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Any and gives

* Any expression formed using any combination of the basic arithmetic operations and extraction of nth roots gives an algebraic number.

Any accelerating electric charge, and therefore any changing electric current, gives rise to an electromagnetic wave that propagates at very high speed outside the surface of the conductor.

Any second-countable space is separable: if is a countable base, choosing any gives a countable dense subset.

* Properly with a generic referent: " Any cow gives milk.

Any meromorphic function f gives rise to a divisor denoted ( f ) defined as

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

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

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 point x gives rise to a continuous function p x from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p x ( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.

Any ξ ∈ gives rise to a canonical vertical vector field X ξ by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ.

Any prime number p gives rise to an ideal pO K in the ring of integers O K of a quadratic field K.

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

Any surface that intersects the wire has current I passing through it so Ampère's law gives the correct magnetic field.

Any ring homomorphism A → B gives a map K 0 ( A ) → K 0 ( B ) by mapping ( the class of ) a projective A-module M to M ⊗ A B, making K 0 a covariant functor.

:( 1 ) Any person who manufactures, sells or hires or offers for sale or hire, or exposes or has in his possession for the purpose of sale or hire or lends or gives to any other person —

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

Any skew semistandard tableau of shape / with positive integer entries gives rise to a sequence of partitions ( or Young diagrams ), by starting with, and taking for the partition places further in the sequence the one whose diagram is obtained from that of by adding all the boxes that contain a value ≤ in ; this partition eventually becomes equal to.

Any attempt to drastically increase this range ( by increasing the angular extent of the true radius lobe nose region ) soon gives rise to problems with excessive rates of valve acceleration etc.

Any nontrivial line arrangement on RP 2 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.

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 symmetric space gives an involutory quandle, where is the result of ' reflecting through '.

Any topological space X gives rise to a category where the objects are the points of X and the morphisms are the homotopy classes of paths.

In his letter, Polding gives Wardell a completely free hand in the design, saying " Any plan, any style, anything that is beautiful and grand, to the extent of our power.

Any premetric gives rise to a preclosure operator cl as follows:

Any and Lie

* 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 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 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 o cannot be realized as an algebraic group ( see identity component ).

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 and .

Any lingering suspicion that this was a trick Al Budd had thought up was dispelled.

Any posse riding down the street to demand Blue Throat's surrender would be wiped out with one deadly burst of fire.

Any organism that falters or misperceives the signals or weakens is done.

Any abilities I may have were achieved in their present shape from experience in sharing in the growth and control of my business, coupled with raising my family.

Any attempt to reconcile this statement of the central issue in the campaign of 1956 with the nature of the man who could not conceive it as the central issue will at least resolve our confusions about the chaotic and misleading results of the earnestness of both doctors and President in a situation which should never have arisen.

Any tragedy, he maintains, has six elements: plot, character, and thought ( the objects of imitation ), diction and melody ( the means of imitation ), and spectacle ( the manner of imitation ).

Any information we have here in Taiwan is at your disposal ''.

He merely said, `` Any good decorator these days can make you a tasteful home ''.

Any approach toward such a system looks toward saving and security.

Any fee so determined shall be entered as a part of such award, and payment thereof shall be made by the Secretary of the Treasury by deducting the amount thereof from the total amount paid pursuant to the award.

Any agreement to the contrary shall be unlawful and void.

Any and all payments required to be made by the Secretary of the Treasury under this title pursuant to any award made by the Commission to the Government of the United States shall be covered into the Treasury to the credit of miscellaneous receipts.

Any claimant whose claim is denied, or is approved for less than the full amount of such claim, shall be entitled, under such regulations as the Commission may prescribe, to a hearing before the Commission, or its duly authorized representatives, with respect to such claim.

Any person who makes application for any such payment shall be held to have consented to all the provisions of this Title.

Any failure to file timely returns or make estimated tax payments when due must be fully explained.

Any alteration of one of these factors is distortion, although we generally use that word only for effects so pronounced that they can be stated quantitatively on the basis of standard tests.

Any irregularity on the piston heads will make it impossible, with normal means, to determine the final combustion chamber volume because the volume displaced by the piston heads cannot be readily computed.

Any fraction of an inch involved in the measurement must be converted to a decimal equivalent to simplify the mathematics.

Any special photographic requirements are also given.

Any house built now below that line without air conditioning will be obsolete in 10 years.

Any reputable French interne can supply you with a dozen similar instances, and I'll presently recount a case out of my own personal experience, but, for the moment, let's resume our catalogue.

Any of the antithyroid drugs, of either type, if given in large enough doses for a long period of time will cause goitre, owing to inhibition of thyroid hormone synthesis, with production of hypothyroidism.

Any other point of intersection between C and Af will be called a tangent point.

Any free elections that were to be held in Poland would have to produce a government in which Moscow had complete confidence, and all pressure from the West for free voting by anti-Soviet elements in Poland would be met by restrictions on voting by these elements.

Any example of grand or exquisite diction may have been created by the poet who compiled numerous lays into the two works we possess or may be due to one of his completely unknown fellow-craftsmen.

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