Then Φ ( x ) = φ.

Then the pullback Φ < sup >*</ sup > F of F by Φ is a multilinear form on V defined by precomposing F with Φ.

Let H and G be Hilbert spaces of dimension n and m respectively, and Φ be a quantum operation taking the density matrices acting on H to those acting on G. Then there are matrices

Then our choice function can choose the least element of every set under our unusual ordering.

Then an arithmetic function a is

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 is a group whose identity element is The group inverse of an arbitrary group element is the function inverse

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 a mathematical procedure is used to find a filter transfer function which can be realized ( within some constraints ) and which approximates the desired response to within some criterion.

Then an " estimator " is a function that maps the sample space to a set of sample estimates.

Let R be a domain and f a Euclidean function on R. Then:

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 the probability density function f *( x ) of the size biased population is

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

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

Then,, and the constant function turns into the identity by substitution.

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 ).

The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.

Then the output is related to the input by the transfer function as

Then because the construction of f was uniform, this is a recursive function of two variables.

Then, its electrical field traces out a helical pattern as a function of time.

Then the function

Then the function

Then, recalling that the likelihood function is defined up to a multiplicative constant, it is just as valid to say that the likelihood function is approximately

Then, on considering the lengths of the intervals to decrease to zero, the likelihood function for an observation from the discrete component is

Then has probability density function.

Then he would get to his feet, as though rising in honor of his own remarkable powers, and say almost invariably, `` Gentlemen, this is an amazing story!!

Then, Jesus indicated that God's forgiveness is unlimited.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.

Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.

Then he thought of Aaron Blaustein standing in his rich house saying: `` God is tired of taking the blame.

Then it is marked on the inside where it comes in contact with the transom, frames, keelson and all the battens.

Then it is replaced and fastened.

Then the chines are rounded off and the bottom is rough-sanded in preparation.

Then, a group of eggs is deposited in a cavity in the beebread loaf and the egg compartment is closed.

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

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.

Then, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.

Then the editorial added prophetically: `` how far they may reach in Asia is yet undetermined, but they fall far short of our dreams of the war conferences ''.

Then she catapults into `` everything and everybody '', putting particular violence on `` everybody '', indicating to the linguist that this is a spot to flag -- that is, it is not congruent to the patient's general style of speech up to this point.

Then comes the time when the last wire is removed and Susie walks out a healthier and more attractive girl than when she first went to the orthodontist.

Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.

Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;