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 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 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 water clock designs were developed independently and some knowledge was transferred through the spread of trade.

Some video formats, such as DV, compress each frame independently using intraframe compression.

Some scholars question whether Frege's negative review of the Philosophy of Arithmetic helped turn Husserl towards Platonism, but he had already discovered the work of Bernhard Bolzano independently around 1890 / 91 and explicitly mentioned Bernard Bolzano, Gottfried Leibniz and Hermann Lotze as inspirations for his newer position.

Some churches founded independently of these lineages also employ this form of church governance.

Some of these languages, e. g., in worlds of fantasy fiction, alternative universes, Earth's future, or alternate history, are presented as distorted versions or dialects of modern English or other natural language, while others are independently designed conlangs.

Some degree of safety was provided by the safety lamp which was invented in 1816 by Sir Humphry Davy and independently by George Stephenson.

Some of the cassettes for the Sing Along machines were recorded by Filipino cover bands for popular songs so that the vocals and accompaniment could be played independently.

Some regions independently developed a character similar to Nasreddin, and the stories have become part of a larger whole.

Some contemporary linguistic philosophers construe " Platonism " to mean the proposition that universals exist independently of particulars ( a universal is anything that can be predicated of a particular ).

Some contemporary linguistic philosophers construe " Platonism " to mean the proposition that universals exist independently of particulars ( a universal is anything that can be predicated of a particular ).

Some followers of Zwingli believed that the Reformation was too conservative, and moved independently toward more radical positions, some of which survive among modern day Anabaptists.

Some early dominant teams did not join a league since they could pull in larger profits independently.

* Some models of sub-controllers ( usually lower cost ) have no memory and processing power to make access decisions independently.

Some of this software ( 3D Topicscape ) works independently from virtual worlds and simply uses 3D to support user " in concept creation, planning, organization, development and actualization ".

Some scammers are part of less organized gangs or are operating independently ; such scammers have reduced access to the above connections, and thus have little success with wealthier investors or business entities attempting to research them, but are still convincing to middle-class individuals and small businesses, and can bilk hundreds of thousands of dollars from such victims.

Some display systems support multiple background layers that can be scrolled independently in horizontal and vertical directions and composited on one another, simulating a multiplane camera.

Some platforms ( e. g., Super NES, Mega Drive / Genesis, Game Boy Advance, Game Boy, Nintendo DS ) provide a horizontal blank interrupt for automatically setting the registers independently of the rest of the program.

Some independently owned BP stations, including former Amoco stations, switched to a different brand due to the public relations fallout as a result of the oil spill.

Some spurgos are similar to Polish pączki, but some specific recipes, such as cottage cheese doughnuts ( varškės spurgos ), were invented independently.

Some British paramedics have gone on to become Paramedic Practitioners, a role that practices independently in the pre-hospital environment in a capacity similar to that of a nurse practitioner, but with more of an acute care orientation.

Some paleontologists believe that it developed independently in monotremes ( or australosphenidans ), rather than being inherited from an ancestor that they share with marsupials and placentals ( or boreosphenidans ); but this idea has critics and the debate is still going on.

Some modern scholars believe that the Gospel of Thomas was written independently of the canonical gospels, and therefore is a useful guide to historical Jesus research.

Some SIGs are associated with various geographic groups, whereas others act independently of official hierarchy.