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

Let R < sup > 2n </ sup > have the basis

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 the open enemy to it be regarded as a Pandora with her box opened ; ;

Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.

`` Let him be now ''!!

Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af be the null space of Af.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

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.

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 us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.

Let this be denoted by Af.

Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

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

Let her out, let her out -- that would be the solution, wouldn't it??

One means to help the birds occurs to me: Let the chimes that ring over Washington Square twice daily, discontinue any piece of music but one.

Dear Sirs: Let me begin by clearing up any possible misconception in your minds, wherever you are.

Let S ( fig. 5 ) be any optical system, rays proceeding from an axis point O under an angle u1 will unite in the axis point O ' 1 ; and those under an angle u2 in the axis point O ' 2.

:: Let no-one deceive you by any means, for that day will not come unless the falling away comes first, and the man of sin is revealed, the son of perdition, who opposes and exalts himself above all that is called God or that is worshiped, so that he sits as God in the temple of God, showing himself that he is God. Thess 2: 3-4

The Supremes, The group that made Ross famous and without which there probably wouldn't be any Diana Ross, scored a handful of hits in the disco clubs without Ross, most notably 1976's " I'm Gonna Let My Heart Do the Walking " and, their last charted single before disbanding, 1977's " You're My Driving Wheel ".

Let f and g be any two elements of G. By virtue of the definition of G, = and =, so that =.

Let a piece of glass and a piece of resin, neither of which exhibiting any electrical properties, be rubbed together and left with the rubbed surfaces in contact.

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 V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

Let π ( x ) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x.

Let be any basis for the column space of and place them as column vectors to form the × matrix.

" We figured we could win if our offense didn't put us into too many holes ", said 35-year old Colts lineman Billy Ray Smith, who was playing in his last NFL game, " Let me put it this way, they didn't put us into any holes we couldn't get out of "

Let αω ∈ I be any such element.

Let X be a set ; the elements of X are usually called points, though they can be any mathematical objects.

At first La Folie charted lower than any other Stranglers studio album, and the first single taken from it, " Let Me Introduce You to the Family ", only charted at No. 42.

Let us therefore assume that this space without doubt belongs to the still undiscovered satellites of Mars, let us also add that perhaps Jupiter still has around itself some smaller ones which have not been sighted yet by any telescope.

Let f be any function from S to P ( S ).

Let K be R, C, or any field, and let V be the set P of all polynomials with coefficients taken from the field K.

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.

was the band's final album to feature any cover songs until 1970's Let It Be ( which included a performance of the traditional folk song " Maggie Mae ").

Cassius tried in all ways to reinforce the soldiers ’ loyalty both with strong speeches (" Let it give no one any concern that he has been one of Caesar's soldiers.

2: 3-12 says, " Let no one deceive you by any means ; for that Day will not come unless the falling away comes first, and the man of sin is revealed, the son of perdition, who opposes and exalts himself above all that is called God or that is worshiped, so that he sits as God in the temple of God, showing himself that he is God.

First there is the quote from 2 Thess. 3-4 :" Let no man deceive you by any means, for unless there come a revolt first, and the man of sin be revealed, the son of perdition, Who opposeth, and is lifted up above all that is called God, or that is worshipped, so that he sitteth in the temple of God, shewing himself as if he were God " which points to the Great Apostasy happening in the future, during the time of anti-Christ.

: Let our position be absolutely clear: An attempt by any outside force to gain control of the Persian Gulf region will be regarded as an assault on the vital interests of the United States of America, and such an assault will be repelled by any means necessary, including military force.