If Af is the operator induced on Af by T, then evidently Af, because by definition Af is 0 on the subspace Af.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.

If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.

If there is a bounded linear operator from X onto Y, then Y is reflexive.

If A is a self-adjoint operator, then is always a real number ( not complex ).

If that steady-state signal was interrupted, it indicated one of two things: either the operator was about to start transmitting, or something else had happened to break the connection — such as a physical break in the telegraph line.

If all layers of an image change regularly but a large number of layer still need to be composited ( such as in distributed rendering ), the commutativity of a compositing operator can still be exploited to speed up computation through parallelism even when there is no gain from pre-computation.

If the operator already has this in place, which is often the case today, the network can be upgraded to EDGE by activating an optional software feature.

If G is a group and X is a set, then a ( left ) group action of G on X is a binary operator:

If negation is cyclic and "∨" is a " max operator ", then the law can be expressed in the object language by ( P ∨ ~ P ∨ ~~ P ∨ ... ∨ ~...~ P ), where "~...~" represents n − 1 negation signs and "∨ ... ∨" n − 1 disjunction signs.

If another operator is found before two operands are found, then the old operator is placed aside until this new operator is resolved.

If there are multiple operations, the operator is given immediately after its second operand ; so the expression written " 3 − 4 + 5 " in conventional infix notation would be written " 3 4 − 5 +" in RPN: first subtract 4 from 3, then add 5 to that.

If it is another extension, the operator places the front cord in the associated jack and pulls the front key backwards to ring the called party.

If the customer knew the number, and the point was direct-dialable, the operator would dial the call.

If the distant city did not have dialable numbers, the operator would dial the code for the inward operator serving the called party, and ask her to ring the number.

If the operator could not get through by dialing the number, she could call the inward operator in the destination city, and ask her to try the number, or to test a line to see if it was busy or out of order.

If an array is used to represent a cycle, it is convenient to obtain the index with a modulo operator, which can result in zero.

If a feature was intended to be located from the center of the part, the operator needs to locate it from the center of the model, not, perhaps, from a more convenient edge or an arbitrary point, as he could when using " dumb " solids.

If she did, he could stand it better in the light.

If she sensed any unusual preoccupation on the part of her mother, she did not comment upon it.

If I hadn't got Nate stopped when I did, my duds'd all be shot plumb to hell!!

If the other pilots were worried, they did not show it.

If you tell him I made a pass at you he might think you misunderstood something I said or did, so instead of just telling him I made a pass, say I tried to date you and that you agreed so you could prove to him what a louse I really am.

If it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.

If life and death did not both present themselves to us, there would be no inscrutability.

If they do as well as they did in 1960 there can be no complaint.

If, in the trustee's judgment, `` reasonable market conditions '' did not prevail during any given year, he was to be allowed to petition the court for an extension of time within the ten-year period.

If one assumes that the average flux did not change between measurements, a mass-distribution curve is obtained which relates the flux of particles larger than a given radius to the inverse 7/2 power of the radius.

If the bottom name in each column did not have a responsible executive identified, the next name above which identified such a responsible executive was substituted.

If we did not mean to say this, why should we be so relieved on finding that the suffering had not occurred??

If so, are you saying, `` Where did the last few years go??

If not, he was willing to accede to William's wishes in any way that did not block his ultimate aim.

If Bridget did get any bundles of cash, the last person who would have rewarded her for services rendered would have been Lizzie Borden.

If that's all the Romans did, it's a surprise to me that Rome fell.

If laying ties on a railroad track, which he once did for $1 an hour, paid more than playing right field for the Yankees, Maris would lay ties on a railroad track.

If working in a zinc mine, which he once did for 87-1/2 cents an hour, paid more than playing center field for the Yankees, Mantle would work in a zinc mine.

If this is true, then the universe today looks just as it did millions of years ago and as it will look millions of years hence, even though the universe is expanding.

a Portago could say, as he did say to me, `` If I die tomorrow, still I have had twenty-eight wonderful years ; ;

If he died before she did, she would never be unable to resist opening it.

If he ever did such a thing again she'd die of shame.

If the authorities remarked on her curious avocation, they did so among themselves.

If a defendant has been convicted and can prove that his lawyer did not adequately handle his case and that there is a reasonable probability that the result of the trial would have been different had the lawyer given competent representation, he is entitled to a new trial.

If that did not work, then Abd al-Rahman would have to be killed.