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Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.
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Many theorems from classical geometry hold true for this spherical geometry as well, but many do not ( see parallel postulate ).
Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.
Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics.
Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
Many theorems, which are modeled after Ramsey's theorem itself, assert that in every partition of a large structured object, one of the classes necessarily contains a large structured subobject, but give no information about which class this is.
Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture.
Many theorems and concepts involving graphs also hold for hypergraphs.
Many such theorems will not do so explicitly, as usually stated in standard mathematical language.
Many of these theorems only apply to corrections calculable in perturbation theory and so instantons, which are not seen in perturbation theory, provide the only corrections to these quantities.
Many forms of these theorems are actually known to be equivalent, so that the assertion that " PIT " holds is usually taken as the assertion that the corresponding statement for Boolean algebras ( BPI ) is valid.
Many other Picard-type theorems can be derived from the Second Fundamental Theorem.
Many important min-max theorems in combinatorics can be expressed in these terms.
Many deep theorems on the structure of finite groups use characters of modular representations.
These properties are essentially weaker versions of strong and supercompact cardinals, consistent with V = L. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts.
Many fixed-point theorems yield algorithms for locating the least fixed point.
Many and which
Many years later I went to see S.K. in England, where he was living at Whiteleaf, near Aylesbury, and he showed me beside his cottage there the remains of the road on which Boadicea is supposed to have travelled.
Many of the individual projects for which development assistance is required call for expenditures over lengthy periods.
Many of the cells and microorganisms which are transparent to visible light, absorb or reflect the much shorter wavelengths of the ultraviolet spectrum.
Many companies have systems, particularly in R & D, which work more or less well, depending upon size and actual belief in the policy on the part of administration, as will be abundantly apparent in subsequent quotations.
Many years later ( on August 3, 1915 ), Lucy Upton wrote Winslow's daughter soon to be graduated from Smith College: `` While I love botany which, after dabbling in for years, I studied according to the methods of that day exactly forty years ago in a summer school, it must be fascinating to take up zoology in the way you are doing.
Many scholars consider modern anthropology as an outgrowth of the Age of Enlightenment, a period when Europeans attempted to study human behavior systematically, the known varieties of which had been increasing since the fifteenth century as a result of the first European colonization wave.
Many amateurs like to specialise in observing particular objects, types of objects, or types of events which interest them.
Many species have an erect woody stem which is covered with persistent dried leaves unless there have been fires, topped by a crown of long thin leaves.
Many of Christie ’ s books and short stories both set in the Middle East and back in England have a decidedly otherworldly influence in which religious sects, sacrifices, ceremony, and seances play a part.
Many parents today purchase “ Baby Signs ” DVDs and books which incorporate ASL signs for infant and toddler communication.
Many of these were part of the corruption indulged in by Scott and the Pennsylvania's president, J. Edgar Thomson, which consisted of inside trading in companies that the railroad did business with, or payoffs made by contracting parties " as part of a quid pro quo ", as biographer David Nasaw writes.
Many arguments from morality are based on morality normativity, which suggest that objective moral truths exist and require God's existence to give them authority.
Many states have Arbor Day although only Victoria has Arbor Week, which was suggested by Premier Dick Hamer in the 1980s.
Many ironclads were built to make use of the ram or the torpedo, which a number of naval designers considered the crucial weapons of naval combat.
Many stories have developed since the founding long ago to become a part of America's folklore and cultural awareness, and non-native American folklore especially includes any narrative which has contributed to the shaping of American values and belief systems.
Many GA-style guitars also have a convex back panel to increase the volume of space in the soundbox without making the soundbox deeper at the edges, which would affect comfort and playability.
Many pop songs are used as anthems, such as Queen's " We Are the Champions ", which is commonly used as a sports anthem.
Many defoliated forest areas were quickly invaded by aggressive pioneer species, such as bamboo and cogon grass, which make it unlikely the forests will be able to regenerate.
Many of the 5200's games appeared simply as updated versions of 2600 titles, which failed to excite consumers.
Many file archivers employ archive formats that provide lossless data compression to reduce the size of the archive which is often useful for transferring a large number of individual files over a high latency network like the Internet.
Many of the SIGs, like SIGGRAPH, SIGPLAN, SIGCSE and SIGCOMM, sponsor regular conferences which have become famous as the dominant venue for presenting innovations in certain fields.
Many Aleuts now live across the state and not only in the Aleutians which has helped to spread the arts of their people and to better develop methods of creating their arts.
Many amplifiers are ultimately slew rate limited ( typically by the impedance of a drive current having to overcome capacitive effects at some point in the circuit ), which sometimes limits the full power bandwidth to frequencies well below the amplifier's small-signal frequency response.
Many were especially critical of him following his remarks in the South Carolina primary, which Obama won.
Many and are
Many of them, moreover, are beginning to complain about the scarcity of Western amusements and to ridicule the old life of the bazaar merchant, the mullah, and the peasant.
Many selections are themselves convincing contributions to this appraisal.
Many of our very best friends are reformers.
Many home-bound subway riders utilizing the Flushing-Main Street express are daily confronted with the sight of the local departing from the Woodside station as their express comes to a stop, leaving them stranded and strained.
Many belly dancers are married, but Serene is one of the few who will admit it.
Many of these experiences are so important that they will be cherished forever by me.
Many hours are given free by the Jaycees to make this and all local pageants outstanding events.
Many adults showing at Westminster today are products of this Class.
Many cells, bacteria, and other microorganisms are transparent to visible light and must be stained for microscopic investigation.
Many industry trade associations are developing campaigns to protect or enhance the share of the consumer's dollar being spent on their particular products.
Many food and beverage companies are already on a highly planned basis.
Many other ( probably nearly all ) snakes at maturity are already more than half their final length.
Many of the new appointees are art collectors.
Many of the features of the homes are the latest modern devices in American homes, but an interesting blend of cultures finds us using Japanese artfulness in our own Western architecture at the same time that the Japanese are adopting Western utility patterns.
Many legislators are already weary and frustrated over the so-far losing battle to block token integration.
Many of the toll-road bonds still are selling at prices that offer the prospect of an annual yield of 4 per cent, or very close to that.
Many others are attracting the traffic needed to push revenues up to the break-even point.
Many of the coal seams in the nationalized British mines are twisting, narrow and very deep.
Many English Catholics are proud of their Catholicism and know that they are in a new ascendancy.
Many of you are familiar, I'm sure, with the story of my early struggles: the fire in January, 1947, that destroyed everything -- even our precious list of subscribers.
Many caecilians and some other amphibians lay their eggs on land, and the newly hatched larvae wriggle or are transported to water bodies.
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