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 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 a system consists of several particles, the total angular momentum about a point can be obtained by adding ( or integrating ) all the angular momenta of the constituent particles.

If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved.

If the game has been scored correctly, the total number of plate appearances for a team should equal the total of that team's runs, men left on base, and men put out.

If the rider is sitting upright, the rider's body creates about 75 % of the total drag of the bicycle / rider combination.

If the player and dealer have the same point total, this is called a " push " and the player typically doesn't win or lose money on that hand.

If more than one wavelength is present ( as is usually the case ), one must sum or integrate over the spectrum of wavelengths present to get the total luminous intensity.

If a source emits a known luminous intensity I < sub > v </ sub > ( in candelas ) in a well-defined cone, the total luminous flux Φ < sub > v </ sub > in lumens is given by

If a 2, 000 ( lb / ac )/ year sequestration rate was achieved on all of cropland in the United States, nearly 1. 6 billion tons of carbon dioxide would be sequestered per year, mitigating close to one quarter of the country's total fossil fuel emissions.

If there is one particular coordinate change that leads to a decrease in the total energy in both directions, the stationary point is a transition structure and the coordinate is the reaction coordinate.

If the player dies, they are still put on the list, but are categorized as " Dead ", with their experience point total listed as at the final killing blow.

If freedmen had total control of money, letters, and law, it seemed it would not be hard for them to manipulate the Emperor.

If one uses the less-stringent 1000 casualties total criterion, there were over 90 civil wars between 1945 and 2007, with 20 ongoing civil wars as of 2007.

If the sheet contains regions where the number of atoms in a ring is different from six, while the total number of atoms remains the same, a topological defect has formed.

If this were generalized to the US population as a whole, it would make deism the fastest-growing religious classification in the US for that period, with the reported total of 49, 000 self-identified adherents representing about 0. 02 % of the US population at the time.

If one hundred six-minute calls are received then the total traffic in that hour is six hundred minutes or 10 Erlangs.

If the team that batted second is substantially behind on first innings total, it may be required to bat again ( i. e., to immediately follow on from its first innings ) in the third innings of the match.

If total cash available is less than cash needs, a deficiency exists.

If all contracts awarded reach their stated target plateau production then this will increase Iraqi production from today's 2. 5 mb / d by 9. 4 mb / d to a total of 11. 9 mb / d, comparable to current Saudi declared capacity of 12. 5 mb / d.

If there is an additional magnitude of value or a loss of value after transformation compared with before then the relation between values ( proportional to labor ) and prices ( proportional to total capital advanced ) is incomplete.

* If the price is between average total cost and average variable cost at the profit-maximizing output, then the firm is said to be in a loss-minimizing condition.

If the photons reside mainly in the higher index material, the confinement is due to total internal reflection.

If two particles have masses and, and velocities and, the total momentum is