Let-Z-be-a-complex-submanifold-of-X-of-dimension-k

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Let and Z

Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are f: X → Z and g: Y → Z.

Let: X Y Z where we assume in the typical case that and.

* Let Z be a random variable that takes the value-1 with probability 1 / 2, and takes the value 1 with probability 1 / 2.

* Let the base field F = Z / 2Z, the field of two elements

Let G be a finite simple undirected graph with edge set E. The power set of E becomes a Z 2 - vector space if we take the symmetric difference as addition, identity function as negation, and empty set as zero.

* A book in which the book itself seeks interaction with the reader ( e. g., Willie Masters ' Lonely Wife by William H. Gass, House of Leaves by Mark Z. Danielewski, or Don't Let the Pigeon Drive the Bus!

Let X, Y, Z be the ambient coordinates in R 3 .

Let G = Z 3 , the cyclic group of three elements with generator a.

Let Z n denote the state in period n ( often interpreted as the size of generation n ), and let X n, i be a random variable denoting the number of direct successors of member i in period n, where X n, i are iid over all n ∈

Let M m be the set of 2 × 2 integral matrices with determinant m and Γ = M 1 be the full modular group SL ( 2, Z ).

Let Z be the closed image of X, and let be the canonical injection.

Let α be a root of f ; we can then form the ring Z.

A binary image is viewed in mathematical morphology as a subset of a Euclidean space R d or the integer grid Z d , for some dimension d. Let E be a Euclidean space or an integer grid, A a binary image in E, and B a structuring element.

Let S be the shift operator on the sequence space ℓ ∞( Z ), which is defined by ( Sx ) i = x i + 1 for all x ∈ ℓ ∞( Z ), and let u ∈ ℓ ∞( Z ) be the constant sequence u i = 1 for all i ∈ Z.

* For any field F let M ( F ) denote the Moufang loop of unit norm elements in the ( unique ) split-octonion algebra over F. Let Z denote the center of M ( F ).

Let Z be the partition function.

Let the third random variable Z be equal to 1 if one and only one of those coin tosses resulted in " heads ", and 0 otherwise.

Let us establish the evident fact that the group Z 3 = Z / 3Z is indeed cyclic.

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 complex

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

Let x 0 , ...., x N-1 be complex numbers.

The saying " Let your Yes be Yes and your No be No " from James 5: 12 is interpolated into a sayings complex from Matthew 5: 34, 37.

Let u, v be arbitrary vectors in a vector space V over F with an inner product, where F is the field of real or complex numbers.

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 be a list of n linearly independent vectors of some complex vector space with an inner product.

Let K be the set C of all complex numbers, and let V be the set C C ( R ) of all continuous functions from the real line R to the complex plane C.

Let be a complex rational function from the plane into itself, that is,, where and are complex polynomials.

Consider an open subset U of the complex plane C. Let a be an element of U, and f: U

Let be an open subset of the complex plane, a point of and a holomorphic function defined on the set.

Let the complex number

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 n and let I be an ideal in this ring.

Let x, y, z be complex numbers, and let a, b be real numbers.

Let q be a prime number, s a complex variable, and define a Dirichlet L-function as

Let be the space of all complex valued Taylor series

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ B ( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

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 X be a projective complex manifold.

:: Let X be a projective complex manifold.

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