Human Knowledge belongs to the World

A Thousand Years

2012-02-19T02:42:00.001-08:00

This is so beautiful. I heart Christina Perri. And I deeply heart this song. I will play the piano to this tune soon. To my loved ones -- I have loved you for a thousand years. I'll love you for a thousand more

Cleaning up MacPorts

2012-02-17T18:23:00.000-08:00

It seems as though MacPorts, when upgrading to a new version of an application, leaves the old version behind. After a while, this will eat up quite a bit of your precious hard disk space. There will probably be a lot of ports installed, and only the ones marked "(active)" are used. The rest may be removed. The following commands will keep those "active" only

port installed > list ; sed -e '1'd list | grep -v "active" | while read i ; do sudo port uninstall --follow-dependents $i ; done

Adding colors to your bash terminal

2012-02-15T05:16:00.001-08:00

Mac Users : This will beautify your console :) . Go to this link. Copy the script and save it as ".bash_profile" in your home directory.

Attached below is my Mac OS Lion's Terminal.

MacPort is very BSD(ish)

2012-02-15T04:54:00.000-08:00

To install opensource applications in your MacOS -- all you need is just a single command! It is equivalent to BSD port and Linux package management.

Firstly, you need to install Xcode then follow by MacPort. Once done, execute your Terminal and install whatever software(s)
available in MacPorts.

To search for a program so called mpg123 (as an example)
port search mpg321

If you want to install mpg321; just type this..
sudo port install mpg321

To delete all intermediates files after installing ....
sudo port clean --all mpg321

To list all outdated packages that you have installed..
port outdated

To just upgrade only one particular application(example mpg321) ..
sudo port upgrade mpg321

To upgrade all outdated packages...
sudo port upgrade outdated

To uninstall program and its dependencies
sudo port -fp uninstall --follow-dependents mpg123

#killall Media, engaged

2012-02-10T05:27:00.000-08:00

With the continuing campaigns for Internet-censoring litigation such as SOPA and PIPA, and the closure of sites such as Megaupload under allegations of 'piracy' and 'conspiracy', the time has come to take a stand against music, film and media companies' lobbyists.

The only way to hit them very hard where it truly hurts... Their profit margins.

Do not buy a single record. Do not download a single song, legally or illegally. Do not go to see a single film in cinemas, or download a copy. Do not buy a DVD in the stores. Do not buy a videogame. Do not buy a single book or magazine.

Wait the 4 weeks to buy them in April, see the film later, etc. Holding out for just 4 weeks will lave a gaping hole in the media and entertainment companies' profits for the 1st quarter. An economic hit which will in turn be observed by governments worldwide as stocks and shares will blip from a large enough loss of incomes.

Cyber War

2012-01-28T04:41:00.000-08:00

Fighting fire with fire very rarely results in anything but destruction. And it’s far easier to destroy something than to create a solution. In real war -- breaking or damaging other people's houses or killing the innocents is completely not right. This goes to cyber war. Example breaking into bank or social networking sites where innocents use it daily is not an effective way to fight back.

If history tells us anything, this escalation will only justify more abusive action on both sides that may ultimately result in a tightly controlled Internet as the only “solution” to the anarchy. Ideas of freedom can win without destroying things. Let’s unite once again to make sure the ACTA/SOPA/PIPA legislation is also defeated or delayed.

kill SOPA

2012-01-28T03:11:00.001-08:00

Stop Online Piracy Act (SOPA), a treaty designed to attack the rights of computer users in some 40-odd countries -- and others later -- is encountering increasing opposition. SOPA threatens, in a disguised way, to punish Internet users with disconnection if they are accused of sharing, and requires countries to prohibit software that can break Digital Restrictions Management (DRM), also known as digital handcuffs.

Those politicians serve the big music and movie companies. They intend to impose what those companies want -- first on 40-odd countries, then on the world. They won't heed civic-minded suggestions offered in a reasonable spirit that assumes their good will; their response to the Wellington Declaration shows that. We will have to stop them. To build a movement to stop them, we need to say, "Join us and fight!"

#Commands executed : for SOPA in `$(ps -o pid,ppid -ax | \ awk "{ if ( \$2 == $pid ) { print \$1 }}")` ; do echo "Kill SOPA and its supporters" kill $SOPA ; done

ALTEC LANSING

2012-01-25T02:06:00.000-08:00

You guys should try this speaker : ALTEC LANSIN EXPRESSIONIST FX3022. It's a double subwoofer. Simply awesome. Go and get it! Trust me, you won't regret for sure.

