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 X be the wedge sum of two spaces K and L, and suppose furthermore that the identified basepoint is a deformation retract of open neighborhoods U ⊂ K and V ⊂ L. Letting A

Let M be an n-dimensional manifold, and let k be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ S there exists a chart ( U ⊂ M, φ: U → R < sup > n </ sup >) containing p such that φ ( S ∩ U ) is the intersection of a k-dimensional plane with φ ( U ).

Let X and Y be Banach spaces, and A ⊂ X, B ⊂ Y a pair of open sets.

Let V be a finite dimensional complex vector space, let H ⊂ Aut ( V ) be an irreducible semisimple complex connected Lie subgroup and let K ⊂ H be a maximal compact subgroup.

Let X and Y be locally convex topological vector spaces, and U ⊂ X an open set.

Let X be an affine algebraic variety embedded into the affine space k < sup > n </ sup >, with the defining ideal I ⊂ k. For any polynomial f, let in ( f ) be the homogeneous component of f of the lowest degree, the initial term of f, and let in ( I ) ⊂ k be the homogeneous ideal which is formed by the initial terms in ( f ) for all f ∈ I, the initial ideal of I.

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 Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

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.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

Let S be a vector space over the real numbers, or, more generally, some ordered field.

* Let A be a commutative ring with unity and let S be a multiplicative subset of A.

Let ( S ,*) be a set S with a binary operation * on it ( known as a magma ).

* Bowman, Durrell S. " Let Them All Make Their Own Music: Individualism, Rush and the Progressive / Hard Rock Alloy ," in Progressive Rock Reconsidered, Kevin Holm-Hudson ( ed ), Routledge, 2002.

Let S be a set of languages that is nontrivial, meaning

Let be a non-negative real-valued function of the interval, and let < math > S =

Let S be the group of all permutations of N, the natural numbers, that fixes all but finitely many numbers then:

Their first album, Let Them Eat Bingo, included the number one single " Dub Be Good to Me ", which caused a legal dispute revolving around allegations of infringement of copyright through the liberal use of unauthorised samples: the bassline was a note-for-note lift from " The Guns of Brixton " by The Clash and the lyrics borrowed heavily from " Just Be Good to Me " by The S. O. S.

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 S be a subgroup of G, and let N be a normal subgroup of G. Then:

Let ( S, f ) be a game with n players, where S < sub > i </ sub > is the strategy set for player i, S = S < sub > 1 </ sub > × S < sub > 2 </ sub > ... × S < sub > n </ sub > is the set of strategy profiles and f =( f < sub > 1 </ sub >( x ), ..., f < sub > n </ sub >( x )) is the payoff function for x S. Let x < sub > i </ sub > be a strategy profile of player i and x < sub >- i </ sub > be a strategy profile of all players except for player i. When each player i < nowiki >