Note that completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one.

Note for completeness that condition ( 1 ) fails: any map t: B → A must map every two-cycle to the identity because the map has to be a group homomorphism, while the order of a two-cycle is 2 which can not be divided by the order of the elements in A other than the identity element, which is 3 as A is the alternating subgroup of, or namely the cyclic group of order 3.

Note that the latter two structures extend the application of these principles beyond mere completeness requirements by introducing an additional operation of negation.

Note also that a join-semilattice which is directed complete is almost a complete lattice ( possibly lacking a least element ) -- see completeness ( order theory ) for details.

( Note: as far as the author is aware of, this criterion of completeness is never used.

Note that can be different from the statement of the logical negation of, depending on the completeness of the inference algorithm and thus also on the formal logic system.

Note that this pressure increase is more than a simple 10: 1 compression ratio would indicate ; this is because the gas is not only compressed, but the work done to compress the gas has also heated the gas and the hotter gas will have a greater pressure even if the volume had not changed.

* In the Star Wars prequels, the galactic capital planet Coruscant has buildings many miles tall, and approaches the completely built-over condition of Trantor in Isaac Asimov's classic Foundation trilogy ( Note: due to this, Coruscant was originally to be named Jhantor but was eventually renamed Coruscant ).

Note, however, that the Secretary has a reporting line to the Board of Directors, regardless of any other reporting lines conferred by concurrent titles.

Note, however, that the Treasurer has a reporting line to the Board of Directors, regardless of any other reporting lines conferred by concurrent titles.

Note that no consistent timeline has yet accommodated every single Conan story.

Note that the non-penalized team has the option to decline any penalty it considers disadvantageous, so a losing team cannot indefinitely prolong a game by repeatedly committing infractions.

Note that the chocolate pictured here is soft, can easily be crumbled by hand, and already has sugar added.

Note: In the United Kingdom the first trilogy has been collected as an omnibus edition titled Corum, Swords of Corum and most recently Corum: The Prince in the Scarlet Robe ( vol.

Note that a mixture with a given composition has one boiling point at a given pressure, when the components are mutually soluble.

Notable jazz bassists from the 1940s to the 1950s included bassist Jimmy Blanton ( 1918 – 1942 ) whose short tenure in the Duke Ellington Swing band ( cut short by his death from tuberculosis ) introduced new melodic and harmonic solo ideas for the instrument ; bassist Ray Brown ( 1926 – 2002 ), known for backing Beboppers Dizzy Gillespie, Oscar Peterson, Art Tatum and Charlie Parker, and forming the Modern Jazz Quartet ; hard bop bassist Ron Carter ( born 1937 ), who has appeared on 3, 500 albums make him one of the most-recorded bassists in jazz history, including LPs by Thelonious Monk and Wes Montgomery and many Blue Note Records artists ; and Paul Chambers ( 1935 – 1969 ), a member of the Miles Davis Quintet ( including the landmark modal jazz recording Kind of Blue ) and many other 1950s and 1960s rhythm sections, was known for his virtuosic improvisations.

Note that the first element in a sequence has the index one and that $ refers to the length ( i. e. total number of elements ) of the sequence.

Note that purnima or pornima is Sanskrit for full moon, which has also become the Malay word for full moon purnama.

// Note this code sample has been mangled ( missing attr init / scope?

Methods PUT and DELETE are defined to be idempotent, meaning that multiple identical requests should have the same effect as a single request ( Note that idempotence refers to the state of the system after the request has completed, so while the action the server takes ( e. g. deleting a record ) or the response code it returns may be different on subsequent requests, the system state will be the same every time ).

Like all subatomic particles, hadrons are assigned quantum numbers corresponding to the representations of the Poincaré group: J < sup > PC </ sup >( m ), where J is the spin quantum number, P the intrinsic parity ( or P-parity ), and C, the charge conjugation ( or C-parity ), and the particle's mass, m. Note that the mass of a hadron has very little to do with the mass of its valence quarks ; rather, due to mass – energy equivalence, most of the mass comes from the large amount of energy associated with the strong interaction.

