Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2023-10-04T12:06:28Zhttps://mathoverflow.net/feedshttps://creativecommons.org/licenses/by-sa/4.0/rdfhttps://mathoverflow.net/q/4558350Mapping spaces in complete Segal spaces and quasi-categoriesChrisLazdahttps://mathoverflow.net/users/136472023-10-04T11:46:00Z2023-10-04T11:46:00Z
<p>Complete Segal spaces and quasi-categories are two common models for the theory of <span class="math-container">$(\infty,1)$</span>-categories, and both are equipped with a natural notion of hom spaces. For complete Segal spaces, which in particular are simplicial spaces <span class="math-container">$X\colon \Delta^{\rm op} \to \mathcal{S}$</span>, we think of <span class="math-container">$X_0$</span> as the space of object, <span class="math-container">$X_1$</span> as the space of morphism &c. Thus if <span class="math-container">$x,y\in X_{0,0}$</span>, then the space of homs should be <span class="math-container">$x\times^h_{X_0} X_1 \times^h_{X_0} y$</span>.</p>
<p>On the other hand, if <span class="math-container">$\mathcal{C}$</span> is a quasi-category, and <span class="math-container">$x,y\in \mathcal{C}_0$</span> are objects in <span class="math-container">$\mathcal{C}$</span>, then the mapping space from <span class="math-container">$x$</span> to <span class="math-container">$y$</span> is defined (for example, there are other versions) by the rule <span class="math-container">${\rm Map}_\mathcal{C}(x,y)_n=\left\{ \sigma\colon \Delta^n\times\Delta^1 \to \mathcal{C} \mid \sigma|_{\Delta^n\times\{0\}}\equiv x,\;\sigma|_{\Delta^n\times\{1\}}\equiv y \right\}$</span>.</p>
<p>If <span class="math-container">$X$</span> is a complete Segal space, then we can associate to it a quasi-category <span class="math-container">$\mathcal{C}(X)$</span> via the rule <span class="math-container">$\mathcal{C}(X)_n:=X_{n,0}$</span>, and this induces an equivalence between the <span class="math-container">$(\infty,1)$</span>-category of complete Segal spaces and that of quasi-categories.</p>
<p>In particular, this functor should match up the hom spaces on both sides - we should have a homotopy equivalence <span class="math-container">${\rm Map}_{\mathcal{C}(X)}(x,y)\simeq x\times^h_{X_0} X_1 \times^h_{X_0} y$</span> whenever <span class="math-container">$X$</span> is a complete Segal space and <span class="math-container">$x,y\in X_{0,0}$</span>. Unfortunately, I'm a bit of a novice in higher category theory and don't know the literature well enough to know where to look for such a result.</p>
<p>So my question is the following: does anyone know where this equivalence of mapping spaces is explicitly proved (or possibly with <span class="math-container">${\rm Map}$</span> replaced with any equivalent notion of mapping spaces for quasi-categories, such as <span class="math-container">${\rm Map}^{\rm R}$</span> or <span class="math-container">${\rm Map}^{\rm L}$</span>)?</p>
https://mathoverflow.net/q/4558341Proof of global Peano existence theorem in ZF?Mikhail Katzhttps://mathoverflow.net/users/281282023-10-04T11:15:57Z2023-10-04T11:15:57Z
<p>By global Peano existence theorem I mean the existence of a maximal interval of solution of a first order ODE <span class="math-container">$x'=f(x,t)$</span> with continuous <span class="math-container">$f$</span>.
The proofs of the global Peano Theorem found in the literature often simply appeal to Zorn’s Lemma; eg. Theorem 4.7 in Ganesh</p>
<blockquote>
<p>S. S. Ganesh, Lecture Notes on Ordinary Differential Equations, Annual Foundation School IIT Kanpur, December 3 - 28, 2007, 34 pp. <a href="https://www.math.iitb.ac.in/%7Esiva/afs07.pdf" rel="nofollow noreferrer">https://www.math.iitb.ac.in/~siva/afs07.pdf</a></p>
</blockquote>
<p>The more careful proofs depend on ADC, usually without mentioning it explicitly. Hale</p>
<blockquote>
<p>J. Hale, Ordinary Differential Equations, 2nd Edition, R. E. Krieger Publ. Co., Florida, 1980</p>
</blockquote>
<p> in his proof of global Peano Theorem (Theorem 2.1, p. 17) writes:</p>
<blockquote>
<p>“...there is a monotone increasing sequence {bn} constructed as above so that the solution <span class="math-container">$x(t)$</span> of (1.1) on <span class="math-container">$[a, b]$</span> has
an extension to the interval <span class="math-container">$[a, b_n]$</span> and <span class="math-container">$(b_n, x(b_n))$</span> is not in <span class="math-container">$V_n$</span>. Since
the <span class="math-container">$b_n$</span> are bounded above, let <span class="math-container">$\omega = \lim_{n\to\infty} b_n$</span>. It is clear that <span class="math-container">$x$</span> has
been extended to the interval <span class="math-container">$[a, \omega)$</span>...” </p>
</blockquote>
<p>What is actually clear is that his construction yields solutions <span class="math-container">$x_n(t)$</span> on <span class="math-container">$[a, b_n]$</span> for each <span class="math-container">$n$</span>, and each
<span class="math-container">$x_n(t)$</span> has extensions to some <span class="math-container">$x_{n+1}(t)$</span>. ADC is needed to justify the
existence of <span class="math-container">$x(t)$</span>.
Similarly Hartman</p>
<blockquote>
<p>P. Hartman, Ordinary Differential Equations, 2nd Edition, SIAM, Philadelphia, 2002</p>
</blockquote>
<p>in the proof of II, 3.1, p. 13, constructs an increasing sequence <span class="math-container">$\{b_n\}$</span> such that any solution on <span class="math-container">$[a, b_n]$</span>
has an extension to a solution on <span class="math-container">$[a, b_{n+1}]$</span>. ADC is needed to justify
the existence of a solution on <span class="math-container">$[a, \omega^+]$</span> for <span class="math-container">$\omega^+ = \lim_{n\to\infty} b_n$</span>. In the proof of III, Lemma 3.1, a key step to the proof of III, 3.1 (Osgood’s Theorem),
ACC is used implicitly to choose the sequence <span class="math-container">$\{u_n(t)\}$</span>.
