By induction we conclude the statement is true for all r ≥ 0.

By the induction hypothesis,

By the end of the year, Alexander was also distinguished with his induction as a Knight of the Order of the Garter.

By Dr. Zaehner, the author of Mysticism, Sacred and Profane, their deliberate induction is regarded as immoral.

By the 1870s, efficient generators that produced alternating current ( alternators ) were available, and it was found that alternating current could power an induction coil directly, without an interrupter.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

By means of mathematical induction, it is easily proven that the above procedure requires the minimal number of moves possible, and that the produced solution is the only one with this minimal number of moves.

By the principle of mathematical induction it follows that the result is true for all natural numbers.

By reasoning using " induction " Bacon meant the ability to gradually generalize a finding based on accumulating data-he advised proceeding by this method ( building a case from the ground up ).

By induction, SO ( n ) therefore has

By Faraday's law of induction we have

By his method of observation and induction as thus explained, his philosophy will be found to be marked off very clearly, on the one hand from the deductive construction of notions of an absolute system, as represented either by Schelling or Hegel, which Cousin regards as based simply on hypothesis and abstraction, illegitimately obtained ; and on the other, from that of Kant, and in a sense, of Sir W. Hamilton, both of which in the view of Cousin are limited to psychology, and merely relative or phenomenal knowledge, and issue in scepticism so far as the great realities of ontology are concerned.

By preventing the clonal expansion of lymphocytes in the induction phase of the immune response, it affects both the cell and the humoral immunity.

By contrast, in an induction motor the rotor rotates at a slower speed than the stator field.

By varying the line frequency with an inverter, induction motors can be kept on the stable part of the torque curve above the peak over a wide range of rotation speeds.

By induction on n it follows that the basis functions are non-negative for all values of and.

By induction, Axioms ( 3 ) and ( 4 ) are equivalent to the single statement

So, i. e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:

By the early 1990s, when his time of eligibility was beginning to run out, many Puerto Ricans, celebrities and ordinary citizens alike, began to campaign for his induction.

By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction:

The same wire ( in practice a coil of wire ) is moved through the same magnetic field at a known speed v. By Faraday's law of induction, a potential difference U is generated across the ends of the wire, which is equal to BLv.

By induction, we assume that for some integer the partially ordered set can be covered by disjoint chains and has at least one antichain of size.

: Proof: By mathematical induction.

By construction, hypothesis testing limits the rate

By the de Broglie hypothesis, we see that

By the above hypothesis, the graph indicates one of the major criticisms of deficit spending as a way to stimulate the economy: rising interest rates lead to crowding out – i. e., discouragement – of private fixed investment, which in turn may hurt long-term growth of the supply side ( potential output ).

By using Song state-gathered statistics, Edward A. Kracke, Sudō Yoshiyuki, and Ho Ping-ti supported the hypothesis that simply because one had a father, grandfather, or great-grandfather who had served as an official of state, it did not guarantee that one would obtain the same level of authority.

By the inductive hypothesis,

By the 18th century the problems with Augustine's idea led Johann Jakob Griesbach to put forward the Griesbach hypothesis, which was that Luke had revised Matthew and that Mark had then written a shorter gospel using material on which both Matthew and Luke agreed ( Matthew → Luke → Mark ).

By this hypothesis, creation of the fictional poetess capitalized on the period's literary fascination with the classical poet Sappho and on a publication ( 1533 ) of poems attributed to Petrarch's " Laura " ( Laura de Sade ; the poems were in fact the work of a descendant of Laura ).

By pointing to differences in organization among other things, Ward Jones Jr. casts doubt on Baswell ’ s earlier hypothesis.

By the second half of the twentieth century, however, observations had failed to confirm this hypothesis, and an alternative hypothesis was proposed that something must instead suppress the diurnal tide.

By the inductive hypothesis R ( r − 1, s ) and R ( r, s − 1 ) exist.

By the inductive hypothesis, it contains either a K < sub > n </ sub >< sub >< sub > i </ sub ></ sub > monochromatically coloured with colour i for some 1 ≤ i ≤ ( c − 2 ) or a K < sub > R ( n </ sub >< sub >< sub > c − 1 </ sub >, n < sub > c </ sub >)</ sub >- coloured in the ' blurred colour '.

By the late 20th century the hypothesis that the Moriori were different from the Māori had fallen out of favour amongst archeologists, who believed that the Moriori were Maori who settled on the Chatham Islands in the 16th century.

Also, since λ is an eigenvalue of A, the dimension Ran ( A − λ I ), r, is strictly less than n. Let A ' denote the restriction of A to Ran ( A − λ I ), By inductive hypothesis, there exists a basis

By the end of the 19th century, there was a hypothesis that the Annals had been fabricated by 15th century Italian author Poggio Bracciolini ( 1380-1459 ).

By taking quantum physics into account, the gravastar hypothesis attempts to resolve contradictions caused by conventional black hole theories.

By that time Hodgkin and Lister had already published research on tissue samples, based on observations made with Lister's innovative microscope, in particular on the " globule hypothesis " of the time which was held in particular by Henri Milne Edwards.

By current standards, the indications seem odd but may be explained by the hypothesis that children may fail to thrive if they have chronically sore throats or severe obstructive sleep apnea ( OSA ).

By concluding that complex life is uncommon, the Rare Earth hypothesis is a possible solution to the Fermi paradox: " If extraterrestrial aliens are common, why aren't they obvious?

By the 1970s, Michael H. Hart's 1979 paper " Atmospheric Evolution, the Drake Equation and DNA: Sparse Life in an Infinite Universe " outlined the first evolutionary model for a habitable zone and hist pessimistic conclusions on the distribution of extraterrestrial life fuelled the Rare Earth hypothesis.

By going through hundreds or thousands of potential similarities between John and Mary, each having a low probability of being true, we may eventually find proof of virtually any hypothesis.

By hypothesis, a welfare recipient is destitute, without funds or assets.