Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2024-08-08T12:54:40Zhttps://mathoverflow.net/feedshttps://creativecommons.org/licenses/by-sa/4.0/rdfhttps://mathoverflow.net/q/4765751Stem extensions and quotients of Schur coversPadraig Ó Catháinhttps://mathoverflow.net/users/275132024-08-08T12:27:01Z2024-08-08T12:27:01Z
<p>Suppose that <span class="math-container">$G$</span> is a finite group, and that <span class="math-container">$\Gamma$</span> is a central extension of <span class="math-container">$G$</span> by <span class="math-container">$A$</span>, that is
<span class="math-container">$$ 1 \rightarrow A \rightarrow \Gamma \rightarrow G \rightarrow 1$$</span>
with the image of <span class="math-container">$A$</span> contained in <span class="math-container">$Z(\Gamma)$</span>. If <span class="math-container">$A$</span> is also contained in <span class="math-container">$[\Gamma, \Gamma]$</span> then this is a stem extension. For a stem extension, it is known that <span class="math-container">$A$</span> is isomorphic to a subgroup of the Schur multiplier of <span class="math-container">$G$</span>, but I have not been able to find a result in the literature relating <span class="math-container">$\Gamma$</span> to a Schur cover of <span class="math-container">$G$</span> unless <span class="math-container">$A$</span> is the full Schur multiplier. Here are some increasingly optimistic questions.</p>
<ol>
<li><p>If <span class="math-container">$G$</span> is perfect and <span class="math-container">$\Gamma$</span> is a stem extension, can we assume that <span class="math-container">$\Gamma$</span> is isomorphic to a quotient of a Schur cover of <span class="math-container">$G$</span>?</p>
</li>
<li><p>If <span class="math-container">$\Gamma$</span> is a stem extension, can we assume that <span class="math-container">$\Gamma$</span> is isomorphic to a quotient of the Schur cover of <span class="math-container">$G$</span>? (There might be multiple non-isomorphic Schur covers.)</p>
</li>
<li><p>If <span class="math-container">$\Gamma$</span> is a central extension, can we relate it the Schur cover of <span class="math-container">$G$</span> in any meaningful way?</p>
</li>
</ol>
<p>The last question is what I'm really interested in. The type of answer I'd like (if such a thing is true) is a proof or reference showing that <span class="math-container">$\Gamma$</span> contains a subgroup which is non-split over <span class="math-container">$B = A \cap [\Gamma, \Gamma]$</span>, that the non-split subgroup is isomorphic to a quotient of the Schur cover, and that <span class="math-container">$\Gamma/B$</span> splits. This seems overly optimistic, so I've asked the first two questions, to hopefully get some control over a more limited situation. Failing that, any references on equivalence of liftings of projective representations would be appreciated.</p>
https://mathoverflow.net/q/4765730When are these base spaces isomorphic?Nicolas Medina Sanchezhttps://mathoverflow.net/users/1663142024-08-08T12:16:31Z2024-08-08T12:16:31Z
<p>Given a smooth manifold <span class="math-container">$\mathcal{M}$</span> that is a fibre bundle over two different base spaces, i.e., there are <span class="math-container">$\Pi_1:\mathcal{M}\rightarrow B_1$</span> and <span class="math-container">$\Pi_2:\mathcal{M}\rightarrow B_2$</span>, if I can prove that the fibers are isomorphic, then necessarily the base spaces are also isomorphic?</p>
https://mathoverflow.net/q/4765725A projectivity property in the category of groupsNeil Stricklandhttps://mathoverflow.net/users/103662024-08-08T11:11:45Z2024-08-08T12:50:19Z
<p>Let <span class="math-container">$F_r$</span> be the free group on <span class="math-container">$r$</span> generators, let <span class="math-container">$G$</span> and <span class="math-container">$H$</span> be finite groups, and let <span class="math-container">$F_r\xrightarrow{\alpha}H\xleftarrow{\beta}G$</span> be surjective homomorphisms. It is then easy to see that we can choose <span class="math-container">$\phi\colon F_r\to G$</span> with <span class="math-container">$\beta\phi=\alpha$</span>, but <span class="math-container">$\phi$</span> need not be surjective, especially if <span class="math-container">$G$</span> is too big. Now suppose in addition that there exists a surjective homomorphism <span class="math-container">$F_r\to G$</span>. Is it then possible to find a surjective homomorphism <span class="math-container">$\phi\colon F_r\to G$</span> with <span class="math-container">$\beta\phi=\alpha$</span>?</p>
<p>I think that this works if <span class="math-container">$G$</span> is nilpotent, because we can then reduce to the case of <span class="math-container">$p$</span>-groups, and use the Frattini quotient to detect surjectivity. However, it would be tidier if I could prove it without any hypothesis. It also works when <span class="math-container">$r=1$</span>, but even that case requires some modular arithmetic and so is not completely trivial. On the other hand, I think that in practice a randomly chosen <span class="math-container">$\phi$</span> will be surjective with high probability.</p>
https://mathoverflow.net/q/4765691Higher Bockstein maps in group cohomologyAntoinehttps://mathoverflow.net/users/4800852024-08-08T09:24:25Z2024-08-08T10:31:42Z
<p>Let <span class="math-container">$p$</span> be an odd prime and <span class="math-container">$n>1$</span>. I am trying to understand why the cohomology ring <span class="math-container">$H^{\ast}(\mathbb{Z}/p^n;\mathbb{F_p})$</span> is given by
<span class="math-container">$$\mathbb{F}_p[y]\otimes\Lambda(x),$$</span>
with <span class="math-container">$|x|=1,|y|=2$</span> and <span class="math-container">$\beta_n(x)=y$</span>, where <span class="math-container">$\beta_n:H^{1}(\mathbb{Z}/p^n;\mathbb{F_p})\to H^{2}(\mathbb{Z}/p^n;\mathbb{F_p})$</span> is the <span class="math-container">$n$</span>th Bockstein map. I have seen a few abstract definitions of <span class="math-container">$\beta_n$</span> as a higher cohomology operation or as a higher differential in the Bockstein spectral sequence. The first questions that come to my mind are:</p>
<p>-Can we describe explicitly <span class="math-container">$\beta_n(x)=y$</span> as a <span class="math-container">$2$</span>-cocycle?</p>
<p>-Is the group <span class="math-container">$E=\mathbb{Z}/p\times_{y}\mathbb{Z}/p^n$</span> determined by <span class="math-container">$y$</span> abelian?</p>
<p>Of course, the latter question would follow from the former.</p>
https://mathoverflow.net/q/4765671Under what conditions do distances from pivot points uniquely identify a point on a manifold?shuhalohttps://mathoverflow.net/users/20822024-08-08T07:46:44Z2024-08-08T07:46:44Z
