Recent Questions - MathOverflow most recent 30 from www.2874565.com 2019-03-12T19:09:57Z http://www.2874565.com/feeds http://www.creativecommons.org/licenses/by-sa/3.0/rdf http://www.2874565.com/q/325284 1 Metric with singularities on Riemann Surfaces and the associated Laplacians Wenzhe http://www.2874565.com/users/87910 2019-03-12T18:42:36Z 2019-03-12T18:42:36Z <p>I have asked this question on Math Stack Exchange </p> <p><a href="https://math.stackexchange.com/questions/3143232/metric-with-singularities-and-associated-laplacian">Metric with singularities and associated Laplacian</a></p> <p>but I have not got any answers/comments, therefore I post this question on the MO. </p> <p>Suppose <span class="math-container">$M$</span> is a compact Riemann surface, and <span class="math-container">$g$</span> is a metric on <span class="math-container">$M$</span> with finitely many singular points. For simplicity let us impose further restrictions on <span class="math-container">$g$</span>, and we suppose in every local neighborhood with coordinate <span class="math-container">$z$</span>, <span class="math-container">$g$</span> is of the form <span class="math-container">\begin{equation} g=f(z)\overline{f(z)}dz d\bar{z}, \end{equation}</span> where <span class="math-container">$f(z)$</span> is a holomorphic function with a power series expansion <span class="math-container">\begin{equation} f(z)=\sum_{m\geq N} a_{m}z^m , N \in \mathbb{Z}. \end{equation}</span> We say <span class="math-container">$g$</span> is singular at the point <span class="math-container">$z=0$</span> if the series expansion of <span class="math-container">$f$</span> has negative powers. In particular we have excluded the case where <span class="math-container">$f$</span> has an essential singularity at <span class="math-container">$z=0$</span>. With respect to such a metric, we can still define the associated Laplacian operator <span class="math-container">$\Delta$</span> on the smooth locus of <span class="math-container">$g$</span>. </p> <p>My question is, are there any results about the spectrum of such a Laplacian? In particular, is it bounded from below? References are welcomed. </p> http://www.2874565.com/q/325283 0 A Wolstenholme typ congruence 1Ichi1 http://www.2874565.com/users/136970 2019-03-12T18:23:58Z 2019-03-12T18:23:58Z <p>I think I can prove the following congruence: For <span class="math-container">$p\geq 5$</span> prime and every <span class="math-container">$n,\nu\in\mathbb{N}$</span> we have <span class="math-container">\begin{align*} 0\equiv\sum_{k=1\atop p\nmid k}^{pn-1}\frac1k \binom{pn(\nu+1)-k-1}{pn\nu-1} \mod p^{2(\operatorname{ord}_p(n)+1)}\mathbb{Z}_p. \end{align*}</span> My question: Is this a known identity or can be obtained by a more general satement?</p> http://www.2874565.com/q/325282 -2 How to calculate coeficients (for margins) from retail selling price [on hold] Make001 http://www.2874565.com/users/136715 2019-03-12T17:42:51Z 2019-03-12T17:42:51Z <p>I offered the client a price of 22USD Per Unit of our Blast Face Sponge. They are mentioning that they need a coefficient of 4.0 because the retailers they sell, request 1.8 - 2.0 ( they earn on higher margins, not on volumes as the market is small).</p> <p>How can I calculate the coefficient to provide the margins this customer is requesting.</p> http://www.2874565.com/q/325279 1 Extension of Erdos-Selfridge Theorem Matt Cuffaro http://www.2874565.com/users/123309 2019-03-12T17:19:23Z 2019-03-12T17:33:55Z <p>Erdos and Selfridge open their paper <a href="https://projecteuclid.org/download/pdf_1/euclid.ijm/1256050816" rel="nofollow noreferrer">"The Product of Consecutive Integers is Never a Power"</a> (1974) with the theorem</p> <blockquote> <p><span class="math-container">$\text{Theorem 1:}$</span> <em>The product of two or more consecutive positive integers is never a power.</em></p> </blockquote> <p><strong>Question:</strong> While I am working through the paper to understand their argument, I wonder: has there been any work extending these results to other rings of integers? <em>i.e.</em> falsifying the equation <em>in general</em> <span class="math-container">$$ (a+1)\cdots(a+k)=b^l $$</span> for (what I suspect) some number field <span class="math-container">$F$</span>, positive <span class="math-container">$a,b\in\mathcal{O}_F$</span>, and <span class="math-container">$k,l\geq 2$</span>.</p> <p><strong>Note</strong>: I know that for <span class="math-container">$F=\mathbb{Q}[\sqrt{5}],\&gt;$</span> <span class="math-container">$a=b=\frac{1}{2}(\sqrt{5}-1),\&gt;k=2,\&gt;l=3$</span>, the equation is true. Perhaps it would invite a reason to restrict the question to fields <span class="math-container">$F=\mathbb{Q}[\sqrt{m}]$</span> where <span class="math-container">$m\not\equiv 1\mod 4.$</span></p> http://www.2874565.com/q/325278 4 $\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects Reid Barton http://www.2874565.com/users/126667 2019-03-12T17:09:56Z 2019-03-12T17:09:56Z <p>Remark 1.30 of Adámek and Rosický, Locally Presentable and Accessible Categories claims that in any locally <span class="math-container">$\lambda$</span>-presentable category, each <span class="math-container">$\mu$</span>-presentable object can be written as a <span class="math-container">$\mu$</span>-small colimit of <span class="math-container">$\lambda$</span>-presentable objects. I've also seen this stated in the literature without any reference given, suggesting it is considered "well-known to experts".</p> <p>However, as Mike Shulman pointed out in a comment on the answer <a href="http://www.2874565.com/a/306129">http://www.2874565.com/a/306129</a>, it is unclear how the argument on pages 35 to 37 of Makkai and Paré cited in Remark 1.30 proves the claim. Not only is it unclear how to apply Lemma 2.5.2 of MP, but the category <span class="math-container">$\mathbf{K}$</span> constructed in its proof, which is the indexing category for the colimit produced by the lemma, has size which is not obviously bounded in terms of the sizes of the input diagrams, because it involves arbitrary morphisms between the given objects, not just ones that appear in the given diagrams.</p> <p>Does anyone know how the claim of Remark 1.30 is to be proved? Alternatively, is there another, perhaps entirely different, proof in the literature?</p> http://www.2874565.com/q/325276 0 The algebraic variety behind a quotient of the ring of regular functions on the complex torus BrianT http://www.2874565.com/users/124592 2019-03-12T17:00:42Z 2019-03-12T17:00:42Z <p>I am trying to understand what algebraic variety stands behind the following quotient.