Let Af denote the form of Af.

Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

Let X denote a Cauchy distributed random variable.

Let w denote the weight per unit length of the chain, then the weight of the chain has magnitude

Let denote the equivalence class to which a belongs.

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set.

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

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Let R denote the field of real numbers.

Let n denote a complete set of ( discrete ) quantum numbers for specifying single-particle states ( for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.

Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.

Let denote the space of scoring functions.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Let us denote the time at which it is decided that the compromise occurred as T.

Let denote the sequence of convergents to the continued fraction for.

Let us denote the mutually orthogonal single-particle states by and so on.

That is, Alice has one half, a, and Bob has the other half, b. Let c denote the qubit Alice wishes to transmit to Bob.

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

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 Q ( x ) denote the number of square-free ( quadratfrei ) integers between 1 and x.

A possible definition of spoiling based on vote splitting is as follows: Let W denote the candidate who wins the election, and let X and S denote two other candidates.

Let π < sub > 2 </ sub >( x ) denote the number of primes p ≤ x such that p + 2 is also prime.

Let be a sequence of independent and identically distributed variables with distribution function F and let denote the maximum.

: Let L be a partially ordered set with the smallest element ( bottom ) and let f: L → L be an order-preserving function.

Let n ( p ; H ) be the smallest number of values we have to choose, such that the probability for finding a collision is at least p. By inverting this expression above, we find the following approximation

Let m be the number of elements in the smallest row of our graph ( m

Let denote the smallest s so that there exists an addition chain

Let p < sub > 1 </ sub > be the smallest prime greater than B < sub > 1 </ sub >, p < sub > 2 </ sub > the next-largest, and so on ; let d < sub > n </ sub > = p < sub > n </ sub > − p < sub > n − 1 </ sub >.

Take the smallest i such that the a < sub > i </ sub > divides some term of g. Let h be the largest ( again with respect to the monomial ordering ) term of g which is divisible by a < sub > i </ sub >, and replace g by g − ( h / a < sub > i </ sub > ) f < sub > i </ sub >.

Let ( P, l ) be the point and connecting line that are the smallest positive distance apart among all point-line pairs.

* Let R be a local ring and M a finitely generated module over R. Then the projective dimension of M over R is equal to the length of every minimal free resolution of M. Moreover, the projective dimension is equal to the global dimension of M, which is by definition the smallest integer i ≥ 0 such that

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

Let k be an integer, and consider the integral

Let k be an integer that counts the steps of the algorithm, starting with zero.

Sylows ' test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n.

Let p be an odd prime and a an integer coprime to p. Then

Let be a ( positive ) integer and consider the elliptic curve ( a set of points with some structure on it ).

Let z be a primitive nth root of unity and let k be a positive integer.

Given an integer n, choose some integer a < n. Let 2 < sup > s </ sup > d = n − 1 where d is odd.

Let denote an integer number ; the next step is to gain the idea of the reciprocal of, not as but simply as.

Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.

Let k be a non-negative integer.

Let n be a positive integer.

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

Let V be a vector space over a field K. For any nonnegative integer k, we define the k < sup > th </ sup > tensor power of V to be the tensor product of V with itself k times:

Let Q denote the set of rational numbers, and let d be a square-free integer ( i. e., a product of distinct primes ) other than 1.

Let " Bankroll " be the amount of money a gambler has at his disposal at any moment, and let N be any positive integer.

Let be a finite Galois extension of number fields with rings of integer.

Let X be a non-negative integer.

Let Inv ( a, b ) denote the multiplicative inverse of a modulo b, namely the least positive integer m such that.

Let Λ be the lattice in R < sup > d </ sup > consisting of points with integer coordinates ; Λ is the character group, or Pontryagin dual, of R < sup > d </ sup >.

* Let k be a positive integer.

Let M be an n-dimensional manifold, and let k be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ S there exists a chart ( U ⊂ M, φ: U → R < sup > n </ sup >) containing p such that φ ( S ∩ U ) is the intersection of a k-dimensional plane with φ ( U ).

Let n be a composite integer with prime factor p. By Fermat's little theorem, we know that for all integers a coprime to p and for all positive integers K: