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curious little things - aggregated feedsenMath Overflow Recent Questions: Strong Nash Equilibria in repeated games
https://mathoverflow.net/questions/311425/strong-nash-equilibria-in-repeated-games
<p>Suppose we have a simultaneous game, that has a strong Nash equilibrium (SNA), i.e. a weak Pareto efficient Nash equilibrium (no deviation of any subset of player brings a benefit to them).</p>
<p>Now suppose we play this game repeatedly. Does the repeated game has a strong Nash equilibrium, too? </p>
<p>I would keep the question about the payoff function of the repeated game as open. Choose whatever adopted payoff works for the answer.</p>
<p>The idea behind the question is as follows: Games that don't have SNA, like prisoner's dilemma, might have those in the repeated scenario, since additional effects like long term strategies come into play.</p>
<p>Based on this, my guess is, that playing the SNA in every game, would also give a SNA in the repeated game.</p>Tue, 25 Sep 2018 16:34:48 -0600Math Overflow Recent Questions: Are quotient varieties local complete intersections?
https://mathoverflow.net/questions/311424/are-quotient-varieties-local-complete-intersections
<p>Let <span class="math-container">$G$</span> be a reductive group acting on the smooth affine variety <span class="math-container">$X$</span> such that the stabilizers are finite. Is it true that the quotient <span class="math-container">$X/G$</span> is a local complete intersection (LCI)? In particular, is the quotient of a smooth affine variety to the algebraic action of a finite group LCI? If no, is there any condition on the action that guarantees such a property? </p>Tue, 25 Sep 2018 16:28:05 -0600Math Overflow Recent Questions: Polyhedral structure related to partitions of an interval?
https://mathoverflow.net/questions/311423/polyhedral-structure-related-to-partitions-of-an-interval
<p>Let <span class="math-container">$I=[0,1]$</span> be the unit interval. Suppose we are given a sequence of nonnegative scalars <span class="math-container">$\{p_i\}_{i=1}^k$</span> such that <span class="math-container">$\sum_{i=1}^k p_i=1$</span>, and denote by <span class="math-container">$p$</span> the vector that consists of these scalars. Let <span class="math-container">${\cal B}(p)$</span> denote the set of disjoint measurable partitions of <span class="math-container">$I$</span> consistent with <span class="math-container">$p$</span>, i.e., <span class="math-container">$\{B_i\}_{i=1}^k \in {\cal B} (p)$</span> if and only if <span class="math-container">$\cup_{i=1}^k B_i = I$</span>, <span class="math-container">$B_i\cap B_j=\emptyset$</span> for <span class="math-container">$i\neq j$</span>, and for all <span class="math-container">$i$</span> we have that <span class="math-container">$B_i$</span> is Lebesgue measurable and satisfies <span class="math-container">$\int_{B_i} dx =p_i$</span>.
Consider the set <span class="math-container">$Z=\left\{z ~|~ z_k = \int_{B_k} x ~ dx ~~\forall k, \mbox{ and } \{B_i\}_{i=1}^k\in {\cal B}(p)\right\}$</span>.</p>
<p>I'm trying to understand the structure of this set <span class="math-container">$Z$</span>. In particular,</p>
<ol>
<li>Is it polyhedral? If so, do the extreme points admit a simple characterization?</li>
<li>Is it possible to characterize <span class="math-container">$Z$</span> by only focusing on <span class="math-container">$\{B_k\}$</span> that constitute intervals?</li>
<li>Is there any reference which studies objects similar to those defined above?</li>
</ol>Tue, 25 Sep 2018 16:26:23 -0600Math Overflow Recent Questions: Published reference on the automorphism group of modular curves $X_1(N)$?
https://mathoverflow.net/questions/311422/published-reference-on-the-automorphism-group-of-modular-curves-x-1n
<p>I wish to cite that the automorphism groups of <span class="math-container">$X_1(N)$</span> have already been completely calculated, and what they are, but I am having difficulty finding this calculation in the literature. </p>
<p>I have found in <a href="https://arxiv.org/pdf/1207.2273.pdf" rel="nofollow noreferrer">multiple [22]</a> <a href="https://arxiv.org/pdf/1308.3267.pdf" rel="nofollow noreferrer">papers [29]</a> on related topics a reference to "F. Momose, Automorphism groups of the modular curves X1(N). Preprint."</p>
<p>I cannot find it. I am wondering if this paper ever made it to publication, or where a copy of it might be? If not, is there a published reference on the topic?</p>Tue, 25 Sep 2018 15:45:54 -0600Math Overflow Recent Questions: Does geometrization of Alexandrov 3-spaces follow from that of 3-orbifolds?
https://mathoverflow.net/questions/311420/does-geometrization-of-alexandrov-3-spaces-follow-from-that-of-3-orbifolds
<p>Galaz-Garcia and Guijarro proved the geometrization of closed (compact, boundaryless) Alexandrov 3-spaces. Part of the strategy was to use the so-called ramified double cover <span class="math-container">$\tilde{X}$</span> of the space <span class="math-container">$X$</span>. This ramified cover is a smooth <span class="math-container">$3$</span>-manifold. Being this the case, the space <span class="math-container">$X$</span> would be isometric to a Riemannian <span class="math-container">$3$</span>-orbifold. </p>
<p>I don't quite follow why then, it's not immediate that the geometrization of <span class="math-container">$X$</span> follows from the geometrization of <span class="math-container">$3$</span>-orbifolds? </p>Tue, 25 Sep 2018 15:38:07 -0600Math Overflow Recent Questions: Traveling at a rate of miles per hour [on hold]
https://mathoverflow.net/questions/311417/traveling-at-a-rate-of-miles-per-hour
<p>Jenny is traveling at a rate that allows him to go 34 miles in 39 minutes.
