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, 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, again, the conditional wave function of subsystem ( I ) is ( up to an irrelevant scalar factor ) equal to and if the Hamiltonian does not contain an interaction term between subsystems ( I ) and ( II ), satisfies a Schrödinger equation.

Then the Hamiltonian for the combined system is

Then when the high energy degrees of freedoms are integrated out, the effective Hamiltonian in the low energy subspace reads

Then, the first order correction to the Hamiltonian is

Then having found the weight of the shortest Hamilton path, determine which edges are in the path by asking for each edge whether there is a Hamiltonian path with weight for the graph modified so that edge has weight ( if there is no such path in the modified graph, then edge must be in the shortest path for the original graph ).

Then the total Hamiltonian of the ensemble is

Then, the Z < sub > 2 </ sub > parity of the number of χ particles is well defined and is conserved in time ( even though the number of χ particles itself depends on the choice of which splitting into a free Hamiltonian and an interacting Hamiltonian we make in the interaction picture, which doesn't even exist for interacting theories ( the number is typically infinite )).

Then, to find the expectation value of the Hamiltonian:

Then the Hamiltonian vector field with Hamiltonian H takes the form

Then a curve γ ( t )=( q ( t ), p ( t )) is an integral curve of the Hamiltonian vector field X < sub > H </ sub > if and only if it is a solution of the Hamilton's equations:

Then one says that the symmetry of the Hamiltonian is realized by the system.

Then the unperturbed Hamiltonian of the atom can be written as

Then under the dipole approximation the interaction Hamiltonian between the atom and the electric field can be expressed as

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

Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then – in the only paid employment he ever had – he took a job as a telegraph operator with the Great Northern Telegraph Company working first in Denmark and then in Newcastle-upon-Tyne, and was soon made a chief operator.

Then the operator

Then the corresponding density operator is:

Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M ′′ of M. This algebra is the von Neumann algebra generated by M.

Conversely, let A be a self-adjoint operator on a Hilbert space H. Then

Then, the Dunkl operator is defined by:

Then it is also a bounded operator from L < sup > r </ sup > to L < sup > r </ sup > for any r between p and q.

Then it is also a bounded operator from to with

Then he got a job with a city insurance company as a computer operator where his free time allowed him to develop his literary skills – publishing an underground paper called the Scandal Sheet.

Then the spectrum for the Laplace-Beltrami operator on is discrete and real, since the Laplace operator is self adjoint with compact resolvent ; that is

Then the multiplier operator associated to this symbol m is defined via the formula

Then T is also a bounded operator from to for any r between p and q.

Then, the operator makes the transition between the scenes using the cross-fader.

Then in the late 1920s, Arthur Chaffin, later owner / operator of the Hite Ferry, and two companions were searching for an alternate route between Green River and Caineville.