<p>Let <span class="math-container">$X$</span> be a smooth manifold of dimension <span class="math-container">$n$</span> equipped with a Riemannian metric.</p>
<p>Suppose that <span class="math-container">$x_1, \dots, x_m$</span> are <em>pivot points</em> on that manifold. We consider the distance functions
<span class="math-container">$$
f_i(x) = d( x_i, x )
$$</span>
and thus assign the distance vector
<span class="math-container">$$
f(x) = \Big( f_i(x) \Big)_{1 \leq i \leq m}
$$</span>
to each point <span class="math-container">$x \in X$</span>.</p>
<p>Over which subsets <span class="math-container">$S \subseteq X$</span> and with how many pivot points is <span class="math-container">$f$</span> injective? Note that the pivot points are known and ordered.</p>
<p>I presume that at least <span class="math-container">$n+1$</span> are needed on an <span class="math-container">$n$</span>-dimensional manifold, and <span class="math-container">$S$</span> is generally not the whole manifold.</p>
https://mathoverflow.net/q/4765651VC-dimension of intersectionABIMhttps://mathoverflow.net/users/368862024-08-08T06:25:46Z2024-08-08T07:46:43Z
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be sets of real-valued functions on <span class="math-container">$X$</span>. Are there any reasonably tight bounds on the VC-dimension of <span class="math-container">$A\cap B$</span> in terms of the VC-dimensions of <span class="math-container">$A$</span> and <span class="math-container">$B$</span>?</p>
https://mathoverflow.net/q/4765641Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?limes_inferiorhttps://mathoverflow.net/users/5346972024-08-08T06:06:52Z2024-08-08T10:02:02Z
<p>My problem is the following: I want to construct <span class="math-container">$n$</span> rays all starting at a point <span class="math-container">$v$</span> that is not in the <span class="math-container">$n$</span>-dimensional ball around <span class="math-container">$0$</span> such that the following is true:</p>
<ul>
<li>The <span class="math-container">$n$</span>-dimensional ball is a subset of the convex cone spanned by those <span class="math-container">$n$</span> rays together.</li>
<li>The rays only intersect the ball in one point, so they are basically tangents on the boundary.</li>
</ul>
<p>Is this possible for <span class="math-container">$n>2$</span>? I know certainly that it is possible for <span class="math-container">$n=2$</span> and maybe even <span class="math-container">$n=3$</span>. However I am not quite sure for even higher <span class="math-container">$n$</span>.</p>
https://mathoverflow.net/q/4765632Composition of two $\mathbb{S}$-modulesSaikathttps://mathoverflow.net/users/1617222024-08-08T05:37:24Z2024-08-08T05:37:24Z
<p>This should be a math.stack question, but I am posting it on math.overflow so that someone who knows the theory of operads might provide some helpful comments and answers. I am reading the book "Algebraic Operads" by J. L. Loday and B. Vallete. I am stuck with the definition of composition of <span class="math-container">$\mathbb{S}$</span>-modules in Sec-5.1.6, pg 99. Below I have written down all the required definitions and have posted my question at the end.</p>
<p>An <span class="math-container">$\mathbb{S}$</span>-module over <span class="math-container">$\mathbb{K}$</span> is a family <span class="math-container">$M = (M(0), M(1), M(2), \ldots, M(N), \ldots)$</span> of right <span class="math-container">$\mathbb{K}[\mathbb{S}_n]$</span>-modules <span class="math-container">$M(n)$</span>.</p>
<p><strong>Tensor product of two <span class="math-container">$\mathbb{S}$</span>-module:</strong></p>
<p>The tensor product of two <span class="math-container">$\mathbb{S}$</span>-module is defined as follows (cf. Section 5.1.4): Let <span class="math-container">$M$</span> and <span class="math-container">$N$</span> be two <span class="math-container">$\mathbb{S}$</span>-module then their tensor product is the <span class="math-container">$\mathbb{S}$</span>-module <span class="math-container">$M \otimes N$</span> defined by
<span class="math-container">$$M \otimes N (n) := \bigoplus_{i+j=n} \mathrm{Ind}^{\mathbb{S}_n}_{\mathbb{S}_i \times \mathbb{S}_j} M(i) \otimes M(j) \cong \bigoplus_{i+j=n} M(i) \otimes M(j) \otimes \mathbb{K}[Sh(i,j)]$$</span>
where <span class="math-container">$\mathrm{Ind}^{\mathbb{S}_n}_{\mathbb{S}_i \times \mathbb{S}_j} M(i) \otimes M(j)$</span> is the induced representation of <span class="math-container">$\mathbb{S}_n$</span> defined in Appendix A.1.3, pg 458. It says the following: Let <span class="math-container">$G$</span> be a group and <span class="math-container">$H$</span> be a subgroup of <span class="math-container">$G$</span>. If <span class="math-container">$M$</span> is a right <span class="math-container">$H$</span>-module, then the induced representation is the following representation of <span class="math-container">$G$</span> (i.e., a left <span class="math-container">$K[G]$</span>-module) <span class="math-container">$$\mathrm{Ind}^G_H M := M \otimes_H \mathbb{K}[G].$$</span></p>
<p><em>Question 1:</em></p>
<p>I am a bit confused with the definition of the tensor product of two <span class="math-container">$\mathbb{S}$</span>-module <span class="math-container">$M$</span> and <span class="math-container">$N$</span>. The definition of <span class="math-container">$\mathbb{S}$</span>-module says it is a family of right <span class="math-container">$\mathbb{K}[\mathbb{S}_n]$</span>-modules. Then by definition <span class="math-container">$M \otimes N (n)$</span> should be a right <span class="math-container">$\mathbb{K}[\mathbb{S}_n]$</span>-module. But the induced representation <span class="math-container">$\mathrm{Ind}^{\mathbb{S}_n}_{\mathbb{S}_i \times \mathbb{S}_j} M(i) \otimes M(j)$</span> is a representation of <span class="math-container">$\mathbb{S}_n$</span>, hence is a left <span class="math-container">$\mathbb{K}[\mathbb{S}_n]$</span>-module and not a right <span class="math-container">$\mathbb{K}[\mathbb{S}_n]$</span>-module. This has confused me, as I am unable to understand how to make <span class="math-container">$\mathrm{Ind}^{\mathbb{S}_n}_{\mathbb{S}_i \times \mathbb{S}_j} M(i) \otimes M(j)$</span> a right <span class="math-container">$\mathbb{K}[\mathbb{S}_n]$</span>-module? Therefore, assuming the definition of tensor product to be
<span class="math-container">$$M \otimes N(n) = \bigoplus_{i+j=n} M(i) \otimes M(j) \otimes \mathbb{K}[Sh(i,j)]$$</span>
I have proceeded further.</p>
<p><em>Question 2:</em></p>
