* Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra.

Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

* Any topologically closed subgroup of a Lie group is a Lie group.

* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.

# 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.

A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.

* Any Lie group gives a Lie groupoid with one object, and conversely.

* Any foliation gives a Lie groupoid.

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 associative algebra A over the field K becomes a Lie algebra over K with the Lie bracket:

* Any Lie group with an infinite group of components G / G < sup > o </ sup > cannot be realized as an algebraic group ( see identity component ).

Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F < sub > 2 </ sub > of the formal group law.

Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan.

* Any member of the genus Eunectes, a group of large, aquatic snakes found in South America

Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.

Any formal or informal group — a family, a church, a club, a business, a trade union — may be said to have government.

" Any attempt to organize the group ... under a single authority would eliminate their independent initiatives, and thus reduce their joint effectiveness to that of the single person directing them from the centre.

Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.

However, in the case of a finitely presented group we know that not all the generators can be trivial ( Any individual generator could be, of course ).

* Any subgroup of a free group is free.

Any group of four students may run for office, but there must always be four students.

Any group that managed to find ways of reasoning effectively would reap benefits for all its members, increasing their fitness.

Any time during a triad conversation, group members can switch seats and one of the co-pilots can sit in the pilot ’ s seat.

Any group can be seen as a category with a single object in which every morphism is invertible ( for every morphism f there is a morphism g that is both left and right inverse to f under composition ) by viewing the group as acting on itself by left multiplication.

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.

* Any two vertices in G can be connected by a unique simple path.

Unsigned, Mushroom ( Andy Vowles ), Daddy G ( Grant Marshall ) and 3D ( Robert Del Naja ) put out " Any Love " as a single, co-produced by Bristol double-act Smith & Mighty.

# Any group G may be regarded as an " abstract " category with one object,, and one morphism for each element of the group.

* Any open set is trivially a G < sub > δ </ sub > set

* Any principal bundle with structure group G gives a groupoid, namely over M, where G acts on the pairs componentwise.

* Any group G can be considered as a one-object category in which every morphism is invertible.

Any group scheme G of finite type is an extension of the connected component of the identity ( i. e., the maximal connected subgroup scheme ) by a constant group scheme.

A graph coloring is an assignment of one of k colors to a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism from G to a complete graph K < sub > k </ sub >: the vertices of K < sub > k </ sub > correspond to the colors of G, and f maps each vertex of G with color c to the vertex of K < sub > k </ sub > that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of K < sub > k </ sub >, and every two distinct vertices of K < sub > k </ sub > are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in K < sub > k </ sub >.

Any canonical transformation involving a type-2 generating function G < sub > 2 </ sub >( q, P, t ) leads to the relations

Let now v and w be two vertices in G. Any spanning tree contains precisely one simple path between v and w. Taking this path in the uniform spanning tree gives a random simple path.

** Any parabolic element of G fixing a boundary point of D is in H.

( 2 ) Any sequence of continuous positive definite functions on G converging to 1 uniformly on compact subsets, converges to 1 uniformly on G.

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

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.

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.

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.