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

Some, she knew, looked upon Thompson almost as a saint, but others read in `` The Hound Of Heaven '' what they took to be the confessions of a great sinner, who, like Oscar Wilde, had -- as one pious writer later put it -- thrown himself `` on the swelling wave of every passion ''.

Some of the marble busts in the park are of young Englishmen who fought and died for Garibaldi.

Some years ago this Class was judged by celebrities who knew nothing of what was required of a Junior's ability to show a dog.

Some of those who question the value of BW have assumed that the only potential would be in the establishment of epidemics.

Some of those who did not initially react with an arm-elevation also associated their behavior in the situation with control factors -- an inability to relinquish control voluntarily.

Some who have written on Utopia have treated it as `` a learned diversion of a learned world '', `` a phantasy with which More amused himself '', `` a holiday work, a spontaneous overflow of intellectual high spirits, a revel of debate, paradox, comedy and invention ''.

Some people have luck and some have no luck and that, whatever people who prefer order say, is the size of it.

Some of them came from people who identified themselves.

Some of the people who speak through my pages are famous ; ;

Some of the New York Philharmonic musicians who live in the suburbs spent yesterday morning digging themselves free from snow.

Some of the General Semantics tradition was continued by Samuel I. Hayakawa, who had a dispute with Korzybski.

Some sources state that following King Edward the Confessor's death in 1066, it was Ealdred who crowned Harold Godwinson as King of England.

" Some scholars theorize that the " we " passages in Acts are just such " handed down " quotations from some earlier source who accompanied Paul on his travels.

Supporters of this view believe that “ to a hypothetical outside reader, presents Christianity as enlightened, harmless, even beneficent .” Some believe that through this work, Luke intended to show the Roman Empire that the root of Christianity is within Judaism so that the Christians “ may receive the same freedom to practice their faith that the Roman Empire afforded the Jews .” Those who support the view of Luke ’ s work as political apology generally draw evidence from the facts that Christians are found innocent of committing any political crime ( Acts 25: 25 ; 19: 37 ; 19: 40 ) and that Roman officials ’ views towards Christians are generally positive.

Some believe that this appeal “ thereby shows Christian ’ s of Luke ’ s day both that their predecessors were innocent before the state and that Paul had no political quarrel with Rome ” but rather with the Jews who were accusing him.

Some believe the revision in question was the work of a single reviser, who in his changes and additions expressed the local interpretation put upon Acts in his own time.

Some Protestant US military chaplains carry the Roman Rite version of the Anointing of the Sick with them for use if called upon to assist wounded or dying soldiers who are Catholics.

Some studies showed that veterans who served in the South during the war have increased rates of cancer, and nerve, digestive, skin and respiratory disorders.

Some NES titles were developed by companies who had licensed their title from a different arcade manufacturer.

Some of which are based on Theosophical interpretations and were notably represented by Mohandas Karamchand Gandhi, who made clear throughout his life and his own commentary on the Gita that it was " an allegory in which the battlefield is the soul and Arjuna, man's higher impulses struggling against evil.

It was Laura Clifford Barney who, by asking questions of ` Abdu ' l-Bahá over many years and many visits to Haifa, compiled what later became the book Some Answered Questions.

Some of Whorf's unpublished manuscripts on spirituality also suggest that he was influenced by the ideas of Helena Blavatsky the founder of the Theosophical Society who wrote about cosmic evolution, a belief that regards reincarnation to be the source of evolution of human races towards continuously higher forms.

Some philosophers who have had more noteworthy theories are Parmenides, Leucippus, Plato, Aristotle, Augustine, Plotinus, Aquinas, Descartes, Spinoza, Leibniz, Hegel, Heidegger, and Sartre.

Some general purpose Bulletin Board Systems had special levels of access that were given to those who paid extra money, uploaded useful files or knew the SysOp personally.

Some of Bede's material came from oral traditions, including a description of the physical appearance of Paulinus of York, who had died nearly 90 years before Bede's Historia Ecclesiastica was written.