Math Education Today

2012-01-23T18:31:00.000-08:00

You have probably heard the parents of primary school childrens complaining about the difficulties their children have had with Mathematics today. Basically, the new mathematics is aiming more towards problem solving rather than focuses alot on computations and routine procedures. This applies not only to primary school but also secondary and post-secondary level. What the use of multiplying big numbers; mass memorization; and others tedious computation? Nowadays we have calculator and computer to compute easily, accurately, efficiently, and save a lot of time. We should tackle complex problems not complex calculations.

Problem solving is crucial in mathematics education because it transcends mathematics. By developing problem solving skills, we learn not only how to tackle math problems, but also how to logically work our way through any problems we may face. The memorizer can only solve problems he has encountered already, but the problem solver can solve problems she has never seen before. The problem solver is flexible; she can diversify. Above all, she can create.

Vim & Microsoft Office on iPad

2012-01-23T17:48:00.000-08:00

I have been a big fan of Vi/Vim editor. And now it is available for iPad/iPhone. Sweet!!!

Vim for iPad/iPhone

One more thing... Free Microsoft Words, Excel, and Powerpoint on iPad can be downloaded here

2012

2012-01-13T06:23:00.000-08:00

nothing new for 2012. It is the same wish that carries me through everyday of every year. So, this is my wish.

Let laughter be the way of life. Learning is the highest aim. Love is the ultimate goal

my white bedroom

2012-01-13T05:52:00.000-08:00

The First Noel

2011-12-25T02:06:00.000-08:00

Elegant(short and simple) way to Proof $$ \frac{d}{dx}(x^n)$$

2011-09-03T16:13:00.000-07:00

Formula = $$ \frac{d}{dx} n^{x} = nx^{n-1} $$
I used to prove this formula using Binomial Theorem. It is pretty tedious if you are weak in algebra.
$$ Proof $$ using Binomial Theorem

Let's plug $$ f(x) = x^{n} $$
$$ f(x) = \lim_{h\to\infty}\frac{(x+h)^{n} - x^{n}}{h} $$
$$ = \lim_{h\to\infty} \frac{\bigg( (x+h)^{n} = \sum_{p=0}^{\infty}\dbinom{n}{p} x^{n-p} h^{p}\bigg) - x^{n}} {h} $$
If you expand and compute the above... you will reach...
$$ = \lim_{h\to\infty} nx^{n-1} + \frac{n(n-1)}{2!}x^{n-2}h+ ........ + nxh^{n-2} + h^{n-1} $$
$$ = nx^{n-1} $$ This completes the proof.

Another elegant way to prove this derivative is using $$ Logarithmic $$ technique. Let's first define $$ y=x^{n} $$

$$ \ln y = \ln x^{n} ...... \underbrace {\ln y = n \ln x }_\textrm{differentiate both sides}$$
$$ \frac{\tilde{y}}{y} = n \frac{1}{x} ..... \tilde{y} = n \frac{1}{x} \otimes y $$ where $$ y = x^{n}$$.

Finally, $$ \frac{d}{dx} n^{x} = nx^{n-1} $$ is proved elegantly.

Another fine tune

2011-04-09T06:29:00.000-07:00

To ALL:
simply enjoy!

I MISS YOU

Proof d/dx (ln x)

2010-11-11T17:46:00.000-08:00

Based on Khanacademy way of proofing $$\ \frac{d}{dx}\left ln(x) = \frac{1}{x} $$ which one of my student showed it to me ... I have another way of proofing this. Almost similar but using imaginary variable.

$$ Proof :$$
Let $$ f(x) = ln(x) $$ then, $$ \frac{ln(x+ \Delta h) - ln(x)}{\Delta h} \,\,= \frac{ln\big( \frac{x+ \Delta h}{x}\big)}{\Delta h} $$ Now we add the imaginary $$ {\color{blue} x}. $$
$$={\color{blue} x}\, ln \frac{\big (\frac{x+ \Delta h}{x} \big )}{{\color{blue} x} \Delta h} \,\, = \frac{1}{x} \, \bigg (\frac{x}{\Delta h} \, ln \bigg (\frac{x + \Delta h}{x} \bigg ) \bigg )$$

$$=\frac{1}{x} \, \bigg ( ln \big( 1 + \frac{1}{\frac{x}{\Delta h}} \big ) \bigg )^{\frac{x}{\Delta h}} \, = \frac{1}{x} \big( (e^{0})=1 \big) \, = \frac{1}{x}$$. This completes the proof

