# All Questions

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### Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint"). I know that all the eigenvalues of matrices in this ...
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### Limits and colimits in a 2-category vs. in an infinity category: the non- (2,1)-case

Suppose that $\mathscr{C}$ is a $2$-category (or more generally a bicategory) which is not a $\left(2,1\right)$-category. Is there any relation between limits and colimits in $\mathscr{C}$ (in the ...
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### Examples of unproven but likely true existential sentence (in the sense of incompleteness)

Some examples of universal statements that are unproven but likely true include the Riemann hypothesis (all non-trivial zeros of the zeta function have real part 1/2) and the Goldbach conjecture (all ...
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### Criteria for abstract polytopes to be convex polytope

Suppose I have an abstract polytope defined by a poset. Are there any methods for determining whether the abstract polytope can be geometrically realized as convex-hull on its set of vertices?
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### Gaussian expectation of an exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation, $$E\left[ \exp(\mathbf{xx}^\top)\right]$$ where $\exp(\cdot)$ is element-wise exponential function (not ...
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### Collatz property implying infinite “fall below” trajectories, is it known?

(this was discovered analyzing Collatz empirically.) a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value. consider ...
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### On a sum statistically independent of its term [on hold]

Suppose $U$ and $V$ are two non-degenerate random variables, say real-valued for simplicity. Suppose further that their sum, $U+V$, and one term, $U$, are statistically independent. This happens when ...
583 views

### Where to find (personal) motivation [on hold]

I think it would be appropriate to make this question CW... It is likely that this question will not survive here on MO for long, but I do hope that the community gives it a chance. I also hope to ...
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### Bi-invariant one forms on compact Lie groups

I'm hoping to learn of a criterion for the existence of a bi-invariant one form on a Lie group $G$. I'm looking for a reference that there are no such one forms son $SU(n)$ (as long as this is in fact ...
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### Complex parameters in the Ritz procedure

I am using the Ritz procedure to write a trial function as the superposition of other admissible functions, with the coefficients being unknown variational parameters to be determined. The variational ...
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### Does the canonical morphism commute with the inverse image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on ...
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### A question about subgroups [on hold]

Is there a group $G$ and a non-abelian subgroup $H$ of $G$ such that $[G:H]=2$, $|Z(H)|>1$ and $C_G(H)=2|Z(H)|$?
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### Concavity of the solution of a parametric implicit function

Suppose $F(x,y;k)=f(x,y)+kg(x,y)=0$ uniquely defines the solution $y(x;k)$ for $x\in \mathbb{D}$, a compact domain, and $0\leq k \leq 1$ is a parameter. We know that for $k=0,1$, $y(x;0)$ and $y(x,1)$ ...
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### Examples of etale group schemes [on hold]

What are important examples of etale group schemes over some field $F$, apart from finite group schemes?
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### Dimension of a similarity class

Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
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### What is known about the Brauer group of an arithmetic surface?

Let $X$ be an arithmetic surface over $\mathbb{Z}$, that is we have $\pi: X\rightarrow Spec(\mathbb{Z})$, $X$ is integral, two-dimensional and regular and $\pi$ is projective and flat. What is known ...
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### Estimating the Variance of a Discrete Normal Distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
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### Degree of a smooth curve in an abelian variety

Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$. Question : Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in ...
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### Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying $b$ is bilinear, ...
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### Stabiliser of the lamination of a free group - Invariant subgraphs

I am studying the paper "Laminations, trees, and irreducible automorphisms of free groups" of Bestvina, Feighn and Handel. But I found a note in the paper "Stabilisers of $\mathbb{R}$-trees with ...
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### Left invertible operators of $B(X,Y)$

Suppose that $X$ and $Y$ are Banach spaces. Is $\{f\in B(X,Y):f\ \text{has a left inverse}\}$ an open subset of $B(X,Y)$?
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### Arbitrarily large $n$ divides $F_n$

Is it true that there exists $n \in \mathbb{N}$ with arbitrarily many prime factors such that $n$ divides $F_n$, where $F_n$ represents the n-th Fibonacci number?
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### Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...
557 views

### Small values of a polynomial evaluated at roots of unity

The MO answer http://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant ...
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### Average distance between houses in the United States (per each of the 50 states) [on hold]

I'm trying to figure out what the average distance is between houses (per state in the U.S.). I can't seem to find it anywhere. My buddy told me about this site so here I am. I'd really appreciate ...
108 views

### Hilbert scheme of an infinitesimal neighborhood of a subvariety

Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...
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### Publishing problem [on hold]

First, I want appreciate your work on this platform, as I have been getting very helpful advice even though I am not a member. My problem is that I have been working on-off on a famous math problem ...
103 views

### Automorphism group of a smooth quadric $Q\subset\mathbb{P}^4$

Let $Q$ be the smooth quadric threefold in $\mathbb{P}^4_{\mathbb{C}}$ defined by the equation $x_0x_4+x_1x_3+x_2^2=0$. Is it true that the automorphism group of $Q$ is $SO(Q;\mathbb{C})$ which is ...
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### Can a lower-order trigonometric polynomials fit the data generated by a higher-order one? [on hold]

Let $x$ be a discrete finite-dimension vector $x$, and write $\hat{x}(\Omega) = \mathcal{F}_{\Omega}x$ where $\mathcal{F}_{\Omega}$ is the linear mapping collecting the Fourier transform of $x$ at the ...
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### A conjecture about strongly regular graphs

Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$ So far I have ...
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### Is any axiom system for sets categorical? [on hold]

$ZF$ define membership by conditions demanding the existence of some constructable right-side-terms $M$ ($x \in M$). Is it meaningsful to ask for a categorical axiom system here? Shouldn't it be ...
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### Clutching functions and Classifying maps

Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C_+X\cup_X C_-X$. Then there are trivialisations of the restrictions $E|_{C_+X}\cong C_+X\times G$, ...
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### boundary density of the Von Koch flake

Given a measurable set $K\subset \mathbb{R}^d$ we consider the occupation ratio $$f_r(x)=vol(K\cap B(x,r))/r^d$$ and especially the asymptotics when $r\to 0$. When $K$ has a fractal boundary and $x$ ...
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### Positivity of the top Lyapunov exponent

I have a general question about the Oseledets Multiplicative Ergodic Theorem. In the context of the MET I'd like to know if there is some reasonably general sufficient condition which implies that the ...
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### Minimum of Random Energy Model (REM) with logarithmically correlated potential

In the paper [FB] (ArXiv, J. Phys. A), the authors analyse a particular Random Energy Model (REM) with logarithmically correlated potential and conjecture in Eq. (2) that the distribution function of ...
### Interpolation between $L_p$ and $B^s_{q,q}$
I am looking for a reference or a direct argument that shows the real interpolation space between $L_p$ and $B^s_{q,q}$ is $B^\alpha_{r,r}$, with the usual conditions on the indices. This result is ...
If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that $$\pi_0(\mathcal{F}) = \mathbb{Z}\, ,$$ i.e. the connected ...