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, 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 of them are obvious, such as the fact that we associate recorded and live music with our responses and behavior in different types of environments and social settings.

Some recreation features, such as scenic values and water interest, also have greater overall value than other interests.

Some areas may provide archeological values such as ancient Indian village sites or hunting areas, caves, artifacts, etcetera.

Some vocational training schools provide such training, but the current need exceeds the facilities.

Some such youngsters rarely smile, or they try to speak with the mouth closed.

Some anthropologists, such as Lloyd Fallers and Clifford Geertz, focused on processes of modernization by which newly independent states could develop.

Some, such as the spadefoot toads, have strong biting jaws and are carnivorous or even cannibalistic.

Some of these have specific adaptations such as enlarged teeth for biting or spines on the chest, arms or thumbs.

Some salamanders adopt defensive poses when faced by a potential predator such as the North American northern short-tailed shrew ( Blarina brevicauda ).

Some jurisdictions have specialized appellate courts, such as the Texas Court of Criminal Appeals, which only hears appeals raised in criminal cases, and the United States Court of Appeals for the Federal Circuit, which has general jurisdiction but derives most of its caseload from patent cases, on the other hand, and appeals from the Court of Federal Claims on the other.

Some adaptations of the Latin alphabet are augmented with ligatures, such as æ in Old English and Icelandic and Ȣ in Algonquian ; by borrowings from other alphabets, such as the thorn þ in Old English and Icelandic, which came from the Futhark runes ; and by modifying existing letters, such as the eth ð of Old English and Icelandic, which is a modified d. Other alphabets only use a subset of the Latin alphabet, such as Hawaiian, and Italian, which uses the letters j, k, x, y and w only in foreign words.

Some alphabets today, such as the Hanuno ' o script, are learned one letter at a time, in no particular order, and are not used for collation where a definite order is required.

Some of the oldest and most widespread stories in the world are stories of adventure such as Homer's The Odyssey.

Some are no more than a widening of a stream valley ; others, such as the Konya Ovasi, are large basins of inland drainage or are the result of limestone erosion.

Some of these churches are known as Anglican, such as the Anglican Church of Canada, due to their historical link to England ( Ecclesia Anglicana means " English Church ").

Some meals are communal, such as fondue, where a pot is set in the middle of the table for each person to dip into.

Some high mountain villages, such as Avoriaz ( in France ) Wengen and Zermatt ( in Switzerland ) are accessible only by cable car or cog-rail trains, and are car free.

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 sects, such as the Universalists, believe in universalism that all souls will ultimately be saved and that there are no torments of hell.

Some, such as Francis Crick in 1994, have attempted a " scientific search for the soul ".

Some of these beliefs, such as Pelagianism and semi-Pelagianism ( see below ) are not considered to be within Arminian orthodoxy and are dealt with elsewhere.

Some Arminians, such as professor and theologian Robert Picirilli, reject the doctrine of open theism as a " deformed Arminianism ".