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

### Example of the Solution of a Quartic Polynomial

Thursday, 30 April 2009### Differentiation Formulae

Wednesday, 29 April 2009Here I summarise the recursive rules for deriving (differentiating) compound expressions.

### Further Calculator Statistics

Tuesday, 28 April 2009This is further to my post on calculator statistics.

If the calculator also stores the six values , and , then it can calculate the linear regression to a parabola.

### MSVC Projects, Source Control and GUIDs

Tuesday, 28 April 2009Further 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 2009Here’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 / mm^{3}

Expressed differently (taking the reciprocal), this is about

≈ 4 fl oz [UK] / mi

≈ 70 ml / km = 70 cc / km = 70 mm^{3} / m

≈ 70,000 μ^{2} = 70,000 μm^{2} = 0.07 mm^{2}

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 2009Here 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 2009There 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 2009To help to remember the kinematic equations, it may be useful to know a little dimensional analysis.

### Catalan and Fibonacci Formulae

Sunday, 26 April 2009Using 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 *n*^{th} Catalan number, is the *n*^{th} Fibonacci number, is the *n*^{th} d-dimensional tetrahedral number, and is the *n*^{th} triangular number.

### Difference Tables

Saturday, 25 April 2009Consider a sequence **C** = (C_{0}, C_{1}, C_{2}, …). Define:

A_{0,j} = C* _{j}*,

A

_{i,j}= A

_{i,j+1}− A

_{i,j}, and

R

_{i}= A

_{i,o}.

This might be pictured:

C_{0} |
C_{1} |
C_{2} |
C_{3} |
C_{4} |
C_{5} |
||||||||||

|| | || | || | || | || | || | ||||||||||

R_{0} |
= | A_{0,0} |
A_{0,1} |
A_{0,2} |
A_{0,3} |
A_{0,4} |
A_{0,5} |
… | |||||||

R_{1} |
= | A_{1,0} |
A_{1,1} |
A_{1,2} |
A_{1,3} |
A_{1,4} |
… | ||||||||

R_{2} |
= | A_{2,0} |
A_{2,1} |
A_{2,2} |
A_{2,3} |
… | |||||||||

R_{3} |
= | A_{3,0} |
A_{3,1} |
A_{3,2} |
… |

**R** = (R_{0}, R_{1}, R_{2}, …) 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).

Therefore .

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.