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 Poincaré duality, the homology class of Z is dual to a cohomology class which we will call, and the cap product can be computed by taking the cup product of and α and capping with the fundamental class of X.

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 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 ).

Whenever any result is sought by its aid, the question will then arise — By what course of calculation can these results be arrived at by the machine in the shortest time?

By then the memory may have been re-assigned to another use, with unpredictable results.

By the late 1970s, these had achieved only mixed results due to flaws in the planning process as well as inadequate funding and a shortage of the skilled managers and technicians needed for implementation.

By 1918, two French mathematicians, Pierre Fatou and Gaston Julia, though working independently, arrived essentially simultaneously at results describing what are now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors ( i. e., points that attract or repel other points ), which have become very important in the study of fractals ( see Figure 3 and Figure 4 ).

* Passive Method – By allowing past karmas to ripen in due course of time and experiencing the results, both good and bad with equanimity.

By submitting a single query to KIS, a user can search a set of remote white pages services and see the results of the search in a uniform format.

By 1939 the Luftwaffe was not much better prepared than its enemies to conduct a strategic bombing campaign, with fatal results during the Battle of Britain.

By the end of August 1889 he had published his results in the Zeitschrift für Physikalische Chemie.

By putting backspin on the ball the Magnus effect results in air travelling over the top of the ball quickly and cleanly whilst air travelling under the ball is turbulent.

By 1986, the party's position appeared to strengthen further with excellent local election results and a thorough rebranding of the party under the direction of Kinnock's director of communications Peter Mandelson.

By assuming that light actually consisted of discrete energy packets, Einstein wrote an equation for the photoelectric effect that agreed with experimental results.

By either adding or subtracting the resulting signals, a lower or upper sideband signal results.

By witnessing currents flowing through point-contacts, similar to what Bardeen and Brattain had accomplished earlier in December 1947, Mataré by June 1948, was able to produce consistent results by using samples of germanium produced by Welker.

By taking pieces of the enemy material to edit together and placing his own narration over the results, Capra gave meaning and purpose to the war with added narrative.

By March 26, 1887, Boys was reporting his results to the Physical Society of London

By forced division of labor Durkheim means a situation where power holders, driven by their desire for profit ( greed ), results in people doing the work they are unsuited for.

By the early 1340s, it was clear that Edward's policy of alliances was too costly, and yielded too few results.

By the time a correct diagnosis has been made, endometrium and other fluids have filled the uterus and fallopian tubes with results similar to retrograde menstruation resulting in endometriosis.

By rule, it is any act other than passing, kicking or successful handing that results in loss of player possession.

By choosing Anthemius, Leo obtained three results: he sent a possible candidate to the Eastern throne far away ; he repulsed Gaiseric's attempt to put a puppet of his on the Western throne ; and he put a capable and proven general with a trained army in Italy, ready to fight the Vandals.

By pooling quite different diseases the results can be leveled out.

By pairing the two dimensions, Eysenck noted how the results were similar to the four ancient temperaments.

This variant of the Burnside problem can also be stated in terms of certain universal groups with m generators and exponent n. By basic results of group theory, the intersection of two subgroups of finite index in any group is itself a subgroup of finite index.

By the end of 1942, the results of the battles of the Coral Sea, Milne Bay and on the Kokoda Track had averted the perceived threat of invasion.

By 1979, he had devised a computer system called UPIC, which could translate graphical images into musical results, wrote Andrew Hugill in 2008.