Similar unacknowledged use of ADC appears on pp. 355–356 in</p>
<blockquote>
<p>J. Kurzweil, Ordinary Differential Equations. Introduction to the Theory of Ordinary Differential Equations in the Real Domain, translated from the Czech
by M. Basch, Studies in Applied Mechanics, 13, Elsevier Scientific Publishing,
Amsterdam, 1986.</p>
</blockquote>
<p>Is there a published proof of global Peano existence that's valid in ZF?</p>
https://mathoverflow.net/q/4558322Parametrizing polynomials with given Galois groupT. Combothttps://mathoverflow.net/users/1273432023-10-04T10:00:42Z2023-10-04T10:08:55Z
<p>Consider a transitive group <span class="math-container">$G \subset S_n$</span>, and the set <span class="math-container">$E$</span> of polynomials in <span class="math-container">$\mathbb{K}[x]$</span> of degree <span class="math-container">$n$</span> with Galois group <span class="math-container">$\subset G$</span>. I am looking for a rational surjective mapping <span class="math-container">$\varphi: \mathbb{K}^{n+1} \rightarrow E$</span>. Is the Zariski closure of <span class="math-container">$E$</span> always of dimension <span class="math-container">$n+1$</span>? Does such rational <span class="math-container">$\varphi$</span> always exists, at least for a field <span class="math-container">$\mathbb{K}$</span> a finite extension of <span class="math-container">$\mathbb{Q}$</span>?</p>
<p>I know that what I am asking is stronger that the inverse Galois problem, but maybe a negative answer is possible. For small groups, a positive answer is interesting. For <span class="math-container">$G=A_3$</span>, it is easy a to build such parametrization on <span class="math-container">$\mathbb{Q}(i\sqrt{3})$</span>. I would be particularly interested in the cases <span class="math-container">$D_5 \subset S_5$</span> and <span class="math-container">$D_7 \subset S_7$</span>.</p>
https://mathoverflow.net/q/4558290Expectation of quotient using exponential concentrationuser18722294https://mathoverflow.net/users/5145172023-10-04T08:09:04Z2023-10-04T08:09:04Z
<p>I want to determin <span class="math-container">$\mathbb{E}_{\alpha}(\frac{X(\alpha)}{q(\alpha) p(\alpha)})$</span>. I know separately the expectation of the nominator respectively the denominator. I know that each <span class="math-container">$q(\alpha)$</span> and <span class="math-container">$p(\alpha)$</span> concentrates towards its mean: <span class="math-container">$P(|q(\alpha)-\mu| \geq \epsilon) \leq \frac{\delta}{\epsilon^2}$</span> where <span class="math-container">$\delta$</span> is exponentially small in the system size. Can I somehow replace <span class="math-container">$q(\alpha)$</span> and <span class="math-container">$p(\alpha)$</span> by its mean in a probabilistic fashion?</p>
https://mathoverflow.net/q/4558280Polynomial equations with linear inequalitiesAlberto Montinahttps://mathoverflow.net/users/42742023-10-04T07:58:05Z2023-10-04T09:54:57Z
<p>Given a set of polynomials equations in variables <span class="math-container">$x_1\dots x_n$</span> and a set of linear inequalities <span class="math-container">$L_k(x_1\dots x_n)\ne0$</span>, is the set of solutions an algebraic set? If it is, what is the corresponding set of polynomials?</p>
<p>PS: if the number of solutions is finite, the answer to the first question is yes. What are the corresponding polynomials?</p>
https://mathoverflow.net/q/4558270relation between $\text{Im}(A)\otimes \text{Im}(B)$ and $\text{Im}(A\otimes B)$Studenthttps://mathoverflow.net/users/1130542023-10-04T07:34:06Z2023-10-04T07:34:06Z
<p>In my previous <a href="https://mathoverflow.net/questions/290956/it-is-true-that-overline-textima-otimes-overline-textimb-subset">question</a>, we have</p>
<blockquote>
<p>If <span class="math-container">$A,B\in \mathcal{L}(H)$</span>, then
<span class="math-container">$$\overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}\subset \overline{\text{Im}(A\otimes B)}.$$</span></p>
</blockquote>
<p>My question now when <span class="math-container">$A, B$</span> are posive operators, do we have
<span class="math-container">$$\text{Im}(A)\otimes \text{Im}(B)=\text{Im}(A\otimes B)?$$</span></p>
https://mathoverflow.net/q/4558260Explicit family of polynomials describing embedded torus in complex projective spacePaul Cussonhttps://mathoverflow.net/users/1436292023-10-04T07:27:25Z2023-10-04T07:27:25Z
<p>This question is cross-posted (with modifications) from <a href="https://math.stackexchange.com/questions/4779029/explicit-family-of-polynomials-describing-embedded-torus-in-complex-projective-s">MSE</a>. The original question is probably unfit for MathOverflow (although a professor I asked said that this is very nontrivial), but I'm hoping my follow-up questions and related discussion are of interest. To give an idea of my weak background, I am currently still going through Griffiths and Harris.</p>
<p>To rehash, let <span class="math-container">$X = \mathbb{C}^n / \Lambda$</span> be a complex torus of complex dimension <span class="math-container">$n$</span>, with the lattice <span class="math-container">$\Lambda$</span> satisfying the Riemann conditions. Knowing that this condition guarantees that <span class="math-container">$X$</span> embeds in <span class="math-container">$\mathbb{CP}^N$</span> for some <span class="math-container">$N$</span>, there exists a family of homogeneous polynomials whose zero locus is the embedded torus.</p>
<p>My original question was to ask for explicit examples of such embedded tori (for <span class="math-container">$n > 1$</span>, as the case of elliptic curves is well known) and their corresponding polynomials. After reviewing the recommended literature I still could not figure out even a simple example, though I'm sure some must be known and there should be a method to directly compute these polynomials given <span class="math-container">$\Lambda$</span>. I did learn in Birkenhake-Lange's textbook that nothing more than cubics are enough.</p>
<p>But it was pointed out to me that there are some subtleties that would make this a difficult task, which lead to my updated questions:</p>
<p>Given <span class="math-container">$X$</span>, what is the smallest integer <span class="math-container">$N$</span> such that <span class="math-container">$X$</span> embeds in <span class="math-container">$\mathbb{CP}^N$</span>? And is this number <span class="math-container">$N$</span> independent of the choice of lattice <span class="math-container">$\Lambda$</span>? I.e. does it only depend on the dimension <span class="math-container">$n$</span> of the torus?</p>
<p>Mention of complete intersection (which I don't know about yet) was brought up, and if I remember correctly, it can be used to argue that any complex dimension <span class="math-container">$2$</span> torus cannot embed in <span class="math-container">$\mathbb{CP}^4$</span>.</p>
<p>At the very least, I would be happy with a reference for a polynomial description of the simplest non-trivial case, <span class="math-container">$n=2$</span>, for some "easy" <span class="math-container">$\Lambda$</span>, just to get me started.</p>
