Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.

Let ( E < sub > n </ sub >) be a sequence of events in some probability space.

Let E be the set of real numbers that can be defined by a finite number of words.

Let p be the nth decimal of the nth number of the set E ; we form a number N having zero for the integral part and p + 1 for the nth decimal, if p is not equal either to 8 or 9, and unity in the contrary case.

Let further P < sub > Alice </ sub > denote the first plaintext block of Alice's message, let E denote encryption, and let P < sub > Eve </ sub > be Eve's guess for the first plaintext block.

E. g., " Let us lie in wait for the just, because he is not for our turn … He boasteth that he hath the knowledge of God, and calleth himself the son of God … and glorieth that he hath God for his father.

Let D, E and F be given on the lines BC, AC and AB so that the equation holds.

Let φ: X → Y be a continuous and absolutely continuous function ( where the latter means that ρ ( φ ( E )) = 0 whenever μ ( E ) = 0 ).

Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.

With features from Destiny's Child, Juelz Santana, Jim Jones, N. O. R. E., and producer Digga, it included the relatively successful singles, " Let Me Know " and " What Means The World To You ".

* Larry E. Williams was a roadie for Neil Diamond before writing Let Your Love Flow.

Let ƒ be measurable, E (| ƒ |) < ∞, and T be a measure-preserving map.

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

Let E and F be two vector bundles over a differentiable manifold M. An R-linear mapping of sections is said to be a kth-order linear differential operator if it factors through the jet bundle J < sup > k </ sup >( E ).

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

* Let Something Good Be Said: Speeches and Writings of Frances E. Willard, ed.

Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes.

Let B equal S < sup > 2 </ sup > and E equal S < sup > 3 </ sup >.

Let A be a set ( of the elements of an algebra ), and let E be an equivalence relation on the set A.

Let us label with s ( s = 1, 2, 3, ...) the exact states ( microstates ) that the system can occupy, and denote the total energy of the system when it is in microstate s as E < sub > s </ sub >.

It features: " The Demon ", " Thunderbolt ", " Dearth ", " Knuckles ", " Star Song ", " Firepower ", " New Waver ", " Space Jam ", " Zoom ", " So Very Sad About Us ", " Phang ", " Speed Racer ", " The Eternal E ", " Hairy Eyeball ", " The Groover ", " Hell Bent for Hell ", " Rachel ", " A Dog's Prayer ", " Blast ", " The Black Rider ", " Slurpee ", " Flipper ", " The Viper ", " Bitch ", " Fried ", " Harmonia ", " U. S. A .", " The Tracer ", " Envelope Woman ", " Plastic Guy ", " Glasgow 3am ", " The Road Is Long ", " Funkified ", " Rigamarole ", " Depresso ", " The Streets Are Hot Tonite ", " Dawn At 16 ", " Spazmatazz ", " Fucker ", " In the Arms of Sheep ", " Speed ", " 77 ", " Me Rock You Snow ", " Feelium ", " Is Alex Milton ", " Rubberman ", " Spacer ", " Rock Me ", " Weeping Willowly ", " Rings ", " So So Pretty ", " Lucky Lad ", " Jackboot ", " Milieu ", " Disconnected ", " Let Your Lazer Love Light Shine Down ", " Phreak ", " Porkbelly ", " Robot Lover ", " Jimmy James ", " America ", " Slinkeepie ", " Dummy Tum Tummy ", " Fakir ", " Jake ", " Camaro ", " Moonkids ", " Make It Fungus ", " V-8 ", and " Die ".

Let the S be a sphere with center O, P a plane which intersects S. Draw perpendicular to P and meeting P at E., Let A and B be any two points in the intersection.

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.

: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )

Let ( A < sub > i </ sub >)< sub > i ∈ I </ sub > be a family of groups and suppose we have a family of homomorphisms f < sub > ij </ sub >: A < sub > j </ sub > → A < sub > i </ sub > for all i ≤ j ( note the order ) with the following properties:

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

Let V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

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

Let F: J → C be a diagram of type J in a category C. A cone to F is an object N of C together with a family ψ < sub > X </ sub >: N → F ( X ) of morphisms indexed by the objects of J, such that for every morphism f: X → Y in J, we have F ( f ) o ψ < sub > X </ sub >

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 J be a directed poset ( considered as a small category by adding arrows i → j if and only if i ≤ j ) and let F: J < sup > op </ sup > → C be a diagram.

Let F: J → C be a diagram.

Let G and H be groups, and let φ: G → H be a homomorphism.

Let T: X → X be a contraction mapping on X, i. e.: there is a nonnegative real number q < 1 such that

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 U be an open subset of R < sup > n </ sup > and f: U → R a function.

Let U, V, and W be vector spaces over the same field with given bases, S: V → W and T: U → V be linear transformations and ST: U → W be their composition.

Let x < sub > t </ sub > be a curve in a Riemannian manifold M. Denote by τ < sub > x < sub > t </ sub ></ sub >: T < sub > x < sub > 0 </ sub ></ sub > M → T < sub > x < sub > t </ sub ></ sub > M the parallel transport map along x < sub > t </ sub >.

Let f: < sup > n </ sup > → be the fitness or cost function which must be minimized.

Let f: D → R be a function defined on a subset D of the real line R. Let I = b be a closed interval contained in D, and let P =

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 < sub > x </ sub > M be the tangent space at x.

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

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 < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ... ) be defined as

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

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 and N be smooth manifolds and be a smooth map.