One of 'e' Logarithm Properties

2010-10-14T17:02:00.002-07:00

The most common logarithm proofs that you can find in most books,internet,wiki,and teachers might have taught you is this.....
Prove $$ \ln (uv) = ln(u) + ln(v) $$

$$Proof:$$
Let $$ln(u)=x \; and \; ln(v)=y$$
Then $$u = e^{x} \; and \; v = e^{y}$$
$$u \* v = e^{x} \* e^{y} = e^{x+y}$$
Therefore, $$e^{ln(u) + ln(v)} = u \*v$$
$$ln(u) + ln(v) = ln(uv)$$. This completes the proof

I have another way of proofing this with calculus ...

$$Proof:$$
Let the variable $$ w=ut $$ in $$ ln(v)=\int^v_1 \frac{1}{t} dt $$
Then $$ dw=u \; dt $$ and the limits of integration $$ t=1 $$ and $$ t=v \Leftrightarrow w=u $$ and $$ w=uv $$
Hence,
$$ ln(v) = \int^{uv}_u \frac{1}{\frac{w}{u}}\frac{1}{u} \; dw = \int^{uv}_u \frac{1}{w} \; dw = \int^{uv}_u \frac{1}{t} \; dt $$
$$ln(u) + ln(v) = \int^u_1 \frac{1}{t} \; dt + \int^{uv}_u \frac{1}{t} \; dt = \int^{uv}_1 \frac{1}{t} \; \\ dt=ln(uv)$$.
The property is proved in different way! Happy proofing!

Fauzi's symphony

2010-10-01T16:53:00.000-07:00

After almost a decade of not playing .... this is all i have now :)

Latex on Blogspot

2010-09-30T02:05:00.000-07:00

Knowledge is only valuable if it is shared

To all those who need to write mathematical related documentation in blogspot and have experience with $$LaTeX $$. Just follow this

click on DESIGN then click again on ADD GADGET then this time choose HTML/JAVASCRIPT ... lastly paste this,

(script src="http://www.watchmath.com/cgi-bin/mathtex3.js" type="text/javascript")(/script)
(script type="text/javascript")
replaceMath( document.body );(/script)

PS: replace the "(" and ")" to "<" and ">"

Examples

\ \x = a_0 + \frac{1}{\displaystyle a_1
+ \frac{1}{\displaystyle a_2
+ \frac{1}{\displaystyle a_3 + a_4}}}

Add another "$" on both ends. The output will be...

$$\ x = a_0 + \frac{1}{\displaystyle a_1
+ \frac{1}{\displaystyle a_2
+ \frac{1}{\displaystyle a_3 + a_4}}}$$

Mathematical Induction

2010-09-12T01:08:00.001-07:00

Mathematical Induction is a good way to start writing proof for beginner too. Mathematical Induction is usually read by students at post-secondary level(Diploma/A-level). Problem is, equations are given to you and you just have to prove $$ (k+1)$$ is true. As a real problem solver, you should explore more and able to create without memorizing!

Example: $$ 1 + 2 + 3 .... + n = \frac{n(n+1)}{2}$$
Before we prove (k+1) .. lets find out why $$\frac{n(n+1)}{2}$$.

If you add the very last number to the first, then move the 2nd last to the 2nd and so on, the result will be ...

Sum them up:
$$\small (n+1) + ((n-1)+2) + ((n-1)+3) .... + ((n-m+1)+m)$$
Result : $$\small \underbrace {(n+1) + (n+1) + (n+1) ...... + (n+1)}_\textrm{n times. so = n(n+1)}$$

Since 2 integers being sum together, we divide them by 2 so, $$\frac{n(n+1)}{2}$$ created. Now you can prove (k+1) for $$\sum_{i=1}^{n}i=\frac{n(n+1)}{2}$$
Now you can prove for (k+1) easily.

Another Example: Let's explore further for $$1^{2} + 2^{2} + 3^{2} + ...... n=?$$. Lets say you can't remember. Time to think creatively and do real math.

