Let, denote a random sample from a distribution having the pdf f ( x, θ ) for γ < θ < δ.

Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )

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

Let γ be a differentiable curve in M with initial point γ ( 0 ) and initial tangent vector X

A connection ∇ on a vector bundle E → M defines a notion of parallel transport on E along a curve in M. Let γ: → M be a smooth path in M. A section σ of E along γ is said to be parallel if

Let γ be the boundary of B ( z < sub > 0 </ sub >, r ), taken with its positive orientation.

Let γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ: → M based at x in M, the connection defines a parallel transport map P < sub > γ </ sub >: E < sub > x </ sub > → E < sub > x </ sub >.

Let γ be the small circle of radius ε, Γ the larger, with radius R, then

Let γ ( s ) be a plane curve, parameterized by its arclength s. The unit tangent vector to the curve is, by virtue of the arclength parameterization,

Let γ be a periodic orbit through a point p and S be a local differentiable and transversal section of φ through p, called Poincaré section through p.

Let ( R, M, φ ) be a differentiable dynamical system with periodic orbit γ through p. Let

Let M be a connected Riemannian manifold and p a point of M. A map f defined on a neighborhood of p is said to be a geodesic symmetry, if it fixes the point p and reverses geodesics through that point, i. e. if γ is a geodesic and then It follows that the derivative of the map at p is minus the identity map on the tangent space of p. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M.

Let γ ( s ) be a regular parametric plane curve, where s is the arc length, or natural parameter.

Let T < sup > a </ sup > be the tangent vector to a given geodesic γ, and X < sup > a </ sup > a vector field along γ connecting it to an infinitesimally near geodesic ( the deviation vector ).

Let M be a pseudo-Riemannian manifold ( or any manifold with an affine connection ) and let p be a point in M. Then for every V in T < sub > p </ sub > M there exists a unique geodesic γ: I → M for which γ ( 0 )

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.