At 5/14/11 05:35 AM, i-am-ghey wrote: all i can think of is if g stays at one sign but does not reach 0, then consider the sets point x, and x+1/n, vary x from 0 to n-1/n, if f(1/n)>f(0), then f(1)>f(0). similarly, if f(1/n)<f(0), then f(1)<f(0). so g must have a zero.

you know that i mean.

At 5/14/11 05:35 AM, i-am-ghey wrote: all i can think of is if g stays at one sign but does not reach 0, then consider the sets point x, and x+1/n, vary x from 0 to n-1/n, if f(1/n)>f(0), then f(1)>0. similarly, if f(1/n)<f(0), then f(1)<0. so g must have a zero.

Simply put:
g(0)+g(1/n)+g(2/n)+...+g((n-1)/n)=f(1)-f (0)=0.
So then if all terms had the same sign, then the result can never equal 0. So either everything is 0, or there is a positive and a negative term in the sum. In that case, intermediate value gives the result.
But I guess that's what you were aiming at.

On a side note, what kind of mathematics were you doing? You finished high school? Do you stude mathematics at university level, or as a spare time interest?

i am studying math at a university.

At 5/14/11 06:45 AM, i-am-ghey wrote: i am studying math at a university.

Ah, I kind of get where your great math skills come from. But it seems these studies suit you very nicely. Some prospects of specialisation you want to go for?

On a side note, an easy one for beginners:
Show that the square of an odd integer is always 1 more than a multiple of 8.

At 5/17/11 06:57 PM, RubberTrucky wrote:

At 5/14/11 06:45 AM, i-am-ghey wrote: i am studying math at a university.

Ah, I kind of get where your great math skills come from. But it seems these studies suit you very nicely. Some prospects of specialisation you want to go for?

On a side note, an easy one for beginners:
Show that the square of an odd integer is always 1 more than a multiple of 8.

i'm kinda interested in real analysis.

and one should note that any odd integer can be written as 2n-1, for n=1,2,3...

one way of doing it is assume (2n-1)^2=8M+1, M is non-negative, and prove it using math induction on n.

alternatively, consider 4n(n-1).

it is clear that the product of two consecutive integers is divisible by 2.

At 5/18/11 09:40 PM, i-am-ghey wrote:
i'm kinda interested in real analysis.

Where I come from, there's about 2 major types of analysis. One is classical analysis (orthogonal polynomials/measure theory/...) and functional analysis (Von Neumann algebras/Quantum Groups/...)
The first one is closely related to solving integral/differential equation problems. the latter is often mixing analysis techniques and applying group theory.

Here's one for you, though.

1.Suppose that a function satisfies that every derivative in x=0 equals 0. Show that the function is constant, or supply a counter example.
2. Consider the differential equation
x''+a(t)x'+b(t)x=0.
suppose x(0)=x'(0)=0.
Show that the solution is given as x(t)=0.

i think one can approach the first question using taylor's expasion of f about x=0.

it's a common way to deal with functions that are n times differentible.

second one is a second order differential equation problem. not sure if i have sufficent knowledge to do it, nut maybe i can find a way.

this is a very tricky one.

for the first question, i can think of an counter-example.

let f(x)=0 x<=0, f(x)=e^(-1/x), x>0.

use induction and l'hopital rule, one can show the derivative at x=0 is 0. but function is not constant.

hey
im studying maths at an A2 level, any hints for mechanics or c4 papers?

At 5/20/11 11:00 PM, Marsupial wrote: hey
im studying maths at an A2 level, any hints for mechanics or c4 papers?

You can state the problem, if you like. Or just clarify what angle you want.

always draw force diagrams no matter if the are required or not. gives you a good picture.

if you are asked to derive a formula, make sure to test it for extreme cases (like theta=0 or 90 degrees). you'll know if you've made any mistakes in your calculations (resolving vectors into perpendicular and parellel components in particular.)

had my c4 exam today got at least 90% i reckon. I'm happy with that as it would mean i only need 60% in mechanics to get an A overall :)
plus might change my degree for next year to mathematical physics instead of physics, what do you guys think?