<p>Let <span class="math-container">$N$</span> be an <span class="math-container">$\mathbb{S}$</span>-module. Then can we tell that the following equality holds? If yes, how should I proceed to prove it? I also could not prove it for the case <span class="math-container">$k=2$</span>.<br />
<span class="math-container">$$N^{\otimes k} (n) = \bigoplus_{i_1 + \cdots + i_k =n} \mathrm{Ind}^{\mathbb{S}_n}_{\mathbb{S}_{i_1} \times \cdots \times \mathbb{S}_{i_k}} \big(N(i_1) \otimes \cdots \otimes N(i_k)\big)
\cong \bigoplus_{i_1 + \cdots + i_k=n} N(i_1) \otimes \cdots \otimes N(i_k) \otimes \mathbb{K}[Sh(i_1,\ldots,i_k)]$$</span></p>
<p><strong>Composition of two <span class="math-container">$\mathbb{S}$</span>-module:</strong></p>
<p>Given two <span class="math-container">$\mathbb{S}$</span>-module <span class="math-container">$M$</span> and <span class="math-container">$N$</span> their composite is the <span class="math-container">$\mathbb{S}$</span>-module <span class="math-container">$$M \circ N(n) := \bigoplus_{k \ge 0} M(k) \otimes_{\mathbb{S}_k} N^{\otimes k}(n)
\cong \bigoplus_{k \ge 0} \Big(M(k) ~~\otimes_{\mathbb{S}_k} \bigoplus_{i_1 + \cdots + i_k =n} \mathrm{Ind}^{\mathbb{S}_n}_{\mathbb{S}_{i_1} \times \cdots \times \mathbb{S}_{i_k}} \big(N(i_1) \otimes \cdots \otimes N(i_k)\big)\Big)$$</span>
Assuming the equality mentioned in "<em>Question 2</em>" holds, we get
<span class="math-container">$$M \circ N(n) = \bigoplus_{k \ge 0} P(k) = \bigoplus_{k \ge 0} \Big(M(k) ~~\otimes_{\mathbb{S}_k} \bigoplus_{i_1 + \cdots + i_k =n} N(i_1) \otimes \cdots \otimes N(i_k) \otimes \mathbb{K}[Sh(i_1,\ldots,i_k)]\Big)$$</span></p>
<p><strong>Example:</strong></p>
<p>The authors have provided an example of the composition of two <span class="math-container">$\mathbb{S}$</span>-modules in 5.1.9, pg 100: Let <span class="math-container">$M$</span> and <span class="math-container">$N$</span> be two <span class="math-container">$\mathbb{S}$</span>-modules with <span class="math-container">$M(0)=0=N(0)$</span> and <span class="math-container">$M(1) = \mathbb{K} = N(1)$</span> then
<span class="math-container">$$M \circ N (2) = M(2) \oplus N(2)$$</span>
<span class="math-container">$$M \circ N(3) = M(3) \oplus \big(M(2) \otimes \mathrm{Ind}^{\mathbb{S}^3}_{\mathbb{S}_2} N(2)\big) \oplus N(3)$$</span></p>
<p><em>Question 3:</em></p>
<p>Assuming the equality in "<em>Question 2</em>" I did the computation for <span class="math-container">$M \circ N(3) = \bigoplus_{k \ge 0} P(k) = P(1) \oplus P(2) \oplus P(3)$</span> (note that since <span class="math-container">$M(0)=0 \implies P(0)=0$</span> and for <span class="math-container">$k > 3$</span> we have <span class="math-container">$P(k) = 0$</span> since <span class="math-container">$i_1 + \cdots + i_k = 3$</span> implies one of <span class="math-container">$i_1,\ldots,i_k$</span> say <span class="math-container">$i_j$</span> is <span class="math-container">$0$</span>, which implies <span class="math-container">$N(i_j)=0$</span>). Now computing <span class="math-container">$P(1)$</span>, <span class="math-container">$P(2)$</span>, and <span class="math-container">$P(3)$</span> we get:
<span class="math-container">$$P(1) = M(1) \otimes_{\mathbb{S}_1} \big(N(3) \otimes \mathbb{K}[Sh(3)]\big) \cong M(1)~ \otimes_{\mathbb{S}_1} N(3) \cong N(3)$$</span>
<span class="math-container">$$P(2) = M(2) \otimes_{\mathbb{S}_2} \Big( \big(N(1) \otimes N(2) \otimes \mathbb{K}[Sh(1,2)]\big) \oplus \big(N(2) \otimes N(1) \otimes \mathbb{K}[Sh(2,1)]\big)\Big)$$</span>
<span class="math-container">$$P(3) = M(3) \otimes_{\mathbb{S}_3} \big(N(1) \otimes N(1) \otimes N(1) \otimes \mathbb{K}[Sh(1,1,1)]\big) \cong M(3) \otimes_{\mathbb{S}_3} \mathbb{K}[\mathbb{S}_3] \cong M(3)$$</span>
My question is how to show <span class="math-container">$P(2) \cong M(2) \otimes\mathrm{Ind}^{\mathbb{S}_3}_{\mathbb{S}_2}N(2)$</span>? The authors have mentioned, "Since <span class="math-container">$\mathbb{S}_2$</span> is exchanging the two summands we get the expected result". I could not decipher the meaning of this.</p>
https://mathoverflow.net/q/4765611Characterize manifolds in Fujiki class $\mathcal C$ by smooth formsTomhttps://mathoverflow.net/users/998262024-08-08T04:03:31Z2024-08-08T04:03:31Z
<p>Let <span class="math-container">$X$</span> be a compact complex manifold, we say <span class="math-container">$X$</span> is in Fujiki class <span class="math-container">$\mathcal C$</span> if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a holomorphic modification) <span class="math-container">$\mu:\tilde X\to X$</span> such that <span class="math-container">$\tilde X$</span> is a compact Kähler manifold. Another characterizaion is <span class="math-container">$X$</span> admits a Kähler current, that is a closed <span class="math-container">$(1,1)$</span>-current <span class="math-container">$T$</span> satisfying <span class="math-container">$T≥εω$</span> for some real number <span class="math-container">$ε>0$</span> and some positive Hermitian form <span class="math-container">$ω$</span> (see for example <a href="https://www.jstor.org/stable/3597177?seq=1" rel="nofollow noreferrer">Demailly-Paun 04</a>, p.1263).</p>
<p>Let <span class="math-container">$X$</span> be a manifold in Fujiki class <span class="math-container">$\mathcal C$</span>, and <span class="math-container">$T$</span> be a Kähler current of <span class="math-container">$X$</span>. Then <span class="math-container">$T$</span> determines a class <span class="math-container">$[T]\in H^{1,1}(X,\mathbb R)$</span>, if we choose a smooths <span class="math-container">$d$</span>-closed <span class="math-container">$(1,1)$</span>-form <span class="math-container">$\tau$</span> representing the same class, i.e. <span class="math-container">$[\tau]=[T]\in H^{1,1}(X,\mathbb R)$</span>, then what preperties does <span class="math-container">$\tau$</span> possess? Obviously, <span class="math-container">$\tau$</span> should not be positive, otherwise, <span class="math-container">$\tau$</span> is a Kähler form. Then what other properties does <span class="math-container">$\tau$</span> have?</p>
https://mathoverflow.net/q/4765593Identification of Fock space and the $L^2$ space of tempered distributionsCBBAMhttps://mathoverflow.net/users/4989312024-08-08T03:39:48Z2024-08-08T09:22:57Z
<p>Let <span class="math-container">$\mathcal{S}'(\mathbb{R}^d)$</span> be the set of tempered distributions over <span class="math-container">$\mathbb{R}^d$</span> and <span class="math-container">$d\phi_C$</span> a Gaussian measure over <span class="math-container">$\mathcal{S}'(\mathbb{R}^d)$</span> with covariance operator <span class="math-container">$C$</span>. Consider the Hilbert space <span class="math-container">$H = L^2(\mathcal{S}'(\mathbb{R}^d), d\phi_C)$</span>. In Glimm & Jaffe's textbook on quantum field theory they give the identification <span class="math-container">$H = \mathcal{F}$</span>
