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 the argument is accepted as essentially sound up to this point, it remains for us to consider whether the patient's difficulties in orienting himself spatially and in locating objects in space with the sense of touch can be explained by his defective visual condition.

:” 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 one considers a small fluorescent light source ( essentially a bright spot ), light coming from this spot spreads out the further out of focus one is.

If, for instance, the algorithmic view is taken, then the reals as constructed here are essentially what classically would be called the computable numbers.

If the lifetime of this transition, τ < sub > 21 </ sub > is much longer than the lifetime of the radiationless 3 → 2 transition τ < sub > 32 </ sub > ( if τ < sub > 21 </ sub > ≫ τ < sub > 32 </ sub >, known as a favourable lifetime ratio ), the population of the E < sub > 3 </ sub > will be essentially zero ( N < sub > 3 </ sub > ≈ 0 ) and a population of excited state atoms will accumulate in level 2 ( N < sub > 2 </ sub > > 0 ).

If the category of " scientist " is limited to those who do scientific research as a profession, then the social role of scientist essentially emerged in the 19th century as part of the professionalization of science.

If the hybrids are infertile, or fertile but less fit than their ancestors, then there will be further reproductive isolation and speciation has essentially occurred ( e. g., as in horses and donkeys.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by J.

* If J = 1, the category with a single object and morphism, then a diagram of type J is essentially just an object of C. A cone to an object X is just a morphism with codomain X.

If the micrometer is in good condition, then they are all so near to zero that the instrument seems to read essentially " dead-on " all along its range ; no noticeable error is seen at any locale.

If they were married before joining the society, their marriages essentially ended when they joined.

If you move " off " an indifference curve traveling in a northeast direction ( assuming positive marginal utility for the goods ) you are essentially climbing a mound of utility.

If a being is essentially omnipotent, then it can also resolve the paradox ( as long as we take omnipotence not to require absolute omnipotence ).

If the server and client support block number wraparound, file size is essentially unlimited.

If a safe pharmacological treatment can be developed – one that increases expression of lysosomal sialidase in neurons without other toxicity – then this new form of therapy could essentially cure the disease.

If it is not, then it can be completed: one can find an essentially unique complete topological ring S which contains R as a dense subring such that the given topology on R equals the subspace topology arising from S.

If the voltage is large enough, essentially all electrons ( and any negative ions ) will be repelled.

If the story related by the Liber pontificalis is essentially correct, the pope probably left Rome on 22 November 545.

If one assumes that the electron is in the ground state, then the van der Waals binding energy is essentially the change of the zero-point energy:

If the sample is a protein, the mean residual weight ( average molecular weight of the amino acids it contains ) is used in place of the molecular weight, essentially treating the protein as a solution of amino acids.

If the BOM character appears in the middle of a data stream, Unicode says it should be interpreted as a " zero-width non-breaking space " ( essentially a null character ).

Stephen Thomas Erlewine of Allmusic gave the album a negative review, remarking that " If you do happen to enjoy their ridiculously stupid jokes, Oi to the World is nearly as fun as anything else in their catalog, even though you might admit their fast, goofy, and essentially tuneless songs are beginning to wear a bit thin.

If all of the petals are essentially identical in size and shape, the flower is said to be regular or actinomorphic ( meaning " ray-formed ").

The conjecture is stated in terms of three positive integers, a, b and c ( whence comes the name ), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d cannot be much smaller than c.