Well, I'm not exactly a people person, but I enjoy teaching:) Haven't done any "serious" teaching yet though, I mean explaining things to other math students.
Actually I think this is the Euclidean Steiner tree problem, since he doesn't have to install a fiber between every pair of cities - he can add hubs in the middle. This is NP-hard.
(Don't know if this is sufficient for what they do in the article though, didn't RTFA)
Thanks for your very informative comment. I seriously wasn't aware of the fact that things used to be that bad. Really makes me feel lucky to live in the 21st century.
I agree that these points make the ideas of the article (as summarised; didn't RTFA;)) completely ridiculous.
I dislike fads as much as you do, but I don't think trans fats are one. Wikipedia confirms it:
A comprehensive review of studies of trans fats was published in 2006 in the New England Journal of Medicine that concludes that there is a strong and reliable connection between trans fat consumption and CHD.
(Answering to myself) just a tiny clarification: Before anyone gets the impression that what I described as "respecting convergence" is a technical term (because I used it twice), the actual name of this property is "f is continuous and has a continuous inverse".
We don't actually need 2-dimensional euclidean space to describe the topological structure of the circle.
There are several different concepts of dimension in mathematics. The one you are probably thinking of is the dimension of a vector space. What we seem to need here is the dimension of a manifold. Intuitively, a n-dimensional manifold is something that locally "looks like" our familiar n-dimensional euclidean space (R^n). You already got that right with the ant example.
Manifolds can be described in different ways. One way is as a certain kind of subset of some higher-dimensional vector space R^m, this is the way you are probably imagining. But it is also possible to describe a manifold without any reference to a surrounding space.
For this we need the concept of a topological space. Informally, a topological space is a set in which we can talk about connectedness, continuity and which sets of points are "a neighborhood" of a given point.
As a topological space, the circle can be seen as the usual interval [0,1] (of real numbers), but with the points 0 and 1 identified (that is, they are considered to be the same point) (usually one would use the analogy "0 and 1 glued together", but this would evoke the intuition of a surrounding space again, which we are trying to avoid:)). For example, the sequence (1/n) converges to 1 (=0), and the path
f(t):= t if 0 <= t < 1, f(t):= t-1 if 1 <= t < 2
is actually continuous in this space (it isn't continuous in the usual topology of [0,1], because f(t) "jumps" from being close to 1 to being zero again, as t approaches 1).
Likewise, topologically a sphere is equivalent to a square (or a disk) with the whole boundary[1] considered to be a single point. A torus is a square with every point on the left edge identified with the corresponding point on the right edge, and every point on the top edge identified with the corresponding point on the bottom edge.
Generally, a n-dimensional topological manifold is defined as a topological space with the following property (+ some technical conditions): For every point on the manifold, you can find a small region U around the point (a "neighborhood"), such that U is topologically the same ("homeomorphic") as a disk/ball or a box[2] in n-dimensional euclidean space. A homeomorphism is essentially a map f which puts the points of one space into one-to-one-correspondence with the points of another space, and respects convergence in the sense that some sequence[3] x_n converges to x if and only if f(x_n) converges to f(x). It can't tear regions apart which are connected, or vice versa.
For example, if we have some point of the sphere, we can take a small neighborhood U of it and map U to a disk in the obvious way. This mapping respects convergence. Thus, the sphere is a 2-dimensional topological manifold.
Now I only described the topological structure; topology is "qualitative" and doesn't talk about concrete distances, angles etc.. If you want to have these, you need a structure called a Riemannian manifold. But I haven't taken a course on differential geometry yet, so I won't talk about that;) But these manifolds can also be constructed without referring to a surrounding space.
I hope I didn't tell you things you already know and that I didn't sound condescending. You are asking good questions and I think you would like topology courses:)
Whether the surrounding spaces are "real" is a matter of philosophy, but as you can see they are not absolutely necessary...
[1]: For the topologists: I'm using "boundary" in the informal sense here; of course the boundary (in the formal sense) of the whole space is always empty. [2]: Actually it doesn't matter whether you require it to be homeomorphic to a ball in R^n or to the whole R^n. [3]: In general it's a net, not a sequence
OCD doesn't mean what you apparently think it means. It is not at all about being unable to resist pleasurable activities.
In a nutshell, it is about intrusive thoughts that something horrible will happen (obsessions) and trying to "counter" it somehow (compulsions). For example touching some object you consider dirty, being afraid that you will get some illness and then washing your hands. Or having a feeling that you have done something "incorrectly" for some reason, and repeating it. etc...
