Here’s a worked example of the solution of a quartic polynomial, as described in my earlier post on solving polynomials algebraically.
Archive for April, 2009
Here I summarise the recursive rules for deriving (differentiating) compound expressions.
Further to my post on build events in MSVC, I’ve noticed that sometimes MSDEV sometimes re-orders the GUIDs in a solution file.
I know a mouse and he hasn’t got a house;
I don’t know why I call him Gerald.
He’s getting rather old, but he’s a good mouse.
I have a simple Python script (sort of attached) that can read these files and sort those GUID sections into order.
Consider cars. A typical fuel efficiency is
≈ 40 mpg (miles / gallon [UK])
≈ 14 km / litre = 14 m / cc = 14 mm / mm3
Expressed differently (taking the reciprocal), this is about
≈ 4 fl oz [UK] / mi
≈ 70 ml / km = 70 cc / km = 70 mm3 / m
≈ 70,000 μ2 = 70,000 μm2 = 0.07 mm2
What’s that? It’s a unit of area.
If you imagine forming a tube of fuel as the car uses it (as the car goes along), then that’s the tube’s cross-sectional area.
Here are some functions you no longer see much.
|coversine||covers(x) = 1 − sin(x)|
|exsecant||exsec(x) = sec(x) − 1|
|hacoversine, cohaversine, havercosine||hacov(x) = covers(x) / 2|
|haversine (half versine)||hav(x) = vers(x) / 2|
|versine (versed sine)||vers(x) = 1 − cos(x)|
There seems to be some confusion about what the primary and secondary colours are. Well, it depends.
To help to remember the kinematic equations, it may be useful to know a little dimensional analysis.
Using the difference table method described in my previous post, I discovered the following formulae.
The first is not new. I do not know whether the last two are:
Here, is the nth Catalan number, is the nth Fibonacci number, is the nth d-dimensional tetrahedral number, and is the nth triangular number.
Consider a sequence C = (C0, C1, C2, …). Define:
A0,j = Cj,
Ai,j = Ai,j+1 − Ai,j, and
Ri = Ai,o.
This might be pictured:
R = (R0, R1, R2, …) is another sequence. We might call C the sequence generated by R and R the generator of C.
It turns out that
The first of these formulae is particularly handy if the the sequence R is finite (or rather, is eventually zero):
Given some unknown sequence, this formula may be used to find empirically the power-series closed form, if it has one.
For example, consider the tetrahedral numbers C = (0, 1, 4, 10, 20, 35, …). We find that R = (0, 1, 2, 1).
This is not a proof of course, but it can be a useful experimental tool. It is also useful for manipulating sum sequences and difference sequences.