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 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 we grasp this opportunity to build an age of productive partnership between the less fortunate nations and those that have already achieved a high state of economic advancement, we will make brighter the outlook for a world order based upon security and freedom.

If a series is drawn then the country already holding the Ashes retains them.

If the sick person wishes to receive the sacrament of penance, it is preferable that the priest make himself available for this during a previous visit ; but if the sick person must confess during the celebration of the sacrament of anointing, this confession replaces the penitential rite A passage of Scripture is read, and the priest may give a brief explanation of the reading, a short litany is said, and the priest lays his hands on the head of the sick person and then says a prayer of thanksgiving over the already blessed oil or, if necessary, blesses the oil himself.

If the next data item is not available when it is needed, the codec has no choice but to produce silence or guess — and if the data is late, it is useless, because the time period when it should have been converted to a signal has already passed.

We must take the offensive and beat Sinn Fein at its own tactics ... If a police barracks is burned or if the barracks already occupied is not suitable, then the best house in the locality is to be commandeered, the occupants thrown into the gutter.

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If however it has already saved its life by self-castration and is again pursued, then it stands up and reveals that it offers no ground for their eager pursuit, and releases the hunters from all further exertions, for they esteem its flesh less.

If a rule has as a singleton on its RHS, add a new rule unless has already been removed through this process.

If the panegyric detailing the ceremony implied that the true center of the Empire was not Rome, but where the Emperor sat ("... the capital of the Empire appeared to be there, where the two emperors met "), it simply echoed what had already been stated by the historian Herodian in the early third century: " Rome is where the emperor is ".

If you be not too much cloyed with fat meat, our humble author will continue the story, with Sir John in it, and make you merry with fair Katharine of France where, for any thing I know, Falstaff shall die of a sweat, unless already a ' be killed with your hard opinions ; for Oldcastle died a martyr, and this is not the man.

If we already created such a " mark ", then we preserve our previous mark and move to the next pixel following the right-hand rule.

If you listened very carefully, I have already stated to you what the third word is.

If you have listened carefully, I have already told you what it is.

If you have listened carefully, I have already told you what it is.

If you have listened carefully, I have already told you what it is.

" If Stobaeus writes correctly, Sotion in the early 1st century CE was already combining the two in the imaginative duo of weeping and laughing philosophers: " Among the wise, instead of anger, Heraclitus was overtaken by tears, Democritus by laughter.

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

If the territory was already on a war footing, the whole society alert and inflamed, then why not direct all this ramped up energy toward something useful?

If a philosophy of Becoming is not possible without change, then ( the potential of ) what is to become must already exist in the present object.

If data which has been compressed lossily is decoded and compressed losslessly, the size of the result can be comparable with the size of the data before lossy compression, but the data already lost cannot be recovered.

If the distinguished name in the add request already exists in the directory, then the server will not add a duplicate entry but will set the result code in the add result to decimal 68, " entryAlreadyExists ".

If a player gets a 6 they can separate chances ( the player can separate 6 on one piece and 3 on the other if they get a 6 and a 3, if the pieces are already out of the house ).

If something was already there with no conflict, he used it without changes in language, he reordered and ruled on where there was conflict, and clarified where context was not given.