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Some and mathematicians
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 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.
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 and define
Some historians define and minimize the Arian conflict as the exclusive construct of Arius and a handful of rogue bishops engaging in heresy ; but others recognize Arius as a defender of ' original ' Christianity, or as providing a conservative response against the politicization of Christianity seeking union with the Roman Empire.
Some states also define assault as an attempt to menace ( or actual menacing ) by placing another person in fear of imminent serious bodily injury.
Some authors also define an alkali as a base that dissolves in water.
Some philosophers deny that the concept of " being " has any meaning at all, since we only define an object's existence by its relation to other objects, and actions it undertakes.
Some toques were created so capoeiristas could communicate with each other within the roda without having to say a word, like Cavalaria, while others were created to define a style, like Regional de Bimba.
Some resource materials define classic cinquains as solely iambic, but that is not necessarily so.
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 health insurance policies also define a medical condition as any illness, injury, or disease except for psychiatric illnesses.
Some sources define egalitarianism as the point of view that equality reflects the natural state of humanity.
Some economy of data is possible where a frame need only rewrite a portion of the pixels of the display, because the Image Descriptor can define a smaller rectangle to be rescanned instead of the whole image.
Remark: Some authors, especially in physics and matrix algebra, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first.
Some closely knit language families, and many branches within larger families, take the form of dialect continua, in which there are no clear-cut borders that make it possible to unequivocally identify, define, or count individual languages within the family.
Some commentators define Modernism as a socially progressive trend of thought that affirms the power of human beings to create, improve and reshape their environment with the aid of practical experimentation, scientific knowledge, or technology.
Some textbooks define an organic compound as one containing one or more C-H bonds ; others include C-C bonds in the definition.
Some national constitutions in effect define their governments ' " core businesses " as being the provision of such things as justice, tranquility, defense, and general welfare.
Some mainstream scholars of Islam define sufism as simply the name for the inner or esoteric dimension of Islam.
" Some structural engineers define a highrise as any vertical construction for which wind is a more significant load factor than earthquake or weight.
Some authorities attempt to define an astronomical winter, often offset, which is based solely on the position of the Earth in its orbit around the sun.
Some people define narcotics as substances that bind at opioid receptors ( cellular membrane proteins activated by substances like heroin or morphine ) while others refer to any illicit substance as a narcotic.
Some simpler Metafont fonts, such as the calligraphic mathematics fonts in the Computer Modern family, use a single pen stroke with a relatively large pen to define each visual " stroke " of the glyphs.
Some textbooks may define the negative binomial distribution slightly differently than it is done here.
Some authors define a modified Arrhenius equation, that makes explicit the temperature dependence of the pre-exponential factor.
Some would prefer to define actions as requiring bodily movement ( see behaviorism ).
Some zoologists define " domestication " as human control over breeding, which can be detected in ancient skeletal samples by changes in the size and variability of ancient horse populations.

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