If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If a union cannot perform this function, then collective bargaining is being palmed off by organizers as a gigantic fraud.

If the Greek letter is used, it is assumed to be a Fourier transform of another function,

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If f is not a function, but is instead a partial function, it is called a partial operation.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.

If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.

If we take the simple valence bond structure and mix in all possible covalent and ionic structures arising from a particular set of atomic orbitals, we reach what is called the full configuration interaction wave function.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

If we cannot stop warfare in our own economic system, how can we expect to abolish it internationally??

If the points aren't thrown all the way, the Turnout cannot be locked, and in turn, the signal cannot be cleared.

`` If we cannot tell evil, horror, and insanity from nonsense, what is the future of humor and comedy ''??

If a litigant chooses to enforce a Federal right in a State court, he cannot be heard to object if he is treated exactly as are plaintiffs who press like claims arising under State law with regard to the form in which the claim must be stated -- the particularity, for instance, with which a cause of action must be described.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If the latter knows nothing about the absent sitter except his name ( given by the experimenter ), he cannot possibly give any clues, conscious or unconscious, far less ask leading questions.

If we cannot make explicit choices, how do we know that our set exists?

If there is an overabundance of food hunted, the arctic fox will bury what the family cannot eat.

If the daughter is an only child or her sisters are deceased and have no living issue, she ( or her heir ) is vested with the title ; otherwise, since a peerage cannot be shared nor divided, the dignity goes into abeyance between the sisters or their heirs, and is held by no one.

If the assets used are not identical ( so a price divergence makes the trade temporarily lose money ), or the margin treatment is not identical, and the trader is accordingly required to post margin ( faces a margin call ), the trader may run out of capital ( if they run out of cash and cannot borrow more ) and go bankrupt even though the trades may be expected to ultimately make money.

If nature cannot err, then there are no paradoxes in it ; to Hobbes, the paradox is a form of the absurd, which is inconsistency: " Natural sense and imagination, are not subject to absurdity " and " For error is but a deception ...

If we cannot stay here alive, then we will stay here dead.

If a pair cannot hit downwards, they will use flat strokes in an attempt to gain the attack.

But this destruction, coupled with a protracted siege of Rain ( 9 – 16 July ), caused Prince Eugene to lament "... since the Donauwörth action I cannot admire their performances ", and later to conclude " If he has to go home without having achieved his objective, he will certainly be ruined.

* Indivisibility of certain goods: If a person wants to buy a certain amount of another's goods, but only has for payment one indivisible unit of another good which is worth more than what the person wants to obtain, a barter transaction cannot occur.

If the remaining array to be searched is reduced to zero, then the key cannot be found in the array and a special " Not found " indication is returned.

If players cannot move either die in a roll, given the position of their checkers then that turn is over and the turn passes to the opponent.

If the Crawford rule is in effect, then another option is the Holland rule, which stipulates that after the Crawford game, a player cannot double until after at least two rolls have been played by each side.

If we look at this situation from far above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.

If a newer software version cannot save files that can be read by the older version, it is not backward compatible with the older version, although it may provide an irreversible upgrade capability for the old files.

If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.

If a bookmaker follows the rules of the Bayesian calculus in the construction of his odds, a Dutch book cannot be made.

If you cannot access the full text, click the first link in this Google search.

If we let them go, they won't stay away, they'll find men to ride with them and they'll be back.

If we look at recent art we find it preoccupied with form.

If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.

If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.

If this practice should take root and spread, the man who submits a manuscript to a publisher will find himself reviewed before he is accepted and publication will become a sort of post-mortem formality.

If any official operation abroad begins to go wrong, we shall look to the ambassador to find out why and to get suggestions for remedial action.

If you elect to use the Standard Deduction or the Tax Table, and later find you should have itemized your deductions, you may do so by filing an amended return within the time prescribed for filing a claim for refund.

If we return to them today, we have no difficulty spotting their weaknesses but we find them still pleasing.

If you are considering a part-time farm where the water must be provided by a well, find out if there is a good well on the farm or the probable cost of having one drilled.

If you find out anything, come on back here and we'll get started on it ''.

If that picture gets around and I find out you had anything to do with it, I'm going to send a couple of my boys around to see you ''.

If the administration ever had any ideas that it could find an acceptable alternative to Prince Souvanna Phouma, whom it felt was too trusting of Communists, it gradually had to relinquish them.

If there is a moral lurking among the shenanigans, it is hard to find.

`` If you find it necessary, Sam, go ahead '', she said, turning on the stair.

If I ever committed suicide, she thought, I would dive straight down from here -- and no one would find me for days.

If the appellate court does find a legal defect in the decision " below " ( i. e., in the lower court ), it may " modify " the ruling to correct the defect, or it may nullify (" reverse " or " vacate ") the whole decision or any part of it.

If they find one they will monitor its progress very thoroughly.

If the heuristic can't find the correct setting, the variable is assigned randomly.

If overall limits on greenhouse gas emission are put into place, cap and trade market mechanisms are purported to find cost-effective ways to reduce emissions.

If A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.

:” If we observe the totality of Pissarro ’ s work, we find there, despite fluctuations, not only an extreme artistic will, never belied, but also an essentially intuitive, purebred art.

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

* In the Tales of Old Dartmoor episode ( recorded in 1956 ) of The Goons radio comedy series, Grytpype-Thynne arranges for the prison to put to sea to visit the Château d ' If in France as part of a plan to find the treasure of the Count of Monte Cristo hid there.

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.