Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

* Given a Gödel numbering of the computable functions, the set ( where is the Cantor pairing function and indicates is defined ) is recursively enumerable.

Given that John of Worcester wrote his chronicle after the eruption of the Canterbury – York supremacy struggle, the story of Ealdred renouncing any claims to Worcester needs to be considered suspect.

* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.

Given a year to complete the work, Gibson undertook the actual writing out of " blind animal terror " at the obligation to write an entire novel – a feat which he felt he was " four or five years away from ".

Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church – Turing thesis.

* June 9 – Royal Assent Given To The Town Of Milford Haven.

* Shay Given – former goalkeeper for both the Republic of Ireland and Newcastle United F. C ..

Given overwhelming evidence of Louis's collusion with the invaders, the verdict was a foregone conclusion – with 693 deputies voting guilty, none for acquittal, with 23 abstaining.

Given that A. florea is more widely distributed and A. andreniformis is considerably more aggressive, honey is – if at all – usually harvested from the former only.

Given the derivatives, wave – particle duality can be assessed as follows.

* The Whole Royal and Magnificent Entertainment Given to King James Through the City of London ( 1603 – 4 ).

Given this information, a timeline of roughly the years 15 – 98 CE can be established for his adult life.

Given this uncertainty, between April – May 2006 at the JINR, a FLNR – PSI team conducted experiments probing the synthesis of this isotope as a daughter in the nuclear reaction < sup > 242 </ sup > Pu (< sup > 48 </ sup > Ca, 3n )< sup > 287 </ sup > Fl.

Given Henry's marital history – banishing his first wife and beheading the second – Mary refused the offer.

Given an adequate supply of food, larvae should pupate and weave a silken cocoon within 1 – 2 weeks after 3 larval stages.

** Masterclasses – Given in public by world renowned filmmakers.

Given that both Jonathan and Peleg Card were scheduled to be tried by the Vermont Committee of Safety ( Peleg died 6 Sep 1777, and it is unknown if the trial was actually held ) it is likely that Philo Card is, in reality, Peleg Card ( 1740 – 1777 ) --- and Stephen ( 1748 – 1798 ) -- ( Source: National Archives of Canada, WO 28 / 10, Reel B-2867, pp. 17 – 18.

Song Jiaoren (; Given name at birth: Liàn 鍊 ; Courtesy name: Dùnchū 鈍初 ) ( 5 April 1882 – 22 March 1913 ) was a Chinese republican revolutionary, political leader and a founder of the Kuomintang ( KMT ).

Given the choice, whether one believes that " the lives of entire peoples and societies have been decisively influenced by exceptional individuals " or, conversely, that whatever happens occurs as a result of impersonal forces oblivious to human intentions – Berlin rejected both options and the choice itself as nonsensical.

Given the subsequent emergence of organized fandom for comic books among adults who grew up reading them during Comics ' Golden Age, it is ironic Wertham at one point in Seduction ( pp. 89 – 90 ) asserts " I have known many adults who have treasured throughout their lives some of the books they read as children.

Given his prestige, Rundstedt was a central figure in the Blomberg – Fritsch Affair which engulfed the German Army in early 1938.

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

** Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.

** Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.

Given the short distances of super special stages compared to the regular special stages and consequent near-identical times for the frontrunning cars, it is very rare for these spectator-oriented stages to decide rally results, though it is a well-known axiom that a team can't win the rally at the super special, but they can certainly lose it.

#( Affine axiom of parallelism ) Given a point A and a line r, not through A, there is at most one line through A which does not meet r.

# Playfair's axiom: Given a line and a point not on, there exists exactly one line containing such that

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms.

#( TR 3 ) Given a map between two morphisms, there is a morphism between their mapping cones ( which exist by axiom ( TR 1 )), that makes everything commute.

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given these facts, algorithm defined by the following pseudocode:

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples into one category or the other.

Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm.

Given a desired backbone conformation, side chain packing can be designed using variations of the dead-end elimination algorithm ; however, attempts to design proteins of novel folds have difficulty using this method due to an absence of plausible backbone models.

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that predicts whether a new example falls into one category or the other.

Given Bluestein's algorithm, such a transform can be used, for example, to obtain a more finely spaced interpolation of some portion of the spectrum ( although the frequency resolution is still limited by the total sampling time ), enhance arbitrary poles in transfer-function analyses, etcetera.

Given an appropriately designed spam filtering algorithm, outbound spam filtering can be implemented with a near zero false positive rate, which keeps customer related issues with blocked legitimate email down to a minimum.

Given decision problems A and B and an algorithm N which solves instances of B, we can use a many-one reduction from A to B to solve instances of A in:

The 1st generalized division algorithm: Given integers,, with, there exist unique integers and with < math > d

Given a polynomial g, polynomials ( f < sub > 1 </ sub >, ..., f < sub > m </ sub >) and an order on the monomials in k ..., x < sub > n </ sub >, we construct the reduction of g modulo f < sub > 1 </ sub >, ..., f < sub > m </ sub > by the following algorithm.

Given a 3D audio " picture " of the scene, a specialised algorithm generates secondary sound waves necessary to recreate it in the particular room and then instructs a large number of speakers ( 300-400 ) to generate the " audio hologram ".

Given a presentation of a group G by generators and relations and a subgroup H of G, the algorithm enumerates the cosets of H on G and describes the permutation representation of G on the space of the cosets.

Given first 12 digits 979-0-9016791-7, the ISMN algorithm evaluates

Given M intervals of the form c ± r ( which means ), the algorithm seeks to find an interval with M − f sources.

Given an incompressible objective function, there is no basis for choosing one algorithm over another.