Linked Questions

71
votes
30answers
11k views

Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
48
votes
29answers
30k views

Why do we teach calculus students the derivative as a limit?

I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students? Something a teacher ...
35
votes
8answers
7k views

Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$. Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ ...
42
votes
5answers
3k views

When does Cantor-Bernstein hold?

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological ...
38
votes
2answers
3k views

Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
20
votes
3answers
4k views

When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy: Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable? In reality, I am interested in ...
7
votes
5answers
2k views

Projection of Borel set from $R^2$ to $R^1$

Hello This should be easy to prove but i have no idea how to do it: If $X \subseteq \mathbb{R}^2$ is borel then $f(X)$ is borel where $f(x,y) = x$ Thanks Tobias
14
votes
6answers
1k views

Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts: $$ \int_a^b f(x)g'(x)ds = ...
27
votes
5answers
2k views

Cocomplete but not complete abelian category

This is a duplicate of the following question to which I did not receive any answer: http://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category Let $\mathfrak C$ be an ...
22
votes
3answers
3k views

Is “compact implies sequentially compact” consistent with ZF?

Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
8
votes
2answers
1k views

When is sin(r \pi) expressible in radicals for r rational?

Perhaps this question will not be considered appropriate for MO - so be it. But hear me out before you dismiss it as completely elementary. As the question suggests, I would like to know when ...
10
votes
2answers
2k views

most general way to generate pairwise independent random variables?

Is there a sort of structure theorem for pairwise independent random variables or a very general way to create them? I'm wondering because I find it difficult to come up with a lot of examples of ...
7
votes
1answer
1k views

Set Theory and V=L

From http://en.wikipedia.org/wiki/Analytical_hierarchy "If the axiom of constructibility holds then there is a subset of the product of the Baire space with itself which is $\Delta^1_2$ and is the ...
5
votes
3answers
287 views

Non-continuous higher differentiability

The standard definition is that a function $f:\mathbb{R}^n\to \mathbb{R}$ is differentiable at a point $x$ if there exists a linear map $\mathrm{d}f_x: \mathbb{R}^n \to \mathbb{R}$ such that $$f(x+h) ...
5
votes
1answer
2k views

Tripartite Graph

I would like to know if there exists a version of the K├Ânig's theorem for tripartite graphs. In other words, let G = (V,T) be a tripartite graph, with V set of vertices ($V$ union of three disjoint ...

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