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curious little things - aggregated feedsenMath Overflow Recent Questions: examples of MF algebras
https://mathoverflow.net/questions/325645/examples-of-mf-algebras
<p>Can anyone show me concrete examples of unital MF algebra and non-unital MF algebra resepctively? Thanks!<img src="https://i.stack.imgur.com/YN4oN.jpg" alt="enter image description here"></p>Sun, 17 Mar 2019 14:46:01 -0600Math Overflow Recent Questions: Power series ring $R[[X_1,\ldots,X_d]]$ over a U.F.D. $R$
https://mathoverflow.net/questions/325643/power-series-ring-rx-1-ldots-x-d-over-a-u-f-d-r
<p>Let <span class="math-container">$R$</span> be a U.F.D. and
<span class="math-container">\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}</span>
Suppose that we have <span class="math-container">$d$</span> elements <span class="math-container">$f_1,\ldots,f_d \in T$</span> and let us consider an ideal <span class="math-container">$J$</span> of <span class="math-container">$T$</span> such that <span class="math-container">$(f_1,\ldots,f_d) \subset J$</span> and the following three conditions<span class="math-container">$\colon$</span>
<span class="math-container">\begin{align*}
&
1. \quad \overline{f_i} \,\colon \overset{{\mathrm{def}}}{=} f_i~{\mathrm{mod}}(X_1,\ldots,X_d)~{\mathrm{is~ a~ prime~ element~ of~}} R ~{\mathrm{for~each}}~ 1 \leq i \leq d\\
&
2. \quad T/(f_1,\ldots,f_d) \phantom{a} {\mathrm{is}}~not~{\mathrm{finite~over}}~ R \\
&
3. \quad T/J \phantom{a} {\mathrm{is~finite~over}}~R.
\end{align*}</span></p>
<h2>Q. Does the following equality hold<span class="math-container">$\colon$$\phantom{A}$${\mathrm{ht}}(J) > d$$\,$</span>?</h2>Sun, 17 Mar 2019 13:54:23 -0600Math Overflow Recent Questions: Structural Stability on Compact $2$-Manifolds with Boundary
https://mathoverflow.net/questions/325642/structural-stability-on-compact-2-manifolds-with-boundary
<p>I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact <span class="math-container">$2$</span>-manifolds with boundary. </p>
<p>Let <span class="math-container">$M^2$</span> be a compact connected 2-manifold and <span class="math-container">$\mathfrak{B}$</span> the space of vector fields with the <a href="https://en.wikipedia.org/wiki/Whitney_topologies" rel="nofollow noreferrer"><span class="math-container">$\mathcal{C}^1$</span> topology</a>. The paper <a href="https://www.sciencedirect.com/science/article/pii/0040938365900182" rel="nofollow noreferrer">"M. M. Peixoto - Structural stability on two-dimensional manifolds"</a> defines structural stability as</p>
<p><a href="https://i.stack.imgur.com/Wyrgq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Wyrgq.png" alt="enter image description here"></a></p>
<p>(By an <span class="math-container">$\varepsilon$</span>-homeomorphism of <span class="math-container">$M^2$</span> onto itself we understand a homeomorphism which moves each point by less than <span class="math-container">$\varepsilon$</span>, using a Riemannian metric. However, I'm not interested in this condition, we can change the word <span class="math-container">$\varepsilon$</span>- homomorphism by just homeomorphism)</p>
<p>Using this definition Peixoto was able to demonstrate the following theorem:</p>
<p><a href="https://i.stack.imgur.com/f9yyB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/f9yyB.png" alt="enter image description here"></a></p>
<blockquote>
<p><strong>My Question:</strong> I would like to know if someone knows a result, like the above, about
Structural Stability in a compact <span class="math-container">$2$</span>-manifold with boundary.</p>
</blockquote>
<hr>
<h2>Just a few comments about my search for the result</h2>
<p>Searching online I found the paper <a href="https://www.sciencedirect.com/science/article/pii/0022039680900832" rel="nofollow noreferrer">Clark Robison - Structural stability on manifolds with boundary</a>, however, the author defines a strange topology using the flows of the vector field, and I don't know if it is the same topology as in the definition 1.</p>
<p>It is also worth to mention that on the paper "M.C. Peixoto and M.M. Peixoto - Structural Stability in the Plane with Enlarged Boundary Condition", the following result was proved. Let <span class="math-container">$G\subset \mathbb{R}^2$</span> be a compact region, such that the boundary <span class="math-container">$L$</span> of <span class="math-container">$G$</span> is a <span class="math-container">$\mathcal{C}^1$</span> simple curve, then <span class="math-container">$X$</span> (a vector field in <span class="math-container">$G$</span>) is structurally stable <span class="math-container">$\Leftrightarrow$</span> <span class="math-container">$X$</span> satisfies conditions <span class="math-container">$A$</span> and <span class="math-container">$B$</span>.
