Some mathematicians, especially in set theory, do not consider the sets and to be part of the relation, and therefore define a binary relation as being a subset of x, that is, just the graph.

Some of the mathematicians who were involved in the selection and presentation of the seven problems were Atiyah, Bombieri, Connes, Deligne, Fefferman, Milnor, Mumford, Wiles, and Witten.

Some even continue to this day to remain a challenge for mathematicians.

Some of the top mathematicians who were or are now permanent professors at the IHÉS include

Some prominent female mathematicians are Hypatia of Alexandria ( ca.

: Some of you may have met mathematicians and wondered how they got that way.

Some notable mathematicians include Archimedes of Syracuse, Leonhard Euler, Carl Gauss, Johann Bernoulli, Jacob Bernoulli, Aryabhata, Brahmagupta, Bhaskara II, Nilakantha Somayaji, Omar Khayyám, Muhammad ibn Mūsā al-Khwārizmī, Bernhard Riemann, Gottfried Leibniz, Andrey Kolmogorov, Euclid of Alexandria, Jules Henri Poincaré, Srinivasa Ramanujan, Alexander Grothendieck, David Hilbert, Alan Turing, von Neumann, Kurt Gödel, Joseph-Louis Lagrange, Georg Cantor, William Rowan Hamilton, Carl Jacobi, Évariste Galois, Nikolay Lobachevsky, Rene Descartes, Joseph Fourier, Pierre-Simon Laplace, Alonzo Church, Nikolay Bogolyubov and Pierre de Fermat.

Some of these mathematicians were influential in arranging the Elements, that Euclid later published.

Some of the greatest physicists have also been creative mathematicians.

Some mathematicians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution.

Some mathematicians, therefore, do not recognize blackboard bold as a separate style from bold: Jean-Pierre Serre, for example, has publicly inveighed against the use of " blackboard bold " anywhere other than on a blackboard, and uses double-struck letters when writing bold on the blackboard, whereas his published works consistently use ordinary bold for the same symbols.

Some mathematicians hold opinions that amount to more nuanced versions of Platonism.

Some mathematicians independently made the related hypothesis ( sometimes incorrectly called the Chinese Hypothesis ) that p is a prime if and only if.

Some mathematicians as late as the sixteenth century continued to view halving as a separate operation, and it often continues to be treated separately in modern computer programming.

Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group.

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature < tt >(−,+,+,+)</ tt > ( Some may also prefer the alternative signature < tt >(+,−,−,−)</ tt >; in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.

Some sources imply that the symbol was first used by Arabic mathematicians.

Some working mathematicians have espoused mathematical realism as they see themselves experiencing naturally-occurring objects.

Some mathematicians and scientists are critical of using closed source mathematical software such as Mathematica for mathematical proofs, because these do not provide transparency, and thus are not verifiable.

Some mathematicians define the Hessian as the determinant of the above matrix.

On 21 November 1930 the declaration of the “ initiative group ” of the Moscow Mathematical Society which consisted of former Luzin's students Lazar Lyusternik and Lev Shnirelman along with Alexander Gelfond and Lev Pontryagin claimed that “ there appeared active counter-revolutionaries among mathematicians .” Some of these mathematicians were pointed out, including the advisor of Luzin, Dmitri Egorov.

Some jokes are based on stereotypes of mathematicians tending to think in complicated, abstract terms, causing them to lose touch with the " real world ".

Some mathematicians objected to his methods of analysis founded on the infinitesimal calculus.

Some historians have found his point of view not to their taste, others have complained that he makes the Tory tradition appear `` contemptible rather than intelligible '', while a sympathetic critic has remarked that the `` intricate interplay of social dynamics and political activity of which, at times, politicians are the ignorant marionettes is not a field for the exercise of his talents ''.

Some historians define and minimize the Arian conflict as the exclusive construct of Arius and a handful of rogue bishops engaging in heresy ; but others recognize Arius as a defender of ' original ' Christianity, or as providing a conservative response against the politicization of Christianity seeking union with the Roman Empire.

Some modern historians theorize that Nero's decision to kill Agrippina was prompted by her plotting to set Gaius Rubellius Plautus ( Nero's maternal second cousin ) or Britannicus ( Claudius ' biological son ) on the throne.

Some historians speculate that it was his conservative powerbase's disapproval of his foreign, non-Orthodox bride, more so than her appearance, that caused Alexei to spurn Charlotte.

Some historians believe that Acts borrows phraseology and plot elements from Euripides ' play The Bacchae.

Some Swedish historians believe the name derives from the cripple secondary god Balder of Nordic mythology.

Some historians have questioned the reliability of some of Bede's accounts.

Herald-Traveler Corp. operated for years under temporary authority from the Federal Communications Commission stemming from controversy over luncheon meetings the newspaper's chief executive had with an FCC commissioner during the original licensing process ( Some Boston broadcast historians accuse the Boston Globe of being covertly behind the proceeding.

Some battles are named for the convenience of military historians so that periods of combat can be neatly distinguished from one another.

Some historians were prepared to go even further, claiming that blitzkrieg was not merely an operational doctrine of the German armed forces but a strategic concept on which the leadership of the Third Reich based its strategic and economic planning.

Some historians have cited the U. S. intervention and bombing campaign ( spanning 1965 – 1973 ) as a significant factor leading to increased support of the Khmer Rouge among the Cambodian peasantry.

Some historians have argued that there is an element of psychological projection in conspiracism.

" Some historians have put forward the idea that more recently the United States has become the home of conspiracy theories because so many high-level prominent conspiracies have been undertaken and uncovered since the 1960s.

Some historians may place Aztlán in Nayarit or the Caribbean while other historians entirely disagree, and make a distinction between legend and the contemporary socio-political ideology.

Some music historians argue that Chicanos of Los Angeles in the late 1970s might have independently co-founded punk rock along with the already-acknowledged founders from British-European sources when introduced to the US in major cities.

The canonization of Saint Udalric, Bishop of Augsburg, by Pope John XV in 999 is the first undoubted example of a papal canonization of a saint from outside Rome ( Some historians maintain that the first such canonization was that of Saint Swibert by Pope Leo III in 804 ).

Some historians, particularly Josephus, claim that Claudius was directed in his actions by the Judaean King Herod Agrippa.

Some historians think Kling's absence was significant enough to prevent the Cubs from also winning a third straight title in 1909, as they finished 6 games out of first place.

Some modern historians think that Caligula suffered from hyperthyroidism.

Some historians who have examined the life of Cotton Mather after Chadwick Hansen ’ s book also seem to yearn for a positive view of Cotton Mather.

Some modern historians deem his piratical reputation unjust, as there is evidence that Kidd acted only as a privateer.

Some of his pamphlets were purported to be written by Scots, misleading even reputable historians into quoting them as evidence of Scottish opinion of the time.

Some historians state that Diocletian, like some emperors before him, adopted Maximian as his filius Augusti, his " Augustan son ", upon his appointment to the throne.

Some historians believe that the story by Petrus Olai refers to a source from the first half of the 15th century, making this the oldest reference to the falling flag.