Lets see ..

$$ \underbrace{\underbrace{1^2 + 2^2}_\textrm{5} + 3^2}_\textrm{14} ... + n^{2}=5, 14, 30, 55, 91, .... $$

$$ \underbrace{\underbrace{1 + 2}_\textrm{3} + 3}_\textrm{6} ....... + n=3, 6, 10, 15, 21 .... $$

Divide them $$\frac{1^2 + 2^2 + 3^2 ... n^2}{1 + 2 + 3 ....... n}=\frac{5, 14, 30, 55, 91 ...}{3, 6, 10, 15, 21 ...}$$

Finally, divide the denominator and numerator $$=\frac{ 3 \, 5 \, 7 \, 9 \, 11 \, 13}{ 3 \, 3 \, 3 \, 3 \, 3 \, 3}$$

Can you see the above number pattern? the numerator are odd integers while denominator are constant 3. Therefore,
$$ \frac{1^2 + 2^2 + 3^2 ... + n^2}{1 + 2 + 3 ....... + n} = \frac{2n+1}{3}$$

$$\frac{1^2 + 2^2 + 3^2 ... + n^2}{\frac{n(n+1)}{2}}=\frac{2n+1}{3}$$

$$1^2 + 2^2 + 3^2 ... + n^2=\frac{n(n+1)(2n+1)}{6}$$ created!

Now you can prove (k+1) for $$\sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1)}{6}$$

Writing Proof for beginner(s)

2010-09-12T01:07:00.001-07:00

I'm glad that some friends and students, outside mathematical field, want to learn the creative way of writing proof. Maybe you can start learning something very basic..like $$ 1+1=2$$. Of cos this sounds so obvious. But how to prove that its true?

Method 1:

$$ Let \, \alpha = 1, then\, (\alpha + \alpha) =$$
$$\alpha(1 + 1) = 2\alpha \Rightarrow$$ By distributive law

You can use any kind of mathematical tools to write proof. Not necessary to be algebra or calculus or geometry etc... writing proof or problem solving is a question that cannot be answered immediately. You will get lost at the very beginning mostly. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In industry, being able to solve a poorly defined problem is much more important to an employer than being able to, say, invert a matrix. A computer can do the latter, but not the former.

So let's try using Set as a tool to solve $$1+1=2$$
Method 2:

$$A = \{a_1, a_2, ...\} B = \{b_1, b_2, ...\} C = \{c_1, c_2, ...\} D = \{r,s\}$$
Then, $$ \alpha + \alpha = \Theta(\{A,B\}) = \Theta(\{a_1, a_2, ... b_1, b_2, ...\})$$
$$2\alpha = \Theta(C * D) = \Theta(\{(c_1,r),(c_2,r)....,(c_1,s),(c_2,s)\})$$

But the function $$ f: \{A;B\} \rightarrow \{C * D\}$$
$$f(x) = \biggl \{ \begin{matrix}(c_i, r) \Rightarrow if \, (x = a_i)\\
(c_i, s) \Rightarrow if \, (x = b_i) \end{matrix}$$

is similarly mapping of $$\{A;B\} \, onto \, (C * D)$$
Hence $$\alpha + \alpha = \Theta (\{A;B\}) = \Theta(\{ C * D\}) = 2\alpha$$.
This completes the proof.

Maple and Mathematica

2010-06-29T04:05:00.000-07:00

I have experienced with both Maple and Mathematica. At the moment I'm using Maple only. It runs efficiently well on Fedora Linux. Plus, the syntax are more like C-programming compare to Mathematica(more confusing). Example.......

Prove that the improper integral $$ \int^{\infty}_{a} \frac{sin(x)}{x^\gamma}\,dx $$ is convergent for $$a > 0 \,\, and\,\, \gamma > 0$$

Maple:

int(sin(x)/x^2,x=1..infinity);

abs(sin(x)/x^2)<=1/x^2; int(1/x^2,x=1...infinity);

Mathematica:

{Intergrate[Sin[x]/x^2,{x,1,Infinity}],

Abs[Sin[x]/x^2]<=1/x^2, Integrate[1/x^2,{x,1,Infinity}]}

From above you can see the syntax differences. Whether it's going to be maple or mathematica or maxima or sage ... the most important --- learn computer algebra system!

My Profile

2010-01-29T21:10:00.000-08:00

Name: Fauzi

Email: fauzspace at gmail dot com

Curious:The Science of Electrical Signal

Interest: mathematics, piano ...