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 Sentinel wins, Master Mold lifts off.

* If G lifts limits of all type J and D has all limits of type J, then C also has all limits of type J and G preserves these limits.

* If G lifts all small limits and D is complete, then C is also complete and G is continuous.

If x is in X and c belongs to the fiber over x ( i. e. ), and is a path with, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x.

the set of homotopy classes of those closed curves γ based at x whose lifts γ < sub > C </ sub > in C, starting at c, are closed curves at c. If X and C are path-connected, the degree of the cover p ( that is, the cardinality of any fiber of p ) is equal to the index ( π < sub > 1 </ sub >( C, c )) of the subgroup p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c )) in π < sub > 1 </ sub >( X, x ).

If the same person lifts a load of 450 N, the equation indicates that the muscle force would reach 5, 000 N, which is at the upper limit of most people ’ s muscle capability.

If a person with a torso weight of 350 N lifts a load of 450 N, from Eq.

If conditions are favorable, then, often even before the original tornado lifts, another wall cloud and occasionally a tornado may form downwind of the old wall cloud, typically to the east or the southeast in the Northern Hemisphere ( vice versa in the Southern Hemisphere ).

If attacked by a vertebrate on land the European Fire-bellied Toad makes a hollow-back and lifts its four legs so that the ventral warning colour is displayed.

If overall limits on greenhouse gas emission are put into place, cap and trade market mechanisms are purported to find cost-effective ways to reduce emissions.

If zeal for orthodoxy caused him to overstep the limits of discretion, his real attitude towards Rome is sufficiently clear.

If the gain is not constant, e. g., by clipping the output signal at the limits of its capabilities, the output signal will be distorted.

If a business can only be advantageously carried on by a large capital, this in most countries limits so narrowly the class of persons who can enter into the employment, that they are enabled to keep their rate of profit above the general level.

# If he's without limits, what limits define GOD apart from who I am ?..

If C is a complete category, then, by the above existence theorem for limits, a functor G: C → D is continuous if and only if it preserves ( small ) products and equalizers.

If the categories C and D have all limits of type J then lim is a functor and the morphisms τ < sub > F </ sub > form the components of a natural transformation

If an Australian de facto couple moves out of a state, they do not take the state with them and the new federal law is tied to the territorial limits of a state.

If the pressure exceeds design limits, the plug pops out to depressurize the pot.

If a vehicle had a ' V ' as the second letter, its mass limits were in line with the eastern states mass limits, which were:

If a vehicle had a ' X ' as the second letter, its mass limits were in line with the western states mass limits, which were:

If the lender is FHA approved and the mortgage is within FHA limits, the FHA provides mortgage insurance that may be more affordable, especially for higher risk borrowers.

If a filename contains only lowercase letters, or is a combination of a lowercase basename with an uppercase extension, or vice-versa ; and has no special characters, and fits within the 8. 3 limits, a VFAT entry is not created on Windows NT and later versions such as XP.

If the Moselle department still existed in its limits of between 1815 – 1871, its population at the 1999 French census would have been 1, 089, 804 inhabitants.

If they were sharp businessmen ( or in some cases businesswomen ), there were no social limits to the wealth they could amass.

If he had married her, he'd have been asking for trouble.

If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.

I showed her the shower and tub, and she said, smiling, `` If you really don't mind, I think I'll get clean in the shower, then soak for a few minutes in your tub.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.

`` If Blue Throat has his way he'll keep us all cooped up in here for days '', he said.

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 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 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 living Jews were unavailable for study, the Bible was at hand.

If he had been `` liquidated '' in some way, he would have become a martyr, a rallying point for people who shared his ideas.

If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.

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 we grasp this opportunity to build an age of productive partnership between the less fortunate nations and those that have already achieved a high state of economic advancement, we will make brighter the outlook for a world order based upon security and freedom.

`` If you become a Baptist, I will not '', Ann informed her husband, but sweeping her threat aside Adoniram continued to search for an answer to the personal dilemma in which he found himself.

Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If only for this modest masterpiece of military history, Blenheim is likely to be read and reread long after newer interpretations have perhaps altered our picture of the Marlborough wars.

If, as Reid says, `` nearly all his poetry was produced when he was not taking opium '', there may be some reason to doubt that he was under its influence in the period from 1896 to 1900 when he was writing the poems to Katie King and making plans for another book of verse.

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 Gorton wanted peace and quiet for his complicated meditations this is where he should have had it.

If she wanted to borrow any sum of money in expecting the arrangements of Congress, it would not become a stranger, unknown to her, to offer himself for that purpose.

If the hardships of the winter at Valley Forge were trying for healthy men, they were, of course, much more so for those not in good health.

If his circumspection in regard to Philip's sensibilities went so far that he even refused to grant a dispensation for the marriage of Amadee's daughter, Agnes, to the son of the dauphin of Vienne -- a truly peacemaking move according to thirteenth-century ideas, for Savoy and Dauphine were as usual fighting on opposite sides -- for fear that he might seem to be favoring the anti-French coalition, he would certainly never take the far more drastic step of ordering the return of Gascony to Edward, even though, as he admitted to the English ambassadors, he had been advised that the original cession was invalid.