# All Questions

**18**

votes

**12**answers

7k views

### How seriously should a graduate student take teaching evaluations?

Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...

**20**

votes

**2**answers

986 views

### State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$

As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
...

**5**

votes

**3**answers

2k views

### Can you efficiently solve a system of quadratic multivariate polynomials?

Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find a common zero of all of these polynomials? In other words, ...

**19**

votes

**5**answers

1k views

### Can one recover a metric from geodesics?

Assume there are two Riemannian metrics on a manifold ( open or closed) with the same set of all geodesics. Are they proportional by a constant? If not in general, what are the affirmative results in ...

**27**

votes

**14**answers

27k views

### Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...

**29**

votes

**2**answers

4k views

### Does the curvature determine the metric?

Hello,
I ask myself, whether the curvature determines the metric.
Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...

**22**

votes

**4**answers

2k views

### When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals)

Sorry if this is a silly question. I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these ...

**22**

votes

**4**answers

3k views

### What is meant by smooth orbifold?

There seems to be some confusion over what the tangent space to a singular point of an orbifold is.
On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth ...

**5**

votes

**1**answer

378 views

### A formula on Kronecker coefficients

Accidentally, I proved the following formula for the Kronecker coefficients using some obscure method.
$$g_{lm^n,mn^l}^{m^{nl}}=1,\ \forall l,m,n\in\mathbb{N},$$
where $n^m$ is the rectangle ...

**17**

votes

**6**answers

3k views

### Geometric meaning of the Euler sequence on \mathbb{P}^n (Example 8.20.1 in Ch II of Hartshorne)

Is there any geometric way to understand the exact sequence in Example 8.20.1 in Ch II of Hartshorne (p. 182), or its dual from theorem 8.13?
Here is the sequence:
$0\to O_{\mathbb{P}^n}\to ...

**12**

votes

**3**answers

865 views

### Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.
...

**16**

votes

**9**answers

4k views

### Should one use “above” and “below” in mathematical writing?

I started thinking about this question because of this discussion:
http://sbseminar.wordpress.com/2010/08/10/negative-value-added-by-journals/
about how journals often change a paper (for the worse) ...

**3**

votes

**0**answers

485 views

### Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...

**21**

votes

**2**answers

6k views

### Introductory text on Galois representations

Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" ...

**12**

votes

**1**answer

6k views

### How many people fully understand the proof of Fermat's Last Theorem?

What is a rough order of magnitude estimate? $$ $$ There is a thread on Meta about this question, http://tea.mathoverflow.net/discussion/567/rapid-closing-of-questions/#Item_0

**7**

votes

**3**answers

1k views

### Does any tensor category correspond to a bialgebra?

I wonder how strong the power of Tannaka philosophy is, and if we accept that a tensor category is a generalized bialgebra, what difficulties we will come up against ?
Edit: Whether most tensor ...

**8**

votes

**1**answer

706 views

### When the Lovász theta-function saturates its upper bound

The Lovász $\vartheta$-function of a graph $G$, $\vartheta(G)$, is well-known to be "sandwiched" between the independence number of the graph, $\alpha(G)$, and the chromatic number of its complement, ...

**1**

vote

**0**answers

110 views

### How to Calculate Minimal Log Discrepancy on a Toric Variety?

Let $\sigma$ be a $3$ dimensional strongly convex rational polyhedral cone on $\mathbb{R}^3$ and $X_\sigma$, the corresponding affine toric $3$-fol. Also, assume that $\Delta$ is an anti-boundary ...

**2**

votes

**2**answers

604 views

### Which numbers appear as discriminants of cubics?

I'm trying to show that all possible splitting fields occur for a class of cubic polynomials, so have started by looking at the discriminants.
Clearly, given that they are products of squares, all ...

**2**

votes

**3**answers

473 views

### Finite / uniquely divisible abelian groups

Is there any counter example for the following statement?
STATEMENT:
Let $0 \to F \to A \to Q \to 0$ be a short exact sequence of abelian groups.
Assume that $F$ is a finite group, and $Q$ is a ...

**7**

votes

**2**answers

200 views

### On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse
On the convexity of element-wise norm 1 of the ...

**2**

votes

**2**answers

490 views

### Multinomial transformation for matrices

Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way:
...

**5**

votes

**3**answers

703 views

### Why do we distinguish the continuous spectrum and the residual spectrum?

As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...

**16**

votes

**5**answers

1k views

### Are there complexity classes with provably no complete problems?

A problem is said to be complete for a complexity class $\mathcal{C}$ if a) it is in $\mathcal{C}$ and b) every problem in $\mathcal{C}$ is log-space reducible to it. There are natural examples of ...

**2**

votes

**1**answer

212 views

### Relation between volume entropy and Hausdorff dim of limit set?

I have a very stupid question: I often see that the volume entropy of a compact Riemmannian manifold with negative curvature coincide with the Hausdorff dim of the limit set or Patterson sullivan ...

**19**

votes

**4**answers

3k views

### The class number formula, the BSD conjecture, and the Kronecker limit formula

If K is a number field then the Dedekind zeta function Zeta_K(s) can be written as a sum over ideal classes A of Zeta_K(s, A) = sum over ideals I in A of 1/N(I)^s. The class number formula follows ...

