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Page "Perfect field" ¶ 8
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Every and reduced
Every 10 percent increase in the price of cigarettes reduced youth smoking by about seven percent and overall cigarette consumption by about four percent.
Every categorical proposition can be reduced to one of four logical forms.
Every five years the NZ Transport Agency will embark on a state highway review to consider whether the existing network should be expanded or reduced, according to traffic flows, changes in industry, tourism and development.
Every reduced word is an alternating product of elements of G and elements of H, e. g.
* Every first time purchaser of an automobile deprived the streetcars operator of income whilst simultaneously created additional traffic congestion which often reduced service speeds and thereby increased their operational costs and making the services less attractive to the remaining users.
Every field officer of his brigade was killed or wounded save two, and his brigade, already sadly reduced by its terrible sacrifices at Chancellorsville, lost in nearly 550 men out of the 1, 350 engaged.
Every field and every polynomial ring over a field ( in arbitrarily many variables ) is a reduced ring.
Every other interchange with major roads along the route ( Toorak, Burke, Tooronga and Warrigal Roads ) was an at-grade intersection controlled by traffic-lights, and because the road was constructed through residential areas, reduced speed limits were also enforced.
* Every time a loan balance is reduced, an equivalent number of bonds are identified by lottery and converted into a different bond also with a one cent par value ( that will convert at par into cash at the next payment date ).
Every new Queen's Counsel created reduced the Serjeants in importance, since even the most junior QC took precedence over the most senior Serjeant.
* Every cyclic shift and the inverse of a cyclically reduced word are cyclically reduced again.
* Every word is conjugate to a cyclically reduced word.

Every and commutative
* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.
* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.
* Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.
* Every localization of a commutative Noetherian ring is Noetherian.
* Every commutative semisimple ring must be a finite direct product of fields.
Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L < sup >∞</ sup >( X ) for some standard measure space ( X, μ ) and conversely, for every standard measure space X, L < sup >∞</ sup >( X ) is a von Neumann algebra.
Every symmetric association scheme is commutative.

Every and is
Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.
Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.
Every taxpayer is well aware of the vast size of our annual defense budget and most of our readers also realize that a large portion of these expenditures go for military electronics.
Every single problem touched on thus far is related to good marketing planning.
Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.
Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available.
Every man in every one of these houses is a Night Rider.
Every library borrower, or at least those whose taste goes beyond the five-cent fiction rentals, knows what it is to hear the librarian say apologetically, `` I'm sorry, but we don't have that book.
Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.
The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.
** Every infinite game in which is a Borel subset of Baire space is determined.
Every natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).
Every line of written text is a mere reflection of references from any of a multitude of traditions, or, as Barthes puts it, " the text is a tissue of quotations drawn from the innumerable centres of culture "; it is never original.
Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.
* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =
Every year, on the last Sunday in April, there is an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth.
Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.

Every and separable
* Every compact metric space is separable.
* Every compact metric space ( or metrizable space ) is separable.
* Every separable metric space is homeomorphic to a subset of the Hilbert cube.
* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.
* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.
* Every separable metric space is isometric to a subset of the
* Every second-countable space is first-countable, separable, and Lindelöf.
* Every subextension of F / k is separable.
* Every finite subextension of F / k is separable.
* Every polynomial over k is separable.
* Every finite extension of k is separable.
* Every algebraic extension of k is separable.
Every separable topological space is ccc.
Every metric space which is ccc is also separable, but in general a ccc topological space need not be separable.
Every second-countable manifold is separable and paracompact.

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