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 he becomes chancellor, Dr. Erhart would make few changes.

If changes were abrupt they were deemed revolutions.

# If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively ( like altitude on a contour map ), then as the eye moves towards the axes ( towards the south-west ), it sees the density of isotherms stay constant, but it sees the density of adiabats grow.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

If one component changes another component's vapor pressure, or if the volatility of a component is dependent on its percentage in the mixture, the law will fail.

If only layer C changes, we should find a way to avoid re-blending all of the layers when computing F. Without any special considerations, four full-image blends would need to occur.

If the spacecraft detects an anomaly, it changes the carrier between four tones, based on urgency.

If the estimate of were not significantly different from 0, we would fail to find evidence that changes in the growth rate and unemployment rate were related.

If the researcher could randomly assign people to different levels of education, the data set thus generated would allow estimation of the effect of changes in years of education on wages.

If this is done on fractals, however, no new detail appears ; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over.

If the primary variable is mass fraction (, given, for example, in ), then the equation changes to:

* If glass flows at a rate that allows changes to be seen with the naked eye after centuries, then the effect should be noticeable in antique telescopes.

If the blood sugar level changes dramatically during evaluation, then the basal rate can be adjusted to increase or decrease insulin delivery to keep the blood sugar level approximately steady.

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.

If the absolute pressure of a fluid stays constant, the gauge pressure of the same fluid will vary as atmospheric pressure changes.

If a force is applied to a particle for a time interval, the momentum of the particle changes by an amount

If momentum is to be conserved in a volume, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum.

If the voltage changes by a large enough amount, an all-or-none electrochemical pulse called an action potential is generated, which travels rapidly along the cell's axon, and activates synaptic connections with other cells when it arrives.

If the defendant accepts these suggestions and changes his penalty proposition, the court approves it and passes the verdict according to the plea agreement.

If we change the orientation of the surface element, the direction of the normal force changes accordingly, but the pressure remains the same.

If Socrates changes, becoming sick, Socrates is still the same ( the substance of Socrates being the same ), and change ( his sickness ) only glides over his substance: change is accidental, whereas the substance is essential.

If the tip is moved across the sample in the x-y plane, the changes in surface height and density of states cause changes in current.

If the change of a sequence of fricatives such that one becomes a stop is dissimilation, then such changes as Proto-Germanic * χs to ( spelled x ) in English would count as a regular sound law: PGmc.