The kind of mathematics I like and dislike

The kind of mathematics I dislike

Those academic mathematics that your teachers teach you.
To follow the guidelines; the steps; which are very mechanical and completely boring. This is the reason why I've stopped teaching. Schools destroy every child's creativity.

The kind of mathematics I like,

Mathematical Proof
Where you don't know where to start to solve a problem; you will get lost at the very beginning; just keep exploring till you find the solution. No specific formulae to be used. Whether to use algebra or calculus or geometry -- it doesn't matter as long it is proved. This is much more playful and very artistic.

Example, If two policemen are escorting a drunk prisoner between them, and both officers go to a cell, then (regardless of the path taken, and the fact that the prisoner may be wobbling about between the policemen) prove that the prisoner must also end up in the cell.

I heart christina perri

iHeart

ENTER

Just another simple tune

Just another tune. Well, for your info, I'm not an active musician. It's just a hobby to kill my boredom. Only fiddle with it once in a blue moon(once per month?). Anyway, I started with Guitar. And I've not been playing guitar for a decade! Kindly refer to this link

Starting from 0:58, it's a song from Christina Perri - Jar of Hearts. I've made a mistake somewhere around 1:53 and I couldn't be bother to retake again. :p Let it be!

My Piano Snippet

From Penguin to Apple

myLinux box

myMac box

RSSI (Radio Signal Strength Indicator)

To monitor your signal strength in MAC OS while tweaking the wireless router; you can run this command in your terminal.

while x=1 ; do /System/Library/PrivateFrameworks/Apple80211.framework/Resources/airport -I | grep CtlRSSI ; sleep 0.5 ; done

From the screenshot below, higher number corresponds to lower signal. "-" means loss. Thus, keep on tweaking!

You can stop this command from refreshing by hitting Control+C.

Cleaning Macro Code

If you look closely to every auto generated VBA code, a lot of unnecessary code taking up space. Emitting them makes your code faster and cleaner. Plus, it is much more easier to understand. Well, I came from UNIX/LINUX background. Unix/Linux coders always want their code to be small, short, simple, and such. That's why Unix-like operating systems are always fast. ;-) Except AIX. It's an ancient unix. :-p



Example

Range("C3:C7").Select
Selecting.Copy
Range("V2").Select
Selection.PasteSpecial Paste := xlValues,

Truncate the above snippet to

Range("C3:C7").Copy
Range("V2").PasteSpecial Paste := xlValues,

And for Font related code, Macro seems to generate unnecessary lines of code which are already consider default. VBA does not need to verify each of these default lines
Example

Range("F1").Select
With Selection.Font
.Name = "Tahoma"
.Size = 10
.Strikethrough = False
.Subscript = False
.OutlineFont = False
.Shadow = False
.Underline = xlUnderlineStyleNone
.ColorIndex = xlAutomatic

Delete those unnecessary lines to only ...

Range("F1").Select
With Selection.Font
.Name = "Tahoma"
.Size = 10

Office Politics

Starbucking with friends. Most of the conversation were all about Office Politics. Boring! Well, this is what I've to say....

If someone stabs you from behind where it really hurts -- be strong and keep on learning!

If favoritism is an issue -- be brave and keep on learning!

When the grass gets greener; it gets better!

Stay Hungry. Stay Foolish - Steve Jobs

A Thousand Years Lyrics

Darling don't be afraid.... I have loved you... for a thousand years...

Learning .......

Currently, I'm delving deeper into ...


1) Computational Statistic, Data Mining and Data Analysis
2) Excel and Visual Basic
3) Cocoa, Mac OS and iOS

A Thousand Years

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 love you for a thousand more

Cleaning up MacPorts

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

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)

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

ALTEC LANSING

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.

The First Noel

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

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.

Proof d/dx (ln x)

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

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

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

Latex on Blogspot

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

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)

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

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

Name: Fauzi

Email: fauzspace at gmail dot com

Curious:The Science of Electrical Signal

Interest: mathematics, piano ...