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If $[X]^2$ is compact then $X$ is finite. Let $\Delta \subseteq X^2$ be the diagonal. It is easy to show that the map $X^2 \setminus \Delta \to [X]^2$ given by $(x,y) \mapsto \{x,y\}$ is continuous, …

I guess I should try and answer that! I don't really have a good answer to the question of why the random method is not mentioned in the book, and I would agree that it might to be the fastest method …

Castelnuovo-Mumford regularity is defined in the general mutligraded case in this paper of Maclagan and Smith. They give the definition in terms of local cohomology and discuss bounding the regularity …

For any $a,b \in X_0$, let $X_1(a,b)$ denote the subspace of $X_1$ lying over $(a,b) \in X_0 \times X_0$. Since $X_0$ is discrete, this iterated fiber product breaks up as a disjoint union of product …

Lev's proof reminded me of two papers, and unless I'm doing something silly, the said conjecture follows as a corollary of those papers. The answer below is just meant to supplement Lev's result, and …

Lev Borisov having put up a complete solution; I'll put up how far I got. In his notation, I was able to show $\frac{\partial f(a)}{\partial a_k} > 0$ when $z^n + a_{n-1} z^{n-1} + \cdots + a_0$ has a …

I believe that for class $2$ and the original question (finding a subgroup $H$ of finite index such that $H_i/H_{i+1}$ is torsionfree for all $i$) the following works. Suppose that $N$ is of class $2 …

They are all continuous with dense images. See e.g. Trèves, Topological Vector Spaces, Distributions and Kernels, p. 272, Theorem 28.2, p. 301, and Remark 28.3, p. 303.

It is a shame that this question is skipped in almost all modern textbooks! Except Nakahara I know of no other modern textbook. There's a general contortion operation, sometimes called the Schouten b …

The following theorem of Kurepa answers the question. Theorem (Kurepa) Suppose that $\kappa$ is regular, $\lambda< \kappa$, and $T$  is a $\kappa$-tree each of whose levels has cardinality less …

In case anyone is interested in situations like this, the answer is yes. The proof relies on an $\epsilon$-perturbation of the following lemma: Suppose $w,x,y\in\mathbb{R}^m_+$ are three vectors on …

There are different models for terrorist or other criminal networks. I can only give references. Roy H. A. Lindelauf (Royal Dutch Defense Academy) et al., "The Influence of Secrecy on the Communicati …

The existence follows from a very general fact about existence of a compatible connection in any principal bundle. That is to say, on one hand you have your vector bundle $E \to M$ endowed with $g$. I …

Alas, the approach through the functions $g_t$ cannot possibly work. That is seen if one takes e.g. $n=2$, $a_1=4/5$, and $a_2=t=3/5$. Based on numerical evidence, the following approach promises to …

A more general solution is $$ f(x,y) = G\left( \dfrac{1-y}{1-x}\right) \exp(x+y) $$ for arbitrary differentiable function $G$. EDIT: To see that this is the general solution, write $f(x,y) = g(x,y) …