SRK Math Thread

[Warrior_s_Dreams]

After finishing real analysis, I never wanted to see that blue book (Rudin) again. On the other hand, abstract algebra is infinitely interesting. Different strokes for different people I guess.

Oh come on… that books a classic :lol:

One thing for me about abstract algebra: I mean it’s cool and all, but does anybody know any applications for it? I think there are some with cryptography and you can model solutions to a rubix cube using permutation groups, but any other ones (or elaboration on the ones I mentioned ) would be nice

[fishjie]

Ok, so I went back to my CV book by Wunsch. So, essentially, the criterion for inclusion into the Mandelbrot set is the following:

1. pick a complex number c
2. plug it into f(z) = z^2 + c
3. define the following sequence:

z_0 = 0
z_1 = f(z_0) = (z_0)^2 + c
z_2 = f(z_1) = (z_1)^2 + c
etc

Now here is the kicker:

if this sequence converges to a finite number, you’re in the Mandelbrot set.

so, for example, c = -2 is in the Mandelbrot set since:

z_0 = 0
z_1 = 4 - 2 = 2
z_2 = 4 - 2 = 2
z_3 = 4 - 2 = 2
so the sequence converges to 2

EDIT If you want to be rigorous about it, pick n = 1, then notice for n > 0 where n is an integer, |z_n - 2| < epsilon for any positive epsilon. It should also be noted that just iterating 4 terms SUGGESTS convergence. It does not PROVE convergence.

Regarding your original question, it can be shown that any convergent sequence is bounded. That was sort of the hidden assumption in the wiki article you cited. However, being bounded isnt enough for convergence. For instance, the sequence sin(n) is bounded at 1 and -1, yet it diverges as n goes to infinity.

Using your example of c = 1 demonstrates an example of something that is not in the Mandelbrot set

z_0 = 0
z_1 = 1
z_2 = 2
z_3 = 5
etc
and so the sequence does not approach a finite number as n grows arbitrarily large. Hence, c = 1 is not in the set.

The nice thing is that it can be shown that if z is in the Mandelbrot set, so is z conjugate. I think this is the reason for its symmetry. What I’m still confused about (in the 20 minutes I spent on this subject) is how come I see one picture symmetrical with respect to the a line on the imaginary plane (say z = i) with the complex plane, and yet in another picture I see the same set rotated 45 degrees to the left.

Now, if you’re in the set, you get assigned a black pixel. What’s interesting is that if you’re not in the set, you get assigned a color depending on how fast the sequence at c diverges. My guess is that the computer uses some O(h) function to compare how fast z_n converges relative to another function, perhaps for instance, 1/h?

Also, it seems as though there is only one Mandelbrot set, but Julia sets do not appear to be unique. A glance at the snippet on Julia sets in this book suggests that one assigns values to both z and c. Furthermore, one need not use z^2 + c. (for example, z^2 + (lambda)z could be used).

Anyways, that should get you started. One other interesting (and computationally efficient) thing: If c is in the Mandelbrot set, so it its conjugate. Hence twice the computational power at about half the cost if you assign this to your algorithm.

yup, we’re on the same page. In terms of my question on convergence, I was more asking along the lines of how a computer would determine whether or not the series converges to a finite number. Cause I wanted to write a fractal generator myself for fun. But, I’ve found quite a couple of mandelbrot java applets on the web with source code, and now that I kinda understand how the fractal works, I should be able to follow along.

Cause the way you can mathematically determine something might not be the same way its coded. This is off topic and has nothing to do with fractals, but for example, computers don’t calculate a large exponent x of a number n by repeatedly multiplying the same n by x number of times. It uses smarter algorithms, such as:
http://en.wikipedia.org/wiki/Exponentiation_by_squaring

And of course, I stumbled upon that when I was reading up on how the RSA encryption algorithm works.
http://en.wikipedia.org/wiki/RSA
Basically, part of the public key contains the product of two large prime numbers. If you can derive those two large prime numbers, you can derive the private key, but factoring out a large product into its two prime factors is not really feasible with today’s computers. Hell even determining if a number is prime is no easy task in itself.

[ejdge]

Just a basic alebraic question:

What is 15x^2*12x^-1 simplified?

