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 conditional variance of X given Y is the Schur complement of C in V:

Then the conditional probability given is a function such that is the conditional expectation of the indicator function for A:

Then the Feynman – Kac formula tells us that the solution can be written as a conditional expectation

Then we rely on the fact that the conditional probability

Then a Yule – Simon distributed variable K has the following geometric distribution conditional on W:

Then the expectation of this random variable X is defined as

Then Professor Marvin Zelen, a statistician and associate of the recently founded Committee for the Scientific Investigation of Claims of the Paranormal ( CSICOP, now known as the Committee for Skeptical Inquiry ( CSI )), proposed in a 1976 article in the same periodical that, in order to eliminate any demographic anomaly, Gauquelin randomly pick 100 athletes from his data-set of 2, 088 and check the birth / planet correlations of a sample of babies born at the same times and places in order to establish a control group, giving the base-rate ( chance ) expectation for comparison ( The 100 random athletes later expanded into a subsample of 303 athletes ).

Then we apply the law of total expectation to each term by conditioning on the random variable X:

Then the expectation in the first-stage problem's objective function can be written as the summation:

Then, if we let and be the expectation of, for any

Then, to find the expectation value of the Hamiltonian:

Then is the non-vanishing vacuum expectation value of the Higgs field.

) 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 X is separable if and only if X ′ is separable.

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

Then all elements of X equivalent to each other are also elements of the same equivalence class.

Then the quotient space X /~ can be naturally identified with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.

Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.

Then the joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

Then for a specific value x of X, the function L ( θ | x )

Then the observation that X

Then the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.

Then X has cardinality at most and cardinality at most if it is first countable.

Then A is dense in C ( X, R ) if and only if it separates points.

Then ρ will be the finest completely regular topology on X which is coarser than τ.

Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.

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

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.