Any genus zero handlebody is homeomorphic to the three-ball B < sup > 3 </ sup >.

Any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues.

Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge.

Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac – Moody algebra:

Any parallel program is composed of simultaneously executing processes, problem decomposition relates to the way in which these processes are formulated.

# 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 sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot – Carathéodory, defined as

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 function itself defines an equivalence relation on according to which if and only if.

* Any Lie group G defines an associated real Lie algebra.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

Article 4 of the GEA of 1995 defines sexual harassment in the workplace as follows: " Any behaviour of a sexual nature or other behaviour attributable to gender which affronts the human dignity of males and females in the workplace.

Any simplicial automorphism φ of X defines a permutation π of Z / n Z such that label ( φ ( M ))

* NAUI defines technical diving as " Any diving beyond the limits of the defined recreational diving limits which is currently set at the following-diving to 40 meters / 130 feet, use of nitrox above 36 %, multiple mix gas diving, penetration diving past the daylight zone and any form of decompression diving ).

Any process that happens regularly in the forward direction of time but rarely or never in the opposite direction, such as entropy increasing in an isolated system, defines what physicists call an arrow of time in nature.

The Law of Imperfect Selection: Any objective rule that defines eligibility for a social transfer program will irrationally exclude some persons.

Any nontrivial line arrangement on RP < sup > 2 </ sup > defines a graph in which each face is bounded by at least three edges, and each edge bounds two faces ; so, double counting gives the additional inequality F ≤ 2E / 3.

4VAC15-20-50 defines a wild animal as " Any member of the animal kingdom, except domestic animals, including without limitation any native, naturalized, or nonnative ( exotic ) mammal, fish, bird, amphibian, reptile, mollusk, crustacean, arthropod or other invertebrate, and includes any hybrid of them, except as otherwise specified in regulations of the board, or part, product, egg, or offspring of them, or the dead body or parts of them.

In England and Wales, the Theft Act 1968 s 3 ( 1 ) defines appropriation as " Any assumption by a person of the rights of an owner "

Any vector field Y on N defines a pullback section φ < sup >*</ sup > Y of φ < sup >*</ sup > TN with ( φ < sup >*</ sup > Y )< sub > x </ sub >

Any numeral system defines the value of all numbers which contain more than one digit, most often by addition of the value for adjacent digits.

The Royal Canadian Mounted Police Gazette, quoting from the Provincial Court of Manitoba, defines these groups as: " Any group of motorcycle enthusiasts who have voluntarily made a commitment to band together and abide by their organizations ' rigorous rules enforced by violence, who engage in activities that bring them and their club into serious conflict with society and the law ".

Any volume pseudo-form ω ( and therefore also any volume form ) defines a measure on the Borel sets by

The British Standards of practice for falsework, BS 5975: 1982, defines falsework as: Any temporary structure used to support a permanent structure while it is not self-supporting.

Any mathematician who considers the mental plane's dimensionality equal to or more than the earthly one defines them as " hyperplanes.

Any human being has at least some kind of spiritual sense, developed through personal reflection or undeveloped but evident from lifestyle and communications, which defines the meta-meanings of human existence, the purpose of life, the meaning of the universe and one's own place in it, and so on.

As an example of the law itself, the State of Michigan defines the offense ( MCL 750. 249 ): " Any person who utters and publishes as true any false, forged, altered or counterfeit record, deed, instrument or other writing specified, knowing it to be false, altered, forged, or counterfeit, with intent to injure or defraud is guilty of uttering and publishing.