<a href="https://i.stack.imgur.com/ADpcb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ADpcb.png" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/5aJ9k.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5aJ9k.png" alt="enter image description here"></a></p>
<hr>
<p>Can anyone help me?</p>Sun, 17 Mar 2019 13:49:04 -0600Math Overflow Recent Questions: How to clacaulte my age on leap year [on hold]
https://mathoverflow.net/questions/325640/how-to-clacaulte-my-age-on-leap-year
<p>I have asked Question this reptaly but not use knowone one had not gave me satsifed answer I have born on 18.03.2000 on 7.30 pm my question is How to clacaulte my age? I have read one website in that website in that they have put who were born on march 18.3.2000 their age will be 16 going ti complete 17will start this is true in information please answer my question</p>Sun, 17 Mar 2019 13:13:44 -0600Math Overflow Recent Questions: The moduli scheme of "$\nu$-canonically embeded curves"
https://mathoverflow.net/questions/325636/the-moduli-scheme-of-nu-canonically-embeded-curves
<p>This is related to the proposition 5.1. of Mumford's GIT.
It states that:</p>
<blockquote>
<p>There is a unique subscheme <span class="math-container">$H$</span> in the Hilbert scheme <span class="math-container">$Hilb_{\mathbb{P}^n}^{P(x)}$</span> such that, for any morphism <span class="math-container">$f : S \to Hilb$</span>, <span class="math-container">$f$</span> factors through <span class="math-container">$H$</span> iff:</p>
<p>i) the induced subscheme <span class="math-container">$\Gamma$</span> in <span class="math-container">$\mathbb{P}^n_S$</span> is a curve of genus <span class="math-container">$g$</span> over <span class="math-container">$S$</span>,</p>
<p>ii) the invertible sheaf <span class="math-container">$\mathscr{O}_{\mathbb{P}}(1)|_{\Gamma}$</span> is isomorphic to <span class="math-container">$\Omega_{\Gamma/S}^{\otimes \nu} \otimes \pi^* \mathscr{L}$</span> for some invertible sheaf <span class="math-container">$\mathscr{L}$</span> on <span class="math-container">$S$</span>,</p>
<p>iii) for every geometric point <span class="math-container">$s$</span> of <span class="math-container">$S$</span>, the fibre <span class="math-container">$\Gamma_s$</span> spans <span class="math-container">$\mathbb{P}_s^n$</span>.</p>
</blockquote>
<p>(Now the induced scheme is <span class="math-container">$\Gamma = W \times_{Hilb} S$</span>, for the universal object <span class="math-container">$W$</span> of the functor <span class="math-container">$Hilb(-)$</span>, and <span class="math-container">$\nu \ge 3$</span>, <span class="math-container">$P(x) = (2\nu x - 1)(g-1)$</span>, <span class="math-container">$n = P(1) -1 $</span>, <span class="math-container">$\pi : \Gamma \to S$</span>.
