Over decades, and across multiple research fields, I've noticed a way to predict I'm on track to make progress. I discover something interesting, only to learn it is already known. As a student, this ...

Yes, the complement of any countable set in $\mathbb{R}^3$ is simply connected, by the Baire category theorem. Say your set is $X = \{x_1, x_2, ... \}$, and let $y$ be any point in $\mathbb{R}^3 \... View answer 60 votes For a picture that launched a thousand papers, I'd nominate the bifurcation diagram of the logistic map. (image via Wikipedia). Answered by Martin M. W. View answer 26 votes Update The Coursera course I recommended long ago has now gone offline, although you can find links to the slides and videos on Hinton's home page. In any case, the field has continued to advance ... View answer Accepted answer 26 votes There's a set of good examples from percolation theory: http://en.wikipedia.org/wiki/Percolation_theory If you create a "random network" with a certain probability p of edges between nodes (see ... View answer 24 votes This picture--for a (7,2) torus knot--shows a geometric pattern you can extend to any (n,2) torus knot. The image is part of Figure 18 in the visually rich paper by Jarke van Wijk and Arjeh Cohen: "... View answer Accepted answer 23 votes This is the perspective of someone who went from a math PhD to the media industry and then to software/tech (and enjoyed it immensely). I can't speak for finance. Don't spend time on classes; instead,... View answer 21 votes You only need one continuous function. There exists a continuous function$f: \mathbb{R} \to \mathbb{R}$with a dense orbit, according to this MathOverflow answer. As in Saúl's construction, you can ... View answer Accepted answer 21 votes Similar issues come up in studying gerrymandering (drawing political districts with partisan objectives), where it's useful to have a measure of how "irregular" a region is. You can read about ... View answer Accepted answer 17 votes One example is in the concept of a Nash equilibrium, whose existence can be proved using various (topological) fixed point theorems. (Google "nash equilibrium proof" for a wide variety of examples... ... View answer Accepted answer 15 votes I think under any reasonable definition there will be only countably many explicit formulas or patterns. So in fact most reals can't be expressed this way. (See also the concept of "computable number."... View answer Accepted answer 15 votes This is a good question. For some reason, terminology in dynamical systems is not standardized at all--and it's interesting to disentangle various definitions. A good book to look at is Differential ... View answer 11 votes You can visualize a solenoid as part of a simple 3D dynamical system. The Wikipedia page on solenoids has excellent illustrations. If you want something even more concrete, and have some "Silly Putty" ... View answer Accepted answer 11 votes Your error bars may be giving you a hint to look more closely at the distribution of your data: it may not be symmetric. For example, if your data is essentially log-normal you could work with the ... View answer 11 votes No, there doesn't exist such a foliation. The existence of any foliation would mean the Euler characteristic is zero, so the surface must be either a torus or a Klein bottle. Foliations for these ... View answer Accepted answer 11 votes To answer your first question, the composition of two time-1 flows won't necessarily be another time-1 flow. One way to see this is to note that when a time-1 flow$\phi_X$has a periodic point$P$(... View answer 10 votes Your intuition is on target, since you mention the Banach fixed point theorem together with the Hartman-Grobman theorem. The proofs I've seen of Hartman-Grobman find the topological conjugacy as the ... View answer Accepted answer 9 votes The reason that the matrices are alike is that, up to a multiplicative factor, each is approximately encoding$Area(T(R_i) \cap R_j))$. This is because the rational points with denominator$q$are ... View answer 9 votes For the specific case you mention of an irrational rotation of the circle, it depends on the rotation number$r$. You can get slow asymptotic convergence for something like $$r=\sum{10^{-n!}}$$ (and ... View answer Accepted answer 7 votes I think the answer is, yes, the graph can be connected. By definition, if the graph G is not connected, then we can find disjoint nonempty open sets A and B, such that G is contained in A union B. In ... View answer 6 votes This is an observation, along with a sketch of a potential construction where (1) does not imply (2) for$\mathbb{R}^3$. Observation. I want to point out an example of a closed set$X \subset \mathbb{...

In all dimensions n>2, you get a non-regular polytope with this property if you take two simplices and glue two faces together. Pairs of vertices will be 1 unit apart, topologically and geometrically, ...

By now you may have more ideas than you need, but heck, here's another non-matrix viewpoint. In the disk model, you can identify your transformations with lists of three distinct points $(a,b,c)$ on ...

I think this question might be better for one of the computer science StackExchange sites. In any case, the literal answer to your question is No. The trouble is that in a conventional force-directed ...

As the questioner notes in a comment, the answer is Yes for n<3. One way to create counterexamples for larger n is to use the work on the Seifert Conjecture. Start with a vector field pointing ...

You need more information to do anything useful. (An upper bound on the global minimum isn't very special--you can sample your function at any point to get one.) Without additional restrictions on ...

Unless I'm misinterpreting the question, the SRB measure is just Lebesgue measure... the cat map is hyperbolic, preserves area, and is topologically transitive. See Theorem 3.10 and the following ...

You can't always approximate by ergodic flows, because ergodic flows might not even exist. For example, on $S^2$ the Poincare-Bendixson theorem rules out ergodic flows, but there are many measure-...

Let $\mu$ be Lebesgue measure on $S^1$, and $\delta_P$ be a point-mass at a point $P \in S^1$. Then there is no flow on $S^1$ whose time averages lead to $\frac{1}{2}(\mu + \delta_P)$. (Consider the ...
(edited to include Willie Wong's idea for $C^0$ case.) This kind of flow can't exist in any dimension. Let $S$ be the unit sphere and $B$ be the open unit ball. If the origin is a global attractor for ...