# 3.3 Nonstandard interpretations (optional)

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How non-standard interpretations can provide insight into tough problems.

## Prime factorization

Note that there are other possible interpretations of $\mathrm{prime}$ . For example, since one can multiply integer matrices,there might be a useful concept ofprime matrices.

For example: Consider only the numbers $F=\{1, 5, 9, 13, \}$ that is, $F=\{k\colon k\in \mathbb{N}\}$ . It's easy to verify that multiplying two of these numbers still resultsin a number of the form $4k+1$ . Thus it makes sense to talk of factoring such numbers:We'd say that 45 factors into $59$ , but 9 is consideredprime since it doesn't factor into smaller elements of $F$ .

Interestingly, within $F$ , we lose unique factorization: $441=9\times 49=21\times 21$ , where each of 9, 21, and 49 are prime, relative to $F$ ! (Mathematicians will then go and look for exactly whatproperty of a multiplication function are needed, to guarantee unique factorization.)

The point is, that all relations in logical formula need to be interpreted. Usually, for numbers, we usea standard interpretation, but one can consider those formulas in different, non-standard interpretations!

## The poincarDisc

A long outstanding problem was that of Euclid's parallel postulate: Given a line and a point not on the line,how many lines parallel to the first go through that point? Euclid took this as an axiom(unable to prove that it followed from his other axioms). Non-Euclidean geometries of Lobachevsky and Riemann took differentpostulates, and got different geometries. However, it was not clear whether these geometrieswere sound whether one could derive two different results that were inconsistent with each other.

Henri Poincardeveloped an ingenious method for showing that certain non-Euclidean geometries are consistentor at least, as consistent as Euclidean geometry.Remember that in Euclidean geometry, the conceptspointandlineare left undefined, and axioms are built on top of them ( e.g. ,two different lines have at most one point in common). While it's usually left to common sense to interpretpoint,line, anda point is on a line, any interpretation which satisfies the axiomsmeans that all theorems of geometry will hold.

The Poincardisc is one such interpretation:pointis taken to meana point in the interior of the unit disc, andlineis taken to meana circular arc which meets the unit disc at right angles. So a statement liketwo points determine a linecan be interpreted as

[*] For any two pointsinside the disc, there is exactly one circular arc which meets the disc at right angles.
Indeed, this interpretation preserves all of Euclid's postulates except for the parallel postulate. You can see thatfor a given line and a point not on it, there are an infinite number of parallel (that is, non-intersecting) lines.

(Note that the distance function is very different within the Poincardisc; in fact the perimeter of the disc is off at infinity.Angles, however, do happen to be preserved.)

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s.
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
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SUYASH
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Ebrahim
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Ebrahim
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s.
Graphene has a hexagonal structure
tahir
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Cied
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
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Uday
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Stotaw
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Azam
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Prasenjit
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Azam
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Prasenjit
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Uday
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Prasenjit
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
how did you get the value of 2000N.What calculations are needed to arrive at it
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