</p> <p>Let <span class="math-container">$I$</span> be the vanishing ideal of some linear subspace <span class="math-container">$\mathbb{C}^k \subset \mathbb{C}^n$</span> in the standard affine variety <span class="math-container">$\mathbb{C}^n$</span>. Then it is elementary that the ring of regular functions on <span class="math-container">$\mathbb{C}^k$</span> is <span class="math-container">$$ \mathbb{C}[u_1,...,u_n] / I. $$</span> So far so good. Let's now consider the algebraic torus <span class="math-container">$(\mathbb{C}^{\times})^n$</span>. Its ring of regular functions is <span class="math-container">$$ \mathbb{C}[u_1,...u_n,u_1^{-1},...,u_n^{-1}]. $$</span> The quotient I am interested in is: <span class="math-container">$$ \mathbb{C}[u_1,...u_n,u_1^{-1},...,u_n^{-1}] / I \mathbb{C}[u_1,...u_n,u_1^{-1},...,u_n^{-1}]. $$</span></p> <p>I was told that this quotient is in fact the ring of regular functions on the intersection <span class="math-container">$\mathbb{C}^k \cap (\mathbb{C}^{\times})^n$</span>, but I have no idea how to prove this. Could someone help ?</p> <p>Thanks you all!</p> http://www.2874565.com/q/325273 2 Convex hull of all rank-1 {-1, 1}-matrices? Tigran Saluev http://www.2874565.com/users/58990 2019-03-12T16:45:28Z 2019-03-12T16:45:28Z <p>Consider set <span class="math-container">$\mathbb{R}^{m\times n}$</span> of <span class="math-container">$m \times n$</span> matrices. I'm particularly interested in properties of polytope <span class="math-container">$P$</span> defined as a convex hull of all {-1,1} matrices of rank 1, that is, <span class="math-container">$$ P = \mathrm{conv}\{uv^T: u\in\{-1,1\}^m, v \in \{-1,1\}^n\}. $$</span> In particular, I would like to know how all (<span class="math-container">$mn-1$</span>)-dimensional faces of this polytope and the corresponding normals can be described. Is there any research on that? Maybe this polytope has a name or belongs to a nontrivial class I could find research on?</p> http://www.2874565.com/q/325272 0 Function classes with high Rademacher complexity gradstudent http://www.2874565.com/users/89451 2019-03-12T16:33:33Z 2019-03-12T16:33:33Z <p>My question is two fold,</p> <ul> <li><p>Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of functions to have a high Rademacher complexity is if it has high packing number in the corresponding pseudo-metric. So one can also ask as to if we know of properties of a function class which will lead it to have a large packing number.)</p></li> <li><p>A possibly commonly occurring situation is that over some domain X, we explicitly know of a function <span class="math-container">$g^* : X \rightarrow \mathbb{R}$</span> and we look at the <span class="math-container">$\epsilon-$</span>ball around <span class="math-container">$g^*$</span> inside some larger function space <span class="math-container">$F$</span>. Is there anything known about conditions when is such an epsilon ball around a specific function of high Rademacher complexity? </p></li> </ul> http://www.2874565.com/q/325271 1 Down to earth, intuition behind a Anabelian group Javier http://www.2874565.com/users/136966 2019-03-12T16:20:27Z 2019-03-12T19:05:49Z <p>An anabelian group is a group that is “far from being an abelian group” in a precise sense: <em>It is a non-trivial group for which every finite index subgroup has trivial center.</em></p> <blockquote> <p>I would like to know what is the motivation behind this definition.</p> </blockquote> http://www.2874565.com/q/325269 2 Topological data of $K3\times T^{2}$ Chequez http://www.2874565.com/users/131104 2019-03-12T15:36:21Z 2019-03-12T15:49:56Z <p>I need some help in order to clarify some topological data of a <span class="math-container">$K3\times T^{2}$</span> Calabi Yau manifold in which <span class="math-container">$K3$</span> part has been obtained as a resolution of a <span class="math-container">$T^{4}/ \mathbb{Z_{2}}$</span> orbifold .</p> <p><strong>1) Number of divisors <span class="math-container">$=19$</span>?</strong> I understand that we will have <span class="math-container">$16$</span> exceptional divisors (one coming from resolving each singularity) and <span class="math-container">$3$</span> ordinary ones (coming from setting each <span class="math-container">$z_{i}=0$</span>; for <span class="math-container">$i=1,2,3$</span>). Is this true?</p> <p><strong>2) Second Chern Classes</strong> I am interested in <span class="math-container">$$c_{2i}=\int_{D_{i}}c_{2}\left ( K3 \times T^{2} \right )=c_{2}\left ( K3 \times T^{2} \right )\mbox{Volume}\left ( D_{i} \right ).$$</span></p> <p>I know that <span class="math-container">$c_{2}(K3)=24$</span>, but I don't know neither <span class="math-container">$c_{2}(T^{2})$</span> nor how to calculate <span class="math-container">$c_{2}({K3\times T^{2}})$</span> where <span class="math-container">$D_{i}$</span> is a divisor (either ordinary or exceptional)</p> <p><strong>3) Intersection numbers</strong> <span class="math-container">$\kappa _{ijk}=\int _{K3 \times T^{2}}D_{i}D_{j}D_{k}$</span>. I just know that <span class="math-container">$\kappa _{ij}=\int _{K3}D_{i}D_{j}=2\delta _{i}^{j}$</span> for the exceptional divisors, but I do not know how to proceed to calculate the triple intersection numbers.</p> <p>Any help or reference is welcome</p> http://www.2874565.com/q/325266 0 Symmetric polynomials in two sets of variables user133644 http://www.2874565.com/users/133644 2019-03-12T14:45:51Z 2019-03-12T14:45:51Z <p>Suppose <span class="math-container">$f(x_1,...,x_m,y_1,...,y_n)$</span> is a polynomial with coefficients in some field which is invariant under permuting the <span class="math-container">$x$</span>'s and the <span class="math-container">$y$</span>'s. Then <span class="math-container">$f$</span> can be generated elementary functions <span class="math-container">$e_k(x_1,...,x_m)$</span> and <span class="math-container">$e_k(y_1,...,y_n)$</span>, or the power sum polynomials <span class="math-container">$p_k(x_1,\ldots,x_m)$</span> and <span class="math-container">$p_k(y_1,\ldots,y_n)$</span>. In this way we are handling the variables <span class="math-container">$x_i$</span> and <span class="math-container">$y_j$</span> independently. My questions is when can <span class="math-container">$f$</span> be expanded in terms of polynomials of <span class="math-container">$x_i$</span> and <span class="math-container">$y_j$</span> together? For instance, can <span class="math-container">$f$</span> be generated by functions of the <span class="math-container">$mn$</span> variables <span class="math-container">$s_{ij}=x_i+y_j$</span>?</p> http://www.2874565.com/q/325265 0 Classification of all equivariant structure on the Möbius line bundles Ali Taghavi http://www.2874565.com/users/36688 2019-03-12T14:37:54Z 2019-03-12T15:36:04Z <p>Is there a classification of all equivariant structures of the Möbius line bundle <span class="math-container">$\ell\to S^1$</span>?.</p> <p>For example the antipodal action of <span class="math-container">$\mathbb{Z}/2\mathbb{Z}$</span> on <span class="math-container">$S^1$</span> cannot be lifted to the total space <span class="math-container">$\ell$</span> to get an equivariant structure. But what about the general case?