https://mathoverflow.net/q/4558250Continuity of the application mapping optimal control and value functionGaetanohttps://mathoverflow.net/users/5100792023-10-04T07:09:21Z2023-10-04T07:09:21Z
<p>Let suppose that <span class="math-container">$u$</span> is the solution of HJB equation and <span class="math-container">$\alpha^*$</span> is the associated optimal control. My question is very simple:</p>
<p>If we know a sequence <span class="math-container">$(u_n,\alpha_n)_{n\in\mathbb{N}}$</span>, such that:</p>
<p><span class="math-container">$$||u_n-u||_\infty\xrightarrow[]{n\to\infty}0.$$</span></p>
<p>Is there a theorem saying that we have then:</p>
<p><span class="math-container">$$||\alpha_n-\alpha^*||_\infty\xrightarrow[]{n\to\infty}0 \text{ ?}$$</span></p>
<p><strong>My question is willingly a bit vague since I don't know what could be the hypothesis to have this kind of approximation.</strong></p>
<p>Thank you very much beforehand for your help!</p>
https://mathoverflow.net/q/4558242Clique-coclique and uncertaintySevahttps://mathoverflow.net/users/99242023-10-04T06:59:42Z2023-10-04T06:59:42Z
<p>The <a href="https://arxiv.org/pdf/0710.2109.pdf" rel="nofollow noreferrer">clique-coclique inequality</a> states that for a graph <span class="math-container">$G$</span> on <span class="math-container">$n$</span> vertices that is either distance-regular or vertex-transitive, the independence number <span class="math-container">$\alpha(G)$</span> and the clique number <span class="math-container">$\omega(G)$</span> satisfy <span class="math-container">$$ \alpha(G) \omega(G)\le n. $$</span> The <a href="https://www.jstor.org/stable/2101892" rel="nofollow noreferrer">uncertainty inequality</a> (well, one of its numerous variations) states that for an abelian group <span class="math-container">$G$</span> of order <span class="math-container">$n$</span>, and any function <span class="math-container">$f\in L(G)$</span>, the support of <span class="math-container">$f$</span> and that of its Fourier transform <span class="math-container">$\hat f$</span> satisfy <span class="math-container">$$ |\mathrm{supp} f||\mathrm{supp}\hat f|\ge n. $$</span></p>
<p>Even though the inequalities go in opposite directions, they manifest a striking similarity. Is this a mere coincidence, or there is a hidden reason for them to be similar?</p>
https://mathoverflow.net/q/4558221On convergence of convex-concave functionsIosif Pinelishttps://mathoverflow.net/users/367212023-10-04T05:47:41Z2023-10-04T06:11:09Z
<p>Let <span class="math-container">$(f_n)$</span> be a sequence of twice differentiable functions on <span class="math-container">$\mathbb R$</span> such that for each <span class="math-container">$n$</span> there exists some <span class="math-container">$x_n\in\mathbb{R}$</span> such that:</p>
<ul>
<li><span class="math-container">$f_n$</span> is strictly convex on <span class="math-container">$(-\infty,x_n)$</span>,</li>
<li><span class="math-container">$f_n$</span> is strictly concave on <span class="math-container">$(x_n, +\infty)$</span>.</li>
</ul>
<p>Suppose also that <span class="math-container">$f_n$</span> uniformly converges to a twice differentiable function <span class="math-container">$f$</span>.</p>
<p>It was <a href="https://mathoverflow.net/q/455798/36721">conjectured</a> that then the sequence <span class="math-container">$(x_n)$</span> will be convergent.</p>
<p>This conjecture was <a href="https://mathoverflow.net/a/455802/36721">disproved</a>.</p>
<p>The OP then asked in a <a href="https://mathoverflow.net/questions/455798/convergence-of-concave-convex-function/455802#comment1180167_455802">comment</a> whether the additional condition that <span class="math-container">$f''$</span> not vanish on any nonempty interval can help.</p>
<p>Below it will be shown that the answer is still negative, even with the latter additional condition.</p>
https://mathoverflow.net/q/4558210Existence and uniqueness of solutions for continuous and directionally differentiable ODETodd Chavezhttps://mathoverflow.net/users/5139462023-10-04T05:35:27Z2023-10-04T05:35:27Z
<p>Given <span class="math-container">$f:\mathbb{R}^n \to \mathbb{R}^n$</span> continuous and directionally differentiable (i.e., such that the directional derivative of <span class="math-container">$f$</span> exists for any direction) at a neighborhood <span class="math-container">$N$</span> of <span class="math-container">$x_0\in\mathbb{R}^n$</span>, consider the ODE
<span class="math-container">\begin{align*}
\dot{x}=f(x)
\end{align*}</span>
with initial condition <span class="math-container">$x(0)=x_0$</span>. By Peano's existence theorem, we know that this Initial Value Problem has at least one solution. Is this solution unique? Directionally differentiable functions are not necessarily locally Lipschitz so Picard's uniqueness theorem does not apply.
If uniqueness does not hold, are there any known counterexamples?</p>
https://mathoverflow.net/q/4558200Spectrum of the convolution of the Maxwell collision kernel with a distributionVasily Ilinhttps://mathoverflow.net/users/1603862023-10-04T05:29:17Z2023-10-04T05:29:17Z
<p>Given the Maxwell collision kernel <span class="math-container">$A(z) = |z|^2I_d - z \otimes z$</span>, where <span class="math-container">$I$</span> denotes the <span class="math-container">$d\times d$</span> identity matrix and <span class="math-container">$z\otimes z = zz^T$</span> is the outer product, it is easy to see that <span class="math-container">$A(z)$</span> has eigen-values <span class="math-container">$0, |z|^2,...,|z|^2$</span>. So the <span class="math-container">$A(z)$</span> is PSD. Now suppose that <span class="math-container">$u$</span> is an absolutely continuous probability distribution on <span class="math-container">$\mathbb{R}^d$</span> with finite second moment. Intuitively, the convolution <span class="math-container">$A*u$</span> should be positive-definite. What is the lower bound on the smallest eigen-value of
<span class="math-container">$$(A*u)(z) = \int A(y-z)u(y)dy?$$</span>
I think it should be expressed in terms of moments of <span class="math-container">$u$</span>, maybe with a dependence on <span class="math-container">$z$</span>.</p>
<p>Here is my attempt at the calculation:
<span class="math-container">$$(A*u)(z) = I_d \int |z-y|^2 u(y) dy - \int (z-y)\otimes (z-y)u(y)dy.$$</span>
The first term is <span class="math-container">$I_d \mathbb{E}\left[|Y-z|^2\right]$</span>, the second moment of <span class="math-container">$|Y-z|$</span>, where <span class="math-container">$Y$</span> is a random variable with law <span class="math-container">$u$</span>. And the second term is <span class="math-container">$\mathbb{E}\left[(Y - z) \otimes (Y-z)\right]$</span>. The smallest eigen-value of their difference is
<span class="math-container">$$\mathbb{E}\left[|Y-z|^2\right] - ||\mathbb{E}\left[(Y - z) \otimes (Y-z)\right]||_{\text{spec}},$$</span>
where <span class="math-container">$||\cdot||_{\text{spec}}$</span> denotes the spectral norm. I am not sure how to find the spctral norm of <span class="math-container">$\mathbb{E}\left[(Y - z) \otimes (Y-z)\right]$</span>.</p>
