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 dance teachers of America make it their business to prepare their young charges for the gymnastics that must come some day if our schools are really responsible, we will be that much ahead.

If, after signing such a declaration, the information is found to be deliberately untrue with the intent to deceive, the applicant may face perjury charges.

If these other charges and currents are comparable in size to the sources producing the above electromagnetic field, then a new net electromagnetic field will be produced.

If convicted of these charges, Lindh could have received up to three life sentences and 90 additional years in prison.

If the accusers were among those named, the defendant was set free and the charges dismissed ; the accusers would face life imprisonment.

If the court finds that presented evidence is not sufficient to support the charges or that a motion to render a judgment without substantial consideration of a case is submitted in violation of the requirements stipulated by the Criminal Procedure Code of Georgia, it shall return the case to the prosecution.

If those six months have passed and the charges have not been reinstated, the statutes of limitation have run out.

If several spatially distributed charges generate such an electric potential, e. g. in a solid, an electric field gradient may also be defined.

If the magnetic-pole distribution is known, then the pole model gives the magnetic field H. Outside the magnet, the field B is proportional to H, while inside the magnetization must be added to H. An extension of this method that allows for internal magnetic charges is used in theories of ferromagnetism.

If the judge finds there is not enough evidence under the standard, the case must be dismissed ( or a subsequent guilty verdict must be vacated and the charges dismissed ).

If they feel a violation of Corps regulations occurred, they will summon the offender to Oa and hold a trial in which the charges are read and the Lantern is allowed to explain his / her / its actions.

If friction tubes are employed when cordite or other smokeless powder charges are used, the erosion of the vent is very rapid unless the escape of the gas is prevented ; in this case T-headed tubes are used.

If the charges on the plates are + q and − q, and V gives the voltage between the plates, then the capacitance C is given by

If three plates are given charges, then the voltage of plate 1 is given by

If the cage is grounded, the excess charges will go to the ground instead of the outer face, so the inner face and the inner charge will cancel each other out and the rest of the cage will retain a neutral charge.

On this basis his biographer concludes: " If Rundstedt had stood trial, it is clear from the Manstein case that he would have been found guilty of some of the charges levelled against him.

If a victim is brought across state lines, federal charges can be laid as well.

If magnetic charges do not exist – or if they do exist but are not present in a region of space – then the new terms in Maxwell's equations are all zero, and the extended equations reduce to the conventional equations of electromagnetism such as ∇• B = 0 ( where ∇• is divergence and B is the magnetic B field ).

If items are returned late, the shop usually charges late fees, which typically accumulate day by day.

If, however, the purchaser is not an end user, but the goods or services purchased are costs to its business, the tax it has paid for such purchases can be deducted from the tax it charges to its customers.

If the back-end load declines the longer the investor holds shares, it is called a contingent deferred sales charges ( or CDSC ).

OFF-NET USAGE If your minutes of use ( including unlimited services ) on other carrier networks (" off-net usage ") during any two consecutive months exceed your off-net usage allowance, AT & T may, at its option terminate your service, deny your continued use of other carriers ' coverage or change your plan to one imposing usage charges for off-net usage.

If a lender charges above the lawful interest rate, a court will not allow the lender to sue to recover the debt because the interest rate was illegal anyway.

If a company exports a product at a price lower than the price it normally charges in its own home market, it is said to be " dumping " the product.

If the " push " is based on the electromagnetic force, this would simply mean the mass accelerates in the opposite direction than what one would expect based on its charge ; for example, an object with negative inertial mass and positive charge would accelerate away from objects with negative charge, and accelerate towards objects with positive charge, the opposite of the normal rule that like charges repel and opposite charges attract.