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 the operators operate on elements from the domain and range, respectively, of the " one-to-one " and " onto " function ƒ.

The Julia set of a function ƒ is commonly denoted J ( ƒ ), and the Fatou set is denoted F ( ƒ ).< ref > Note that for other areas of mathematics the notation can also represent the Jacobian matrix of a real valued mapping between smooth manifolds .</ ref > These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.

Note that in particular for ƒ < sub > 2 </ sub > if we set c equal to e, the base of the natural logarithm, then we get that e < sup > x </ sup > is a transcendental function.

Note that in an infinite dimensional space, we can have a bilinear form ƒ for which is injective but not surjective.

) Note that the same set of points would not have, as an accumulation point, any point of the open unit interval ; hence that space cannot be compact.

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

Note that any valid SGML or XML document that references an external subset in its DTD, or whose body contains references to parsed external entities declared in its DTD ( including those declared within its internal subset ), may only be partially parsed but cannot be fully validated by validating SGML or XML parsers in their standalone mode ( this means that these validating parsers will not attempt to retrieve these external entities, and their replacement text will not be accessible ).

Although siblinghood is symmetric ( if A is a sibling of B, then B is a sibling of A ) and transitive on any 3 distinct people ( if A is a sibling of B and C is a sibling of B, then A is a sibling of C, provided A is not C ( Note that " is a sibling of " is NOT a transitive relation, since A R B, and B R A implies A R A by transitivity )), it is not reflexive ( A cannot be a sibling of A ).

Note that the ICF and ZAAP are essentially identical to CP, but distinguished for software cost control: they are slightly restricted such they cannot be used to run arbitrary operating systems, and thus do not count in software licensing costs ( which are typically based on the number of CPs ).

Note that if n < sub > 1 </ sub > and n < sub > 2 </ sub > are the same, the antisymmetric expression gives zero, which cannot be a state vector as it cannot be normalized.

Note that some metres have an overall rhythmic pattern to the line that cannot easily be described using feet.

Note that, although capitalization can aid in recognizing named entities in languages such as English, this information cannot aid in determining the type of named entity, and in any case is often inaccurate or insufficient.

Note that inferior courts cannot evade binding precedent of superior courts, but a court can depart from its own prior decisions.

( Note that the propagation of the prion depends on the presence of normally folded protein in which the prion can induce misfolding ; animals which do not express the normal form of the prion protein cannot develop nor transmit the disease.

Note that with this construction multiple preorders "" can give the same relation "<", so without more information, such as the equivalence relation, "" cannot be reconstructed from "<".

Note: Christian Scientists, Quakers and the Salvation Army cannot be classified as specifically opposing infant baptism, since they generally do not observe baptism in any form.

Since we've assumed that we can infallibly identify programs for computing squares, we can determine whether t, which depends on a and i, is such a program, and that for every a and i ; thus we have obtained a program that decides whether program a halts on input i. Note that our halting-decision algorithm never executes t, but only passes its description to the squaring-identification program, which by assumption always terminates ; since the construction of the description of t can also be done in a way that always terminates, the halting-decision cannot fail to halt either.

Note: As in all track-and-field events, results cannot count towards records if they are Wind assisted (> 2. 0 m / s ).

Signal processors, such as four-phase modems, cannot change the DSR, but the modulation rate depends on the line modulation scheme, in accordance with Note 4.

**** Note that this axiom precludes the possibility that the consumer cannot decide, and that a consumer is able to make this comparison with respect to every conceivable bundle of goods.

Note that this leaves the fuel at atmospheric pressure, and therefore it cannot travel into a throat which has been pressurized by a supercharger mounted upstream ; in such cases, the entire carburetor must be contained in an airtight pressurized box to operate.

Note the distinction with virtual memory systems ; in this case, the entirety of a process's addressable space must be in core memory, must be contiguous, and its size cannot be larger than the real memory capacity.

Note that one formulation of AC is that the Cartesian product of a family of nonempty sets is nonempty ; but since the empty set is most certainly compact, the proof cannot proceed along such straightforward lines.

Note that and can be perpendicular, but cannot be parallel.

Note that a hazard cannot be played unless one's opponent is moving, except for the Speed Limit.

( 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: 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 extent of these interests and how available they will be for the public to enjoy.

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

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.

Most European domestic power supplies run at 230 V, so the current drawn by a particular European appliance ( in Europe ) will be less than for an equivalent American one ( in the United States ).< ref group =" Note "> The formula for power is given by

Note that this premise uses the phrase " is not ", a form of " to be "; this and many other examples show that he did not intend to abandon " to be " as such.

Note that strengthened negations may be compatible with weakened forms of AC.

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 that the RGB color may be premultiplied, hence saving the additional multiplication before RGB in the equation above.

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

Note that this expression is not well-defined for all time series or processes, because the variance may be zero ( for a constant process ) or infinite.

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 that the requirement that the maps be continuous is essential ; if X is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded.

( Note that K is the SI unit for temperature ( kelvin ) and should not be confused with k, the SI prefix for kilo.

Note that this calendar will normally need to add a 53rd week to every 5th or 6th year, which might be added to December or might not be, depending on how the organization uses those dates.

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

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.