Exam Letter, May 2008

With a triggering material from Terence Tao, a mathematics professor:

Among chess players, it is generally accepted that one of the most effective ways to improve one’s skill is to continually play against opponents which are slightly higher rated than you are. In mathematics, the opponents are unsolved or imperfectly understood mathematical problems, concepts, and theories, rather than other mathematicians; but the principle is broadly the same. (more)

I believe that every test, exam, or academic hurdle of sorts is an opportunity to learn and to improve ourselves.

Our answers - or solutions or responses - are like the swing of a racket to hit the ball.

The questions are like a "stress test" - seeing how far we can go without failing. Each correct answer reinforces our understanding.

Logic gives order to the knowledge fragments in our minds. Patterns and sequences become obvious. We become better at handling the ball.

But of course we would also encounter questions which we don't fully know the answer.

Because, otherwise, we'd be stuck in yesterday's ceiling. Yes, we might have attained 100% yesterday. But that is then just a nice number. It doesn't translate into reality.

Reality is when we face the unknown: new discoveries, new learnings, new advancement. Who needs to be stuck in yesterday?

And now, while time still allows us to heighten our ceilings as maximally as possible - to expand our horizon as distantly as achievable - let us then fulfill it.

Potential is what we haven't done yet.

Don't waste it.

Happy SWOT VAC.

Past, Present & Future

For lack of a better title...

What's Past

Two good friends from Dunedin came over to visit us here in Melbourne, about 4 weeks ago.

How sweet!





What's Present

My first Uni assignment (1,ooo words) on medical ethics got marked and returned this week.
So I feel like showing the feedback (without being a show-off).
It's almost like a present; the several days of labour reaped excellent harvest. A present in the present.
For the glory of God!



What's Future

Winter's coming.

After the Sem exam, about 4 weeks of break are coming.

I'd probably delve a bit into the Big Bang Theory, black holes, nebulae, ..., quarks, hadrons, bosons, etc.

For lack of better ideas on what to spend a few million seconds on.

Poetic Math

An excerpt from "Words of a Mathematical Mind", written as a script for a Math choral speaking contest in September 2005. It did not win any prize.

...

Take this simple question:
2x + 5 = 11 [Two x plus five equals eleven]
- (What?) - 2x + 5 = 11
It’s an equation
Solved by subtraction, and division or multiplication
Or try a function
f(x) = x³ –x² –x
With which you can do differentiation
Or maybe integration
And various other operations
Flowing smooth in calculating motion


Beyond theory, into reality
Far and wide across the air
From currency to astronomy
Doubtless to say, numbers are there
From symmetry to geometry
And all that you thought were merely lines
From intensity, to luminosity
Lighting up the golden mines
Into the riches, wealth, economy
Numbers dictating every little penny

...

Reading this again makes me think: Why go all the trouble to make this?
But then again I also think: What new things to try out in this world we're living in?

A ship at port would never sink;
But that's not what ships are made for.

Signal Transduction

Viewer discretion is advised
Editor's note: Each line is equivalent to 1 minute



Starring


1. GPCR = G-protein-coupled receptor (a.k.a. 7-helix GPCR)

2. G-protein = GTP-binding protein

3. AC = adenylyl cyclase

4. cAMP = cyclic adenosine monophosphate "second messenger"

5. PKA = protein kinase A (cAMP-dependent protein kinase)

*G-protein is heterotrimeric and is composed of three subunits:
G alpha (G_α), G beta (G_β), and G gamma (G_γ).

Also:
GTP/GDP = guanosine tri/diposphate
ATP = adenosine triphosphate

"How does the hormone adrenaline - "fight or flight hormone" - increase blood glucose?"


Hormone
{adrenaline}

1↓binds to and alters the conformation of

Receptor
{7hGPCR}

2↓activates [GTP displaces GDP, and binds to G_α]

G_αs.GTP
"G-alpha-s, GTP"

3↓detaches from G-protein, and activates

AC

4↓catalyzes conversion of ATP to

cAMP

5↓binds to and activates

PKA


Now PKA has more than just one intracellular target protein.
Examples:

---(i)---
PKA

6↓enters nucleus, and activates

KERB

7↓↓↓many steps

alters gene expression


---(ii)---
PKA

6↓activates

Phosphorylase kinase

7↓activates glycogen phosphorylase b to form

Glycogen phosphorylase a

8↓catalyzes conversion of glycogen to form

Glucose 1-phosphate

9↓↓↓many steps

Glucose


So, there you have it (at first-semester level).