where <span class="math-container">$\mathcal{F}$</span> denotes a Fock space representation. I am having some trouble understanding this identification. Usually Fock space is given as a (symmetric) direct sum
<span class="math-container">$$\mathcal{F} = \bigoplus_{n=0}^\infty \mathcal{F}_n$$</span>
but what are the spaces <span class="math-container">$\mathcal{F}_n$</span> when working with this particular choice of <span class="math-container">$H$</span>? How can one understand the Fock space representation of <span class="math-container">$H$</span> here?</p>
<p>I am interested in this because Glimm & Jaffe define <span class="math-container">$\mathcal{F}_n$</span> as the <span class="math-container">$n$</span> particle subspace of <span class="math-container">$\mathcal{F}$</span>, but more importantly they define Wick monomials as an orthogonal projection onto the <span class="math-container">$\mathcal{F}_n$</span>. That is, they define
<span class="math-container">$$:\phi(f_1)\cdots\phi(f_n): = E_n \phi(f_1) \cdots \phi(f_n)$$</span>
where <span class="math-container">$\phi \in \mathcal{F}$</span> (and is thus a tempered distribution) and <span class="math-container">$E_n: \mathcal{F} \rightarrow \mathcal{F}_n$</span> is an orthogonal projection.</p>
https://mathoverflow.net/q/4765571Statistics of random Voronoi S-tessellationsQidong Hehttps://mathoverflow.net/users/1730542024-08-08T02:10:34Z2024-08-08T02:10:34Z
<p>Given a locally finite set of points <span class="math-container">$\{x_{1},x_{2},\dots\}\subset\mathbb{R}^{d}$</span>, the Voronoi cell of a point <span class="math-container">$x_{i}$</span>, denoted by <span class="math-container">$C(x_{i})$</span>, consists of all the points in <span class="math-container">$\mathbb{R}^{d}$</span> that are closer to <span class="math-container">$x_{i}$</span> than to any other point <span class="math-container">$x_{j}$</span>. The partition of the whole space <span class="math-container">$\mathbb{R}^{d}$</span> by the Voronoi cells <span class="math-container">$C(x_{i})$</span> is referred to as a Voronoi tessellation.</p>
<p>A natural extension of this construction involves replacing the points <span class="math-container">$x_{1},x_{2},\dots$</span> by mutually disjoint, bounded, convex sets <span class="math-container">$K_{1},K_{2},\dots\subset\mathbb{R}^{d}$</span> so that the (generalized) Voronoi cell of the set <span class="math-container">$K_{i}$</span>, denoted by <span class="math-container">$C(K_{i})$</span>, consists of all the points in <span class="math-container">$\mathbb{R}^{d}$</span> that are closer to <span class="math-container">$K_{i}$</span> than to any other set <span class="math-container">$K_{j}$</span>. In the reference book <em>Stochastic Geometry and its Applications</em>, the partition of <span class="math-container">$\mathbb{R}^{d}$</span> by the cells <span class="math-container">$C(K_{i})$</span> is referred to as a Voronoi S-tessellation.</p>
<p>A lot is known about the statistics of random Voronoi tessellations when the underlying points <span class="math-container">$x_{i}$</span> are sampled from a Poisson point process.
Statistical quantities such as the mean volume of a typical Voronoi cell have been explicitly computed as functions of the intensity <span class="math-container">$\lambda$</span> of the Poisson point process (see Chapter 9 of <em>Stochastic Geometry and its Applications</em>).
In contrast, much less seems to be known about random Voronoi S-tessellations, in particular, when the sets <span class="math-container">$K_{i}$</span> are taken to be identical disks (d=2) or balls (d=3), as one would get by sampling from a Gibbs point process with hard-sphere interactions (again at some intensity <span class="math-container">$\lambda$</span>).
Indeed, in this case, online searches yield mostly simulation results and non-rigorous, heuristic arguments.</p>
<p>I am wondering if there is any known rigorous result on the statistics of such random Voronoi S-tessellations (e.g., the mean volume of a typical Voronoi cell) or any rigorous method that may be relevant, especially those applicable to large <span class="math-container">$\lambda\gg1$</span>.</p>
https://mathoverflow.net/q/4765543rational homology of SO(2,1) over number fieldsClaudio Bravohttps://mathoverflow.net/users/5346762024-08-08T01:18:04Z2024-08-08T01:18:04Z
<p>Let <span class="math-container">$\mathrm{SO}(2,1)$</span> be the special orthogonal group defined by the quadratic from <span class="math-container">$q(x,y,z)=x^2+y^2-z^2$</span>.
This is a connected non-simpy connected algebraic group.</p>
<p>Now, let <span class="math-container">$F$</span> be a number field, and let <span class="math-container">$G=\mathrm{SO}(2,1)(F)$</span> be the (abstract) group of <span class="math-container">$F$</span>-points of <span class="math-container">$\mathrm{SO}(2,1)$</span>.</p>
<p>My question is the following: What is known on the rational homology groups <span class="math-container">$H_*(G,\mathbb{Q})?$</span> Is it true that <span class="math-container">$H_p(G,\mathbb{Q})=\lbrace 0 \rbrace$</span>, for all <span class="math-container">$p>>0$</span>?</p>
<p>In a geometric case, i.e., assuming that <span class="math-container">$F=\mathbb{C}$</span>, Is there an integer <span class="math-container">$N>0$</span> such that <span class="math-container">$H_p(G,\mathbb{Q})=\lbrace 0 \rbrace$</span>, for all <span class="math-container">$p>N$</span>? Is <span class="math-container">$N=3$</span>?</p>
<p>In the article entitled "the homology of <span class="math-container">$\mathrm{SL}_2(F[t,t^{-1}])$</span>", K. P. Knudson proves that <span class="math-container">$H_p(\mathrm{SL}_2(F),\mathbb{Q})=\lbrace 0 \rbrace$</span>, for all <span class="math-container">$p \geq 2r+3s+1$</span>, where <span class="math-container">$r$</span> (resp. <span class="math-container">$s$</span>) is the number of real (resp. conjugate pairs of complex) embeddings of <span class="math-container">$F$</span>.
In order to prove this fact, Knudson uses a result due to Borel and Yang (see the article "The rank conjecture for number fields") on the rational cohomology of <span class="math-container">$\mathrm{SL}_2(F)$</span>. However, the Borel & Yang's result holds for simply connected groups (thus it does not directly apply for <span class="math-container">$\mathrm{SO}(2,1)$</span>).