What is her average speed, in miles per hour?</p>Tue, 25 Sep 2018 15:16:39 -0600Math Overflow Recent Questions: When does an optimization problem with symmetric constraints has symmertic optimal solution?
https://mathoverflow.net/questions/311416/when-does-an-optimization-problem-with-symmetric-constraints-has-symmertic-optim
<p>Consider the optimization problem
<span class="math-container">\begin{array} \
\min\text{ / } \max &f(x_1,\cdots,x_n) \\ \text{subj. }& g(x_1,\cdots,x_n)=\text{constant} ,\end{array}</span>
where <span class="math-container">$g$</span> is a symmetric function, i.e. <span class="math-container">$g(x_1,\cdots,x_n)=g(\sigma(x_1,\cdots,x_n))$</span> where <span class="math-container">$\sigma\in\text{Symm}\{x_1,\cdots,x_n\}$</span>.
There is a principle (sometimes called Purkiss principle <a href="https://www.jstor.org/stable/2975573?seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">see here</a>) which states, under some mild assumptions, that if <span class="math-container">$f$</span> is also symmetric, then the optimal solution is of form <span class="math-container">$x_1=\cdots=x_n$</span>(see, also <a href="https://mathoverflow.net/questions/58721/when-does-symmetry-in-an-optimization-problem-imply-that-all-variables-are-equal">other post</a>).</p>
<p><strong>Q.</strong> Are there other versions of Purkiss principle that address aforementioned problem, meaning not necessarily symm. objective function but symmetric constraints? </p>Tue, 25 Sep 2018 15:12:02 -0600Math Overflow Recent Questions: Hodge decomposition of the symmetric product of a curve
https://mathoverflow.net/questions/311415/hodge-decomposition-of-the-symmetric-product-of-a-curve
<p>Let X be a smooth projective connected curve over <span class="math-container">$\mathbb{C}$</span> and let <span class="math-container">$n>1$</span> be an integer. Let <span class="math-container">$Y= Sym^n_X$</span> be the <span class="math-container">$n$</span>-th symmetric product of <span class="math-container">$X$</span>.</p>
<blockquote>
<blockquote>
<p>Is there, for every <span class="math-container">$i$</span>, a nice formula for the Hodge decomposition of <span class="math-container">$H^i(Y,\mathbb{C})$</span>?</p>
</blockquote>
</blockquote>
<p>If not, what part of the Hodge diamond can be described easily?</p>Tue, 25 Sep 2018 15:08:25 -0600Math Overflow Recent Questions: Derived equivalences and the Coxeter polynomial
https://mathoverflow.net/questions/311413/derived-equivalences-and-the-coxeter-polynomial
<p>Let <span class="math-container">$A$</span> be a quiver algebra with an acyclic quiver and primitive idempotents <span class="math-container">$e_i$</span>.
The Cartan matrix <span class="math-container">$C_A$</span> of <span class="math-container">$A$</span> is defined as the matrix with entries <span class="math-container">$dim(e_i A e_j)$</span> and the Coxeter matrix <span class="math-container">$\phi_A$</span> of <span class="math-container">$A$</span> is defined as <span class="math-container">$\phi_A=-C_A^{-1} C_A^T$</span>. The Coxeter polynomial of <span class="math-container">$A$</span> is defined as characteristic polynomial of <span class="math-container">$\phi_A$</span>. The Coxeter polynomial is a derived invariant and thus derived equivalent algebras share the same Coxeter polynomial.</p>
<p>Is the following true:</p>
<blockquote>
<p>A is derived equivalent to a path algebra <span class="math-container">$KQ$</span> of Dynkin type if and only if it has the same Coxeter polynomial as <span class="math-container">$KQ$</span>? </p>
<p>In case the answer is no, is this true in case <span class="math-container">$A$</span> is additionally a Nakayama algebra with a linear quiver?</p>
</blockquote>
<p>I think this is at least true when <span class="math-container">$A$</span> is a Nakayama algebra with a linear quiver (corresponding to a Dyck path) and there is computational way using trivial extensions and representation-finiteness of those trivial extensions to test it, but it gets very ugly when <span class="math-container">$Q$</span> is of type <span class="math-container">$D_n$</span> (<span class="math-container">$E_n$</span> can be done with the computer and indeed, for <span class="math-container">$E_6$</span> it is true. It is also true for Dynkin type <span class="math-container">$A_n$</span> and <span class="math-container">$n \leq 8$</span> and Dynkin type <span class="math-container">$D_n$</span> for <span class="math-container">$n \leq 6$</span>).</p>Tue, 25 Sep 2018 15:07:05 -0600Math Overflow Recent Questions: Is continuity preserved under norm operations
https://mathoverflow.net/questions/311411/is-continuity-preserved-under-norm-operations
<p>Let <span class="math-container">$F$</span> be a continuously differenable function over <span class="math-container">$\mathbb{R}$</span>. Let <span class="math-container">$\Omega$</span> be
a bounded subset of <span class="math-container">$\mathbb{R}^2$</span>. Assume that for every <span class="math-container">$w\in L^2(\Omega)$</span> then <span class="math-container">$v(x)=F(w(x))$</span>, <span class="math-container">$x\in \Omega$</span>, is also in <span class="math-container">$L^2(\Omega)$</span>. Can we say that <span class="math-container">$$\|F(w_1(\cdot)+\eta w_2(\cdot))\|_{L^2(\Omega)}$$</span>
is continuous in <span class="math-container">$\eta$</span>?</p>Tue, 25 Sep 2018 14:56:01 -0600