We will denote the values of f{t} on different components by Af.

We will use the notation to denote the multiplicative inverse of, it is defined exactly when and are coprime ; the following construction explains why the coprimality condition is needed.

We call the set C the pair of A and B, and denote it

We usually denote this set using set-builder notation as

We write hom ( a, b ) ( or hom < sub > C </ sub >( a, b ) when there may be confusion about to which category hom ( a, b ) refers ) to denote the hom-class of all morphisms from a to b. ( Some authors write Mor ( a, b ) or simply C ( a, b ) instead.

We can see this from the Bayesian update rule: letting U denote the unlikely outcome of the random process and M the proposition that the process has occurred many times before, we have

We use the shorthand notation to denote the joint probability of by.

We denote the subgroup of principal fractional ideals by Prin ( R ).

We are now left with the word tawdry to denote, essentially, the inferior garb worn by the poor and not a general put-down for St Ives itself, which remains a charming and attractive small river town.

We know from Plutarch that Xenocrates, if he did not explain the Platonic construction of the world-soul as Crantor after him did, nevertheless drew heavily on the Timaeus ; and further that he was at the head of those who, regarding the universe as unoriginated and imperishable, looked upon the chronological succession in the Platonic theory as a form in which to denote the relations of conceptual succession.

Suppose X is a normed vector space over R or C. We denote by its continuous dual, i. e. the space of all continuous linear maps from X to the base field.

We denote the canonical probability distributions for the Hamiltonian and the trial Hamiltonian by and, respectively.

We denote the eigenstates of by.

We denote the diagonal components of the density matrices for the canonical distributions for and in this basis as:

We denote the unitary operators we get by U ( a, A ), and these give us a continuous, unitary and true representation in that the collection of U ( a, A ) obey the group law of the inhomogeneous SL ( 2, C ).

We denote this map by Ad < sub > g </ sub >:

We will denote this space by C. We define

We can rephrase the above proof, using partitions, which we denote as:

We denote the spectrum of a commutative ring A by Spec ( A ).

" is reporting the sentence " We shall all sin from time to time " ( assuming the archbishop is including himself in the proposition ), where shall is used to denote simple futurity.

We end up with a sequence of numbers which denote the height of each throw to be made.

Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. We denote the dual cell ( to be defined precisely ) corresponding to S by DS.

We spoke of the need for advertising, and I agreed that the deep dive would be most useful for publicity.

We would like to do that too ''.

We would establish no censorship.

We would be ignoring the special circumstances of other countries.

We would all meet at ten o'clock at the Kehl bridge, five miles from Strasbourg, and march triumphantly across into Germany.

`` We would see it that way, but it was glorious then.

We can't think of anyone else who would want to separate serious candidates from other candidates, either.

We have no right to criticize them, as they realize they would be sitting ducks in a nuclear war.

We will know, and He will know, to whom it is rendered, what the birds would ask:

We wish the President would remember that `` fiscal responsibility '' was the battle-cry of the party that lost the election.

We would lose our export markets and deny ourselves the imports we need.

We would be crippled by reduced output, industrial decline, widespread unemployment ''.

We would write to one another and make a definite plan.

We would attend a film and, later on, I stated, we might go to the Mayflower Coffee Shop or Child's or Toffenetti's for waffles.

We would have the means to seek out and destroy the enemy's force -- whether it were fixed or mobile.

We quickly ran into the same trouble that plagued Bill Ruger in his first experiments: Three or four bullets would be placed well in a six-inch bull at 100 yards and then, unaccountably, one could stray far out of the group.

We would have preferred, however, to have had the rest of the orchestra refrain from laughing at this and other spots on the recording, since it mars an otherwise sober, if not lofty, performance.

We found no obvious effects due to preferred orientation of the crystallites in this sample nor would we expect to on the basis of the shape found from electron microscopic examination.

We expected, in short, that most of the world would make itself over in our image and that it would be relatively simple, from such a position, to deal with the localized aberrations of the Soviet Union ''.

In many cases that statement -- `` We break even on our downtown operation and make money on our branches '' -- would be turned around if the cost analysis were recalculated on terms less prejudicial to the old store.

We always thought we would die with our boots on.

We may now take up for consideration a hard case which seems to require either no action employing economic pressure or else action that would seem to violate the principles set forth above.

We discussed the candle and decided the hypothetical other bum would have left it burning to light his way to the window and because he'd have no reason to blow it out.

President Kennedy's invitation to the Spanish-born master said, `` We feel your performance as one of the world's greatest artists would lend distinction to the entertainment of our guests ''.