## Thursday, 23 September 2010

### 23-9-10

1. Calculus (pure math) ，覺得自己基本功做不好
2. Co-geom(China AL) ，就算DSE唔考都要學好佢
3. Abstract Algebra -- M. Artin的那本Algebra還有陶哲軒的實分析
4. E&M   不知怎的，AL的EM睇極唔明

Osu! - 已經做好被作弊佬爆上黎搶overall #1的準備，同時刷25000PC去做全破achievements，自己現在map > play
CG - 努力練習中，目標係CW33/34出現

## Tuesday, 21 September 2010

Question 1: rather easy...
i)yes since x=+/- sqrt (k) y, no matter it's positive or negative it can be written in forms of x=cy.
ii)yes for the same reason.
iii)no except x=y (bonus for x=y) since log x = k log y, x=y^k.
iv, v)By contradiction, i.e., when x=cy, f(x)=kf(y), f(cy)=kf(y) is solvable (Bonus for solvable)  i.e., unique numbers of x,y is obtainable so x,y is no longer a variables.
b)So far I can only find three characteristics:
1)fixed point 0: f(0)=kf(0).
2)Strictly increasing: k^nf(y)=f(c^n)y
3)Everywhere differentiable: Proof left for readers.
For geometrical approach the obvious way is "parametic equations".

Question 2: hard
a)It must be multivariate functions.
b,c) They can be non-partial variations since the comoponent of the product has the same nature.
d)They can be non-partial variations as long as the degree of the function is unique.
e,f)Kept open.

## Monday, 20 September 2010

### Drafts of answers to the "question of applied maths" Part 2

My answer: As mentioned in (a) the main prupose of score is to compare, one of the example will be the examinations and tests. (1+1) Since "result" shows an integrated review of the process, and different results can't be directly compared, score instead of results shows "how good does it performed" instead of "how can it perform". Moreover not just only the overall results are quantificated, "advantages" and "disadvantages" of performance is quantificated. As a result, extra infromation from the score is obtained through the process. (5)
The most important factor that decides f is the reversibility of the performance. (1) For example, scores in a competition or examination is not reversable but scores in a game is reversable through Save and Load Method. (1)
First we consider the case that "the process is not reversable".
It means that only one results will be put into f(x). (1) The output of f(x) is directly put into comparason. Assume all factor (elements of input set) is partially proportional to the score obtained. As a result, when one of the factor goes to the "good" side (is expected to make an increase in score), it must be strictly increasing. As a result, "assessment quality" is nearly equivlent to the score obtained. (3)
When the process is reversable, we have to consider one more factor: the overall score is counted as sum of each play or best play among the many trials. (2)
Case 1: The process is reversable, but only the best one is counted. (e.g. sports, musical game)
In this case the "final score obtained" will be man{f(x1), f(x2)...f(xn)}.
Assume the score that can be obtained by a player with his best performance or highest obtainable score be f(k)=S.
When number of plays is infinite-ly high, it is expected the the score obtained will be S. And within a finite number of plays, a trend that tends to S is also expected.  As result, the growth rate of the score shows the importance of repeated exercise on the process.(2)
For example, the exercise brings improvements on the performance at the rate of (x^1/2), f can be a quardratic function so that the overall exercise is directly proportional to the score obtained.
Another exmaple is the musical game, when a broken combo is completed, the scores multiplies geometrically. However the probabilty of complete one more broken combo is much smaller than the original performance, which is approximately "quadratic effort" to get "quardratic score". (2)
Case 2: The process is reversable, and there's another column as the "overall score" of score obtained for each process.
In this case, the final score will be Σf(xi).
As long as the score obtained is positive and is bounded, to a certain extent an infinite number of play is far important than score obtained in each process. (1)
However in reality we can't play an infinite number of the process, which enable us to find the equilibrium between number of plays and quality of plays. (2)
Case i: Quality >> Quantity
In this case we assume that quantity is negligible when compared to quality. A tiny change in quality will lead to an irretrievable change in total score. That is, f(x)>>f(x-Δx) or even f(x)>>kf(x-Δx).
However "quality" can be improved in terms of "quantity". i.e., through lots of exercise the quality can be improved. This surely violates the irretrievability of quality. Therefore the improvement of quality through quantity can't be directly shown in the score obtained, the only solution to f is that, the final score is the mean of the previous score. For example, a win rate of 70% in 1000 plays is much better than 60% in 10000 plays. (Of course, when quantity is not large enough comparason in this case is meaningless under disturbances of small chaos.)
Case ii: Quality > Quantity
In this case quality is important than quantity where the difference can be offsets by more plays. Considering the improvement of quality by increasing quantity f can be (growth rate of quality by quantity)(growth rate of quality).
For the case Quality =< Quantity it's similar to case ii.
When Quality is much less important than Quantity, where 2min(f)>max(f). It's quite obvious that whenever max(f) > min (f), there exist m>n such that m min(f) < n max (f). Then we can conclude that min (f) = max (f), i.e., f is a constant. When the score shows somehow like "play count", quality is not important at all. (8)
d)Now extend the co-domain of the process into an n-space vector. Give examples that how f process and affect the coming decision of controlling invariate x. Discuss the feasibility to put a continous movement of controller (reality player) in the computer / processor, and after a strict process (i.e., f(x) is fixed), to give the expected results. (Hint: Chaos Theory may be involved.) (43M)

