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Any and discrete

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

Any discrete variable is a segmentation.

Any discrete emotion can be used in a persuasive appeal ; this may include jealousy, disgust, indignation, fear, blue, disturbed, haunted, and anger.

* Any finite abelian group, with the discrete topology.

Any discrete subset of Euclidean space is countable, since the isolation of each of its points ( together with the fact the rationals are dense in the reals ) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many.

Any group can be given the discrete topology.

Any Fuchsian group ( a discrete subgroup of SL ( 2, R )) is a Kleinian group, and conversely any Kleinian group preserving the real line ( in its action on the Riemann sphere ) is a Fuchsian group.

* Any countable discrete group with property ( T ) is finitely generated.

* Any discrete group is locally compact.

* Any discrete space is a 0-dimensional manifold.

Any and valuation

Any continuous valuation v on K n that is invariant under rigid motions can be represented as

Any continuous valuation v on K n that is invariant under rigid motions and homogeneous of degree j is a multiple of W n-j .

* Any field is a valuation ring.

Any and ring

* Any ring A can be considered as a Z-algebra in a unique way.

* 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 ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.

Any legal wrestler is open to attack from any direction at any time, including when they are downed, as long as they are within the ring area enclosed by the ring ropes.

Any type or number of topologies may be used -- star, bus, ring, etc.

* Any ring, thought of as a category with only one object, is a preadditive category.

Any ( non-graded ) ring A can be given a gradation by letting A 0

Any number system that forms a commutative ring — for instance, the integers, the real numbers, and the complex numbers — can be extended to a wheel in which division by zero is always possible ; however, in such a case, " division " has a slightly different meaning.

Any ring can be seen as a preadditive category with a single object.

* Any commutative ring becomes a *- ring with the trivial involution.

* Any ring is also a semiring.

* Any ring R can be considered as a one-object preadditive category ; the category of left modules over R is the same as the additive functor category Add ( R, Ab ) ( where Ab denotes the category of abelian groups ), and the category of right R-modules is Add ( R op , Ab ).

Any quotient of a ring by a maximal ideal is a simple ring.

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

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

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 two-sided ideal of a ring R is an R-bimodule.

* Any module over a commutative ring R is automatically a bimodule.

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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