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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 >.
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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 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.
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 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
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 ( 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 and be
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 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 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 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 it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let and ring
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.
Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.
Let M and N be ( left or right ) modules over the same ring, and let f: M → N be a module homomorphism.
Let denote the ring of integers ; that is, let be the set of integers equipped with its natural operations of addition and multiplication.
Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:
* The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I < sub > n </ sub > be the ideal of all continuous functions f such that f ( x ) = 0 for all x ≥ n. The sequence of ideals I < sub > 0 </ sub >, I < sub > 1 </ sub >, I < sub > 2 </ sub >, etc., is an ascending chain that does not terminate.
Let k be a field ( such as the rational numbers ) and K be an algebraically closed field extension ( such as the complex numbers ), consider the polynomial ring kX < sub > n </ sub > and let I be an ideal in this ring.
Let F be a field and p ( X ) be a polynomial in the polynomial ring F of degree n. The general process for constructing K, the splitting field of p ( X ) over F, is to construct a sequence of fields such that is an extension of containing a new root of p ( X ).
Let k be an algebraically closed field and let A < sup > n </ sup > be an affine n-space over k. The polynomials ƒ in the ring k ..., x < sub > n </ sub > can be viewed as k-valued functions on A < sup > n </ sup > by evaluating ƒ at the points in A < sup > n </ sup >, i. e. by choosing values in k for each x < sub > i </ sub >.
Let k be the field of complex numbers C. Let A < sup > 2 </ sup > be a two dimensional affine space over C. The polynomials f in the ring Cy can be viewed as complex valued functions on A < sup > 2 </ sup > by evaluating ƒ at the points in A < sup > 2 </ sup >.
Let again k be the field of complex numbers C. Let A < sup > 2 </ sup > be a two dimensional affine space over C. The polynomials g in the ring Cy can be viewed as complex valued functions on A < sup > 2 </ sup > by evaluating g at the points in A < sup > 2 </ sup >.
* Let k be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k over k.
Let V be an algebraic set defined as the set of the common zeros of an ideal I in a polynomial ring over a field K, and let A = R / I be the algebra of the polynomials over V. Then the dimension of V is:
On November 21, 2009, Flair returned to the ring as a heel on the " Hulkamania: Let the Battle Begin " tour of Australia, losing to Hulk Hogan in the main event of the first show by brassknuckles.
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