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 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.

This permits an alternative definition for the ( general ) radical of an ideal I in R. Define Rad ( I ) as the preimage of N ( R / I ), the nilradical of R / I, under the projection map R → R / I.

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

Define a map

Define a projection map π < sub > ρ </ sub >: E → X by π < sub > ρ </ sub >() = π ( p ).

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 a map

) Define a function ƒ from the nonzero complex numbers to the nonzero real numbers by

Define a data matrix, X < sup > T </ sup >, with zero empirical mean ( the empirical ( sample ) mean of the distribution has been subtracted from the data set ), where each of the n rows represents a different repetition of the experiment, and each of the m columns gives a particular kind of datum ( say, the results from a particular probe ).

Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite ( n-1 )- face ( facet ).

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.

** Thomas Newman and Peter Gabriel for Define Dancing, from WALL-E performed by various artists.

Define if for all and let denote the probability of a jump from to.

" The ability to name and classify the word and social concepts Ammi calls " The Power to Define ", which in the wrong hands is " one of the greatest weapons that can be used to control men and nations ," but is the key to salvation from past oppression.

Define a function d as the identity function over all elements over the manifold M, excepting a small neighbourhood H belonging to M. Over H d comes to differ from identity by a smooth function.

Define y ( t ) to be the distance of the vehicle heading North from the origin and x ( t ) to be the distance of the vehicle heading West from the origin.

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.

Define a function K recursively on the ordinals as follows:

Define the inverse different or Dedekind's complementary module as the set of x ∈ K such that tr ( xy ) is an integer for all y in O < sub > K </ sup >, then I is a fractional ideal of K containing O < sub > K </ sup >.

: is a basis for the ring of integers of K. Define a quadratic form

Define ν < sub > p </ sub >( n ) as the exponent of the highest power of the prime p that divides n. I. e.

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, where is the discrete valuation corresponding to the ideal P < sub > i </ sub >.

* Define a ≤ b as a < b or a =

* Define a b as " not b < a " ( i. e., take the inverse complement of the relation ), which corresponds to defining a ~ b as " neither a < b nor b < a "; these relations and ~ are in general not transitive ; however, if they are, ~ is an equivalence ; in that case "<" is a strict weak order.

Define the twin prime constant C < sub > 2 </ sub > as

Define C < sub > 1 </ sub > as the full subcategory of C consisting of those objects X of C for which ε < sub > X </ sub > is an isomorphism, and define D < sub > 1 </ sub > as the full subcategory of D consisting of those objects Y of D for which η < sub > Y </ sub > is an isomorphism.

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 now a strictly increasing chain < math >( A_

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

* 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.

Define a binary relation on the set of-terms such that if and only if ; and let denote the equivalence class of terms containing ; and let where is the set of terms based on the symbol set.

The essence of the Mediator Pattern is to " Define an object that encapsulates how a set of objects interact.

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

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

:: 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 an application interface using a set of pre-defined components ( forms, buttons, and so on )

Define to be the set of all consumption bundles strictly preferred to by consumer i, and let V be the sum of all.

* Define the File, a set of declarative macros used in IBM DOS / VSE