The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0.

By studying the limit of the manifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group: at large times the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thin piece is a graph manifold, but this extra complication is not necessary for proving just the Poincaré conjecture.

By interpreting the upper half of the complex plane as a model of the hyperbolic plane ( the Poincaré half-plane model ) Ford circles can also be interpreted as a tiling of the hyperbolic plane by horocycles.

By the Poincaré lemma, the θ < sup > i </ sup > locally will have the form dx < sup > i </ sup > for some functions x < sup > i </ sup > on the manifold, and thus provide an isometry of an open subset of M with an open subset of R < sup > n </ sup >.

By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Poincaré.

By projective duality, an ellipse can be defined also as the envelope of all lines that connect corresponding points of two lines which are related by a projective map.

By wave-particle duality the effect can be analyzed purely in terms of waves though not as conveniently.

By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups.

This follows, in part, from the fact the covariant Hom functor Hom ( N, –): C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.

By duality, the same is true for join-prime and join-irreducible elements.

By the Serre duality, this is the same as:

By planar duality it became coloring the vertices, and in this form it generalizes to all graphs.

Here it is easy to see that if some linear combination of the x < sub > i </ sub > and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group T other than the trivial character takes the value 1 on P. By Pontryagin duality we have T ′ contained in the kernel of χ, and therefore not equal to T.

By focusing all of one's being on one specific practice, duality is transcended.

By the In-Betweener's own admission, he represents duality itself, specifically in respect to concepts such as good and evil ; reason and emotion ; truth and illusion ; and life and death.

By the duality of networks, it can be proved that the number of essentially series networks is equal to the number of essentially parallel networks.

By the principle of duality, the closure operator satisfies the identities:

By projective duality, the existence of an ordinary line in a set of non-collinear points in RP < sup > 2 </ sup > is equivalent to the existence of an ordinary point in a nontrivial arrangement of finitely many lines.

By duality, this results also says that any finite nontrvial set of points on the plane has at least three ordinary lines.

By the end of the recording sessions, The Gift was faced with a double album, where two distinctive but complementary environments surfaced ( one more intimate and introspective, the other more euphoric ) and in which duality was the key word.

By projective duality, G ( 1, 2 ) is isomorphic to P < sup > 2 </ sup >, so we may give it homogenous coordinates.

Jahrhundert ’, Zeitschrift für Musikwissenschaft, iv, 257 – 65 ), who employed it in reference to half of his perceived binarism between that 17th-century dance music which was in fact danced to, and that which was written in dance forms but was actually abstract music intended only for listening ( By extension, this duality applied to the dichotomy between any music for a specific purpose, Gebrauchsmusik, and that for none but the pleasure or edification of listening, Vortragsmusik ).

By applying the functor F to this sequence, one obtains a chain complex ; the homology H < sub > n </ sub > of this complex depends only on F and X and is, by definition, the n-th derived functor of F, applied to X.

By the homotopy axiom, one has that is also a functor, called the homology functor, acting on hTop, the quotient homotopy category:

By dualizing the homology chain complex ( i. e. applying the functor Hom (-, R ), R being any ring ) we obtain a cochain complex with coboundary map.

By taking the fundamental homology class of these manifolds, this yields a bilinear product on homology.

By studying a site with known historical occupations and then excavating it to establish prehistoric activity, it was reasoned that by using analogy and homology based on the historical data, theories could be postulated about the past society that had used the site long before the historical records were made.

By virtue of their high degree of homology, the new gene copies that came in to existence following the gene duplication naturally tend to either unequal crossover or unidirectional gene conversion events.

By virtue of his self-reliance, his individualism and his freedom from external restraint, the private eye is a perfect embodiment of the middle class conception of liberty, which amounts to doing what you please and let the devil take the hindmost.

By 2008, sales in the US reached $ 14. 6 billion, the biggest selling drugs in the US by therapeutic class.

By convention, the bicyclic β-lactams are numbered starting with the position occupied by sulfur in the penams and cephems, regardless of which atom it is in a given class.

By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them ; we are studying the relationships between various classes of mathematical structures.

By contrast, Marx's dialectical materialism considers history as a product of material class struggle in society.

By re-defining class as a relationship that changed over time, Thompson proceeded to demonstrate how class was worthy of historical investigation.

By the next century, gardens became available to the middle class as well.

By the 1830s mathematics, physics, chemistry, and biology had emerged with world class science, led by Alexander von Humboldt ( 1769 – 1859 ) in natural science and Carl Friedrich Gauss ( 1777 – 1855 ) in mathematics.

By this system, the children were free people and could inherit property, thus originating a class of " mulattos " with property and some with wealthy fathers.

By the mid 1890s, the Golden Dawn was well established in Great Britain, with membership rising to over a hundred and including every class of Victorian society.

By 2000 the social structure included a politically active working class, a primarily clerical middle class, and an upper bracket consisting of managers, entrepreneurs, and professionals.

By " ideology ", Marx and Engels meant ideas that reflect the interests of a particular class at a particular time in history, but which contemporaries see as universal and eternal.

By contrast, when Bertrand Russell writes, in The Principles of Mathematics, " A class [...] is neither a predicate nor a class-concept, for different predicates and different class-concepts may correspond to the same class.

By the 20th century, the saloon, or lounge bar, had become a middle-class room — carpets on the floor, cushions on the seats, and a penny or two on the prices, while the public bar, or tap room, remained working class with bare boards, sometimes with sawdust to absorb the spitting and spillages, hard bench seats, and cheap beer.

By the end of the 19th century, a new educated social class which transcended the divisions of race and caste was emerging as a result of British attempts to nurture a range of professionals for the Ceylon Civil Service and for the legal, educational, and medical professions.

By contrast, the cut-and-paste transposition mechanism of class II TEs does not involve an RNA intermediate.

By 1892 an entirely new class of warship, the torpedo boat destroyer ( TBDs ), was invented to counter them.

By 1822 he was in the class of the director Nicolò Zingarelli, studying the masters of the Neapolitan school and the orchestral works of Haydn and Mozart.

By his own description Vitruvius served as a ballista ( artilleryman ), the third class of arms in the military offices.

By 1855 the term had passed into common usage by upper class whites, and was common usage among all Southerners, regardless of race, throughout the rest of the 19th century.

By 1499, however, Cardinal Francisco Jiménez de Cisneros grew frustrated with the slow pace of the efforts of Granada's first Archbishop, Fernando de Talavera, to convert non-Christians to Christianity and undertook a program of forced Christian baptisms, creating the Converso ( convert ) class for Moors and Jews.

By the late 18th century Lyme Regis had become a popular seaside resort, especially after 1792 when the outbreak of the French Revolutionary Wars made travel to the European mainland dangerous for the English gentry, and increasing numbers of wealthy and middle class tourists were arriving there.