We can work with the <span class="math-container">$\mathrm{Spin}(2,1)$</span> group, but I think that there are some technical difficulties.</p>
<p>Thanks in advance :)</p>
https://mathoverflow.net/q/4765531Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper boundSnake Eyeshttps://mathoverflow.net/users/3864962024-08-08T00:53:26Z2024-08-08T08:43:52Z
<p><span class="math-container">$\DeclareMathOperator\grade{grade}$</span>Let <span class="math-container">$R$</span> be a commutative Noetherian ring. For an ideal <span class="math-container">$I$</span> of <span class="math-container">$R$</span>, let <span class="math-container">$\grade(I,R)$</span> be the maximal length of an <span class="math-container">$R$</span>-regular sequence in <span class="math-container">$I$</span>.</p>
<p>My question is: If <span class="math-container">$\sup\{\grade(\mathfrak m, R)\mid \mathfrak m \in \operatorname{MaxSpec}(R)\}$</span> is finite, then does <span class="math-container">$R$</span> have finite Krull dimension?</p>
https://mathoverflow.net/q/4765520Compute $\int_{-\infty}^\infty \exp\left(-a_1 x^2 + a_2 x + a_3 \exp(a_4 x) \right) \text{d}x$ numericallyriantihttps://mathoverflow.net/users/5346822024-08-08T00:32:17Z2024-08-08T00:32:17Z
<p>Would anyone suggest how to compute the following integral fast and accurately using numerical approximation methods, other than the Monte Carlo method?</p>
<p><span class="math-container">$$\int_{-\infty}^\infty \exp\left(-a_1 x^2 + a_2 x + a_3 \exp(a_4 x) \right) \text{d}x$$</span></p>
<p>where <span class="math-container">$a_1 > 0$</span>, and <span class="math-container">$a_2, a_3, a_4 \in \mathbb{R}$</span></p>
<p>Thanks</p>
https://mathoverflow.net/q/4765350Does nice behavior near a singular point force solution to be in Frobenius series?Jack Buttcanehttps://mathoverflow.net/users/950022024-08-07T17:05:51Z2024-08-08T09:15:32Z
<p>I have a pair of partial differential operators <span class="math-container">$\Delta_1$</span> and <span class="math-container">$\Delta_2$</span> in <span class="math-container">$y_1, y_2$</span> formed from constants, multiplication by <span class="math-container">$y_1$</span> or <span class="math-container">$y_2$</span> and derivatives in the form <span class="math-container">$y_1 \frac{\partial}{\partial y_1}$</span>, <span class="math-container">$y_2 \frac{\partial}{\partial y_2}$</span>. I know that there are exactly 12 Frobenius series solutions <span class="math-container">$\Delta_1 f_j = \Delta_2 f_j=0$</span>,
<span class="math-container">$$ f_j(y_1,y_2) = y_1^{i a_{j,1}} y_2^{i a_{j,2}} \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty b_{j,n_1,n_2} y_1^{n_1} y_2^{n_2}, \qquad j=1,\ldots,12 $$</span>
with <span class="math-container">$a_{j,k} \in \mathbb{R}$</span>, <span class="math-container">$b_{j,n_1,n_2} \in \mathbb{C}$</span>, <span class="math-container">$b_{j,0,0}=1$</span>.
I have another function <span class="math-container">$g(y_1,y_2)$</span>, defined by an integral which also satisfies <span class="math-container">$\Delta_1 g = \Delta_2 g = 0$</span> and near the singular point <span class="math-container">$y_1=y_2=0$</span> has
<span class="math-container">$$ g(y_1,y_2) = \sum_{j=1}^{12} c_j y_1^{i a_{j,1}} y_2^{i a_{j,2}}+O(y_1+y_2). $$</span>
If I knew that the space of all solutions has dimension 12, I could conclude that
<span class="math-container">$$ g(y_1,y_2) = \sum_{j=1}^{12} c_j f_j(y_1, y_2). $$</span>
I cannot show the dimension is 12; it may well be false.
Is this condition on the behavior of <span class="math-container">$g$</span> near the singular point enough to conclude it lies in the span of the Frobenius series?
Something to the effect of solutions that don't lie in the span of the Frobenius series must be badly behaved near the singular point?</p>
https://mathoverflow.net/q/4765291Distribution class closed under convolution counterexample?japalmerhttps://mathoverflow.net/users/5102062024-08-07T14:58:26Z2024-08-08T06:26:15Z
<p>Define the class of probability density functions: <span class="math-container">$p \in \mathcal{C}$</span> iff <span class="math-container">$p(x)=p(|x|)$</span>, and <span class="math-container">$\log p(\!\sqrt{x})$</span> is convex on <span class="math-container">$[0,\infty)$</span>.</p>
<p>Conjecture: if <span class="math-container">$p,q \in \mathcal{C}$</span>, then <span class="math-container">$p * q \in \mathcal{C}$</span> (where <span class="math-container">$*$</span> denotes convolution).</p>
<p>This would be similar to the closure of the class of log concave densities (Polya frequency functions <span class="math-container">$\mathrm{PF}_2$</span>) under convolution, but applying to heavy-tailed densities.</p>
<p>Can anyone come up with a counterexample (or proof)?</p>
https://mathoverflow.net/q/4765211Primality of divisor sumsStanley Yao Xiaohttps://mathoverflow.net/users/108982024-08-07T14:03:29Z2024-08-08T05:16:38Z
<p>Let <span class="math-container">$k \geq 2$</span> be an integer. Put <span class="math-container">$[k] = \{1, \cdots, k\}$</span>. Let <span class="math-container">$\mathcal{P} = \{p_1, \cdots, p_k\}$</span> be a set of <span class="math-container">$k$</span> primes. For every subset <span class="math-container">$S \subseteq [k]$</span> put <span class="math-container">$d_S = \prod_{j \in S} p_j$</span>. The empty product is equal to <span class="math-container">$1$</span>, by convention.</p>
<p>For every <span class="math-container">$k \geq 2$</span>, does there always exist a set of primes <span class="math-container">$\mathcal{P}$</span> of cardinality equal to <span class="math-container">$k$</span> such that for every partition <span class="math-container">$A \sqcup B = [k]$</span> we have <span class="math-container">$d_A + d_B = 2^k q$</span>, where <span class="math-container">$q$</span> is an odd prime?</p>
https://mathoverflow.net/q/4765166A stable splitting of linear surjectionsConnor Malinhttps://mathoverflow.net/users/1345122024-08-07T12:37:30Z2024-08-08T10:23:26Z
<p>Some computations I've been doing in Weiss calculus predict the following stable splitting of the space of linear surjections: <span class="math-container">$\Sigma^\infty_+ \mathrm{Sur}(\mathbb{R}^n,\mathbb{R}^{m_1+m_2})$</span>
as the wedge sum <span class="math-container">$\bigvee_{i \leq n} (\Sigma^\infty_+\mathrm{Sur}(\mathbb{R}^i,\mathbb{R}^{m_1}) \wedge \Sigma^\infty_+\mathrm{Sur}(\mathbb{R}^{n-i},\mathbb{R}^{m_2}) \wedge \Sigma^\infty_+ O(n)^\vee)_{h(N(O(i) \times O(n-i)))} \wedge \mathrm{Ad}^{O(n)}.$</span></p>
<p>Here <span class="math-container">$ N(O(i) \times O(n-i))$</span> is the normalizer in <span class="math-container">$O(n)$</span> and <span class="math-container">$\mathrm{Ad}^{O(n)}$</span> is the adjoint representation of <span class="math-container">$O(n)$</span>. My question is whether or not this splitting actually exists.</p>
<p>It seems possible that this splitting is equivalent to <a href="https://math.mit.edu/%7Ehrm/papers/stable-splittings.pdf" rel="nofollow noreferrer">Miller's stable splitting</a> of Stiefel manifolds, and there are similar formulas one can write down for any partition <span class="math-container">$m_1+m_2+\dots +m_k$</span> which are also predicted to be true. Note that a similar formula does hold if we replace linear surjections by surjections of finite sets.</p>
