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( However, rotations are more subtle in high dimensions, and visualizing high-dimensional spaces remains difficult, even for experienced mathematicians.
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However and rotations
However, spinors transform well under the infinitesimal orthogonal transformations ( like infinitesimal rotations or infinitesimal Lorentz transformations ).
However, in a certain sense rotations and reflections contain finer geometrical information than can be expressed in terms of their actions on vectors.
However, it is common to write " there are seven reflecting tetrominos (= 19 total ) up to rotations ".
However their rotations are not known or not known enough.
However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.
However, one can also consider improper rotations, i. e. a mirror-reflection possibly followed by a proper rotation.
However, qualities such as refined technique, speed and flexibility have become increasingly important in the modern era as the legal throwing area has been reduced from 90 to 34. 92 degrees and throwing technique involves three to four controlled rotations.
However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles ( in numerous variants ), both because this is conceptually simpler, and because one can build a combination of three gimbals to produce rotations in three dimensions.
However, linear molecules only have two rotations because rotations along the bond axis do not change the positions of the atoms in the molecule.
However, a closer inspection tells us that this is only true if we consider rotations confined to the plane.
However and are
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, in the context of legal and civic policy, these controversies are less than novel.
However, despite the insight of many of his observations, his own conclusions are open to suspicion because of his failure to employ at all times the correct research methods.
However, there are still several types of calls that necessitate the use of telephone operators.
However, he states unequivocally `` the scandals in business are far less significant than the scandals in labor ''.
However, few are taxed, and the owners and location of most boats are unknown to the assessors on the date of assessment of town valuations.
However, the surface temperature gradient can produce erroneous vapor-pressure measurements for the bulk liquid helium unless precautions are taken to isolate the tube ( which passes through the surface to the vapor pressure bulb ) from the liquid helium surface.
However, if their United States income is not subject to the withholding of tax on wages, their returns are due June 15, 1962, if they use a calendar year, or the 15th day of the 6th month after the close of their fiscal year.
However, the aircraft which we have today are tied to large, `` soft '' airfields.
However, some formulas are used more than others.
However, there is also much to be gained by making use of the abilities of the local people who are available and interested in recreation.
However, there are photographs of suspended drops of cyclohexanol phthalate ( viscosity 155 poises ) suspended in corn syrup of 71 poises in a paper by Mason and Bartok.
However, any oxygen nonequivalence would shorten either the already extremely short Af interlayer distance of 2.55 A or the non-hydrogen-bonded Af interlayer interactions which are already quite short at 2.58 Aj.
However, as so many of the records are not certainly based on newborn snakes, these data must be taken tentatively ; ;
However, if these procedures are applied more often, conditioned emotional responses are temporarily abolished.
However, it cannot be presumed, informal though the structure of the population seems, that there are not well-defined roles within the system.
However, the differences in procedure are fundamental.
However, the regionally differentiated results, which appear below in tables, are interesting evidence of the problems of developing self-government under even the most favorable circumstances.
However, if Federal funds are used, it would be entirely appropriate to train workers for jobs which could be obtained elsewhere as well as for jobs in the area of chronic unemployment.
However, it is not known to either the union or the public precisely how much of a cost increase is caused by a given change in the basic wage rate, although the companies are presumed to have reliable estimates of this magnitude.
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, modesty and triviality are different qualities.
`` However, evidence that the Axis powers are making significant preparations indicative of such an intention is being reported with increasing frequency from a variety of sources.
However and more
However, there is a more profound consideration to this proposed merger than profit and loss.
However, its modern one-story layout is designed to increase our production capacity, permit more efficient manufacturing, and substantially reduce current repair and maintenance costs.
However, because of Church laws, lately more stringently enforced, which forbid the marriage of cousins closely related consanguineously, a means of facilitating the goal of in-group relations may be that of recourse to illegitimate unions.
However well chosen and cleverly arranged, such memorabilia unfortunately amounted to more of an interruption than an auxiliary to the evening's main business, which ( considering the talent at hand ) should probably have been the gathering of fresh samples of the Chicago style.
However, he thinks we should love God more than ourselves and our neighbours, and more than our bodily life — since the ultimate purpose of loving our neighbour is to share in eternal beatitude: a more desirable thing than bodily well being.
However, in contexts where " amoeboid " is defined more loosely, there are many amoeboid species that are in the Excavata clade.
However, such apparent simplifications can perversely make a script more complicated.
However Juhanen ( 1992 ) is skeptical about an affiliation of Japanese to Altaic languages, while Róna-Tas ( 1998: 77 ) remarked that a relationship between Altaic and Japanese, if it ever existed, must be more remote than the relationship of any two of the Indo-European languages.
However it is in its proper sense, in which it indicates national character and the subjection due to that character, that the word is more important.
However, anxiety should not be confused with fear, it is more of a dreaded feeling about something which appears intimidating and can overcome an individual.
However, this has now been replaced by a more professional exhibition space incorporating laser, video, slide shows, audio commentaries, and some interactive elements.
However, as more and more deaths quickly ensue, it becomes apparent that there is an epidemic.
However a comparable performing digital system is more complex and can require more bandwidth than its analog counterpart.
However, the availability of commercially farmed abalone has allowed more common consumption of this once rare delicacy.
However, Odysseus proves to be more eloquent, and the council gives him the armor.
However, Wolfram points out that at the time " Clovis got no farther than the Seine ; only after several more years did the Franks succeed in occupying the rest of the Gallo-Roman buffer state north of the Loire.
However, Amasis was later faced with a more formidable enemy with the rise of Persia under Cyrus who ascended to the throne in 559 B. C. E.
However, more vital propositions, such as Hume's argument for the role of habit in a theory of knowledge, are retained.
However, further experience takes these equal chances, and forces the imagination to observe that certain chances arise more frequently than others.
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