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curious little things - aggregated feedsenMath Overflow Recent Questions: Examples of schemes $S$ with $H^{3}(S,\mathbb G_{m})=0$
https://mathoverflow.net/questions/295859/examples-of-schemes-s-with-h3s-mathbb-g-m-0
<p>The examples I have are: $S$ is equal to the spectrum of a global field; or a proper non-empty open subscheme of the spectrum of the ring of integers $\mathcal O_{K}$ of a number field $K$ (proper means $S$ is not all of ${\rm Spec}\,\mathcal O_{K}$); or $S$ is a non-empty open subscheme of a smooth, complete and irreducible curve over a finite field. Can anyone supply other examples, please?</p>Thu, 22 Mar 2018 08:03:18 -0600Math Overflow Recent Questions: Tensor product of hyperfinite von Neumann algebras
https://mathoverflow.net/questions/295858/tensor-product-of-hyperfinite-von-neumann-algebras
<p>Let $(M_n,\tau_n)_{n\geq 1}$ be a non-commutative probability space. Suppose each $M_n$ is hyperfinite. Is it true that $\overline{\otimes}_{n=1}^\infty M_n$ is again hyperfinite? How to prove or disprove this?</p>Thu, 22 Mar 2018 07:18:08 -0600Math Overflow Recent Questions: Geometric fundamental group and algebraically closed residue field
https://mathoverflow.net/questions/295856/geometric-fundamental-group-and-algebraically-closed-residue-field
<p>my questions relates to the following talk of Tsuji: </p>
<p><a href="https://www.youtube.com/watch?v=2brDj26phP0" rel="nofollow noreferrer">https://www.youtube.com/watch?v=2brDj26phP0</a> </p>
<p>At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not work if $k$ is not algebraically closed. I do not understand this remark at all. </p>
<p>For those of you that do not want to have a look at the video here is a quick summary of the part in question:</p>
<p>The assumptions are as follows: $K$ a complete valuation field of mixed characteristic (0,p) with ring of integers $\mathcal{O}_K$ uniformizer $\pi$ and residue field $k$ (<em>algebraically closed</em>). $A$ an semistable algebra i.e. $$ A \underset{étale}{\leftarrow} \mathcal{O}_K[T_1,\cdots, T_c, \cdots, T_d]/(T_1 T_2 \cdots T_c-\pi)$$ let $\mathcal{K}=Frac(A)$ and $\overline{\mathcal{K}}$ an algebraical closure with $\overline{K} \subset \overline{\mathcal{K}}$. For every finite extension $\mathcal{K} \subset \mathcal{L} \subset \overline{\mathcal{K}}$ denote by $A_{\mathcal{L}}$ the integral closure of $A$ in $\mathcal{L}$. Define the set $$ S:=\big\{\mathcal{L} \, \vert \, A_{\mathcal{L}}[\frac{1}{T_1\cdots T_d}] /A[\frac{1}{T_1\cdots T_d}] \, \text{étale}\big\}$$ such that $Spec(A)$ is connected and $Spec(A/\pi A)\neq 0$ and $Spec(A/\sum_{i \in I} T_iA)$ is irreducible or empty for all $I \subset\{1,\cdots,d\}$. Set $\mathcal{K}^{ur}=\cup_{\mathcal{L}\in S} \mathcal{L}$.
He then states that $$ Gal(\mathcal{K}^{ur}/\mathcal{K}\overline{K})= \pi_1(A[1/T_1\cdots T_d] \otimes _{\mathcal{O}_K} \overline{K},Spec(\overline{K})).$$ Why is this not true if $k$ is not algebraically closed? Any help would be very much appreciated.</p>Thu, 22 Mar 2018 06:57:24 -0600Math Overflow Recent Questions: Can every Banach space with the Schur property embed into $L_{1}(\mu)$ for some $\mu$?
https://mathoverflow.net/questions/295854/can-every-banach-space-with-the-schur-property-embed-into-l-1-mu-for-some
<p>In 1974, W. B. Johnson and E. Odell observed that there are subspaces $X$ of $L_{1}$ with the Schur property. In 1980, J. Bourgain and H. P. Rosenthal constructed a subspace $X$ of $L_{1}$ such that $X$ has the Schur property, but $X$ is not isomorphic to a subspace of $l_{1}$. Hence, I have the first question as follows:</p>
<p>Question 1. Is every Banach space with the Schur property isomorphic to a subspace of $L_{1}(\mu)$ for some measure $\mu$?</p>
<p>Moreover, W. B. Johnson and E. Odell gave natural non-trivial conditions that a subspace of $L_{p}$ embeds into $l_{p}$ for $1<p<\infty, p\neq 2$. But </p>
<p>Question 2. Are there conditions that a subspace of $L_{1}$ embeds into $l_{1}$ ?</p>
<p>Thank you!</p>Thu, 22 Mar 2018 06:41:06 -0600Math Overflow Recent Questions: Is this morphism of regular local rings regular?
