## Monday, December 21, 2015

### Another fan awakens

While rewatching Star Wars episodes 1-6, my wife remarked that Han Solo can be roughly translated into Chinese as 單身 (Han sounds like 漢 and solo means single which is 單身) which has the meaning of a bachelor, a suitable way to describe his character.
She also wished that she has the Jedi's mind control powers.  I told her that she already has that power. For instance, when she says "You will take out the trash" and waves her hand, I take the trash outside to the garbage bin.

## Thursday, December 17, 2015

### The fan awakens

As I mentioned in this post, I have been a longtime fan of Star Wars, as Return of the Jedi was released when I was a teenager. To prepare ourselves to watch the new Star Wars movie, my family and I have been watching the first 6 episodes again over the last couple of weeks.  It has been years since I have seen some of them, and it brought back some nice memories.  I always chuckled at some of names that are used, for instance Admiral Ackbar who looks like a cuttlefish, is from the species Mon Calamari.  Christopher Lee, whose is famous for playing Count Dracula, plays a character called Count Dooku.  And midichlorian sounds like mitochondrion, which are microscopic parts in cells responsible for supplying components for generating energy in the cell .

One thing I remember from my childhood is that there is one part of the Star Wars theme song that sounds exactly like part of a deodorant commercial jiggle I saw on a Venezuelan TV Channel.

## Thursday, November 19, 2015

### Compression of random data

Over the years, several companies have purported to come up with algorithms that can compress any files, even files of random-looking data.  See here for a review of such claims over the years. Compression here means that the compressed file is of a smaller size than the original file.  The problem with this is that if you keep applying the algorithm to the compressed file over and over again, you would reach a file of the smallest size: a file of a single bit.  Since there are only two possible files of a single bit ("0" or "1"), this simply cannot be the case.  This uses the pigeonhole principle and a similar argument of the impossibility of such an algorithm is as follows.  There are 2n different bit strings of n bits.  On the other hand, there are at most 2n-1 different bit strings of n-1 or less bits.  The pigeonhole principle implies it is not possible to compress all n bits strings to a smaller number of bits.

Based on these arguments, Mike Goldman proposed the following challenge (excerpted from comp.compression faq):

Mike Goldman <whig@by.net> makes another offer:

I will attach a prize of $5,000 to anyone who successfully meets this challenge. First, the contestant will tell me HOW LONG of a data file to generate. Second, I will generate the data file, and send it to the contestant. Last, the contestant will send me a decompressor and a compressed file, which will together total in size less than the original data file, and which will be able to restore the compressed file to the original state. With this offer, you can tune your algorithm to my data. You tell me the parameters of size in advance. All I get to do is arrange the bits within my file according to the dictates of my whim. As a processing fee, I will require an advance deposit of$100 from any contestant.  This deposit is 100% refundable if you meet the challenge.

This appears to be an easier challenge since the compression and decompression algorithm can be tuned to the data generated.  Incidentally, in the following exchange, an attempt is made to win the challenge by exploiting how data is stored in a computer disk system and using the filename as additional storage by splitting the data into many files.

There are several aspects that makes Mike Goldman's challenge possibly solvable and not subject to the constraints of the pigeonhole principle.   First of all, the compression algorithm can be tuned to the data.  In other words, the compression algorithm only needs to compress only one single file.  It doesn't need to compress all files and thus the pigeonhole principle does not apply.  Secondly, the challenge did not prescribe what computer language the decompressor can be written in.  Can it be written in C, Perl, Python or Matlab?  Using a high level language essentially provides the decompressor with complex subroutines that are callable with simple calls, and thus some of the compression can be "hidden" in this way.

## Thursday, October 22, 2015

### Back to the future of energy storage

Yesterday was "Back to the Future" day, the day when Michael J. Fox's character arrived into the future from the movie "Back to the Future II".  The time machine he used was a DeLorean using a flux capacitor requiring 1.2 Gigawatts of power.  I have written a blog post about flux capacitors a couple of years ago and mused that flux capacitors can be considered as normal capacitors and that a large enough supercapacitor today can generate that amount of power if discharged quickly enough. There were endless discussions about whether the prediction about the future in the movie was accurate or not and one comparison to the DeLorean was the Tesla electric car, especially the Tesla Model X with its Falcon wing doors. While the Tesla cannot fly and still needs roads, it does have autopilot capabilities.  A fully loaded Tesla comes with a 90kWh battery and if it all discharges within a quarter of a second, it would generate the 1.2 Gigawatts needed for time travel.

## Monday, September 28, 2015

### Supermoon lunar eclipse

Yesterday was a "supermoon" lunar eclipse, when the full moon is closest to earth and a lunar eclipse is happening at the same time.  At the beginning of the evening it was cloudy, but luckily the sky cleared up and we have a great view of the moon outside our garage.  We took some nice pictures of the eclipse:

The next supermoon lunar eclipse will occur in 18 years.

