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 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 churches founded outside the Anglican Communion in the late 20th and early 21st centuries, largely in opposition to the ordination of openly homosexual bishops and other clergy are usually referred to as belonging to the Anglican realignment movement, or else as " orthodox " Anglicans.

Some of the late 90s / early 2000s games were under development by other companies at one time, but rights to the game programs and all of the existing code was bought and finished by other developers.

Some time late in 2000, Haznawi traveled to the United Arab Emirates, where he purchased traveler's cheques presumed to have been paid for by Mustafa al-Hawsawi.

Some 640 structures burned before a late afternoon sea breeze stopped its progress, allowing firefighters to put it out.

Some scholars argue that conspiracy theories once limited to fringe audiences have become commonplace in mass media, contributing to conspiracism emerging as a cultural phenomenon in the United States of the late 20th and early 21st centuries, and the possible replacement of democracy by conspiracy as the dominant paradigm of political action in the public mind.

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.

" Some believe that the newly globalised economy of the late 20th and early 21st century has facilitated this process through the use of new information technology.

Some engagements in late 20th and early 21st century guerrilla wars involved mounted troops, particularly against partisan or guerrilla fighters in areas with poor transport infrastructure.

Some believe that the large amount of economic liberalization starting from the late 1970s to present, indicates that the CPC has transitioned to endorse economic neoliberalism.

Some locations in northern Britain, late 9th and early 10th centuries.

Some supported extending the NEP, as Lenin had suggested late in his life, or ending it and replacing it with a planned economy, a position Lenin held when he initiated NEP.

Some of these men were executed as early as 83 or 85 however, lending little credit to Tacitus ' notion of a " reign of terror " late in Domitian's reign.

Some conventional destroyers were completed in the late 1940s and 1950s which built on wartime experience.

Some researchers have taken up speculative views, such as Jerzy Strzelczyk, who assumed that in the light of contemporary concepts and habits of marriage of that time ( when as a rule marriages were contracted with teenage girls ) is assumed that Dobrawa had passed her early youth, so, it's probable that she was in her late teens or twenties.

Some evolutionary biologists, such as Richard Lewontin and the late Stephen Jay Gould have employed dialectical materialism in their approach, playing a precautionary heuristic role in their work.

* Some see parallels between James and 1 Peter, 1 Clement, and the Shepherd of Hermas and take this to reflect the socio-economic situation Christians were dealing with in the late 1st or early 2nd century.

Some of the most expensive fighters such as the F-14 Tomcat, F-22 Raptor and F-15 Eagle were employed as all-weather interceptors as well as air superiority fighter aircraft, while commonly developing air-to-ground roles late in their careers.

Some scholars from the Jesus Seminar argue that the birth narratives of Luke and Matthew are a late development in gospel writing about Jesus.

Some disco songs incorporated sounds produced with synthesizers and drum machines, and some compositions were entirely electronic ; examples include Giorgio Moroder's late 1970s productions such as Donna Summer's hit single " I Feel Love " from 1977, Cerrone's Supernature ( 1977 ), Yellow Magic Orchestra's synth-disco-pop productions from their self-titled album ( 1978 ), Solid State Survivor ( 1979 ), and several early 1980s disco-pop productions by the Hi-NRG group Lime.

Some insulator manufacturers stopped making glass insulators in the late 1960s, switching to ceramic materials.

Some historians argue that it had ceased to exist by 1791 ; others that it was still operating as late as 1813.

Some Vietnamese and Chinese minorities remain, particularly in the towns, but many left in two waves-after independence in the late 1940s and again after 1975.

Some of the late Lombard structures of the 9th and 10th century have been found to contain elements of style associated with Romanesque architecture and have been so dubbed " first Romanesque ".

Some books ( guidebooks in particular ) suggest that mazes on cathedral floors originated in the medieval period as alternatives to pilgrimage to the Holy Land, but the earliest attested use of the phrase " chemin de Jerusalem " ( path to Jerusalem ) dates to the late 18th century when it was used to describe mazes at Reims and Saint-Omer.

Some see it ( by definition ) as a late modern ( in the sense of late modernity ) conceptualization.