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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.
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Then and presheaf
Then a C-valued presheaf on X is the same as a contravariant functor from O ( X ) to C. This definition can be generalized to the case when the source category is not of the form O ( X ) for any X ; see presheaf ( category theory ).
Then K < sub > X </ sub > is the sheaf associated to the presheaf K < sub > X </ sub >< sup > pre </ sup >.
Then and on
Then he hunkered down on the heels of his handmade boots, peered into the orderly chaos of axle, shock absorber, and spring.
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 calmly and carefully slugged the remaining five shots into the venomous head -- caught in the wicker back of the chair, the eyes dead on him as the life finally went out of the brute.
Then when Miss Langford was on the end of the line of girls, Jack, in the middle of the line, gave an extra hard pull and the young teacher sprawled backwards, sitting down hard, her dress flying over her head.
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 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 followed a period in which he wrote reviews for The New York Times Book Review, The Commonweal, Commentary, had a small piece in Partisan Review, and moved on to Hudson, The Village Voice, and Exodus.
Then, to conclude on an indeterminate note: `` Nevertheless, if fallout increased substantially, or remained high for a long time, it would become far more important as a potential health hazard in this country and throughout the world ''.
Then came their bathroom, and then a bedroom that, judging by the photographs on the walls, must belong to Mme Cestre.
Then Rector, attired in his best blue serge suit, sat in a chair out on the lawn, in the shade of a tree, smoking a cigarette and waiting.
Then and X
) 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 compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).
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 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.
* 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 the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.
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.
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