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Page "Congruence relation" ¶ 24
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By and first
By 1853, the new partnership announced the precision vernier caliper as the first fruit of their joint efforts.
By this scheme, pulling one signal to clear locks all the other switch and signal levers in safe positions until the first signal is again restored to normal.
By comparing reaction cells sealed from the same manifold temperature dependency corresponding to activation energies ranging from 11 to 18 Af was observed while dependence on the first power of the light intensity seemed to be indicated in most cases.
By the time the child first attacks the actual problem of reading, he is completely familiar and at ease with all of the elements of words.
By this procedure rinsing progresses in two stages, first by dilution until the time when the drains are separated and thereafter by displacement of the soil-bearing liquor by clean rinse water, since soiled liquor squeezed from the specimens at the nip passes directly to waste from the suds box drains.
By convention in some law reports, the appellant is named first.
By 1865, following the American Civil War, stocks in small industrial companies, such as iron and steel, textiles and chemicals were first sold by curbstone brokers.
By the first wife he had Mustafa III and by the second wife he had Abdul Hamid I.
By the 1850s when they first came into sustained contact by outside groups, there were estimated 7, 000 Adamanese, divided into the following major groups:
By his first wife, whose name is not known, he had three children:
By his first wife, Eschiva of Ibelin, he was the father of Hugh I of Cyprus and was crowned in Nicosia on September 22, 1197.
By 473 BC, after the death of Phrynichus, one of his chief rivals, Aeschylus was the yearly favorite in the Dionysia, winning first prize in nearly every competition.
By the 1950s, the development of civil jets grew, beginning with the de Havilland Comet, though the first widely-used passenger jet was the Boeing 707, because it was much more economical than other planes at the time.
By the 1st century BC, Aediles were elected in July, and took office on the first day in January.
By the 19th century orchestral music in Europe had standardized the string section into the following homogeneous instrumental groups: first violins, second violins, violas, cellos, and double basses.
By the end of 1898, Western pilgrims started coming to Akka on pilgrimage to visit ` Abdu ' l-Bahá ; this group of pilgrims, including Phoebe Hearst, was the first time that Bahá ' ís raised up in the West had met ` Abdu ' l-Bahá.
By the first century BC brass was available in sufficient supply to use as coinage in Phrygia and Bithynia, and after the Augustan currency reform of 23 BC it was also used to make Roman dupondii and sestertii.
By " computerizing " this method of communications, the name of the first BBS system was born: CBBS-Computerized Bulletin Board System.
By October 2007 bluetongue had become a serious threat in Scandinavia and Switzerland and the first outbreak in Denmark was reported.
By 1977, in its first edition, Disk / Trend, a leading hard disk drive industry marketing consultancy segmented the industry according to MBs ( decimal sense ) of capacity.
By 1985, construction was completed and the World Financial Center saw its first tenants.
By an accident, a painter misinterpreted the name and painted " Ski-Doo " on the first prototype.
The 1523 " Turin map " of the islands was the first to refer to them as Los Lagartos, meaning alligators or large lizards, By 1530 they were known as the Caymanes after the Carib word caimán for the marine crocodile, either the American or the Cuban crocodile, Crocodylus acutus or C. rhombifer, which also lived there.
By 1972, the economic progress of Allende's first year had been reversed, and the economy was in crisis.
By the middle of the decade, some of the 1950s generation of American comedians, such as Jerry Lewis, went into decline, while Peter Sellers found success with international audiences in his first American film The Pink Panther.

By and isomorphism
By setting K = ker ( f ) we immediately get the first isomorphism theorem.
By associating the element a in G with the inner automorphism ƒ ( x ) = x < sup > a </ sup > in Inn ( G ) as above, one obtains an isomorphism between the quotient group G / Z ( G ) ( where Z ( G ) is the center of G ) and the inner automorphism group:
By the first isomorphism theorem we then have that
By contrast, we have the natural isomorphism < ref group =

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 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 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:

0.204 seconds.