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Page "Poincaré duality" ¶ 10
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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 simplex
Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. We denote the dual cell ( to be defined precisely ) corresponding to S by DS.
Let be the vertices of a simplex in a vector space A.
Let be a semi-infinite Pascal's m-simplex and its n < sup > th </ sup > component, a finite ( m − 1 )- simplex with edge length n.

Let and T
* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )
Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.
Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.
Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.
Let us denote the time at which it is decided that the compromise occurred as T.
Let T be a linear operator represented by the matrix
Let V and W be vector spaces ( or more generally modules ) and let T be a linear map from V to W. If 0 < sub > W </ sub > is the zero vector of W, then the kernel of T is the preimage of the zero subspace
Let T: X → X be a contraction mapping on X, i. e.: there is a nonnegative real number q < 1 such that
Let T be a set consisting of all infinite sequences of 0s and 1s.
Let K be a closed subset of a compact set T in R < sup > n </ sup > and let C < sub > K </ sub > be an open cover of K. Then
Let the curve be a unit speed curve and let t = u × T so that T, u, t form an orthonormal basis: the Darboux frame.

Let and containing
Let be a maximally consistent set of-formulas containing witnesses.
Let denote the number of triangles containing the vertex.
Let M be the intersection of all subgroups of the free Burnside group B ( m, n ) which have finite index, then M is a normal subgroup of B ( m, n ) ( otherwise, there exists a subgroup g < sup >-1 </ sup > Mg with finite index containing elements not in M ).
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 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.
Following this, she recorded her second album for Word, Let There Be Love, containing a variety of tracks from up-tempo to an old Irish hymn sung in Gaelic.
After another five-year layoff, Zevon signed with industry veteran Danny Goldberg's Artemis Records and again rebounded with the mortality-themed 2000 release Life'll Kill Ya, containing the hymn-like " Don't Let Us Get Sick " and an austere version of Steve Winwood's 1980s hit " Back in the High Life Again ".
Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( 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 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
* is minimal: Let be any transitive relation containing, we want to show that.
Let if and only if ( where denotes the largest connected subset containing ).
The group's biggest breakthrough came with the album Keep It Live ( 1982 ), containing the hit " Let It Whip " — which reached No. 1 on the R & B charts and won a Grammy Award for Best R & B Performance by a Duo or Group with Vocals.
Let k be a field and L and K be two extensions of k. The compositum, denoted KL is defined to be where the right-hand side denotes the extension generated by K and L. Note that this assumes some field containing both K and L. Either one starts in a situation where such a common over-field is easy to identify ( for example if K and L are both subfields of the complex numbers ); or one proves a result that allows one to place both K and L ( as isomorphic copies ) in some large enough field.
Let M be an algebra of bounded operators on a Hilbert space H, containing the identity operator and closed under taking adjoints.
Let M be an n-dimensional manifold, and let k be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ S there exists a chart ( U ⊂ M, φ: U → R < sup > n </ sup >) containing p such that φ ( S ∩ U ) is the intersection of a k-dimensional plane with φ ( U ).
Let C and D be formulae each containing at least one instance of the subformula A:
* Let K < sup > a </ sup > be an algebraic closure of K containing L. Every embedding σ of L in K < sup > a </ sup > which restricts to the identity on K, satisfies σ ( L ) = L. In other words, σ is an automorphism of L over K.
Let X be a vector field on M of class C < sup > r − 1 </ sup > and let p ∈ M. An integral curve for X passing through p at time t < sub > 0 </ sub > is a curve α: J → M of class C < sup > r − 1 </ sup >, defined on an open interval J of the real line R containing t < sub > 0 </ sub >, such that
Let denote the vector space containing all matrices with rows and columns with entries in.
Let us suppose that somebody ( whom I am calling “ Charles ”) makes just the following five assertions containing the name “ Caesar .”
The single was released in Sweden with ' Let Go ' as the lead track and containing the b-sides from the first two UK CDs.
Let Ω < sup > 1, 0 </ sup > be the space of complex differential forms containing only's and Ω < sup > 0, 1 </ sup > be the space of forms containing only's.
Let be a finite set, containing n objects or letters.

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