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 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 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 degree ( Q ) ≤ 4 then the multiple roots of P may be found algebraically.

Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is automatic.

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

If k is not algebraically closed, for example the field of real numbers, the picture becomes more complicated because of the existence of non-linear irreducible polynomials.

If the field F is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, " curves " with no points on them.

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m ( m − 1 )/ 2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.

If F is algebraically closed, this is equivalent to a curve of genus zero ; however, the field of all real algebraic functions defined on the real algebraic variety x < sup > 2 </ sup >+ y < sup > 2 </ sup > = − 1 is a field of genus zero which is not a rational function field.

For instance, consider the result: " If E is a field with the property that every nonconstant polynomial with coefficients in E has a root in E, then E is algebraically closed.

" Despite its simplicity, it suggests a deeper conjecture: " If is an algebraic extension and if every nonconstant polynomial with coefficients in F has a root in E, is E algebraically closed?

If σ, ..., σ are the distinct K-linear field embeddings of L into an algebraically closed field containing K ( where n is the degree of the extension L / K ), then

If K is given inside an algebraically closed field C, then the conjugates can be taken inside C. Usually one includes α itself in the set of conjugates.

) If every such " finitely-solvable " system is itself solvable, then we call the module M algebraically compact.

If R is an associative algebra with 1 over some field k, then every R-module with finite k-dimension is algebraically compact.

* If F is algebraically closed and char ( F ) does not divide | G |, then the number of irreducible characters of G is equal to the number of conjugacy classes of G. Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of G ( and they even divide the index of the center of G in G if ).

If the characteristic of the field does not divide the order of G, then there is an inner product defined on this space defined by where | G | denotes the order of G. The set of irreducible characters of G forms an orthogonal basis, and if K is a splitting field for G, for instance if K is algebraically closed, then the irreducible characters form an orthonormal basis.

If the field K is not algebraically closed, the theorem can fail.

If K is formally real and Ω is an algebraically closed field containing K, then there is a real closed subfield of Ω containing K. A real closed field can be ordered in a unique way.

If the field is algebraically closed of characteristic 0 and the algebra is finite dimensional then all Cartan subalgebras are conjugate under automorphisms of the Lie algebra, and in particular are all isomorphic.

If is a linear Lie algebra ( a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space V ) over an algebraically closed field, then any Cartan subalgebra of is the centralizer of a maximal toral Lie subalgebra of ; that is, a subalgebra consisting entirely of elements which are diagonalizable as endomorphisms of V which is maximal in the sense that it is not properly included in any other such subalgebra.

* If V is an algebraic set in K < sup > n </ sup >, for an algebraically closed field K, then the Morley rank of V is the same as its usual Krull dimension.

If, on the other hand, the numbers are chosen so as to make exp ( z < sub > 1 </ sub >),..., exp ( z < sub > n </ sub >) all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a strengthening of Baker's theorem.

If the Weyl tensor is algebraically special at some, there is a useful set of conditions, found by Lluis ( or Louis ) Bel and Robert Debever, for determining precisely the Petrov type at.