Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2016-06-26T08:29:53Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/2430800Find equidistant points on surface of sphereuser94347http://mathoverflow.net/users/943472016-06-26T05:02:38Z2016-06-26T05:02:38Z
<p>Given a sphere, find the maximum number of points can be placed on the surface of the sphere such that all are equidistant from each other.</p>
<p>I am not at all good in maths. Please let me know what can be the approach for this? Is there any kind of formula for this?</p>
http://mathoverflow.net/q/2430780Self-contained Numbersapc89http://mathoverflow.net/users/890242016-06-26T03:23:40Z2016-06-26T03:23:40Z
<p>Given any natural number $a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the following set $S = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a <em>self-contained</em> number as any natural number which satisfies the next rule:
$$\forall k\leq n,\exists\sigma\subset S\setminus\{(a_{k},k)\}; a_{k} = \sum_{s\in\sigma}(-1)^{p_{s}}\|s - s_{2}\|,\,\text{where}\, p_{s}\in\{0,1\}$$
It's worth mentioning here that $s_{2}$ is the projection on the second coordinate. For instance, the number 123 is self-contained since we can rewrite its digits as: 1 = 3 - 2, 2 = 3 - 1 and 3 = 1 + 2. On the other hand, the number 102 is not self-contained given that 2 $\neq$ 1 + 0 and 2 $\neq$ 1 - 0. Since the definition's been made clear, I would like to ask if anyone could provide me a criterion to identify them quickly or how to generate all of them. Thank you in advance.</p>
http://mathoverflow.net/q/2430761Left adjoint to Double Nerve?user84563http://mathoverflow.net/users/845632016-06-26T02:05:05Z2016-06-26T03:16:22Z
<p>The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor $N^2:2Cat\rightarrow sSSet$ from 2-categories to bisimplicial sets have a similar left adjoint? I think I've seen a vague reference to it somewhere but nothing explicit. </p>
http://mathoverflow.net/q/2430751Evaluation of sum of factorialsMatt Majichttp://mathoverflow.net/users/942002016-06-26T01:28:08Z2016-06-26T02:59:05Z
<p>Is there an evaluation of this sum (possibly involving gamma functions)? $k$ and $n$ are natural numbers and $x$ is real with $0<x<1$.
$$ \sum_{\substack{k=0\\n-k\text{ even}}}^n \frac{(-1)^{(n-k)/2}}{k+x} \frac{(n+k)!}{k!(\frac{n+k}{2})!(\frac{n-k}{2})!}$$
Any help is much appreciated.</p>
http://mathoverflow.net/q/2430741Approximate unit in C*-algebra with additional propertiestruebaranhttp://mathoverflow.net/users/240782016-06-26T00:22:02Z2016-06-26T00:22:02Z
<p>In the book about K-theory (a friendly approach) by Wegge and Olsen I came across the following notion: in Lemma 16.4 authors assume that the $C^*$-algebra $A$ possess "(commuting) approximate unit $(a_n)_n$ so that $a_{n+1}a_n=a_n$". In the next lemma the only assumption about $A$ is that $A$ is $\sigma$-unital, which means that there is some countable approximate unit. However the authors claim that in this case we may assume that this approximate unit is as above. </p>
<blockquote>
<p>Why we can assume that this approximate unit have this additional property (having assumed that $A$ is just $\sigma$-unital? </p>
</blockquote>
http://mathoverflow.net/q/2430721$G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$?PhysicsMathhttp://mathoverflow.net/users/826452016-06-26T00:14:27Z2016-06-26T01:59:17Z
<p>As summarized in the title, suppose there is an isomorphism between $G_1 \times G_2$ and $G_3$, is it always true that $BG_1 \times BG_2$ is homotopy equivalent to $BG_3$? If it is not always true could you give a counterexample and the condition for it to be true?</p>
<p>Similarly suppose there is an isomorphism between $G_1 \rtimes G_2$ and $G_3$, is it always true that $BG_1 \times BG_2$ is homotopy equivalent to $BG_3$? If it is not always true could you give a counterexample and the condition for it to be true?</p>
http://mathoverflow.net/q/2430690Difference of if-them statements in fuzzy logic and fuzzy control?user2243865http://mathoverflow.net/users/579282016-06-25T23:33:52Z2016-06-25T23:33:52Z