</p> <p>In particular, let <span class="math-container">$\phi_{\theta}$</span> be the irrational rotation of the circle by <span class="math-container">$\theta$</span>. Can the action of <span class="math-container">$\mathbb{Z}$</span> on <span class="math-container">$S^1$</span> given by <span class="math-container">$n.x=\phi_{\theta}^n(x)$</span> be lifted to an action on the total space of the Möbius bundle to give us an equivariant bundle? </p> http://www.2874565.com/q/325264 7 Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension? Alison Miller http://www.2874565.com/users/422 2019-03-12T14:22:14Z 2019-03-12T17:49:41Z <p>I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:</p> <p><b>Lemma:</b> if L/K is an abelian extension of number fields, then there are infinitely many primes of K that do not split competely in L.</p> <p>Of course this is implied by Cebotarev's density theorem, but the adelic proof uses only algebra/topology and finiteness of class number/Dirichlet's units theorem.</p> <p>There is a well-known elementary proof, <a href="http://www.2874565.com/questions/15220/is-there-an-elementary-proof-of-the-infinitude-of-completely-split-primes">(see eg this MO question)</a> that there are infinitely many primes that <em>are</em> split in L/K. I was wondering whether there is also an elementary argument for infinitude of non-split primes in the extension? (As usual the notion of "elementary" is flexible, but I'm looking for something that uses a minimum of machinery.)</p> <p>One possibility would be to distill the adelic proof into something algebraic, although this seems hard. Another option would be to look for ideals of O_K that are not norms from O_L: any such ideal must have a factor which does not split completely.</p> <p>One of the answers to the MathOverflow question linked above does mention the paper <em>Primes of degree one and algebraic cases of ?ebotarev's theorem</em> of Lenstra and Stevenhagen, which gives an elementary proof under the assumption that L contains a nontrivial ray class field of K. But it seems that one still needs to prove the first inequality in some form to use this.</p> http://www.2874565.com/q/325263 0 Limit of a function, given the recurrence relation Darkrai http://www.2874565.com/users/120427 2019-03-12T14:11:30Z 2019-03-12T15:08:00Z <blockquote> <p>Let <span class="math-container">$f(n)$</span> be a function defined for <span class="math-container">$n\ge 2$</span> and <span class="math-container">$n\in N$</span> which follows the recurrence(for <span class="math-container">$n\ge 3$</span>) <span class="math-container">$$\displaystyle f(n)=f(n-1) +\frac {4\cdot (-1)^{(n-1)} \cdot \left(\sum_{d \vert (n-1)} \chi (d)\right) }{n-1}$$</span> where <span class="math-container">$d\vert (n-1)$</span> means <span class="math-container">$d$</span> divides <span class="math-container">$(n-1)$</span> Also assume that <span class="math-container">$f(2)=-4$</span>.</p> <p>Where I define <span class="math-container">$$\chi(d) = \begin{cases} 1, &amp; \text{if $d=4k+1$ where $k$ is a whole number} \\ -1, &amp; \text{if $d=4k+3$ where $k$ is a whole number} \\ 0, &amp; \text {if $d$ is even natural number} \end{cases}$$</span>. Then find <span class="math-container">$$\lim_{n\to \infty} f(n)$$</span></p> </blockquote> <p>First of all this is not at all an assignment or homework problem. It is just a question I came up with, when I was playing with a limit consisting of tedious geometry. </p> <p>Second thing, I tried to find an explicit formula for the function but it seems impossible for me. Also I tried to use the recurrence and guess the approaching value. But the function I guess approaches to some limit (which I don't know) very slowly and hence I am not able to guess the limit. </p> <p>Any guidance and help towards the solution would be quite helpful. </p> http://www.2874565.com/q/325261 1 Norm quadrics and their motives masa M http://www.2874565.com/users/124965 2019-03-12T12:53:56Z 2019-03-12T16:32:01Z <p>Let <span class="math-container">$k$</span> be a field of characteristic <span class="math-container">$\neq 2$</span> and <span class="math-container">$\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$</span> a Pfister form over <span class="math-container">$k$</span>. Denote by <span class="math-container">$Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$</span> the projective quadric of dimension <span class="math-container">$2^{n-1}-1$</span> given by the equation <span class="math-container">$\langle\!\langle a_{1},\cdots,a_{n-1}\rangle\!\rangle=a_{n}t^{2}$</span>. Let <span class="math-container">$P_{\underline{a}}$</span> denote the quadric given by the equation <span class="math-container">$\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle=0$</span>.</p> <p><strong>Question:</strong> If <span class="math-container">$P_{\underline{a}}$</span> has a <span class="math-container">$k$</span>-rational point then does <span class="math-container">$Q_{\underline{a}}$</span> have a <span class="math-container">$k$</span>-rational point?</p> <p>This question is Lemma 4.2 in “Motivic cohomology with <span class="math-container">$\mathbb{Z}/2$</span>-coefficients” by V. Voevodsky. I do not understand this proof; for any rational point <span class="math-container">$p$</span> of <span class="math-container">$P_{\underline{a}}$</span>, why can he say that there exists a linear subspace <span class="math-container">$H$</span> of dimension <span class="math-container">$2^{n-1}-1$</span> which lies on <span class="math-container">$P_{\underline{a}}$</span> and passes through <span class="math-container">$p$</span>?</p> http://www.2874565.com/q/325259 0 Euclidean projection onto certain convex set yon http://www.2874565.com/users/44552 2019-03-12T12:47:27Z 2019-03-12T17:03:39Z <p>Consider the closed convex set</p> <p><span class="math-container">$$ C = \{ x \in R^n : \alpha \| x \| + \langle c, x \rangle \leq 0 \}, $$</span></p> <p>for constants <span class="math-container">$\alpha &gt; 0$</span>, <span class="math-container">$c \in R^n$</span>. </p> <p>My question is whether the Euclidean projection <span class="math-container">$x \mapsto \arg \min_{y \in C} \|y-x \|$</span> admits a simple closed form solution as it does for the second order cone</p> <p><span class="math-container">$$ K = \{ (x, t) \in R^{n+1} : \alpha \|x\| + t \leq 0 \}. $$</span></p> http://www.2874565.com/q/325258 0 Classifying map of a simple circle bundle BrianT http://www.2874565.com/users/124592 2019-03-12T11:49:49Z 2019-03-12T16:26:54Z <p>Let <span class="math-container">$\mathbb{K}_0 \subset \mathbb{K}$</span> be two tori (subtori of <span class="math-container">$(S^1)^n$</span>). We suppose that <span class="math-container">$\mathbb{K}_0$</span> is obtained from <span class="math-container">$\mathbb{K}$</span> by the following procedure: consider, on the lie algebra <span class="math-container">$\text{Lie}(\mathbb{K})$</span> of <span class="math-container">$\mathbb{K}$</span>, a linear function <span class="math-container">$p : \text{Lie}(\mathbb{K}) \to \mathbb{R}$</span>. If <span class="math-container">$p$</span> is "rational", in the sense that <span class="math-container">$\ker p \subset \text{Lie}(\mathbb{K})$</span> closes into a torus via the exponential map, then we can define <span class="math-container">$\mathbb{K}_0$</span> as the codimension <span class="math-container">$1$</span> subtorus of <span class="math-container">$\mathbb{K}$</span> with Lie algebra <span class="math-container">$\ker p$</span>.