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 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 X be the unit Cartesian square ×, and let ~ be the equivalence relation on X defined by ∀ a, b ∈ (( a, 0 ) ~ ( a, 1 ) ∧ ( 0, b ) ~ ( 1, b )).

Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).

* Let be the set of ordered pairs of integers with not zero, and define an equivalence relation on according to which if and only if.

Let the symbols have the prescribed meaning, does or does not the equivalence still hold?

Let A be a set ( of the elements of an algebra ), and let E be an equivalence relation on the set A.

: Let X be a projective scheme over C. Then the functor associating the coherent sheaves on X to the coherent sheaves on the corresponding complex analytic space X < sup > an </ sup > is an equivalence of categories.

Let S a multiplicatively closed subset of R, i. e. for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S < sup >− 1 </ sup > M, is defined to be the following module: as a set, it consists of equivalence classes of pairs ( m, s ), where m ∈ M and s ∈ S. Two such pairs ( m, s ) and ( n, t ) are considered equivalent if there is a third element u of S such that

Thus Q defines an inner product on dom T. Let H < sub > 1 </ sub > be the completion of dom T with respect to Q. H < sub > 1 </ sub > is an abstractly defined space ; for instance its elements can be represented as equivalence classes of Cauchy sequences of elements of dom T. It is not obvious that all elements in H < sub > 1 </ sub > can identified with elements of H. However, the following can be proved:

Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector ; in other words all the vectors in N get mapped into the equivalence class of the zero vector.

Let D be the vector space of rational divisor classes on V, up to algebraic equivalence.

Let p be a point of M. Consider the space consisting of smooth maps defined in some neighborhood of p. We define an equivalence relation on as follows.

Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V. The set Hom ( Y, X ) of morphisms of algebraic spaces is then defined by the condition that it makes the descent sequence

Let us also assume the existence of only one class or type of service, all of which is supplied at the same voltage, phase, etc. to residential, commercial, and industrial customers.

Let us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable.

Let be a Gödel numbering of the computable functions ; a map from the natural numbers to the class of unary ( partial ) computable functions.

Let be the mean of the values in associated with class c, and let be the variance of the values in associated with class c. Then, the probability of some value given a class,, can be computed by plugging into the equation for a Normal distribution parameterized by and.

Let us assume that H < sub > f </ sub > is in this time complexity class, and we will attempt to reach a contradiction.

Let be a category and let be a class of morphisms of.

Let be a category, H a class of morphisms of ; the category is said to have enough H-injectives if for every object X of, there exist a H-morphism from X to an H-injective object.

Let Rp ( A, a ), meaning " the set a represents the class A ," denote a binary relation defined as follows:

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 M be a Banach manifold of class at least C < sup > 2 </ sup >.

* 2000: Spellemannprisen in the class Rock, for the album Let Them Eat Cake

When the nuclear-powered Arktika class icebreaker | Arktika class NS 50 Let Pobedy | NS 50 Let Pobedy was put into service in 2007, it became the world's largest icebreaker.

Let let a function of class and let The h-translation of, that is the function

Let be a class of combinatorial structures.

Let C be a curve of class m and let I and J denote the circular points at infinity.

Let P < sub > 1 </ sub >, P < sub > 2 </ sub >, …, P < sub > m </ sub > be given as foci of a curve C of class m. Let P be the product of the tangential equations of these points and Q the product of the tangential equations of the circular points at infinity.

Let σ be a signature consisting only of a unary function symbol f. The class K of σ-structures in which f is one-to-one is a basic elementary class.