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??

Let φ: M × R → M be the one-parameter semigroup of local diffeomorphisms of M induced by the vector flow of Y and denote φ < sub > t </ sub >( p ) := φ ( p, t ).

The frame bundle F ( E ) can be given a natural topology and bundle structure determined by that of E. Let ( U < sub > i </ sub >, φ < sub > i </ sub >) be a local trivialization of E. Then for each x ∈ U < sub > i </ sub > one has a linear isomorphism φ < sub > i, x </ sub >: E < sub > x </ sub > → R < sup > k </ sup >.

Let ( f < sub > 1 </ sub >, …, f < sub > k </ sub >) be another smooth local frame over U and let the change of coordinate matrix be denoted t ( i. e. f < sub > α </ sub >

Let τ < sub > 1 </ sub > and τ < sub > 2 </ sub > be two topologies on a set X and let B < sub > i </ sub >( x ) be a local base for the topology τ < sub > i </ sub > at x ∈ X for i

Let R be a ( Noetherian, commutative ) regular local ring and P and Q be prime ideals of R. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra.

Let Q = ƒ ( P ) and let t be a local uniformizing parameter at P ; that is, t is a regular function defined in a neighborhood of Q with t ( Q ) = 0 whose differential is nonzero.

Let A ⊆ R → S be homomorphisms where R is not necessarily local ( one can reduce to that case however ), with A, S regular and R finitely generated as an A-module.

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 finite Galois extension of local fields with group and finite residue fields.

Let ( R, m ) be a Noetherian local ring with maximal ideal m, and let M be a finitely-generated R-module.

Haru later meets a diverse group of allies, including Hamrio Musica, grandson of a local blacksmith ; Let Dahaka and Julia, two who appear human but are in fact of the Dragon Race ; Griffon Kato, a strange blue creature and Plue's friend ; Ruby, a penguin and a casino owner ; Belnika, a mage who can control the substance Etherion ; and Niebel, Sieg's close friend.

Let be a local section of orthonormal bases.

Let the local coordinates be called.

Let E be a vector bundle of fibre dimension k over a differentiable manifold M. A local frame for E is an ordered basis of local sections of E.

Let e =( e < sub > α </ sub >)< sub > α = 1, 2 ,..., k </ sub > be a local frame on E. This frame can be used to express locally any section of E. For suppose that ξ is a local section, defined over the same open set as the frame e, then

Let be a local field with discrete valuation and let

By 1992, they started a new band, The Jim Jims, with guitarist Daniel Herring .< ref name =" JJJ "> The Jim Jims provided a cover version of The Velvet Underground's " Heroin " on Check This Action ... Let the Fun Begin, a compilation CD of local Geelong bands released in 1992.

Let m be the dimension of M and, in some local chart, consider the standard coordinate vector fields

7 .↑ Pierre Gauthier Vendean linguist ( from Saint-Vincent-sur-Jar ), professeur at the university of Nantes, ( in Langue et littérature: La langue régionale: Les parlers vendéens dans l ’ espace linguistique poitevin-saintongeais, in: Vendée, Encyclopédie Bonneton: written with Guy Perraudeau ), using the Vendean example, restates in 2003: « Let ’ s recall first that the Vendée, before being a department, was under the Ancien Régime ( pre-revolution monarchy ), what we called « Bas-Poitou » ( Lower Poitou ) and to understand what are the Vendean dialects, their origins, their life, their future, it ’ s necessary to situate them in a bigger linguistic, cultural and historic area, the one bordered by the Loire and the Gironde on one part, the Atlantic Ocean and the Massif Central on another part, where are still alive in rural areas local dialects with a sufficient coherence to constitute a minority language, the Poitevin-Saintongese one .”

Let γ be a periodic orbit through a point p and S be a local differentiable and transversal section of φ through p, called Poincaré section through p.

Let denote the set of all local sections whose domain contains.