https://mathoverflow.net/q/47650111Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-spaceIlya Gekhtmanhttps://mathoverflow.net/users/1829552024-08-07T08:49:17Z2024-08-08T02:23:11Z
<p>Consider the group <span class="math-container">$\operatorname{PSL}(2,\mathbb C)$</span> acting by Möbius transformations of the Riemann sphere. It is known that this action can be extended to an action on the unit ball which is isometric with respect to the hyperbolic metric. To prove this you write each Möbius transformation as a product of inversions in spheres and show that each of these act by isometries.</p>
<p>My question: can anyone point out a reference where this is done rigorously and explicitly (or explain the computation)? Everywhere I read the proof that inversions in spheres act by isometries on the unit ball with the hyperbolic metric is left as an "exercise" which I cannot do. I am teaching a class on this but need to understand myself...</p>
https://mathoverflow.net/q/4764714Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?Arbujahttps://mathoverflow.net/users/878562024-08-06T17:58:37Z2024-08-08T01:34:30Z
<p>Suppose <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> is Borel. Let <span class="math-container">$\text{dim}_{\text{H}}(\cdot)$</span> be the Hausdorff dimension, and <span class="math-container">$\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$</span> be the Hausdorff measure <em>in its dimension</em> on the Borel <span class="math-container">$\sigma$</span>-algebra.</p>
<blockquote>
<p><strong>Question:</strong> If <span class="math-container">$G$</span> is the graph of <span class="math-container">$f$</span>, is there an <em>explicit</em> <span class="math-container">$f$</span> such that:</p>
<ol>
<li>The function <span class="math-container">$f$</span> is everywhere surjective (i.e., <span class="math-container">$f[(a,b)]=\mathbb{R}$</span> for all non-empty open intervals <span class="math-container">$(a,b)$</span>)</li>
<li><span class="math-container">$\mathcal{H}^{\text{dim}_{\text{H}}(G)}(G)=0$</span></li>
</ol>
</blockquote>
<p>Note, not all everywhere surjective <span class="math-container">$f$</span> satisfy 2. of the question. For example, consider the <a href="https://en.wikipedia.org/wiki/Conway_base_13_function" rel="nofollow noreferrer">Conway base-13 function</a>. Since it's zero almost everywhere, <span class="math-container">$\text{dim}_{\text{H}}(G)=1$</span>, and <span class="math-container">$\mathcal{H}^{\text{dim}_{\text{H}}(G)}(G)=+\infty$</span>.</p>
<blockquote>
<p><strong>Optional</strong>: If such an <span class="math-container">$f$</span> exists, does <span class="math-container">$f$</span> have other interesting properties?</p>
</blockquote>
https://mathoverflow.net/q/4764133What are necessary and/or sufficient conditions for a Dirichlet series to admit analytic continuation?Stanley Yao Xiaohttps://mathoverflow.net/users/108982024-08-05T22:44:42Z2024-08-08T00:03:53Z
<p>Let <span class="math-container">$A = \{a(n)\}_{n \geq 1}$</span> be a sequence of complex numbers. By normalizing, we may as well assume that <span class="math-container">$|a(n)| \leq 1$</span> for all <span class="math-container">$n \geq 1$</span>. Under this assumption, the Dirichlet series</p>
<p><span class="math-container">$\displaystyle D_A(s) = \sum_{n=1}^\infty \frac{a(n)}{n^s}$</span></p>
<p>converges absolutely on the half-plane <span class="math-container">$\mathbb{H}_{>1} = \{z \in \mathbb{C} : \Re(z) > 1\}$</span>.</p>
<p>What is the most general statement, in terms of conditions on the sequence <span class="math-container">$A$</span>, which guarantees that <span class="math-container">$D_A(s)$</span> can be analytically continued to give a meromorphic function on the whole complex plane?</p>
<p>Certain examples show that conditions might be quite subtle. For example, if</p>
<p><span class="math-container">$\displaystyle a(n) = \begin{cases} 1 & \text{if } n \text{ is prime} \\ 0 & \text{otherwise} \end{cases}$</span></p>
<p>then <span class="math-container">$D_A(s)$</span> cannot be analytically continued.</p>
https://mathoverflow.net/q/4763321Using topology for proving periodicityG. Panelhttps://mathoverflow.net/users/1599402024-08-04T15:26:37Z2024-08-08T08:51:41Z
<p>Let <span class="math-container">$f\in\mathcal{C}\big(\mathbb{R},\mathbb{C}^\ast\big)$</span> a continuous function with modulus <span class="math-container">$r$</span> satisfying: <span class="math-container">$f(t)=r(t)e^{it}$</span>. Assume that the image of <span class="math-container">$f$</span> is homeomorphic to the unit circle.</p>
<p><strong>Question:</strong> Is <span class="math-container">$f$</span> a <span class="math-container">$2\pi$</span>-periodic function?</p>
https://mathoverflow.net/q/4762761Bounds for ground set of Steiner system (inverse EKR style problem)Tuatarianhttps://mathoverflow.net/users/5280952024-08-03T12:48:49Z2024-08-08T11:21:55Z
<p>Imagine we have <span class="math-container">$r$</span> subsets of a ground set <span class="math-container">$S$</span>, each of size <span class="math-container">$k$</span>, such that each set of size <span class="math-container">$l$</span> is contained in at most one of the <span class="math-container">$r$</span> sets. What can we say about the minimum value of <span class="math-container">$|S|$</span>? I am primarily interested in lower bounds but any info about upper bounds would be helpful as well</p>
https://mathoverflow.net/q/4747101Prove or disprove that the matrix equation of the form $AX+XA^{-T}=0$ has a nonsingular anti-symmetric solution $X$White Cathttps://mathoverflow.net/users/5319702024-07-09T09:09:54Z2024-08-08T12:01:40Z
<p>I’m trying to prove that for <span class="math-container">$A=J_n(i)$</span>, that is, the Jordan block matrix corresponding to the eigenvalue <span class="math-container">$i$</span> of size <span class="math-container">$n$</span>, where <span class="math-container">$n$</span> is even, the matrix equation <span class="math-container">$AX+XA^{-T}=0$</span> has a nonsingular anti-symmetric solution <span class="math-container">$X$</span>.</p>
<p>I have tried it on small values of <span class="math-container">$n (= 2,4,6)$</span> by brute force computations and proved that it is true.</p>
<p>Any ideas on how can I prove this for an arbitrary even <span class="math-container">$n$</span>? Ideas or suggestions would suffice. Thank you so much!</p>
https://mathoverflow.net/q/4531322Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequencySimple Conjugatehttps://mathoverflow.net/users/5108062023-08-20T17:37:54Z2024-08-07T23:05:50Z
<p>let <span class="math-container">$$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$</span></p>
<p>I'm studying fractal geometry and I've encountered the following question: Is there an <span class="math-container">$\alpha$</span>-regular measure giving <span class="math-container">$X$</span> positive measure?</p>
<p>Now, I've noticed that the Lebesgue measure of <span class="math-container">$X$</span> must be zero, simply by looking at its complement, and by the LLN the limit <span class="math-container">$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}a_{i}$</span> exists almost surely, hence <span class="math-container">$X^{c}$</span> has Lebesgue measure 0. I'm not sure how to proceed, and would be happy for any suggestions.
