http://mathoverflow.net/questions/10147/when-does-local-invertibility-imply-invertibility

Generally, local invertibility does not imply invertibility. However, for differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ then surjectivity and local invertibility do imply invertibility.

As well as being the most obvious, it's also the only (non-contrived) case that I can think of. Are there any more?

Specifically, I'm looking for examples of spaces $X$ which are at least topological spaces (but may be more structured) and subsets of endomorphisms on $X$ for which local invertibility implies invertibility.

http://mathoverflow.net/questions/10146/good-books-on-problem-solving-math-olympiad

Hello, I would want all book tips you could think of regarding Problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example would be the books from The art of problem solving, Engels book and Paul Zeits book. Books on certain topics, say geometry is also appreciated!

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From the shelves, I swept into the wastebasket or, onto the floor in its vicinity, heaps of circulars, separata, a displaced ecologist's report on the ravages committed by a bird of some sort, the Ozimaya Sovka ("Lesser Winter-Crop Owl"?), and the tidily bound page proofs (mine always came in the guise of long, horribly slippery and unwieldy snakes) of picaresque trash, full of cricks and punts, imposed on me by proud publishers hoping for a rave from the lucky bastard.

С полок я смел в корзину для бумаг и на пол вблизи нее кипы циркуляров, разрозненные оттиски, статью перемещенного эколога касательно опустошений, учиненных какой-то птичкой по имени “озимая совка” (“Lesser Winter-Croр Owl”?), и опрятно переплетенные гранки (мои всегда приходили в обличии длинных, жутко скользких и неподатливых змей) полной натужных каламбуров авантюрной дребедени, присылаемой мне гордыми издателями в надежде на восторженный отзыв везучего сукина сына.

http://grani.ru/Society/Science/m.172836.html

http://mathoverflow.net/questions/10126/need-a-reference-for-this

Let $G$ be a finite group and $\chi$ be an irreducible character of $G$ (characteristic zero algebraically closed base field). I f$H$ is the kernel of $\chi$ then the irreducible representations of $G/H$ are exactly all the irreducible constituents of all tensor powers $\chi^n$.

1. Do you know any reference for this theorem?

2. Is it also working in positive characteristic?

3. Is it also working for some infinite groups? (maybe some special classes: reductive, Lie type, etc)

Thank you very much!

http://mathoverflow.net/questions/10128/when-is-a-isomorphic-to-a3

this is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a counterexample? you can also ask this in other categories as well, for example rings. if you restrict to boolean rings, the question becomes a topological one which makes you think about fractals: let $X$ be stone space such that $X \cong X + X + X$, does it follow that $X \cong X + X$ (here + means disjoint union)?

http://mathoverflow.net/questions/10131/subspace-topology-for-functors

let $X : Ring \to Set$ be a functor (a Z-functor in the language of demazure, gabriel) and $V \subseteq X$ a subfunctor. assume that $U \subseteq V$ is an open subfunctor. does then exist an open subfunctor $W \subseteq X$ such that $U = V \cap W$? if $X$ and $V$ are schemes, this should be true.

http://mathoverflow.net/questions/10127/sum-of-radical-ideals

let $A$ be a ring and endow the closed subsets of $Spec A$ with the grothendieck topology of finite covers. it would be nice if $V \mapsto A/I(V)$ defines a sheaf. this is not true in general and is related (equivalent?) to the following pure algebraic question:

in which rings $A$ are the radical ideals closed under sum?

if $A$ is a noetherian integral domain of dimension $0$ or $1$, it's true. you cannot omit integral here (consider $k[x,y]/(x^2 y + y^2)$ localized at $(x,y)$), nor the dimension (in $k[x,y]$, consider $(x^2 + y)+(y) = (x^2,y)$). are there other interesting examples/counterexamples or approaches for general classification?