https://mathoverflow.net/questions/295852/is-this-morphism-of-regular-local-rings-regular
<p>Let $R$ be a regular, local $\mathbb{Q}$-algebra with a regular system of parameters $x_1, \dotsc, x_n$, and let
$$f \colon \mathbb{Q}[X_1, \dotsc, X_n]_{(X_1, \dotsc, X_n)} \rightarrow R$$
be the map given by $X_i \mapsto x_i$. Then $f$ is flat (for instance, by Bourbaki, cf. EGA III, 0.10.2.2). </p>
<p>Is $f$ a regular morphism, that is, are the (geometric) fibers of $f$ regular? Certainly, the closed fiber is regular, but how about the others?</p>
<p>The answer should be yes, and I would appreciate an argument for this. In the case when $R$ is excellent, the positive answer seems to be a special case of EGA IV, 7.9.8, but a less contrived argument (and one that would not use an additional excellence assumption) would be greatly appreciated.</p>Thu, 22 Mar 2018 06:15:23 -0600Math Overflow Recent Questions: Why Is The Category Of Sets A Grothendieck Topos? [on hold]
https://mathoverflow.net/questions/295851/why-is-the-category-of-sets-a-grothendieck-topos
<p>Why Is The Category Of Sets A Grothendieck Topos? How can just object and relation lead to a Grothendieck Topos which {!} 'suggests the possibility of synthesis of algebraic geometry, topology, and arithmetic' [ReS, The vision]. </p>Thu, 22 Mar 2018 06:11:08 -0600Math Overflow Recent Questions: Role of polyhedral domain in convergence of finite element method
https://mathoverflow.net/questions/295849/role-of-polyhedral-domain-in-convergence-of-finite-element-method
<p>I am reading <a href="http://www.mathematik.uni-muenchen.de/~diening/archive/diening_kreuzer-AFEM2007.pdf" rel="nofollow noreferrer">a paper by Diening and Kreuzer</a> where they consider the convergence of finite element approximations for $p$-Laplace equation when using a certain algorithm.</p>
<p>In the paper, they assume that the domain $\Omega$ is polyhedral. It is not clear to me why exactly this is assumed. <strong>What is the significance of the polyhedral domain assumption for results of this type?</strong></p>
<p>My first guess is that this is related to discretization of the boundary $\partial \Omega$. If it is not piecewise linear, then different meshes will cover different domains, which hopefully approximate $\Omega$ in some way.</p>
<ul>
<li>Is there a standard technique or result that allows extending the types of results Diening and Kreuzer get to non-polyhedral domains? I might want to use different meshes on the same domain.</li>
<li>In particular, can the error caused by discretizing a domain with non-flat boundary with triangular elements be controlled in some way?</li>
<li>Or is there an actual problem with convergence of FEM in non-polyhedral (say, Lipschitz) domains?</li>
</ul>Thu, 22 Mar 2018 05:53:12 -0600Math Overflow Recent Questions: A Liouville theorem for a uniformly elliptic equation in divergence form
https://mathoverflow.net/questions/295841/a-liouville-theorem-for-a-uniformly-elliptic-equation-in-divergence-form
<p>I would like to know if there exists a Liouville theorem for solutions $u : \mathbb{R}^n \to \mathbb{R}$ of uniformly elliptic equations of the kind
$$
D_i \left( a_{ij} D_j u \right) + b_i D_i u = 0.
$$
I assume the coefficients $a_{ij},b_i \in C^{\infty}(\mathbb R^n) \cap L^{\infty}(\mathbb{R}^n)$. </p>
<p>Any hint/reference would be highly appreciated! </p>Thu, 22 Mar 2018 04:31:16 -0600Math Overflow Recent Questions: Alfred van der Poorten--rational functions paper
https://mathoverflow.net/questions/295840/alfred-van-der-poorten-rational-functions-paper
<p>Does anybody has a copy of the following paper:
Alfred van der Poorten, Some facts that should be better known, especially about rational functions; Number Theory and Applications”, Richard A. Mollin (ed.), Kluwer Academic Publishers, Dordrecht, 1989?</p>Thu, 22 Mar 2018 04:24:01 -0600Math Overflow Recent Questions: What is suitable function approximation for this case? [on hold]
https://mathoverflow.net/questions/295839/what-is-suitable-function-approximation-for-this-case
<p>I have following expression <br>
$\frac{P(x, t+dt)}{(dt)^2} $<br>
And i need to get rid of division by (dt)², any ideas how to approximate function?</p>Thu, 22 Mar 2018 04:04:41 -0600