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Let and R
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 and be
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 and fixed
Let f and g be functions, with and fixed.
Let us suppose also that a choice of positive roots Φ < sup >+</ sup > has been fixed.
Let L < sub > n </ sub > be the space of all complex n × n matrices, and let adX be the linear operator defined by adX Y = for some fixed X ∈ L < sub > n </ sub >.
Let be a fixed number, and let be the set of pairs of numbers whose product is at least.
Let be the set of non-negative integers ( natural numbers ), let n be any fixed constant, and let be the set of-tuples of natural numbers.
Let V be a ( fixed ) positive definite matrix of size p × p.
Let xyz system be fixed while the XYZ system rotates.
Let P be the point of application of the force F and let R be the vector locating this point in a fixed frame.
Let K be a fixed commutative ring.
Let be a fixed prime number.
Let H be a fixed hyperplane through the origin in C < sup > n + 1 </ sup >.
Let R be a ring and let Mod < sub > R </ sub > be the category of modules over R. Let B be in Mod < sub > R </ sub > and set T ( B ) = Hom < sub > R </ sub >( A, B ), for fixed A in Mod < sub > R </ sub >.
Let the likelihood function be considered fixed ; the likelihood function is usually well-determined from a statement of the data-generating process.
Let f be a fixed integrable function and let T be the operator of convolution with f, i. e., for each function g we have
Let be the subalgebra of fixed points in under the adjoint action of G, so that for instance
Let δ < sub > x </ sub > denote the Dirac measure centred on some fixed point x in some measurable space ( X, Σ ).
Let a set of basic propositional connectives be fixed ( for instance, in the case of superintuitionistic logics, or in the case of monomodal logics ).
Let N > 1 be a fixed integer and consider the polynomials f < sub > 1 </ sub >, ..., f < sub > N </ sub > in variables X < sub > 1 </ sub >, ..., X < sub > N </ sub > with coefficients in an algebraically closed field k ( in fact, it suffices to assume k = C ).
The minimal separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an ( a, b )- separator S can be regarded as a predecessor of another ( a, b )- separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two ( a, b )- separators in ' G '.
Let A be a set consisting of N distinct i-element subsets of a fixed set U (" the universe ") and B be the set of all ( i − r )- element subsets of the sets in A.
Let have fixed load factor.
Let be a fixed homology theory.

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