[Black_Jesus]

Just a basic alebraic question:

What is 15x^2*12x^-1 simplified?

I’m more interested in talking about different kinds of math you like as opposed to this being a homework thread.

Looks like you’ll be failing Math and Reading this semester.

[ckrazy]

Looks like you’ll be failing Math and Reading this semester.

Double whammy.

Double kill?

[rsigley]

180x

[runaround]

Oh come on… that books a classic :lol:

One thing for me about abstract algebra: I mean it’s cool and all, but does anybody know any applications for it? I think there are some with cryptography and you can model solutions to a rubix cube using permutation groups, but any other ones (or elaboration on the ones I mentioned ) would be nice

For reasons that weren’t covered in my undergraduate classes, the description of electronic band structures in 3 dimensional k space uses group theory to designate the directional axes. The wiki article makes it sound like it has something to do with symmetry groups in physics, which of course I know nothing about.

Also, I’ve heard that some of the more esoteric branches of control engineering require a solid understanding of group theory to even know what’s going on. What little I’ve heard suggests that the application areas for this sort of thing are in aerospace engineering and robotics–subjects which I don’t know anything about. What little control theory I’ve seen in my circuit analysis textbooks sticks with LTI systems, which aren’t (to my knowledge) areas of active research anymore in control engineering.

[Victron]

I’ll be taking Calculus II next fall since I’ve decided to change my major something related to math. Either Mathematics, comp sci, comp engineer, physics, etc. Always got good grades in math throughout high school and college but I was majoring business and kinesiology then. Now I can’t really stand both of it and decided math is for me since I like to grind out math problems and take great pride of it.

The problem is I took Calculus I like 3 years ago. I barely remember all that Derivitives shit yet alone still remember how to graph sin, cos, and tan problems. My question is how related is Calc II to Calc I ?

[thurst]

anybody here know anything about operations research? next to accounting that subject always kinda intersted me after taking one class in it. ti seems to be a mainly population/supply modeling though.

[Shazay]

Snip

You need to know your shit for Calc II. Its easily the hardest course up to differential equations. A review of derivatives should suffice but whats to come is going to require practice.

[ResolutePower]

I’ll be taking Calculus II next fall since I’ve decided to change my major something related to math. Either Mathematics, comp sci, comp engineer, physics, etc. Always got good grades in math throughout high school and college but I was majoring business and kinesiology then. Now I can’t really stand both of it and decided math is for me since I like to grind out math problems and take great pride of it.

The problem is I took Calculus I like 3 years ago. I barely remember all that Derivitives shit yet alone still remember how to graph sin, cos, and tan problems. My question is how related is Calc II to Calc I ?

If you’re switching into pure math (or most math faculty programs) then derivatives will be the least of your worries. You’ll be forced to take real math, not the watered down versions that science and engineering students have to take. Cauchy, Weierstrass and Lagrange will be your new best friends.

[Warrior_s_Dreams]

I’ll be taking Calculus II next fall since I’ve decided to change my major something related to math. Either Mathematics, comp sci, comp engineer, physics, etc. Always got good grades in math throughout high school and college but I was majoring business and kinesiology then. Now I can’t really stand both of it and decided math is for me since I like to grind out math problems and take great pride of it.

The problem is I took Calculus I like 3 years ago. I barely remember all that Derivitives shit yet alone still remember how to graph sin, cos, and tan problems. My question is how related is Calc II to Calc I ?

Calc II is related to Calc I in that you rely on your knowledge of differentiation rules to solve problems involving integrals. If you dont remember what the Fundamental Theorem of Calculus is, you’re in for a world of hurt. Even still, you should retake calculus I because you will lag behind in Calc II if you dont. Later on in the course, you will be rewriting functions in terms of their derivatives (Taylor Series). When you are forced to use integration by parts, you will need differentiation. When you to U-substution, you will need differentiation. When you find arc length, you need differentiation. Surface area requires differentiation. If your professor decides to do differential equations in your Calc II course, you need differentiation.

If you forgot trig, you’re also in trouble. Doing trig integrals requires using certain trig identities. Doing trig substitution requires you to use trigonometry (if you dont change the variables, but that requires differentiation again).

