This came up on the front page of ST a while back:

This is bad. It seems that the word "abstract" has been stripped of its original meaning and used as an sweeping term for modern art. Maybe I haven't been to any art conventions lately, but I have hardly seen any piece in the Singapore Biennales which I can call "abstract" with a clear conscience and a straight face.

Most of the works, however, can be labeled as "postmodern". Now this is a word which can be abused much any way you like, because not even artists can agree on what it is really about. Allow me to make clear this distinction between "abstract" and "postmodern":

Abstract Art

1. MONDRIAN Piet: Still Life with Gingerpot I - Cubist

2. Ibid.: Still Life with Gingerpot II - Simplified (both 1912)

These pieces are very easy to understand. Basically, they are still-life drawings of kitchenware drawn in a very interestingly sloppy manner. This shift of styles towards ingenious sloppiness or simpler form is what we call Abstraction.

Postmodern Art

3. ČERNÝ David: Mimi Kampa (2000)

Postmodern art happens when the artist hates you.

## Thursday, April 28, 2011

## Tuesday, April 26, 2011

### Real Analysis

So the exam was over on Monday. I am still not sure if I will need to do an S/U on the module, but if I ever get so much as a B+ I would be content and hell I'll drag Surya out to one of those green spaces left around school and dance around with him in joy. (FYI: Surya is the other guy from Engineering who took this module for fun.)

Artist's Impression

Bidding for this module was foolhardiness from the start. Learning the methods of Real Analysis has been an uphill task, but explaining its contents to interested passers-by was also difficult, so I'll do just that.

The Meaning of Real Analysis

Calculus, the crown jewels of MA1505, was invented independently by different folks in the 14th to 17th centuries. Apparently where caluclus started, it was mathematically sloppy, but got away with it for being so useful. Nonetheless it was met with attacks from other scholars, among them my old friend Bishop Berkeley. In time, the properties of the calculus-universe would be gradually built up and moulded by those willing to clean up Newton's and Leibniz's mess, and the cleaning process became the field we call Real Analysis.

Formalities: Definition, Theorems and Proofs

Because it looks base-wards from calculus, rather than to the frontiers of math, most of the concepts we tackle are simple and intuitive. This sadly does not make them any easier. It does, however, mean that we have to scrutinise our every preconception to all that we think we know to be true. This is achieved by definitions made precise with numerical gauges and basic rules of algebra. A case in point:

Continuity of a function is

MA1505: "that you can draw its graph in one go without having to lift your pencil from the paper."

MA2108: "if ∀ε>0 ∃δ>0 such that ∣x-a∣<δ ⇒ ∣f(x)-f(a)∣<ε, then f is continuous on a."

So instead of drawing a continuous line without lifting the pencil from the paper to prove a point, you broke everything up into numbers and shifted them around and prayed hard that they agreed with you.

This may seem frivolous at first, but as the lectures go on we saw the definition spew out quite some interesting conclusions. For example, a function can be continuous at only one point. Trying to show it with pencil and paper is quite pointless, but you will be able to show, with definition, that something like

f(x) = {x

does just that at x=0.

Also I mentioned Thomae's Popcorn function some entries back. You guys totally should look at it again. The idiot is continuous on all irrational x and discontinuous on all rational x. It is so crazy I still find it hard to swallow, even if I am able to prove it back to front.

The start of the module was the hardest because we were having our hands tied up behind out backs and mulling over things like a+b = b+a like we were newborns. Things got better later on when we found out certain theorems have been proven so that we can use them with impunity. But then there were other things to worry about.

Aesthetics: Niceness

This is to recall the things I picked up last semester about mathematicians being artists inside. William Byers wrote about mathematical beauty. David Hilbert complained about finding not enough imagination in his students. Leung Man Chun played the Concierto de Aranjuez before every lecture.

Artists would be familiar to the fact that mathematics can be, and are often used to make art. In particular, I have come across a section of the shelf on Medical/Science library (5th floor) with material on Splines, NURBs and other cool inscrutable stuff we find mentioned and used all the time in 3D modeling software. Apparently meshSmooth, spline curves, and modeling by contour uses Real Analysis. This is so interesting

I would like to take this opportunity to show off my "modeling by contour"

In Real Analysis, niceness is a real thing, and is mathematically defined. A continuous function is a nice mapping of ℝ on ℝ, and a uniformly continuous function is a nice continuous function. And a properly divergent sequence diverges quite neatly also. And together they bask your notes in a warm glow.