</p> <p>We denote <span class="math-container">$EG \to BG$</span> the universal principal bundle associated with a Lie group <span class="math-container">$G$</span>, and will consider cohomology with coefficients in <span class="math-container">$\mathbb{C}$</span>. I am trying to understand the classifying map of the circle bundle <span class="math-container">$$ \pi: B\mathbb{K}_0 \overset{\mathbb{K} / \mathbb{K}_0}{\longrightarrow} B\mathbb{K}, $$</span> or more precisely its induced map in cohomology. Below are a few arguments in this direction. </p> <hr> <p>First of all, the cohomology <span class="math-container">$H^*(B(\mathbb{K} / \mathbb{K}_0), \mathbb{C})$</span> can be identified with <span class="math-container">$$ \mathbb{C}[(\text{Lie}(\mathbb{K}) / \text{Lie}(\mathbb{K}_0))^*] \simeq \mathbb{C}[p]. $$</span> Thus, the classifying map in cohomology is of the form <span class="math-container">$$ f : \mathbb{C}[p] \to H^*(B \mathbb{K}, \mathbb{C}). $$</span></p> <p>Let now <span class="math-container">$I$</span> denote the vanishing ideal of <span class="math-container">$\text{Lie}(\mathbb{K}) \otimes \mathbb{C}$</span>, that is, the ideal generated by the polynomials on <span class="math-container">$\mathbb{R}^n \otimes \mathbb{C}$</span> which vanish on <span class="math-container">$\text{Lie}(\mathbb{K})$</span>, and denote <span class="math-container">$I_0$</span> that of <span class="math-container">$\text{Lie}(\mathbb{K}_0) \otimes \mathbb{C}$</span>. If <span class="math-container">$(u_1,...,u_n)$</span> is a basis for <span class="math-container">$\mathbb{R}^{n*}$</span>, then we have <span class="math-container">$$ H^*(B \mathbb{K}) \simeq \mathbb{C}[u_1,...,u_n] / I, \quad H^*(B \mathbb{K}_0) \simeq \mathbb{C}[u_1,...,u_n] / I_0. $$</span></p> <p>Now, the two above cohomology groups are related by the Gysin sequence <span class="math-container">$$ ... \longrightarrow H^*(B \mathbb{K}) \overset{\cup f(p)}{\longrightarrow} H^{*+2}(B \mathbb{K}) \overset{\pi^*}{\longrightarrow} H^{*+2}(B \mathbb{K}_0) \overset{\pi_*}{\longrightarrow} H^{*+1}(B \mathbb{K}) \longrightarrow ... \ . $$</span> In particular, the image of the cup-product by <span class="math-container">$f(p)$</span> is the kernel of the projection <span class="math-container">$$ \mathbb{C}[u_1,...,u_n] / I \to \mathbb{C}[u_1,...,u_n] / I_0. $$</span></p> <p><strong>Claim:</strong> <span class="math-container">$\cup f(p)$</span> is the multiplication by <span class="math-container">$p$</span> !</p> <p>Could someone help me ? Thanks to all for your help!</p> http://www.2874565.com/q/325251 1 Extension of Verma modules James Cheung http://www.2874565.com/users/110229 2019-03-12T10:10:32Z 2019-03-12T16:31:24Z <p>The category <span class="math-container">$\mathcal{O}$</span> is the category of all finitely generated, locally <span class="math-container">$\mathfrak{b}$</span>-finite and <span class="math-container">$\mathfrak{h}$</span>-semisimple <span class="math-container">$\mathfrak{g}$</span>-modules, where <span class="math-container">$\mathfrak{g}$</span> is a complex semisimple Lie algebra with Cartan subalgebra <span class="math-container">$\mathfrak{h}$</span> and Borel subalgebra <span class="math-container">$\mathfrak{b}$</span> containing <span class="math-container">$\mathfrak{h}$</span>.</p> <p>Let <span class="math-container">$M(\lambda)$</span> be the Verma module with highest weight <span class="math-container">$\lambda$</span>, <span class="math-container">$L(\lambda)$</span> be its unique simple quotient. </p> <p><strong>My Question:</strong> Does <span class="math-container">$\text{Ext}^i_{\mathcal{O}}(M(\lambda),M(\mu))=0$</span> imply that <span class="math-container">$\text{Ext}^i_{\mathcal{O}}(M(\lambda),L(\mu))=0$</span>?</p> http://www.2874565.com/q/325247 0 Why does this numerical scheme work on this nonlinear PDE? ThomasD http://www.2874565.com/users/136938 2019-03-12T07:41:18Z 2019-03-12T15:43:44Z <p>i am currently solving a nonlinear PDE of mixed parabolic/hyperbolic type of the Form</p> <p><span class="math-container">\begin{align*} \frac{\partial}{\partial x} \left(A \frac{\partial p}{\partial x} \right) +\frac{\partial}{\partial y} \left(B \frac{\partial p}{\partial y} \right)=\frac{\partial F(p)H}{\partial t} \end{align*}</span></p> <p>which is solved for p with the finite Volume Method. A,B and H are known. This results in a large system of equations <span class="math-container">\begin{align} \underline{\underline{A}}\, \underline{ p} = \underline{R}(p) \end{align}</span> As you can see, the right hand side is dependent on p. The nonlinearity is adressed with a fixed point iteration or Newton-Raphson Iteration. The F-p Curve looks like a logistic function <a href="https://en.wikipedia.org/wiki/Logistic_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Logistic_function</a> </p> <p>However, a regular fixed point iteration does not work, simply calculating new right hand sides and substituting them on the RHS will not result in a convergent solution. A working solution strategy is substituting F(p) with <span class="math-container">\begin{align} F(p)=\Pi p\end{align}</span> The newly introduced variable is a prefactor of p, so it can be dragged to the left hand side into the system matrix. So now the system matrix is dependent on p. This solution converges nicely, but of course at the cost of a nonlinear system matrix which makes the computation quite expensive. So my question is: Why does this converge, but the nonlinear RHS does not? And are there other solution strategies that do not need a nonlinear system matrix?</p> <p>Edit: <span class="math-container">$\Pi$</span> is simply calculated by dividing the last calculated <span class="math-container">$F$</span> by the last calculated <span class="math-container">$p$</span></p> http://www.2874565.com/q/325225 3 Do smooth manifolds admit unique cubical structures? Tim Campion http://www.2874565.com/users/2362 2019-03-11T23:19:40Z 2019-03-12T15:08:42Z <p>It seems to me that a smooth manifold should admit the structure of a cubical complex by Morse theory, since handle attachments seem to be perfectly cubical maps.</p> <p>Is this cubical structure "essentially unique," whatever that means cubically? Or is it non-unique <strike> as in the simplicial case </strike> (PL structures are unique for smooth manifolds -- the Hauptvermutung fails only for more general spaces, such as topological manifolds!)