* 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 )

Let A ⊆ R → S be homomorphisms where R is not necessarily local ( one can reduce to that case however ), with A, S regular and R finitely generated as an A-module.

Let R ⊆ S be a map of complete local domains, and let Q be a height one prime ideal of S lying over xR, where R and R / xR are both regular.

:: Finite Field Kakeya Conjecture: Let F be a finite field, let K ⊆ F < sup > n </ sup > be a Kakeya set, i. e. for each vector y ∈ F < sup > n </ sup > there exists x ∈ F < sup > n </ sup > such that K contains a line

Let F be a functor from the category of ringed spaces to the category of sets, and let G ⊆ F. Suppose that this inclusion morphism G → F is representable by open immersions, i. e., for any representable functor Hom (−, X ) and any morphism Hom (−, X )→ F, the fibered product G ×< sub > F </ sub > Hom (−, X ) is a representable functor Hom (−, Y ) and the morphism Y → X defined by the Yoneda lemma is an open immersion.

Let C be a category, and let c be an object of C. A sieve S on c is a subfunctor of Hom (−, c ), i. e., for all objects c ′ of C, S ( c ′) ⊆ Hom ( c ′, c ), and for all arrows f: c ″→ c ′, S ( f ) is the restriction of Hom ( f, c ), the pullback by f ( in the sense of precomposition, not of fiber products ), to S ( c ′).

Let G ⊆ ℂ < sup > n </ sup > be a complex domain and f: G → ℂ be a C < sup > 2 </ sup > ( twice continuously differentiable ) function.

Let V be a vector space, and let F be the family of linearly independent subsets of V. Then F is a family of finite character ( because a subset X ⊆ V is linearly dependent iff X has a finite subset which is linearly dependent ).

Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain.

Let exactly 1'' '' of `` A '' extend beyond `` B '' and use a square to check your angle to exactly 90 degrees.

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

Gloria Gaynor ( born September 7, 1949 ) is an American singer, best known for the disco era hits ; " I Will Survive " ( Hot 100 number 1, 1979 ), " Never Can Say Goodbye " ( Hot 100 number 9, 1974 ), " Let Me Know ( I Have a Right )" ( Hot 100 number 42, 1980 ) and " I Am What I Am " ( R & B number 82, 1983 ).

Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD.

Let us call the particles A and B.

Let the input power to a device be a force F < sub > A </ sub > acting on a point that moves with velocity v < sub > A </ sub > and the output power be a force F < sub > B </ sub > acts on a point that moves with velocity v < sub > B </ sub >.

Let us take a hypothetical single scan line, with B representing a black pixel and W representing white:

Let A, B, C and D be the hexes that make up a rhombus, with A and C being the non-touching pair.

# Let e1 be an edge that is in A but not in B.

Let us call them A, B and C.

Other chart hits by White included " Never, Never Gonna Give Ya Up " (# 2 R & B, # 7 Pop in 1973 ), " Can't Get Enough of Your Love, Babe " (# 1 Pop and R & B in 1974 ), " You're the First, the Last, My Everything " (# 1 R & B, # 2 Pop in 1974 ), " What Am I Gonna Do with You " (# 1 R & B, # 8 Pop in 1975 ), " Let the Music Play " (# 4 R & B in 1976 ), " It's Ecstasy When You Lay Down Next to Me " (# 1 R & B, # 4 Pop in 1977 ) and " Your Sweetness is My Weakness " (# 2 R & B in 1978 ).

# Let B < sub > 1 </ sub >, B < sub > 2 </ sub > be base elements and let I be their intersection.

* Let B be a base for X and let Y be a subspace of X.

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 the two horses be horse A and horse B.

Let B contains all the sentences of A except ¬ φ.

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??