I don't understand what a exponent is!

I kid, I kid.

At 6/16/11 04:28 PM, Marsupial wrote: had my c4 exam today got at least 90% i reckon. I'm happy with that as it would mean i only need 60% in mechanics to get an A overall :)

usually the actual score in an exam is 10-15% lower than the expected score. i believed i have gotten around 95% advanced level pure math exam but in fact i scored 84% after checking my marks, 5% above the cut-off score for A grade. that's for careless mistakes and poor presentation i was unaware of.

i have heard of a person who were confident he could get at least 70% while his actual mark is 45%

plus might change my degree for next year to mathematical physics instead of physics, what do you guys think?

you might give it a try if you have a solid math background, especially in statistics and calculus.

I'm taking summer school because I failed math, I'm very good at it, especially fractions. It's when it comes to algebra that really fucks with me. It's all online so that makes it easier. Basically everyone had to start at the VERY basics of math like 2+2, I blew right through all of it because it was so easy. Right now I'm working on adding and subtracting fractions, which is still easy. I'm fine with everything except algebra. When it gets to shit like finding the value of n is when I start to screw up.

At 6/16/11 10:24 PM, i-am-ghey wrote:

At 6/16/11 04:28 PM, Marsupial wrote: had my c4 exam today got at least 90% i reckon. I'm happy with that as it would mean i only need 60% in mechanics to get an A overall :)

usually the actual score in an exam is 10-15% lower than the expected score. i believed i have gotten around 95% advanced level pure math exam but in fact i scored 84% after checking my marks, 5% above the cut-off score for A grade. that's for careless mistakes and poor presentation i was unaware of.

Well with absoultely no mistakes i reckon i got something like 95 as there was only one part of a question that i didn't do fully so dropped couple marks there plus i checked most of the questions after and found that I didn't spot any mistakes, even with the 8 mark binomial megaexpansion, I don't really make many stupid errors anymore which comes from months of practise. However it is feasible that i might have missed out a "c" here or there when integrating so i'm not too sure but i reckon I should still get around 90% or above :)

i have heard of a person who were confident he could get at least 70% while his actual mark is 45%

yeah thats happened to me a couple times but i reckon i just saiid I was at 70 to make myself feel better about the exam when in fact i subconcsiously knew I had failed it
In january I thought I had gotten a D in c3 maths which was a particularly tough paper but turns out I managed a B at 76%

plus might change my degree for next year to mathematical physics instead of physics, what do you guys think?

you might give it a try if you have a solid math background, especially in statistics and calculus.

Yeah I am interested in physics but I do have a certain passion for maths and I do enjoy it, my physics tutor studies mathematical physics and he has a solid knowledge in both and is doing his phd now so i'll definetly look into that

I'm not surprised with that post above. Maths clubs in any shape or form all over the world are suspectible to trolling

was there any open ended questions in the paper apart from direct calculations?

in hong kong, the advanced level physics exam consists of three separate papers: conventional questions, multiple choice questions, and essay type questions.

most people lose a lot of marks (say 40%) from essays even though they know the answers to most questions, because they missed a lot of cruicial points, same for open-ended questions in the other paper. some get minor mark deductions here and there because they didn't show all the steps in the calculations.

at any rate, you should not be too confident, and try to get as many points as possible in the coming paper.

At 6/17/11 08:45 AM, i-am-ghey wrote: was there any open ended questions in the paper apart from direct calculations?

in hong kong, the advanced level physics exam consists of three separate papers: conventional questions, multiple choice questions, and essay type questions.

most people lose a lot of marks (say 40%) from essays even though they know the answers to most questions, because they missed a lot of cruicial points, same for open-ended questions in the other paper. some get minor mark deductions here and there because they didn't show all the steps in the calculations.

at any rate, you should not be too confident, and try to get as many points as possible in the coming paper.

not sure what you mean by open-ended questions but I doubt there were any, pretty standard paper, our specifications are probably quite different anyway

mm i guess but I'm solid at mechanics got full marks in my last mock and i got an A in the physics mechanics paper

At 6/17/11 08:45 AM, i-am-ghey wrote: was there any open ended questions in the paper apart from direct calculations?

in hong kong, the advanced level physics exam consists of three separate papers: conventional questions, multiple choice questions, and essay type questions.

