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 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, the hypothesis implies that M < sup > k </ sup > T

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 1962 Glock had persuaded the management of the BBC to increase the orchestra's budget to allow for joint principals in the string sections, to attract top musicians who could play in the BBC SO without having to give up their solo or chamber careers.

By 2018, the Clear Skies Act will supposedly allow 3 million tons more NOx through 2012 and 8 million more by 2020, for SO < sub > 2 </ sub >, 18 million tons more through 2012 and 34 million tons more through 2020.

By an auspicious choice of the parameter n < sub > 0 </ sub >, we will arrive at a contradiction.

By construction, h extends to the whole upper hemisphere of S < sup > n </ sup >, and as such is null-homotopic.

By definition, a and b can be written as multiples of c: a = mc and b = nc, where m and n are natural numbers.

By using a set of n flexible rods, one can arrange that every rod touches every other rod.

By 1960, having been influenced more by jazz music than blues, Brown began incorporating jazz styled arrangements in his music, with Brown naming the Famous Flames hits " I'll Go Crazy " and " Think " as examples of his changing style away from more traditional forms of R & B and rock ' n ' roll.

By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.

By the age of 17, he had resided at 22 different locations and attended 12 different schools .< ref group =" n ">

On May 3, 2011, it was announced that Patti Smith is one of the winners of the Polar Music Prize: " By devoting her life to art in all its forms, Patti Smith has demonstrated how much rock ’ n ' roll there is in poetry and how much poetry there is in rock ’ n ' roll.

By 1973, I knew that what was needed was some pure, stripped down, no bullshit rock ' n ' roll.

By construction, d * e = 1 mod φ ( n ).

By the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n tends to infinity.

By the central limit theorem, this distribution approaches the normal distribution as n increases.

By extension, the cross-covariance matrix between two random vectors X and Y ( X having n elements and Y having p elements ) is the n × p matrix

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.

On the other hand, if T is written as a unary number ( a string of n ones, where n = T ), then it only takes time n. By writing T in unary rather than binary, we have reduced the obvious sequential algorithm from exponential time to linear time.

By definition, ( ℓmn ) denotes a plane that intercepts the three points a < sub > 1 </ sub >/ ℓ, a < sub > 2 </ sub >/ m, and a < sub > 3 </ sub >/ n, or some multiple thereof.

By replacing n with m / M, and subsequently introducing density ρ = m / V, we get:

By defining this relation as valid for all integer n and nonzero b, it follows that