https://mathoverflow.net/q/4558089When is $\mathrm{gcd}(k,p^k-1)=1$ true?Martin Brandenburghttps://mathoverflow.net/users/28412023-10-03T22:29:30Z2023-10-04T09:14:52Z
<p>Let <span class="math-container">$p$</span> be a prime. Is there a classification of the numbers <span class="math-container">$k \geq 1$</span> such that <span class="math-container">$\gcd(k,p^k-1)=1$</span>? If not, can we at least produce an explicit infinite subset? What is known about these <span class="math-container">$k$</span>?</p>
<p>For the lack of a better name, let me call <strong><span class="math-container">$k$</span> good for <span class="math-container">$p$</span></strong> if <span class="math-container">$\gcd(k,p^k-1)=1$</span>. Many numbers are good for <span class="math-container">$p=2$</span>. This seems to be related but not identical to the OEIS sequence <a href="https://oeis.org/A049093" rel="nofollow noreferrer">A049093</a>. If <span class="math-container">$p>2$</span>, all good numbers for <span class="math-container">$p$</span> are odd. Most odd numbers are good for <span class="math-container">$3$</span>; for <span class="math-container">$k \leq 1000$</span> the only <span class="math-container">$24$</span> exceptions are <code>39, 55, 117, 165, 195, 253, 273, 275, 351, 385, 429, 495, 507, 585, 605, 663, 715, 741, 759, 819, 825, 897, 935, 975</code>. There is no OEIS sequence for these. For <span class="math-container">$p=5$</span> it is similar, but for <span class="math-container">$p > 5$</span> the story becomes different.</p>
<p><em>Background.</em> I can prove a certain theorem in abstract algebra for all pairs <span class="math-container">$(p,k)$</span> with <span class="math-container">$\gcd(k,p^k-1)=1$</span>. Now, I wonder how many cases I have already covered.</p>
https://mathoverflow.net/q/4558073Girth 5 graphs with diameter 2David Woodhttps://mathoverflow.net/users/259802023-10-03T22:23:03Z2023-10-04T05:16:26Z
<p>Is there an infinite class of graphs with diameter 2, girth 5, and minimum degree at least 2?</p>
<p>Girth 5 is necessary, since otherwise complete bipartite graphs are an answer. Minimum degree at least 2 is necessary, since otherwise stars are an answer.</p>
<p>The Hoffman–Singleton graph (which has 60 vertices) is the largest graph I know with the properties in the question.</p>
<p>The question is interesting if "minimum degree at least 2" is replaced by "minimum degree at least 3", but I think it makes little difference because degree 2 vertices are highly restricted in a graph with diameter 2.</p>
<p>This question is related (but different) to the question <a href="https://mathoverflow.net/questions/355283/girth-and-diameter-of-a-graph-with-minimum-degree-at-least-3">Girth and diameter of a graph with minimum degree at least 3</a>.</p>
https://mathoverflow.net/q/4558010Can the Constructible Universe be built in absence of Unions and Power?Zuhair Al-Joharhttps://mathoverflow.net/users/953472023-10-03T20:26:04Z2023-10-04T11:28:18Z
<p>Can <span class="math-container">$L$</span> be built in</p>
<p><a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory" rel="nofollow noreferrer"><span class="math-container">$\sf ZF$</span></a> <span class="math-container">$\sf-Regularity-Union-Power+ Boolean \ Union$</span>?</p>
<p>We know that <span class="math-container">$L$</span> can be built in <span class="math-container">$\sf KP$</span>, but here we don't have Set Union.</p>
<p>If the answer is to the negative, then would adding <span class="math-container">$\sf Cartesian \ products; Ordinal \ Union$</span> and <span class="math-container">$\sf Global \ Choice$</span>, enable a positive answer?</p>
<p><strong>Ordinal union</strong> stipulates that if <span class="math-container">$X$</span> is a set of ordinals, then set union of <span class="math-container">$X$</span> exists.</p>
<p>If we can recover <span class="math-container">$L$</span> in the latter theory, then if we add the axiom of <span class="math-container">$\sf Successor \ cardinals$</span> (i.e. for every cardinal <span class="math-container">$\kappa$</span> there is a cardinal <span class="math-container">$\lambda$</span> such that <span class="math-container">$\kappa < \lambda$</span>), would that lead to <span class="math-container">$L \models \sf ZFC$</span>?</p>
https://mathoverflow.net/q/4557992Root system terminologyErichttps://mathoverflow.net/users/5144982023-10-03T20:21:09Z2023-10-04T08:46:53Z
<p>Let <span class="math-container">$\Phi$</span> be a <a href="https://en.wikipedia.org/wiki/Root_system" rel="nofollow noreferrer">root system</a>. In a paper I'm writing, I need to work with subsets <span class="math-container">$\Phi' \subset \Phi$</span> satisfying the following two conditions:</p>
<ol>
<li><p>For all <span class="math-container">$\lambda_1,\lambda_2 \in \Phi'$</span> and <span class="math-container">$c_1,c_2 \geq 0$</span> such that <span class="math-container">$c_1 \lambda_1 + c_2 \lambda_2 \in \Phi$</span>, we have <span class="math-container">$c_1 \lambda_1 + c_2 \lambda_2 \in \Phi'$</span>.</p>
</li>
<li><p>For all <span class="math-container">$\lambda \in \Phi'$</span>, we have <span class="math-container">$-\lambda \notin \Phi'$</span>.</p>
</li>
</ol>
<p>One example would be a choice of positive roots.</p>
<p>Is there a term for such subsets? I'm not an expert in root systems or Lie theory, so I might be missing something obvious.</p>
https://mathoverflow.net/q/4557890How is this 3d curve called which looks like the infinity sign?mathoverflowUserhttps://mathoverflow.net/users/1659202023-10-03T17:21:49Z2023-10-04T09:04:31Z
<p>How is this 3d curve called, which looks like a 2d lemniscate but is 3d? (Is there a known parametric equation for it?)</p>
<p><a href="https://i.stack.imgur.com/FJXkz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FJXkz.png" alt="Rydberg_dihedral_kernel_KPCA_visualization_dim_3" /></a></p>
<p>It was generated with <a href="https://sagecell.sagemath.org/?z=eJztVt-P2jgQfkfif3DZh9qQhE3CVu3qeKgOaR9OJ52EVKEidmUSA4bESW3Dbv_7G_8IEJbtPfThpKo8mPHMN589M_bEK1mVSO0KRqWIcpZVZV0prnklEAdRavQXk4IV__z5udvxGrEv6--IKiTqbqfbydkK7XaYBkty3-0g-Emm91KgeFjSF6vv9xNnufGmdZZbw2MyxLS_JA3PWtLyb6rxbnfBVdBymVM0u0cl1ZK_4Pncr4lWlUQUcYFmCysvnXzkzPMre1PfpFkFLJQM4H8JiDDpe86TL9-wXNLis1gXTIFDMGuIbtBG61rdDyFKvYmUptmOvWQbKtYsgjwOv-2ZMolUw_Tuw93HT7fDnGkmSy6o0GG1CtX3smQQTBa6mMJnrjfhXnAd5pyuK0ELt9QDGp9nBs-I30ItudD4wU8nAHuIuDgwqRhuYyYNxoAmkayeFT6qIlrXTOQ4VPsSTyaEQHDPPNtovm6TNCw-icdSgGXOF1Fe6adaVvk-M9j5dkGG3iIqWWLSt0o_ISQSmCC-QvzdeItYoRiKbQG5KaA0ecQFExDtICautNtrllfF-kLlhEGm_odKHQ6Q3pxD_HOMebAlwYFK3MueegOlJQZGqyVuBrfDzgi5jDsOdoOEXMbstAvyw6Kwl5pCLY-1ORzmdpXFf6Tak7-1JjQHjd3FaGfb9YfW1fjVzyv6yQMb2Q6KryTzC8t0JX_3mXbdDi4rP1O2t8r0ZpVsof_YLnyRbvw9jpMAhhSGxAzxyEhmSEc9gN4YFUR-C-It_CMzf0z6Bv6YoBAZMU7NMLJKM4zQwOBShxudcEkLl45aeufq9NY_sf6p8R9c4tKWatSs2t5N2qiSc-7QczshbYSRF5JGkzQav4PYZePXi6rbMWdGnLXIuySI79Lmws6g7k3vFP4gfwXdfLZwE3fIe2LcO9oN49QwfvUkZ7gpQoCckpPBX5Oe6xZA3QuudJApueZCJXSHe3R74eU-meb0g9sVP3CyHq8fRNPTp6Fx0hC-gpBKZjZ3fEBi8WTfl4IJrcZpgHbWMn5fS_vw3GuWvz-jmT2diHIgOqONVlyfrPj1t6gVvIuhRRepDa3ZOaaGFeoKgKqNPMMUBWAKLtibCFzXg6IgwF49Y_Iv4X060Q==&lang=sage&interacts=eJyLjgUAARUAuQ==" rel="nofollow noreferrer">this Sagemath - script</a>, where you can zoom in 3d in your browser.