Th nice thing about physics/math, I found was the fact that it didn't rely on open essays. I mean, here's the situation, the potential, now calculate the maximal density and the average velocity of the system kind of things I'm fine with. But to go towards "now write a 25 page treatment of global environment issues and planning on solutions" never really got on to me. far too practical and engineery to my taste.

Marsupial, I wonder what kind of mathematical physics you're interested in. I know there's stuff like Quantum theory and Quantum information, relying on Operator theory. Or statistical mechanics, with all its ensembles and measure theory. There's, I know, a lot to be said about geometry and Symplectic systems, as well as Riemannian Geometry in relativity. Finally, dynamic systems and a lot of differential equation theory.

Funny thing, as a physicist, try to know what you talk about mathematically. It's a funny issue over here. A lot of physicist pay so much attention on 'covariant', 'contravariant' and 'invariant' vectors. I always found that helluva confusing. I stick with vector fields and differential forms and just know they are practically the same. The fact that a lot of physicists bother about making that distinction is a bit silly to us.

C4 exam on Monday.

Gonna wreck that shit.

Been nailing some practice papers at 95%+ it's so damn easy.

I got my average problem question scores back today. Did okay in the 2 maths modules, 6.9/10 in maths IV (vector calculus and Fourier series) and 7.3/10 in maths V (2nd order partial differential equation and matrices). Not great, but they'll do. I think I've done better than that in both of the exams, and the problems only make up 15% of my grade, the exam is the other 85%.

At 6/17/11 07:03 PM, SomaGuye wrote: C4 exam on Monday.

Gonna wreck that shit.

Been nailing some practice papers at 95%+ it's so damn easy.

I got my best mark in the entire A-Level in C4. It's a nice paper.

At 6/17/11 08:07 PM, TheMaster wrote:
I got my best mark in the entire A-Level in C4. It's a nice paper.

Subject?

At 6/17/11 08:40 PM, RubberTrucky wrote:

At 6/17/11 08:07 PM, TheMaster wrote:
I got my best mark in the entire A-Level in C4. It's a nice paper.

Subject?

C4 is part of the second year of the maths A-level. If I remember rightly, it's mostly integration, with a little bit of trig in there, but it was a couple of years ago now so I can't quite remember which bits of the course were in which modules.

At 6/17/11 06:52 PM, RubberTrucky wrote:

lot of physicist pay so much attention on 'covariant', 'contravariant' and 'invariant' vectors. I always found that helluva confusing.

LOL i called them upper and lower myself because i can't remember which is which.

that's interesting.

i never realise there is such a HUGE difference in terms of syllabus. even ordinary differential equations are not included in HKALE pure math, let alone partial differential equation. we never learn anything about multivariable calculus until year one in undergraduate studies. do you guys need to know hyperbolic sines and cosines and stuff like that as well?

but of course the exam questions in hong kong is less standard and much harder. needs more problem solving skills than knowledge.

At 6/17/11 10:32 PM, i-am-ghey wrote: that's interesting.

i never realise there is such a HUGE difference in terms of syllabus. even ordinary differential equations are not included in HKALE pure math, let alone partial differential equation. we never learn anything about multivariable calculus until year one in undergraduate studies. do you guys need to know hyperbolic sines and cosines and stuff like that as well?

We don't do PDEs until undergraduate either, the two maths modules I've just finished are part of my physics degree. There is a little bit of ordinary differential equations in the optional "Further Maths" A-Level, which I did take, but very few universities actually require it.

