Then they were given 1/2 to 1-1/2 avocados per day as a substitute for part of their dietary fat consumption.

) 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 the pullback of the covector determined by g ( denoted dg ) is given by

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, in the later account found in the Syriac Doctrine of Addai, a painted image of Jesus is mentioned in the story ; and even later, in the account given by Evagrius, the painted image is transformed into an image that miraculously appeared on a towel when Christ pressed the cloth to his wet face.

Then, he wore a Harlequin cap, given to successful cricketers at Oxford.

Then N < sub > x </ sub > is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net.

Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

Then the total number of i-type organisms after the first round of reproduction, given as, is

Then by the law of moment of momentum, the torque on the fluid is given by:

⟨ 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 the solution of the Laplace equation inside the sphere is given by

Then, Pandulf's nephew Pandulf II was given Benevento when Otto II partitioned Landulf IV's territory, with Landulf IV keeping Capua.

Then he ordered 2 golden pieces to be given to every householder in Constantinople and 200 pounds of gold ( including 200 silver pieces annually ) to be given to the Byzantine Church.

Then we imagine the rays are parallel ( and given by two angles ), and we let the colour of a point on the surface be determined by the angle between this direction and the slope of the surface at the point.

Then the formula Cold ( r )→ High ( t ) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book ( see Chapter 7 ) where fuzzy control is represented as a theory of Hájek's basic logic.

Then the n-th series coefficient c < sub > n </ sub > is given by:

Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.

Then given the Hamiltonian operator, the equation to satisfy for all functions ( with associated multiplication operator ) is

* Then, to return a new sample given the most recent sample, we proceed as follows:

Then follow the characters of their family and given names.

Then there exist integers x and y such that

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Then 2Z has two cosets in Z ( namely the even integers and the odd integers ), so the index of 2Z in Z is two.

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field ( an instance of the Hilbert – Speiser theorem ).

Then, for some positive integers m and n, a < sup > n </ sup > and b < sup > m </ sup > are in I.

Then there exist positive integers a and b such that e = a / b where clearly b > 1.

Then, which is the integers, and.

Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.

Then consists of integers between and.

Then, since is assumed rational, there must exist, positive integers such that

Then for any left R-module M, the sets of the form, for all x in M and all positive integers n, form a base for a topology on M that makes M into a topological module over the topological ring R.

Suppose that p is an odd prime number and a is a quadratic residue modulo p that is nonzero mod p. Then Hensel's lemma implies that a has a square root in the ring of p-adic integers Z < sub > p </ sub >.

Then, by choosing appropriate bases for the integers of each of these fields, the above theorem implies the following:

Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

Let a, b, and c be elements of G. Then:

Then b < sub > 0 </ sub > is the value of p ( x < sub > 0 </ sub >).

Then the image set f ( I ) is also an interval, and either it contains f ( b ), or it contains f ( a ); that is,

It is frequently stated in the following equivalent form: Suppose that is continuous and that u is a real number satisfying or Then for some c ∈ b, f ( c ) = u.

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, the chance that the first letter typed is ' b ' is 1 / 50, and the chance that the second letter typed is a is also 1 / 50, and so on.

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

Then there is an exact sequence relating the kernels and cokernels of a, b, and c:

Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to.

Let vectors and let Then the area of the parallelogram generated by a and b is equal to.

Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

Denote the orthocenter of triangle ABC as H, denote the sidelengths as a, b, and c, and denote the circumradius of the triangle as R. Then

Then, if a, a ′, b and b ′ are alternative settings for the detectors,

Suppose a vertex joins three units with spin numbers a, b, and c. Then, these requirements are stated as:

Then A is an algebra over K if the following identities hold for any three elements x, y, and z of A, and all elements (" scalars ") a and b of K:

g < sub > 1 </ sub >( x ), a ≤ x ≤ b. Then

g < sub > 2 </ sub >( x ), a ≤ x ≤ b. Then

Then the element c = ba is nilpotent ( if non-zero ) as c < sup > 2 </ sup > = ( ba )< sup > 2 </ sup > = b ( ab ) a = 0.