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 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 jurisdictions hold this as an absolute right, and in its absence, a sentence may potentially be overturned, with the result that a new sentencing hearing must be held.

Some belief systems, such as those in the Abrahamic tradition, hold that the dead go to a specific plane of existence after death, as determined by a god, gods, or other divine judgment, based on their actions or beliefs during life.

Some Rabbis hold that it must have been carried off to Babylon, while others hold that it must have been hidden lest it be carried off into Babylon and never brought back.

Some traditions hold that Aristobulus of Britannia, one of the Seventy Disciples, was the brother of Barnabas.

Some groups of individuals who hold basic Protestant tenets identify themselves simply as " Christians " or " born-again Christians ".

Some virtue ethicists hold that consequentialist theories totally disregard the development and importance of moral character.

Some writers, such as James-Charles Noonan, hold that, in the case of cardinals, the form used for signatures should be used also when referring to them, even in English ; and this is the usual but not the only way of referring to cardinals in Latin .< ref > An Internet search will uncover some hundreds of examples of " Cardinalis Ioannes < surname >", examples modern and centuries-old ( such as this from 1620 ), and the phrase " dominus cardinalis Petrus Caputius " is found in a document of 1250.

Some dancers prefer to hold the arms much lower, similar to conventional ballroom positioning.

Some Christian denominations hold that salvation depends upon transformational faith in Jesus, which expresses itself in good works as a testament ( or witness ) to ones faith for others to see ( primarily Eastern Orthodox Christianity and Roman Catholicism ), while others ( including most Protestants ) hold that faith alone is necessary for salvation.

Some deists rejected the claim of Jesus ' divinity but continued to hold him in high regard as a moral teacher ( see, for example, Thomas Jefferson's famous Jefferson Bible and Matthew Tindal's Christianity as Old as the Creation ).

Some hold that knowledge of any kind can only be gained a priori, while others claim that some knowledge can also be gained a posteriori.

Some forms of Buddhism hold belief in cycles in which life span of human beings changes according to human nature.

Some Shaivites hold the view that he is incessantly destroying and creating the world.

Some scientists hold that H. floresiensis was a modern H. sapiens with pathological dwarfism.

Some foundationalists, such as St. Augustine of Hippo and Alvin Plantinga, hold that all of our beliefs rest ultimately on beliefs accepted by faith.

Some hold that singing in the Spirit is identified with singing in tongues in, which they hold to be " spiritual or spirited singing ", as opposed to " communicative or impactive singing " which Paul refers to as " singing with the understanding ".

Some of these mass relations hold approximately, but most don't ( see Georgi-Jarlskog mass relation ).

Some scholars hold that the Soviet Union was a hybrid entity containing elements common to both multinational empires and nation states.

Some Asian countries hold bilateral agreements to irradiate exotic fruits for quarantine purposes and export it to the USA.

Some who hold this theory regard the Limbo of Infants as a state of maximum natural happiness, others as one of " mildest punishment " consisting at least of privation of the beatific vision and of any hope of obtaining it.

Some sets include racks to hold tiles ( if they are shaped small or differently ).

* Some scholars hold that there has been extensive editorial reshaping of the stories and statements within the Mishnah ( and later, in the Talmud.