Let Af denote the form of Af.

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 Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

Let C be a plane curve ( the precise technical assumptions are given below ).

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 x < sub > t </ sub > be a curve in a Riemannian manifold M. Denote by τ < sub > x < sub > t </ sub ></ sub >: T < sub > x < sub > 0 </ sub ></ sub > M → T < sub > x < sub > t </ sub ></ sub > M the parallel transport map along x < sub > t </ sub >.

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

Let C be a non-singular algebraic curve of genus g over Q.

Let C be a positively oriented, piecewise smooth, simple closed curve in the plane < sup > 2 </ sup >, and let D be the region bounded by C. If L and M are functions of ( x, y ) defined on an open region containing D and have continuous partial derivatives there, then

Let E → M be a vector bundle with covariant derivative ∇ and γ: I → M a smooth curve parameterized by an open interval I.

Let γ be a differentiable curve in M with initial point γ ( 0 ) and initial tangent vector X

Let C be a simple closed curve on a sphere of radius 1.

A connection ∇ on a vector bundle E → M defines a notion of parallel transport on E along a curve in M. Let γ: → M be a smooth path in M. A section σ of E along γ is said to be parallel if

Let X be a curve of genus g defined over the finite field with p elements.

Let be the pink curve, and let be the red one.

Let P = ( x, y ) be a point on a given curve with A = ( x, 0 ) its projection onto the x-axis.

Let P = ( r, θ ) be a point on a given curve defined by polar coordinates and let O denote the origin.

Let γ: → M be a differentiable curve with γ ( 0 )

Indeed, let K be an imaginary quadratic field with class field H. Let E be an elliptic curve with complex multiplication by the integers of K, defined over H. Then the maximal abelian extension of K is generated by the x-coordinates of the points of finite order on some Weierstrass model for E over H.

Let the elliptic curve E have no complex multiplication.

Let r ( t ) be a curve in Euclidean space, representing the position vector of the particle as a function of time.

Let s ( t ) represent the arc length which the particle has moved along the curve.

Let E be an elliptic curve with integer coefficients in a Neron minimal model.

Suppose that the Fermat equation with exponent &# 8467 ; ≥ 3 had a solution in non-zero integers a, b, c. Let us form the corresponding Frey curve E. It is an elliptic curve and one can show that its discriminant Δ is equal to 16 ( abc )< sup > 2 &# 8467 ;</ sup > and its conductor N is the radical of abc, i. e. the product of all distinct primes dividing abc.