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 mathematicians and historians have wrongfully claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss, which is untrue-Gauss himself appreciated Lobachevsky's published works very highly, but they never had personal correspondence between them prior to the publication.

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 scholars have objected to the language of the speeches as too Lukan in style to reflect anyone else's words.

Some delegates objected to the establishment of the Politburo, and in response, the Politburo became responsible to the Central Committee, and Central Committee members could participate in Politburo sessions with a consultative voice, but could not vote unless they were members.

Some, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true.

Some commentators have criticized poststructuralism for being radically relativistic or nihilistic ; others have objected to its extremity and linguistic complexity.

Some have objected to this phrasing, which is an elaboration of referring to training as " human capital ", either for the same reason that phrase is objectionable, or on the grounds that it implies that the human in which the knowledge is " invested " is a resource to be exploited.

Some have objected to what they think is a harsher political and social climate, especially towards immigrants and Muslims.

Some sources claim Godard didn't trust Hollywood and refused ; Richard Benton claimed that Godard wanted to shoot the film in New Jersey in January and took offense when would-be producer Norah Wright objected that was unreasonable considering the story took place in Texas with its year round warm environment while her partner, Elinor Jones, claimed they did not believe Godard was right for the project in the first place.

Some archaeological authors have objected that Arthur and his restorers were not discovering the palace and civilization as it was, but were creating a modern artefact based on contemporary art and architecture.

Some politically active libertarians objected to his attitude of non-participation in politics, an attitude he himself changed later.

Some senators objected to furnishing Santa Anna financial assistance.

Some clinicians objected to this, characterizing the new diagnostic system as an " unwieldy conglomeration of disparate models that cannot happily coexist " and may have limited usefulness in clinical practice.

Some priests objected on the grounds that if they were to switch, the religious implements they utilised would require re-consecration, at not insignificant expense.

Some groups and individuals ( including the Greens / Green Party USA ) have objected to " corporate personhood.

Some professionals objected to this move, as they felt that places given to amateurs would come at the expense of the LPGA Tour's less successful professionals, who need to play regularly to make a living.

Some contemporaries, notably William Cobbett, objected to Rush's extreme use of bloodletting.

Some of the American servicemen from the American South in the Services Club objected to Māori soldiers also using the Club, and on 3 April 1943 began stopping Māori soldiers from entering.

Some Holocaust survivors objected to Spiegelman making a comic book out of their tragedy.

Some difficulties arose with the Lords of the Admiralty, who objected to a junior officer debating the navy publicly in the House of Commons.

Some modern Muslims have objected to the term, saying that the term was not used by Muhammad himself or his earlier followers, and that the religion teaches the worship of God alone ( see shirk and tawhid ) and not Muhammad or any other of God's prophets, according to Muslim beliefs.

Some local residents objected to the North Stand because of the shadow cast over the cemetery, which Celtic believed was necessary to install 4, 000 seats in the North Stand and to complete the overall project.

Some Puritans objected to all ornament and sought to abolish choirs, hymns, and, inasmuch as liturgy itself was rejected, devotionals.

Some members of Parliament objected because Townshend's plan was expected to generate only ₤ 40, 000 in yearly revenue, but he explained that once the precedent for taxing the colonists had been firmly established, the program could gradually be expanded until the colonies paid for themselves.

Some of the Aboriginals objected to Brant giving land grants to whites in this reserve area.

Some Romans objected to the war against Parthia.

Some other critics also objected to the film's depictions of violence and sexuality.