The steps via which adrenaline increases blood glucose. □

NobelPrize.org

Credits:
A/Prof M. A. Bogoyevitch (Week 10 lecturer)
Nelson & Cox, Lehninger Principles of Biochemistry (4th ed.)
Signal Transduction in Cells - NobelPrize.org

"God created Mathematics (III)"

Where all this takes us.

Isn't it amazing how much mathematics has grown over the years? A theorem that's only one line (with the equation being only one half line actually) takes 150 pages of proof. Talk about elegance!

From numbers and quantification, to the conceptualization, formalization and abstraction of real situations - from what is seen to what is unseen - human ideas fuel the advancement of math as a field of academia.

Is it worthwhile to study Mathematics then?

Humans created Mathematics.

But God created humans.

See where this is taking us?

(And, to study Math is a way of manufacturing action from our faith - to glorify God because of the beauty of His creation.)

To me, God is somewhat like an axiom. It's taken to be true without needing any proof. If Set Theory relies on the Axiom of Existence and Axiom of Choice, then our whole existence in this world relies on the Axiom that God is God.

In long series of logical reasoning that eventually ask where all this began, my answer is: God is the "uncaused cause".

God doesn't need a beginning.
The beginning needs God.

God doesn't need a reason.
Reasoning needs God.

For otherwise, in who else's image would we get human reasoning ability and the freedom of choice?


For now, I rest my case. QED.

All glory to God.

P/S: Now, a friend of mine did point out: "It's funny that you mentioned God created Mathematics. Because 1 ≠ 3." To that I respond: God created Math, not the other way round. God is not limited by His creation.

"God created Mathematics (II)"

A sequel to the previous.

xⁿ + yⁿ = zⁿ

Just look at that equation (more at: MathWorld).

At first glance, most of us would probably see 3² + 4² = 5², which is an example of Pythagoras' Theorem.

But for cases where the integer n = 2 or greater, are there any solutions? (By the way, x, y, and z are nonzero integers.)

A 17th-century French lawyer (and mathematician) named Fermat said that there aren't. This statement became known as Fermat's Last Theorem.

It was proven true only hundreds of years later, around 1993 to 1995 by Andrew Wiles, a British mathematician.

When asked whether his proof is the same as Fermat's, Wiles responded:

There's no chance of that. Fermat couldn't possibly have had this proof. It's 150 pages long. It's a 20th-century proof. It couldn't have been done in the 19th century, let alone the 17th century. The techniques used in this proof just weren't around in Fermat's time.

I took a look at the mathematics being used in the proof, and it made me dumbstruck. There was just too much math jargon that I didn't bother comprehending the material fully.

The few new words that got stuck in my mind are elliptic curves, Taniyama-Shimura conjecture, and Galois representations, although I know _nothing_ about them.

To be continued.

"God created Mathematics (I)"

Apologies for not having updated this blog in a long while. Two friends have mentioned it, and now I'm beating the old inertia to start afresh a new one.

Why do people study Math? In school, we're exposed to the idea of something being "more" than others (which are "less"). Like some people are taller - they have "more" height than others. Or perhaps we'd assign something more specific to denote that height - like 6 feet, or 183 cm.

Numbers, therefore, play a part as a scale for comparison in everyday human life.

And as we go into more schooling - and delve into more Math - we get exposed to the idea of unknowns. Some unknowns can become known. Some require that other unknowns be known first. Like if we know that three people have a total of 24 DVD's, and if we know that two of them have 4 each, we can tell - or calculate, deduce, conclude - how many the third person has.

The equation to represent this problem could be:
4+4+x = 24

And the solution steps:
8+x = 24
8+x-8 = 24-8
x = 16

It's algebra (where everybody writes about x, y, z and all other letters instead of unknown number one, unknown number two, unknown number three and so on...because it's shorter).

Mathematics also has an intimate relationship with logic, as is taught in secondary schools. Like:

All herbivores eat plants.
I am a herbivore.
Therefore I eat plants.

The first two statements are called premises, and the third is called the conclusion.

Let me introduce to you the contrapositive, which is also a valid argument:

All herbivores eat plants.
I do not eat plants.
Therefore I am not a herbivore.

It follows this format
Statement: "If P, then Q"
Contrapositive: "If not Q, then not P"

I was quite fascinated by this fact when I first encountered it last year!

And where else does Math take us? There are a lot of places.
The three mentioned above are number theory, linear algebra and logic.

Some more that I've superficially encountered:
Graph theory
-path and circuits, Hamiltonian and Eulerian

Functions
-quadratic, cubic, hyperbolic

Statistics
-median, quartiles, t-tests, ANOVA

To be continued.