Archive for April, 2009

Example of the Solution of a Quartic Polynomial

Thursday, 30 April 2009

Here’s a worked example of the solution of a quartic polynomial, as described in my earlier post on solving polynomials algebraically.



Differentiation Formulae

Wednesday, 29 April 2009

Here I summarise the recursive rules for deriving (differentiating) compound expressions.


Further Calculator Statistics

Tuesday, 28 April 2009

This is further to my post on calculator statistics.

If the calculator also stores the six values \Sigma x^3, \Sigma x^4 and \Sigma x^2y, then it can calculate the linear regression to a parabola.


MSVC Projects, Source Control and GUIDs

Tuesday, 28 April 2009

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.


Cross-section of Fuel

Monday, 27 April 2009

Here’s another, brief, post on the subject of dimensional analysis; it concerns a curious interpretation of dimensions. [Previous posts concerned statistics and kinematics.]

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.

Forgotten Functions

Sunday, 26 April 2009

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)

Colour Mixing

Sunday, 26 April 2009

There seems to be some confusion about what the primary and secondary colours are. Well, it depends.


Dimensional Analysis of the Kinematic Equations

Sunday, 26 April 2009

To help to remember the kinematic equations, it may be useful to know a little dimensional analysis.


Catalan and Fibonacci Formulae

Sunday, 26 April 2009

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:

\bigtriangleup_d(n) = \dbinom{n+d-1}{d} = \sum_{k=1}^{d} \dbinom{d-1}{k-1} \dbinom{n}{k};

C_n = \sum_{i=0}^{n} \dbinom{n}{i} \bigtriangleup(F_{i-1});

\sum_{i=0}^{n-1} C_i = \sum_{i=0}^{n} \dbinom{n}{i} \bigtriangleup(F_{i-2}).

Here, C_n = \dfrac{1}{n+1}\dbinom{2n}{n} is the nth Catalan number, F_n is the nth Fibonacci number, \bigtriangleup_d(n) is the nth d-dimensional tetrahedral number, and \bigtriangleup(n) = \bigtriangleup_2(n) is the nth triangular number.

Difference Tables

Saturday, 25 April 2009

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:

C0 C1 C2 C3 C4 C5
|| || || || || ||
R0 = A0,0 A0,1 A0,2 A0,3 A0,4 A0,5
R1 = A1,0 A1,1 A1,2 A1,3 A1,4
R2 = A2,0 A2,1 A2,2 A2,3
R3 = A3,0 A3,1 A3,2

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

C_n = \sum_{k=0}^{n} \dbinom{n}{k} R_k;

R_n = \sum_{k=0}^{n} (-1)^{n-k} \dbinom{n}{k} C_k.

The first of these formulae is particularly handy if the the sequence R is finite (or rather, is eventually zero):

C_n = \sum_{k} R_k \dbinom{n}{k}.

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).

Therefore C_n = \dbinom{n}{1} + 2 \dbinom{n}{2} + \dbinom{n}{3} = \dbinom{n+2}{3}.

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.