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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, in the case of algebraic curves it is very common not to restrict the curve to having points only defined over the real numbers.
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, 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 ).

However and properties
However, the proposed correlation of the many interrelated properties of crystals will reveal discrepancies in the recorded data and suggest areas for reinvestigation.
Experiments have been conducted to attempt the synthesis of ununennium ( Uue ), which is likely to be the next member of the group, but they have all met with failure .< ref name =" link "> However, ununennium may not be an alkali metal due to relativistic effects, which are predicted to have a large influence on the chemical properties of superheavy elements.
Dmitri Mendeleev claimed he arranged his tables in order of atomic weight (" Atomgewicht ") However, in deference to the observed chemical properties, he violated his own rule and placed tellurium ( atomic weight 127. 6 ) ahead of iodine ( atomic weight 126. 9 ).
However, expressing these properties as axioms requires use of second-order logic.
However, some alloys can also have their properties altered by heat treatment.
However, there were also properties which could not be explained, including the non-spherical distribution of charge in certain nuclei.
However, the argument concludes that the conceptual challenge of bare particulars leaves a bundle of properties and nothing more as the only possible conception of an object, thus justifying bundle theory.
However, colloidal suspensions of higher-volume fraction form colloidal gels with viscoelastic properties.
However this was later limited to adaptations of animated properties.
However, the work of Friedrich Hund, Robert Mulliken, and Gerhard Herzberg showed that molecular orbital theory provided a more appropriate description of the spectroscopic, ionization and magnetic properties of molecules.
However, these causal properties can't be detected by anyone outside the mind, otherwise the Chinese Room couldn't pass the Turing test — the people outside would be able to tell there wasn't a Chinese speaker in the room by detecting their causal properties.
However, HPLC techniques exist that do utilize affinity chromatogaphy properties.
However, glass making involves several steps of the ceramic process and its mechanical properties are similar to ceramic materials.
However, by identifying clusters of observable properties that are associated with core examples of clitics on the one hand, and core examples of affixes on the other, one can pick out a battery of tests that provide an empirical foundation for a clitic / affix distinction.
However she had been corresponding with her uncle Raymond, King and holder of family properties in Antioch where he was seeking further protection from the French crown.
However, in robust statistics, statistical theory goes on to consider the balance between having good properties, if tightly defined assumptions hold, and having less good properties that hold under wider conditions.
However, because of hafnium's neutron-absorbing properties, hafnium impurities in zirconium would cause it to be far less useful for nuclear-reactor applications.
However, many of the minor substances in honey can be affected greatly by heating, changing the flavor, aroma, or other properties, so heating is usually done at the lowest temperature possible for the shortest amount of time.
However the magnetic properties of the core material cause several side effects which alter the behavior of the inductor and require special construction:
However, many of the physical properties of water and ice are controlled by the formation of hydrogen bonds between adjacent oxygen and hydrogen atoms.
However, it is usually accompanied by physical or chemical changes that permanently degrade the material's insulating properties.
However, it bears repeating that none of the above is a defining quality of the infinitive ; infinitives do not have all these properties in every language, as it is shown below, and other verb forms may have one or more of them.
However, it is an empirical fact that microscopic particles of the same species have completely equivalent physical properties.

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