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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.
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Some and mathematicians
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 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 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 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 and especially
Some alphabets disregard tone entirely, especially when it does not carry a heavy functional load, as in Somali and many other languages of Africa and the Americas.
Some are willing to travel to the other designated areas, especially during slow periods of the day.
Some birds, especially corvids and parrots, are among the most intelligent animal species ; a number of bird species have been observed manufacturing and using tools, and many social species exhibit cultural transmission of knowledge across generations.
Some carbohydrates ( especially after condensation to oligo-and polysaccharides ) contain less carbon relative to H and O, which still are present in 2: 1 ( H: O ) ratio.
Some biological agents ( especially smallpox, plague, and tularemia ) have the capability of person-to-person transmission via aerosolized respiratory droplets.
Some kinds of raw beans, especially red and kidney beans, contain a harmful toxin ( lectin phytohaemagglutinin ) that must be removed by cooking.
Some constitutions, especially written constitutions, also act as limiters of state power, by establishing lines which a state's rulers cannot cross, such as fundamental rights.
Some are styled " national " banks, such as the National Bank of Ukraine ; but the term " national bank " is more often used by privately owned commercial banks, especially in the United States.
Some calves are ear tagged soon after birth, especially those that are stud cattle in order to correctly identify their dams ( mothers ), or in areas ( such as the EU ) where tagging is a legal requirement for cattle.
Some canid species have also been trapped and hunted for their fur and, especially the gray wolf and the red fox, for sport.
Some of the most elaborate columns in the ancient world were those of the Persians, especially the massive stone columns erected in Persepolis.
Some displays, especially newer LCD monitors, replace the traditional anti-glare matte finish with a glossy one.
Some players perform with the sides of one, two, or three fingers, especially for walking basslines and slow tempo ballads, because this is purported to create a stronger and more solid tone.
Some of the main topics of the field are moral responsibility, moral development, moral character ( especially as related to virtue ethics ), altruism, psychological egoism, moral luck, and moral disagreement.
Some roads, especially in the Sinai and southeastern part of the country, are off-limits to foreigners.
Some aficionados who prefer class-A amplifiers also prefer the use of thermionic valve ( or " tube ") designs instead of transistors, especially in Single-ended triode output configurations for several claimed reasons:
Some arthropods however, especially large insects with tracheal respiration, expand their new exoskeleton by swallowing or otherwise taking in air.
Some Directors, especially more established ones, take on many of the roles of a Producer, and the distinction between the two roles is sometimes blurred.
Some authors use, or Cpq, instead of, and, or Epq, instead of, especially in contexts where is used for other purposes.
Some people use oral sex as a means of preserving virginity, especially in the case of adolescent girls who fellate their boyfriends to create and maintain intimacy while preserving their virginity, avoiding pregnancy, or both.
Some people interpret the Bible, especially the New Testament, as teaching belief in, and practice of, faith healing.
Some electrics, especially those with odd-shaped bodies, have one or both strap buttons on the back of the body.
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light.