The same goes for me.

EE with minor in math. May be able to help out

Well, since the OP is gone: What sort of math are you two into?

I guess i would consider myself somewhat of a mathemetician. I am definitely younger, but fairly proficient for my age. (I have a decent understanding of anything short of PDE's)

I'm studying maths at the moment, but am unfamiliar with continuum theory.

@abgemacht: I'm a master student, so that's fifth year. I did a bachelor thesis in probability theory (stochastic processes mainly), but my interest goes out to topology and differential geometry. How about you?

Besides, do you guys know what is meant by "continuum theory"? I would guess he referws to calculus, but I'm not sure. Wikipedia directs "continuum theory" to topology.

You probably mean the 'Continuum hypothesis'. There is the denumerable set of natural numbers. Then there is the non-denumerable set or real numbers. There is no bijective functions from reals to natural numbers. The question, is there some set X, that is smaller than the set of reals, but bigger than the natural numbers? The Continuum hypothesis suspects that the answer is no. However, this is more related to logic than mathematics. The CH has been proven independent from ZFC logic. In that sense, that saying that it is true or false, both don't lead to an inconsistency.

@bas

I had to take a course on stochastic processes last semester for my masters and it almost killed me. I know it's really important stuff, though, so more power to you.

My research is in modeling of nanoelectroncis. So, it's some DiffEq and a lot of linear algebra and numerical analysis. Although, there will always be a special place in my heart for that abstract algebra course I took as an undergrad.


Yeah, "continuum theory" isn't clear to me, either. Wikipedia links it to a bunch of different things. If he ever comes back, it would be nice to have a clarification.

yes, by "continuum theory" i meant the study of continua (conected compact metric spaces), and don't know much more, though it seems interesting.

this post is just cos I was wondering if I could find some mathematicians around here, they are rare, and I like when I find someone (I mean, not in the school, I'm studying maths myself)

@abgemacht: Stochastic processes can also be pretty cool. For example, I read on wikipedia today that the graph of a Brownian motion (sort of a mathematician's version of diffusion) is almost surely 1.5-dimensional. How cool is that?

If you liked the abstract algebra course, then I'd definitely recommend you to try some more "abstract math" classes. Not because it's important or useful, but just becuase it's fun. Besides, it really pays to have a mathematical midset when doing physics or engineering.

@principians: What makes you so interested in continuum theory? is it something you have worked on before, or something you would like to learn?

@bas

I don't disagree that it isn't cool, I'm just saying the course almost killed me. Never had so much trouble in math my whole life. Which is a shame, because statistical mechanics can be very important in my field.

Yeah, if I ever have time, I'll try to take another abstract course, but as a masters student yourself, I'm sure you know how hard it is to take courses that don't relate at all to your work.

I'm finishing up my PhD studies in Electrical and Computer Engineering, specifically focusing on the field of information theory, which is the deep mathematical study of fundamental performance limits in communication systems.
I have taken the standard applied math courses in ECE that cover signal processing, stochastic processes, classification/estimation theory, communications theory, etc.
I have also taken graduate level math courses, real analysis and probability theory, which have covered measure theory.

@principians:
It seems that are you talking about the following:
http://en.wikipedia.org/wiki/Continuum_(topology)
http://en.wikipedia.org/wiki/Metric_space
What specifically do you want to learn about the following topics?

They are simply formal mathematical definitions part of a larger body of theory.
A metric space is basically a set of mathematical objects (e.g., a set points on the 2 real plane) and meaningful distance metric associated with those objects (e.g., euclidean distance). If you have a set of objects and you can find a meaningful (i.e., having the properties of non-negativity, symmetry, zero for identical objects, and triangle inequality) distance metric associated with those objects, then you have yourself a metric space. The definition is quite general. Any subset of points from the 2 plane plus the euclidean distance (L-2 distance) is a well-defined metric space. Also, the euclidean distance could be replaced with a wide array of other distances metrics, e.g., taxi-cab distance (L-1 distance), etc., and you would still have a metric space.
Imprecisely, people usually refer a function as a being a metric or not, however, technically, one must consider the function with respect to a set upon which it is measuring. That is why there is this formal definition of metric space which talks about both a set and a function.

