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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 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 :=
Let T < sub > ij </ sub > := e < sub > ij </ sub >( 1 ) be the elementary matrix with 1's on the diagonal and in the ij position, and 0's elsewhere ( and i ≠ j ).
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 ).
Let A :=

R and :=
The latter example leads to a generalization of modules over rings: If C is a preadditive category, then Mod ( C ) := Add ( C, Ab ) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of ( left ) R-modules.
* a left principal ideal of R is a subset of R of the form Ra :=
P := YOUNGESTOFFSPRING ( FATHER ( FATHER ( R )));
M := YOUNGESTOFFSPRING ( MOTHER ( MOTHER ( R )));
Then k < sub > x </ sub > := R < sub > x </ sub >/ m < sub > x </ sub > is a field and m < sub > x </ sub >/ m < sub > x </ sub >< sup > 2 </ sup > is a vector space over that field ( the cotangent space ).
* Net present value of a net cash flow, R ( t ), is given by the one-form w ( t ) := ( 1 + i )< sup >− t </ sup > where i is the discount rate.
It is a vector space over the residue field k := R / m.
We compute the singular cohomology of X with coefficients in R := Z < sub > 2 </ sub >.
First we choose a peripheral vertex x and set R := (
left ( R ) := T
level ( R ) := level ( R ) + 1

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