At 6/17/11 10:32 PM, i-am-ghey wrote: that's interesting.

i never realise there is such a HUGE difference in terms of syllabus. even ordinary differential equations are not included in HKALE pure math, let alone partial differential equation. we never learn anything about multivariable calculus until year one in undergraduate studies. do you guys need to know hyperbolic sines and cosines and stuff like that as well?

I'm always confused when people talk abu-out graduate/undergraduate/...

Math curriculum from the age of 13:

13-14-> Elements of calculations (PEMDAS), handling expressions with abstract letters, planar geometry with area of circle and so on.
14-15-> linear equations, goniometry, basic graphs
15-16-> geometry of similarity and congruence, full trigonometry, quadratic equations, geometry of circle
16-17-> systems of linear equations, continuity, limits, logic, logarithms
17-18-> differentiation/integration, elements of affine geometry, conic sections, probability theory/ statistics (also, some elements of mathematical reasoning, like induction)

University: things have changed, but this is how I've done it.

Year 1
Analysis: rigorous integration and differential equations, limits and Cauchy sequences, theory of differentiation and integration, multivariate functions (differentiation and continuity), metric spaces, introduction of special functions (goniometric, goniometric hyperbolic, exponential function, Gamma/Beta function)
Linear Algebra: Introduction of vectorspaces, matrices and notions of inproduct spaces
Geometry: Affine Geometry and theory of curves (Frenet frame)
Logic/Algebraic structures: Elements of logic languages and integer moduli spaces

Year 2:
Statistics and probability theory: A global overiew of probability theory (Bayes and such) and manipulation of statistic distributions and estimators
Algebra: Theory of groups, rings and algebras (isomorphism theories, normal subgroups and ideals, UFD, PID) , theory of diagonalisation theorems for normal operators, Gauss-Jordan theory
Analysis: Lebesgues integration, Cauchy theorem for complex functions, elements of Banach spaces.
Geometry: Projective spaces, theory of algebraic curves, surfaces in Euclidean space
Topology: (never took this course) Open sets, closed sets, continuous functions, connected and compact sets,...

Year 3 and year 4 (I've taken this within the new system, so i just run down the courses I've picked)
Differential Geometry: theory of surfaces more in depth (theorema egregium, differential frames, Hopf theorem,..)
Riemannian Geometry: expansion of the above-> Riemann curvature tensor, Desch symmetric spaces, Schur theorem,...
Submanifolds: Theory of submanifolds in standard spaces, in spheres and Kahler geometry
Advanced geometry: Quantification, Fedosov and Hodge cohomology and *-products, geometry of fibre bundles.
Differential Topology: Brouwer degree, winding number, theorem of Gauss-Bonnet

Algebra: Theory of Gallois, Multinomial rings (Groebner basis), Hilbert's Nullstellenszatz on algebraic curves.
Algebraic geometry: Algebraic curves and surfaces with a stress on the interaction of algebra and algebraic maps
Operator theory: Hilbert spaces and operators on it, theory of Gelfand, theorem of Stone, study of spectra.
Functional analysis: traceclass operators, theory of Banach/Hilbert spaces, topologies, theorem of Krein-Milman, Hahn-Banach theorem
Algebraic topology: Homotopy fundamental groups, Homology and cohomology theory

but of course the exam questions in hong kong is less standard and much harder. needs more problem solving skills than knowledge.

Chinese students here, stand out a lot. Other students are out there partying and having fun in their time off, however Chinese students usually just keep on going and you rarely ever see a Chinese student not busy studying. There seems to be a very heavy responsibility on performing and obtaining the best possible grades where you come from.
We have pretty high standards when it comes to exams, as I've heard stories comparing with other universities. But it will never rival the demands made in a few Asian universities. I'm pretty confident that some high level American university would be a cake walk to you.

Sat C4, It was pretty easy, noticeably harder than average but had absolutely no problems with it.

sat m1 today, was pretty much a breeze, there was one tricky question but I think I nailed it as I got an angle that looked appropriate to the diagram. I wouldn't usually judge my answer on this but the angle was near a right-angle and I got 82 degrees so i assumed that was correct