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 is also surjective, then the Banach space X is called reflexive.

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 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 the United States were to punish a visiting diplomat without sufficient grounds, U. S. representatives in other countries could receive harsher treatment.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.

Chinese immigrants were blamed for importing the opium-smoking habit to the U. S. The 1903 blue-ribbon citizens ' panel, the Committee on the Acquirement of the Drug Habit concluded, " If the Chinaman cannot get along without his dope we can get along without him.

“ The U. S. Fish and Wildlife Service-funded Ivory-billed Woodpecker searches will continue through the 2008-09 search season ,” says Laurie Fenwood, Ivory-billed Woodpecker Recovery Team Coordinator for the U. S, Fish and Wildlife Service .… If no birds are confirmed, the Cornell Lab of Ornithology will not send an organized team into the field next year.

If I manage to leave for the U. S., then I will be over there.

If a judge determines that there is sufficient evidence to believe that the defendant committed the crime, it is said that the defendant is " held to answer " or " bound over " ( in U. S. jurisdictions ).

If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories ( T, U, V, W, Z ) count fourfold, I counts twice, and X counts only once.

If one considers a version of QCD with N < sub > f </ sub > flavors of massless quarks, then there is a global ( chiral ) flavor symmetry group SU < sub > L </ sub >( N < sub > f </ sub >) × SU < sub > R </ sub >( N < sub > f </ sub >) × U < sub > B </ sub >( 1 ) × U < sub > A </ sub >( 1 ).

If an electron of energy E is incident upon an energy barrier of height U ( z ), the electron wave function is a traveling wave solution,

If the bias is small, we can let U − E ≈ φM in the expression for κ, where φM, the work function, gives the minimum energy needed to bring an electron from an occupied level, the highest of which is at the Fermi level ( for metals at T = 0 kelvins ), to vacuum level.

If U. S. officials can " help clean up his image " and lift the ban on arms sales to the Panamanian Defense Force, Noriega will "' take care of ' the Sandinista leadership for us.

But they soon had another US No. 2 hit with a cover of Simon & Garfunkel's " A Hazy Shade of Winter " from the soundtrack of the film Less Than Zero, and the melancholic " If She Knew What She Wants ", written by Jules Shear, reached the U. S. Top 30 and the German Top 20.

The U. S. Supreme Court explained this, in U. S. Public Workers v. Mitchell: " If granted power is found, necessarily the objection of invasion of those rights, reserved by the Ninth and Tenth Amendments, must fail.

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.