Simplifying non-trivial root expressions

129 days ago

This was a surprising pair of identities I stumbled on in college.

167be10ac38834e0d3493cce6841d7646b9dfccd

and

c3baaa8b26e39aaf81dcc9ee0492cfd05857ce06

So, for example, with a=10 and b=2, 8cf03a9fc049419d85c28ec95b88054299ae0cc1

I don’t remember the source of the question, but I suspect it was someone with knowledge of this relation.

I’m very curious whether this is part of a larger class of mathematical expressions involving roots that can be simplified.

Here’s a proof:
f71b19cdb2dc1af8331e0cdd32dd7c4c80c31a5e
 
6ba189a5f9bc88eb54175d2c6eba285a9cec4e32
 
8956774fb7cebd1affe297b1e72b49802c50de97
 
cff6bd32c9a48d7826683e2d4753cfd92a71ba66
 
1021c6b5adb867a12775ed18b10c010b9408f1d5
 
80fce0f5fa5ebdd3857726668eb3441d6f69f815

Solving for m and n:

a77a14002b55328d044275c83d7b764b70eebdbb   e3d25a3f776fce9a4d1b73a79ee60b686d5450c9

Interestingly, this can be used in a telescoping identity when a=k, b=1:

6ae782b9bb859f9597c83626a69f1e5b9e8702e0

Comment

---

Fibonacci-like sequences that converge to 0

1754 days ago

I was curious about whether Fibonacci-like sequences could converge to 0, with a proper choice of starting terms and the usual recurrence relationship:

69c34c11ab0976d233bc7fee8af206d1aeb338ba

It turns out that they can, if the ratio of the second term to the first is 7249246bfa7088a808ebe568b18dd326c782dc4c, or 1-φ, approximately -0.618.

I found this using the closed form for Fibonacci-like sequences

25e243f0a1759585683430bdfcc60e927f296abc and solving for 510842f7d1fc6a07b50c4521dd7ee6a633628623.

Here are the first few elements of the sequence:
381e6a8cefe0b5429ab77c6b4bc1ca2e90ee483c

Interestingly, the integer part are the Lucas numbers and the multiples of the square root are the Fibonacci numbers, and their ratio converges to the square root of 5, based on their closed forms.

Comment

---

cor: a sample correlation game

1761 days ago

If you know what sample correlation is, you might enjoy Cor: A Sample Correlation Game, which is available here

Comment

---

truecrypt on the BeagleBone Black (Debian)

1971 days ago

As I’m using Revision B, I ended up downloading source to a mount point on the sdcard (/mnt/sdcard) as it’s too easy to fill up the 2GB root device otherwise.

Download and unpack truecrypt-7.1a-source

Download and unpack wxWidgets

export SDCARD=/mnt/sdcard
cd ${SDCARD}
wget http://prdownloads.sourceforge.net/wxwindows/wxWidgets-2.8.12.tar.gz
tar xvfz wxWidgets-2.8.12.tar.gz
export WX_HOME=${SDCARD}/wxWidgets-2.8.12/

Download PKCS11 headers

mkdir ${SDCARD}/pkcs
cd ${SDCARD}
wget 'ftp://ftp.rsasecurity.com/pub/pkcs/pkcs-11/v2-20/pkcs*.h'
export PKCS11_INC=${SDCARD}/pkcs

Build truecrypt

cd ${SDCARD}/truecrypt-7.1a-source
make NOGUI=1 WX_ROOT=${WX_HOME} wxbuild
make NOGUI=1 WXSTATIC=1

Copy the binary to the system

sudo cp Main/truecrypt /usr/bin
chown bin.bin /usr/bin/truecrypt
chmod 0755 /usr/bin/truecrypt

Comment

---

MediaWiki and the Case of the Sporadic LaTeX Failure

2279 days ago

I recently upgraded to MediaWiki 1.21 for my personal wiki, and I was annoyed to see that LaTeX support was removed, though it and MathJax are available as extensions.

Once I installed it, I was vexed by sporadic failures in typesetting — texvc would succeed in producing an image about 20% of the time.

Eventually, I found out why — the memory limits on the wfShellExec call were set pretty low, and so most of the time it would fail to malloc the memory it needed.

Increasing the global default for shell calls in LocalSettings.php fixed this:

$wgMaxShellMemory = 402400;

If you’re having the same problem, give it a try.

Comment

---

« Older