?</p> <p><strong>EDIT:</strong> In the interest of concreteness, let's say that a <em>cubical complex</em> consists of:</p> <ul> <li><p>a (normal [1]) <a href="https://ncatlab.org/nlab/show/cubical+set" rel="nofollow noreferrer">cubical set</a> <span class="math-container">$X$</span></p></li> <li><p>such that each nondegenerate <span class="math-container">$n$</span>-cell <span class="math-container">$\sigma \in X_n$</span> is uniquely determined (up to automorphism [2]) by its 0-skeleton.</p></li> </ul> <p>One subtlety is that there are presheaf categories which may be called "categories of cubical sets" -- one may or may not have one or both <a href="https://ncatlab.org/nlab/show/connection+on+a+cubical+set" rel="nofollow noreferrer">connections</a>, and one may or may not have <em>symmetries</em> (aka "exchanges" aka "extensions") which allow one to permute the axes of a cube, and may or may not have <em>reversals</em>, which allow one to flip the direction of an edge or higher cube. See <a href="http://emis.ams.org/journals/TAC/volumes/11/8/11-08abs.html" rel="nofollow noreferrer">Grandis and Mauri</a> for a discussion. For the purposes of this definition, let's assume we've fixed one of these categories of cubical sets -- and why not just plain cubical sets (the version described at the link).</p> <p>Define as usual the <em>star</em> of a nondegenerate cell <span class="math-container">$\sigma \in X_n$</span> to be the set of cells <span class="math-container">$\tau \in X_m$</span> containing <span class="math-container">$\sigma$</span>, the <em>closure</em> of a set <span class="math-container">$S$</span> of cells of <span class="math-container">$X$</span> to be the smallest cubical subcomplex containing <span class="math-container">$S$</span>, and the <em>link</em> of a nondegenerate cell <span class="math-container">$\sigma \in X_n$</span> to be the closure of the star of <span class="math-container">$\sigma$</span> minus the star of the closure of <span class="math-container">$\sigma$</span>. Say that <span class="math-container">$X$</span> is <em>PL</em> if </p> <ul> <li>the link of any <span class="math-container">$n$</span>-cell is topologically a sphere.</li> </ul> <p>Here we're using the geometric realization functor from cubical sets to topological spaces, which sends the <span class="math-container">$n$</span>-cube to <span class="math-container">$[0,1]^n$</span>.</p> <p>I'm not quite sure when to say that two cubical complexes are "equivalent", though.</p> <p>[1] The word "normal" just means that the automorphism group of the <span class="math-container">$n$</span>-cube acts freely on the set of nondegenerate <span class="math-container">$n$</span>-cells of <span class="math-container">$X$</span>. This automorphism group is trivial unless we have symmetries and / or reversals, so the "normal" condition is vaccuous if we're considering e.g. plain cubical sets.</p> <p>[2] Again, this only matters if we have symmetries and /or reversals.</p> http://www.2874565.com/q/325105 72 What are some noteworthy "mic-drop" moments in math? Mark S http://www.2874565.com/users/8927 2019-03-10T19:20:58Z 2019-03-12T17:40:23Z <p>Oftentimes in math the <strong>manner</strong> in which a solution to a problem is announced becomes a significant chapter/part of the lore associated with the problem, almost being remembered more than the manner in which the problem was solved. I think that most mathematicians as a whole, even upon solving major open problems, are an extremely humble lot. But as an outsider I appreciate the understated manner in which some results are dropped.</p> <p>The very recent example that inspired this question:</p> <ul> <li>Andrew Booker's recent <a href="https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf" rel="noreferrer">solution</a> to <span class="math-container">$a^3+b^3+c^3=33$</span> with <span class="math-container">$(a,b,c)\in\mathbb{Z}^3$</span> as <span class="math-container">$$(a,b,c)=(8866128975287528,-8778405442862239,-2736111468807040)$$</span> was publicized on Tim Browning's <a href="https://pub.ist.ac.at/~tbrownin/" rel="noreferrer">homepage</a>. However the homepage has merely a single, austere line, and does not even indicate that this is/was a semi-famous open problem. Nor was there any indication that the cubes actually sum to <span class="math-container">$33$</span>, apparently leaving it as an exercise for the reader.</li> </ul> <p>Other examples that come to mind include:</p> <ul> <li>In 1976 after Appel and Hakken had proved the Four Color Theorem, Appel <a href="http://historyofmathematics.org/wp-content/uploads/2013/09/2004-Walters.pdf" rel="noreferrer">wrote</a> on the University of Illinois' math department blackboard "Modulo careful checking, it appears that four colors suffice." The statement "Four Colors Suffice" was used as the stamp for the University of Illinois at least around 1976.</li> <li>In 1697 Newton famously offered an "anonymous solution" to the Royal Society to the <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/Brachistochrone.html" rel="noreferrer">Brachistochrone problem</a> that took him a mere evening/sleepless night to resolve. I think the story is noteworthy also because Johanne Bernoulli is said "recognized the lion by his paw."</li> <li>As close to a literal "mic-drop" as I can think of, after noting in his 1993 lectures that Fermat's Last Theorem was a mere corollary of the work presented, Andrew Wiles famously <a href="https://wild.maths.org/andrew-wiles-and-fermats-last-theorem" rel="noreferrer">ended his lecture</a> by stating "I think I'll stop here."</li> </ul> <blockquote> <p>What are other noteworthy examples of such announcements in math that are, in some sense, memorable for being <strong>understated</strong>? Say to an outsider in the field?</p> </blockquote> <p>Watson and Crick's famous ending of their DNA paper, "It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material," has a bit of the same understated feel...</p> http://www.2874565.com/q/325052 16 How many random walk steps until the path self-intersects? Joseph O'Rourke http://www.2874565.com/users/6094 2019-03-10T03:01:48Z 2019-03-12T18:28:34Z <p>Take a random walk in the plane from the origin, each step of unit length in a uniformly random direction. </p> <blockquote> <p><strong><em>Q</em></strong>. How many steps on average until the path self-intersects?</p> </blockquote> <p>My simulations suggest ~<span class="math-container">$8.95$</span> steps. <hr /> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <a href="https://i.stack.imgur.com/zWEDH.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zWEDH.jpg" alt="RandLines_8"></a> <br /> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <a href="https://i.stack.imgur.com/1bR7H.