Question: What is "N"?
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The following natural hybrids involving "N. bongso" have been recorded.
"N. bongso" × "N. gymnamphora" "N. bongso" × "N. singalana" "N. bongso" × "N. talangensis" In his 1928 monograph "The Nepenthaceae of the Netherlands Indies", B. H. Danser mentioned what he thought was the hybrid "N. bongso" × "N. pectinata"; however, this plant material is now known to represent "N. densiflora", a species described by Danser 12 years later. 
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A model of the Peano axioms is a triple ("N", 0, "S"), where "N" is a (necessarily infinite) set, 0 ∈ "N" and "S" : "N" → "N" satisfies the axioms above.
Dedekind proved in his 1888 book, "What are numbers and what should they be" () that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic.
In particular, given two models ("N""A", 0"A", "S""A") and ("N""B", 0"B", "S""B") of the Peano axioms, there is a unique homomorphism "f" : "N""A" → "N""B" satisfying formula_38 and it is a bijection.
The second-order Peano axioms are thus categorical; this is not the case with any first-order reformulation of the Peano axioms, however. 
By means of an intense series of mathematical analogies that some physicists have termed "magical," Werner Heisenberg wrote out an equation that is the quantum mechanical analog for the classical computation of intensities.
The equation below appears in his paper of 1925.
Its general form is as follows:  formula_2 This general format indicates that some term C is to be computed by summing up all of the products of some group of terms A by some related group of terms B.
There will potentially be an infinite series of A terms and their matching B terms.
Each of these multiplications has as its factors two measurements that pertain to sequential downward transitions between energy states of an electron.
This type of rule differentiates matrix mechanics from the kind of physics familiar in everyday life because the important values are where (in what energy state or "orbital") the electron begins and in what energy state it ends, not what the electron is doing while in one or another state.
The formula looks rather intimidating, but if A and B both refer to lists of frequencies, for instance, all it says to do is perform the following multiplications and then sum them up:Multiply the frequency for a change of energy from state n to state n-a by the frequency for a change of energy from state n-a to state n-b.
and to that add the product found by multiplying the frequency for a change of energy from state n-a to state n-b by the frequency for a change of energy from state n-b to state n-c,<br>and so forth: <br>Symbolically that is: <br>f(n, n-a) * f(n-a,n-b)) +<br>f(n-a,n-b) * f(n-b,n-c) +<br>etc.
It would be very easy to do each individual step of this process for some measured quantity.
For instance, the boxed formula at the head of this section gives each needed wavelength in sequence.
The values calculated could very easily be filled into a grid as described below.
However, since the series is infinite, nobody could do the entire set of calculations.
Heisenberg originally devised this equation to enable himself to multiply two measurements of the same kind, so it happened not to matter in which order they were multiplied.
Heisenberg noticed, however that if he tried to use the same schema to multiply two variables, such as momentum, p, and displacement, q, then "a significant difficulty arises."
It turns out that multiplying a matrix of p by a matrix of q gives a different result from multiplying a matrix of q by a matrix of p. It only made a tiny bit of difference, but that difference could never be reduced below a certain limit, and that limit involved Planck's constant, h. More on that later.
Below is a very short sample of what the calculations would be, placed into grids that are called matrices.
Heisenberg's teacher saw almost immediately that his work should be expressed in a matrix format because mathematicians already were familiar with how to do computations involving matrices in an efficient way.
formula_3 (Equation for the conjugate variables momentum and position) Matrix of pMatrix of qThe matrix for the product of the above two matrices as specified by the relevant equation in Heisenberg's 1925 paper is:Where:<br>A=p(n︎←n-a)*q(n-a︎←n-b)+p(n︎←n-b)*q(n-b︎←n-b)+p(n︎←n-c)*q(n-c︎←n-b)+...B=p(n-a︎←n-a)*q(n-a︎←n-c)+p(n-a︎←n-b)*q(n-b︎←n-c)+p(n-a︎←n-c)*q(n-c︎←n-c)+...C=p(n-b︎←n-a)*q(n-a︎←n-d)+p(n-b︎←n-b)*q(n-b︎←n-d)+p(n-b︎←n-c)*q(n-d︎←n-d)+...and so forth.
If the matrices were reversed, the following values would result:A=q(n︎←n-a)*p(n-a︎←n-b)+q(n︎←n-b)*p(n-b︎←n-b)+q(n︎←n-c)*p(n-c︎←n-b)+...<br>B=q(n-a︎←n-a)*p(n-a︎←n-c)+q(n-a︎←n-b)*p(n-b︎←n-c)+q(n-a︎←n-c)*p(n-c︎←n-c)+...<br>C=q(n-b︎←n-a)*p(n-a︎←n-d)+q(n-b︎←n-b)*p(n-b︎←n-d)+q(n-b︎←n-c)*p(n-d︎←n-d)+...and so forth.
Note how changing the order of multiplication changes the numbers, step by step, that are actually multiplied. 
The following natural hybrids involving "N. bicalcarata" have been recorded.
"N. ampullaria" × "N. bicalcarata" ("N. ampullaria" × "N. gracilis") × "N. bicalcarata" [="N. × trichocarpa" × "N. bicalcarata"] "N. bicalcarata" × "N. gracilis" [="N. × cantleyi"] "N. bicalcarata" × "N. mirabilis" (including "N. bicalcarata" × "N. mirabilis" var.
"echinostoma") "N. bicalcarata" × "N. rafflesiana" ?
("N. bicalcarata" × "N. rafflesiana") × "N. mirabilis" var.
"echinostoma" None of these hybrids are known to be colonised by "C. schmitzi" as is the parent species "N. bicalcarata".