In general I'm trying to find the Hausdorff dimension of <span class="math-container">$X$</span>.</p>
https://mathoverflow.net/q/44581912Normal curvature of Veronese embeddingAnton Petruninhttps://mathoverflow.net/users/14412023-04-30T12:48:51Z2024-08-08T01:59:52Z
<blockquote>
<p>Is it true that the Veronese embedding minimizes the maximal normal curvature among all smooth embeddings <span class="math-container">$\mathbb{R}\mathrm{P}^n\hookrightarrow \mathbb{S}^N$</span>?</p>
</blockquote>
<p><strong>Remarks</strong></p>
<ul>
<li><p>The <a href="https://en.wikipedia.org/wiki/Veronese_surface" rel="nofollow noreferrer">Veronese embedding</a> <span class="math-container">$\mathbb{R}\mathrm{P}^n\hookrightarrow \mathbb{S}^{(n+3)\cdot n/2}$</span> describes a minimal surface. Its induced metric tensor <span class="math-container">$=2\times$</span>(canonical metric on <span class="math-container">$\mathbb{R}\mathrm{P}^n$</span>). It sends geodesics of <span class="math-container">$\mathbb{R}\mathrm{P}^n$</span> to circles with curvature <span class="math-container">$\kappa_n=\sqrt{\tfrac{n-1}{n+1}}$</span> in the sphere; in particular, its normal curvature <span class="math-container">$\kappa_n$</span> at every point in every direction.</p>
</li>
<li><p>The same question can be asked about <span class="math-container">$\mathbb{C}\mathrm{P}^n$</span> and <span class="math-container">$\mathbb{H}\mathrm{P}^n$</span>.</p>
</li>
<li><p>The value <span class="math-container">$\kappa_n$</span> can be visualized the following way. Suppose <span class="math-container">$d_n$</span> is the spherical distance between vertices of an inscribed equilateral <span class="math-container">$n$</span>-simplex in <span class="math-container">$\mathbb{S}^{n-1}$</span>. Then <span class="math-container">$\kappa_n$</span> is the curvature of a spherical circle of radius <span class="math-container">$d_n/2$</span>. Note that <span class="math-container">$d_n\to \tfrac\pi2$</span> and <span class="math-container">$\kappa_n\to 1$</span> as <span class="math-container">$n\to\infty$</span>.</p>
</li>
<li><p>I can solve the case <span class="math-container">$n=2$</span>. Now <a href="https://anton-petrunin.github.io/veronese/RP2.pdf" rel="nofollow noreferrer">it is written</a>; any comments are welcome.</p>
</li>
<li><p>In fact any submanifold with normal curvature <span class="math-container">$<\kappa_2=\tfrac1{\sqrt{3}}$</span> is diffeomorphic to the standard sphere. (In this case the curvature of the induced metric is <span class="math-container">$\tfrac14$</span>-pinched; so it is sufficient to show that the submanifold is simply connected.)</p>
</li>
<li><p>This question is from my <a href="https://arxiv.org/abs/2304.00886" rel="nofollow noreferrer">paper (see 1.4)</a>, and it was motivated by Gromov's <a href="https://arxiv.org/abs/2212.06122" rel="nofollow noreferrer">"Isometric immersions with controlled curvatures"</a>.</p>
</li>
</ul>
https://mathoverflow.net/q/4142164Find all integer solutions to the following easy-looking Diophantine equationsBogdan Grechukhttps://mathoverflow.net/users/890642022-01-19T14:50:34Z2024-08-08T06:05:50Z
<p>In general, it is not clear <a href="https://mathoverflow.net/questions/410297">What does it mean to solve an equation?</a> in integers. In this question, let us assume that an equation
<span class="math-container">$$
P(x_1,\dots,x_n)=0
$$</span>
is solved if we have proved that its integer solution set <span class="math-container">$S \subset {\mathbb Z}^n$</span> can be represented as the finite union <span class="math-container">$S=S_1 \cup \dots \cup S_m$</span>, where each <span class="math-container">$S_i$</span> is either a polynomial family or a family defined by recurrence relations. Here, <span class="math-container">$S \subset {\mathbb Z}^n$</span> is a polynomial family if there exists polynomials <span class="math-container">$P_1,\dots,P_n$</span> in some variables <span class="math-container">$u_1,\dots,u_k$</span> and integer coefficients such that <span class="math-container">$(x_1,\dots,x_n) \in S$</span> if and only if there exists integers <span class="math-container">$u_1,\dots,u_k$</span> such that <span class="math-container">$x_i=P_i(u_1,\dots,u_k)$</span> for <span class="math-container">$i=1,\dots,n$</span>.</p>
<p>Following Zidane <a href="https://mathoverflow.net/questions/316708/">What is the smallest unsolved Diophantine equation?</a> , let us define size <span class="math-container">$H$</span> of the equation <span class="math-container">$P=0$</span> as a result of substitution 2 instead of all variables, absolute values instead of all coefficients, and evaluating.</p>
<p>All equations with <span class="math-container">$H \leq 8$</span> are easy to solve. However, simple-looking equation <span class="math-container">$xy-zt=1$</span> with <span class="math-container">$H=9$</span> has been open for decades. In 2010, Vaserstein<sup>1</sup> <a href="https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n2-p07-p.pdf" rel="nofollow noreferrer">https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n2-p07-p.pdf</a> proved that</p>
<ul>
<li>The solution set to <span class="math-container">$xy-zt=1$</span> is a polynomial family with <span class="math-container">$46$</span> parameters.</li>
</ul>
<p>As a corollary of this, Vaserstein solved the following families of equations</p>
<ul>
<li><span class="math-container">$xy-zt=D$</span> for any integer <span class="math-container">$D$</span>;</li>
<li><span class="math-container">$yz=x^2+D$</span> for any integer <span class="math-container">$D$</span>;</li>
<li><span class="math-container">$x_1x_2+x_3x_4+Q(x_5,\dots,x_n)=D$</span> for quadratic form <span class="math-container">$Q$</span> and integer <span class="math-container">$D$</span>.</li>
</ul>
<p>In addition, the following equations/families has been solved:</p>
<ul>
<li>Equations in the form <span class="math-container">$dyz=ax^2+bx+c$</span> for integers <span class="math-container">$a,b,c,d$</span> has been solved in the answer to question <a href="https://mathoverflow.net/questions/412451/">Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$</a></li>
<li>All equations with <span class="math-container">$H \leq 12$</span>.</li>
</ul>
<p>I was also able to solve all equations with <span class="math-container">$H=13$</span> except of the following ones.
