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Page "Product rule" ¶ 57
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On and other
On the other hand, howsomever, maybe you wouldn't either.
On the other hand, we cannot regard artistic invention as pure, uncaused, and unrelated to the times in which it occurs.
On the other hand, the fictional detective does not break strikes or handle divorce cases ; ;
On the other hand, if he wishes to continue in his chosen profession, he must abandon his own code and sacrifice his precious individualism.
On the other hand, Molesworth was naturally assailed in the Tory press.
On the other hand, Arnold's `` The unplumbed, salt, estranging sea '', taken in its context, certainly does so.
On the other hand, Dr. Ezra Styles recorded the following testimony of John Angell, the last disciple of Gorton: ``
On the other hand, he did not want to offend Edward either, and he found himself in a very difficult position.
On the other hand, the consensus of opinion is that, used with caution and in conjunction with other types of evidence, the native sources still provide a valid rough outline for the English settlement of southern Britain.
On the other side are the Celtic survivalists who have taken a tack divergent from both these schools of nineteenth century thought.
On the other hand, the bright vision of the future has been directly stated in science fiction concerned with projecting ideal societies -- science fiction, of course, is related, if sometimes distantly, to that utopian literature optimistic about science, literature whose period of greatest vigor in the late nineteenth and early twentieth centuries produced Edward Bellamy's Looking Backward and H. G. Wells's A Modern Utopia.
On the other side of the church was a quiet, well-kept house with shutters and recently painted.
On the other hand, it is no interference with sovereignty to point out defects where they exist, such as that a plan calls for factories without power to run them, or for institutions without trained personnel to staff them.
On the other hand, the new authority of $3,825 million proposed for missile procurement ( excluding research and construction ) in 1961 is $581 million higher than for 1960.
On the other side of the Golden Horn I rode through Eminonu Square, with Yeni Cami, or the New Mosque, which dates from the Seventeenth Century, just across from the entrance to the bridge.
On the other hand, out-of-sight does not lead to out-of-mind when children cannot be easily observed and you have to make a long trek to reach the pool.
On the other hand, the process of obsoleting an old product and introducing the new one is usually mighty expensive.
On the other hand, a similar attack might have been made on City B whose population was known to be lousy.
On the other hand, there are a few antithyroid drugs of this same general type, such as resorcinol, possessing no reducing activity and possibly acting through formation of a complex with molecular iodine.
On the other hand, the temporary reduction in hypothalamic excitability through the injection of a barbiturate into the posterior hypothalamus causes a lessening in frequency and amplitude of cortical strychnine spikes until the hypothalamic excitability is restored.
On the other hand, the null space of Af and the null space of Af together span V, the former being the subspace spanned by Af and the latter the subspace spanned by Af and Af.
) On the other hand, even when the binomial model does not describe well the physical phenomenon being studied, the binomial model may still be used as a baseline for comparative purposes ; ;
On the other hand, the women class members appeared to reach a far greater understanding than have women members in other sections that it is more natural for males as a group to view sex as sex rather than always associating it with love as most women seem to do.

On and hand
On the one hand, he does not work for a large agency, but is almost always self-employed.
On the one hand, the Public Health Service declared as recently as October 26 that present radiation levels resulting from the Soviet shots `` do not warrant undue public concern '' or any action to limit the intake of radioactive substances by individuals or large population groups anywhere in the Aj.
On such occasions he had not had the courage to look at the face above the hand, whatever face it might be.
On the other hand, in a more favorable vein, general business activity should receive some stimulus from rising Federal spending, and the reduction in business inventories has probably run a good part of its course.
On the one hand, the major European nations had to maintain vis-a-vis each other an emphasis upon sovereignty, independence, formal equality -- thus insuring for themselves individually an optimal freedom of action to maintain the `` flexibility of alignment '' that the system required and to avoid anything approaching a repetition of the disastrous Napoleonic experience.

