Specifically, let G be a simply-connected Lie group with Lie algebra.

Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as

Specifically, let G be a group and let X and Y be two associated G-sets.

Specifically, let X be a separable normed space and B the closed unit ball in X < sup >∗</ sup >.

Specifically, the programs Front Row and Cover Flow, utilized in conjunction with the Apple Remote, let users easily browse through and view any multimedia content stored on their Macs.

The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let Δ < sub > K </ sub > denote discriminant of K, let r < sub > 1 </ sub > ( resp.

Specifically, let X and Y be a topological spaces with A a subspace of Y.

Specifically, let L be a tame oriented knot or link in Euclidean 3-space ( or in the 3-sphere ).

Specifically, let LGdenote the space of continuous maps

Specifically, if ƒ is an invertible function with domain X and range Y, then its inverse ƒ < sup >− 1 </ sup > has domain Y and range X, and the inverse of ƒ < sup >− 1 </ sup > is the original function ƒ.

Specifically, if F is a vector space of linear functionals on X which separates points of X, then the continuous dual of X with respect to the topology σ ( X, F ) is precisely equal to F.

Specifically, a ( locally small ) complete category C has an initial object if and only if there exist a set I ( not a proper class ) and an I-indexed family ( K < sub > i </ sub >) of objects of C such that for any object X of C there at least one morphism K < sub > i </ sub > → X for some i ∈ I.

Specifically, the symbol F in a formal language is a functional symbol if, given any symbol X representing an object in the language, F ( X ) is again a symbol representing an object in that language.

Specifically, if you can prove that for every X ( or every X of a certain type ), there exists a unique Y satisfying some condition P, then you can introduce a function symbol F to indicate this.

Specifically if γ is a closed loop at c such that p < sub >#</ sub >() = 1, that is p o γ is null-homotopic in X, then consider a null-homotopy of p o γ as a map f: D < sup > 2 </ sup > → X from the 2-disc D < sup > 2 </ sup > to X such that the restriction of f to the boundary S < sup > 1 </ sup > of D < sup > 2 </ sup > is equal to p o γ.

Specifically, if X is a 1-dimensional CW-complex, the attaching map for a 1-cell is a map from a two-point space to X,.

Specifically, if X and Y are CW-complexes, then one can form a CW-complex X × Y in which each cell is a product of a cell in X and a cell in Y, endowed with the weak topology.

Specifically, assume we have a nonsingular variety X, and n hypersurfaces Z < sub > 1 </ sub >, ..., Z < sub > n </ sub > which have local equations f < sub > 1 </ sub >, ..., f < sub > n </ sub > near x for polynomials f < sub > i </ sub >( t < sub > 1 </ sub >, ..., t < sub > n </ sub >), such that the following hold:

Specifically, if two manifolds, X and Y, intersect transversely in a manifold M, the homology class of the intersection is the Poincaré dual of the cup product of the Poincaré duals of X and Y.

Specifically, if X and Y are the representation spaces of two linear representations of G then a linear map f: X → Y is called an intertwiner of the representations if it commutes with the action of G. Thus an intertwiner is an equivariant map in the special case of two linear representations / actions.

For example, after having established that the set X contains only non-empty sets, a mathematician might have said " let F ( s ) be one of the members of s for all s in X.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

Let X be a nonempty set, and let.

" Let X be the unit Cartesian square ×, and let ~ be the equivalence relation on X defined by ∀ a, b ∈ (( a, 0 ) ~ ( a, 1 ) ∧ ( 0, b ) ~ ( 1, b )).

Let X be a topological space, and let x < sub > 0 </ sub > be a point of X.

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Let X be a measurable space, let μ be a measure on X, and let N be a measurable set in X.

To find out which move to make, let X be the Nim-sum of all the heap sizes.

Let φ be a net on X based on the directed set D and let A be a subset of X, then φ is said to be frequently in ( or cofinally in ) A if for every α in D there exists some β ≥ α, β in D, so that φ ( β ) is in A.

For any topological space X, let C ( X ) denote the family of real-valued continuous functions on X and let C *( X ) be the subset of bounded real-valued continuous functions.