**13**

votes

**1**answer

1k views

### Double Referencing in arXiv

I am writing two separate paper that are closely related. When I try to submit to arXiv, is it possible for each paper to refer to the other paper with an arXiv link, rather then putting a newer ...

**11**

votes

**2**answers

1k views

### Is there a “disjoint union” sigma algebra?

I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
For an indexed family of sets ...

**5**

votes

**1**answer

1k views

### Fubini Study Metric and Einstein constant

Hi all,
it is well known that the complex projective space with the fubini study metric is Einstein, but what is the explicit value, i.e. for which $\mu$ does $Ric=\mu g$ hold?
Moreover, I would ...

**3**

votes

**2**answers

1k views

### *-homomorphisms between matrix algebras

Edited question:
Are there any other non-trivial *-homomorphisms between matrix algebras apart from the unitary homomorphisms?
Original question:
Does there exist a surjective (but not bijective) ...

**10**

votes

**2**answers

896 views

### Good reference for globally formulated calculus of variations on Riemannian manifolds?

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant ...

**5**

votes

**2**answers

2k views

### Galois group of a product of irreducible polynomials

Hello
Suppose given a polynomial $P=Q_1\cdots Q_k$ of degree $n$, where each $Q_i$ is irreducible. Suppose also that I know the Galois group $G_i$ (over the rationals) of each irreducible factor ...

**8**

votes

**1**answer

860 views

### Results about existence/uniqueness of solution to Euler-Lagrange equations?

While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:
What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...

**20**

votes

**1**answer

1k views

### Rationality of intersection of quadrics

Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking ...

**2**

votes

**0**answers

84 views

### Non-clean fiber products

Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g.
When do fibre products of smooth manifolds ...

**4**

votes

**2**answers

789 views

### Relation between Hausdorff dimension and Bowen's equation

I am reading the paper Hausdorff dimension for Horseshoes, by McCluskey and Manning. In the following theorem
Theorem:
Let $\Lambda$ be a basic set for a $C^1$ axiom A diffeomorphism $f:M^2\to ...

**7**

votes

**0**answers

298 views

### Proof of Lomnicki and Ulam on Infinite Product Probability Spaces

Given an arbitrary, nonempty family $(\Omega_i,\Sigma_i,\mu_i)_{i\in I}$ of probability spaces, there exists a probability measure $\mu$ on $\otimes_i\Sigma_i$ such that for every finite set ...

**10**

votes

**2**answers

1k views

### Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.

**10**

votes

**1**answer

599 views

### Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?

There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations ...

**2**

votes

**1**answer

114 views

### PA proves that functions are total

Is there a total recursive function $f:N \to N$ such that for no $\Sigma_1$ formula $\phi(x,y)$ which defines it (i.e., defines its graph), is it true that PA proves that "$\phi$ defines a total ...

**2**

votes

**3**answers

530 views

### Groups of Rational Points on Gaussian Circles

Let a gaussian circle $C_R$ be any circle defined by the equation:
$$x^2+y^2 = R, (x,y) \in \mathbb{R}^2$$, where $R$ is the norm of a gaussian integer ($R=a^2+b^2, (a,b) \in \mathbb{Z}^2$). IF $R$ ...

**5**

votes

**1**answer

4k views

### Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution?

**0**

votes

**0**answers

74 views

### Prove that k ≤ log2N [on hold]

I have the following problem and I don't know where to star:
Let n ≥ 2 and let n = p1p2...pk be its prime factorization, where the primes are not necessarily distinct. Prove that k ≤ log2N (hint: ...

**7**

votes

**1**answer

138 views

### Is the word problem decidable for free finitely generated self-square groups?

A self-square group is a group with extra structure, which encodes the fact that the group is isomorphic to its own direct square.
To be exact, the group $G$ has a special element $1$, a unary ...

**7**

votes

**0**answers

62 views

### Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ ...

**4**

votes

**5**answers

1k views

### What axioms are stronger than the Axiom of choice?

What other axioms in set theory are stronger than AC ? I mean what are those axioms that will imply AC ?

**1**

vote

**0**answers

68 views

### Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...

**8**

votes

**1**answer

2k views

### How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?

I know that the answer is $\mu_p \times \mathbb{Z}/p\mathbb{Z}$ if $E$ is ordinary, and $\alpha_p$ if $E$ is supersingular, where $\mu_p$ and $\alpha_p$ are the kernels of Frobenius on $\mathbb{G}_m$ ...

**4**

votes

**3**answers

890 views

### Example of symplectic and hamiltonian diffeomorphism on $S^2$ and $T^2$

Hello everybody. For a purpose of consolidation of some result I am trying to set down, I need to construct an example to sustain the theory and I am looking for symplectic and Hamiltonian ...

**4**

votes

**2**answers

349 views

### Lipschitz continuity of singular values

How smooth are the singular values of a matrix F in terms of entries of F? I am hoping for Lipschitz continuity, but was not able to find it.