Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of R < sup > n </ sup > to the manifold.

Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if

* Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.

Every smooth surface S has a unique affine plane tangent to it at each point.

Every Sámi settlement had its seita, which had no regular shape, and might consist of smooth or odd-looking stones picked out of a stream, of a small pile of stones, of a tree-stump, or of a simple post.

Every compact smooth manifold of dimension 2n, which has only handles of index ≤ n, has a Stein structure provided n > 2, and when n = 2 the same holds provided the 2-handles are attached with certain framings ( framing less than the Thurston-Bennequin framing ).

Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.

Every sufficiently smooth DAE is almost everywhere reducible to this semi-explicit index-1 form.

* Every smooth Hilbert manifold can be smoothly embedded onto an open subset of the model Hilbert space.

* Every continuous plurisubharmonic function can be obtained as a limit of monotonically decreasing sequence of smooth plurisubharmonic functions.

Every camp has one or two directors ; the job of the director is to make the camp run smooth and mostly safe without losing the free-spirited and cosy experience of the camp.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

Every simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal of R. By the above paragraph, any quotient R / m is a simple module.

Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.

Every Boolean ring R satisfies x ⊕ x

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

* Every left ideal I in R is finitely generated, i. e. there exist elements a < sub > 1 </ sub >, ..., a < sub > n </ sub > in I such that I = Ra < sub > 1 </ sub > + ... + Ra < sub > n </ sub >.

* Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.

Every year since 1982, the W. C. Handy Music Festival is held in the Florence / Sheffield / Muscle Shoals area, featuring blues, jazz, country, gospel, rock music and R & B.

Every adult citizen of this small settlement signed the small petition ; E. K Dyer and his wife, William Johnson, Joseph Otis and his wife, Hiram Walker and his wife, Joseph Pease and R. H. Valentine.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

Every bijective function g has an inverse g < sup >− 1 </ sup >, such that gg < sup >− 1 </ sup > = I ;

Every atom across this plane has an individual set of emission cones .</ p > < p > Drawing the billions of overlapping cones is impossible, so this is a simplified diagram showing the extents of all the emission cones combined.

Every ketose will have 2 < sup >( n-3 )</ sup > stereoisomers where n > 2 is the number of carbons.

Every aldose will have 2 < sup >( n-2 )</ sup > stereoisomers where n > 2 is the number of carbons.

Every time an MTA receives an email message, it adds a < tt > Received </ tt > trace header field to the top of the header of the message, thereby building a sequential record of MTAs handling the message.

* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.

< li > Every positive definite matrix is invertible and its inverse is also positive definite.

Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n, and with the exception of < var > n </ var > = 1, < var > n </ var > must end in 0, 2, 3, 5, 7, or 8.

* Every linear combination of its components Y = a < sub > 1 </ sub > X < sub > 1 </ sub > + … + a < sub > k </ sub > X < sub > k </ sub > is normally distributed.

Every sedenion is a real linear combination of the unit sedenions 1, < var > e </ var >< sub > 1 </ sub >, < var > e </ var >< sub > 2 </ sub >, < var > e </ var >< sub > 3 </ sub >, ..., and < var > e </ var >< sub > 15 </ sub >,

Every inner automorphism is indeed an automorphism of the group G, i. e. it is a bijective map from G to G and it is a homomorphism ; meaning ( xy )< sup > a </ sup >

Every previous group 17 element has seven electrons in its valence shell, forming a valence electron configuration of ns < sup > 2 </ sup > np < sup > 5 </ sup >.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers: