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

Some Anglicans accept that anointing of the sick has a sacramental character and is therefore a channel of God's grace, seeing it as an " outward and visible sign of an inward and spiritual grace " which is the definition of a sacrament.

Some operations may change the state of the ADT ; therefore, the order in which operations are evaluated is important, and the same operation on the same entities may have different effects if executed at different times — just like the instructions of a computer, or the commands and procedures of an imperative language.

Public discourse ranged in tone from organized arguments by tobacconist and medical practitioner John Williams, who posited that " several arguments proving that inoculating the smallpox is not contained in the law of Physick, either natural or divine, and therefore unlawful ," to more slanderous attacks, such as those put forth in a pamphlet by Dr. William Douglass of Boston entitled The Abuses and Scandals of Some Late Pamphlets in Favour of Inoculation of the Small Pox ( 1721 ), on the qualifications of inoculation's proponents.

Some experiences were therefore rejected out of hand, and some unpleasant experiences endured in the present to ensure a better life in the future.

Some critics, such as Jerry Coyne ( professor of evolutionary biology at the University of Chicago ) and Eugenie Scott ( a physical anthropologist and executive director of the National Center for Science Education ) have argued that the concept of irreducible complexity, and more generally, intelligent design is not falsifiable, and therefore, not scientific.

Some Kuwaitis are originally from Persia and other non-Arab countries, however, they assimilated with the Arabs of Kuwait after acquiring Kuwaiti citizenship and are therefore Arabized Arabs.

Some followers of Marx concluded, therefore, that a communist revolution would inevitably occur.

Some North American fans, already attached to particular spellings, took great umbrage at Schodt's renditions, forgetting that in the original Japanese most character and mecha names are written in katakana, and that there were, therefore, no " official spellings.

Some people use the phrase " naturalistic fallacy " or " appeal to nature " to characterize inferences of the form " This behaviour is natural ; therefore, this behaviour is morally acceptable " or " This property is unnatural ; therefore, this property is undesireable.

Some physicists have explored the notion that if the dimensionless physical constants had sufficiently different values, our Universe would be so radically different that intelligent life would probably not have emerged, and that our Universe therefore seems to be fine-tuned for intelligent life.

Some historians, mainly of the late 19th century and before, therefore speculated ( on no evidence ) that he must have traveled overland to the mouth of the Loire or the Garonne.

Some have argued that logic dictates, therefore, that the jar acts as a prison for Elpis as well, withholding it from men.

Some pollutants are biodegradable and therefore will not persist in the environment in the long term.

Some historians, social scientists and anthropologists have, however, argued that their disciplines already describe psychological motivation and that Psychohistory is not, therefore, a separate subject.

Some of the sources say that Descartes denied the animals could feel pain, and therefore could be used without concern.

Some manufacturers therefore built transmitters that supplied + 5 V and-5 V and labeled them as " RS-232 compatible ".

Some philosophers see language as representing directly entities that already exist in the objective world, and that categorization is therefore not generally variable but to some extent pre-given.

Some, therefore, of the miserable remnant, being taken in the mountains, were murdered in great numbers ; others, constrained by famine, came and yielded themselves to be slaves for ever to their foes, running the risk of being instantly slain, which truly was the greatest favour that could be offered them: some others passed beyond the seas with loud lamentations instead of the voice of exhortation ... Others, committing the safeguard of their lives, which were in continual jeopardy, to the mountains, precipices, thickly wooded forests, and to the rocks of the seas ( albeit with trembling hearts ), remained still in their country.

Some in the UFO community, such as nuclear physicist Stanton Friedman, say there is no basis for the search and it is therefore unscientific.

Some scenes of the Mesolithic, however, can be typed and therefore, judging from their various modifications, are fairly clear.

Some, like Louis Jacobs, argue that the main body of the Gemara is not simple reportage of conversations, as it purports to be, but a highly elaborate structure contrived by the Saboraim, who must therefore be regarded as the real authors.

Some scholars therefore believe that it is an interpolation, while others have argued that it is original.

Some critics were dissatisfied by his choice of poem order and omissions in the book and some feminists argued that Hughes had essentially driven her to suicide and therefore should not be responsible for her literary legacy.