<p>I'm reading up on fuzzy sets, logic, and controllers. I understand the basic concepts, but am confused about the interpretation of if-then statements in fuzzy logic.</p>
<p>The fuzzy control literature (for example, <a href="http://www2.ece.ohio-state.edu/~passino/FCbook.pdf" rel="nofollow">Passino and Yurkovich's book (pdf)</a>) treats fuzzy if-then statements as normative "rules", to make decisions. For example, in a automobile cruise-control system, where the goal is to maintain speed X, a rule might be "if speed is slightly lower than target and acceleration is zero, press gas lightly". </p>
<p>However, in the more mathematical literature, if-then statements are used for knowledge representation and inference. For example, "if apple is green and hard, it is not ripe." Given two inputs of the apple's color and hardness, you infer it is ripe, but it says nothing about what decision you should make.</p>
<p>I have a math background so the latter case makes sense to me. How should I think about the normative case? Is a fuzzy controller defined as a device that tries to achieve "truth" among it's rules? Or am I missing something entirely?</p>
http://mathoverflow.net/q/2430670Concentration of matrix norms under random projection.Anirbithttp://mathoverflow.net/users/388522016-06-25T23:00:32Z2016-06-25T23:00:32Z
<p>Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables. </p>
<ul>
<li>Are there cases where one can been able to quantify $P_G [ \vert \vert \vert X \vert \vert - \vert \vert GX \vert \vert \vert > t ] $ ? </li>
</ul>
<p>(choose any matrix norm for which something like this can be shown!) </p>
<hr>
<p>In the above I am trying to quantify how much some matrix norm deviates under doing such a projection. I am thinking of this as doing a compression of the row size from $p$ to $s << p$. </p>
<ul>
<li>Are there other ways known of doing such a compression on the rows preserving some norm of the matrix? </li>
</ul>
http://mathoverflow.net/q/2430653Do the normal forms for braid groups really conceal information about better than randomly applying the braid relations?Joseph Van Namehttp://mathoverflow.net/users/222772016-06-25T21:58:07Z2016-06-25T21:58:07Z
<p>In braid based cryptography, one typically wants to conceal the way a certain braid $b$ has been obtained. One therefore puts $b$ into some normal form. Since every braid has a unique normal form, the normal form of $b$ only reveals the braid $b$ without revealing any other information about how the braid $b$ was obtained. Therefore, normal forms conceal the method which the braid has been obtained. However, do braid normal forms really conceal information about the braids any better than randomly applying the braid laws?</p>
<p>Let $b$ be a positive braid. Let $H$ be the set of all positive braid words which are equivalent to $b$. Now say that two braid words $w,w'$ are $\simeq$-equivalent if $w'$ can be obtained from $w$ by applying only the identities $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for $|i-j|>1$. Let $V=H/\simeq$. Let $E$ be the set of all pairs $\{\mathbf{x},\mathbf{y}\}\in V$ where there exists representatives $x\in\mathbf{x},y\in\mathbf{y}$ such that $y$ can be obtained from $x$ by applying the relation $\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$. What is the random walk mixing time for the graph $(V,E)$? Does it take longer for a computer to calculate a normal form for the braid $b$ or to take a nearly random walk on $[b]$ up to the mixing time?</p>
http://mathoverflow.net/q/2430647What is the smallest density of a metrizable space without countable separation?Taras Banakhhttp://mathoverflow.net/users/615362016-06-25T21:48:17Z2016-06-26T04:31:18Z
<p>A Tychonoff space $X$ is defined to have <em>countable separation</em> if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\in X$ and $y\in bX\setminus X$ there is a set $U\in\mathcal U$ containing exactly one point of the doubleton $\{x,y\}$. The class of spaces with countable separation is very rich: for every compact Hausdorff space $K$ the family of subspaces of $K$ having countable separation is a $\sigma$-algebra, closed under the Suslin $A$-operation. This notion was indroduced by Kenderov, Kortezov and Moors in 2001 (at least). </p>
