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.

The sentiment is summarized in a line from Ovid's Amores I. 1. 27 Sex mihi surgat opus numeris, in quinque residat-" Let my work rise in six steps, fall back in five.

Let the directrix be the line x = − p and let the focus be the point ( p, 0 ).

Let this point be called G. Point G is also the midpoint of line segment FQ:

Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is, so when it bounces off, its angle of inclination must be equal to.

Let us take a hypothetical single scan line, with B representing a black pixel and W representing white:

Prince Hans-Adam II of Liechtenstein, speaking to the International Institute for Strategic Studies on 25 January 2001, argued for a more flexible approach to territorial integrity, in line with historical norms, saying: Let us accept the fact that states have lifecycles similar to those of human beings who created them.

Let K be the set C of all complex numbers, and let V be the set C < sub > C </ sub >( R ) of all continuous functions from the real line R to the complex plane C.

For similar reasons, RCA also substituted altered versions for two other tracks: " Let Me In ", changing the line " you shut your door ; you know where " to " you shut your door ; now it ain't fair.

Although there are a few occasions in which members of one orchestra join the other, the full forces do not join together until the latter part of the last movement, when the tenor and baritone sing the final line of Owen's poem " Strange Meeting " (" Let us sleep now …") as " In Paradisum deducant " (" Into Paradise lead them ...") is sung first by the boys ' choir, then by the full choir ( in 8-part canon ), and finally by the soprano.

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.

Loudon Wainwright wrote ' Wine with dinner ', about excessive drinking, which includes the line, ' Let us spend a lost weekend under the volcano '.

Let f: D → R be a function defined on a subset D of the real line R. Let I = b be a closed interval contained in D, and let P =

Let 2a be the distance measured along the apse line from periapsis P to apoapsis A, as illustrated in the equation below:

:* The title of Go Let It Out may be a reference to the line in " Hey Jude ", " So let it out and let it in, hey, Jude, begin ( 2000 )

Let A, B, C denote the vertex angles of the reference triangle, and let x: y: z be a variable point in trilinear coordinates ; then an equation for the Euler line is

In a surprising ending, where Richard refuses to be captured and leaps down to his death with the " wrong " closing line " Let us to't pell-mell ; if not to heaven, then hand-in-hand to hell ", his falling into the inferno is followed by the eerily upbeat tune " I'm Sitting On The Top Of The World " ( Ray Henderson, Joe Young and Sam Lewis ) in the classic version sung by Al Jolson.

Let him continue our family line.

Let be the symmetry subgroup of for the medium.

Let us define the covariance matrix of the noise, reminding ourselves that this matrix has Hermitian symmetry, a property that will become useful in the derivation:

Let M be a connected Riemannian manifold and p a point of M. A map f defined on a neighborhood of p is said to be a geodesic symmetry, if it fixes the point p and reverses geodesics through that point, i. e. if γ is a geodesic and then It follows that the derivative of the map at p is minus the identity map on the tangent space of p. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M.

Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn ; AD and BC intersect chord PQ at X and Y correspondingly.

Let the angle bisector of angle A intersect side BC at a point D. The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC:

Let P < sub > 0 </ sub > be the point where AD and CF intersect.

Let us look in on one of these nerve centers -- SAC at Omaha -- and see what must still happen before a wing of B-52 bombers could drop their Aj.

Let us, like the French, have outdoor cafes where we may relax, converse at leisure and enjoy the passing crowd.

Let me say at the outset that the music sounded as sparkling on the last playing as it did on the first.

Let the body down slowly, taking at least five seconds for the letting down.

Let us not try to key them out at this stage of the game, and let us just call them Bombus.

Let us look at the operator Af.

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 suppose that the animal did not in fact fall into the trap and did not suffer at all, but that we mistakenly believe it did, and say as before that its suffering was an evil thing.

Let us look at the heavy-electrical-goods industry in which General Electric, Westinghouse and a number of other manufacturers were recently convicted of engaging in a conspiracy to rig prices and allocate the market.

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let the function g ( t ) be the altitude of the car at time t, and let the function f ( h ) be the temperature h kilometers above sea level.

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 ).

Ancient resources as well as recent archaeological evidence suggest that, at one point, Caligula had the palace extended to annex this structure. When several kings came to Rome to pay their respects to him and argued about their nobility of descent, he cried out " Let there be one Lord, one King ".

Peaking at number six in the UK, the album yielded the hits " Day-In, Day-Out " ( his 60th single ), " Time Will Crawl ", and " Never Let Me Down ".

It is unclear which version of the paradox is stronger .< ref group =" Note "> Let be the number of civilizations ( per unit volume ) that can be seen at a radius.

Let be the number of particles at position at time.

Let A ( k ) be its Fourier transform at time 0:

Another notable ( and defiant ) phrase in the speech was also spoken in German, " Lass ' sie nach Berlin kommen " (" Let them come to Berlin ")-- addressed at those who claimed " we can work with the Communists ", a remark which Nikita Khrushchev scoffed at only days later.

Ashcroft composed a paean called " Let the Eagle Soar " which he sang at the Gordon-Conwell Theological Seminary in February 2002, which was satirically featured in Michael Moore's 2004 movie Fahrenheit 9 / 11 and has been frequently mocked by comedians such as David Letterman, Stephen Colbert and David Cross, to name a few.