And a curve of genus <span class="math-container">$g$</span> is a proper smooth morphism whose geometric fibres are connected curves and of genus <span class="math-container">$g$</span>.)</p>
<p>And in Deligne-Mumford's "the irreducibility of the space of curves of given genus", the authors says:</p>
<blockquote>
<p>Following standard arguments (the proof of 5.1. of GIT), it is easy to prove that there is a subscheme <span class="math-container">$H \subset Hilb_{\mathbb{P}^{5g - 6}}^{P(x)}$</span> of "all" tri-canonically embedded stable curves.
To be precise, there is an isomorphism of functors:
<span class="math-container">$$Hom(S, H) \cong \{ \text{stable curve } \pi : C \to S, \text{plus isomorphisms } \mathbb{P}(\pi_*(\omega_{C/S}^{\otimes 3})) \cong \mathbb{P}_S^{5g-6} \}.$$</span></p>
</blockquote>
<p>This is the analogy of the first statement for <span class="math-container">$\nu = 3$</span>.</p>
<p>But for me, these are a priori slightly different.
I think that, if (ii) and (iii) is equivalent to <span class="math-container">$\mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu})) \cong \mathbb{P}_S^{n}$</span>, these two statements are the same.
But I can't show this.
So my question is:</p>
<blockquote>
<p>Let <span class="math-container">$f : S \to Hilb_{\mathbb{P^n}}^{P(x)}$</span> be a morphism and <span class="math-container">$\pi : \Gamma \to S$</span> the corresponding scheme. Assume this <span class="math-container">$\pi$</span> is an <span class="math-container">$S$</span>-curve.
Then (ii) and (iii) of 5.1. iff <span class="math-container">$\mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu})) \cong \mathbb{P}_S^{n}$</span> over <span class="math-container">$S$</span>?</p>
</blockquote>
<p>Here is what I tried so far:
Let <span class="math-container">$i : \Gamma \to \mathbb{P}$</span> be the immersion, <span class="math-container">$p : \mathbb{P} \to S$</span> the projection.
Then (iii) is quivalent to the canonical map <span class="math-container">$p_*\mathscr{O}_{\mathbb{P}}(1) \to \pi_* i^* \mathscr{O}_{\mathbb{P}}(1)$</span> is an isomorphism.
So for "only if", it's sufficient to show that <span class="math-container">$\mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu} \otimes \pi^* \mathscr{L})) \cong \mathbb{P}(\pi_*(\Omega_{\Gamma/S}^{\otimes \nu})) $</span>.
And for the converse, I showed that (iii) holds.</p>
<p>Thank you very much!</p>Sun, 17 Mar 2019 12:28:46 -0600Math Overflow Recent Questions: Induction principle from naturality
https://mathoverflow.net/questions/325633/induction-principle-from-naturality
<p>The Church encoding of the sum type <span class="math-container">$A + B$</span> goes like that:</p>
<p><span class="math-container">$$\prod_{X:\mathsf{Set}_{\mathcal{U}}} (A\to X)\to (B\to X) \to X$$</span></p>
<p>But it lacks an induction principle.</p>
<p>According to this <a href="https://homotopytypetheory.org/2018/11/26/impredicative-encodings-part-3/" rel="nofollow noreferrer">blog</a> article by <a href="https://mathoverflow.net/users/49/mike-shulman">Mike Shulman</a>, one can recover the induction principle by extending the above encoding with a naturality condition:</p>
<p><span class="math-container">$
\sum_{\alpha:\prod_{X:\mathsf{Set}_{\mathcal{U}}} (A\to X)\to (B\to X) \to X} \prod_{X,Y:\mathsf{Set}_{\mathcal{U}}} \prod_{f:X\to Y} \prod_{h:A\to X} \prod_{k:B\to X} f(\alpha_X(h,k)) = \alpha_Y(f\circ h, f\circ k)
$</span></p>
<p>In Coq syntax (with the option <code>-impredicative-set</code>), it gives:</p>
<pre><code>Variables A B : Set.
