Let's discuss Mathematics

[Ammar]

Are derivatives of vector valued functions tensors as well?

E.g. derivative of a fn from R^2 -> R is a vector. 2nd derivative is a matrix. 3rd derivative = tensor?

You can see it like that. If you consider the derivates as Tensors you can write the multidimensional variant of the Taylor expansion as

f(x+h) = f(x) + f'(x)(h) + (1/2) * f''(x)(h,h) + (1/6)f'''(x)(h,h,h)...

Remember my point about the rank n-Tensor being a multilinear real-valued function of n-vectors.

If you write H as the Hesse-Matrix you get the Tensor corresponding to f'' as

f''(v,w) = v * H * w^t.

In the above formula you evaluate it for v=h=w though.

 

[SS-18 ICBM]

What kind of mathematics is needed to describe chaotic systems? Will we ever develop systems of equations that can reasonably model the movements of weather systems? What about markets? And others?

 

[ParadigmShifter]

Orbits, dynamical systems, difference equations and metric spaces mainly.

The problem with modeling the weather etc. is not the equations but the fact that an extremely small error measuring your variable (such as round off, which is inevitable since we can't measure a variable to infinite accuracy) means over time the error is compounded and wildly different results appear, this is the butterfly effect, see "bifurcation".

 

[SS-18 ICBM]

But more powerful computers and instruments would make the figures more accurate? Is it even possible to reach the level of accuracy required for effective modeling of such systems?

 

[ParadigmShifter]

No, it's not possible. The nature of a chaotic system is that there are regions where infinitesimally small discrepancies in measured vs. actual value result in huge differences over time. Try zooming into to a point near a boundary edge of the mandelbrot set for example.

 

[ParadigmShifter]

Here's a good link about chaos anyway

http://hypertextbook.com/chaos/

 

[SS-18 ICBM]

An essay about teaching math as an art.

 

[Mise]

Hey guys, I got another small maths question.

How do I divide a rectangle into a series of contiguous smaller rectangles such that there is no space between them, and the area of each rectangle is proportional to some value associated with each rectangle?

E.g. if I have a large rectangle representing Europe, and the small rectangles represent countries in Europe, with their areas being proportional to their population, what algorithm or theorem or whatever do I need to read up on on wiki?

 

[ParadigmShifter]

Surely it is a trivial problem if there are no constraints on tile width and height... order the ratios in descending order, halve the box in the proportion of max:remaining, continue with the remaining values, etc.

If you need a specific arrangement (e.g. position or tiles relative to others, or fixed tile sizes), it get's difficult quickly I think. (i.e you probably have to try all possible arrangements to find the best one).

Here's something I dug up googling for "box packing mathematics"

http://www.plambeck.org/oldhtml/mathematics/klarner/boxpacking/index.htm

 

[Mise]

Ahh, that's really easy, lol. Great, I can do that in Excel I think... Cheers!

 

[ParadigmShifter]

If you only divide the rectangle in one direction, it's just the pie-chart algorithm (EDIT: cut down a radius and made into a rectangle of course)

 

[dutchfire]

(Thumbs up for using excel!)

 

[Ayatollah So]

Here's one for y'all math experts. What's wrong with the following argument?

From this website:

Clearly zero is a very special number, being the first natural number.

One is pretty special too, being the multiplicative identity.

Two is the only even prime.

Three is the lowest odd prime...

Clearly lots of numbers are special. This led me to propose the following theorem:

Theorem: Every natural number (0, 1, 2, ...) is a special number.

Proof:

Let’s posit that the set of nonspecial natural numbers is nonempty, and deduce a contradiction.

If there exists a nonspecial natural number then there must be a lowest nonspecial natural number.

What an unusual property! The lowest nonspecial natural number!

Whatever number has that unusual property must be kinda... special.

Therefore the lowest nonspecial natural number is special.

Therefore, if the set of nonspecial natural numbers is nonempty then it contains a special number.

That is clearly nonsensical, therefore the set of nonspecial natural numbers is empty.

Therefore all natural numbers are special, QED.

And yet I can't find anything special about 7920687935872092847630945767548023. But it must be special somehow!

There's some interesting commentary on the linked site, but I think they missed the weakest point in the argument - and I'm curious how many people will pick on the aspect I would pick on.

 

[ParadigmShifter]

Well special isn't well defined and he seems to argue degrees of membership in the set of special numbers which is in fuzzy logic territory which isn't classical set theory.

"Clearly lots of numbers are special" is poor form as well.

All he has really proven is that the set of natural numbers has no maximum element, via the axiom if N is a natural number then so is N+1, and from the well-ordering of the naturals (so every non-empty set has a least element).

 

[Mise]

I find it interesting that there are no consecutive numbers in that sequence. Highly non-random, IMO. No 1's either!

 

[ParadigmShifter]

Also, if we say that a number is the smallest non-special number hence making it special, what is special about the next largest non-special number? It can't be the least because we already found it. So if this becomes the new least special number what is special about the number we claimed was the least before?

I think that probably nails it.

EDIT: Basically, the argument only makes sense if the number of non-special numbers is precisely 1 (or zero).

 

[dutchfire]

Analysis and Linear Algebra exams tomorrow.
Anyone want to say anything interesting about tensor products, complex inproducts, solving systems of linear differential equations, Zermelo-Fraenkel set theory, real numbers, differentiability, Riemann integrability or other such nonsense?

 

[ParadigmShifter]

I know about real numbers and differentiability! I was pretty good at analysis and have a textbook available

What's a complex inproduct? Is it the dot product over a complex vector space?

EDIT: Presumably since inner product is the proper name for dot product, and I couldn't find any info googling for complex inproduct.

EDIT2: Anyway, get as many past papers as you can, look for questions likely to come up this year based on previous history, and cram heavily in those areas

 

[dutchfire]

I know about real numbers and differentiability! I was pretty good at analysis and have a textbook available

What's a complex inproduct? Is it the dot product over a complex vector space?

EDIT: Presumably since inner product is the proper name for dot product, and I couldn't find any info googling for complex inproduct.

Well, both subjects aren't that hard for me, but I fear it will be very difficult to remember everything, normally I make my exercises with my textbooks at hand so I don't have to remember all those details and definitions.

A complex inproduct (that's probably not the correct English term) is a generalization of the dot product over a complex vector space.
So basically < , > is a complex dot product when:
f_w: V->C, v-> <v,w> is lineair for every w in V.
<v, w>*=<w,v>
<v, v> is an element of R and <v,v> > 0 if v=/=0 and <0,0>=0

@edits:

Anyway, get as many past papers as you can, look for questions likely to come up this year based on previous history, and cram heavily in those areas

Lineair Algebra has got a new teacher this year, and the course has been changed a bit. Analysis is an oral exam. So no dice.

 

[ParadigmShifter]

Yeah, that's the standard inner product over C.

Read my 2nd edit for advice on the exam

EDIT NINJA: Write out definitions and such you are looking up often. It will become ingrained if you do it enough times. Keep doing it until you can write down the definition and it agrees with the book exactly.

More complicated stuff such as long proofs, try and write down a summary of what each step/section is doing or aiming for as well, easier to remember than a load of algebraic manip.

EDIT NINJA RETURNS: Good luck with the exams anyway! Sounds like everyone is in the same boat and they normalise the exam scores I presume?