Then there exists a finite set of integer coefficients satisfying the equation:

Then the interpolating polynomial is formed as above using the topmost entries in each column as coefficients.

Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

Then every cohomology class in H < sup > 2k </ sup >( X, Z ) ∩ H < sup > k, k </ sup >( X ) is the cohomology class of an algebraic cycle with integral coefficients on X.

Then every non-torsion cohomology class in H < sup > 2k </ sup >( X, Z ) ∩ H < sup > k, k </ sup >( X ) is the cohomology class of an algebraic cycle with integral coefficients on X.

Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.

Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaves on X.

Then, for all coefficients λ + ( 1 − λ ) =

Then, since there are no powers of x on the left, all of the coefficients on the right must be zero, from which it follows that

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

More precisely, let F be a field, and let F be the ring of polynomials in one variable, X, with coefficients in F. Then each f < sub > i </ sub > lies in F. ∂< sub > X </ sub > is the derivative with respect to X.

Then, instead of driving toward objectives or impeding anomalies, she or he works on the metasystem, rules, potential coefficients, categorizations, information architectures, etc.

Then the GKW operator acts on the Taylor coefficients as

Then its coefficients generate a proper ideal I, which by Krull's theorem ( which depends on the axiom of choice ) is contained in a maximal ideal m of R. Then R / m is a field, and ( R / m ) is therefore an integral domain.

Then comparing the coefficients:

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, with staring eyes and lips drawn thin, Miriam said to the young woman, `` You are ugly -- uglier than you used to be, and you were always very ugly.

Then people wonder why Russian pupils are more advanced than American students.

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

Then, of course, there are those of us who either do not want or need or cannot afford another car.

Then, too, European drivers have reputations for being somewhat crazy on the road and some Americans are not particularly keen on getting mixed up with them.

Then there was the caterer's ad which read: `` are you getting married or having an affair??

Then there are the trustees and officers of the great educational foundations, who inevitably exert an influence on educational decisions by their support or refusal to support various educational programs, experiments, and demonstrations.

Then, too, misplaced or jammed-together teeth are prone to trapping food particles, increasing the likelihood of rapid decay.

Then there are a pair of old biddies played by Grace Carney and Sibly Bowan who may be right off the shelf of stock Irish characters, but they put such a combination of good will and malevolence into their parts that they're quite entertaining.

Then, after many more billions of years, when all the galaxies are whistling toward a common center, this movement will slow down and reverse itself again.

Then we have informed a large number of our crew that when they reach the United States, they will be punished but that in the meanwhile, they may run loose and are expected to perform their jobs in good order.

Then Frankie said: `` What are you gonna do ''??

In a few of her novels, including Death Comes as the End and And Then There Were None, there are multiple victims.

Seven stories are inspired by a nursery rhyme: And Then There Were None by Ten Little Indians ; One, Two, Buckle My Shoe by One, Two, Buckle My Shoe ; Five Little Pigs by This Little Piggy ; Crooked House by There Was a Crooked Man ; A Pocket Full of Rye by Sing a Song of Sixpence ; Hickory Dickory Dock by Hickory Dickory Dock, and Three Blind Mice by Three Blind Mice.

) There are also numerous instances where the killer is not brought to justice in the legal sense but instead dies ( death usually being presented as a more ' sympathetic ' outcome ), for example Death Comes as the End, And Then There Were None, Death on the Nile, Dumb Witness, The Murder of Roger Ackroyd, Crooked House, Appointment with Death, The Hollow, Nemesis, Cat Among the Pigeons, and The Secret Adversary.

Schweitzer notes that St. Paul apparently believed in the immediacy of the " Second Coming of Jesus ": " Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord " ( 1 Thessalonians 4. 17 ).

Then looking at him darkly Zeus who gathers the clouds spoke to him :' Do not sit beside me and whine, you double-faced liar. To me you are the most hateful of all gods who hold Olympos. Forever quarrelling is dear to your heart, wars and battles .… And yet I will not long endure to see you in pain, sinceyou are my child, and it was to me that your mother bore you. But were you born of some other god and proved so ruinouslong since you would have been dropped beneath the gods of the bright sky.

Then we are told that P1 is not sped up, so S1 = 1, while P2 is sped up 5 ×, P3 is sped up 20 ×, and P4 is sped up 1. 6 ×.

Then he came, in his words, to " the theory of symmetry: all governments and regimes to a first approximation are bad, all peoples are oppressed, and all are threatened by common dangers.

Then the results obtained are compared with the expected results.

Then the beads are pressed in between the warp threads.

Then the runes are shaved off and the shavings are mixed with mead and sent abroad so that Æsir have some, Elves have some, Vanir have some, and Men have some, these being beech runes and birth runes, ale runes, and magic runes.