How to prepare for this module

Being a Math student helps. Having taken MA1100 Fundamentals of Mathematics also helps, because your teachers will assume you picked up from there. If you're one of my ilk, it helps to understand that this is no maths that you have seen before. It is useless to think of anything as "plug in formulas to get a number" (even though I have guiltily indulged in "plug in theorems to get a proof" from time to time). It is helpful to remind yourself that nothing is static, everything is uncertain, and that skepticism is the rule before where the fatal QED lies. Having a lively mind helps, so taking up some sport would be a good idea.

Postscript

Exam Qn 3 b) iii)

Ohmigod, look at this guy, just look at him.

Bidding for this module was foolhardiness from the start. Learning the methods of Real Analysis has been an uphill task, but explaining its contents to interested passers-by was also difficult, so I'll do just that.

The Meaning of Real Analysis

Calculus, the crown jewels of MA1505, was invented independently by different folks in the 14th to 17th centuries. Apparently where caluclus started, it was mathematically sloppy, but got away with it for being so useful. Nonetheless it was met with attacks from other scholars, among them my old friend Bishop Berkeley. In time, the properties of the calculus-universe would be gradually built up and moulded by those willing to clean up Newton's and Leibniz's mess, and the cleaning process became the field we call Real Analysis.

Formalities: Definition, Theorems and Proofs

Because it looks base-wards from calculus, rather than to the frontiers of math, most of the concepts we tackle are simple and intuitive. This sadly does not make them any easier. It does, however, mean that we have to scrutinise our every preconception to all that we think we know to be true. This is achieved by definitions made precise with numerical gauges and basic rules of algebra. A case in point:

Continuity of a function is

MA1505: "that you can draw its graph in one go without having to lift your pencil from the paper."

MA2108: "if ∀ε>0 ∃δ>0 such that ∣x-a∣<δ ⇒ ∣f(x)-f(a)∣<ε, then f is continuous on a."

So instead of drawing a continuous line without lifting the pencil from the paper to prove a point, you broke everything up into numbers and shifted them around and prayed hard that they agreed with you.

This may seem frivolous at first, but as the lectures go on we saw the definition spew out quite some interesting conclusions. For example, a function can be continuous at only one point. Trying to show it with pencil and paper is quite pointless, but you will be able to show, with definition, that something like

f(x) = {x

^{2}if x is rational, 0 if x is irrational}does just that at x=0.

Also I mentioned Thomae's Popcorn function some entries back. You guys totally should look at it again. The idiot is continuous on all irrational x and discontinuous on all rational x. It is so crazy I still find it hard to swallow, even if I am able to prove it back to front.

The start of the module was the hardest because we were having our hands tied up behind out backs and mulling over things like a+b = b+a like we were newborns. Things got better later on when we found out certain theorems have been proven so that we can use them with impunity. But then there were other things to worry about.

Aesthetics: Niceness

This is to recall the things I picked up last semester about mathematicians being artists inside. William Byers wrote about mathematical beauty. David Hilbert complained about finding not enough imagination in his students. Leung Man Chun played the Concierto de Aranjuez before every lecture.

Artists would be familiar to the fact that mathematics can be, and are often used to make art. In particular, I have come across a section of the shelf on Medical/Science library (5th floor) with material on Splines, NURBs and other cool inscrutable stuff we find mentioned and used all the time in 3D modeling software. Apparently meshSmooth, spline curves, and modeling by contour uses Real Analysis. This is so interesting

In Real Analysis, niceness is a real thing, and is mathematically defined. A continuous function is a nice mapping of ℝ on ℝ, and a uniformly continuous function is a nice continuous function. And a properly divergent sequence diverges quite neatly also. And together they bask your notes in a warm glow.

How to prepare for this module

Being a Math student helps. Having taken MA1100 Fundamentals of Mathematics also helps, because your teachers will assume you picked up from there. If you're one of my ilk, it helps to understand that this is no maths that you have seen before. It is useless to think of anything as "plug in formulas to get a number" (even though I have guiltily indulged in "plug in theorems to get a proof" from time to time). It is helpful to remind yourself that nothing is static, everything is uncertain, and that skepticism is the rule before where the fatal QED lies. Having a lively mind helps, so taking up some sport would be a good idea.

Postscript

Exam Qn 3 b) iii)

Ohmigod, look at this guy, just look at him.

## Sunday, April 17, 2011

### Freshman Journal 1.2.13

**A: Real Analysis**

Cue from Bartle and Sherbert (2000) Chapters 2, 3, 4 and 5 (corresponding to school notes 1, 2, 4, 5). Chapter 3 (Infinite Series) is sadly not represented in textbook.

Summary:

Chapter 1: ℝ

Chapter 2: Sequences

Chapter 3: Series

Chapter 4: Limits of functions

Chapter 5: Continuous functions

Modus Operandi: Mop up unanswered questions in tutorials (without reference to answers), finish textbook exercises.

