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 S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

If the ratio a '/ a be sufficiently constant, as is often the case, the above relation reduces to the condition of Airy, i. e. tan w '/ tan w = a constant.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

* If the operation is commutative, ab = ba, then the value depends only on

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If the sets A and B are equal, this is denoted symbolically as A = B ( as usual ).

If a problem can be shown to be in both NP and co-NP, that is generally accepted as strong evidence that the problem is probably not NP-complete ( since otherwise NP = co-NP ).

If the user pressed keys 1 + 2 = 3 simultaneously the letter " c " appeared.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

If κ is an infinite cardinal number, then cf ( κ ) is the least cardinal such that there is an unbounded function from it to κ ; and cf ( κ ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ ; more precisely

If the disk was not otherwise prepared with a custom format, ( e. g. for data disks ), 664 blocks would be free after formatting, giving 664 × 254 = 168, 656 bytes ( or almost 165 kB ) for user data.

This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x < sup > 2 </ sup > = 2, yet no rational number has this property.

If y = f ( x ) is differentiable at a, then f must also be continuous at a.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

If G is any subgroup of GL < sub > n </ sub >( R ), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups.

If n ≥ 2, then the group GL ( n, F ) is not abelian.

If V is a vector space over the field F, the general linear group of V, written GL ( V ) or Aut ( V ), is the group of all automorphisms of V, i. e. the set of all bijective linear transformations V → V, together with functional composition as group operation.

If V has finite dimension n, then GL ( V ) and GL ( n, F ) are isomorphic.

If F is a finite field with q elements, then we sometimes write GL ( n, q ) instead of GL ( n, F ).

If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into GL ( n, C ).

A representation of a Lie group G on a vector space V ( over a field K ) is a smooth ( i. e. respecting the differential structure ) group homomorphism G → Aut ( V ) from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into GL ( n, K ).

* If G is a matrix Lie group ( i. e. a closed subgroup of GL ( n, C )), then its Lie algebra is an algebra of n × n matrices with the commutator for a Lie bracket ( i. e. a subalgebra of ).

If the underlying field is R or C and GL ( V ) is considered as a Lie group, then this construction makes the Grassmannian into a smooth manifold.

If the ground field k is arbitrary and GL ( V ) is considered as an algebraic group, then this construction shows that the Grassmannian is a non-singular algebraic variety.

Roy Thomas writes, " If we lost the original GL, we gained the Earth-Two Robotman ; if we dropped Jay ( Flash ) Garrick, we picked up on Johnny Quick ; Liberty Belle could stand in for Wonder Woman till more super-powered ladies came along.

If M is connected then the holonomy group depends on the basepoint x only up to conjugation in GL ( k, R ).

If G ≤ GL < sub > n </ sub > is a smooth closed-subgroup that acts irreducibly on affine-space over, then G is reductive.

If G = GL < sub > n </ sub >( R ), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.