* 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 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 A ⊆ B be an extension of commutative rings.

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 the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

Let R be a fixed commutative ring.

Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.

The Beatles ' 1968 track " Back in the U. S. S. R " references the instrument in its final verse (" Let me hear your balalaikas ringing out / Come and keep your comrade warm ").

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.

Let R denote the field of real numbers.

Let R be an integral domain.

Let R be a domain and f a Euclidean function on R. Then:

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 us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable.

Let R be the quadratic mean ( or root mean square ).

Let R be a ring and G be a monoid.

* Let R :=

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

Let R < sup > 2n </ sup > have the basis

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

Let V be a vector space over a field K, and let be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field.

Let R be the set of all sets that are not members of themselves.

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 and V be two open sets in R < sup > n </ sup >.

: 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 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 =