( Note: The Turkish word " ocak " has the double meaning of home and fireplace, like the Spanish " hogar ".

Note: Ringette distances itself from hockey as it has its own set of rules and is closely related to a mix of lacrosse and basketball.

Note that, in the first part of the second premise, the present-tense verb, " putting ," has been changed to the infinitive, " to put.

Note: Work has started in 2006 on a city wide WiFi project called Hotcity.

Note that one denomination dominates the non-Mormon section of the movement: Community of Christ, which has about 250, 000 members.

Note that not every probability distribution has a defined mean ; see the Cauchy distribution for an example.

Note: A number of other plants have been suggested by biblical scholars, e. g., Most notably Ginseng which looks similar to the mandrake root and has fertility enhancing properties, for which it was picked by Reuben in the Bible.

Note that although Be < sub > 2 </ sub > has a bond order of 0 according to MO, there is experimental evidence of a highly unstable Be < sub > 2 </ sub > compound having a bond length of 245 pm and bond energy of 10kJ / mol.

Note that E ( at 2f ) has an equal-size, real and inverted image ; I ( at f ) has its image at infinity ; and K ( at f / 2 ) has a double-size, virtual and upright image.

Note that this is subtly different to the ' thanks ' list in the original BBC Model B.

( Note, however, that a GC stacking interaction with the next base pair is geometrically different from a CG interaction.

Note that this is different from the recursive languages which can be decided by an always-halting Turing machine.

Note that the points must all be different numbers for them to count towards the fire bet.

: Note: Individual casinos may pay some of these bets at different payout ratios than those listed below.

Years are usually counted from a particular starting point, usually called the epoch, with era referring to the particular period of time ( Note the different use of the terms in geology ).

Note that pure application of this method may provide only a partial ordering on the strings, since different strings can represent the same number ( as with " 2 " and " 2. 0 ", or when scientific notation is used, " 2e3 " and " 2000 ").

Note: This study only involved male subjects which may be significant as sex steroids may have different effects in the responses to cortisol administration.

Note: In Australia, this is different.

Note that this is different from the original Elfengold published by White Wind.

Note that the aspectual systems of certain dialects of English, such as African-American Vernacular English ( see for example habitual be ), and of creoles based on English vocabulary, such as Hawaiian Creole English, are quite different from those of standard English, and often distinguish aspect at the expense of tense.

Note the map indicates the contemporary coast lines, which are different, in parts of the English Channel and the North Sea, from those of Roman Britain.

Note that the orange circles appear of different sizes, even though equal.

Note that different requirements apply to the other related concepts ( cryptographic hash functions, checksums, etc.

: is a spherical harmonic function of degree ℓ and order m. Note that the generalized Laguerre polynomials are defined differently by different authors.

Note the different check digits in each.

( Note: this study-cycle follows a different schedule than the regular one ; contains extensive archives in English ).

( Note: Astronomy should not be confused with astrology, which assumes that people's destiny and human affairs in general are correlated to the apparent positions of astronomical objects in the sky -- although the two fields share a common origin, they are quite different ; astronomers embrace the scientific method, while astrologers do not.

Note this does not restrict dormant forms of life to these environments, so this argument only contradicts the widest interpretation of panspermia — that life is ongoing and is spread across many different environments throughout the Universe — and presupposes that any life needs those elements, which the proponents of alternative biochemistries do not consider certain.

Note that " secure " here has a precise mathematical meaning, and there are multiple different ( meaningful ) definitions of what it means for an encryption scheme to be " secure ".

Note that an eight-dimensional vector can be specified in many different ways depending on what basis is chosen for the space.

Note that the term " automatic revolver " has two different meanings.

* The probability a set will have features different and features the same is ( Note: d = 0 all four features the same is impossible as no two cards are identical ).