set computer science , a data type that is a collection of unique values Set C , an implementation of Set computer science in the C language Standard Template Library STL Chemistry SET, single electron transfer Psychology Set psychology , a set of expectations which shapes perception or thought Technology Set, to make become solid see Solidification A television setSET, Secure Electronic Transaction , a standard protocol for securing credit card transactions over insecure networks SET, Single Electron Transistor , a device to amplify currents in nanoelectronics SET, single ended triode , a type of electronic amplifier SET Awards, The Science, Engineering & Technology Student of the Year Awards Set command , a command for manipulating ..., a Quality Assurance job title in some software companies Saw set , the process of setting the teeth of a saw so each tooth protrudes to the side of the blade The arts Theatre Theatrical scenery Set construction , construction of scenery for theatrical, movie and video production Dancing Set, the basic square formation in Square dance square dancing Music DJ set or DJ mix , a musical performance by a DJ Set theory music , dealing with concepts for categorizing musical objects and describing their relationships Set music , a collection of discrete entities, for example pitch sets, duration sets, and timbre sets Set Thompson Twins album Set Thompson Twins album Set Alex Chilton album Set Alex Chilton album Set list , a list of songs to be performed at a concert Fiction Set or Father Set , a fictional ... Sport and games A chess setSet, a signal used in American football Set darts , when one player wins three legs Set game , a card game Set, a unit of play in tennis Set, a term for three of a kind in poker Set, a unit of play in volleyball formerly game Set, a group of repetitions in weight training ... more details
The term semi infinite has several related meanings in various branches of pure and applied mathematics . It typically describes objects which are infinite or unbounded in some but not all possible ways. In ordered structures and Euclidean spaces Generally, a semi infiniteset is bounded in one direction, and unbounded in another. For instance, the natural numbers are semi infinite considered as a subset of the integers similarly, the interval mathematics intervals math c, infty math and math infty,c math and their closed counterparts are semi infinite subsets of math R math . Half space s are sometimes described as semi infinite regions. Semi infinite regions occur frequently in the study of differential equations . ref Bateman, http projecteuclid.org euclid.bams 1183492736 Transverse seismic waves on the surface of a semi infinite solid composed of heterogeneous material , Bull. Amer. Math ... HeatDiffusionInASemiInfiniteRegion Heat Diffusion in a Semi Infinite Region accessed ... semi infinite metal bar. A semi infinite integral is an improper integral over a semi infinite interval. More generally, objects indexed or parametrised by semi infinite sets may be described as semi infinite. ref Cator, Pimentel, http arxiv.org abs 1001.4706v3 A shape theorem and semi infinite geodesics ... ness properties, not cardinality or measure mathematics measure properties semi infinite sets are typically infinite in cardinality and measure. In optimisation Main Semi infinite programming Many optimisation mathematics optimisation problems involve some set of variables and some set of constraints. A problem is called semi infinite if one but not both of these sets is finite. The study of such problems is known as semi infinite programming . ref Reemsten, R ckmann, http books.google.ca books?id sJgX5jQZnQcC&lpg PP1&ots gvz6 MY t1&dq semi infinite 20programming&pg PP1 v onepage&q&f false Semi infinite Programming , Kluwer Academic, 1998. ISBN 0 7923 5054 5 ref References references ... more details
In group theory , an area of mathematics , an infinite group is a group mathematics group , of which the forgetful functor underlying set contains an infinitesetinfinite number of element mathematics element s. Examples R , Lie group General linear group Category Group theory de Unendliche Gruppe nl Oneindige groep ru ... 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
The Paradoxes of the Infinite , is a mathematical work by Bernard Bolzano on the theory of Set mathematics sets . It was published in 1851, three years after Bolzano s death, by a friend. The work contained many interesting results in set theory. Bolzano expanded on the theme of Galileo s paradox, giving more examples of correspondences between the elements of an infiniteset and proper subsets of infinite sets. In the work he also coined the term set . References citation last Burton first David date 1997 title The History of Mathematics An Introduction publisher McGraw Hill edition Third pages 592 Category History of mathematics Category Infinity Category Logic literature Category Set theory ... more details
Doctorwhobook title Infinite Requiem series Virgin New Adventures number 36 featuring Seventh Doctor br Bernice Summerfield Bernice cover Infinite Requiem.jpg writer Daniel Blythe publisher Virgin Books isbn ISBN 0 426 20437 9 pages date March 1995 preceding Set Piece Doctor Who Set Piece following Sanctuary Doctor Who Sanctuary Infinite Requiem is an original novel written by Daniel Blythe and based on the long running United Kingdom British science fiction on television science fiction television series Doctor Who . It features the Seventh Doctor and Bernice Summerfield Bernice . A prelude to the novel, also penned by Blythe, appeared in Doctor Who Magazine 223. DWspinoff External links http www.drwhoguide.com whona36p.htm Infinite Requiem Prelude Doctor Who RG id who na36 title Infinite Requiem http members.lycos.co.uk cloisterlibrary infi.htm The Cloister Library Infinite Requiem TardisLibrary id 369 Reviews OG review id na 36 title Infinite Requiem DWRG id infi title Infinite Requiem Category Virgin New Adventures Category 1995 novels Doctorwho stub ... 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 ... 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 ... zeta function, the values of the Dedekind zeta function at integers encode at least conjecturally ... to other L functions For the case in which K is an abelian extension of Q , its Dedekind zeta ... 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 ... have different class numbers, so the Dedekind zeta function of a number field does not determine its ... and L functions Category Algebraic number theory 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
Image Infinite broom.svg right Standard infinite broom In topology , the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of Connected space connectedness . The closed infinite broom is the Closure topology closure of the infinite broom, and is also referred to as the broom space . ref Chapter 6 exercise 3.5 of Citation last Joshi first K. D. title Introduction to general topology publisher John Wiley & Sons, Inc. location New York isbn 978 0852264447 id MathSciNet id 0709260 year 1983 ref Definition The infinite broom is the subset of the Euclidean plane that consists of all closed line segment s joining the Origin mathematics origin to the point 1, 1 n as n varies over all positive integer s, together with the interval mathematics interval , 1 on the x axis. ref name CIT Citation unused data p. 139 last1 Steen first1 Lynn Arthur author1 link Lynn Arthur Steen last2 Seebach first2 J. Arthur Jr. author2 link J. Arthur Seebach, Jr. title Counterexamples in Topology origyear 1978 publisher Dover Publications, Inc. location Mineola, NY edition Dover Publications Dover reprint of 1978 isbn 978 0 486 68735 3 id MathSciNet id 1382863 year 1995 ref The closed infinite broom is then the infinite broom together with the interval 0, on the x axis. In other words, it consists of all closed line segments joining the origin to the point 1, 1 n or to the point 1, 0 . ref name CIT Properties Both the infinite broom and its closure are Connected space connected , as every open set in the plane which contains the segment on the x axis must intersect slanted segments. Neither are locally connected . Despite the closed infinite broom being arc connected , the standard infinite broom is not path connected . ref name CIT See also Comb space References references Category Topological spaces ... more details
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 , 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. 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 1 sqrt rm i tau eta tau . , math More generally, suppose math a, b, c, d , math are integers with math ad bc 1 , math , so that math tau mapsto frac a tau b c tau d math is a transformation belonging to the modular group . We may assume that either math c 0 , math , or math c 0 , math ...,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 Dedekind sum math ... functions 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 zh ... more details
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