# All Questions

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### Determinant of a tensor product [on hold]

Let V and W be two vector spaces over a field of characteristic zero. Give a formula for the top exterior power of V tensor W.
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### Stable unions without stable images

A regular category is one with finite limits and pullback-stable images (i.e. (regular epi, mono) factorizations). A coherent category is a regular category that also has pullback-stable finite ...
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### Doubling metrics, doubling measures, Lebesgue density

As stated in this question, Lebesgue differentiation theorem holds on locally doubling space? and proved here, http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf the Lebesgue differentiation theorem (...
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### How can i integral of this function? [on hold]

I want to know how can i solve this function. $\int (1-y^d)^n \, dy$ Is it possible to solve it? If you know the method, please teach me.
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### Is the category of prederivators cartesian closed?

The question is in the title. ${\bf PDer} = Fun({\bf Cat}^\text{op}, {\bf CAT})$ is obviously cartesian since $\bf CAT$ is. The usual argument for presheaf categories does not apply directly since 1-...
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### Strict/strong functors are co/reflective inside lax functors, the coendy way

Bozapalides' remarks on lax presheaves show that the category $[{\cal A}^\text{op}, {\bf Cat}]$ is reflective and coreflective inside the category of lax functors, lax natural transformations and ...
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### How to compute bounding coefficients for McDiarmid's inequality?

I am trying to understand the proof in Sec. A2 of Gretton et al.. To make the question self-contained, I summarize below the key ingredients. At the end of the post, I state my question. Given a ...
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### Kähler classes for surfaces of general type with $c_1^2=3c_2$

Given a smooth, compact complex surface with ample canonical bundle satisfying $c_1^2=3c_2$, is it true that every Kahler class is a multiple of $c_1$? This seems to be the case for fake projective ...
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### Does a vector belongs to a simplicial subcone when it belong to cone with more than n generators?

Assume $x_{0}\in \text{cone}(a_{1},\dots,a_{N})$, where $a_{i}\in \mathbb{R}^{n}_{+}$ ($a_{i}\in \mathbb{R}^{n}$, and $a_{i}\geq 0$) for $i=1,\dots,N$ (i.e., $x_0$ lies in the cone generated by $a_{i}$...
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### a modular character problem [on hold]

Let $B\in$Bl$(G|D)$ and suppose that $\sigma\in$Aut$(G)$ fixes every $\chi\in$Irr$(B)$. If $d\in D$, show that $d$ and $d^\sigma$ are $G$-conjugate. It is a problem from Navarro's book "characters and ...
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### Quotient of Cohen-Macaulay ring

Let $A$ be a Cohen-Macaulay ring, and $I$ an ideal of $A$. What can we say about $\operatorname{depth}(A/I)$? I know that $\operatorname{depth}(A/I)\le \dim(A/I)$.
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### De Rham Cohomology in positive characteristic

This question is about the possible existence of a "compactified" algebraic De Rham cohomology in positive characteristic. Namely, one knows that, for a smooth, but not proper, variety $U$ over a ...
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### When is a functor a right derived functor?

Suppose we have Grothendieck abelian categories $\mathcal{A}, \mathcal{B}$. Suppose also we have given an exact functor of triangulated categories $$F \colon D(\mathcal{A}) \to D(\mathcal{B})$$ ...
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### What are the most important mathematical prerequisites for machine learning? [closed]

Next week I like to start the machine learning class with Andrew Ng and now I like to brush up on some mathematical topics. My inquires let me to some recommendations: Linear Algebra: matrices ...
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### What is the structure group of the Hopf fibration $S_1\rightarrow S_3 \stackrel{p}\rightarrow S_2$?

I am studying fiber bundles and have thoroughly reviewed the famous example of the Möbius strip. In that example, I learned how to discover that the structure group of the Möbius strip fiber bundle ...
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### BV functions with values in metric space

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### Existence of solution to first order pde [closed]

Let $U : = \big(-\frac 12, \frac 12 \big)^2 \setminus B_R(0)$ for some sufficiently small $R > 0$. I would like to prove the existence of a solution $\rho = \rho (x_1, x_2)\in C^1(U)$ to the ...
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### Eigenvalues of the Laplacian modulo primes

Let $d\geq 1$ be an integer and consider $M = \mathbb{R}^d / \mathbb{Z}^d$. Let $\Delta$ be the Laplace operator on $M$. Then the eigenvalues are $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$ ...
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### Small set such that $\{1 , \ldots , n\} \cdot A = \mathbb{Z} / p \mathbb{Z}$

Let $p$ be a large prime and $n < p$. What is the smallest size of a set $A \subset \mathbb{Z} / p \mathbb{Z}$ such that $A \cdot \{1 , \ldots , n\} = \mathbb{Z} / p \mathbb{Z}$? Here $\cdot$ ...
### Some examples of $\mathbb Q$-Gorenstein smoothing
I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition. Definition. For a normal projective surface $X$ with quotient ...
Let $X$ be a compact metric space, and let $$\nu_s(X):=\sup\limits_{\varepsilon>0} \inf\limits_{\mathcal{E}} \sum\limits_{E \in \mathcal E} \mathrm{diam}(E)^s$$ be the $s$-dimensional Hausdorff ...