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However and case
However, in this case as elsewhere it was necessary to arrive at a single standard to be applied to all situations, representing an averaging of conditions, and thus to fix particular points in time which would be considered the dividing points between daytime and nighttime conditions.
However, in this case the district manager was led to see the errors of his ways.
However implausible this is, it is certainly the case that Aristotle's rigid separation of action from production, and his justification of the subservience of slaves and others to the virtue – or arete – of a few justified the ideal of aristocracy.
However, certain critical interlocutory court orders, such as the denial of a request for an interim injunction, or an order holding a person in contempt of court, can be appealed immediately although the case may otherwise not have been fully disposed of.
However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.
However, to make a working computer, users still had to add a case, power supply transformers, power switch, ASCII keyboard, and composite video display.
However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori.
However, in drafting opinions, the Chief Justice enjoys additional influence in case disposition if in the majority through his power to assign who writes the opinion.
However, this is not necessarily the case.
However, this is not the case all the time.
However, in the case of a creditor's petition, the Statement of Affairs will rarely be filed on the same day the court order is made.
However, Edwards states that more of the changes suggested by High Anglicans were implemented ( though by no means all ) and Spurr comments that ( except in the case of the Ordinal ) the suggestions of the ' Laudians ' ( Cosin and Matthew Wren ) were not taken up possibly due to the influence of moderates like Sanderson and Reynolds.
" However, held the Cadillac court, " one who manufactures articles dangerous only if defectively made, or installed, e. g., tables, chairs, pictures or mirrors hung on the walls, carriages, automobiles, and so on, is not liable to third parties for injuries caused by them, except in case of willful injury or fraud ,"
However, research using an artificial gut has shown that in the specific case of carotenoids a greater proportion is absorbed from cooked vegetables than from raw vegetables.
However, throughout history there have been exceptions, with Mutt and Jeff being an early ( if not the earliest ) case in which the creator owned his works.
However, this need not be the case.
However, as is often the case with the interpretation of scripture, there is dispute on this matter.
However, if they showed treasonous inclinations, the Emperor did punish them with just force, as in the case of Polybius and Pallas ' brother, Felix.
" However, in the case of the so-called Nilo-Hamitic languages ( a concept he introduced ), it was based on the typological feature of gender and a " fallacious theory of language mixture.
However, Constantius ' actions in this regard may not have been so much to do with Jewish religion as Jewish business ; apparently, it was often the case that privately owned Jewish businesses were in competition with state-owned businesses.
However, during 2006-8, several reviews of case reports and pilot studies have failed to confirm this effect, collectively indicating no statistically significant interaction between daily consumption of 250 mL cranberry juice and warfarin in the general population.
) However, this has generally not been the case throughout the rest of history for circulation coins made of precious metals.
However, because fiat money is backed by government guarantee of a certain amount of goods and services, where the value of this is in turn determined by free market currency exchange rates, similar to the case for the international market exchange values which determines the value of metals which back commodity money, in practice there is very little economic difference between the two types of money ( types of currencies ).
However, it should be noted that any individual, even a healthy adult in middle age, can experience a severe case, and each person's case should be measured by the loss of fluids, preferably in consultation with a doctor or other health worker.

However and algebraic
However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not algebraically independent.
However, most modern algebraic geometry texts starting with Alexander Grothendieck's foundational EGA use the convention in this article .< ref > A notable exception to modern algebraic geometry texts following the conventions of this article is Commutative algebra with a view toward algebraic geometry / David Eisenbud ( 1995 ), which uses " h < sub > A </ sub >" to mean the covariant hom-functor.
However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.
However, the resulting algebraic structure is not a field, and should not be expected to behave like one.
However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory.
If the base field F is the field R of real numbers, then x < sup > 2 </ sup > + y < sup > 2 </ sup > = − 1 defines an algebraic extension field of R ( x ), but the corresponding curve considered as a locus has no points in R. However, it does have points defined over the algebraic closure C of R.
However, the user also has the option of typing in moves in algebraic notation, ideal for laptop users with only the built in touchpad as their mouse.
However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its " inverse " is not necessarily the identity mapping.
However, a purely algebraic theory was first developed by, with Armand Borel as one of its pioneers.
However, for univariate algebraic equations, the number of solutions is finite and all solutions, are contained in any algebraically closed field containing the coefficients, for example, the field of complex numbers in case of equations over the rationals.
and e would therefore be algebraic of degree p. However e is transcendental, so Ω must be irrational.
: However, the definition of a correspondence in algebraic geometry is not completely standard.
However, the converse is true if X and Y contain only algebraic numbers and their size is at least 2 × 2.
However it can also be used to suggest particular test-statistics that might be of interest or to suggest simplified tests — for this one considers algebraic manipulation of the ratio to see if there are key statistics in it related to the size of the ratio ( i. e. whether a large statistic corresponds to a small ratio or to a large one ).
However, the Garden of Eden theorem can be generalized beyond Euclidean spaces, to cellular automata defined on the elements of an amenable group or a sofic group ; the proof of this generalization uses the Ax – Grothendieck theorem, an analogous relation between injectivity and bijectivity in algebraic geometry.
However, the algebraic properties of this structure are somewhat different from those of classical propositional logic in that the principle of distributivity fails.
However, under certain conditions algebraic solutions can be obtained ; for example, the equation can be solved as
However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions ( quotients of algebraic expressions ), and expressions that contain irrational numbers, such as √ 2 / 2 ( see square root of 2 ) and π / 4 ( see proof that π is irrational ).

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