Define f ( x ) = 1 for all nonzero x.

Define f ( n ) = | n |, the absolute value of n.

Define f ( a + bi ) = a < sup > 2 </ sup > + b < sup > 2 </ sup >, the squared norm of the Gaussian integer a + bi.

Define f ( a + bω ) = a < sup > 2 </ sup > − ab + b < sup > 2 </ sup >, the norm of the Eisenstein integer a + bω.

Define f := f < sub > 1 </ sub > ∘ p < sub > 1 </ sub > + f < sub > 2 </ sub > ∘ p < sub > 2 </ sub >.

Define f: R < sup > n </ sup > → R by f ( x ) = x < sup > T </ sup > M x.

Define f: l < sup >∞</ sup > → l < sup >∞</ sup > as the shift f ( x < sub > 1 </ sub >, x < sub > 2 </ sub >,...)

Define the function f: S → < nowiki ></ nowiki > pointwise by

If the image of f is not dense, then there is a complex number w and a real number r > 0 such that the open disk centered at w with radius r has no element of the image of f. Define g ( z ) = 1 /( f ( z ) − w ).

Define the Lefschetz number Λ < sub > f </ sub > of f by

To see this, let X be an H-space with identity e and let f and g be loops at e. Define a map F: × → X by F ( a, b )

Define f < sup > n </ sup > as the n-th iterate of f, where n is a non-negative integer, by:

Define the linear map T: V → V point-wise by Tx = Mx, where on the right-hand side x is interpreted as a column vector and M acts on x by matrix multiplication.

Define size ( x ) to be the size of the tree rooted at x ( the number of descendants of x, including x itself ).

Define a relation R < sub > L </ sub > on strings by the rule that x R < sub > L </ sub > y if there is no distinguishing extension for x and y.

Define a subbase of open sets G < sub > x </ sub > for any integer x to be G < sub > x </ sub > = O < sub >

* Define a distance function between any point x of M and any non-empty set Y of M by:

Define, for all x in E,.

Define e < sup > x </ sup > by the limit

Define e < sup > x </ sup > as the value of the infinite series

* Attribute Types — Define an object identifier ( OID ) and a set of names that may be used to refer to a given attribute, and associates that attribute with a syntax and set of matching rules.

* Name Forms — Define rules for the set of attributes that should be included in the RDN for an entry.

* Content Rules — Define additional constraints about the object classes and attributes that may be used in conjunction with an entry.

* Define to be the minimum number of moves required to transfer n disks using r pegs

Define C < sub > n </ sub >( X ) for natural n to be the free abelian group formally generated by singular n-simplices in X, and define the boundary map

Define G to be the greatest common divisor of P and P ’.

Define to be the position of the cart.

Let W be a linear subspace of V. Define the symplectic complement of W to be the subspace

Consider T to be a differentiable multilinear map of smooth sections α < sup > 1 </ sup >, α < sup > 2 </ sup >, ..., α < sup > q </ sup > of the cotangent bundle T * M and of sections X < sub > 1 </ sub >, X < sub > 2 </ sub >, ... X < sub > p </ sub > of the tangent bundle TM, written T ( α < sup > 1 </ sup >, α < sup > 2 </ sup >, ..., X < sub > 1 </ sub >, X < sub > 2 </ sub >, ...) into R. Define the Lie derivative of T along Y by the formula

Let M be a smooth manifold and an open cover of M. Define the disjoint union with the obvious submersion.

# Define a set of Web service orchestration concepts that are meant to be used by both the external ( abstract ) and internal ( executable ) views of a business process.

Define the 1-skeleton to be the identification space obtained from the union of the 0-skeleton, 1-cells, and the identification of points of boundary of 1-cells by assigning an identification mapping from the boundary of the 1-cells into the 1-cells.

Let X be a vector field on M. Choose any right-invariant lift to the tangent bundle of P. Define

Define γ < sub > i </ sub > to be three if α < sub > i </ sub > is odd, and zero otherwise, and define β < sub > i </ sub >

Define the th residual to be

Define an n-set to be a set V with cardinality.

Define the Weyl algebra W ( V ) to be

and let L * be the adjoint of L. Define the elliptic operator Δ = LL * + L * L. As in the de Rham case, this yields the vector space of harmonic sections

:: Kakeya set conjecture: Define a Besicovitch set in R < sup > n </ sup > to be a set which contains a unit line segment in every direction.

Define to be the cylinder of length 1, radius, centered at the point a ∈ R < sup > n </ sup >, and whose long side is parallel to the direction of the unit vector e ∈ S < sup > n-1 </ sup >.

Define an ( n, k )-Besicovitch set K to be a compact set in R < sup > n </ sup > containing a translate of every k-dimensional unit disk which has Lebesgue measure zero.