Any one-dimensional equation of order is equivalent to an-dimensional first-order system ( as described in Ordinary differential equation # Reduction to a first order system ), but not necessarily vice versa.

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

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

* 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 ring of characteristic n is a ( Z / nZ )- algebra in the same way.

* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.

* Any commutative ring R is an algebra over itself, or any subring of R.

Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.

Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.

Any algebra whose elements are idempotent is also power-associative.

:“ Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T ( V ) to A .”

: Any linear transformation f: V → A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T ( V ) to A as indicated by the following commutative diagram:

* Any algebra over a commutative ring K may be regarded as a purely even superalgebra over K ; that is, by taking A < sub > 1 </ sub > to be trivial.

* Any Z or N-graded algebra may be regarded as superalgebra by reading the grading modulo 2.

Any affine group scheme is the spectrum of a commutative Hopf algebra ( over a base S, this is given by the relative spectrum of an O < sub > S </ sub >- algebra ).

Any animal could when travelin' fast, be sent heels over head by the simple process of overtakin' the brute, seizin' its tail, and givin' the latter a pull to one side.

" Any threat of war Clovis could make would only be effective if they were neighbors ; " it is nowhere written that Syagrius was handed over in 486 or 487.

Any player may declare the game over at any time during his turn if either of two conditions is true: one chain has 41 or more tiles, or there is at least one chain on the board and every chain on the board has 11 or more tiles.

Any season ticket holder could put themselves forward for election, with a certain number of nominations, and votes were cast by all season ticket holders over the age of 18.

Any and all profits made from book deals or any movies about Lindh's experience will be automatically handed over to the federal government.

Any list of these would have to include the Night of the Living Houses ( said to be the first comic strip to enter the collection of the Louvre ) wherein Nemo and a friend are chased down a city street by a gang of tenement houses on legs ; the Walking Bed, in which Nemo and Flip ride over the rooftops on the increasingly long limbs of Nemo's bed ( see illustration ); and the Befuddle Hall sequence, wherein Nemo and his friends attempt to find their way out of a funhouse environment of a Beaux-Arts interior turned topsy-turvy.

Any Hohenstaufen in Sicily was bound to have claims over the cities of Lombardy, and as a check to Manfred, Urban IV introduced Charles of Anjou into the equation to place the crown of the Two Sicilies in the hands of a monarch amenable to papal control.

Any component of target velocity perpendicular to the line of sight cannot be determined by using the Doppler effect alone, but it can be determined by tracking the target's azimuth over time.

In July 2006, a book, ' Peace at Any Price: How the World Failed Kosovo ', written by two former senior staffers at UNMIK, exposed a catalogue of errors and incompetence in the institution over its seven year history.

# Any relatively high-bandwidth transmission over a short period.

Any Transport over MPLS-Cisco Systems.

Any large body of water upwind will impact lake-effect precipitation to the lee of a downwind lake by adding moisture or pre-existing lake-effect bands, which can re-intensify over the downwind lake.

Any production over 100 g must be notified to the OPCW and a country can have a stockpile of no more than one tonne of these chemicals.

Any person over the age of majority and of sound mind ( having appropriate mental capacity ) can draft his own will with or without the aid of a lawyer.

Any quadratic polynomial over the complex numbers ( polynomials of the form where,, and ∈ ) can be factored into an expression with the form using the quadratic formula.

Any amount over and above that required to maintain the capital at the beginning of the period is profit.

Any number of transputers could be connected together over even longish links ( tens of metres ) to form a single computing " farm ".

Any member of the priesthood who presides over a congregation can, and often is, known as " pastor " or ( if an elder ), " presiding elder ".

Any two adjacent conductors can be considered a capacitor, although the capacitance will be small unless the conductors are close together for long distances or over a large area.

Any police entry of an individual's home always requires a warrant ( for either search or arrest ), absent exigent circumstances, or the free and voluntary consent of a person with reasonably apparent use of or control over the property.

# Any sequence satisfying the recurrence relation can be written uniquely as a linear combination of solutions constructed in part 1 as λ varies over all distinct roots of p ( t ).