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

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

Let n be the number of points and d the number of dimensions.

Let X be a finite set with n elements.

Let A be the arithmetic mean and H be the harmonic mean of n positive real numbers.

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 x < sub > 1 </ sub >, ..., x < sub > n </ sub > be the sizes of the heaps before a move, and y < sub > 1 </ sub >, ..., y < sub > n </ sub > the corresponding sizes after a move.

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

Let n be the length of a statement in Presburger arithmetic.

Let ( q < sub > 1 </ sub >, w, x < sub > 1 </ sub > x < sub > 2 </ sub >... x < sub > m </ sub >) ( q < sub > 2 </ sub >, y < sub > 1 </ sub > y < sub > 2 </ sub >... y < sub > n </ sub >) be a transition of the GPDA

Let be an oriented smooth manifold of dimension n and let be an n-differential form that is compactly supported on.

:: Let n = 0

Let be a random sample of size n — that is, a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ < sup > 2 </ sup >.

Let M be an n × n Hermitian matrix.

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 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.

In early November 2005, Staind released the limited edition 2-CD / DVD set of Chapter V. The set included several rarities and fan favorites — music videos ; a complete, 36-page booklet with exclusive artwork ; an audio disc with an acoustic rendition of " This is Beetle "; the original, melodic rendition of " Reply "; the previously released B-side singles " Novocaine " and " Let It Out "; and live versions of " It's Been Awhile " and " Falling ", among many others.

Let ( X, d ) be a non-empty complete metric space.

Let us suppose that L is a complete lattice and let f be a monotonic function from L into L. Then, any x ′ such that f ′( x ′) ≤ x ′ is an abstraction of the least fixed-point of f, which exists, according to the Knaster – Tarski theorem.

It was backed by the more traditional " Let The Juke Box Keep On Playing ," complete with fiddle, " Western Boogie " bass line, steel guitar and weepy vocal.

Among the more complete rehearsed songs that have been featured on bootlegs are " Watching Rainbows ", " Commonwealth ", Suzy's Parlor ( published under the name Suzy Parker ), and " The Palace of the King of the Birds " ( later recorded but not released by McCartney as " Castle of the King of the Birds "); " All Things Must Pass ", " Let It Down ", " Isn't It A Pity ", and " Hear Me Lord '", later released by Harrison ; " Gimme Some Truth " and " Oh My Love ", later released by Lennon ; and " Teddy Boy " and " Hot as Sun ", later released by McCartney.

Portions of the rooftop concert were seen in the Let It Be film and the Anthology documentary, and three tracks were used for the Let It Be album, while the complete recording has been bootlegged.

: Let L be a complete lattice and let f: L → L be an order-preserving function.

Some gadgets from the James Bond film series are featured in the game and are often used to complete particular mission objectives ; for example, in one level the electromagnetic watch from Live and Let Die is used to acquire a jail cell key.

Let us continue on in the direction in which we now are moving, and we shall see ... not a re-establishment of the Union, but a complete and final disruption of it.

Let R ⊆ S be a map of complete local domains, and let Q be a height one prime ideal of S lying over xR, where R and R / xR are both regular.

Let R → S be a local homomorphism of complete local domains.

Let be a complete lattice, with infimum and minimum symbolized by and, respectively.

Let be a complete lattice, with infimum and minimum symbolized by and, respectively.

Let q be a positive integer, and let G be the complete bipartite graph K < sub > q, q < sup > q </ sup ></ sub >.

Let and complete this to get an orthonormal basis at.

Let be a complete n-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound

It gives a complete history of The Beatles ' recording sessions, from 4 September 1962, shortly after the Beatles had returned to England from Germany, to 2 April 1970, when their final album, Let It Be, was mixed.

One of Kert's last recordings was the 1987 2-CD studio cast album of the complete scores of two George and Ira Gershwin musicals: Of Thee I Sing and its sequel Let ' Em Eat Cake.

: Let L be a complete partial order, and let f: L → L be a Scott-continuous ( and therefore monotone ) function.

The CD reissue of the year 2000 revised the cover art including a different concert photo of Elvis, omitted the track " Let Me Be There ", due to its presence on Elvis: As Recorded Live on Stage in Memphis, and it added the complete album From Elvis Presley Boulevard, Memphis, Tennessee as tracks 10-19-in effect compiling the Graceland sessions rather than reissuing the album.

More audio tracks from Liévin are available on the " In Your Room " singles (" Policy of Truth ", " World in My Eyes ", " Death's Door ", " Never Let Me Down Again " and " Fly on the Windscreen "), and the complete soundtrack of Liévin is now available on the re-release on 2DVD set of Devotional.

If the exaltation of the cross has been performed, after the veneration of the cross the augmented litany, " Have mercy on us, O God …," is not pronounced, as is usual at a vigil ; instead the supplicatory litany is immediately said: " Let us complete our morning prayer unto the Lord.

Let the Fire Burn, a complete acoustic album, was released on April 24, 2012.