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 the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If, on the other hand, they opted for representation, it had to be representation per se -- representation as image pure and simple, without connotations ( at least, without more than schematic ones ) of the three-dimensional space in which the objects represented originally existed.

If so, they may have originally comprised residual Celtic elements in central eastern Europe such as the Cotini, who formed a Celtic enclave in the Germanic-speaking zone and are described by Tacitus as iron-ore miners working as tributaries of the powerful Quadi Germanic people.

If the air mass is colder than the ground below it, it is labeled k. If the air mass is warmer than the ground below it, it is labeled w. While air mass identification was originally used in weather forecasting during the 1950s, climatologists began to establish synoptic climatologies based on this idea in 1973.

If the western city wall was originally to the east of the Church of the Holy Sepulchre, then the western hill, on which it is sited, would have been advantageous to an enemy.

If, for example, a cello player is assigned a gavotte that was originally written for harpsichord, this gives the student insight in how to play the piece.

If they formed in their current locations, their orbits must originally have been circular ; otherwise accretion ( the coalescence of smaller bodies into larger ones ) would not have been possible because the large relative velocities between planetesimals would have been too disruptive.

The evil demon problem originally motivated skepticism, but can be resuited to object to reliabilist accounts as follows: If our experiences are controlled by an evil demon, it may be the case that we believe ourselves to be doing things that we are not doing.

If restarted, mining at White Cliffs would occur in the same area it originally proliferated on Great Barrier.

If the composition of the rocks was originally similar, they may be very difficult to distinguish from one another if the metamorphism has been great.

If done correctly it is very hard for audiences to tell what sounds were added and what sounds were originally recorded ( location sound ).

If the rock was originally banded or foliated ( as, for example, a laminated sandstone or a foliated calc-schist ) this character may not be obliterated, and a banded hornfels is the product ; fossils even may have their shapes preserved, though entirely recrystallized, and in many contact-altered lavas the vesicles are still visible, though their contents have usually entered into new combinations to form minerals that were not originally present.

If originally of similar composition they may be very difficult to distinguish from one another if the metamorphism has been great.

Podkayne of Mars is a science fiction novel by Robert A. Heinlein, originally serialised in Worlds of If ( November 1962, January, March 1963 ), and published in hardcover in 1963.

Category: Works originally published in If ( magazine )

Former Easterhouse guitarist Ivor Perry was brought in to replace Marr, and the band recorded some new material with him which was never completed, including an early version of " Bengali in Platforms " that was originally intended as the B-side of " Stop Me If You Think You've Heard This One Before ".

Category: Works originally published in If ( magazine )

If unsuccessful, the champion received $ 1, 000 ( originally $ 500 ) for each correct answer they had given in the first half of the round.

* " If This Goes On —" ( 1940 ; originally published in Astounding Science Fiction )

If it was originally part of a marriage chest, or cassone, it was perhaps an uneasy subject for a festive wedding commemoration.

The album featured covers of a 1981 Diana Ross song " Work That Body ", co-written by Paul Jabara and " If You Were a Woman and I Was a Man ", originally recorded by Bonnie Tyler.

If a microscope did not originally have a mechanical stage it may be possible to add one.

If we had moved it 2 tenths then the second pair and only the second would be in alignment since these are the only ones which were originally misaligned by that amount.

The reflective soprano solo " If God be for us " ( originally written for alto ) quotes Luther's chorale Aus tiefer Not.