Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

Let R be a fixed commutative ring.

Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.

The Beatles ' 1968 track " Back in the U. S. S. R " references the instrument in its final verse (" Let me hear your balalaikas ringing out / Come and keep your comrade warm ").

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 R denote the field of real numbers.

Let R be an integral domain.

Let R be a domain and f a Euclidean function on R. Then:

Gloria Gaynor ( born September 7, 1949 ) is an American singer, best known for the disco era hits ; " I Will Survive " ( Hot 100 number 1, 1979 ), " Never Can Say Goodbye " ( Hot 100 number 9, 1974 ), " Let Me Know ( I Have a Right )" ( Hot 100 number 42, 1980 ) and " I Am What I Am " ( R & B number 82, 1983 ).

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 R be the quadratic mean ( or root mean square ).

Let R be a ring and G be a monoid.

* Let R :=

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 V be a vector space over a field K, and let be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field.

Let R be the set of all sets that are not members of themselves.

Let ( M, g ) be a Riemannian manifold and ƒ: M < sup > m </ sup > → R < sup > n </ sup > a short C < sup >∞</ sup >- embedding ( or immersion ) into Euclidean space R < sup > n </ sup >, where n ≥ m + 1.

Let U and V be two open sets in R < sup > n </ sup >.

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 >.

Let P < sub > F </ sub > be the domain of a prefix-free universal computable function F. The constant Ω < sub > F </ sub > is then defined as

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 M be a smooth manifold and let f ∈ C < sup >∞</ sup >( M ) be a smooth function.

Let e be the error in b. Assuming that A is a square matrix, the error in the solution A < sup >− 1 </ sup > b is A < sup >− 1 </ sup > e.

Let us for simplicity take, then < math > 0 < c =- 2a </ math > and.

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 r be a non zero real number and let the r < sup > th </ sup > power mean ( M < sup > r </ sup > ) of a series of real variables ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ... ) be defined as

Let t and s ( t > s ) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s < sup > 2 </ sup > equals half the harmonic mean of c < sup > 2 </ sup > and t < sup > 2 </ sup >.

Let M be a 2n × 2n matrix with real entries.

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2n, so its cohomology groups lie in degrees zero through 2n.

Let ( x < sub > i </ sub >( s ), u ( s ), p < sub > i </ sub >( s )) be a curve in R < sup > 2n + 1 </ sup >.

Let ( M, g, ω, J ) be an almost Hermitian manifold of real dimension 2n and let ∇ be the Levi-Civita connection of g. The following are equivalent conditions for M to be Kähler:

Genesis 1: 9 " And God said, Let the waters be collected ". Letters in black, < font color ="# CC0000 "> niqqud in red </ font >, < font color ="# 0000CC "> cantillation in blue </ font >

* Let D < sub > 1 </ sub > and D < sub > 2 </ sub > be directed sets.

Let us have more benches and fewer forbidden areas around fountains and gardens.

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

Let us now put some flesh on the theoretical bones we have assembled by giving illustrations of roleplaying used for evaluation and analysis.

Let this generation have theirs.

Onkelos, a righteous convert and authoritative commentator of the same period, explained the verse, " Let Reuben live and not die ..." ( Deuteronomy 33: 6 ) to mean that Reuben should merit the World to Come directly, and not have to die again as result of being reincarnated.

They have appeared together in a number of films, including Edie & Pen ( 1996 ), American Perfekt ( 1997 ) and Let the Devil Wear Black ( 1999 ).

Let Marcion then exhibit, as gifts of his god, some prophets, such as have not spoken by human sense, but with the Spirit of God, such as have both predicted things to come, and have made manifest the secrets of the heart ; let him produce a psalm, a vision, a prayer -- only let it be by the Spirit, in an ecstasy, that is, in a rapture, whenever an interpretation of tongues has occurred to him ; let him show to me also, that any woman of boastful tongue in his community has ever prophesied from amongst those specially holy sisters of his.

Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD.

Let ( A < sub > i </ sub >)< sub > i ∈ I </ sub > be a family of groups and suppose we have a family of homomorphisms f < sub > ij </ sub >: A < sub > j </ sub > → A < sub > i </ sub > for all i ≤ j ( note the order ) with the following properties:

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

' Let us obey the Biblical injunction: you of course, have the choice of natural means ; but as for me, I am afraid that there is no course open to me but the scientific way.

... Let the tears of agony we have shed, and the gloom of mourning throughout my command bear witness.

Indeed, the great mercy of God who desires that all men should be saved, and Jesus ' tenderness toward children which caused him to say: ' Let the children come to me, do not hinder them ,' allow us to hope that there is a way of salvation for children who have died without Baptism.

* Let A have spectral resolution

When the owners of the field come, they will say, ' Let us have back our field.

Let those who have been accustomed unjustly to wage private warfare against the faithful now go against the infidels and end with victory this war which should have been begun long ago.

Let those who for a long time, have been robbers, now become knights.

Let those who have been fighting against their brothers and relatives now fight in a proper way against the barbarians.

Let those who have been serving as mercenaries for small pay now obtain the eternal reward.

Let those who have been wearing themselves out in both body and soul now work for a double honor.

We didn't move at the beginning, but he says, ' Let me have these seats.