By considering the propagation of the negative energy modes of the electron field backward in time, Ernst Stueckelberg reached a pictorial understanding of the fact that the particle and antiparticle have equal mass m and spin J but opposite charges q.

By considering circumscribed spheres, the notion of Delaunay triangulation extends to three and higher dimensions.

By spring 1942, he was considering involvement in war work.

By considering NRM members innocent " victims " of psychological coercion these theories open the door for psychological treatments.

By contrast, socialism emphasises distributing output among trades as required by each trade, not necessarily considering how trades organise internally.

By selling vast quantities of shrimp and catfish to the U. S., Vietnam triggered antidumping complaints by the U. S., which imposed tariffs in the case of catfish and was considering doing the same for shrimp.

By 1942 Rockefeller had become increasingly impatient that his purchased property might never be added to the park, and wrote to the Secretary of the Interior Harold L. Ickes that he was considering selling the land to another party.

By the mid-1980s, therefore, it had become essential that the PNDC demonstrate that it was actively considering steps towards constitutionalism and civilian rule.

By 1702, Victor Amadeus was considering changing allegiance to the emperor again having entered secret correspondence with the emperor who promised him the Duchy of Montferrat.

By considering the impact of emotions, cognitive errors, irrational preferences, and the dynamics of group behavior, behavioral finance offers succinct explanations of excess market volatility as well as the excess returns earned by stale information strategies .... cognitive errors may also explain the existence of market inefficiencies that spawn the systematic price movements that allow objective TA analysis methods to work.

By 1925 she was considering how to prepare a laboratory guide in that field but " was persuaded by colleagues to write an advanced text " ( quoted in Hutchinson, p. 107 ).

By only considering geodesic distances or other isometric properties as done in spectral shape analysis, it is possible to retrieve all cats in a database of animals independent of the pose.

By trying the killer move before other moves, a game playing program can often produce an early cutoff, saving itself the effort of considering or even generating all legal moves from a position.

By considering bets on more than one event de Finetti could justify additivity.

By considering a coordinate transformation on a manifold as a map from the manifold to itself, the transformation of covariant indices of a tensor are given by a pullback, and the transformation properties of the contravariant indices is given by a pushforward.

By 08: 00 the Russians were occupying redoubts 1, 2 and 3, and, considering it too close to the enemy, had razed redoubt No. 4.

By 1925 he was considering appointing Stewart to the Senate, to remove him from active political involvement, but was handicapped by the absence of any other Alberta representation in his cabinet.

By then considering perturbations by various mass terms they were able to calculate the superpotential in the presence of arbitrary numbers of colors and flavors, valid even when the theory is no longer weakly coupled.

By February, the opposition lords were considering accusing the Duke of York of high treason, which resulted in the king proroguing parliament on 24 February in order to protect his brother.

By 1985, their campaign had acquired sufficient traction for Congress to begin considering passing protectionist laws.

By considering the three possible outcomes ( Red Sox, Yankees, some other team ), you will see that regardless of which of the three outcomes eventuates, you lose.

By considering Riemann surface with boundary, we can couple it to the d = 1 conformal theory instead of coupling d = 2 theory to d = 0.

By not considering whether the person could afford the payments if they were to increase in the future, many of these loans may have put the borrowers at risk for default.

By considering the occurrences of values from, one has an alternate way to refer to, through.

By applying this general principle, a great number of complex compounds of osmium, ruthenium, iridium, and rhenium, with triphenylphosphine, triphenylarsine, and triphenylstibine have been obtained in this laboratory during the past few years.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

By the 1990s, many of these bands and artists had disbanded, were no longer performing, or were being carried by independent labels because their music tended to be more lyrically complex ( and often more controversial ) than mainstream Christian pop.

By 1951, 87 subroutines in the following categories were available for general use: floating point arithmetic ; arithmetic operations on complex numbers ; checking ; division ; exponentiation ; routines relating to functions ; differential equations ; special functions ; power series ; logarithms ; miscellaneous ; print and layout ; quadrature ; read ( input ); nth root ; trigonometric functions ; counting operations ( simulating repeat until loops, while loops and for loops ); vectors ; and matrices.

By allowing these functions ( and constants ) to be complex numbers, trigonometric functions and their inverses become included in the elementary functions ( see trigonometric functions and complex exponentials ).

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 ).

By analogy, stone arches are irreducibly complex — if you remove any stone the arch will collapse — yet we build them easily enough, one stone at a time, by building over centering that is removed afterward.

By contrast, Shakespeare's King John, a relatively anti-Catholic play that draws on The Troublesome Reign for its source material, offers a more " balanced, dual view of a complex monarch as both a proto-Protestant victim of Rome's machinations and as a weak, selfishly motivated ruler ".

By 1912, the discovery of vitamins was a boost for Marmite, as the spread is a rich source of the vitamin B complex ; vitamin B < sub > 12 </ sub > is not naturally found in yeast extract, but is added to Marmite during manufacture.

By the beginning of 20th century the booming and more complex mining sector in mineral-rich countries provided only slight benefits to local communities in terms of sustainability.

By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis for where runs from to, such that each of the corresponding-dimensional subspaces is-invariant.

By successively dividing out factors x − a, one sees that any polynomial with complex coefficients can be written as a constant ( its leading coefficient ) times a product of such polynomial factors of degree 1 ; as a consequence the number of ( complex ) roots counted with their multiplicities is exactly equal to the degree of the polynomial.

By a general property of the Fourier transform, that symmetry means it is the complex conjugate of:

By that time, Howard M. Thompson, the owner of Metagaming, was not happy with the TFT work done by Steve Jackson, stating that it was too complex and had taken too long.

By and large, the theories of existentialism assert that conscious reality is very complex and without an " objective " or universally known value: the individual must create value by affirming it and living it, not by simply talking about it or philosophising it in the mind.

By use of variable engine strokes from a complex crankshaft, Atkinson was able to increase the efficiency of his engine, at the cost of some power, over traditional Otto-cycle engines.

By 1972, the 8600 had reached a dead end ; the machine was so incredibly complex that it was impossible to get one working properly ; even a single faulty component would render the machine non-operational.

By many accounts, her childhood was somewhat complex.

By the end of the 19th century, a large underground complex had also been built which served as quarters for the troops inside the city.

By complex cooperative action the proteases may proceed as cascade reactions, which result in rapid and efficient amplification of an organism's response to a physiological signal.

By the late Devonian, land plants had stabilized freshwater habitats, allowing the first wetland ecosystems to develop, with increasingly complex food webs that afforded new opportunities.

By the time the enlarged project was completed in 1676, the river front measured 196 metres and the complex had fifteen courtyards, the largest being the cour d ' honneur (" court of honour ") for military parades.

By listening at the door of the room where Adolphe conducted his classes, Georges learned to sing difficult songs accurately from memory, and developed an ability to identify and analyse complex chordal structures.