Note that this definition cannot be expressed in the language of set theory itself.

Note that with this definition, two elements a and b may very well have several greatest common divisors, or none at all.

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 however that Kierkegaard's meaning of " nihilism " differs from the modern definition in the sense that, for Kierkegaard, levelling led to a life lacking meaning, purpose or value, whereas the modern interpretation of nihilism posits that there was never any meaning, purpose or value to begin with.

Note that the example net given above on the neighborhood system of a point x does indeed converge to x according to this definition.

Note that an alternate definition of preorder requires the relation to be irreflexive.

Note that < em > finite </ em > in this definition is essential.

Note that the tensor rank of a matrix can also mean the minimum number of simple tensors necessary to express the matrix as a linear combination, and that this definition does agree with matrix rank as here discussed.

Note, however, that this definition is not universal.

Note that this definition is tied to the critical density of the present cosmological era: the critical density changes with cosmological time, but the energy density due to the cosmological constant remains unchanged throughout the history of the universe.

Note: if as in the simplified definition, then is automatically 1, and the order of modulo is automatically.

Note that the definition is somewhat ambiguous because the user of a program can be another program and, if the latter is a portion of the set of program that is required to be interoperable, it might well be that it does need to have knowledge of the characteristics of other units.

Note that when this definition is used, the connection between the NA and the acceptance angle of the fiber becomes only an approximation.

Note that with this definition, unlike SNR, a SINAD reading can never be less than 1 ( i. e. it is always positive when quoted in dB ).

Note that some references may use the reciprocal as the definition.

Note that the enriched structure on hom-sets is a consequence of the three axioms of the first definition.

Note that this definition of the negative binomial distribution does not easily generalize to a positive, real parameter r.

Note that the definition of PSPACE-completeness is based on asymptotic complexity: the time it takes to solve a problem of size n, in the limit as n grows without bound.

Then x is an interior point of S if there exists a neighbourhood of x which is contained in S. Note that this definition does not depend upon whether neighbourhoods are required to be open.

Note that by definition, a linear combination involves only finitely many vectors ( except as described in Generalizations below ).

Note that whereas the Reynolds number and Grashof number are subscripted with a length scale variable, the Prandtl number contains no such length scale in its definition and is dependent only on the fluid and the fluid state.

Note that this definition of V depends on the gauge choice for the vector potential A ( the gradient of any scalar field can be added to A without changing B ).

Note that this definition is very different from that given by Weberling.

" Note that in a cryptic clue, there is almost always only one answer that fits both the definition and the wordplay, so that when you see the answer, you know it is the right answer-although it can sometimes be a challenge to figure out why it is the right answer.

In radar applications, it ranges from 12-18 GHz according to the formal definition of radar frequency band nomenclature in IEEE Standard 521-2002 .< ref > Note that in the band 11. 2 – 12 GHz the working definitions of K < sub > u </ sub > band and X band overlap ; satellite communications engineers would generally regard frequencies above 11. 2 GHz as being part of the K < sub > u </ sub > band .</ ref >

Note that is positive while is negative ; in any reversible work-extracting process, entropy is overall not increased, but rather is moved from a hot ( high-entropy ) system to a cold ( low-entropy one ), decreasing the entropy of the heat source and increasing that of the heat sink.

Note that the above equation is flawed — as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to the third law of thermodynamics.

* Maximum entropy classifier ( aka logistic regression, multinomial logistic regression ): Note that logistic regression is an algorithm for classification, despite its name.

Note, however, that if you were looking at an isolated system which had reached equilibrium long in the past, so that any departures from equilibrium were the result of random fluctuations, then the backwards prediction would be just as accurate as the forward one, because if you happen to see the system in a nonequilibrium state it is overwhelmingly likely that you are looking at the minimum-entropy point of the random fluctuation ( if it were truly random, there's no reason to expect it to continue to drop to even lower values of entropy, or to expect it had dropped to even lower levels earlier ), meaning that entropy was probably higher in both the past and the future of that state.

Note, however, that the above entropic uncertainty function is distinctly different than the quantum Von Neumann entropy represented in phase space.

Note that these extra bits of entropy will not apply to other mmap () segments, only libraries.

Note: If 1/2-inch panel board is used inside and out, or 5/8-inch one side and 3/8-inch the other, and 1/8-inch glass is used, stock lumber in Af, Af, and Af can be used in making the glass panels.

Note that the acid can be the charged species and the conjugate base can be neutral in which case the generalized reaction scheme could be written as HA < sup >+</ sup > H < sup >+</ sup > + A.

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, however, that the fifth article did not completely deny perseverance of the saints ; Arminius, himself, said that " I never taught that a true believer can … fall away from the faith … yet I will not conceal, that there are passages of Scripture which seem to me to wear this aspect ; and those answers to them which I have been permitted to see, are not of such as kind as to approve themselves on all points to my understanding.

Note that wav files can also be encoded with the gsm codec.

Note that arrows between categories are called functors, subject to specific defining commutativity conditions ; moreover, categorical diagrams and sequences can be defined as functors ( viz.

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

Note that for any fixed w the function f ( x ) = F ( w x ) is computable ; thus the universality property states that all computable functions of one variable can be obtained in this fashion.

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

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 ").

These polar unit vectors can be expressed in terms of Cartesian unit vectors in the x and y directions, denoted i and j respectively :< ref > Note: unlike the Cartesian unit vectors i and j, which are constant, in polar coordinates the direction of the unit vectors u < sub > r </ sub > and u < sub > θ </ sub > depend on θ, and so in general have non-zero time derivatives .</ ref >

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 that this document type declaration can only appear after the optional XML declaration, and before the document body, if the document syntax conforms to XML.

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 if the operations are performed in a colour space where γ is not equal to 1 then the operation will lead to non-linear effects which can potentially be seen as aliasing artifacts ( or ' jaggies ') along sharp edges in the matte.

: Note that the equivalence relation generated in this manner can be trivial.

Note that by reversing the equation, we can swap < math > < </ math > and, for example:

Note that one can also define a contravariant functor as a covariant functor on the dual category.

Note: Many of the essays found in these works have been individually translated and can be found in the English collections.

It is unclear which version of the paradox is stronger .< ref group =" Note "> Let be the number of civilizations ( per unit volume ) that can be seen at a radius.

Hunters use the clues to try to find the benchmark ; the benchmark can be logged as Found, Not Found, Note, or Destroyed.

Note that due to the reduced number of colors in the image, there are obvious display issues. GIF is palette-based: the colors used in an image ( a frame ) in the file have their RGB values defined in a palette table that can hold up to 256 entries, and the data for the image refer to the colors by their indexes ( 0 – 255 ) in the palette table.

Note also that ' br ' is an empty element in that, while it may have attributes, it can take no content and it may not have an end tag.

Note that the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to n − 1, i. e..