The background lies in the following formula of a positive semidefinite matrix, which then is processed with Kernel-PCA to be visualized in 3dim and is <a href="https://mathoverflow.net/questions/396276/simplex-invariants?noredirect=1&lq=1">related to this question</a> and <a href="https://mathoverflow.net/questions/455470/a-rkhs-interpretation-of-the-rydberg-formula-for-hydrogen-and-an-application-for">this question</a>.</p>
<p>In the book "Matrices and Graphs in Geometry" by Miroslav Fiedler, we have the following generalisation of the sum of angles equals <span class="math-container">$\pi$</span> in triangles:</p>
<p><a href="https://i.stack.imgur.com/0fiWk.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0fiWk.png" alt="theorem_dihedral_kernel" /></a></p>
<p>In this case, which we are looking at, we have:</p>
<p><span class="math-container">$$G_n^{-1} = (d_1^T,d_2^T,\dots,d_n^T)$$</span></p>
<p>where</p>
<p><span class="math-container">$$G_n = (1/\max(i,j)^2)_{1 \le i,j \le n}$$</span></p>
<p>is a Gram matrix and <span class="math-container">$d_{n+1} := \sum_{i=1}^n d_i$</span>. Aftwerwards we treat the entries in the positive semidefinite matrix above as a p.s.d. kernel and thus can visualize its entries <span class="math-container">$1,\cdots,n+1$</span> with KernelPCA to get the picture above.</p>
<p>Visualized (with KernelPCA) in 2dim, it looks like this:</p>
<p><a href="https://i.stack.imgur.com/i5rIF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/i5rIF.png" alt="Rydberg_dihedral_kernel_KPCA_visualization_dim_2" /></a></p>
<p><strong>Edit</strong>:
Changing the kernel to <span class="math-container">$K(a,b) = \frac{k(a,b)}{\sqrt{k(a,a)k(b,b}} = \frac{\min(a,b)}{\max(a,b)}$</span> will give us <a href="https://sagecell.sagemath.org/?z=eJztV1GP2jgQfkfiP_jYh9qQBEjY6ro6HqpDtw-nk6oiVaiIRSYxwZA4qW2yu_31HccJJLB71akPJ1Xlwdjjb2Y8840nyVZmKVKHhFEpvIiFWZpnimueCcRhKjX6m0nBkg9_vu92Kok4pvkzogqJvNvpdiK2RZJGWJC7bgfBTzJ9lALlMouOoca55ClbR7zgKpMKYKTWOhwwdTa12k2lR_ubYUqfyq1-329v-n1MBxsyxHEYWYRxfVo9BEOMwQCxMGIMGH-NY6VclNCTj_o0saTpP1Tjw-EikISmm4iixR1KqZb8CS-X1cnRNpOIIi7QYlXON3buSC7i6cfZXyfjSRbfX9nflskPef7sJVzQJK5zDujUQl45iAHg_34ar6QOnxmIoiYDlRv1RZpTwg4lA_jfAML1-5WXsy7fsUjS5L2IE6ZAwVmcqEQ7rXN1N4Qk652nNA0P7CncUREzD2ps-OXIlCkyNQxu397-_m40jJhmErihQrvZ1lXPacogvNC1UbqPXO_co-DajTiNM8iWdXWPpk3m8KJi-wbKTmh8Xy1nALv3uCiYVAy3MbMaY0AzT2aPCp9EHs1zJiLsqmOKZzNCILhHHu40j9tGZhdlVpMDO0u-8qJMr-sbAZL9CmrY7ohMppj0S2G1IMAUJohvEf9tukcsUQyNS0q5oVSaPOKECYh2MCaW7P1LO1dkfaJyxiBT_wNTRQHpjTjEv8SYO3viFFTiXrjuDZSW2NxLIyV2BZezXBFyGffYOQx8chmzla7Iv5LCnnIKXJ64KYpl6WX1nVRXxl_zCY1TV5eqnW3bO1tX42evV_SDBVu3qOtkfmKhzuSvPtPmrbBZ-RHaXqPpVZZKov_YryqSbqp7PPYdGAIYfDOMJ2ZmhmDSA-iNEUHkI5iO4B-Z9YPfN_AHH7nITMeBGSal0AwTNDC4wOImZ5zfwgWTltyqWnmp75f6gdEfXOKClmhSe22fJqhFftO2W9m2k6CeTKqJX0v8WlKdYGyz8fNF1e2YmhHnmgkm75zgdlRf2AXwXvdOAZVkpZ9Bulys7MKWeU9Me46o9o3NubH5uTLTwM0RAuScXG2c37jAes-pG1z7RQzP4Q4wHjNR0AR6BSYNQ9WN60E_kXmWPH_PTo170YjtXtbEdUebv6hCJXSrOzS60LKPcHMbQe0FPVAqNa5f0ObnR1WtpIEOBQlOmTnc6WUfi3X5LSCY0GoaOOhQ7kzf5LL8SDhqFr1pmFmsz4YiMNQw6225Pu_i62djK3gbQ8ucp3Y0Zw1MA52DrzwDFdXWcRT_yqYj77ahliQAhhdt1oY2EDhJBnlOwGH2iMk3ocq-jw==&lang=sage&interacts=eJyLjgUAARUAuQ==" rel="nofollow noreferrer">the following surface in 3d</a>:</p>
<p><a href="https://i.stack.imgur.com/fNmuD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fNmuD.png" alt="visualization_rydberg_angle_kernel_kpca_3dim_pic_1" /></a></p>
<p><a href="https://i.stack.imgur.com/jRNrM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jRNrM.png" alt="visualization_rydberg_angle_kernel_kpca_3dim_pic_2" /></a></p>
<p><a href="https://i.stack.imgur.com/Ch5fz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ch5fz.png" alt="visualization_rydberg_angle_kernel_kpca_3dim_pic_3" /></a></p>
<p>I would also be intersted to know how this surface is called, if it has a name.</p>
https://mathoverflow.net/q/4557752Exact decay for solutions of fractional Laplacian equationsadiazhttps://mathoverflow.net/users/1119992023-10-03T13:43:31Z2023-10-04T11:41:09Z
<p>If <span class="math-container">$U$</span> is the unique radially decreasing solution of
<span class="math-container">\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s U+ U &=U^p &&\text{ in } \mathbb{R}^N\\
U&>0 &&\text{ in } \mathbb{R}^N\\
U(|x|)&\to 0 &&\text{ as } |x| \to \infty
\end{aligned}
\right.