Slashdot Mirror
Well, I'm not exactly a people person, but I enjoy teaching :) Haven't done any "serious" teaching yet though, I mean explaining things to other math students.
Probably not hard-wired, but it is amazing how much intuition about completely abstract concepts you develop if you work with them for years.
Actually I think this is the Euclidean Steiner tree problem, since he doesn't have to install a fiber between every pair of cities - he can add hubs in the middle. This is NP-hard.
(Don't know if this is sufficient for what they do in the article though, didn't RTFA)
Thanks for your very informative comment. I seriously wasn't aware of the fact that things used to be that bad. Really makes me feel lucky to live in the 21st century.
;)) completely ridiculous.
I agree that these points make the ideas of the article (as summarised; didn't RTFA
(Answering to myself) just a tiny clarification: Before anyone gets the impression that what I described as "respecting convergence" is a technical term (because I used it twice), the actual name of this property is "f is continuous and has a continuous inverse".
*embarassed* I hope I'm not being too much of a smartass. I'm just really bored...
We don't actually need 2-dimensional euclidean space to describe the topological structure of the circle.
There are several different concepts of dimension in mathematics. The one you are probably thinking of is the dimension of a vector space. What we seem to need here is the dimension of a manifold. Intuitively, a n-dimensional manifold is something that locally "looks like" our familiar n-dimensional euclidean space (R^n). You already got that right with the ant example.
Manifolds can be described in different ways. One way is as a certain kind of subset of some higher-dimensional vector space R^m, this is the way you are probably imagining. But it is also possible to describe a manifold without any reference to a surrounding space.
For this we need the concept of a topological space. Informally, a topological space is a set in which we can talk about connectedness, continuity and which sets of points are "a neighborhood" of a given point.
As a topological space, the circle can be seen as the usual interval [0,1] (of real numbers), but with the points 0 and 1 identified (that is, they are considered to be the same point) (usually one would use the analogy "0 and 1 glued together", but this would evoke the intuition of a surrounding space again, which we are trying to avoid
Likewise, topologically a sphere is equivalent to a square (or a disk) with the whole boundary[1] considered to be a single point. A torus is a square with every point on the left edge identified with the corresponding point on the right edge, and every point on the top edge identified with the corresponding point on the bottom edge.
Generally, a n-dimensional topological manifold is defined as a topological space with the following property (+ some technical conditions):
For every point on the manifold, you can find a small region U around the point (a "neighborhood"), such that U is topologically the same ("homeomorphic") as a disk/ball or a box[2] in n-dimensional euclidean space. A homeomorphism is essentially a map f which puts the points of one space into one-to-one-correspondence with the points of another space, and respects convergence in the sense that some sequence[3] x_n converges to x if and only if f(x_n) converges to f(x). It can't tear regions apart which are connected, or vice versa.
For example, if we have some point of the sphere, we can take a small neighborhood U of it and map U to a disk in the obvious way. This mapping respects convergence. Thus, the sphere is a 2-dimensional topological manifold.
Now I only described the topological structure; topology is "qualitative" and doesn't talk about concrete distances, angles etc.. If you want to have these, you need a structure called a Riemannian manifold. But I haven't taken a course on differential geometry yet, so I won't talk about that
I hope I didn't tell you things you already know and that I didn't sound condescending. You are asking good questions and I think you would like topology courses
Whether the surrounding spaces are "real" is a matter of philosophy, but as you can see they are not absolutely necessary...
[1]: For the topologists: I'm using "boundary" in the informal sense here; of course the boundary (in the formal sense) of the whole space is always empty.
[2]: Actually it doesn't matter whether you require it to be homeomorphic to a ball in R^n or to the whole R^n.
[3]: In general it's a net, not a sequence
No offense, but this seems to come up in *every* discussion about gaming addiction on /.
OCD doesn't mean what you apparently think it means. It is not at all about being unable to resist pleasurable activities.
In a nutshell, it is about intrusive thoughts that something horrible will happen (obsessions) and trying to "counter" it somehow (compulsions). For example touching some object you consider dirty, being afraid that you will get some illness and then washing your hands. Or having a feeling that you have done something "incorrectly" for some reason, and repeating it. etc...
Wikipedia Article
(By the way, it wasn't particularly bad in your posting, I just used this opportunity to finally comment on this ;))