Note however that both in algebra and model theory the binary operations considered are defined on all of S × S.

Note that this statement is only true when the refractive indices of both materials are real numbers, as is the case for materials like air and glass.

* Note: Contains both the original version ( Japanese audio with English subtitles only ) and the edited G version ( English audio only ).

* Note: Contains both the Japanese version and uncut English dub and audio commentary with Steve Ryfle and Stuart Galbraith IV.

Note: The above definition of group velocity is only useful for wavepackets, which is a pulse that is localized in both real space and frequency space.

) Note that not all airports are assigned codes in both systems — for example, airports that do not have airline service may not need an IATA code.

To this plan both Britain and France gave a general assent, and the Andrássy Note was adopted as the basis of negotiations.

Note that there is no contradiction among these two results ; both computations are correct.

Note that from the perspective of the smaller-mass object — from the moon, in the preceding example — a spacecraft might appear to orbit in an irregular path about the or point, but from the perspective above the orbital plane, it becomes clear that both the smaller mass and the spacecraft are orbiting the larger mass ( or more precisely, all of the objects are in orbit around the barycenter of the system ); they simply have overlapping orbital paths.

Note: the president is both the chief of state and head of government

On April 23, 1998, the Ministry of Foreign Affairs of the Republic of the Marshall Islands issued an official Circular Note, denouncing representatives of both " EnenKio " and " Melchizedek " for making fraudulent representations.

Note the coats of arms both bear on their clothing — Malcolm wears the Lion of Scotland, which historically was not used until the time of his great-grandson William I of Scotland | William the Lion ; Margaret wears the supposed arms of Edward the Confessor, her grand-uncle, although the arms were in fact concocted in the later Middle Ages.

( Note that in both of those examples those quantities would be maximized rather than minimized ).

( Note: Lava flows refer to both actively flowing lava and the solidified landforms they produce.

Note the intersection of the two circles — the distinction between short a, i and u is made by both speakers, but Arabic lacks the mid articulation of short vowels, while Hebrew lacks the distinction of vowel length.

Note that both " persons and estates " are to be protected from the arbitrary power of any magistrate, inclusive of the " power and will of a legislator.

Note that in both three-and four-level lasers, the energy of the pumping transition is greater than that of the laser transition.

Note that an enzyme that displays both PLA1 and PLA2 activities is called a Phospholipase B.

Note that both arrows point in the same direction.

Note that the letter ' h ' in both the transcription and transliteration forms should logically be omitted.

Note that only two sequences of moves by O ( both starting with center, top-right, left-mid ) lead to a draw, with the remaining sequences leading to wins from X.

Note that in this equation, both force and displacement are vector quantities.

Note that both the limit and the colimit functors are covariant functors.

Note that the weighted ski is on-edge and forming an arc ; if both legs were parallel the skis would begin to smoothly carve.

Note 2: A data transmission circuit may transfer information in ( a ) one direction only, ( b ) either direction but one way at a time, or ( c ) both directions simultaneously.

( Note that it is not the probabilities of photons appearing at various points along the detection screen that add or cancel, but the amplitudes.

Note that establishing this chain of logical implication ( or quantifying the relevant probabilities ) makes this form logically valid.

Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure.

Note that if we used the actual real-world probabilities, every security would require a different adjustment ( as they differ in riskiness ).

This article discusses computing probabilities for many commonly occurring events in the game of Texas hold ' em and provides some probabilities and odds < ref name =" odds " group =" Note "> The odds presented in this article use the notation x: 1 which translates to x to 1 odds against the event happening.

Note: Since we ’ re assuming that the voting population is large, it is reasonable and permissible to think of the probabilities as unchanging once a voter is selected for the sample.

Note also that if Af, then Af is divisible by the polynomial p, because Af contains each Af as a factor.

Note that we need not know the value of p, for the experiment to be binomial.

Note that factors p < sup > 0 </ sup > =

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

: Note the difference in the style of the subscripts: k and p are abbreviations of the words kinetic and potential, whereas p ( italic ) is the symbol for the physical quantity pressure rather than an abbreviation of the word " pressure ".

The tree is rotated on the edge joining p with its parent g, then rotated on the edge joining x with p. Note that zig-zig steps are the only thing that differentiate splay trees from the rotate to root method introduced by Allen and Munro prior to the introduction of splay trees.

Note that S is at least the length of the meridian from p to q:

Note that the bef2c compiler is not correct since it does not handle either ' p ' or string mode, but it would not be impossible to make it do so ( although the C language might not be well-suited for this ).

" Note " and " Part " are used to refer to footnotes and parts ( when parts are specifically designed ) within the same piece ; " p ." and " pp. " are used to refer to other pages within the same piece.

( Note that the conditions under which the cancellation law holds are quite strict, and this explains why Fermat's little theorem demands that p be a prime in order to make a general case for all n. For example, 2 × 2 ≡ 2 × 5 ( mod 6 ), but we cannot conclude that 2 ≡ 5 ( mod 6 ), since 6 is not prime.

Note the crucial feature: the unknown parameter p interacts with the data x only via the statistic T ( x )

Note that T is a function of p alone, while V is a function of q alone.

:• Note: Four of the dedicatees ( Turczyński, Gruenberg, Sirota, Closson ) were master class pupils of Busoni in Vienna, Jul 1908 ( Beaumont, 1987, p. 91 ), while Blanchet had been a pupil at a master class in Weimar, Jul-Sep, 1899 ( Dent ).

Note that Aut ( p ) and are naturally isomorphic in this case ( as a group is always naturally isomorphic to its opposite through ).

Fear and Trembling Note p. 78

In particular, given a point p in a Riemannian manifold, it is always possible to find metrics conformal to the given metric g for which the Ricci tensor vanishes at p. Note, however, that this is only pointwise assertion ; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.

Note that this implies that M and N must have the same dimension at p.

" ( p. 85 ) Note that Bill Runs Above did not mention the lyrics of the songs, and while they are no doubt important, most lyrics of most songs employ vocables, syllable sounds such as " ya ", " hey ", and " loi " ( p. 86 ).