But enough about integrals. If you seriously want to learn calculus well, you must have an intimate knowledge of limits, on which the notion of derivative and integral rest, which is what you learn in Calc I.

Regardless, make sure you have precalculus under your belt. If you don’t, then you should retake precalculus because professors assume you already know that stuff and if you dont, then you are really going to be hurting later on.

[Victron]

Thanks for the tips guys,

Looking back in my transcript, I took Trig in Fall 2005, Pre-calc Spring 2006, Calc 1 in Fall 2006. I also took History of Mathematics, Physics I and stats not too long ago. All of those classes I got all high B’s

I rather not take Calc all over again since I’ve been in college since 2005 and I kinda wanna finish. Shit, I don’t even have my major declared yet. Oh well, lets hope I have a strong will to finish :china::

You guys ever seen the movie Pi ? Remember the part when the guy was talking about fibonacci sequence ? That shit just excited me. How nerdy is that :looney:

Great thread btw :tup:

EDIT: here’s the link to the scene i was talking about

[media=youtube]FcN2i9GKsFQ[/media]

[Warrior_s_Dreams]

Thanks for the tips guys,

Looking back in my transcript, I took Trig in Fall 2005, Pre-calc Spring 2006, Calc 1 in Fall 2006. I also took History of Mathematics, Physics I and stats not too long ago. All of those classes I got all high B’s

I rather not take Calc all over again since I’ve been in college since 2005 and I kinda wanna finish. Shit, I don’t even have my major declared yet. Oh well, lets hope I have a strong will to finish :china::

You guys ever seen the movie Pi ? Remember the part when the guy was talking about fibonacci sequence ? That shit just excited me. How nerdy is that :looney:

Great thread btw :tup:

EDIT: here’s the link to the scene i was talking about

[media=youtube]FcN2i9GKsFQ[/media]

I don’t know who your advisor is, but if I were you, I would ditch them like I would the plague. because they have failed you.

Assuming you had enough courses to get your electives out of the way, you’ve still got at least 2 years to go, and thats if you take more than 2 classes a semester, and mind you, 2 math courses are plenty.

Here, take this diagnostic test (without using a calculator)

http://www.math.ncku.edu.tw/~mhchen/fall09/diagnostic.pdf

If you had problems with any of those, you are in serious remedial need.

To be quite blunt, you have some tough times ahead of you. You took precalc in 2006. If you can’t rattle off sin and cos graphs off the top of your head, you should seriously question why you want to pursue a math-related major. I don’t know if you’ve taken a physics class that is for physics majors (as opposed for non-majors), but if you don’t know basic concepts like vectors, vector equations, curve integrals, derivatives, trigonometry, etc… you are already peaced out for free. Hopefully your CS skills are better. Otherwise, you are going to be playing catchup for at least a year ON TOP of the courses you are taking.

[MaxVandalism315]

sudoku!

[Warrior_s_Dreams]

If you like sudoku… and you like abstract algebra… then you might like this

Mathematics of Sudoku

[MaxVandalism315]

^ head explodes

[fishjie]

oh my god fractals are amazing. thanks to this Warrior’s Dreams posts, i’ve gone off and made my own mandlebrot set generator. easy, because everyone and their mother has coded one so there were plenty of helpful examples. then i went off and did a julia set too. the wikipedia article on julia set made no sense to me, but luckily there were enough web sites that dumbed it down significantly that now its starting to make sense, at least from a layman’s perspective.

So with the mandlebrot set, you have z(n) = z(n-1)^2 + c. You set z(0), the orbit, fixed at 0, and then make c variable, where c is a complex number. on the y axis you have your imaginary component, and on the x is your real. For any point c that remains bounded, you color it black. For any point that doesn’t, you color it white. Technically, fractals are black and white, but you get the psychedelic colors by coloring based on how many iterations it takes to escape to infinity.

