# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### How to calculate error for 3D-trilateration [closed]

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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Modifying tensor to be positive definite everywhere [on hold]

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 ...
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### 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 ...
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### 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*} ...
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### Pull back of a semistable vector bundle to a product is semistable?

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 ...
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### Wild automorphisms of profinite groups

Is there a profinite group $G$, a continuous automorphism $\alpha$ of $G$ and a topologically finitely generated closed subgroup $H \leq G$ such that $\alpha(H) \lneq H$ ? Note that if an ...
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### Hoeffding's lemma for unbounded r.v with bounded exponential map

Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty$ for all $\lambda \in [-c,c]$. Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: ...
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### Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
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+100

### “The” natural double complex associated to a principal $G$-bundle?

Let $\pi: P \to M$ be a principal $G$-bundle. We have the associated adjoint bundle $ad(P)= P \times_{ad} \mathfrak g$ whose sections correspond to infinitesimal guage trasformations. Consider the ...
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### number of maximal subgroups of the symmetric group

What is the asymptotics of the number of the maximal subgroups of $S_n$ (as a function of $n$)? This must be written down somewhere... EDIT I am actually more interested in the number of conjugacy ...
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### Goldbach for certain classes of $n$

Asked on MSE without response here. $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$. The Wiki article on the Goldbach conjecture states that In 1975, ...
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### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$. I am interested in the following questions: How ...
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I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on. In particular ...
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### On the number of divisors in a given range

Given $\alpha\in\Bbb N$, can there be more than $(\log N)^4$ divisors (composites allowed) of $N$ in $\big[\frac\alpha2,\alpha\big]$ when $\sqrt N\in\big[\frac\alpha2,\alpha\big]$? What is the ...
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### On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$ $L^{1}/H^{1}_{0}$ is ...
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### Did Bishop, Heyting or Brouwer take partial functions seriously?

The partial μ-recursive functions which may or may not be provably total seem to have some direct relation to the initial motivations for intuitionistic mathematics. (Following Kronecker, one ...
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### The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = ...
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### A PDE problem on the Heat-Like differential equations

I came across the following questions in part of my work: Consider the Heat-Like equation of the form $\frac{\partial u}{\partial t}=\hat{H}u + f(x,t)u + g(x,t)$ where $\hat{H}$ is a Sturm-Liouville ...
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### Is there a function with a fixed point but does not satisfy the Banach contraction principle? [closed]

This is my question Is there a function which does not satisfy the Banach contraction principle, but has a fixed point? thank
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+50

### Cartesian square root of a measure preserving action

Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such ...
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### Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes

Let $C_1,C_2,\ldots,C_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C_i|=|K \cap C_i|$ for ...
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### Homological stability for orthogonal groups

In Vogtmann's paper "Spherical posets and homological stability for $O_{n,n}$" it is shown that for all fields different than the field $F_2$ with two elements the homology groups of the orthogonal ...
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### a question on warped product

This question is on J.Cheeger and Tobias H.Colding's paper "lower bound on Ricci curvature and the almost rigidity of warped products". For a warped product $M=(a,b)\times_f N^{n-1}$ with metric $g$. ...
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### A Graph-Theory Related Question

Let $n$ be a positive integer and partition a grid of $4n$ by $4n$ unit squares into $4n^2$ squares of sidelength $2$. (The squares with sidelength $2$ have all of their sides on the gridlines of the ...
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### Test correlation between 3 variables by hand [closed]

Do the three areas (courtside, lower deck, upper deck) differ in soda sales per hour? Courtside Lower Deck Upper Deck 38 35 11 42 37 25 40 ...
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### General approaches to extension theorems as Caratheodory

I would like to know if there are some general studies about extension-like theorem, in the sense which i'm going to describe. This paragraph is not rigorous; I just would like the idea to be clear. I ...
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### Is there a shelling of a (threshold)shifted complex, such that any partial shelling is still (threshold)shifted?

first the relevant definitions: A complex $\Delta \subset 2^{[n]}$ is a family of subsets of $[n]$ that is closed downwards, i.e. if $A \subset B$ and $B \in \Delta$, then $A \in \Delta$. A complex ...
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### Cubic, divisor of rational function $x/z$? [closed]

Let $k$ be a field, and let $a \neq 0$, $1 \in k$. Let $C = V(y^2z - x(x-z)(x - az))$. What is the divisor of the rational function $\psi([x, y, z]) = x/z \in k(C)$?
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### $C = V(x^3 - xz^2 - y^2z)$, linear equivalence [closed]

Let $C = V(x^3 - xz^2 - y^2z) \subset \mathbb{P}^2(\mathbb{C})$. Let $p_0 = [0, 1, 0]$, $p_1 = [0, 0, 1]$, $p_2 = [1, 0, 1]$, $p_3 = [-1, 0, 1]$. I have two questions. Is $2p_0$ linearly equivalent ...