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 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 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 government scientists say privately that the figure probably is closer to 80 megatons, and that the full 50-megaton bomb that Khrushchev mentioned may still be detonated.

Some early scientists such as Georg Ernst Stahl ( 1659-1734 ) and Francisque Bouillier ( 1813 – 1899 ) had supported a form of animism which life and mind, the directive principle in evolution and growth, holding that all cannot be traced back to chemical and mechanical processes, but that there is a directive force which guides energy without altering its amount.

Some results of Steiner's research have been investigated and supported by scientists working to further and extend scientific observation in directions suggested by an anthroposophical approach.

Some scientists believe it has to do with a kind of neural map that the brain has of the body, which sends information to the rest of the brain about limbs regardless of their existence.

Some scientists argue that megafauna cryptids are unlikely to exist undetected in great enough numbers to maintain a breeding population, and are unlikely to be able to survive in their reported habitats due to issues of climate and food supply.

Some political scientists define a civil war as having more than 1000 casualties, while others further specify that at least 100 must come from each side.

Some contemporary philosophers and scientists ( e. g. David Deutsch in The Fabric of Reality ) employ virtual reality in various thought experiments.

Some scientists have predicted that unless significant measures ( such as seeking out and protecting old growth forests that have not been disturbed ) are taken on a worldwide basis, by 2030 there will only be 10 % remaining, with another 10 % in a degraded condition.

Some scientists even concluded that life may have begun on Earth in hydrothermal vents far under the ocean's surface.

Some scientists and engineers have expressed reservations about nuclear power, including: Barry Commoner, S. David Freeman, John Gofman, Arnold Gundersen, Mark Z. Jacobson, Amory Lovins, Arjun Makhijani, Gregory Minor, Joseph Romm and Benjamin K. Sovacool.

Some scientists consider Homo rudolfensis, a group larger bodied group of fossils with similar morphology to the original H. habilis fossils to be a separate species while others consider them to be part of H. habilis-simply representing species internal variation, or perhaps even sexual dimorphism.

Some scientists have proposed moving this species out of Homo and into Australopithecus due to the morphology of its skeleton being more adapted to living on trees rather than to moving on two legs like Homo sapiens.

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

Some non-software industries are beginning to use techniques similar to those used in free software development for their research and development process ; scientists, for example, are looking towards more open development processes, and hardware such as microchips are beginning to be developed with specifications released under copyleft licenses ( see the OpenCores project, for instance ).

Some scientists hope that the approval process of cisgenic plants might be simpler than that of proper transgenics, but it remains to be seen.

Some scientists speculated they saw a " little man " ( homunculus ) inside each sperm.

Some scientists argue that since the head and appendages are the greatest areas for heat loss from the body, the ability to grow long hair on the crown of the head as well as the face provides a distinct advantage in a cold climate.

Some important work IUPAC has done in these fields includes standardizing nucleotide base sequence code names ; publishing books for environmental scientists, chemists, and physicists ; and leading the way in improving education in science.

Some scientists dispute IQ entirely.

Some scientists say that the 2011 Japanese nuclear accidents have revealed that the nuclear industry lacks sufficient oversight, leading to renewed calls to redefine the mandate of the IAEA so that it can better police nuclear power plants worldwide.

Some scientists, including members of the Space Studies Institute, argue that the vast majority of mankind eventually will live in space and will benefit from doing this.

Some former Soviet scientists said they were actually hampered by Fuchs's data, because Beria insisted that their first bomb (" Joe 1 ") should resemble the American plutonium bomb (" Fat Man ") as much as possible, even though the scientists had discovered a number of improvements and different designs for a more efficient weapon.

Some have not been followed up by scientists yet.

Some lost cities whose locations have been rediscovered have been studied extensively by scientists.