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curious little things - aggregated feedsenMath Overflow Recent Questions: Matrix decomposition in a specific form
https://mathoverflow.net/questions/331834/matrix-decomposition-in-a-specific-form
<p>Can we prove that for any real valued <span class="math-container">$d\times d$</span> matrix <span class="math-container">$A$</span>, <span class="math-container">$A$</span> can be decomposed to finite product of such matrices</p>
<p><span class="math-container">$A=\prod_{i=1}^n (I+R_i)$</span></p>
<p>where <span class="math-container">$I$</span> is the identity matrix and <span class="math-container">$rank(R_i)=1$</span>.</p>
<p>As far as I know, if there is <span class="math-container">$LDU$</span> or <span class="math-container">$LU$</span> decomposition to <span class="math-container">$A$</span>, then we can easily figure out each <span class="math-container">$R_i$</span>.</p>Fri, 17 May 2019 15:50:54 -0600Math Overflow Recent Questions: Does the pure motive determine the Voevodsky motive?
https://mathoverflow.net/questions/331832/does-the-pure-motive-determine-the-voevodsky-motive
<p>I do not quite understand the construction of Voevodsky motives yet. Let <span class="math-container">$k$</span> be a field (possibly not algebraically closed), <span class="math-container">$X$</span> be a connected smooth projective <span class="math-container">$k$</span>-scheme. Does the motive of <span class="math-container">$X$</span> in the abelian category of pure numerical motives determine its Voevodsky motive? The coefficients are either <span class="math-container">$\mathbb{Z}$</span> or <span class="math-container">$\mathbb{Q}$</span> (the answer should address both). Please provide a reference.</p>Fri, 17 May 2019 15:33:06 -0600Math Overflow Recent Questions: Box counting dimension of a set and Lipschitz functions
https://mathoverflow.net/questions/331831/box-counting-dimension-of-a-set-and-lipschitz-functions
<p>If <span class="math-container">$f$</span> is Lipschitz, then the following holds for the Hausdorff dimension:
<span class="math-container">$$\dim_H f(A) \le \dim_H A.$$</span></p>
<p>Is the same true for the box counting dimension?</p>Fri, 17 May 2019 15:15:18 -0600Math Overflow Recent Questions: Smooth projective varieties with equal integral Voevodsky motives and different fundamental groups
https://mathoverflow.net/questions/331830/smooth-projective-varieties-with-equal-integral-voevodsky-motives-and-different
<p>Let <span class="math-container">$k$</span> be an algebraically closed field, <span class="math-container">$DM$</span> be the category of <span class="math-container">$\mathbb{Z}$</span>-motives over <span class="math-container">$k$</span>. Are there two smooth projective <span class="math-container">$k$</span>-schemes that have isomorphic motives in <span class="math-container">$DM$</span> but have non-isomorphic etale fundamental groups? Note that for <span class="math-container">$\mathbb{Q}$</span>-motives the question <a href="https://mathoverflow.net/a/331687/138661">has been addressed</a> on MO.</p>Fri, 17 May 2019 15:14:18 -0600Math Overflow Recent Questions: Expected value of a random variable conditioned on a positively correlated event
https://mathoverflow.net/questions/331829/expected-value-of-a-random-variable-conditioned-on-a-positively-correlated-event
<p>I have a random variable <span class="math-container">$x \in [a, b]$</span> with PDF <span class="math-container">$f(x)$</span> and an event <span class="math-container">$E$</span> which satisfies the following property for any <span class="math-container">$x'<b$</span>.</p>
<p><span class="math-container">$$\Pr[E|x > x'] \geq \Pr[E]$$</span></p>
<p>My question is whether or not the following inequality holds.</p>
<p><span class="math-container">$$\int_{a}^{b} uf(u)\Pr[E|x=u]du \geq \Pr[E]\int_{a}^{b} uf(u)du$$</span></p>Fri, 17 May 2019 15:01:46 -0600Math Overflow Recent Questions: Finite generation of the image of the induced homomorphism on homotopy groups of H-spaces
https://mathoverflow.net/questions/331828/finite-generation-of-the-image-of-the-induced-homomorphism-on-homotopy-groups-of
<p>Let <span class="math-container">$f:X\rightarrow Y$</span> be a map of <span class="math-container">$H$</span>-spaces such that image of homology groups <span class="math-container">$H_i(X,\mathbb{Z})$</span> for <span class="math-container">$i\geq 1$</span> under <span class="math-container">$f_*$</span> are finitely generated. Does this imply that the image of homotopy groups under <span class="math-container">$f_*$</span> are also finitely generated?