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1bR7H.jpg" alt="RandLines_11"></a> <br /> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <sup> Red: origin. Top: the <span class="math-container">$8$</span>-th step self-intersects. Bottom: the <span class="math-container">$11$</span>-th step self-intersects. (Not to same scale.) </sup> <hr /> I suspect this is known in the SAW literature (SAW=Self-Avoiding Walk), but I am not finding this explicit number.</p> <p>Related: <a href="http://www.2874565.com/q/23583/6094">self-avoidance time of random walk</a>.</p> <p><em>Added</em>. Here is a histogram of the number of steps to self-intersection. <hr /> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <a href="https://i.stack.imgur.com/eVHTa.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/eVHTa.jpg" alt="Histogram"></a> <br /> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <sup> <span class="math-container">$10000$</span> random trials. </sup></p> <hr /> http://www.2874565.com/q/324867 35 Ultrafilters as a double dual Adam P. Goucher http://www.2874565.com/users/39521 2019-03-07T15:35:11Z 2019-03-12T17:26:10Z <p>Given a set <span class="math-container">$X$</span>, let <span class="math-container">$\beta X$</span> denote the set of ultrafilters. The following theorems are known:</p> <ul> <li><span class="math-container">$X$</span> canonically embeds into <span class="math-container">$\beta X$</span> (by taking principal ultrafilters);</li> <li>If <span class="math-container">$X$</span> is finite, then there are no non-principal ultrafilters, so <span class="math-container">$\beta X = X$</span>.</li> <li>If <span class="math-container">$X$</span> is infinite, then (assuming choice) we have <span class="math-container">$|\beta X| = 2^{2^{|X|}}$</span>.</li> </ul> <p>These are reminiscent of similar claims that can be made about vector spaces and double duals:</p> <ul> <li><span class="math-container">$V$</span> canonically embeds into <span class="math-container">$V^{\star \star}$</span>;</li> <li>If <span class="math-container">$V$</span> is finite-dimensional, then we have <span class="math-container">$V = V^{\star \star}$</span>;</li> <li>If <span class="math-container">$V$</span> is infinite-dimensional, then (assuming choice) we have <span class="math-container">$\dim(V^{\star \star}) = 2^{2^{\dim(V)}}$</span>.</li> </ul> <p>This suggests that the operation of taking the collection of ultrafilters on a set can be viewed as a double iterate of some 'duality' of sets. Can this be made precise: that is to say, is there some notion of a 'dual' of a set <span class="math-container">$X$</span>, <span class="math-container">$\delta X$</span>, such that the following are true?</p> <ul> <li>The double dual <span class="math-container">$\delta \delta X$</span> is (canonically isomorphic to) the set <span class="math-container">$\beta X$</span> of ultrafilters on <span class="math-container">$X$</span>;</li> <li>If <span class="math-container">$X$</span> is finite, then <span class="math-container">$|\delta X| = |X|$</span> (but not canonically so);</li> <li>If <span class="math-container">$X$</span> is infinite, then (assuming choice) <span class="math-container">$|\delta X| = 2^{|X|}$</span>.</li> </ul> <p>Apart from the tempting analogy between <span class="math-container">$\beta X$</span> and <span class="math-container">$V^{\star \star}$</span>, further evidence for this conjecture is that <span class="math-container">$\beta$</span> can be given the structure of a monad (the 'ultrafilter monad'), and monads can be obtained from a pair of adjunctions.</p> http://www.2874565.com/q/324847 6 When is a fold monomorphic/epimorphic dremodaris http://www.2874565.com/users/66017 2019-03-07T10:48:13Z 2019-03-12T15:23:27Z <p>Given a functor <span class="math-container">$F : \mathcal C \to \mathcal C$</span> with initial algebra <span class="math-container">$\alpha : FA \to A$</span>, and another algebra <span class="math-container">$\xi : FX \to X$</span>, we obtain a unique morphism <span class="math-container">$\mathsf{fold}~\xi : A \to X$</span> such that <span class="math-container">$\mathsf{fold}~\xi \circ \alpha = \xi \circ F(\mathsf{fold}~\xi)$</span>.</p> <p>I am looking for a sufficient condition - ideally phrasable as <span class="math-container">$P(F) \wedge Q(\xi)$</span> rather than <span class="math-container">$R(F, \xi)$</span> - which guarantees that:</p> <ol> <li><span class="math-container">$\mathsf{fold}~\xi$</span> is monomorphic,</li> <li><span class="math-container">$\mathsf{fold}~\xi$</span> is epimorphic.</li> </ol> <p>(I'm looking for separate conditions for 1 and for 2.)</p> <p>I would expect:</p> <ol> <li><span class="math-container">$F$</span> preserves monomorphism and <span class="math-container">$\xi$</span> is monomorphic</li> <li><span class="math-container">$F$</span> preserves epimorphism and <span class="math-container">$\xi$</span> is epimorphic</li> </ol> <p>but I can't seem to prove this.</p> <p>EDIT: I'm also interested in the dual question (which, mathematically, of course just has dual answers). I mention this anyway, because Valery Isaev's answer uses local finite presentability, which is something that might not have been answered had I asked the dual question. So feel free to use concepts that are only dually familiar. In particular, I'm willing to assume that <span class="math-container">$F^{op}$</span> is polynomial (or similar in a less powerful category, e.g. <span class="math-container">$FX$</span> is a coproduct of powers of <span class="math-container">$X$</span> in <span class="math-container">$\mathcal C^{op}$</span>).</p> http://www.2874565.com/q/324812 3 The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve MCS http://www.2874565.com/users/120369 2019-03-06T21:15:01Z 2019-03-12T18:57:43Z <p>Working over the complex numbers, consider a function <span class="math-container">$F\left(x,y\right)$</span> and a curve <span class="math-container">$C$</span> defined by <span class="math-container">$F\left(x,y\right)=0$</span>. </p> <p>I know that to construct the Jacobian variety associated to <span class="math-container">$C$</span>, one integrates a basis of global holomorphic differential forms over the contours of the curve's homology group. I'm looking for information that is oriented toward actually computing things for given concrete examples; everything I've seen so far, however, has been uselessly abstract or non-specific. Note: I'm new to this—I'm an analyst who knows next to nothing about algebra and even less about differential geometry or topology.</p> <p>In my quest for a sensible answer, I turned to a <a href="https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACR0014.0001.001" rel="nofollow noreferrer">H.F. Baker's</a> wonderful (though densely written) text from the start of the 20th century. Just reading through the first few pages makes it <em>abundantly</em> clear that there is a general procedure for constructing a basis of holomorphic differential forms for a given curve. Ted Shifrin's comment on this <a href="https://math.stackexchange.com/questions/2726917/holomorphic-1-form-on-projective-curve?rq=1">math-stack-exchange problem</a> only makes me more certain than ever that the answers I seek are out there, somewhere.