**B: Engineering Maths II**

Cue from Farlow et al. (2007) Chapters 1-6 and 8.

The textbook syllabus is matched to the school notes as follows:

(School notes : Textbook sections)

Part 1. ODE and applications

Chapter 1 ODEs: 1.3, 2.2, 4.2-5

Chapter 2 Oscillations: 4.1, 4.6, 4.7

Chapter 3 Modelling: 1.1, 2.3-5

Chapter 4 Laplace Transform: 8.1-3

Part 2. Linear Algebra and applications

Chapter 5 Matrices: 3.1, 3.3

Chapter 6 Linear Transformations: 5.1-4

Chapter 7 ODE Systems: 6.2, 6.4

Part 3. PDE

Chapter 8 PDEs: refer to Advanced Engineering Math textbook by Kreyzig

Modus Operandi: Finish remaining textbook exercises, start past-year papers, create help-sheet based on compiled summary.

**C: Engineering Physics II**

Broad subtopics covered in First-year Engineering Physics, NUS:

1) Classical Mechanics (PC1431)

2) Thermodynamics (PC1431)

3) Electromagnetism

4) Optics

5) Quantum Mechanics

6) Atomic Physics

Modus Operandi: One-page summary per field + detailed overview of derivations (although to derive all that stuff in 4) or 5) is a bit trying)

**D: Electrical Engineering**

It turns out the content is really quite sparse and can be taken down to one page per chapter. Modus Operandi: copy everything down to a pocket-size (A6) notebook, then read on the bus.

**E: German Language**

A scheme of organising new vocabulary in the process of being perfected:

1. Nouns (organise into feminine / masculine / neuter)

1.i. by categories: ideas / things/ quantities and events / people / places

1.ii. by verb endings -ung / -eit / -ion / -ie / -ik (all with plural -en; female only)

1.iii. By gender-specific subcategory: e.g. [quantities], [days], [languages]

1.iv. By etymology: I use only [International] for nouns

1.v. By plural forms: e.g. ["-e], [e], [n]

1.vi. By composite roots: e.g. [-tasche(n)], [-pl(ä)tz(e)]

2. Verbs

2.i. Category 1: Modal (e.g. wollen, können, müssen)

2.ii. Category 2: Simple native (e.g. essen, schlafen)

2.iii. Category 3: Compound native (e.g. be.zahlen, ver.kaufen)

2.iv. Category 4: Cognate - native with English correspondent (e.g. trinken)

2.v. Category 5: International - Romance etymologies (e.g. notieren)

2.vi. Category 6: Separable (e.g. vorbei/gehen, wieder/sehen)

"." and "/" are used for clarity.

3. Prepositions

3.i. Accusative prepositions (e.g. bis, durch, entlang)

3.ii. Dative prepositions (e.g. aus, bei, mit)

3.iii. Variable-case prepositions (e.g. an, in, auf)

* note some prepositions can/must come after instead of before its subject

4. Others (these are not subdivided further)

4.i. Conjunctions

4.ii. Adjectives

4.iii. Adverbs

### A person to look up to

He may not be a genius, or know everything

He does not need to be great, to be a model of success

He does not need you to feel small around him.

He is a person who wakes the sleeping giants

He exists not to be adulated, but to be surpassed.

He does not need to be great, to be a model of success

He does not need you to feel small around him.

*The person you look up to, he lights a fire in you.*He is a person who wakes the sleeping giants

He exists not to be adulated, but to be surpassed.

## Thursday, April 07, 2011

### Für der dritten Vokabeltest

On nouns eg. Leute, Ferien for which only a plural form exists:

If a noun exists only in the plural form, does it have a gender?

If a tree falls in the forest and no one is around to hear it, does it still make a sound?

On nouns for which the plural form does not conform to any of the 8 regular forms

This ist ziemlich interessant. Notieren Sie that such words tend to be of foreign origin, in particular from Greek or Lateinisch. Let's have a look.

die Praxis - Praxen: Ancient Greek πρᾶξις

das Stadion - Stadien: Ancient Greek στάδιον

das Zentrum - Zentren: Latin centrum - Ancient Greek κέντρον

die Basis - Basen: Ancient Greek βάσις auch via Latin

das Taxi - Taxen: borrowed word short for taxicab, portmanteau of taximeter and cabriolet - taximeter is from French taximètre, which is from German (!) Taxameter - Taxameter is coined from medieval Latin taxa, meaning tax - taxa comes from proto-Indo-European, let's not go there.

die Cafeteria - Cafeterien or Cafeterias: borrowing from the American Spanish, which in turn is a borrowing from French, which borrowed from Turkish, and then Arabic etc.

