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 summing from k

Then we could have written a formula of degree k which is equivalent to φ, namely.

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

Then letting y < sub > k </ sub > =

Then between 80k-125 k years ago, modern humans began appearing in the middle east.

⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.

Then f: A < sub > 1 </ sub > × A < sub > 2 </ sub > → X is a morphism and f ∘ i < sub > k </ sub >

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.

The other class of Dedekind rings which is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k of regular functions on C is a Dedekind domain.

Then any model of B is a field of characteristic greater than k, and ¬ φ together with B is not satisfiable.

Then we could define, which grows much faster than any for finite k ( here ω is the first infinite ordinal number, representing the limit of all finite numbers k ).

We can use this fact to prove part of a famous result: for any prime p such that p ≡ 1 ( mod 4 ) the number (− 1 ) is a square ( quadratic residue ) mod p. For suppose p = 4k + 1 for some integer k. Then we can take m = 2k above, and we conclude that

Then p < sub > k </ sub > converges monotonically to π ; with p < sub > k </ sub >-π ≈ 10 < sup >− 2 < sup > k + 1 </ sup ></ sup > for k ≥ 2. s

Then a < sub > k </ sub > converges quartically against 1 / π ; that is, each iteration approximately quadruples the number of correct digits.

Then p < sub > k </ sub > converges quartically to π ; that is, each iteration approximately quadruples the number of correct digits.

Then a < sub > k </ sub > converges quintically to 1 / π ( that is, each iteration approximately quintuples the number of correct digits ), and the following condition holds:

Then every cohomology class in H < sup > 2k </ sup >( X, Z ) ∩ H < sup > k, k </ sup >( X ) is the cohomology class of an algebraic cycle with integral coefficients on X.

Then converges to with probability 1 as.

Let P and P < sub > 1 </ sub >, P < sub > 2 </ sub >, ... be probability measures on some set S. Then P < sub > n </ sub > converges weakly to P if and only if

Then sequence in converges to some point if it converges pointwise for each.

The two definitions can be shown to be equivalent as follows: define the closure of a set S in X to be the set of all points x such that some net that converges to x is eventually in S. Then that closure operator can be shown to satisfy the axioms of a preclosure operator.

Then, the numerical solution converges to the exact solution as if and only if the method is zero-stable.

Then converges in law to a Ornstein – Uhlenbeck process as tends to infinity.

Then, until the likelihood converges to a stable value ( this is usually accomplished by setting a small threshold value and running the algorithm until it increases by less than that threshold value ), do the following at each node:

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Schweitzer notes that St. Paul apparently believed in the immediacy of the " Second Coming of Jesus ": " Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord " ( 1 Thessalonians 4. 17 ).

Then Chris Broad scored three hundreds in successive Tests and bowling successes from Graham Dilley and Gladstone Small meant England won the series 2 – 1.

Then we are told that P1 is not sped up, so S1 = 1, while P2 is sped up 5 ×, P3 is sped up 20 ×, and P4 is sped up 1. 6 ×.

Then Satan will be put into the " bottomless pit " or abyss for 1, 000 years, known as the Millennial Age.

Then Crapsey decided to make the criterion a stanza of five lines of accentual-syllabic verse, in which the lines comprise, in order, 1, 2, 3, 4, and 1 stresses and 2, 4, 6, 8, and 2 syllables.

Then, considering all three colour channels, and assuming that the colour channels are expressed in a γ = 1 colour space ( that is to say, the measured values are proportional to light intensity ), we have:

: Then the estimated value of the weight θ < sub > 1 </ sub > is

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, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Then, on 1 July 1948, Tirana called on all Yugoslav technical advisors to leave the country and unilaterally declared all treaties and agreements between the two countries null and void.

Then since the first bit of a non-zero binary significand is always 1 it need not be stored, giving an extra bit of precision.

Then divide 18 by 12 to get a quotient of 1 and a remainder of 6.

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 find Tabor ( Hosea 5: 1 ), Shechem ( Hosea 6: 9 the Revised Version ( British and American )), Gilgal and Bethel ( Hosea 4: 15, 9: 15, 10: 5, 10: 8, 10: 15, 12: 11 ).

Then, on paper, they bought $ 200 worth of each, for a total bet of $ 1, 000, using the prices on September 29, 1980, as an index.