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 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 I could make myself feel the same way

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 we were creating a wholly new society, we could insist that our social, political, economic and philosophic institutions foster rather than hamper man ; ;

If you had screamed right there in the street where we stood, I could not have felt more fear.

If Robinson was a liar and a slanderer, he was also a very canny gentleman, for nothing that Pike could do would pry so much as a single word out of him.

If Woodruff could not furnish a strong explanation, the governor insisted that he lower his prices in accord with the scale printed in The Advocate.

If an atom bomb in 1945 could destroy an entire city surely the atomic arsenal we now have is more than adequate to fulfill any military objective required of it.

If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.

If the crummy bastard could write!!

If only Simms Purdew could do that, whatever the thing he remembered and told.

If there is time after the warning, the basement shielding could be improved substantially by blocking windows with bricks, dirt, books, magazines, or other heavy material.

If launched in a careful but determined way within the next few weeks, the Peace Corps could have several hundred persons in training this summer for placement next fall.

If we stop thinking in terms of tremendous multimegaton nuclear weapons and consider employing much smaller nuclear weapons which may be more appropriate for most important military targets, it would seem that the B-52 or B-70 could carry a great many small nuclear weapons.

If we could use all the small airfields we have in this country, we could disperse our strategic aircraft by a factor of 10 or more.

If notched for the battens, they would require more work, be weakened and limber holes would have to be bored so that bilge water could flow through.

`` If I could only think of something at the studio, near me, to absorb his boundless energy '', I said.

If only state funds were used to pay for the vocational education, it could be argued that the state should not have to bear the cost of vocational training which would benefit employers in other states.

If it could be shown that judgments of good and bad were not judgments at all, that they asserted nothing true or false, but merely expressed emotions like `` Hurrah '' or `` Fiddlesticks '', then these wayward judgments would cease from troubling and weary heads could be at rest.

If, however, the figure to be discerned were complicated, composed of several interlocking subfigures, and so on, even the tracing process failed him, and he could not focus even relatively simple shapes among its parts.

If they opted for illusion, it could only be illusion per se -- an illusion of depth, and of relief, so general and abstracted as to exclude the representation of individual objects.

If you look at a reading meant for someone else, you will probably see that many of the items could be considered as applicable to you, even when you were not in the picture at all!!