I'm afraid that I can't really answer this one since this is far mor difficult. However we can surely answer no to the second half of the question provided that random decision is made by processor. If random decision is made, the "score" obtained after a certain process (f) must differs between a more preferred result and a less one. As a result the difference in cabability cause different decision in later stage of the play.

Last part of the question : Paper-scissor-stone is a very simple game that paper < scissor < stone < paper. We can see that there's no stright order between the three signs. Inversely the "win rate" can't reflect which of the signs were used most.
i)Show that when your opponent plays perfectly random, you chance to win is 1/3. (8M)
ii)Show the relationship between inversability of score and information and the inordered nature of the process. (12M)
iii)Account the possibility to create a mathematical tool towards cyclic inequalities system (i.e., a>b>c>a or more variables, not the inequalities involving cyclic functions) to process such scores. (30M)

part d,e is kept open...

## Sunday, 19 September 2010

### Drafts of answers to the "question of applied maths" Part 1

due to horribly long answer (over 1000 wds), I decided to present my answer (not model answer) into 2 parts.

part a,b: marking scheme provided

"score given" by another person
a)Define the term "score" in different perspectives. (15M)
-Another expression of "result"; simplified and quantificated - with explaination, each 2+3M
Score is defined as the integrated expression of a results after a certain process.(2 - "another expression") Results can be expressed in many forms in terms of visibility or sound (2 - Ex of another expression), but score is a rather simplied version of a results (2 - "simplicity"), which is presented with a number or a rank (set). (3 - Ex of simplicity)
At the same time one of the important difference between results and score comes from the compatability of scores. (2 - "quantification") Under the same system and process, scores are comparable while comparason is not so obvious as score (1 - Ex of another expression) .We can say that score is the quantification of results, with the help of ordered field (most probably real number) we can compare much more easily. (3 - Ex of quantification)
b)b)Consider the following cases, select an appropiate function f(x), state its domain and co-domain as well as the brief explaination to the process f. You may select f as a overall score (insert x and directly takes f(x) as the score) and f as a partial score (f as a partial score, while the total score is equal to sum of f(x_i).)
i)Score = hits within a period of time
ii)RPG game final results
iii)Rhythm game, score from each hits is proportional to the combo and inversely proportional to the time offsets error. (12M)
Expected answer : explaination (1) + PS/OS (1) + domain (1) + co-domain (1)
i)
f(x)=x, since ability is strictly equivlent to the hitting speed. (1)
Obviously this is OS, or PS = 1, also f: N→N. (1+2)
ii)
f: N^n →N, each number in the n-space vector divided by the corresponding growth factor and take their absolute value or mean. (2)
The idea comes from the limited growth chance. It's also a OS system, PS can't exist in this case. (1+1)
iii)
g = sum of f and f = x^2/e where x is the current combo and e as a measurement of error, it's obviously an PS system. (1)
f: N^2→Q, g: Q→Q, if integer function is considered than g: Q→N. (2)
This calculation alogarithm is obvious to sum up points from each hits to get an overall score. (1)
c)With reference to your answer in (a), state the nature of score and what can score show. Furthermore discuss, with examples, the compatability of score and its meaning, i.e., the growth factor / f'(x) VS the number of plays, cumulative scores, etc. (30M)
Nature and meaning of score (8M) - Comparason
Compatability (22M) - Reversibility of scores (e.g. S/L, replay) VS play count VS Play performance