<span class="math-container">$$
x^3 + 1 = yz
$$</span>
<span class="math-container">$$
x^2y=z^2 \pm 1
$$</span>
<span class="math-container">$$
x^2y=tz+1
$$</span>
<span class="math-container">$$
x^2 + y^2 \pm 1 = zt
$$</span>
<span class="math-container">$$
x^2 \pm 1 = yzt
$$</span>
<span class="math-container">$$
x_1x_2x_3+x_4x_5=1
$$</span></p>
<p>For each of the listed equations, the <strong>question</strong> is to find all integer solutions. Specifically, check whether the set of all integer solutions is a finite union of polynomial families and/or families defined by recurrence relations. You do not need to write the resulting families explicitly, because, as example <span class="math-container">$xy-zt=1$</span> indicates, they may be quite complicated.</p>
<p>See here <a href="https://mathoverflow.net/questions/400714">Can you solve the listed smallest open Diophantine equations?</a> for a version of this question where we only want to check whether any integer solution exists, and here <a href="https://mathoverflow.net/questions/411958">On the smallest open Diophantine equations: beyond Hilbert's 10 problem</a> for a version where we also check whether the solution set is finite or infinite (and find all solutions if there are finitely many).</p>
<p><sup>1</sup><em>Vaserstein, Leonid</em>, <a href="https://doi.org/10.4007/annals.2010.171.979" rel="nofollow noreferrer"><strong>Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups</strong></a>, Ann. Math. (2) 171, No. 2, 979-1009 (2010). <a href="https://zbmath.org/?q=an:1221.11082" rel="nofollow noreferrer">ZBL1221.11082</a>, <a href="https://www.jstor.org/stable/20752233" rel="nofollow noreferrer">JSTOR</a>.</p>
https://mathoverflow.net/q/3840278A formula in Ramanujan's lost notebook and its connection with Chudnovsky series for $1/\pi$Paramanand Singhhttps://mathoverflow.net/users/155402021-02-15T12:02:45Z2024-08-08T07:11:33Z
<p>While studying Berndt's <em>Ramanujan's Lost Notebook Vol. 2</em>, page 369 (<a href="https://doi.org/10.1007/b13290_16" rel="nofollow noreferrer">chapter on Springerlink</a>), I found that Ramanujan gave values of a certain expression <span class="math-container">$$\frac{1}{\sqrt{Q_n}}\left(\sqrt {n} P_n-\frac{6}{\pi}\right)\tag{1}$$</span> for a few integer values of <span class="math-container">$n$</span>. Most notable among them is the value for <span class="math-container">$n=163$</span> with <span class="math-container">$$\frac{1}{\sqrt{Q_n}}\left(\sqrt {n} P_n-\frac{6}{\pi}\right)=362\sqrt{\frac{3}{3335}}\tag{2}$$</span> Let's now define the symbols used above to get full context needed for the question. We have by definition
<span class="math-container">\begin{align}
P(q) & =1-24\sum_{j=1}^{\infty} \frac{jq^j} {1-q^j}\tag{3a}\\
Q(q) & =1+240\sum_{j=1}^{\infty} \frac{j^3q^j} {1-q^j}\tag{3b}\\
R(q) & =1-504\sum_{j=1}^{\infty} \frac{j^5q^j} {1-q^j}\tag{3c}
\end{align}</span>
and <span class="math-container">$$P_n=P(-e^{-\pi\sqrt {n}}), Q_n=Q(-e^{-\pi\sqrt {n}}), R_n=R(-e^{-\pi\sqrt{n}})\tag{4}$$</span> These expressions famously appear in the general series for <span class="math-container">$1/\pi$</span> given by Chudnovsky brothers <span class="math-container">$$ \frac{1}{\pi} = \frac{1}{\sqrt{-j_{n}}}\sum_{m = 0}^{\infty}\frac{(6m)!}{(3m)!(m!)^{3}}\frac{a_{n} + mb_{n}}{j_{n}^{m}}\tag{5}$$</span> where
<span class="math-container">\begin{align}
j_{n} &= 1728\frac{Q_{n}^{3}}{Q_{n}^{3} - R_{n}^{2}}\tag{6a}\\
b_{n} &= \sqrt{n(1728 - j_{n})}\tag{6b}\\
a_{n} &= \frac{b_{n}}{6}\left\{1 - \frac{Q_{n}}{R_{n}}\left(P_{n} - \frac{6}{\pi\sqrt{n}}\right)\right\}\tag{6c}
\end{align}</span>
We can see that Ramanujan's expression <span class="math-container">$(1)$</span> is related to <span class="math-container">$a_n$</span> in <span class="math-container">$(6c)$</span>.</p>
<p>Berndt obtained the value in <span class="math-container">$(2)$</span> by using a table of values of <span class="math-container">$a_n, b_n$</span>. Berndt's book does not indicate the procedure to obtain these numbers.</p>
<p>My question is whether Ramanujan had some inkling of the Chudnovsky series or not. At least the available literature does not give any explicit details regarding this. Also it is not clear how he computed the value of expression <span class="math-container">$(1)$</span> for <span class="math-container">$n=163$</span>. Is there any way to evaluate this without using <span class="math-container">$a_n, b_n$</span>?</p>
<hr />
<p>It is interesting to observe however that the Chudnovsky series <span class="math-container">$(5)$</span> can be obtained using the approach described by Ramanujan in his 1914 paper <em>Modular Equations and Approximations to <span class="math-container">$\pi$</span></em> (<a href="http://ramanujan.sirinudi.org/Volumes/published/ram06.html" rel="nofollow noreferrer">Link</a>).</p>
<p><strong>Update</strong>: In the above paper Ramanujan gives a technique to evaluate <span class="math-container">$P(q^2)$</span> in closed form for <span class="math-container">$q=e^{-\pi\sqrt{n}} $</span> for many integer values of <span class="math-container">$n$</span>. Using the identity <span class="math-container">$$2P(q^2)-P(-q)=\left(\frac{2K}{\pi}\right)^2(1-2k^2)$$</span> one can then calculate the value of <span class="math-container">$P(-q) $</span> in closed form. I tried to do the calculations for <span class="math-container">$n=11$</span> but it seems too formidable and I couldn't complete it so far. Also the paper by Ramanujan does not have the formulas related to <span class="math-container">$n=43,67,163$</span> and thus it appears that he did develop some formulas for these values after his 1914 paper, but somehow failed to record them in his lost notebook and instead gave direct evaluation of expression <span class="math-container">$(1)$</span> for some values of <span class="math-container">$n$</span>.</p>
https://mathoverflow.net/q/2914853An Abelian quotient ring by soclekarparvarhttps://mathoverflow.net/users/488892018-01-26T15:35:42Z2024-08-07T23:07:23Z
<p>Let $R$ be a ring with identity whose (right) socle $S$ contains its nilpotent elements. Is it necessarily true that the quotient $R/S$ is an abelian ring? ( By an abelian ring I mean a ring whose idempotents are central.)
For example, $R$ may be a reduced ring -such as a domain-which does trivialy satisfies the hypothesis.
Thanks for any answer!</p>
https://mathoverflow.net/q/2780452Mapping between NotationsSSequencehttps://mathoverflow.net/users/1123852017-08-05T20:01:46Z2024-08-08T00:41:50Z
<p>As in my other question, it is assumed that the (total) function describing a given notation is denoted as $address:p\rightarrow \Bbb{N}$ and assumed to be bijective. </p>
<p>Suppose we are given two notations $N_1$ and $N_2$ for some $p \in \omega{_C}{_K}$(Church-Kleene). Denote the mapping from $N_1$ to $N_2$ as $P{_1}{_2}:\Bbb{N} \rightarrow \Bbb{N}$.</p>
<p>If we wanted to be more formal, we could say that consider the two functions $address1:p\rightarrow \Bbb{N}$ and $address2:p\rightarrow \Bbb{N}$ corresponding to $N_1$ and $N_2$ respectively. Then $P{_1}{_2}$ is defined by:
$$P{_1}{_2}(address1(x))=address2(x) \qquad for \; all \; x < p$$</p>
<p>Now assume that the less-than relations corresponding to $N_1$ and $N_2$ are computable. My question simply is the following: Is the function $P{_1}{_2}$ always Total-computable? That is, computable using an oracle-program that has access to the set representing "indexes of total recursive functions". </p>
<p>If no, then what would be the example for it (also generally, what would be the suitable upper-bound in that case). If yes, then can someone give a reference where a complete or partial proof(if there is one) for a reasonable upper-bound(or lack of it) is provided.</p>
<p>P.S.
I posted this question on Math.SE (<a href="https://math.stackexchange.com/questions/2380838/mapping-between-notations">https://math.stackexchange.com/questions/2380838/mapping-between-notations</a>). After no replies/comments since few days after posting it, I am posting it here. Hopefully that isn't a problem. </p>