<p>It can be shown that each metrizable space of density at most continuum has countable separation. On the other hand, by tranfinite induction it is possible to construct a metrizable space of density $\beth_\omega$ without countable separation (this follows from the fact that each Tychonoff space $(X,\tau)$ of cardinality $|X|\ge|\tau^\omega|>\mathfrak c$ contains a subspace $Y$ without countable separation).</p>
<p>Let us recall that $\beth_\omega=\sup_{n\in\omega}\beth_n$ where $\beth_0=\omega$ and $\beth_{n+1}=2^{\beth_n}$, so $\beth_\omega$ is rather large. The key property of the cardinal $\kappa=\beth_\omega$ in this context is that $|\kappa^\omega|=2^\kappa>\mathfrak c$.</p>
<p><strong>Question:</strong> What is the smallest density of a metrizable space without countable separation? Can it be equal to $\mathfrak c^+$? Or it is always $\ge \aleph_\omega$? </p>
http://mathoverflow.net/q/2430630If a Weyl element preserves a root, then it has a representative which preserves the root space?Qing Zhttp://mathoverflow.net/users/134662016-06-25T21:00:25Z2016-06-26T07:26:35Z
<p>Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\alpha\in \Sigma$, let $U_\alpha$ be the root space of $\alpha$. Denote $x_\alpha:F\rightarrow U_\alpha$ the fixed isomorphism.</p>
<p>Let $\alpha\in \Sigma$ be a root, and $w\in W$ be a Weyl element such that $w(\alpha)=\alpha$. My question is: is it true that $w$ has a representative $\dot w\in G$ such that
$$\dot w x_\alpha(r)\dot w^{-1}=x_\alpha(r),\forall r\in F?$$</p>
<p>If this is false in general, in what cases it is true?</p>
<p>Thanks in advance.</p>
http://mathoverflow.net/q/2430620Basic Definition and Notations in RWREodakimkihttp://mathoverflow.net/users/943282016-06-25T20:41:52Z2016-06-26T01:57:29Z
<p>From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ together with the $\sigma-$algebra $\mathcal{F}$ generated by cylinder functions forms a measurable space $(\Omega, \mathcal{F})$. We define a probability measure $P$ on $(\Omega, \mathcal{F})$ and a time homogeneous Markov chain $\{X_n\}_{n \ge 0}$ with transition probabilities
\begin{equation}
Q_{\omega} (X_{n+1} = k|X_n=v) =\omega_v (k)
\end{equation}
$Q_{\omega}$ is called the quenched law of the RW, and the marginal of $\mathbb{P}:=P\otimes Q_{\omega}$ on $V^{\mathbb{N}}$ is the annealed law of the RW. I intentionally used the letters $Q$ and $k$ above.</p>
<p>1) Is this what I should understand from this definition:</p>
<p>$\omega$ is simply an element of the set $\Omega$, and $P$ is a (probability) measure on $(\Omega, \mathcal{F})$. Using $\Omega$ as an index set, $Q_{\omega}$ is the probability measure on $(E_{\omega}, \Sigma_{\omega})$ and $X_n=X_n(\omega, z)$ is a $V-$valued measurable function, i.e., $X_n: (E_{\omega}, \Sigma_{\omega}, Q_{\omega}) \to (V,\mathcal{S})$ for each $\omega \in \Omega$.</p>
<p>2) The environment $\omega$ can be viewed as a sequence of random variables. How
can I connect these two definitions?</p>
<p>Thanks for your time and comments in advance. </p>
http://mathoverflow.net/q/2430610Ring structure on cohomology of groupsmayer_vietorishttp://mathoverflow.net/users/942972016-06-25T20:22:39Z2016-06-25T20:22:39Z
<p>Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter by $H^{∗}(G;A)$. However, I never read that for an arbitrary $G$-module we can define a "product" on cohomology groups that gives the structure of a graded ring, whilst we are talking about the associative graded-commutative ring (given by the so-called cup product) whether $A=Z,F$, with the latter being an arbitrary field and both they've been treated in that case as trivial GG-modules. Why we don't have ring structure in the case where the coefficients are given by non-trivial $G$-modules?</p>
<p>P.S.