Record sum : Set := {
type :> forall (X : Set), (A -> X) -> (B -> X) -> X;
natural :
forall (X Y : Set) (f : X -> Y) (inl : A -> X) (inr : B -> X),
f (sum_carrier X inl inr) =
sum_carrier Y (fun a => f (inl a)) (fun b => f (inr b))
}.
</code></pre>
<p>One can the define the constructors <code>inl</code> and <code>inr</code>:</p>
<pre><code>Definition inl (a : A) : sum :=
{| type := fun X l r => l a; natural := ltac:(reflexivity) |}.
Definition inr (b : B) : sum :=
{| type := fun X l r => r b; natural := ltac:(reflexivity) |}.
</code></pre>
<p>Then the induction principle goes like that:</p>
<pre><code>Definition sum_ind (P : sum -> Set)
(Hleft : forall a, P (inl a)) (Hright : forall b, P (inr b)) (s : sum) :
P s.
</code></pre>
<p>However I cannot figure out how to use the naturality of <code>s</code> to conclude <code>P s</code>, even by daring using the infamous option <code>-type-in-type</code> allowing me to instantiate the morphism <code>f</code> with <code>P</code>.</p>
<p>How does the added naturality condition allow for proving the induction principle?</p>Sun, 17 Mar 2019 11:45:44 -0600Math Overflow Recent Questions: Box dimension of the graph of an increasing function
https://mathoverflow.net/questions/325632/box-dimension-of-the-graph-of-an-increasing-function
<p>This <a href="https://mathoverflow.net/questions/304573/hausdorff-dimension-of-the-graph-of-an-increasing-function">Hausdorff dimension of the graph of an increasing function</a> shows that:</p>
<blockquote>
<p>Let <span class="math-container">$f$</span> be a continuous, strictly increasing function from <span class="math-container">$[0,1]$</span> to
itself with <span class="math-container">$f(0)=0, f(1)=1$</span>. Then <span class="math-container">$dim_H \; G = 1$</span> where <span class="math-container">$G$</span> is the graph of <span class="math-container">$f$</span>. </p>
</blockquote>
<p>I have at hand the Casino function, described as follows in Massopoust's <em>Interpolation and Approximation with Splines and Fractals</em>:</p>
<blockquote>
<p>Let <span class="math-container">$X = [0,1] \times \mathbb{R}$</span>, <span class="math-container">$N = 4$</span> and <span class="math-container">$Y = \{(x_v,y_v):0 = x_0 < \ldots x_N = 1, 0 = y_0 < \ldots < y_N = 1\}$</span>. Define an IFS by <span class="math-container">$f_i(x,y) =
\begin{pmatrix}
x_i-x_{i-1} & 0 \\
0 & y_i - y_{i-1}
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
+
\begin{pmatrix}
x_{i-1} \\
y_{i-1}
\end{pmatrix}
$</span> for <span class="math-container">$i = 1, \ldots, N$</span>.</p>
<p>The associated RB operator <span class="math-container">$T$</span> is contractive and its unique fixed point is called a Casino function <span class="math-container">$c:[0,1] \to [0,1]$</span>. These functions are monotone increasing and therfore <span class="math-container">$dim_H \; graph(c) = \dim_B \; graph(c) = 1$</span>.</p>
</blockquote>
<p>I was wondering how can I show that <span class="math-container">$dim_B \; graph(c) = 1$</span> and whether there is a general argument establishing:</p>
<blockquote>
<p>Let <span class="math-container">$f$</span> be a continuous, strictly increasing function from <span class="math-container">$[0,1]$</span> to
itself with <span class="math-container">$f(0)=0, f(1)=1$</span>. Then <span class="math-container">$dim_B \; G = 1$</span> where <span class="math-container">$G$</span> is the graph of <span class="math-container">$f$</span>. </p>
</blockquote>
<p>I don't find an argument stablishing <span class="math-container">$dim_B \; G \le 1$</span>.</p>Sun, 17 Mar 2019 11:40:19 -0600Math Overflow Recent Questions: Estimator example cube
