# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.

Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian.

* Any smooth manifold is a diffeological space.

Any Calabi – Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi – Yau manifold.

Any smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology.

# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.

Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system.

Any covering space of a differentiable manifold may be equipped with a ( natural ) differentiable structure that turns p ( the covering map in question ) into a local diffeomorphism – a map with constant rank n.

Any vector bundle V over a manifold may be realized as the pullback of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle ; these universal Chern classes in turn can be explicitly written down in terms of Schubert cycles.

Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields.

* Any open manifold admits a ( non-complete ) Riemannian metric of positive ( or negative ) curvature.

Any “ nice ” space such as a manifold or CW complex is semi-locally simply connected.

Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.

* Any manifold with constant sectional curvature is an Einstein manifold — in particular:

Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.

Any manifold is an n-handlebody, that is, any manifold is the union of handles.

Any handlebody decomposition of a manifold defines a CW complex decomposition of the manifold, since attaching an r-handle is the same, up to homotopy equivalence, as attaching an r-cell.

Any compact manifold has isoperimetric dimension 0.

* Any compact symplectic manifold admits a symplectic Lefschetz pencil ( Donaldson 1999 ).

Any symplectic manifold ( or indeed any almost symplectic manifold ) has a natural volume form.

Any similar canister with the labels reading " dog spray " and / or " bear spray " is regulated under the Pest Control Products Act-while legal to be carried by anyone, it is against the law if its use causes ' a risk of imminent death or serious bodily harm to another person ' or harming the environment and carries a penalty up to a fine of $ 500, 000 and jail time of maximum 3 years.

Any break in the skin carries a risk of infection.

Any sandy component or quartz component of the original ore automatically carries through the smelting process as silicon dioxide.

Any cross between red-flowered and white-flowered snapdragons, give an intermediate and heterozygous phenotype with pink flowers, that carries both the dominant and recessive alleles.

President Woodrow Wilson regarded " hyphenated Americans " with suspicion, saying, " Any man who carries a hyphen about with him carries a dagger that he is ready to plunge into the vitals of this Republic whenever he gets ready.

Any legislation pending in either house at the end of a session that is not the end of a legislative term of office continues and carries over to the next legislative session.

Any certified product in the US or Canada carries a kosher symbol on the label.

* Cargo ship: Any sort of ship or vessel that carries cargo, goods, and materials from one port to another, including general cargo ships ( designed to carry break bulk cargo ), bulk carriers, container ships, multipurpose vessels, and tankers.

Any practitioner who carries out implant treatment, whether in the surgical insertion or the final provision of the prosthesis, must be adequately trained.

Any mutational event that disrupts the superhelical structure of DNA carries with it the potential to compromise the genetic stability of a cell.

# Disturbed natural forest – Any forest type above that has in its interior significant areas of disturbance by people, including clearing, felling for wood extraction, anthropogenic fires, road construction, etc.

# Disturbed natural forest – Any forest type above that has in its interior significant areas of disturbance by people, including clearing, felling for wood extraction, anthropogenic fires, road construction, etc.

Any self-replicating mechanism which does not make a perfect copy will result in the creation of different variants and thus be subject to natural selection as the variants which are better at persisting in their environment will outlive and outreproduce variants which are not so suited to their environment.

For example natural suited blackjacks pay 2 to 1 instead of 1. 5 to 1, Any five card 21 automatically pays 2-1, pairs of aces cannot be split, all other pairs can be split, including 4s, 5s, and 10s, only one split is allowed, doubling is allowed on hard 8, 9, 10 and 11 only, and insurance can be taken any time the dealer shows an ace, no matter what cards the player holds.

* Any use whatsoever of matrimony exercised in such a way that the act is deliberately frustrated in its natural power to generate life is an offense against the law of God and of nature, and those who indulge in such are branded with the guilt of a grave sin.

Any dielectric material develops a dielectric polarization ( electrostatics ) when an electric field is applied, but a substance which has such a natural charge separation even in the absence of a field is called a polar material.

Any apparent inequalities were independent of natural organization, and had their cause in the unequal desire for instruction.

# Any assignment of the elements of a formal language to natural numbers in such a way that the numbers can be manipulated by an algorithm to simulate manipulation of elements of the formal language.

Any binary relation R on a set A can generate a natural partial order on the R-cycle equivalence classes of set A, so that xRy implies ≥.

Any oxidizable material present in a natural waterway or in an industrial wastewater will be oxidized both by biochemical ( bacterial ) or chemical processes.

Any visitor to the long-since dilapidated ruins of Stari Bar ( Old Bar ), one of the world ’ s largest fortified archaeological sites, will be impressed by the beauty of the natural surroundings into which it blends, dignifying the majestic Mount Rumija which towers above it.

Any law in violation of the constitution or " natural law " which underlay it, was void.

" Any person applying for membership had to demonstrate they had made a new discovery in the natural sciences.

Any of the illuminations of the target ( i. e., Source 1a, 1b, or 3a ), and the returning radiation, can be affected by the atmosphere, or other natural phenomena such as the ocean, between source and target, or between target and sensor.

Any regular curve may be parametrized by the arc length ( the natural parametrization ) and from the point of view of a bug on the curve that does not know anything about the ambient space, all curves would appear the same.

Any potential limitation posed by the natural world was felt to be surpassed using human ingenuity.

Any hypothetical partition of natural numbers into interesting and dull sets seems to fail.

Any group scheme admits natural left and right actions on its underlying scheme by multiplication and conjugation.

Any bounded operator L that has finite rank is a compact operator ; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting.

Any natural scene or color photograph can be optically and physiologically dissected into three primary colors, red, green and blue, roughly equal amounts of which give rise to the perception of white, and different proportions of which give rise to the visual sensations of all other colors.

Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers.

Any of several related theories for the parsing of natural language qualify as constituency grammars, and most of them have been developed from Chomsky's work, including

Any breed can compete, but the Arabian generally dominates the top levels because of the breed's stamina and natural endurance abilities.