For the julia set, you have z(n) = z(n-1)^2 + c. This time, you fix C at whatever value, and then make the orbit, z(0) variable. There are an infinite number of julia sets, since the set of all complex numbers between -2 and 2 is uncountably infinite (countably? i forget the difference). And since there’s an infinite number of complex numbers, you can zoom in on these sets an infinite number of times, and there will always be new detail. you will see self similar shapes that are slightly similar yet different no matter how far you zoom. INFINITE COMPLEXITY!!!11111

And here’s the crazy kicker. The mandlebrot set is the set of all c values for which the corresponding julia set is connected. Connected means that the “black” points are clumped together. Whereas the disconnected julia sets means that all the black points are isolated, and these are when you get the crazy fractal dust images. I’ve been getting high and looking at crazy ass spirals and shit.

Its also interesting to note that these systems were discovered by Julia and Fatou, but back then computers didn’t even exist. They were trying to graph the points by hand, lol. Mandlebrot was the one who decided to graph them using a computer. I wonder if Julia and Fatou realized how insane the pictures would be?

Anyway, its really cool that generating these things doesn’t require any math beyond adding, subtracting, and multiplying complex numbers. I know the actual math behind complex dynamics is insane.

I just had a two questions.

1. I asked earlier in this thread how to determine if something is in the set. From what I’ve read, and how I’ve coded it, when the real or imaginary component of Z(n), where Z(n) = Z(n-1)^2 + C, ever exceeds the value 2, then its escaped to infinity and not in the set. Why? Is there some simple proof I’m missing? Essentially, its saying that if the Z(n) is ever outside the unit circle corresponding to radius 2, it escapes.

2. Whats the proof that the mandlebrot set is the set of all C values corresponding to connected julia sets? Is it like 30 pages long or something?

edit:

so naturally i’ve been obsessing over fractals even though its completely irrelevant to my job, but its so much fun. here are some cool vids.

The Colors of Infinity documentary hosted by Arthur C fucking Clark, with guests such as mandlebrot himself, and stephen hawking!!!11
[media=youtube]qB8m85p7GsU"[/media]

animating the julia set based on various c values scribbled on a mandlebrot set:
[media=youtube]xmzwe7q-1K0"[/media]

mandlebrot zoom!!!
[media=youtube]gEw8xpb1aRA"[/media]

[HoneyBBQGrundle]

1. I asked earlier in this thread how to determine if something is in the set. From what I’ve read, and how I’ve coded it, when the real or imaginary component of Z(n), where Z(n) = Z(n-1)^2 + C, ever exceeds the value 2, then its escaped to infinity and not in the set. Why? Is there some simple proof I’m missing? Essentially, its saying that if the Z(n) is ever outside the unit circle corresponding to radius 2, it escapes.

2. Whats the proof that the mandlebrot set is the set of all C values corresponding to connected julia sets? Is it like 30 pages long or something?

1. I don’t know, but geometrically it probably has something to do with how squaring complex numbers works. You square the absolute value and then double the angle it makes with the positive x axis. For example, the number 8 units away from the origin that forms an angle of 60 degrees would turn into the number 64 units away with angle 120 degrees.

Now adding c makes things more complicated. Presumably this could bring the number back into a small area near the origin, but intuitively it seem like it would be hard to keep doing this if c is fixed. The sequence z_n would keep changing angles and adding c only slides the point along a fixed vector. Whether it’s impossible or not isn’t obvious to me though.

2. I have no idea. I know less than you about this stuff and I probably shouldn’t have even answered your first question without thinking it over more, but it may be helpful at least as a plausibility argument.

[Warrior_s_Dreams]

oh my god fractals are amazing. thanks to this Warrior’s Dreams posts, i’ve gone off and made my own mandlebrot set generator.
I just had a two questions.

1. I asked earlier in this thread how to determine if something is in the set. From what I’ve read, and how I’ve coded it, when the real or imaginary component of Z(n), where Z(n) = Z(n-1)^2 + C, ever exceeds the value 2, then its escaped to infinity and not in the set. Why? Is there some simple proof I’m missing? Essentially, its saying that if the Z(n) is ever outside the unit circle corresponding to radius 2, it escapes.

2. Whats the proof that the mandlebrot set is the set of all C values corresponding to connected julia sets? Is it like 30 pages long or something?

I’m happy that it worked for you. I’ll reply more in detail in a couple of days (since I’m prepping for finals as we speak), but to answer your first question, yes, there is a proof of it, which I can demonstrate later. The second one seems like an interesting side project. I’ll look into it later.