3
votes
2answers
773 views
Variant of Fermat’s last theorem
By Fermat's last theorem, the equation $u^3+v^3=w^3$ has no solutions
in positive integers $u,v,w$. Now consider the following variant : call $\rho(x)$
the distance between $x$ and …
7
votes
1answer
388 views
Hopf algebras as cohomology of $\mathbb{CP}^\infty$, $\Omega S^3$ and related $H$-spaces
Let me begin by a couple of questions :
Consider a graded abelian group $V=\oplus_{i\geq 0} V_i$ such that $V_{2i}=\mathbb{Z}$ and $V_{\textrm{odd}}=0$. What are the possible …
9
votes
1answer
409 views
Waring’s problem for matrices
Probably a well-know question, but I haven't solved it, so I'll ask.
I can show that every matrix in $M_2(\mathbb{R})$ is the sum of two squares of matrices in $M_2(\mathbb{R})$.
…
11
votes
2answers
674 views
Estimate rate of real correct/wrong from 4 answers quiz.
I recently read that one in ten students think the first man on the moon was Buzz Lightyear, a Toy story cartoon. I'm not here to discuss the data in itself, rather, this reading g …
1
vote
0answers
333 views
Does the dual Banach space $B(\ell^\infty)$ has weak* normal structure?
Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if
$$
\sup_{y\in K} ||x-y||=diam(K).
$$
where $diam(K)$ denotes the …
7
votes
2answers
647 views
number fields generated by units of number fields
Which number fields are generated by the units of some number field? That is, if $K$ is a number field and $U(K)$ its group of units, the field $k = \mathbb{Q}(U(K))$ is a subfield …
5
votes
1answer
539 views
Serre spectral sequence with spectra
A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason …
0
votes
3answers
575 views
Explicitly constructing an infinite set with particular size
I would like to preface by saying that I have no significant experience working with set theory, so I'm probably making an intuitive mistake. I have figured out where the mistake …
3
votes
2answers
170 views
Finite categories and partial orders
I'm studying category theory for the first time in a very succint book for computer scientists (I'm not actually a computer scientist, I'm a physicist, but my interest in cat theor …
1
vote
2answers
585 views
a small questions about hopf theorem
The famous hopf theorem says that a smooth map from a oriented closed dimension p manifold to S^{p} is homotopic if and only if f and g have the same brower degree. To prove the th …
0
votes
1answer
361 views
How to restore the original formula from a binomial-like expansion?
I encountered with a recursive formula of the following kind:
$$A(0,x)=1$$
$$A(n,x)= \sum _{j=0}^{n-1} \binom{n-1}{j} A(n-j-1,x) \sum _{k=0}^{x-1} A(j,k)$$
The sum terms can be …
1
vote
1answer
487 views
A question in R.C.Penner’s paper about Teichmuller space
In R.C.Penner "Decorated Teichmuller theory of boarded surface", on Page 7 and 8, it says that (without proof) the Teichmuller space of surface with $s$ labelled punctures and $r$ …
7
votes
1answer
260 views
Ergodic splitting in L_p
I have a curiosity on the Ergodic decomposition given by the von Neumann's theorem:
$$L^2(X,\Sigma,\mu)=L^2(X,\Sigma_T,\mu)\oplus\overline{\{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu)\ …
5
votes
1answer
249 views
Multiple factors of the character of a representation
In algebra, various objects admit a unique decomposition into irreducible elements. For instance integers $n\ge1$, univariate polynomials $p\in k[X]$ (even multivariate ones), or c …
6
votes
1answer
427 views
Invariant quadratic forms of irreducible representations
Let $G$ be a finite group, and $k$ be a field of characteristic zero (not necessarily algebraically closed!). Let $\rho : G \to \mathrm{End}_k \left(k^n\right)$ be a irreducible re …