Let's just forget the last two

If a noun exists only in the plural form, does it have a gender?

If a tree falls in the forest and no one is around to hear it, does it still make a sound?

On nouns for which the plural form does not conform to any of the 8 regular forms

This ist ziemlich interessant. Notieren Sie that such words tend to be of foreign origin, in particular from Greek or Lateinisch. Let's have a look.

die Praxis - Praxen: Ancient Greek πρᾶξις

das Stadion - Stadien: Ancient Greek στάδιον

das Zentrum - Zentren: Latin centrum - Ancient Greek κέντρον

die Basis - Basen: Ancient Greek βάσις auch via Latin

das Taxi - Taxen: borrowed word short for taxicab, portmanteau of taximeter and cabriolet - taximeter is from French taximètre, which is from German (!) Taxameter - Taxameter is coined from medieval Latin taxa, meaning tax - taxa comes from proto-Indo-European, let's not go there.

die Cafeteria - Cafeterien or Cafeterias: borrowing from the American Spanish, which in turn is a borrowing from French, which borrowed from Turkish, and then Arabic etc.

Let's just forget the last two

### Some more things that need clarifying

Concerning (apparently) obscure fields of study

Clarification: If you ask an astronomer to read your star signs, you are asking for a beating with a blunt instrument which you may not even know the name to.

Clarification: Linguistics is not equal to foreign languages.

To add: My Arts GEM shall be a linguistics module, and so will my SS if there ever is one

"Why would anyone want to take a minor in Mathematics?"

Clarification: Mathematics is actually awesome

**1. Astronomy**Clarification: If you ask an astronomer to read your star signs, you are asking for a beating with a blunt instrument which you may not even know the name to.

**2. Linguistics**Clarification: Linguistics is not equal to foreign languages.

To add: My Arts GEM shall be a linguistics module, and so will my SS if there ever is one

**3. Mathematics**"Why would anyone want to take a minor in Mathematics?"

Clarification: Mathematics is actually awesome

## Monday, April 04, 2011

### Songs from 3 April 2011

El Silencio

Federico García Lorca

As heard in El Jardin de la Vida y la Muerte

Oye, hijo mío, el silencio.

Es un silencio ondulado,

un silencio,

donde resbalan valles y ecos

y que inclina las frentes

hacia el suelo.

Listen, my son: the silence / It is a rolling silence, / a silence / where valleys and echoes slip / and it bends foreheads / down to the ground. // Life and death? / La Vida y la Muerte? / Eros and Thanatos

Raghupati Raghava Raja Ram

Vishnu Digambar Paluskar

As heard in Satyagraha

रघुपति राघव राजाराम, पतित पावन सीताराम

सीताराम सीताराम, भज प्यारे तू सीताराम

ईश्वर अल्लाह तेरो नाम, सब को सन्मति दे भगवान

Raghupati raaghav raajaaraam, patit paavan sitaram

Siitaaraam, sitaram, bhaj pyaare tu sitaram

Iishvar Allah tero naam, sab ko sanmati de bhagavaan

Chief of the house of Raghu, Lord Rama, uplifters of those who have fallen, Sita and Rama,

Sita and Rama, Sita and Rama, O beloved, praise Sita and Rama,

God and Allah are your names, Bless everyone with this wisdom, Lord.

Federico García Lorca

As heard in El Jardin de la Vida y la Muerte

Oye, hijo mío, el silencio.

Es un silencio ondulado,

un silencio,

donde resbalan valles y ecos

y que inclina las frentes

hacia el suelo.

Listen, my son: the silence / It is a rolling silence, / a silence / where valleys and echoes slip / and it bends foreheads / down to the ground. // Life and death? / La Vida y la Muerte? / Eros and Thanatos

Raghupati Raghava Raja Ram

Vishnu Digambar Paluskar

As heard in Satyagraha

रघुपति राघव राजाराम, पतित पावन सीताराम

सीताराम सीताराम, भज प्यारे तू सीताराम

ईश्वर अल्लाह तेरो नाम, सब को सन्मति दे भगवान

Raghupati raaghav raajaaraam, patit paavan sitaram

Siitaaraam, sitaram, bhaj pyaare tu sitaram

Iishvar Allah tero naam, sab ko sanmati de bhagavaan

Chief of the house of Raghu, Lord Rama, uplifters of those who have fallen, Sita and Rama,

Sita and Rama, Sita and Rama, O beloved, praise Sita and Rama,

God and Allah are your names, Bless everyone with this wisdom, Lord.

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