Let-M-be-a-smooth-manifold-of-dimension-n-for-in

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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 < sup >∞</ sup >( M ) vanishing at x, and let I x < sup > 2 </ sup > 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 < sup >∞</ sup >( M ) be a smooth function.

Let r be a non zero real number and let the r < sup > th </ sup > power mean ( M < sup > r </ sup > ) 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 < sup >− 1 </ sup > 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 < 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 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 smooth

Let g be a smooth function on N vanishing at f ( x ).

Let be an oriented smooth manifold of dimension n and let be an n-differential form that is compactly supported on.

Suppose we have an n-dimensional oriented Riemannian manifold, M and a target manifold T. Let be the configuration space of smooth functions from M to T. ( More generally, we can have smooth sections of a fiber bundle over M .)

Let x = x ( u, v, w ), y = y ( u, v, w ), z = z ( u, v, w ) be defined and smooth in a domain containing, and let these equations define the mapping of into.

Let be the sheaf of germs of smooth functions on M × M which vanish on the diagonal.

Let C be a positively oriented, piecewise smooth, simple closed curve in the plane < sup > 2 </ sup >, and let D be the region bounded by C. If L and M are functions of ( x, y ) defined on an open region containing D and have continuous partial derivatives there, then

Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2n, so its cohomology groups lie in degrees zero through 2n.

Let M be a smooth manifold.

Let E → M be a vector bundle with covariant derivative ∇ and γ: I → M a smooth curve parameterized by an open interval I.

Let M be a smooth manifold.

Let M be a smooth manifold and an open cover of M. Define the disjoint union with the obvious submersion.

Let X be a smooth projective variety where all of its irreducible components have dimension n. Then one has the following version of the Serre duality: for any locally free sheaf on X,

Let E → M be a smooth vector bundle over a differentiable manifold M. Denote the space of smooth sections of E by Γ ( E ).

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

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