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Let and M

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Let M be a smooth manifold and let x be a point in M. Let T x M be the tangent space at x.

Let M be a smooth manifold and let x be a point in M. Let I x be the ideal of all functions in C ∞( M ) vanishing at x, and let I x 2 be the set of functions of the form, where f i , g i ∈ I x .

Let M be a smooth manifold and let f ∈ C ∞( M ) be a smooth function.

Let r be a non zero real number and let the r th power mean ( M r ) of a series of real variables ( a 1 , a 2 , a 3 , ... ) be defined as

Let M be a ( pseudo -) Riemannian manifold, which may be taken as the spacetime of general relativity.

Let M and N be ( left or right ) modules over the same ring, and let f: M → N be a module homomorphism.

Let M be an n × n Hermitian matrix.

Let P − 1 DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues.

Let ( M, g ) be a Riemannian manifold and ƒ: M m → R n a short C ∞- embedding ( or immersion ) into Euclidean space R n , where n ≥ m + 1.

Let M and N be smooth manifolds and be a smooth map.

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 the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

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 commutative

Let R be a fixed commutative ring.

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

Let g be a Lie algebra, h a maximal commutative Lie subalgebra consisting of semi-simple elements ( sometimes called Cartan subalgebra ) and let V be a finite dimensional representation of g. If g is semisimple, then g = g and so all weights on g are trivial.

Let A be a commutative Banach algebra, defined over the field C of complex numbers.

Let A be a commutative ring.

Let P be a finitely generated projective module over a commutative ring R and X be the spectrum of R. The rank of P at a prime ideal in X is the rank of the free-module.

Let R be any ring, not necessarily commutative.

Let R be a commutative ring and let A and B be R-algebras.

Let be a commutative ring with 1, e. g. ( Instead we can take to be a field and can replace by the field ).

Let K be a fixed commutative ring.

Let A be a superalgebra over a commutative ring K. The submodule A 0 , consisting of all even elements, is closed under multiplication and contains the identity of A and therefore forms a subalgebra of A, naturally called the even subalgebra.

Let R be a commutative superring.

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 A be a commutative Noetherian ring with unity.

Let R be a commutative ring.

Let R and S be commutative rings and φ: R → S a ring homomorphism.

Let R be a commutative ring with prime characteristic p ( an integral domain of positive characteristic always has prime characteristic, for example ).

Let R be a commutative ring with unity, and let M, N and L be three R-modules.

Let R be a commutative ring with identity 1.

* Let R ⊂ S be an integral extension of commutative rings, and P a prime ideal of R. Then there is a prime ideal Q in S such that Q ∩ R = P. Moreover, Q can be chosen to contain any prime Q 1 of S such that Q 1 ∩ R ⊂ P.

Let A ⊆ B be an extension of commutative rings.

Let A and B be two commutative rings with unity, and let f: A → B be a ( unital ) ring homomorphism.

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