Then the Cauchy product is defined by a discrete convolution as follows.

Then, for the class of scores which are discrete convolutions, configurations related to each other by translation of one protein by an exact lattice vector can all be scored almost simultaneously by applying the convolution theorem.

Then we can solve for f ′.

Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy – Riemann equations ( 1a ) and ( 1b ) at that point.

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

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

Then the image set f ( I ) is also an interval, and either it contains f ( b ), or it contains f ( a ); that is,

It is frequently stated in the following equivalent form: Suppose that is continuous and that u is a real number satisfying or Then for some c ∈ b, f ( c ) = u.

Then the pair (( A < sub > i </ sub >)< sub > i ∈ I </ sub >, ( f < sub > ij </ sub >)< sub > i ≤ j ∈ I </ sub >) is called an inverse system of groups and morphisms over I, and the morphisms f < sub > ij </ sub > are called the transition morphisms of the system.

* 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.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

Then ( X × Y, ( π < sub > 1 </ sub >, π < sub > 2 </ sub >)) is a terminal morphism from Δ to the object ( X, Y ) of D × D. This is just a restatement of the above since the pair ( f, g ) represents an ( arbitrary ) morphism from Δ ( Z ) to ( X, Y ).

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

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate f < sub > s </ sub > ( in samples per unit time ) is:

Then we can translate this system into fuzzy program in such a way that f is the interpretation of a vague predicate Good ( x, y ) in the least fuzzy Herbrand model of this program.

Then f is called an immersion if its derivative is everywhere injective.

Then Little Billy began shouting orders to round up the ponies and fill the water buckets and for the cooks to hurry up with the meal.

Then she turned the station wagon around and headed it back down the hill, with the village as her ostensible destination.

Then, with a shrug of pretended indifference, she took a compact from her purse and went through the motions of fixing her make-up.

Then, as he doubled, gasping, vomiting the breakfast he had so lately eaten, Lord straightened him with an uppercut.

Then, with a glory that almost wiped out the deep, downward sags in her careworn face, Matilda leaned over the wheel and shouted to Hez, who was stumbling along in the heat and the dust on the opposite side of the wagon `` Pa!!

Then, and only then, with the Jacksons and Dan as their true guests of honor, did the Harrows take time to catch up on the news.

Then he took off his wet boots and dropped down into the water to talk with the beasts, needing their comfort more than they needed his.

Then the Bonaventure seemed to disintegrate with a roar of live steam, geysers of sparks and flames, and a dense cloud of black-and-orange smoke.

Then she saw Ramey and her face was misshapen with bewilderment.

Then they were tumbling again, and the big man reached into the same pocket he had gone for earlier, and came up with a vicious switchblade.

Then suddenly we found ourselves in the middle of another fight, an irrational, an indecent, an undeclared and immoral war with our strongest ( and some had thought noblest ) ally.

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, after overtures to accept a settlement and go through with a divorce, Miriam gave a ghastly echo of Mrs. Micawber by suddenly stating, `` I will never leave Mr. Wright ''.

Then Miriam varied the senseless psychological warfare by suddenly withdrawing a suit for separate maintenance that had been pending, and asking for divorce on the grounds of cruelty, with the understanding that Wright would not contest it.

Then, in some way, this lack of faith in the cavalry became mixed up in his mind with the dragging effect of wagon trains and was hardened into a prejudice.

My own stern hand has rent the ancient bond, And thereof shall the ending not have end: But not for me, that loved her, to be fond Lightly to please me with a newer friend Then hold it more than bravest-feathered song, That I affirm to thee, with heart of pride, I knew not what did to a friend belong Till I stood up, true friend, by thy true side ; ;

Then I spoke at the ninetieth birthday party of W. E. Burghardt Du Bois, who embarked on a fictional trilogy at eighty-nine and who, with The Crisis, had created a Negro intelligentsia that had never existed in America before him.

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 a full-time planning office will be established in Rome to work with a five-member Georgia Tech research staff for development of an area planning and industrial development program.

Then the Communese reply came back from many mouthpieces with striking consistency.

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 Angelina turned and with an easy grace walked toward the kitchen.

Then she fell asleep again as soddenly as a person with fever, and when she awoke it was dark outside and the clarity was back in her eyes.

Then, abandoning the studies in the face of their promising outlook for all concerned, B. & O. entered on-again-off-again negotiations with C. & O. which resulted in the present situation.