\end{equation}</span>
Is it possible to show that <span class="math-container">$\lim_{|x|\to \infty} (1+ |x|^{N+2s})U(|x|)=l(>0)$</span> where <span class="math-container">$s\in (0, 1)$</span> and <span class="math-container">$N\geq 2$</span> and <span class="math-container">$1<p<\frac{N+2s}{N-2s}.$</span> Thank you.</p>
https://mathoverflow.net/q/455773-1Suggestions for journals in motivic homotopy theoryAlexey Dohttps://mathoverflow.net/users/4823982023-10-03T13:33:53Z2023-10-04T08:48:39Z
<p>I am a novice PhD student working in algebraic geometry, especially motivic homotopy theory and singularity theory. I just finished my first text and my advisor suggests me to find a suitable journal to submit (in case I can't, he will suggest). He told me that I should have a look at a particular journals ranking; for instance, the one the australian math soc and my potential paper could match a journal of rank A-B and then I can also try to maximize the ratio = level of the journal/level of the my article.</p>
<p>Since I have no experience on this, I hope that some of you could give me some journals which are suitable for submitting a paper in motivic homotopy theory and singularity theory. And then I can analyze these ones more carefully.</p>
<p>Thank you in advance.</p>
https://mathoverflow.net/q/4557392How many configurations of tubes are there?seldom seenhttps://mathoverflow.net/users/22252023-10-02T22:48:27Z2023-10-04T06:13:42Z
<p>Can <span class="math-container">$n$</span> disjoint lines in <span class="math-container">$\boldsymbol R^3$</span> be knotted? No... Let <span class="math-container">$X_n$</span> be the configuration space of <span class="math-container">$n$</span> disjoint lines in <span class="math-container">$\boldsymbol R^3$</span>. It is not hard to see that <span class="math-container">$X_n$</span> is path connected: Let <span class="math-container">$L^1,...,L^n$</span> be <span class="math-container">$n$</span> disjoint lines. First wiggle them a little so that no two of them are parallel, and choose coordinates so that none of them are parallel to the <span class="math-container">$y-z$</span> plane. Parametrize <span class="math-container">$L^i=L^i_0$</span> as <span class="math-container">$s\mapsto (s, b_i + s\cdot v_i)$</span>, where <span class="math-container">$b_i,v_i\in\boldsymbol R^2$</span>. For each <span class="math-container">$t\in\boldsymbol R_{\geq 0}$</span> shift the <span class="math-container">$L$</span> to the left by <span class="math-container">$t$</span> units and squeeze by <span class="math-container">$1/(t+1)$</span> in the <span class="math-container">$y-z$</span> directions, so that <span class="math-container">$L^i_t$</span> is parametrized by <span class="math-container">$(s, (b_i+t\cdot v_i)/(t+1) + s\cdot (v_i/(t+1)))$</span>. In the limit <span class="math-container">$t\mapsto\infty$</span>, <span class="math-container">$L^i_{\infty}$</span> is parametrized by <span class="math-container">$s\mapsto (s,v_i)$</span>, the lines are all parallel, and the problem reduces to path connectivity of the space of <span class="math-container">$n$</span> points in the plane. Fair enough.</p>
<p>What happens if the lines are thickened up a little? Let <span class="math-container">$Y_n$</span> be the configuration space of cylinders of radius 1 in <span class="math-container">$\boldsymbol R^3$</span>. The obvious adaptation of the argument above fails because we aren't allowed to shrink the cylinders. Path connectivity of <span class="math-container">$X_n$</span> seems to depend on the fact that we can push whatever rats nest <span class="math-container">$L$</span> is arranged in out to infinity, but to do it the lines have to become arbitrarily close to one-another.</p>
<p>Questions: Is <span class="math-container">$Y_n$</span> connected? How many connected components does it have? I tried for a while to find a naive, combinatorial proof that <span class="math-container">$X_n$</span> is connected but everything I try implies connectivity of <span class="math-container">$Y_n$</span>.</p>
https://mathoverflow.net/q/4557251If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?Lorenzo Pompilihttps://mathoverflow.net/users/2720402023-10-02T17:29:49Z2023-10-04T09:45:54Z
<p>Let <span class="math-container">$\mathcal H^1(\mathbb R^n)$</span> be the real Hardy space (as in <a href="https://www.jstor.org/stable/j.ctt1bpmb3s" rel="nofollow noreferrer">Stein's "Harmonic Analysis"</a>, Chapter 3). It is well known that <span class="math-container">$\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$</span> and its elements have zero mean.</p>
<p>I would like to know: does a function <span class="math-container">$f\in L^1\cap L^\infty(\mathbb R^n)$</span> with <span class="math-container">$\int_{\mathbb R^n} f(x)\,dx=0$</span> belong to <span class="math-container">$\mathcal H^1(\mathbb R^n)$</span> assuming it decays fast enough at infinity? For example, is polynomial decay
<span class="math-container">$$ |f(x)|\leq (1+|x|)^{-M} $$</span>
for some <span class="math-container">$M>0$</span> large enough a sufficient condition, together with the zero mean condition? If not, what about exponential decay, etc...? A Related question would be: is any Schwartz function with zero mean also in <span class="math-container">$\mathcal H^1(\mathbb R^n)$</span>?</p>
<hr />
<p>What I know is that any function in <span class="math-container">$L^q(\mathbb R^n)$</span>, <span class="math-container">$q>1$</span> with compact support and zero mean belongs to <span class="math-container">$\mathcal H^1(\mathbb R^n)$</span> (and the <span class="math-container">$L^q$</span> assumption can be relaxed to <span class="math-container">$L\log L$</span>). I had a quick look at Stein's book and Grafakos' book, but I did not find anything like what I am looking for. I thought that if no condition like that exists, it must be something well known.</p>
https://mathoverflow.net/q/4556921Property of some permutations of non-negative integers such that $a(n)<2^k$ iff $n<2^k$Notamathematicianhttps://mathoverflow.net/users/2319222023-10-02T08:34:15Z2023-10-04T10:45:55Z
<ul>
<li><p>Let
<span class="math-container">$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$</span></p>
</li>
<li><p>Let
<span class="math-container">$$
f(n) = 2^{\ell(n)}
$$</span></p>
</li>
<li><p>Let <span class="math-container">$q_1(n)$</span> and <span class="math-container">$q_2(n)$</span> be an arbitrary self-inverse permutations of non-negative integers (that is, <span class="math-container">$q_i(q_i(n)) = n$</span>) such that <span class="math-container">$q_i(n)<2^k$</span> iff <span class="math-container">$n<2^k$</span>.</p>
</li>
<li><p>Let <span class="math-container">$p_1(n)$</span> and <span class="math-container">$p_2(n)$</span> be a permutations of non-negative integers such that <span class="math-container">$p_i(n)<2^k$</span> iff <span class="math-container">$n<2^k$</span>. Here
<span class="math-container">$$
p_1(n) = q_1(p_2(q_2(n) - f(n)) + f(n)), \\
p_1(0) = 0, \\
p_2(n) = q_2(p_1(q_1(n) - f(n)) + f(n)), \\
p_2(0) = 0
$$</span></p>
</li>
</ul>
<p>I conjecture that
<span class="math-container">$$p_1(p_2(n)) = n.$$</span></p>
<p>At least it works for any combination of <a href="https://oeis.org/A054429" rel="nofollow noreferrer">A054429</a> (complement all but the most significant bit in binary expansion of <span class="math-container">$n$</span>), <a href="https://oeis.org/A059893" rel="nofollow noreferrer">A059893</a> (reverse the order of all but the most significant bit in binary expansion of <span class="math-container">$n$</span>) and <a href="https://oeis.org/A059894" rel="nofollow noreferrer">A059894</a> (complement and reverse the order of all but the most significant bit in binary expansion of <span class="math-container">$n$</span>).</p>
<p>Here is the PARI/GP prog to check it numerically:</p>
<pre><code>bc(n) = if(n == 0, 0, 3*2^logint(n, 2) - n - 1)