On and simplicity
On the other hand, others bemoaned the austere Roman culture during his papacy ; Giorgio Vasari in 1567 spoke of a time when " the grandeurs of this place reduced by stinginess of living, dullness of dress, and simplicity in so many things ; Rome is fallen into much misery, and if it is true that Christ loved poverty and the City wishes to follow in his steps she will quickly become beggarly ...".
On the other hand, to achieve this simplicity, thin clients sometimes lag behind thick clients ( PC Desktops ) in terms of extensibility.
On the other hand, the computational simplicity of the Elo system has proven to be one of its greatest assets.
On April 12, 2010, the Pulitzer Prize Board awarded Williams a posthumous special citation that paid tribute to his " craftsmanship as a songwriter who expressed universal feelings with poignant simplicity and played a pivotal role in transforming country music into a major musical and cultural force in American life.
During his Oxford years he wrote Justitia Divina ( 1653 ), an exposition of the dogma that God cannot forgive sin without an atonement ; Communion with God ( 1657 ), Doctrine of the Saints ' Perseverance ( 1654 ), his final attack on Arminianism ; Vindiciae Evangelicae, a treatise written by order of the Council of State against Socinianism as expounded by John Biddle ; On the Mortification of Sin in Believers ( 1656 ), an introspective and analytic work ; Schism ( 1657 ), one of the most readable of all his writings ; Of Temptation ( 1658 ), an attempt to recall Puritanism to its cardinal spiritual attitude from the jarring anarchy of sectarianism and the pharisaism which had followed on popularity and threatened to destroy the early simplicity.
On account of this emphasis on simplicity, stability, and a combination of linear and circular movements, proponents claim the style is more practical for self-defense than most other martial arts.
On Easter day 1949, seven brothers committed themselves to a life following Christ in simplicity, celibacy and community.
On January 18, 2011, the LA Times reported after considering a new theme based on DC Comics superhero sidekicks, the park opted for simplicity and will rename the Little Flash coaster to Road Runner Express.
On the other hand Marc Théodore Bourrit, the contemporary of these three men, was rather a curious and inquisitive traveller than a scientific investigator, and charms us even now by his genial simplicity as contrasted with the austerity and gravity of the three writers we have mentioned.
On its Catalogue website Muji states that " at the heart of Muji design is the Japanese concept of ' Kanketsu ', the concept of simplicity ", aiming to " bring a quiet sense of calm into strenuous everyday lives.
On the other hand Marc Théodore Bourrit, the contemporary of these three men, was rather a curious and inquisitive traveller than a scientific investigator, and charms us even now by his genial simplicity as contrasted with the austerity and gravity of the three writers we have mentioned.
On his return he began to lecture and soon gained a wide reputation for the simplicity of his style and the precision of his views.

On and algebra
He was much interested, too, in universal algebra and elliptic functions, his papers " Preliminary Sketch of Biquaternions " ( 1873 ) and " On the Canonical Form and Dissection of a Riemann's Surface " ( 1877 ) ranking as classics.
On the other hand, other sites are used, and the Grothendieck topos has taken its place within homological algebra.
On the Lie algebra level, there is a corresponding linear mapping from the Lie algebra of G to End ( V ) preserving the Lie bracket.
On the other hand, a finite-dimensional representation of G can be used to generate a *- subalgebra of C ( G ) which is also a Hopf *- algebra.
On the other hand it is simple to construct a group algebra, the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means of Pontryagin duality.
On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not locally symmetric: one of them, known as Berger's first criterion, is a consequence of the Ambrose – Singer theorem, that the curvature generates the holonomy algebra ; the other, known as Berger's second criterion, comes from the requirement that the connection should not be locally symmetric.
Viewing a variety V and its homomorphisms as a category, a subclass U of V that is itself a variety is a subvariety of V implies that U is a full subcategory of V, meaning that for any objects a, b in U, the homomorphisms from a to b in U are exactly those from a to b in V. On the other hand there is a sense in which Boolean algebras and Boolean rings can be viewed as subvarieties of each other even though they have different signatures, because of the translation between them allowing every Boolean algebra to be understood as a Boolean ring and conversely ; in this sort of situation the homomorphisms between corresponding structures are the same.
On the Iwasawa algebra associated to a normal element of Cp, Proc.
On the other hand, for any fixed cardinal κ, there is a free ( or universal ) κ-complete Boolean algebra generated by any given set.
On the algebra of logic.
On the exterior algebra of differential forms over a smooth manifold, the exterior derivative is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero.
On vector fields, it is an example of a Lie bracket ( vector fields form the Lie algebra of the diffeomorphism group of the manifold ).

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