Few tips that might be helpful:</p>
<p>This is true for <span class="math-container">$\pi_1$</span> since it is Abelian.</p>
<p>This is true rationally.</p>
<p>This fact is not true: If <span class="math-container">$f_*$</span> is zero on homologies then it is so on homotopy groups. <a href="https://math.stackexchange.com/q/3229869">Here</a> is the counterexample if you like to see it.</p>Fri, 17 May 2019 14:58:41 -0600Math Overflow Recent Questions: Papers containing Ihara avoidance arguments
https://mathoverflow.net/questions/331826/papers-containing-ihara-avoidance-arguments
<p>I am trying to understand some of the recent research in number theory. There is apparently a certain lemma, called Ihara's lemma, which can be established in some contexts and is unknown in other contexts. Occasionally, one can still prove its consequences unconditionally. This acrobatics is called Ihara avoidance. What are some important papers containing arguments like this? Also, what are the papers containing Ihara avoidance-type argument that are technically easy to understand? </p>
<p>I feel like I will never be able to penetrate this sea of indices so if there is some
easy paper which relies on that idea, I could try to understand it well and other papers then would become less scary. </p>Fri, 17 May 2019 14:37:57 -0600Math Overflow Recent Questions: Generalization of Cauchy's eigenvalue interlacing theorem?
https://mathoverflow.net/questions/331825/generalization-of-cauchys-eigenvalue-interlacing-theorem
<p>Cauchy's Interlacing Theorem says that given an <span class="math-container">$n \times n$</span> symmetric matrix <span class="math-container">$A$</span>, let <span class="math-container">$B$</span> be an <span class="math-container">$(n-1) \times (n-1)$</span> principal submatrix of it, then the eigenvalues of <span class="math-container">$A$</span> and those of <span class="math-container">$B$</span> interlace.</p>
<p>Using this property, one can obtain a lower bound on the <span class="math-container">$k$</span>-th largest eigenvalue of a <span class="math-container">$t \times t$</span> principal submatrix of <span class="math-container">$A$</span>, using the <span class="math-container">$(k+n-t)$</span>-th largest eigenvalue of <span class="math-container">$A$</span>. This lower bound is best possible, for example when <span class="math-container">$A$</span> is diagonal. But for many interesting (fixed) matrices, such bound is usually far from being optimal. For example, let <span class="math-container">$$A=\begin{bmatrix}
0 & 1 & 1 & 0\\
1 & 0 & 0 & 1\\
1 & 0 & 0 & 1\\
0 & 1 & 1 & 0
\end{bmatrix}$$</span></p>
<p>The eigenvalues of <span class="math-container">$A$</span> are <span class="math-container">$2, 0, 0, -2$</span>. So if we would like to bound from below the largest eigenvalue of its <span class="math-container">$3 \times 3$</span> principal submatrix using Cauchy's Theorem, we only get a lower bound of <span class="math-container">$0$</span>. However it is straightforward to check that it is always at least <span class="math-container">$\sqrt{2}$</span>.</p>
<p>I am wondering if there is a more "quantitative" Interlacing Theorem, say if your matrix satisfies some additional properties (non-negative, binary, etc.), then one can obtain a better lower bound on the <span class="math-container">$k$</span>-th largest eigenvalue of a <span class="math-container">$t \times t$</span> principal submatrix of <span class="math-container">$A$</span>?</p>Fri, 17 May 2019 14:20:41 -0600Math Overflow Recent Questions: Learning maths for machine learning with high school freshman background [on hold]
https://mathoverflow.net/questions/331822/learning-maths-for-machine-learning-with-high-school-freshman-background
<p>I am from Europe and I visit the "High School" there. Let me walk you through what we did already in school:</p>
<ul>
<li>Linear Functions</li>
<li>Fraction equations (Both Algebra and Graphical Solutions)</li>
<li>Power laws</li>
</ul>
<p>These are the main algebra things. Ofc we solved equations and all that.
But I am into machine learning and I know how to code the models and all that. And now I would just like to know how I should continue with my journey ? </p>
<p>Thanks in advance.
Leon</p>Fri, 17 May 2019 14:10:46 -0600Math Overflow Recent Questions: Asking SnapPy for core curves after surgery
https://mathoverflow.net/questions/331820/asking-snappy-for-core-curves-after-surgery
<p>Suppose I give SnapPy a cusped hyperbolic 3-manifold (using, say, the link editor) and specify some filling. SnapPy can then provide a presentation of the fundamental group of the filled manifold. Can it tell me what the core curve of the added solid torus is, as a word in the fundamental group?</p>Fri, 17 May 2019 13:45:00 -0600