However, if there is no " obvious " Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.

However, S may be bounded as subset of R < sup > n </ sup > with the lexicographical order, but not with respect to the Euclidean distance.

However, unlike Archimedes ' method, which was based on Euclidean geometry, Newton's and Leibniz's integral calculus did not have a rigorous foundation.

However, it is sometimes denoted in order to emphasize its role as the first Euclidean space.

However S < sup > 2 </ sup > × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.

However, these facts are counterintuitive: we do not normally expect open sets, other than connected components, to be compact, and for affine varieties ( for example, Euclidean space ) we do not even expect the space itself to be compact.

However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections.

However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained.

However, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space ( where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions ) or to non-Euclidean spaces such as hyperbolic space.

However, there is a polynomial-time approximation scheme ( PTAS ) for Euclidean Steiner trees, i. e., a near-optimal solution can be found in polynomial time.

However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point p can be identified naturally ( by translation ) with the tangent space at a nearby point q.

However, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel, the theorem remains valid in the Euclidean plane.

In particular, if the map is a continuous bijection ( a homeomorphism ), so that the two spaces have the same topology, then their i-th homotopy groups are isomorphic for all i. However, the real plane has exactly the same homotopy groups as a solitary point ( as does a Euclidean space of any dimension ), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces.

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, when a prominent surveyor finds a Triangle with more than 180 degrees, he is fired from his job and generally considered a crackpot, since such a construction is not possible in Euclidean geometry.

However, the blocks in Salt Lake City became irregular during the late 19th century when the LDS Church lost authority over growth and before the adoption of zoning ordinances in the 1920s.

However, this may vary depending on zoning.

However, zoning restrictions may deny an owner any economically viable use of his land.

However, new zoning laws and planning difficulties posed by Glasgow City Council look set to push many of these long-established facilities out of the city in the near future.

However, Sullivan has been subject to several court rulings for zoning violations on the property, including one in 1991 and again in 1999.

However, strong planning and zoning has maintained much of the 19th and early 20th century charm of the town, especially along its famous mile-long Main Street.

However, Los Angeles has become more dense over the past half-century, principally due to small lot zoning and a high demand for housing due to population growth.

However, the city announced in October 2009 that the property would go to auction with no minimum bid, and that zoning regulations would be relaxed for any buyer in order to spark development.

However, Los Angeles, California's inclusionary zoning ordinance for rental housing was invalidated in 2009 by the California Court of Appeal for the Second Appellate District because it directly conflicted with a provision of the state's Costa-Hawkins Rental Housing Act of 1996 which specifically gave all landlords the right to set the " initial rental rate " for new housing units.

However, the height and density of existing housing is generally lower than is permitted under the current zoning.

However, the U. S. Supreme Court has upheld the constitutionality of zoning laws on a very broad basis, even though such laws all post-date the 1787 Constitution.

However, the City ’ s use of zoning as a job retention tool did not achieve its goal, and manufacturing has continued to decline at the same pace after the zoning was enacted as it did before the preservation measures were in place.

However, at least one court has applied the RLUIPA in an eminent-domain case because the authority to condemn the property came from the city's zoning scheme.

However, he revealed that colonial officials had abruptly changed the zoning of the Discovery Bay project, and gave it to new developers because they feared it would fall into the hands of the former Soviet Union.

However, due to recently imposed zoning regulations, most of these operations have been moved to the surrounding new town of Ayetharyar.

However, in recent decades, for what doubtless are multiple reasons, an unannounced but nonetheless readily observable shift has occurred in both facets of national activity.

However, as a practicing historian, he, himself, has left few clues to the amount of professional scholarship that he used when writing history.

However, it seems axiomatic that the government has an obligation `` to exercise its mandate reasonably, equitably and with full regard for the disruptions which it inevitably causes ''.

However, because this vulnerability is mutual, it is to the advantage of neither side to destroy the opponent's cities, at least so long as the opponent has nuclear weapons with which to effect reprisal.

However, that is not all he has to do.

However, there are relatively few such political constituencies, and, as has been pointed out, there is seldom a clear-cut distinction between the educational interests of one social class and those of another.

However, electrostatic focus, important for many applications, has not been realized for these devices.

However, readers who accept Freud's findings and believe that he has solved completely the mystery of dreams, should ponder over the following words in his Interpretation Of Dreams, Chapter 1::

However, by cultivating a wine dealer and accepting his advice, one will soon enough ascertain whether he has any knowledge of wines ( as opposed to what he may have been told by salesmen and promoters ) and, better yet, whether he has a taste for wine.

However much football has been over-emphasized, the public likes to measure its collegiate favorites by the scoreboard, so, while Yale need never give its record a thought again since outscoring its opponents 694 to 0 in the season of 1888, Dartmouth had to wait until its championship team of 1925 for national recognition.

However, as Erickson has noted, the individual's failure to develop preliminary identities during infancy and childhood need not be irreversibly deterministic with respect to a given area of identity diffusion in his ( or her ) adolescence.

However, if the child has been constantly surrounded, during nursery and early school age, by peer groups ; ;

In a long commentary which he has inserted in the published text of the first act of the play, he says at one point: `` However, that experience never raised a doubt in his mind as to the reality of the underworld or the existence of Lucifer's many-faced lieutenants.

However, while Apollo has a great number of appellations in Greek myth, only a few occur in Latin literature, chief among them Phoebus ( ; Φοίβος, Phoibos, literally " radiant "), which was very commonly used by both the Greeks and Romans in Apollo's role as the god of light.

However, the monophyly of Cabozoa has been refuted and the monophylies of Archaeplastida and Chromalveolata are currently strongly challenged.

However, only a very small percentage of all these people has remained in Angola, either for personal reasons ( intermarriage ) or as professionals ( e. g. medical doctors ).

However, as has been pointed out by Alexis Manaster Ramer and Paul Sidwell ( 1997 ), Strahlenberg actually opposed the idea of a closer relationship between the languages which later became known as " Altaic "..

However, the method has been criticised as containing possible sources of error.

However, metal formation for condensed-phase astatine has also been suggested.

However, astatine has a few metallic characteristics as well, such as plating out on a cathode, coprecipitating with metal sulfides in hydrochloric acid, and forming a cation in strong acidic solutions.

However, the hydrogen-1 atom has no neutrons and a positive hydrogen ion has no electrons.

However, it is the editor who has " the power to impose the dominant definition of the writer and therefore to delimit the population of those entitled to take part in the struggle to define the writer ".

However, this has now been replaced by a more professional exhibition space incorporating laser, video, slide shows, audio commentaries, and some interactive elements.