https://mathoverflow.net/questions/325629/estimator-example-cube
<p>We make subsequent throws of a fake cubic cube for which the probability of falling out six is 1/6 - epsilon, the probability of falling out of one is 1/6 + epsilon and the others eyes drop out with the same probability 1/6. Provide a consistent and unbalanced estimator epsilon parameter and calculate its variance. I need help i start my trip with this and it is so hard for me :/</p>Sun, 17 Mar 2019 11:04:53 -0600Math Overflow Recent Questions: Hausdorff dimension of the boundary of fibres of Lipschitz maps
https://mathoverflow.net/questions/325624/hausdorff-dimension-of-the-boundary-of-fibres-of-lipschitz-maps
<p>Let <span class="math-container">$f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$</span> be a Lipschitz map.</p>
<blockquote>
<p>Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of <span class="math-container">$f$</span>? I.e. do we have an upper bound for
<span class="math-container">$$ \sup_{y\in \mathbb{R}^{n-k}} \dim_H(\partial f^{-1}(\{y\})) ?$$</span> </p>
</blockquote>
<p>Theorem 2.5 in [1] tells us, that for almost every <span class="math-container">$y\in \mathbb{R}^{n-k}$</span> we have that <span class="math-container">$\dim_H(f^{-1}(y))\leq k$</span>. This tells us
<span class="math-container">$$ \text{essup}_{y\in \mathbb{R}^{n-k}} \dim_H(\partial f^{-1}(\{y\})) \leq k.$$</span>
Can we pass to the supremum? And are there even better bounds? I mean, I used <span class="math-container">$\partial f^{-1}(\{y\})\subseteq f^{-1}(\{y\})$</span> as <span class="math-container">$f$</span> is continuous and the monotonicity of the Hausdorff dimension, but I guess that one can do better than this.</p>
<p><strong>[1] G. Alberti, S. Bianchini, G. Crippa,</strong> Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples.
<em>Ann. Sc. Norm. Super. Pisa Cl. Sci.</em> (5) 12 (2013), no. 4, 863–902. </p>Sun, 17 Mar 2019 10:17:04 -0600Math Overflow Recent Questions: Generating Machin-like formulas with inverse hyperbolic tangents for logarithms
https://mathoverflow.net/questions/325623/generating-machin-like-formulas-with-inverse-hyperbolic-tangents-for-logarithms
<p><a href="https://en.wikipedia.org/wiki/Machin-like_formula" rel="nofollow noreferrer">Machin-like formulas</a> for <span class="math-container">$\pi$</span> have the following general form:
<span class="math-container">$$c_{0} \frac{\pi}{4}=\sum_{n=1}^{N} c_{n} \arctan \frac{a_{n}}{b_{n}}$$</span>
Recently browsing through this question <a href="https://mathoverflow.net/questions/125629/machin-like-formulas-for-logarithms">here</a>, I really became very curious in finding out Machin Type formula's for logarithms,of course using the inverse hyperbolic tangent function. </p>
<p>Now, its relatively easy and well known to find out Machin Type formulas for <span class="math-container">$\pi$</span> using complex numbers the original equation can be written as <span class="math-container">$$(1+i)^{c_{0}}=\prod_{n=1}^{N}\left(b_{n}+a_{n} i\right)^{c_{n}}$$</span> and then suitable <span class="math-container">$a_n$</span> and <span class="math-container">$b_n$</span> could be found using algorithms like branch and bound search, with arbitrary small <span class="math-container">$\frac{a_n}{b_n}$</span> for efficient computation.</p>
<p>Now what I am really curious about is the find whether there are methods for obtaining such a series for <span class="math-container">$\log k$</span> which could be in the form of <span class="math-container">$$c_{0} \log k=\sum_{n=1}^{N} c_{n} \tanh ^{-1}\frac{a_{n}}{b_{n}}$$</span>
For arbitary <span class="math-container">$k$</span>?</p>Sun, 17 Mar 2019 10:02:13 -0600