Note that any matrix obtained by multiplying by a complex scalar λ determines the same transformation, so a Möbius transformation determines its matrix only up to scalar multiples.

Note that the connection coefficients become Möbius transformations on the triangle maps.

Note that if we regard the product as a vector space, then B is not a linear transformation of vector spaces ( unless or ) because, for example.

Note that one can always normalize the weights by making the following transformation on the weights.

Note: Recently, another foundation to construct the DFT without the Hohenberg – Kohn theorems is getting popular, that is, as a Legendre transformation from external potential to electron density.

Note that energy can be exchanged with the flow in an isentropic transformation, as long as it doesn't happen as heat exchange.

Note that when a transformation decreases symmetry, a transformation of the same kind ( the inverse ) obviously for some patterns increases the symmetry.

Note that natural transformations can be composed: if μ ( X ): F ( X ) → G ( X ) is a natural transformation from the functor F: C → D to the functor G: C → D, and η ( X ): G ( X ) → H ( X ) is a natural transformation from the functor G to the functor H, then the collection η ( X ) μ ( X ): F ( X ) → H ( X ) defines a natural transformation from F to H. With this composition of natural transformations ( known as vertical composition, see natural transformation ), D < sup > C </ sup > satisfies the axioms of a category.

Note that any distribution can be converted to canonical form by rewriting as and then applying the transformation.

Note that, even if the ' drastic set of assumptions ' does not work for some particular polynomial P, P can be transformed into a related polynomial Q for which the assumptions are correct, e. g. by adding a suitable complex number to give distinct roots distinct magnitudes if necessary ( which it will be if some roots are complex conjugates ), and then repeatedly applying the root squaring transformation used in Graeffe's method enough times to make the smaller roots significantly smaller than the largest root ( and so, clustered in comparison ); the Graeffe's method approximation can be used to start the new iteration for Laguerre's method.

: Note that the transformation law for 2-jets is second order in the coordinate transition functions.

: Note that the transformation law is second order in the coordinate transition functions.

Note that in the limit, the gauge sector approaches one of the pure gauge transformation, which is the same as the pure gauge transformation to which the BPST instanton approaches as at t = 0, hence establishing the connection between the sphaleron and the instanton.

Note: According to 5 Steps To A 5 AP Psychology, transduction means the transformation of stimulus energy to the electrochemical energy of neural impulses.

Note that though the ideal gas itself contains no additional energy, the compressed gas does exert an increased pressure.

Note that chemists often write H < sup >+</ sup >( aq ) and refer to the hydrogen ion when describing acid-base reactions but the free hydrogen nucleus, a proton, does not exist alone in water, it exists as the hydronium ion, H < sub > 3 </ sub > O < sup >+</ sup >.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

* Note: Botswana does not have a westernmost point as the western section is formed by the 22nd meridian of longitude east of Greenwich.

Note that the associative property does not hold for expressions that include non-linear operators, such as the antilinear time reversal operator in physics.

Note that traditional Berkeley DB also supports a Java API, but it does so via JNI and thus requires an installed native library.

* Note: technically Chad does not have an easternmost point, the northern section of the border being formed by the 24 ° of latitude

Note: in Greek, the term for " consul " is " ypatos ", which translates as " supreme one ", and hence does not necessarily imply a joint office.

Note that the validity of a conditional proof does not require that the CPA is actually true, only that if it is true it leads to the consequent.

Note that in many cases, digital cable telephone service is separate from cable modem service being offered by many cable companies and does not rely on Internet Protocol ( IP ) traffic or the Internet.

Note that this does not require 100 % awareness from the onset of the data warehousing effort, no master plan is required upfront.

Note that a dominatrix does not necessarily dominate a male partner ; a dominatrix may well have female submissives, nor must the role of a dominatrix involve pain toward the submissive ; her domination can be verbal, involving humiliating tasks and servitude.

( Note that anti-trust regulation does take into account cases when only large companies are viable because of the nature of an industry, but favors many participants over few, whenever possible.

