Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd.

Then the energy of the vacuum is exactly E < sub > 0 </ sub >.

Then, p < sup > 2 </ sup > is the fraction of the population homozygous for the first allele, 2pq is the fraction of heterozygotes, and q < sup > 2 </ sup > is the fraction homozygous for the alternative allele.

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Then X is reflexive if and only if each X < sub > j </ sub > is reflexive.

Then the cotangent space at x is defined as the dual space of T < sub > x </ sub > M:

Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.

Then the complex derivative of ƒ at a point z < sub > 0 </ sub > is defined by

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.

Then the overall runtime is O ( n < sup > 2 </ sup >).

Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.

Then, dividing the units of energy ( such as eV ) by a fundamental constant that has units of velocity ( M < sup > 0 </ sup > L < sup > 1 </ sup > T < sup >-1 </ sup >), facilitates the required conversion of using energy units to describe momentum.

Then, removing the catalyst would also result in reaction, producing energy ; i. e. the addition and its reverse process, removal, would both produce energy.

Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).

Then is a derivation and is linear, i. e., and, and a Lie algebra homomorphism, i. e.,, but it is not always an algebra homomorphism, i. e. the identity does not hold in general.

Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ).

Then, ~ 150, 000 years later ( i. e. around 50, 000 years ago ), sub-groups of this population began to expand our species ' range to regions outside of, and ( later ) within, this continent ( Tishkoff, 1996 ).

Then a general definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism that has an inverse, i. e. there exists a morphism with and.

Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

Then the data are passed through high-performance restartable dithering engine which is used regardless of monitor bit depth, i. e. also for 24 bits per pixel colour.

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i. e., for all a, b and c in P, we have that:

Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ).

Then it was pointed out that " seeing the most submarines " depended not only on the number of submarines present, but also on the number of eyes looking ; i. e., patrol density.

Suppose a partially ordered set P has the property that every chain ( i. e. totally ordered subset ) has an upper bound in P. Then the set P contains at least one maximal element.

Then any vector in R < sup > 3 </ sup > is a linear combination of e < sub > 1 </ sub >, e < sub > 2 </ sub > and e < sub > 3 </ sub >.

Then these dummies could be selected with mouse, and next character from the password " e " is typed, which replaces the dummies " asdfsd ".

Then that researcher's Bradford multiplier b < sub > m </ sub > is 2 ( i. e. 10 / 5 ).

Then it is used as: General Strike of a city, i. e., " General Strike in Florence ", or a General Strike in a whole country or province, for the purpose of gaining political rights, i. e., the right to vote ; as in Belgium, or Sweden.

Then, the markedness of the ruler comes into play: it is " anchored " at point A, and slided and rotated until one mark is at point C, and one at point D, i. e., CD = AB.

Then each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.

For about three decades immediately before 1902 it was negative, reaching − 6. 64 s. Then it increased to + 63. 83 s at 2000.

Then one need only check the records in each bucket T against those in buckets T where k ranges between − m and m.

For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h

Then, we can show that if the game starts with n spots, it will end in no more than 3n − 1 moves and no fewer than 2n moves.

Then, because P ( A ) and P ( A < nowiki >'</ nowiki >) are the only two possibilities and are also mutually exclusive, P ( A < nowiki >'</ nowiki >) = 1 − P ( A ).

Then we have 3 − (− 2 ) = 3 + 2 = 5.

Then n is palindromic if and only if a < sub > i </ sub > = a < sub > k − i </ sub > for all i. Zero is written 0 in any base and is also palindromic by definition.

Then a will be a root of the formal derivative P ’, with multiplicity m − 1.

Then the powers z, z < sup > 2 </ sup >, ... z < sup > n − 1 </ sup >, z < sup > n </ sup >

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

Then T has a chi-squared distribution with n − 1 degrees of freedom.

Then Q ( x ) has a zero α of multiplicity r, and in the partial fraction decomposition, r of the partial fractions will involve the powers of ( x − α ).

Then he tied 2nd – 4th in Pärnu with 4. 5 / 7 (+ 3 = 3 − 1 ).

Then in 1951, he triumphed again at Moscow, URS-ch19, with 12 / 17 (+ 9 = 6 − 2 ), against a super-class field which included Efim Geller, Petrosian, Smyslov, Botvinnik, Yuri Averbakh, David Bronstein, Mark Taimanov, Lev Aronin, Salo Flohr, Igor Bondarevsky, and Alexander Kotov.

Then the tautological projection R < sup > n + 1, 1 </ sup > −

Then, for all coefficients λ + ( 1 − λ ) =

− 9x < sup > 2 </ sup >) Then, " bring down " the next term from the dividend.

P is a polynomial of degree n − r. Then we have the following:

Then ∂< sub >*</ sub >() is the class of ∂ u in H < sub > n − 1 </ sub >( A ∩ B ).

Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P ( W ) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space ( the subspace

Then one uses the analyticity of x ↦ x < sup > α − 1 </ sup > to conclude that