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 we try to study T using characteristic values, we are confronted with two problems.

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

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 T is the total `` length '' of the process, its feed state may be denoted by a vector p(T) and the product state by p(Q).

* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If T is a good symmetry of the dynamics, then

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 we sit here reading editorials and looking at public-opinion polls and other reports that cross our desks, we should realize that this is raw, undigested opinion expressed in the absence of leadership.

If it could be shown that judgments of good and bad were not judgments at all, that they asserted nothing true or false, but merely expressed emotions like `` Hurrah '' or `` Fiddlesticks '', then these wayward judgments would cease from troubling and weary heads could be at rest.

If it pleases symphony audiences as a light, jolly piece, a series of impressions musically expressed, it succeeds.

If A is expressed as an N × N matrix, then A < sup >†</ sup > is its conjugate transpose.

If it is the fraction of impinging atoms getting through the foil which is measured, the result can still be expressed in terms of the equivalent stopping cross section of the atoms.

If the rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction, and has a unique eventually repeating infinite decimal expansion.

If the difference in vapor pressure between the two components A and B is large ( generally expressed as the difference in boiling points ), the mixture in the beginning of the distillation is highly enriched in component A, and when component A has distilled off, the boiling liquid is enriched in component B.

If there is a direct object, the indirect object can be expressed by an object pronoun placed between the verb and the direct object.

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 too many of them are expressed, that particular chip will not work.

If the key signature indicates G-sharp, a local flat before a G makes it G-flat ( not G natural ), though often this type of rare accidental is expressed as a natural, followed by a flat () to make this clear.

If the orbiting periods were in this relation, the mean motions ( inverse of periods, often expressed in degrees per day ) would satisfy the following

If a point X is located on a parabola which has focal length and if is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from and as follows, assuming they are all expressed in the same units.

If the court satisfies itself that the defendant fully acknowledges the consequences of the plea agreement, and he / she was represented by the defense council, his / her will is expressed in full compliance with the legislative requirements without deception and coercion, also if there is enough body of doubtless evidence for the conviction and the agreement is reached on legitimate sentence-the court approves the plea agreement and renders guilty judgment.

If we acknowledge the fact that we only can measure a probability with some error of measurement attached, we still get into problems as the error of measurement can only be expressed as a probability, the very concept we are trying to define.

He privately expressed concern, noting in the margin of his 1844 Essay, " Better begin with this: If species really, after catastrophes, created in showers world over, my theory false.

If it were rational, it could be expressed as a fraction a / b in lowest terms, where a and b are integers, at least one of which is odd.

President Hugo Chávez has repeatedly expressed his disapproval of the practice of kidnapping stating on 14 April that, " If I were a guerrilla, I wouldn't have the need to hold a woman, a man who aren't soldiers ... Free the civilians who don't have anything to do with the war.

“ If food is treated as a code, the messages it encodes will be found in the pattern of social relations being expressed.

" If we mistakenly confuse precision with accuracy, then we might be misled into thinking that an explanation expressed in precise mathematical or graphical terms is somehow more rigorous or useful than one that takes into account particulars of history, institutions or business strategy.

If these genes are not expressed, the cell cycle does not continue, effectively inhibiting cell division.

If the net gain expressed in dB is negative, it is also called the " net loss ".

If the net gain is expressed as a ratio, and the ratio is less than unity, a net loss is indicated.