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 after calling, we are left with a function that takes a single argument and returns another function, not a function that takes two arguments.

( Note that we are imagining the vector field to be like the velocity vector field of a fluid ( in motion ) when we use the terms flow, sink and so on.

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

Alice now discards three of her own cards, and replaces them with three from the top of the deck ( Note: in a player-dealt casino game there is often a rule that the dealer must discard before picking up the deck, but this is a home game so we won't worry about such details ).

Note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer.

) Note that we have omitted the Π < sub > j </ sub > n < sub > j </ sub > term, because each single-particle state can appear only once in a fermionic state.

Note that we have to restrict the sum to ordered values of m < sub > 1 </ sub >, ..., m < sub > N </ sub > to ensure that we do not count each multi-particle state more than once.

Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.

#* 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: 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 this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model ; for example, as a model of planetary motion.

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 that we require the domains of G < sub > 2 </ sub >, G < sub > 3 </ sub > to be broad enough to make the above properties meaningful.

Note we have g ( 1 ) = 1.

Note that here we are not concerned about the number of computational steps, or the size of the computer memory used.

Note that the shells exist for both protons and neutrons individually, so that we can speak of " magic nuclei " where one nucleon type is at a magic number, and " doubly magic nuclei ", where both are.

A subset T is totally ordered if for any s, t in T we have s ≤ t or t ≤ s. Such a set T has an upper bound u in P if t ≤ u for all t in T. Note that u is an element of P but need not be an element of T. An element m of P is called a maximal element ( or non-dominated ) if there is no element x in P for which m < x.

Note that, if the language is infinite, the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number n is " already " produced for a number which is less than n. If it already is produced, use the output for input n + 1 instead ( recursively ), but again, test whether it is " new ".

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 for a Euclidean function that is so established there need not exist an effective method to determine q and r in ( EF1 ).

) 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.

Note that these treatments need to be prescribed and monitored carefully to avoid harmful side-effects such as neuroleptic malignant syndrome with the antipsychotic medications.

Note that the non-inverting input of the operational amplifier will need a path for DC to ground ; if the signal source might not give this, or if that source requires a given load impedance, the circuit will require another resistor — from input to ground.

Note that S and T need not be subgroups.

Note that a system need not include source code derived in any way from AT & T Unix to meet the specification.

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 the signal-space points for BPSK do not need to split the symbol ( bit ) energy over the two carriers in the scheme shown in the BPSK constellation diagram.

" Note the distinction between states, and the need to clearly mention their status as GMO ( Genetically Modified Organism ) which makes them unacceptable to most organic food buyers.

Note however that a right inverse of F ( i. e. a functor G such that FG is naturally isomorphic to 1 < sub > D </ sub >) need not be a right ( or left ) adjoint of F. Adjoints generalize two-sided inverses.

Note that you will need a font that supports the Unicode IPA Extensions to see the IPA characters.

( Note that Japanese has no articles, and the different word order obviates any need for the relative pronoun who.

Note however that the existence of f has to be assumed from the start ; two objects B and B ′ that simply have isomorphic sub-and factor objects need not themselves be isomorphic ( for example, in the category of abelian groups, B could be the cyclic group of order four and B ′ the Klein four-group ).

Note that if usual data compression algorithms are used, files that are similar when uncompressed may be very different when compressed, and thus the entire file will need to be transferred – local changes in uncompressed files yield global changes in compressed files.

Note that the vertex degrees must be finite, but need not be bounded: it is possible to have one vertex of degree 10, another of degree 100, a third of degree 1000, and so on.

Note, however, that at the undergraduate, post-graduate and doctoral levels within these countries, there need not be any standardised process via which mathematicians of differing ability levels can be tested or examined.

Note that we need two delta functions since there are two roots to

After the outbreak of the Second World War in Europe in 1939, Wallis saw a need for strategic bombing to destroy the enemy's ability to wage war and he wrote a paper entitled " A Note on a Method of Attacking the Axis Powers ".

Note that one may need to require ' to be absolutely integrable with respect to the weight ' in order for this integral to be finite.

Note that the above examples show that the sum of diagonalizable matrices need not be diagonalizable.

Note that since u, v need to be continuous, the assumption on positivity does not diminish the generality.

Note: Corporations need to be conscious of and consistent with how internal consultant costs are accounted for on both a project and organizational level to evaluate cost effectiveness.