Let V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

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.

Let P ( α ) be a property defined for all ordinals α.

Let us consider two patterns with the same step p, but the second pattern is turned by an angle α.

Let be the coordinates of a rotation by α around the axis as previously described.

Let α and β be two differential forms on M, and let X and Y be two vector fields.

Let be an arbitrary real m-dimensional column vector of such that |||| = | α | for a scalar α.

Let α be the next n digits of the radicand, and β be the next digit of the root.

Let α be the phase of the first input and β be the phase of the second.

Let ( f < sub > 1 </ sub >, …, f < sub > k </ sub >) be another smooth local frame over U and let the change of coordinate matrix be denoted t ( i. e. f < sub > α </ sub >

Let R be the radius of the circle, θ is the central angle in radians, α is the central angle in degrees, c the chord length, s the arc length, h the height of the segment, and d the height of the triangular portion.

Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

Let f < sub > 1 </ sub >: ( X < sub > 1 </ sub >, α < sub > 1 </ sub >) → ( Y < sub > 1 </ sub >, β < sub > 1 </ sub >) and f < sub > 2 </ sub >: ( X < sub > 2 </ sub >, α < sub > 2 </ sub >) → ( Y < sub > 2 </ sub >, β < sub > 2 </ sub >) be morphisms of motives.

Let K be a field and L a finite extension ( and hence an algebraic extension ) of K. Multiplication by α, an element of L, is a K-linear transformation

* Let α be a real number greater than 1 and there exists a non-zero real number λ such that

Let α be a root of f ; we can then form the ring Z.

Let e =( e < sub > α </ sub >)< sub > α = 1, 2 ,..., k </ sub > be a local frame on E. This frame can be used to express locally any section of E. For suppose that ξ is a local section, defined over the same open set as the frame e, then

# Let an angle ( h, k ) be given in the plane α and let a straight line a ′ be given in a plane α ′.

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 us for simplicity take m = k as an example.

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

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

Let the directrix be the line x = − p and let the focus be the point ( p, 0 ).

Let be a non-negative real-valued function of the interval, and let < math > S =

Let us assume the bias is V and the barrier width is W. This probability, P, that an electron at z = 0 ( left edge of barrier ) can be found at z = W ( right edge of barrier ) is proportional to the wave function squared,

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

LET x = rnd * 20! Let the value ' x ' equal a random number between ' 0 ' and ' 20 '

LET y = rnd * 20! Let the value ' y ' equal a random number between ' 0 ' and ' 20 '

Let A =

:: Let n = 0

:: Let repeat = TRUE

Let ( S, f ) be a game with n players, where S < sub > i </ sub > is the strategy set for player i, S = S < sub > 1 </ sub > × S < sub > 2 </ sub > ... × S < sub > n </ sub > is the set of strategy profiles and f =( f < sub > 1 </ sub >( x ), ..., f < sub > n </ sub >( x )) is the payoff function for x S. Let x < sub > i </ sub > be a strategy profile of player i and x < sub >- i </ sub > be a strategy profile of all players except for player i. When each player i < nowiki >

Let X = " to make something that its maker cannot lift ".

* Let TQBF =

* The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I < sub > n </ sub > be the ideal of all continuous functions f such that f ( x ) = 0 for all x ≥ n. The sequence of ideals I < sub > 0 </ sub >, I < sub > 1 </ sub >, I < sub > 2 </ sub >, etc., is an ascending chain that does not terminate.

# Let p = ( p < sub > 1 </ sub >, p < sub > 2 </ sub >) and q = ( q < sub > 1 </ sub >, q < sub > 2 </ sub >) be elements of W, that is, points in the plane such that p < sub > 1 </ sub > = p < sub > 2 </ sub > and q < sub > 1 </ sub > = q < sub > 2 </ sub >.

Let β be the constant bearing from true north of the loxodrome and be the longitude where the loxodrome passes the equator.

Let φ be the sentence β (< u >#( β )</ u >).

Let the variance-covariance matrix for the observations be denoted by M and that of the parameters by M < sup > β </ sup >.

Let GF ( p < sup > m </ sup >) be a field with p < sup > m </ sup > elements, and β an element of it such that the m elements

Let a rigid object move along a regular curve described parametrically by β ( t ).

Let α and β be partial transformations of a set

Let the acute angle between X and Y be α, Y and Z be β.