br(n) = my(A = binary(n)); fromdigits(concat(1, Vecrev(vector(#A - 1, i, A[i+1]))), 2)
bcr(n) = my(A = binary(n)); fromdigits(concat(1, Vecrev(vector(#A - 1, i, 1 - A[i+1]))), 2)
p1(n) = if(n == 0, 0, my(A = 2^logint(n, 2)); bc(p2(br(n) - A) + A))
p2(n) = if(n == 0, 0, my(A = 2^logint(n, 2)); br(p1(bc(n) - A) + A))
test(n) = p1(p2(n)) == n
</code></pre>
<p>I also conjecture that if we go backwards from the resulting pair <span class="math-container">$p_1(n)$</span> and <span class="math-container">$p_2(n)$</span> to the original pair <span class="math-container">$q_1(n)$</span> and <span class="math-container">$q_2(n)$</span>, then the last one is unique (that is, there are no two different pairs of <span class="math-container">$q_1(n)$</span> and <span class="math-container">$q_2(n)$</span> which result to the same pair of <span class="math-container">$p_1(n)$</span> and <span class="math-container">$p_2(n)$</span>).</p>
<p>Is there a way to prove it? If the last conjecture is true, is there a way to go backwards and find unique original pair of <span class="math-container">$q_1(n)$</span> and <span class="math-container">$q_2(n)$</span> when the resulting pair <span class="math-container">$p_1(n)$</span> and <span class="math-container">$p_2(n)$</span> is known?</p>
https://mathoverflow.net/q/4556571Misunderstanding the definition of kernel in digraphsvidyarthihttps://mathoverflow.net/users/1002312023-10-01T15:10:51Z2023-10-04T09:46:05Z
<p>By <a href="https://doi.org/10.1016/S0012-365X(98)00091-0" rel="nofollow noreferrer" title="zbMATH review at https://zbmath.org/0955.05049">Borodin–Kostochka–Woodall '97</a> paper, the first paragraph says that directed odd cycles do not have kernels. But, I don't get this. Like, consider any <span class="math-container">$\lfloor\frac{n}{2}\rfloor$</span> set of independent vertices in an odd cycle, and we orient every other vertex in the remaining set of vertices to one of the vertices in the independent set. So, don't we get a kernel? What exactly is missing here? Any light on this regard? Thanks beforehand.</p>
https://mathoverflow.net/q/4555740Temporal evolution of a globally hyperbolic spacetimeBastam Tajikhttps://mathoverflow.net/users/5033632023-09-30T07:35:42Z2023-10-04T05:44:31Z
<p>Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal).</p>
<p>For Simplicity assume that there's no matter <span class="math-container">$T_{\mu\nu} =0$</span>. And the spacetime satisfies Einstein's Field Equations.</p>
<p>Given a Cauchy hypersurface <span class="math-container">$S$</span> at initial time <span class="math-container">$t=t_0$</span>,</p>
<p><strong>the main question is</strong>:</p>
<p><em>Can there exist any solution where the evolution of at least a single point <span class="math-container">$p \in S$</span> is smooth BUT <em><strong>nowhere analytic in time</strong></em>?</em></p>
<p>The initial data is that of the <a href="https://en.m.wikipedia.org/wiki/Cauchy_problem" rel="nofollow noreferrer">Cauchy data</a> and <span class="math-container">$S$</span> is supposed to be <a href="https://arxiv.org/abs/gr-qc/0306108" rel="nofollow noreferrer">spacelike</a>.</p>
<p><strong>Assume</strong> <span class="math-container">$p \in U \subsetneq S$</span> where <span class="math-container">$U$</span> is an open subset of <span class="math-container">$S$</span> for the initial data set.</p>
<p>In case <span class="math-container">$U$</span> should not be even dense in <span class="math-container">$M$</span> or fulfill any other topological constraints, it must be clarified.</p>
<hr />
<p>Physical motivation for the <strong>interested reader</strong>:</p>
<p>A smooth but no-where analytic evolution function for a Cauchy well-posed problem is my personal way to formulate <strong>indeterminism</strong>.</p>
<p>In case such instances exist, specially in 3+1 dimensions then even in case of Hyperbolic spacetimes there can exist irreducible chaos which is problematic for the classical limit of the relevant quantum system and can challenge the whole quantization of gravity.</p>
https://mathoverflow.net/q/4548629Reconstruction of commutative differential graded algebrasWalterfieldhttps://mathoverflow.net/users/3172532023-09-19T02:04:22Z2023-10-04T11:46:37Z
<p>Let <span class="math-container">$k$</span> be an algebraically closed field of characteristic <span class="math-container">$0$</span>.</p>
<p>Let <span class="math-container">$A,B$</span> be commutative differential graded algebras (cdga) over <span class="math-container">$k$</span> such that <span class="math-container">$H^{i}(A)=H^{i}(B) =0 \ (i>0)$</span>.</p>
<p>Here, differentials are defined cohomologically, i.e,
<span class="math-container">$$
\cdots \rightarrow A^{i-1} \overset{d_A}{\rightarrow} A^i \overset{d_A}{\rightarrow} A^{i+1} \overset{}{\rightarrow} \cdots
$$</span>
where <span class="math-container">$d_A$</span> is the differential of <span class="math-container">$A$</span>.
In other word, <span class="math-container">$A,B$</span> are connective.</p>
<p>We also denote the dg categories of dg <span class="math-container">$A$</span>-modules and dg <span class="math-container">$B$</span>-modules by <span class="math-container">$D_{dg}(A),D_{dg}(B)$</span>, respectively.</p>
<p><strong>Question</strong>
If <span class="math-container">$D_{dg}(A),D_{dg}(B)$</span> are quasi-equivalent as dg categories, then <span class="math-container">$A, B$</span> are quasi-isomorphic as cdgas ?</p>
<p><strong>Edit(10/4)</strong>
Also, are there any candidate of conditions for cdgas for the reconstruction theorem(including the version below) to hold?</p>
<p>Any comments and references are welcome. Thank you !</p>
<p>Please take a look at the useful comments.</p>
<p>The same question is in <a href="https://math.stackexchange.com/q/4771259/952389">MSE</a>.</p>
<p><strong>Edit (a variant of the question, 9/20)</strong>:</p>
<p>Let <span class="math-container">$\tilde{D}_{dg}(A),\tilde{D}_{dg}(B)$</span> be dg-derived categories, i.e, <span class="math-container">$\tilde{D}_{dg}(A) = D_{dg}(A)/ Ac(A), \tilde{D}_{dg}(B) = D_{dg}(B)/ Ac(B)$</span>,
where <span class="math-container">$Ac(A) \subset D_{dg}(A), Ac(B) \subset D_{dg}(B)$</span> are the full sub dg categories of acyclic complexes of <span class="math-container">$D_{dg}(A), D_{dg}(B)$</span>, respectively.</p>
<p>If <span class="math-container">$\exists F: \tilde{D}_{dg}(A) \rightarrow \tilde{D}_{dg}(B)$</span> are quasi-equivalent of dg categories and <span class="math-container">$H^0(F)$</span> is a monoidal functor between monoidal categories <span class="math-container">$H^0(\tilde{D}_{dg}(A)), H^0(\tilde{D}_{dg}(B))$</span>, then <span class="math-container">$A,B$</span> are quasi-isomorphic ?</p>
<p><strong>Edit (classical case, 9/21)</strong>:</p>
<p>When we consider two commutative rings <span class="math-container">$A,B$</span>, where we regard them as cdgas in degree <span class="math-container">$0$</span>.
Even in this case, is the question true ?</p>
<p><strong>Edit (10/4)</strong>
Can my series of questions be solved from Toen's dg Morita theory or Lurie's theory of higher algebra?
Or is this exactly what should be studied in the future?</p>
https://mathoverflow.net/q/4487332When does the sum of squares of initial primes equal a triangular number?Antoine Balanhttps://mathoverflow.net/users/958452023-06-12T23:41:39Z2023-10-04T08:49:10Z
<p>Let <span class="math-container">$(p_i)$</span> be the sequence of prime numbers. Can we solve the equation:
<span class="math-container">$$\sum_{i=1}^k p_i^2=\frac{n(n+1)}{2}$$</span>
in <span class="math-container">$(k,n)$</span>? Note that <span class="math-container">$(7,36)$</span> is solution. Is this the unique solution?</p>
https://mathoverflow.net/q/4482574Parameter-free effective cardinalsReflecting_Ordinalhttps://mathoverflow.net/users/1702862023-06-05T14:39:43Z2023-10-04T07:25:14Z
<p>In the paper "Effective cardinals and determinacy in third order arithmetic" by Juan Aguilera, effective cardinals is defined.
