* Let H be a group, and let G be the direct product H × H. Then the subgroups

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).

Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).

Let f and g be any two elements of G. By virtue of the definition of G, = and =, so that =.

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Let a, b, and c be elements of G. Then:

Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.

Let R be a ring and G be a monoid.

Let this point be called G. Point G is also the midpoint of line segment FQ:

Let F and G be a pair of adjoint functors with unit η and co-unit ε ( see the article on adjoint functors for the definitions ).

Let ( G ,.

Let G and H be groups, and let φ: G → H be a homomorphism.

Let G be a group.

Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

Let G be a group.

Let N and K be normal subgroups of G, with

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

Let A ⊆ R → S be homomorphisms where R is not necessarily local ( one can reduce to that case however ), with A, S regular and R finitely generated as an A-module.

Let R ⊆ S be a map of complete local domains, and let Q be a height one prime ideal of S lying over xR, where R and R / xR are both regular.

:: Finite Field Kakeya Conjecture: Let F be a finite field, let K ⊆ F < sup > n </ sup > be a Kakeya set, i. e. for each vector y ∈ F < sup > n </ sup > there exists x ∈ F < sup > n </ sup > such that K contains a line

Let A ⊆ B be an extension of commutative rings.

Let F be a functor from the category of ringed spaces to the category of sets, and let G ⊆ F. Suppose that this inclusion morphism G → F is representable by open immersions, i. e., for any representable functor Hom (−, X ) and any morphism Hom (−, X )→ F, the fibered product G ×< sub > F </ sub > Hom (−, X ) is a representable functor Hom (−, Y ) and the morphism Y → X defined by the Yoneda lemma is an open immersion.

Let C be a category, and let c be an object of C. A sieve S on c is a subfunctor of Hom (−, c ), i. e., for all objects c ′ of C, S ( c ′) ⊆ Hom ( c ′, c ), and for all arrows f: c ″→ c ′, S ( f ) is the restriction of Hom ( f, c ), the pullback by f ( in the sense of precomposition, not of fiber products ), to S ( c ′).

Let V be a vector space, and let F be the family of linearly independent subsets of V. Then F is a family of finite character ( because a subset X ⊆ V is linearly dependent iff X has a finite subset which is linearly dependent ).

Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain.

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.

Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

Let P < sub > F </ sub > be the domain of a prefix-free universal computable function F. The constant Ω < sub > F </ sub > is then defined as

Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

Let M be a smooth manifold and let f ∈ C < sup >∞</ sup >( M ) be a smooth function.

Let e be the error in b. Assuming that A is a square matrix, the error in the solution A < sup >− 1 </ sup > b is A < sup >− 1 </ sup > e.

Let us for simplicity take, then < math > 0 < c =- 2a </ math > and.

Let r be a non zero real number and let the r < sup > th </ sup > power mean ( M < sup > r </ sup > ) of a series of real variables ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ... ) be defined as

Let t and s ( t > s ) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s < sup > 2 </ sup > equals half the harmonic mean of c < sup > 2 </ sup > and t < sup > 2 </ sup >.

Let n be the number of points and d the number of dimensions.

Let X be a finite set with n elements.

Let A be the arithmetic mean and H be the harmonic mean of n positive real numbers.

Let n denote a complete set of ( discrete ) quantum numbers for specifying single-particle states ( for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.

Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.

Let x < sub > 1 </ sub >, ..., x < sub > n </ sub > be the sizes of the heaps before a move, and y < sub > 1 </ sub >, ..., y < sub > n </ sub > the corresponding sizes after a move.

Let s = x < sub > 1 </ sub > ⊕ ... ⊕ x < sub > n </ sub > and t = y < sub > 1 </ sub > ⊕ ... ⊕ y < sub > n </ sub >.

Let n be the length of a statement in Presburger arithmetic.

Let ( q < sub > 1 </ sub >, w, x < sub > 1 </ sub > x < sub > 2 </ sub >... x < sub > m </ sub >) ( q < sub > 2 </ sub >, y < sub > 1 </ sub > y < sub > 2 </ sub >... y < sub > n </ sub >) be a transition of the GPDA

Let be an oriented smooth manifold of dimension n and let be an n-differential form that is compactly supported on.

:: Let n = 0

Let be a random sample of size n — that is, a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ < sup > 2 </ sup >.

Let M be an n × n Hermitian matrix.