Connectedness and compactness are properties that a metric space might or might not have. In general, these properties are somewhat deep and hard to explain without getting too technical, but for simple examples of metric spaces where the set in question is a subset of the 2 real plane and the metric is the euclidean distance, connectedness and compactness are fairly intuitive notions. Loosely speaking, such a space is connected if the entire set looks like "one continuous blob" of points. Compactness means that the set must be bounded and contains all of its limit points.

A continuum, as you said, is simply a compact and connected metric space.

@yebellz

Have you done anything with quantum information theory? I have some people in my group doing that. It's crazy shit.

Connectedness isn't hard to explain. A space is disconnected if there is a proper subset which does not have a boundary. It is the definition of the boundary that's sort of hard to grasp. However, in a metric space, that's also easy:

\partial U:=\{p\in X:\inf_{u\inU}d(p,u)=0\}.

The above is Latex code. I suppose you are familiar with it?

@abge: no i have not done anything with quantum info theory, which is quite a beast of its own as it is the generalization of classical info theory to be applied to quantum systems, where the physical representation of information has fundamentally different properties.

@bas: Of course, whether something is "diffcult" or "easy" to explain or whether a concept is "deep" or "shallow" is quite subjective. I say that the concepts of connectedness and compactness are somewhat "deep" (in the context of the casual environment of this web forum) and are certainly difficult to explain without getting into the technical details of what are points of closure, etc.

Here is the definition of connectedness that you give (reworded and corrected to note the null set exception):
Every subset (besides the set itself and the null set) has a non-empty boundary.

Since a set has a non-empty boundary if and only if the set is both open and closed, understanding this definition basically comes back to understanding the "deeper" concepts of open and closed sets.

As you said, the concept of boundary is "sort of hard to grasp" as well. Perhaps, you best illustrated this difficulty by giving an incorrect definition for the boundary.

What you gave is the definition of the closure of the set U. The boundary is the closure U with the interior of U removed. Without having to define what interior means, we can alternatively define boundary as a the intersection of the closure of U with the closure of the complement of U.

closure
\bar{U} := \{p\in X:\inf_{u\inU}d(p,u)=0\}

boundary
\partial U := \bar{U} \cap \overline{(X \ U)}

I'm like yebellz and abgemacht. A computer engineer degree with a math minor (it has been over a decade since I got it, though). My engineering degree is focused on software and AI specifically, and math focus was mostly linear algebra with a touch of graph theory thrown in.
I'd have to look up and study the theory you are looking at, but I can be someone to bounce thoughts off of as well...

I think what I'm saying is that there are enough math specialists here that you can ask questions and generation discussion, principians.

@yebellz: You're right (I should have seen this, shame on me).

I don't think you need to understand the deeper concepts, even if they are there. For example, you can teach a first year student what continuity is without ever referring to deeper concepts. I bet many people on this site know the epsilon-delta definition of continuity. But most people don't know the equivalent statement "inverse image of open is open".

My point is that in the case of metric spaces, you don't need to know what topology is. You can define the boundary intuitively as "all points that are arbitrary close to U and to its complement" (close of course referring to the metric). Then you say a space is disconnected if there is a proper (not X and not the empty set) subset which has no (meaning empty) boundary. The only thing that's a bit strange here is the infimum.

Compactness is a different story. Even the version you quote (totally bounded+complete) is rather unwieldy to write down formally, let alone work with it. I myself like the characterisation "every infinite subset has a limit point", but then you'd have to also explain what a limit point is.

Besides, you may have noted I'm not very technical and precise in my mathematical statements. I try to do intuitive math, but not everyone likes that. The guys grading my exercises sure don't :p