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By the uniqueness theorem for Fourier expansions, we must then equate the Fourier coefficients term by term, giving
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By and uniqueness
By contrast, popular science emphasizes uniqueness and generality, taking a tone of factual authority absent from the scientific literature.
By reversing this construction, John Milnor and Sergei Novikov ( among many others ) were able to answer questions about the existence and uniqueness of high-dimensional manifolds: this is now known as surgery theory.
By and theorem
has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.
By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.
By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
By setting K = ker ( f ) we immediately get the first isomorphism theorem.
* By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.
By " satisfactory " one would mean at least the equivalent of Plancherel theorem.
By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.
By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.
By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius.
By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.
By Rice's theorem, the 1-halting problem is undecidable.
By the corollary to the recursion theorem, there is an index such that returns.
By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.
By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.
By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.
By the central limit theorem, this distribution approaches the normal distribution as n increases.
By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.
By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once.
By the rank-nullity theorem, a system of n vectors in k dimensions ( where all dimensions are necessary ) satisfies a ( p = n − k )- dimensional space of relations.
By Tychonoff's theorem we have that is compact since is, so the closure of in is a compactification of.
By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).
By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.
By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.
By the divergence theorem Gauss's law can alternatively be written in the differential form:
By and for
By then Hez could see for himself, and so could the others.
By prevailing over other claimants for the loyalties of men, the nation-state maintained an adequate measure of certainty and order within its territorial borders.
By subduing disparate lesser groups the nation has, to some degree at least, broadened the capacity for individual liberty.
By her eighteenth birthday her bent for writing was so evident that Papa and Mamma gave her a Life Of Dickens as a spur to her aspiration.
By this time, as we shall see, the Tories were already planning to `` punish '' Steele for his political writing by expelling him from the House of Commons.
By a combination of music, lighting effects, and narration, famous events that have transpired in these locations are evoked and re-created for large audiences usually to considerable acclaim.
By comparison, Stone Harbor bird sanctuary's allies seem less formidable, for aside from the Audubon Society, they are mostly the snowy, common and cattle egrets and the Louisiana, green, little blue and black-crowned herons who nest and feed there.
Suggest the following twenty-first-century amendment: By moving the term `` Republic '' to lower case, substituting the modern phrase, `` move ahead '' for the stodgy `` keep '', and by using the Postmaster's name on every envelope ( in caps, of course, with the `` in spite '' as faded as possible ), the slogan cannot fail.
By law this is 75% for the Virgin Islands, Guam and Puerto Rico.
By law this is 70% for the Virgin Islands, Guam and Puerto Rico.
By using the appropriate version any one of these factors can be determined for any combination of the other three.
By automobile from New York, for example, you can take a one or two-day tour to Annapolis, Maryland to see the colonial homes and the U.S. Naval Academy ( where you can shoot the dress parade on Wednesdays ) ; ;
By 1937 he had clarified his intentions to serve his people: `` I have striven for clarity and melodious idiom, but at the same time I have by no means attempted to restrict myself to the accepted methods of harmony and melody.
By the use of various weights, data for a force-rate of shear graph can be obtained.
Serum potassium at this time was 3.8 mEq. per liter, and the hemoglobin was 13.9 gm. By Dec. 1, 1958, the weakness in the pelvic and quadriceps muscle groups was appreciably worse, and it became difficult for the patient to rise unaided from a sitting or reclining position.
By holding out prospects for external capital assistance, the United States can provide strong incentives to prepare for the concerted economic drive necessary to achieve self-sustaining growth.
By permitting freshman students we might extend the opportunity for such a course to some individuals who otherwise might never get to take it.
By early summer, he wrote from Laramie that he was suffering from the wound inflicted in the ambush and was in a bad way financially, so Pels sent him a draft for $100, warning that it was still not wise for him to return.
) By its greater corporeal presence and its greater extraneousness, the affixed paper or cloth serves for a seeming moment to push everything else into a more vivid idea of depth than the simulated printing or simulated textures had ever done.
By adding a systematic analysis with symbols to the typed transcripts of interviews, they have supplied a new set of techniques for the therapist.
By airline from Concord to Burlington is a distance of about 150 miles, counting a slight deviation for the stop at either Barre or Montpelier.
By tradition, a red wine should be served at approximately room temperature -- if anything a little cooler -- and be aged enough for the tannin and acids to have worked out and the sediment have settled well.
I grinned at him, handed him a couple of dollars and said, `` By the time you get the parking charge figured up, there should be a cigar in it for you ''.
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