## Tuesday, July 28, 2015

### Closest integer square

Given a real number $x\geq 0$, consider the following 2 functions: $f(x)$ is the closest square of an integer to $x$, and $g(x)$ is the integer closest to $\sqrt{x}$.
In other words, $f(x) = y$ where $y = m^2$ for an integer $m$ and $|x-y| \leq |x-n^2|$ for all integers $n$.  In case of a tie, i.e. $x$ (resp. $\sqrt{x}$) is midway between two successive squares (resp. integers), we define $f(x)$ (resp. $g(x)$) to be the smallest such square (resp. integer).

The question we like to ask is: is $g(x)^2 = f(x)$ true?  For arbitrary real numbers $x$, the answer is no.

The midpoint between two successive integers $a$ and $a+1$ is $a+\frac{1}{2}$ whose square is $a^2+a+\frac{1}{4}$.  On the other hand, the midpoint between two successive squares $a^2$ and $(a+1)^2$ is $a^2+a+\frac{1}{2}$.  It is interesting to note that the difference between the 2 midpoints is always $\frac{1}{4}$ regardless of what $a$ is (this is true even if $a$ is not an integer).
Graphically this is illustrated as:

This means that for $a^2+a+\frac{1}{4} < x < a^2+a+\frac{1}{2}$, $g(x) = \sqrt{f(x)}+1$.  For instance $2.375$ is closer to $1$ than to $4$, but $\sqrt{2.375} = 1.541...$ is closer to  $2$ than to $1$.

On the other hand if
$a^2 \leq x < a^2+a+\frac{1}{4}$ or $a^2+a+\frac{1}{2} < x \leq (a+1)^2$,
then $g(x)^2 = f(x)$.

This analysis also shows the following facts:

• If $x$ is an integer, then evaluating $f(x)$ and $g(x)$ do not result in a tie, as the 2 midpoints above are not integers.
• If $x$ is an integer, then $g(x)^2 = f(x)$ as the interval $[a^2+a+\frac{1}{4}, a^2+a+\frac{1}{2}]$ does not contain any integers.

## Saturday, January 24, 2015

### Trigonometry of regular polgons

In school we learned that the exterior angle of a regular n-sided polygon is $360/n$ degrees or $2\pi/n$ radians.  Similarly the interior angle of a regular n-side polygon is $180(n-2)/n$ degrees or $(n-2)\pi/n$ radians.  It can be shown that the trigonometry functions (cos, sin, tan, ...) of a rational multiple of $\pi$ is an algebraic number, i.e, a root of a polynomial with integer coefficients.  Gauss showed that $n$ being a power of 2 multiplied by distinct Fermat primes (i.e. primes of the form $2^{2^m}+1$) is sufficient to ensure the polygon can be constructed with straightedge and compass and Wantzel showed that this is also necessary (http://mathworld.wolfram.com/TrigonometryAngles.html).  This implies that the expression of $\sin(k\pi/n)$, $\cos(k\pi/n)$, etc. can be written explicitly using radicals. Some expressions for these angles are given in (http://en.wikipedia.org/wiki/Exact_trigonometric_constants) for angles where the number of minimal nesting of radicals is 2 or less.

Other values of $n$ for which the minimal nesting of radicals is 2 or less include $n=40$ and $n=120$.  For instance for $n=40$, we have:

$\sin\left(\frac{\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2- \sqrt{2}} \sqrt{5+\sqrt{5}}$
$\cos\left(\frac{\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5+\sqrt{5}}$
$\sin\left(\frac{3\pi}{40}\right) = \left(\frac{-\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5-\sqrt{5}}$
$\cos\left(\frac{3\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5-\sqrt{5}}$
$\sin\left(\frac{7\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2+\sqrt{2}} - \frac{\sqrt{2}}{8} \sqrt{2- \sqrt{2}} \sqrt{5-\sqrt{5}}$
$\cos\left(\frac{7\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5-\sqrt{5}}$
$\sin\left(\frac{9\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2- \sqrt{2}} \sqrt{5+\sqrt{5}}$
$\cos\left(\frac{9\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5+\sqrt{5}}$
$\sin\left(\frac{11\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5+\sqrt{5}}$
$\cos\left(\frac{11\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5+\sqrt{5}}$
$\sin\left(\frac{13\pi}{40}\right) = \left(\frac{1+\sqrt{5}}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5-\sqrt{5}}$
$\cos\left(\frac{13\pi}{40}\right) = \left(\frac{\sqrt{5}+1}{8}\right) \sqrt{2+\sqrt{2}} - \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5-\sqrt{5}}$
$\sin\left(\frac{17\pi}{40}\right) = \left(\frac{1+\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5-\sqrt{5}}$
$\cos\left(\frac{17\pi}{40}\right) = \left(\frac{-\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+\sqrt{2}} \sqrt{5-\sqrt{5}}$
$\sin\left(\frac{19\pi}{40}\right) = \left(\frac{\sqrt{5}-1}{8}\right) \sqrt{2-\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2+ \sqrt{2}} \sqrt{5+\sqrt{5}}$
$\cos\left(\frac{19\pi}{40}\right) = \left(\frac{1-\sqrt{5}}{8}\right) \sqrt{2+\sqrt{2}} + \frac{\sqrt{2}}{8} \sqrt{2-\sqrt{2}} \sqrt{5+\sqrt{5}}$