I'm curious about its little variation, parameter-free effective cardinal.
For any ordinal <span class="math-container">$\alpha$</span>, we say that <span class="math-container">$\alpha$</span> is a parameter-free effective cardinal, if there is an infinite ordinal <span class="math-container">$\beta<\alpha$</span> such that <span class="math-container">$\alpha$</span> is the supremum of the order types of parameter-free <span class="math-container">$\Sigma_1(L_\beta)$</span>-well orderings of <span class="math-container">$\beta$</span>, or <span class="math-container">$\alpha$</span> is a limit of the previous kind of ordinals.</p>
<p>Let the <span class="math-container">$\alpha$</span>th p.f.e. cardinal be <span class="math-container">$\delta_\alpha$</span>.</p>
<p>I noticed that for the least ordinal that is <span class="math-container">$\Sigma^1_1$</span>-reflecting on <span class="math-container">$\Sigma^1_1$</span>-reflecting ordinals, the next two p.f.e. cardinal after it are not admissible, by a simple modification of proposition 13 of J. Aguilera's paper "<a href="https://arxiv.org/abs/1906.11769v1" rel="nofollow noreferrer">The order of reflection</a>". At the same time, this phenomenon do not happen for effective cardinals before the non-locally-countable ordinals appears, as Aguilera noted.</p>
<p>Some questions:</p>
<ol>
<li><p>Is there an ordinal <span class="math-container">$\alpha$</span> such that there is an ordinal <span class="math-container">$\delta_\alpha<\beta<\delta_{\alpha+1}$</span> s.t. there is no parameter-free <span class="math-container">$\Sigma_1(L_{\delta_\alpha})$</span>-well ordering of <span class="math-container">$\delta_\alpha$</span> with order type <span class="math-container">$\beta$</span>? If it exists, what's the least one of them?</p>
</li>
<li><p>Is it true that for any ordinal smaller than the least <span class="math-container">$\Sigma^1_1$</span>-reflecting ordinal, it is a p.f.e. cardinal iff it is admissible or a limit of admissible ordinals (except <span class="math-container">$\omega$</span>)?</p>
</li>
<li><p>For any ordinal <span class="math-container">$\alpha$</span>, what's the least admissible ordinal such that none of the next <span class="math-container">$\alpha$</span> p.f.e. cardinals of it is admissible? What if "admissible" changed to "effective cardinal"?</p>
</li>
<li><p>For the least ordinal <span class="math-container">$\alpha$</span> that is <span class="math-container">$\Sigma^1_1$</span>-reflecting on <span class="math-container">$\Sigma^1_1$</span>-reflecting ordinals, is it true that <span class="math-container">$\delta_{\alpha+2}$</span> is the largest ordinal that <span class="math-container">$L_\alpha$</span> is <span class="math-container">$\Sigma_1$</span> stable to?</p>
</li>
</ol>
https://mathoverflow.net/q/1688668Models of PRA/EFA with induction on $X$ but not $\omega^X$Eliezer Yudkowskyhttps://mathoverflow.net/users/383302014-06-02T19:38:57Z2023-10-04T08:30:03Z
<p>As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that is, $\omega^{\omega^\cdots}$ for $N$ layers of $\omega$. See e.g. the second answer to <a href="https://mathoverflow.net/questions/138875/why-do-stacked-quantifiers-in-pa-correspond-to-ordinals-up-to-epsilon-0">Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?</a>.</p>
<p>If this is so then by Godel's Completeness Theorem there must be models of (if I have this right) $PRA+I(\Pi^0_{N+1})$ within which $\omega \uparrow\uparrow N$ is well-ordered, but $\omega \uparrow\uparrow (N+1)$ is not well-ordered.</p>
<p>What would these nonstandard models look like? Is there any intuitive way to describe them via ultrapowers or something similar? Or can we say something about their structure, the way that $\mathbb{N} + \mathbb{Z}\cdot\mathbb{Q}$ is the structure for countable nonstandard models of full first-order arithmetic?</p>
<p>For concreteness: Is there any more specific way to describe a nonstandard model of PRA in which the Ackermann function is not complete, beyond "The Ackermann function is not complete"? Hopefully via some construction which will extend to "Kirby-Paris hydras of height $N+2$(?) always terminate, but some of height $N+3$(?) don't"? Obviously every primitive recursive function $f_n$ with $n \in \mathbb{N}$ is complete for every standard and nonstandard input, while the Ackermann function is incomplete for some input $\varpi$ which is a nonstandard number ($\varpi > 0, \varpi > 1, \dots$), likewise with non-terminating hydras where at least one branch has nonstandard width, but beyond this I cannot visualize anything about what the nonstandard model looks like.</p>
<p>(Understanding this would provide a direct, non-Godelian argument for why induction on $N+1$ quantifiers is <em>necessary</em> as well as <em>sufficient</em> to prove well-ordering of $\omega \uparrow\uparrow N$. It would also show in a direct, non-Godelian way that $PA$ cannot prove the well-ordering of $\epsilon_0$, since this would require induction on infinite quantifiers.)</p>
https://mathoverflow.net/q/1220600The description of Hurwitz groupsKlim Puhovhttps://mathoverflow.net/users/106262013-02-17T14:12:36Z2023-10-04T07:26:40Z
<p>Let <span class="math-container">$G$</span> be a Hurwitz group, i.e. the automorphism group of some <a href="https://en.wikipedia.org/wiki/Hurwitz_surface" rel="nofollow noreferrer">Hurwitz surface</a> <span class="math-container">$C$</span>. Then <a href="https://en.wikipedia.org/wiki/Hurwitz%27s_automorphisms_theorem" rel="nofollow noreferrer">Hurwitz's automorphisms theorem</a> shows that the quotient map of <span class="math-container">$C$</span> by <span class="math-container">$G$</span> has ramification points of indices <span class="math-container">$2$</span>, <span class="math-container">$3$</span> and <span class="math-container">$7$</span>. My question is how to deduce that <span class="math-container">$G$</span> is generated by elements <span class="math-container">$x$</span> and <span class="math-container">$y$</span> satisfying <span class="math-container">$x^2=y^3=(xy)^7=1$</span>?</p>
https://mathoverflow.net/q/7641316Torsion subgroups in families of twists of elliptic curves Giuseppehttps://mathoverflow.net/users/137412011-09-26T12:29:59Z2023-10-04T11:29:26Z
<p>Here is the short version:</p>
<blockquote>
<p>Fix an elliptic curve $E/\mathbb{Q}$. How does the torsion structure $E_d(\mathbb{Q})_{tors}$ vary, as $E_d$ runs through the quadratic twists of $E$?</p>
</blockquote>
<p>Here is the longer version:</p>
<p>I have been playing with SAGE this morning. I inserted the elliptic curve ('11a1') $$E : y^2 + y = x^3 - x^2 - 10x - 20$$ which has rational torsion subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. I then computed its quadratic twist $E_d$ for all squarefree $d$ up to 2000, and observed $E_d(\mathbb{Q})_{tors}$ was always trivial. </p>
<blockquote>
<p>Can it be that, in this particular family of quadratic twists, all but one of the curves have trivial torsion? Is this a general phenomenon? </p>
</blockquote>
<p>(I ran this experiment for several other curves $E$ and got the same impression; that all but one of the curves in a family of twists have the same torsion structure.) </p>