</p> <p>Broadly speaking, my goals are as follows. In all of these, my aim is to be able to use the answers to these questions to compute various specific examples, either by hand, or with the assistance of a computer algebra system. So, I'm looking for formulae, explanations and/or step-by-step procedures/algorithms, and/or pertinent reference/reading material.</p> <p>(1) In the case where <span class="math-container">$F$</span> is a polynomial, what is/are the procedure(s) for determining a basis of holomorphic differential 1-forms over <span class="math-container">$F$</span>? If the procedure varies depending on certain properties of <span class="math-container">$F$</span> (say, if <span class="math-container">$F$</span> is an affine curve, or a projective curve, or of a certain form, or some detail like that), what are those variations?</p> <p>(2) In the case where <span class="math-container">$F$</span> is a polynomial of <span class="math-container">$x$</span>-degree <span class="math-container">$d_{x}$</span>, <span class="math-container">$y$</span>-degree <span class="math-container">$d_{y}$</span>, and <span class="math-container">$C$</span> is a curve of genus <span class="math-container">$g$</span>, I know that the basis of holomorphic differential 1-forms for <span class="math-container">$C$</span> will be of dimension <span class="math-container">$g$</span>. In the case, say, where <span class="math-container">$C$</span> is an elliptic curve, with:</p> <p><span class="math-container">$$F\left(x,y\right)=4x^{3}-g_{2}x-g_{3}-y^{2}$$</span></p> <p>the classical <em>Jacobi Inversion Problem</em> arises from considering a function <span class="math-container">$\wp\left(z\right)$</span> which parameterizes <span class="math-container">$C$</span>, in the sense that <span class="math-container">$F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)$</span> is identically zero. Using the equation: <span class="math-container">$$F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)=0$$</span> we can write: <span class="math-container">$$\wp^{-1}\left(z\right)=\int_{z_{0}}^{z}\frac{ds}{4s^{3}-g_{2}s-g_{3}}$$</span> and know that the multivaluedness of the integral then reflects the structure of the Jacobian variety associated to <span class="math-container">$C$</span>.</p> <p>That being said, in the case where <span class="math-container">$C$</span> is of genus <span class="math-container">$g\geq2$</span>, and where we can write <span class="math-container">$F\left(x,y\right)=0$</span> as: <span class="math-container">$$y=\textrm{algebraic function of }x$$</span> nothing stops us from performing the exact same computation as for the case with an elliptic curve. Of course, this computation must be wrong; my question is: <em>where and how does it go wrong</em>? How would the parameterizing function thus obtained relate to the "true" parameterizing function—the multivariable Abelian function associated to <span class="math-container">$C$</span>? Moreover, how—if at all—can this computation be modified to produce the correct parameterizing function (the Abelian function)?</p> <p>(3) My hope is that by understanding both (1) and (2), I'll be in a position to see what happens when these classical techniques are applied to <em>non-algebraic</em> plane curves defined but with <span class="math-container">$F$</span> now being an analytic function (incorporating exponentials, and other transcendental functions, in addition to polynomials). Of particular interest to me are the transcendental curves associated to exponential diophantine equations such as: <span class="math-container">$$a^{x}-b^{y}=c$$</span> <span class="math-container">$$y^{n}=b^{x}-a$$</span></p> <p>That being said, I wonder: has this already been done? If so, links and references would be much appreciated.</p> <p>Even if it has, though, I would still like to know the answers to my previous questions, even if it's merely for my personal edification alone.</p> <p>Thanks in advance! </p> http://www.2874565.com/q/324243 1 Lehmer’s totient problem 喻 良 http://www.2874565.com/users/14340 2019-02-27T05:51:14Z 2019-03-12T16:57:54Z <p>Euler’s totient function <span class="math-container">$\varphi$</span> is a function defined over <span class="math-container">$\mathbb{N}$</span> so that <span class="math-container">$\varphi(n)=|\{m\mid m&lt;n\wedge (m,n)=1\}|$</span>.</p> <p>Now <a href="https://en.wikipedia.org/wiki/Lehmer%27s_totient_problem" rel="nofollow noreferrer">Lehmer’s totient problem</a> asks whether <span class="math-container">$n$</span> is prime iff <span class="math-container">$\varphi(n)$</span> divides <span class="math-container">$n-1$</span>. </p> <p>I am curious whether the question can be expressed as a question in ring language. More specifically, whether there is a firs order formula <span class="math-container">$\psi(x)$</span> in the ring language so that <span class="math-container">$\mathbb{Z}\models \psi(n)$</span> if and only if <span class="math-container">$\varphi(n)$</span> divides <span class="math-container">$n-1$</span>. </p> <p><strong>Remark</strong> As James pointed out, the question has a positive answer. But what I really want is an algebraic answer. Probably the question was not in a proper shape. How about this: </p> <p>Whether there is a first order formula <span class="math-container">$\psi(x,y)$</span> in the ring language so that there is theory <span class="math-container">$A$</span> extending axioms of ring theory in a proper way (which is not necessary consistent with theory of <span class="math-container">$\mathbb{Z}$</span>) so that </p> <p>(1). <span class="math-container">$ A\vdash\forall x\exists y\psi(x,y)\wedge (\forall x \forall y(\psi(x,y)\rightarrow( y\mbox{ divides } x-1\leftrightarrow x\mbox{ is prime})) ) $</span>; and</p> <p>(2). For all <span class="math-container">$m$</span> and <span class="math-container">$n$</span>, <span class="math-container">$\mathbb{Z}\models \psi(n,m)$</span> if and only if <span class="math-container">$m=\varphi(n)$</span>.</p> http://www.2874565.com/q/284855 17 What is the status of the 4-dimensional Smale Conjecture? Rieendstac http://www.2874565.com/users/99280 2017-10-31T06:07:34Z 2019-03-12T16:59:28Z <p>4-dimensional Smale conjecture claims the following:</p> <p>The inclusion $SO(5)$ → $SDiff(S^4)$ is a homotopy equivalence.</p> <p>or Does $Diff(S^4)$ have the homotopy-type of $O(5)$ ?.</p> <p>The inclusion $SO(n + 1$) → $SDiff(S^n)$ is a homotopy equivalence for n = 1 (trivial proof), n = 2 [1004,Smale,1959,Proc. Amer. Math. Soc.], n = 3 [464,Hatcher, 1983,Ann. of Math.], and is not a homotopy equivalence for n ≥ 5 [41,Antonelli, Burghelea, &amp; Kahn,1972,Topology] and [164,Burghelea &amp; Lashof,1974,Trans. Amer. Math. Soc.].