## Wednesday, 15 September 2010

### Question on variations

F.4,5 level? yeah~
Question 1 direct variations
a)For each of the following conditions, does it imply that x is directly proportional to y? Assume x,y are real.
i)x^2 varies directly to y^2. (4M)
ii)x^3 varies directly to y^2. (4M)
iii)log x varies directly to log y. (5M + 1 Bonus)
iv)tan x varies directly to tan y. (5M + 2 Bonus)
v)sin x varies directly to cos y. (5M + 2 Bonus)
b)In a more general way, when does x proportional to y if f(x) varies to f(y) is given? Explain your answer in terms of:
i)Algebrical method (11M)
ii)Geometry apporach (15M)
Question 2 partial variations
a)State the nature of functions that is used in partial variations. (3M)
b)If the function is symmetrical, what's its special properties? Explain. (6M + 1 Bonus)
c)If the function is cyclic, what's its special properties? Explain. (8M + 1 Bonus)
d)If the function is alternative, what's its special properties? Explain. (6M + 1 Bonus)
e)If f'(x) is directly proportional to f'(y) where f is a polynomial, find the corresponding types of variation between x and y. (7M)
f)For the relationship x=f(y,z), where f is a polyomial, find the relationship between x and (δf/δy+δf/δz). (13M)
Full marks: 100 Time limits : 75 minutes or 1.5 hour

## Saturday, 11 September 2010

### Question on Applied Maths

Pure mathematics is a strict science that "approximation" is pointless in most of the proof, and some approximation tools like big O notation (O(f(x))) is bonded to give an accurate limits on the behavior of a function.
However, we can see that approximation in the field of physics and mathsmatics is just powerful, like the formula E=mc^2, is done by approximation. The following question is quite interesting in the sense that it is approximation does give a general view of the whole problem:

a)Define the term "score" in different perspectives. (15M)

b)Consider the following cases, select an appropiate function f(x), state its domain and co-domain as well as the brief explaination to the process f. You may select f as a overall score (insert x and directly takes f(x) as the score) and f as a partial score (f as a partial score, while the total score is equal to sum of f(x_i).)
i)Score = hits within a period of time
ii)RPG game final results
iii)Rhythm game, score from each hits is proportional to the combo and inversely proportional to the time offsets error. (12M)

c)With reference to your answer in (a), state the nature of score and what can score show. Furthermore discuss, with examples, the compatability of score and its meaning, i.e., the growth factor / f'(x) VS the number of plays, cumulative scores, etc. (30M)

d)Now extend the co-domain of the process into an n-space vector. Give examples that how f process and affect the coming decision of controlling invariate x. Discuss the feasibility to put a continous movement of controller (reality player) in the computer / processor, and after a strict process (i.e., f(x) is fixed), to give the expected results. (Hint: Chaos Theory may be involved.)  (43M)

If possible I will present my answer (does not means that it is model answer) for part a,b,c.

### Proposed Notes List

Mathematics
-Tricks on quadratic graphs and polynomials
-Trigonometry transformation
-Algebra of M2
Physics
-Gravitation
-Gas (Gas Laws, Ideal Gas Law, Kinetic Theory)
-Wave
-Electromagnetics
Chemistry
-Redox and electrolysis
-Chemistry of carbon compounds
-Introduction to physical chemistry (if it is taught)
Economics
-Efficiency and Equity
-National Income and Price level

That list is tough enough...

## Friday, 10 September 2010

### 偽．SIMC回憶錄1.2

Day 0 12:00, SG

*_*

(待續)

*嘛, 我懶了...很多東西都沒寫出來呢

## Tuesday, 7 September 2010

### 06-09-10

1)筆記； 理化經濟最好還有數學
Progress: Chem : 10% Other 0%
2)SIMC回憶錄+SIMC problem analysis PART II
Progress: 5% (1.2 50%)
3)Pilot
Progress: unknown (2.2 50%)