Note that materialism does not necessarily imply egoism, as indicated by Karl Marx, and the many other materialists who espoused forms of collectivism.

( Note, chart does not include population statistics.

Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.

Note that Reynolds does not object to the Marxist use of the term feudalism.

* * Note: Guinea does not have a northern-most point, this section of the border being formed by a straight horizontal line

* * Note: Guinea-Bissau does not have a northern-most point, the border here being formed by a straight horizontal line

Note that chair does not change form between " the chair is here " ( subject ) and " I saw the chair " ( direct object ), a distinction made by word order and context.

Note that ƒ cannot be extended to a homomorphism of rings ( from the complex numbers to the real numbers ), since it does not preserve addition:

Note that this " high temperature " approximation does not distinguish between fermions and bosons.

* * Note: Morocco does not have a southern-most point, the border being formed by a straight horizontal line

Note: In cases where the optical design of a microscope produces an appreciable lateral separation of the two beams we have the case of classical interference microscopy, which does not result in relief images, but can nevertheless be used for the quantitative determination of mass-thicknesses of microscopic objects.

( Note that while Mercury does allow impure functionality, it is not necessary in most software, and serves primarily as a way of calling foreign language code.

Note that just because a player is described as being " at bat " in this sense, he will not necessarily be given an at bat in his statistics ; the phrase actually signifies a plate appearance ( assuming it is eventually completed ).

Note: Concentrations of Cr ( VI ) in US municipal drinking water supplies reported by EWG are within likely, natural background levels for the areas tested and not necessarily indicative of industrial pollution ( CalEPA Fact Sheet ), as asserted by EWG.

Note also that the terms are not always treated as mutually exclusive ; there is not necessarily anything contradictory in the statement that " the language of the Pennsylvania Dutch is a dialect of German ".

Note that support of ' levels ' by a PBX is not necessarily incremental.

Note also that the unraveling of the chunks can be done in any place in the literate program text file, not necessarily in the order they are sequenced in the enclosing chunk, but as is demanded by the logic reflected in the explanatory text that envelops the whole program.

Note that parameters of a function are not necessarily in parentheses but separated by spaces.

Note that the term byte also had this ambiguity ; at one time, byte meant a set of bits but not necessarily 8.

Note that reporting the beliefs of others in a non-fiction format is not necessarily an endorsement of the ultimate veracity of those beliefs, it is simply saying it is true that people believe them ( for such topics as mythology, religion ).

Note that this method does not reduce total radiated energy, and therefore systems are not necessarily less likely to cause interference.

Note: The English is not necessarily a direct translation of the Japanese.

Note that the aperture stop is not necessarily the smallest stop in the system.

Note that one can support Causal Determinism without necessarily supporting Logical Determinism and vice versa ( depending on one's views on the nature of time, but also randomness ).

Note that this concept of boundedness has nothing to do with finite size, and that a subset S of a bounded poset P with as order the restriction of the order on P is not necessarily a bounded poset.

Note that any present day null constraint on the time variation of alpha does not necessarily rule out time variation in the past.

Note however, that even byte-by-byte inspection will not necessarily uncover such information since it can be easily concealed by encryption, steganography, or simple unfamiliarity.

Note that video monitoring does not necessarily guarantee that a human response is made to an intrusion.

Note: Because of the many variations in the rules of Skat, the rules below are necessarily general, although rules not found in official German tournament play are marked as such.

Note that this does not necessarily mean he is committing an offside offence ; it only becomes an offence if the ball were to be played to him at this moment, whether or not he is in an offside position when he receives the ball, as he could receive the ball in an onside position but he'd still have committed an offside offence.

" Note that this is not necessarily the first album released by an artist.

Note that this zonation is based on flora ; animals do not necessarily follow the same biogeographic boundaries.

Note that the approximate point spectrum and residual spectrum are not necessarily disjoint ( however, the point spectrum and the residual spectrum are ).

( Note that this hypothesis does not necessarily contradict the scavenging hypothesis: both subsistence strategies could have been in use – sequentially, alternating or even simultaneously.