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 F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

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 the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.

If F. tularensis were used as a weapon, the bacteria would likely be made airborne for exposure by inhalation.

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 F and G are ( covariant ) functors between the categories C and D, then a natural transformation η from F to G associates to every object X in C a morphism in D such that for every morphism in C, we have ; this means that the following diagram is commutative:

If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.

If certain coordinate systems are used, for instance, polar-toroidal coordinates ( common in plasma physics ) using the notation ∇ × F will yield an incorrect result.

If φ is a scalar valued function and F is a vector field, then

If F is clear from context then Ω < sub > F </ sub > may be denoted simply Ω, although different prefix-free universal computable functions lead to different values of Ω.

If we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

If F ( r ) represents gravity, it is a negative term proportional to 1 / r < sup > 2 </ sup >, so the net acceleration in r in the rotating frame depends on a difference of reciprocal square and reciprocal cube terms, which are in balance in a circular orbit but otherwise typically not.

If a vector field F with zero divergence is defined on a ball in R < sup > 3 </ sup >, then there exists some vector field G on the ball with F = curl ( G ).

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.

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 he sees the heroic in a Sartoris or a Sutpen, he sees also -- and he shows -- the blind and the mean, and he sees the Compson family disintegrating from within.

If Robinson was a liar and a slanderer, he was also a very canny gentleman, for nothing that Pike could do would pry so much as a single word out of him.

If, the editors sometimes, dozed and printed pretentious, New, York-mind, dross, they, also printed, Malraux,, Silone,, Chiaromonte,, Gide, Bellow,, Robert Lowell, Francis Fergusson, Mary McCarthy, Delmore Schwartz, Mailer, Elizabeth Hardwick, Eleanor Clark,, and a host of, other good writers.

If they also defeat the school bill, the GOP task force won't have much research to do.

If you are a nonresident alien and a resident of Puerto Rico, your return is also due June 15, 1962, or the 15th day of the 6th month after the close of your fiscal year.

If your child is under 19 or is a student you may also claim an exemption for him if he qualifies as your dependent, even though he earns $600 or more.

If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.

If the mechanics are similar, we can also infer that the thermodynamics will also be similar.

If word classes differ in their resistance or liability to stem replacement within meaning slot, it is conceivable that individual meanings also differ with fair consistence trans-lingually.

If Cynewulf was literate, the Beowulf poet may have been also, and so may the final redactor of The Iliad and The Odyssey.

If high accuracy is required in preflight leveling, it is usually necessary to integrate or doubly integrate the accelerometer outputs ( this also minimizes the noise problem ).

If the farm is in an industrial area where wages are high, farm labor costs will also be high.

If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.

If you know that he ( God ) is just ( righteous ), know that everyone also who does what is just ( righteous ) has been born of him.

If it was intentional, it was now also unnecessary ; ;

If an affidavit is notarized or authenticated, it will also include a caption with a venue and title in reference to judicial proceedings.

If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.

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

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

If there are proteins left in the shell matrix, it is also possible that they can trigger an allergic ( asthmatic ) attack.

If necessary, motoring slowly around the location of the anchor also helps dislodge it.

If the Onemen Bay is considered part of the river, it also receives the Velikaya River ( Chukotka ) from the south and the Kanchalan River from the north.

If this etymology is combined with the tradition reported by Geoffrey of Monmouth stating that Ambrosius Aurelianus ordered the building of Stonehenge – which is located within the parish of Amesbury ( and where Ambrosius was supposedly buried ) – and with the presence of an Iron Age hill fort also in that parish, then it may be tempting to connect Ambrosius with Amesbury.

If an alpha channel is used in an image, it is common to also multiply the color by the alpha value, to save on additional multiplications during compositing.