Solving Math’s Assignment with Sage

Someone sent me this question:

Solve for the currents in the circult of Figure 2, if E(t)=5H(t-2) and the initial currents are zero. [Hint : Use Lapalce transform to solve this problem.]

So, to solve it, form mesh analysis of two loops. Then, convert them from time domain to complex domain with Laplace transform. Next, solve I1 and I2 with normal algebra. Then only inverse I1 and I2 back to time domain.

Of cause, if you familiar with Sage, you can solve it within 30min (or lesser?).

t = var('t')
s = var('s')
I1 = var('I1')
I2 = var('I2')

E(t) = 5*unit_step(t-2)

E(s) = E(t).laplace(t, s); E(s)
# >> 5*e^(-2*s)/s

equation = [
	-E(s) + I1*20*s + 10*(I1-I2) == 0,
	10*(I2-I1) + I2*30*s + I2*10 == 0 ]

solution = solve(equation, I1, I2); solution
# >> [[I1 == 1/2*(3*s + 2)*e^(-2*s)/(6*s^3 + 7*s^2 + s), I2 == 1/2*e^(-2*s)/(6*s^3 + 7*s^2 + s)]]

# Note that Sage cannot inverse-Laplace time-delay function. So, taking out e^(-2*s)
I1(s) = 1/2*(3*s + 2)/(6*s^3 + 7*s^2 + s)
I2(s) = 1/2/(6*s^3 + 7*s^2 + s)

i1_temp(t) = I1(s).inverse_laplace(s, t); i1_temp
# >>t |--> -1/10*e^(-t) - 9/10*e^(-1/6*t) + 1

i2_temp(t) = I2(s).inverse_laplace(s, t); i2_temp
# >> t |--> 1/10*e^(-t) - 3/5*e^(-1/6*t) + 1/2

# Referring to Table. For G(s)= e^(as)F(s), the inverse is g(t) = f(t-a).
u(t) = unit_step(t)
i1(t) = u(t-2) * ( -1/10*e^(-(t-2)) - 9/10*e^(-1/6*(t-2)) + 1 ) # Answer for i1
i2(t) = u(t-2) * ( 1/10*e^(-(t-2)) - 3/5*e^(-1/6*(t-2)) + 1/2 ) # Answer for i2

p1 = plot(i1(t), 0, 10, color='blue', legend_label='i1(t)')
p2 = plot(i2(t), 0, 10, color='red', legend_label='i2(t)')
show(p1 + p2)

And, the final answers are:

i_1(t) =u(t)\left( \frac{-1}{10}e^{-(t-2)} -\frac{9}{10}e^{-(t-2)/6} +1 \right)
i_2(t) =u(t)\left( \frac{1}{10}e^{-(t-2)} -\frac{3}{5}e^{-(t-2)/6} +\frac{1}{2} \right)

Current I1 and I2 vs time.

Bye SAINT2012

I happened to be the last batch of speakers in SAINT2012 conference. Next year it will be merged with COMPSAC as “New COMPSAC”, which will be held in Kyoto in 2013. The name of “SAINT” will not be used anymore.

Badge in COMPSAC/SAINT2012

Happened to know that I have to teeeaaaach in the coming short-semester. It seems like I am getting far far away from my PhD completion. I blame noone but myself. When will I have the gut to tender resignation letter? When will I have the time to do nothing but my own research?

Ephesus, Turkey.

Visit some old cities of Turkey during the SAINT2012 conference. Tortured by the sunshine, and was amazed at the legacy of Ephesus.

Good Bye, Osaka

Had 3 months (October to Jan 2012) research attachment in Imase Lab of Faculty of Information Science and Technology, Osaka University.

It’s time to say good bye.

Thanks for all the guidance and care. I will miss you all >_< ……………

A memorial.

p/s: 残念desu, I have yet to visit any MEIDO CAFE.

ChromiumOS/Chromebook

I must say, I like the idea of  ChromiumOS (or Chromebook).

My girl spends most of her leisure time on movies and Internet – surfing the web, chatting in Facebook, watching movies, etc. And Chromebook just happens to suit her lifestyle.

The Chrombook does a little thing more than just a browser-machine. Besides surfing the net with Chrome, you can play music and video in Chromebook without connected to Internet. There is a “file manager” in Chrome that allows you to browse the files inside the solid-state harddisk or the external SD card.

So, why Chromebook? Well, it has 8-second boot-up time, hassle-free user experience, no annoying viruses or patches, thousands of web-apps and games. They are installation-free. You just need to sign-in and used.  Unfortunately, the sales of Chromebook is not doing well in US (2011). However, believe me or not, Chromebook will be a big thing after html5 is mature, perhaps after 2015.

#

The following are the snapshot of ChromiumOS that running in VirtualBox.

ChromiumOS - login screen
ChromiumOS - Webstore

Uniqueness of Mathematics

A very meaningful sentence from Introduction to Proofs and Real Analysis, written by Richard C. Penney:

“There is a fundamental difference between mathematics and other sciences. In most sciences, one does experiments to determine laws. A “law” will remain a law, only so long as it is not contradicted by experimental evidence. Newtonian physics was accepted as valid until it was contradicted by experiment, resulting in the discovery of the theory of relativity.

Mathematics, on the other hand, is based on absolute certainty. A mathematician may feel that some mathematical law is true on the basis of, say, a thousand experiments. He/she will not accept it as true, however, until it is absolutely certain that it can never fail. Achieving this kind of certainty requires constructing a logical argument showing the law’s validity–i.e. constructing a proof.”

Something in My Mind

Let A and B be two events of an experiment.

Given a statement:
A implies B but B does not imply A (or simply A implies B).

e.g.
A: Otaku.
B: Watch Anime.
That is: Otaku watches Animes. But those who watch Anime may not be an Otaku.

In Set Theory, we may say A is inside B but B is not inside A. Or A \subset B.

To put them into conditional probability:

\Pr(B|A)=\frac{\Pr(B \cap A)}{\Pr(A)}=\frac{\Pr(A)}{\Pr(A)}=1

\Pr(A|B)=\frac{\Pr(A \cap B)}{\Pr(B)}=\frac{\Pr(A)}{\Pr(B)}\leq 1

 

Thus, the statement: “A implies B but B does not imply A” may be written as:

\Pr(B|A)=1 and \Pr(A|B) \leq 1.

 

 

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Gaia disagree with my misuse of Mathematics notation.