</p> <p>I looked everywhere but I could not find anything. Is this problem still open? Thanks.</p> http://www.2874565.com/q/63423 95 Checkmate in $\omega$ moves? Johan Wästlund http://www.2874565.com/users/14302 2011-04-29T15:13:54Z 2019-03-12T19:09:03Z <blockquote> <p>Is there a chess position with a finite number of pieces on the infinite chess board $\mathbb{Z}^2$ such that White to move has a forced win, but Black can stave off mate for at least $n$ moves for every $n$?</p> </blockquote> <p>This question is motivated by a question <a href="http://www.2874565.com/questions/27967/decidability-of-chess-on-an-infinite-board">posed here</a> a few months ago by Richard Stanley. He asked whether chess with finitely many pieces on $\mathbb{Z}^2$ is decidable.</p> <p>A compactness observation is that if Black has only short-range pieces (no bishops, rooks or queens), then the statement "White can force mate" is equivalent to "There is some $n$ such that White can force mate in at most $n$ moves". </p> <p>This probably won't lead to an answer to Stanley's question, because even if there are only short-range pieces, there is no general reason the game should be decidable. It is well-known that a finite automaton with a finite number of "counters" can emulate a Turing machine, and there seems to be no obvious reason why such an automaton could not be emulated by a chess problem, even if we allow only knights and the two kings. </p> <p>But it might still be of interest to have an explicit counterexample to the idea that being able to force a win means being able to do so in some specified number of moves. Such an example must involve a long-range piece for the losing side, and one idea is that Black has to move a rook (or bishop) out of the way to make room for his king, after which White forces Black's king towards the rook with a series of checks, finally mating thanks to the rook blocking a square for the king.</p> <p>If there are such examples, we can go on and define "mate in $\alpha$" for an arbitrary ordinal $\alpha$. To say that White has a forced mate in $\alpha$ means that White has a move such that after any response by Black, White has a forced mate in $\beta$ for some $\beta&lt;\alpha$. </p> <p>For instance, mate in $\omega$ means that after Black's first move, White is able to force mate in $n$ for some finite $n$, while mate in $2\omega + 3$ means that after Black's fourth move, White will be able to specify how many more moves it will take until he can specify how long it will take to mate.</p> <p>With this definition, we can ask exactly how long-winded the solution to a chess problem can be:</p> <blockquote> <p>What is the smallest ordinal $\gamma$ such that having a forced mate implies having a forced mate in $\alpha$ for some $\alpha&lt;\gamma$? </p> </blockquote> <p>Obviously $\gamma$ is infinite, and since there are only countably many positions, $\gamma$ must be countable. Can anyone give better bounds?</p> http://www.2874565.com/q/41784 15 Roots of permutations Fedor Petrov http://www.2874565.com/users/4312 2010-10-11T13:48:08Z 2019-03-12T15:51:49Z <p>Consider the equation <span class="math-container">$x^2=x_0$</span> in the symmetric group <span class="math-container">$S_n$</span>, where <span class="math-container">$x_0\in S_n$</span> is fixed. Is it true that for each integer <span class="math-container">$n\geq 0$</span>, the maximal number of solutions (the number of square roots of <span class="math-container">$x_0$</span>) is attained when <span class="math-container">$x_0$</span> is the identity permutation? How far may it be generalized?</p> http://www.2874565.com/q/15220 36 Is there an "elementary" proof of the infinitude of completely split primes? François G. Dorais http://www.2874565.com/users/2000 2010-02-14T00:40:27Z 2019-03-12T16:32:00Z <p>Let <span class="math-container">$K$</span> be a Galois extension of the rationals with degree <span class="math-container">$n$</span>. The <a href="http://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem" rel="nofollow noreferrer">Chebotarev Density Theorem</a> guarantees that the rational primes that split completely in <span class="math-container">$K$</span> have density <span class="math-container">$1/n$</span> and thus there are infinitely many such primes. As Kevin Buzzard pointed out to me in a <a href="http://www.2874565.com/questions/15151/product-of-all-fp-p-prime/15157#15157">comment</a>, there is a simpler way to see that there are infinitely many rational primes that split completely in <span class="math-container">$K$</span>, namely that the <a href="http://en.wikipedia.org/wiki/Dedekind_zeta_function" rel="nofollow noreferrer">Dedekind zeta-function</a> <span class="math-container">$\zeta_K(s)$</span> has a simple pole at <span class="math-container">$s = 1$</span>. While this result is certainly much easier to prove than Chebotarev's Theorem, it is still not an elementary proof.</p> <blockquote> <p>Is there a known elementary proof of the fact that there are infinitely many rational primes that split completely in <span class="math-container">$K$</span>?</p> </blockquote> <p>Selberg's elementary proof of Dirichlet's Theorem for primes in arithmetic progressions handles the case where <span class="math-container">$\text{Gal}(K/\mathbb{Q})$</span> is Abelian. I don't know anything about the general case. Since Dirichlet's Theorem is stronger than required, it is possible that an simpler proof exists even in the Abelian case.</p> <p><em>Remarks on the meaning of elementary</em>. I am aware that there is no uniformly recognized definition of "elementary proof" in number theory. While I am not opposed to alternate definitions, my personal definition is a proof which can be carried out in first-order arithmetic, i.e. without quantification over real numbers or higher-type objects. Obviously, I don't require it to be explicitly formulated in that way &mdash; even logicians don't do that! Odds are that whatever you believe is elementary is also elementary in my sense.</p> <p>Kurt Gödel <a href="http://plato.stanford.edu/entries/goedel/#SpeUpThe" rel="nofollow noreferrer">observed</a> that proofs of (first-order) arithmetical facts can be much, much shorter in second-order arithmetic than in first-order arithmetic. This observation explains some of the effectiveness of analytic number theory, which is implicitly second-order. In view of Gödel's observation, it is possible that we have encountered arithmetical facts with a reasonably short second-order proof (i.e. could be found in an analytic number theory textbook) but no reasonable first-order proof (i.e. the production of any such proof would necessarily exhaust all of our natural resources). The above is unlikely to be such, but it is interesting to know that beasts of this type could exist...</p> 山西福彩快乐十分钟
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