# All Questions

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### Non-flat $R \subseteq S$, which is integral, separable, $R$ is a noetherian (not integrally closed) integral domain

On ramification theory in noetherian rings, of Auslander and Buchsbaum say: "Chapter 4 is devoted to showing that under various conditions if $S$ is unramified over $R$, then $S$ is $R$-projective. ...
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### Proving inequation with ceilings in Finite Field of characteristic $p$

Take $ui = pt_i +j_i$ where $p$ is a prime number and $u(p-r) \equiv 1$ $(\mbox{mod p})$ for positive integers $1 \le i, r, j_i\le p-1$ and $t_i \ge 0$. How can I prove that: ...
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### A quasicompact space with a net that contains no convergent strict subnet

If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...
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### higher direct images of relative canonical sheaf plus a fractional divisor

For a map $f: Y \rightarrow X$ branched over simple normal crossing divisor $B=\sum_iB_i$, do we know of similar local freeness property for higher direct image of relative canonical sheaf plus a ...
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### Thom Class of tensor bundles

Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...
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### Higher algebra and terminology about 2-objects

It is well known that one way to build higher category theory is to use some induction process, where an $n$-category has as $0$-cells some $n-1$ categories, such that for two $0$-cells $\mathcal{A}$ ...
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### Nemytskii/superposition operator without separability of Banach space?

Let $T:[0,1] \times X \to \mathbb{R}$ be a nonlinear map where $X$ is a Banach space. Suppose that $T$ is a Caratheodory map, so that $t \mapsto F(t,x)$ is measurable and $x \mapsto F(t,x)$ is ...
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### Riemann-Roch formula for nodal curves

Let $X$ be an irreducible, reduced, projective curve over an algebraically closed field, with at worst nodes as singularities. Let $\mathcal{F}$ be a trivial vector bundle on $X$ of rank $r$. Consider ...
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### On Cantor sets every map is $C^{\infty}$

For a fixed Cantor set $K\subset [0,1]$ and a continuous function $g:[0,1]\to \mathbb R.$ Is it always possible to find a $C^{\infty}$ map $f:[0,1]\to \mathbb R$ such that $g$ and $f$ coincide in $K?$ ...
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### How to obtain a permutation of a tensor product? [on hold]

I am looking for a way to efficiently compute a re-ordered kronecker product from the result of another kronecker product. For example, consider $$F=A\otimes B\otimes C\otimes D\otimes E$$ from the ...
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### Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
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### Continuity of Induced Functional Structures [on hold]

Bredon's Topology and Geometry gives definition of Induced Functional Structure as follows: Suppose $F_x$ is a functional structure on space $X$ and let $f:X\to Y$ be a map. Then the induced ...
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### Distribution of infinity-norm over the unit sphere

I need to compute probabilities of the form $P( \Vert X \Vert_\infty < r ),$ where $X$ is a random variable of dimension $n$, drawn with a uniform distribution on the unit sphere ...
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### Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group

Recently in a paper we get the following result: Let a discrete group $\Gamma$ act on a discrete abelian group $G$ by group automorphisms. Every irreducible unitary representation $\pi$ of ...
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### Distribution of trivial subset sums

Suppose I have a set $S$ of $n$ integers picked independently, uniformly at random from $[-L, L].$ Let $z(S)$ be the number of subsets of $S$ which sum to zero. The question is: what is the ...
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### Must a proof of the asymptotic Goldbach conjecture be effective to imply GRH?

It was shown by Hardy and Littlewood that GRH (i.e. the Generalized Riemann Hypothesis for Dirichlet L-functions) implies that every large enough odd number is the sum of three primes. Later on (circa ...
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### Which journals publish research announcements?

Perhaps, somebody asked this already, excuse me in this case. Can anybody advise mathematical journals that publish research announcements? (I mean little papers without proofs.) It sometimes ...
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### schatten 1-norm of rank $k$ matrix

I am looking for a high-probability lower bound for the following rank-$k$ matrix $$X = u_1 v_1^T + u_2 v_2^T + \cdots + u_k v_k^T,$$ where $u_1,\dots,u_k,v_1,\dots,v_k$ are independent $N(0,I_n)$ ...
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### Geometric generic fibre

This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE. Question 1: Are the fibres of a family of complex varieties ...
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### Counter examples for strengthening Whitehead's theorem?

Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, ...
At the end of the following paper, Erdos asked if there is a family $F$ of entire functions of size continuum such that for every $z \in \mathbb{C}$, $\{f(z) : f \in F\}$ has size less than continuum. ...
Let $K$ be a local field of characteristic $p$, where $p$ is a prime number greater than 2. In particular, $(x+y)^p=x^p+y^p$ for $x,y\in K$. The fake Heisenberg group is defined to be  ...