Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

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 g be a smooth function on N vanishing at f ( x ).

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

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

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

Let x < sub > 0 </ sub >, ...., x < sub > N-1 </ sub > be complex numbers.

Let now x ' and y ' be tuples of previously unused variables of the same length as x and y respectively, and let Q be a previously unused relation symbol which takes as many arguments as the sum of lengths of x and y ; we consider the formula

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 the quantities u and v be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions.

Let u, v be arbitrary vectors in a vector space V over F with an inner product, where F is the field of real or complex numbers.

Let the curve be a unit speed curve and let t = u × T so that T, u, t form an orthonormal basis: the Darboux frame.

Let m < sub > 1 </ sub > and m < sub > 2 </ sub > be the masses, u < sub > 1 </ sub > and u < sub > 2 </ sub > the velocities before collision, and v < sub > 1 </ sub > and v < sub > 2 </ sub > the velocities after collision.

Let Y = u ( X < sub > 1 </ sub >, X < sub > 2 </ sub >, ..., X < sub > n </ sub >) be a statistic whose pdf is g ( y ; θ ).

Let u = x < sup > 3 </ sup >.

Let y be a function given by the sum of two functions u and v, such that:

Let x: y: z be a variable point in trilinear coordinates, and let u

Let vectors < u > a </ u >, < u > b </ u >, < u > c </ u > and < u > h </ u > determine the position of each of the four orthocentric points and let < u > n </ u > = (< u > a </ u > + < u > b </ u > + < u > c </ u > + < u > h </ u >) / 4 be the position vector of N, the common nine-point center.

Let the input power to a device be a force F < sub > A </ sub > acting on a point that moves with velocity v < sub > A </ sub > and the output power be a force F < sub > B </ sub > acts on a point that moves with velocity v < sub > B </ sub >.

Let the ship moves with velocity v. In the ship reference frame, capturing the 4 hydrogen we losing a momentum:

Let A be an m × n matrix, with column vectors v < sub > 1 </ sub >, v < sub > 2 </ sub >, ..., v < sub > n </ sub >.

Let v be the last vertex before u on this path.

Here is Leibniz's argument: Let u ( x ) and v ( x ) be two differentiable functions of x.

Let f = uv and suppose u and v are positive functions of x.

Let u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers.

Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )

Let v be an arbitrary vector in V. There exist unique scalars such that:

Let S be the number of particles in the swarm, each having a position x < sub > i </ sub > ∈ < sup > n </ sup > in the search-space and a velocity v < sub > i </ sub > ∈ < sup > n </ sup >.

Proof: Let ( v, λ ) be an eigenvector-eigenvalue pair for a matrix A.

Let s be the function mapping the sphere to itself, and let v be the tangential vector function to be constructed.