Then the periodic Bernoulli functions P < sub > n </ sub > are defined as

Then all the terms in the asymptotic series can be expressed in terms of elementary functions.

Then there are computable functions and.

Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question.

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

Then the functions may be added pointwise to produce a group homomorphism.

Then, in 1978 an effort was made to consolidate the several singular functions ; FEMA was created to house civil defense and disaster preparedness under one roof.

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.

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Then one faces the problem of expression sin ( θ ) and cos ( θ ) as functions of x.

Then the subspace of consisting of functions φ such that all functional derivatives of at φ are zero, that is:

Then, for all measurable real-or complex-valued functions f and g on S,

Then it fulfilled a number of functions: for example, Gödöllő's first pharmacy was opened in it in 1814.

Then the quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms.

Then for all Riemann mapping functions F ( z )

Let be the algebra of functions defined on the manifold M. Then

Then one checks that these two functions are in fact both ring homomorphisms.

Then, the design of the compact kitchen unit “ UNIREX ” was designed in which one single large appliance had more functions, from the conservation to the cooking phase.

Then the assemblies are dipped into a bath of molten salt ( typically NaCl, KCl and other compounds ) which functions both as heat transfer medium and flux.

Then each critical function and or / process must have its own contingency plan in the event that one of the functions / processes ceases or fails.

Then the theorem states that for analytic functions f, if

Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists.

Now let F ′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated to E with fibre F ′ is a bundle E ′ with a local trivialization subordinate to the cover U < sub > i </ sub > whose transition functions are given by

Then π < sub > ρ </ sub >: E → X is a fiber bundle with fiber F and structure group G. The transition functions are given by ρ ( t < sub > ij </ sub >) where t < sub > ij </ sub > are the transition functions of the principal bundle P.

Then, all but blind, he said there was nothing in Back to Methuselah --, -- `` G.B.S. ought to have known that '', -- and `` I look at my bookshelves despairingly, knowing that I can have nothing more to do with them ''.

Then you can do the finishing touches at your leisure.

Then during washing, the greasy soil rolls back at the edges so that emulsified droplets can disengage themselves from the sorbed oil mass, with the aid of mechanical action, and enter the aqueous phase.

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 the enthusiasm and energy of all elements can be channeled to produce cumulative progress toward a common objective.

Then, after I'm back, another fifty so you can put some mileage on yourself and have a solid alibi somewhere while I take care of your seat cover boy ''.

Then, with the hymn writer of old, you can say:

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

Then we can express the variables as the sum of the ( time averaged ) mean field () that varies in space and a small fluctuating field () that varies in space and time.

Then the momentum equation can be expressed as

Then the mass balance equation can be written as

Then the mass balance equation can be expressed as

Then one can rewrite the field in the form

Then nodules of blue earth have to be removed and an opaque crust must be cleaned off, which can be done in revolving barrels containing sand and water.

Then, in 1828, Friedrich Wöhler published a paper on the synthesis of urea, proving that organic compounds can be created artificially.

Then the bra can be computed by normal matrix multiplication.

Then the reservists have to serve up to three weeks a year and can be called up to serve two weeks during a non-military crisis.

Then we can solve for f ′.

Then it can be shown that one can write a thermodynamic version of the above calorimetric rules:

Then, according to Adkins ( 1975 ), it can be shown that one can write a further thermodynamic version of the above calorimetric rules:

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 each point p of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains p.

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 >.