# All Questions

15 views

### Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$

I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ...
18 views

### Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$

Is there anything known about uniqueness of classical solutions to $$\partial_t u -u\Delta u=0\quad u(0,\cdot)=1$$ on smooth domains $[0,T]\times D$ without boundary conditions? I know that ...
25 views

### Integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of varieties of general type is a rational numbers?

It is known that the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau varieties is a rational numbers. My question is on moduli space of varieties of ...
25 views

### How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan. In his lecture, he uses the following results: Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in ...
12 views

### Regularity on Neumann problem on polygonal domain

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity). Let $\Omega$ denote a cube in $R^n$ and consider ...
85 views

### family of polynomials with square discriminant

The title pretty much sums it up: do people know of nice parametrized families of polynomials (with integer coefficients) with square discriminant. I should say that one such family consists of ...
13 views

### Compact embedding and fractional Sobolev spaces in unbounded domain

It is known that there exists a compactness results involving fractional sobolev spaces in bounded domain. What about unbounded domain? More precisely, Under which conditions, we can extend the ...
45 views

### Information and intuition packed in the Chern character for coherent sheaves

even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting. Let us consider a smooth projective algebraic variety ...
153 views

### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...
9 views

### How to calculate error for 3D-trilateration [on hold]

I'm developing a 3D positioning system that uses four anchor nodes of known location to position a fifth node of unknown location. I'm calculating the position by using trilateration, MATLAB code ...
82 views

### What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...
19 views

### Algorithm: In every vertex whose distance from $v_i$ is not greater that $d_i$ place $r_i$ objects [on hold]

You are given a tree with $N$ $(1 \le N \le 10^5)$ vertices and $N - 1$ edges. Weight of edge won't exceed 200. Design an algorithm to do $Q$ $(1 \le Q \le 10^5)$ operations of two types as fast as ...
31 views

### Convergence of local stable manifolds

This question is a kind of local version of a previous post (MO224171). Let $\ f_n \$ be a sequence of Morse functions on $\mathbb{R}^d$, converging, in the $C^\infty$-topology, to a limit Morse ...
52 views

### Name for the variety of preimages of a finite morphism

If $f:X\to Y$ is a finite morphism of degree $d$ between two varieties, you get a closed subset of the symmetric product $X^{(d)}$ (or perhaps rather the Hilbert scheme $X^{[d]}$), defined as the ...
84 views

### Bound that random walk stays within with constant probability?

For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$. What is the bound, in terms of $n$, such that if the ...
202 views

### rational numbers and triangular numbers

This question is an offshoot of Ratio of triangular numbers. Suppose $ka(a+1)=nb(b+1)$, where $k,n >1$ are relative prime integers, and $a,b \geq 0$ are integers. Which $k,n$ pairs have no solution ...
50 views

### Modifying tensor to be positive definite everywhere

Consider a (0,2)-tensor. It is known that it is positive definite somewhere and it is negative definite otherwise. Is there a theory how to "make" that tensor positive definite everywhere, while ...
50 views

### Euler Characteristic of simple sheaves

Let $X$ be a projective curve over a field $K$ (any characteristic). Let $\mathcal{F}$ be a coherent simple sheaf (In the sense, that $\mathcal{F}$ doesn't have non-trivial subsheaves). What is the ...
15 views

### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} ...
Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be an ample line bundle on $X$. Let $F$ be a $\mu_L$ semistable rank 2 vector bundle on $X$ (semistability in the sense of ...