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 G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

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 G is a group, and g is a fixed element of G, then the conjugation map

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 in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable.

Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.

If f is an element of G ( x, y ) then x is called the source of f, written s ( f ), and y the target of f ( written t ( f )).

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

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 day should ever come that foreign invaders swarm ashore along the Gulf Coast '', the account reads, `` they can count on heavy opposition from a group of commando-trained telephone employees -- all girls.

If coffee is sold at the cafeteria, let a few workers in each department get it for the whole group.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

If the joining of monomers takes place at such a carbon atom, the free hydroxy group of the pyranose or furanose form is exchanged with an OH-side-chain of another sugar, yielding a full acetal.

If the amino group is removed from an amino acid, it leaves behind a carbon skeleton called an α-keto acid.

If a bubble touches identically-colored bubbles, forming a group of three or more, those bubbles — as well as any bubbles hanging from them — are removed from the field of play, and points are awarded.

If the Bastarnae remained an identifiable group, it is highly likely that they participated in the vast Gothic-led migration, driven by Hunnic pressure, that was admitted into Moesia by emperor Valens in 376 and eventually defeated and killed Valens at Adrianople in 378.

If the carbonyl group is an aldehyde, the monosaccharide is an aldose ; if the carbonyl group is a ketone, the monosaccharide is a ketose.

If there is a tie, we must consider the atoms at distance 2 from the stereocenter — as a list is made for each group of the atoms bonded to the one directly attached to the stereocenter.

If the addition of a functional group takes place at the less substituted carbon atom of the double bond, then the electrophilic substitution with acids is not possible.

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in " standard form " meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.

If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).

If a database cannot tolerate even such smaller probability of data loss, then the RAID group itself is replicated ( mirrored ).

* If we think of as the group of real numbers under addition, then the direct product still consists of.

If the ether is part of a more complex molecule, it is described as an alkoxy substituent, so-OCH < sub > 3 </ sub > would be considered a " methoxy -" group.

If there is another functional group at a carbon, it may be named with the Greek letter, e. g., the gamma-amine in gamma-aminobutanoic acid is on the third carbon of the carbon chain attached to the carboxylic acid group.