Excuse me if you find the question simple, but I am looking desperately for an answer. Thank you!</p>
http://mathoverflow.net/q/243059-1Closed form formula for fill rate given a discrete distribution? [on hold]user94335http://mathoverflow.net/users/943352016-06-25T19:51:36Z2016-06-25T20:02:32Z
<p>I'm wondering whether there is a closed form way to obtain good estimates for <a href="https://www.google.com/search?q=fill+rate&oq=fill+rate&aqs=chrome.0.0l6.894j0j7&sourceid=chrome&ie=UTF-8" rel="nofollow">fill rate</a> given a discrete distribution of demand. </p>
<p>I created a simple monte carlo simulation to see if I could see any patterns. I throw a dice 100,000 times and maintain stock values from 1 to 6. This results in this mardown table (Cut and paste in <a href="http://dillinger.io/" rel="nofollow">http://dillinger.io/</a> to see it fully formatted):</p>
<p>| Stocking Level | Total Demand | Total Demand Serviced | Total Demand Serviced / Total Demand |
|:--------------:|:------------:|:---------------------:|:------------------------------------:|
| 1 | 3500082 | 1000000 | 0.285707591993559 |
| 2 | 3503039 | 1834191 | 0.5235999370832012 |
| 3 | 3499088 | 2499394 | 0.7142986972605433 |
| 4 | 3497891 | 2998013 | 0.8570916017680368 |
| 5 | 3498174 | 3331758 | 0.9524277523073467 |
| 6 | 3500243 | 3500243 | 1.0000000000000000 |</p>
<p>Here's the last column:</p>
<p>Total Demand Serviced / Total Demand |
:------------------------------------:|
0.285707591993559 |
0.5235999370832012 |
0.7142986972605433 |
0.8570916017680368 |
0.9524277523073467 |
1.0000000000000000 |</p>
<p>So the first value is roughly 1 / 3.5, which is E[min(X,s)] / E[X], but what about the rest of the values? </p>
<p>Thoughts?</p>
<p>TIA,
Ole</p>
http://mathoverflow.net/q/2430572Resolution of the ideal of the Abel-Jacobi image of a curve?ritahttp://mathoverflow.net/users/106102016-06-25T19:42:41Z2016-06-25T20:17:55Z
<p>Let $C$ be a complex curve of genus $g\ge 2$ and let $a\colon C\to J(C)$ be the Abel-Jacobi map. Is there a finite resolution of the ideal $\mathcal I_{a(C)}$ whose terms are sums line bundles of the form $\mathcal O_{J(C)}(m \Theta)$? </p>
<p>I think I remember seeing something of this type years ago, but I haven't been able to find anything related on the web. Maybe I just imagined it. </p>
http://mathoverflow.net/q/2430530Can some exotic sphere be diffeomorphically embedded into some $R^n$?Arnold Neumaierhttp://mathoverflow.net/users/569202016-06-25T18:06:53Z2016-06-25T21:17:24Z
<p>Can some (or perhaps every) exotic sphere be diffeomorphically embedded into some $R^n$? How does such an embedding (if it exists) look like? I.e., what are the equations for a particular embedding?</p>
<p>Note that the answer to the related question <a href="http://mathoverflow.net/questions/134479/">Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?</a> is only about immersions, which leaves my question open.</p>
http://mathoverflow.net/q/243051-4A property of minimal prime ideals [on hold]rostamihttp://mathoverflow.net/users/943312016-06-25T17:44:00Z2016-06-25T23:15:32Z
<p>Let $R$ be a commutative ring with $1$, and let $p$ be a minimal prime ideal of $R$. If $p\subseteq I_1+ I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $p\subseteq I_1 $ or $p\subseteq I_2 $?</p>
http://mathoverflow.net/q/2430490Exact formula for computing n-step transition probability of random walks with self-transitionsyeliqseuhttp://mathoverflow.net/users/943272016-06-25T17:30:50Z2016-06-26T05:18:43Z
<p>Consider a semi-infinite random walks $X_n$, $n=0,1,2,\ldots$, whose state space is a set of consecutive integers and whose one-step transition probabilities are $P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}$. Assume that $P_{00}=1-p$ and $P_{ii-1}=q>0$, $P_{ii}=r>0$, $P_{ii+1}=p$ for $i\geq1$; $p+q+r=1$. Consider that the process starts at state $i$ at time $0$. Let $P_{ij}^{n}$ be the probability that the process is at state $j$ at time $n$. I would like to know whether there is an exact formula for computing the $n$-step transition probability $P_{ij}^n$? Or at least $P_{00}^n$? I need an explicit formula rather than the asymptotic analysis of the random process as $n\rightarrow \infty$.</p>
<p>I came across the Karlin-McGregor representation formula in [1] which shows that</p>
<p>\begin{equation}
P_{ij}^n=\pi_j\int_{-1}^1x^nQ_i(x)Q_j(x)\mathrm{d}\psi(x)
\end{equation}
for some polynomials $Q_n(x)$ ($Q_0(x)=1$) and measure function $\psi(x)$. But I was not able to find an explicit formula of $Q_n(x)$ and $\psi(x)$ for the case the self-transition probability $r>0$. Does anyone know whether $P_{ij}^n$ can be explicitly calculated for this case?</p>
<p>[1] Karlin, Samuel; McGregor, James. Random walks. Illinois J. Math. 3 (1959), no. 1, 66--81. <a href="http://projecteuclid.org/euclid.ijm/1255454999" rel="nofollow">http://projecteuclid.org/euclid.ijm/1255454999</a>.</p>
http://mathoverflow.net/q/2430453Bijection modeling isomorphism of infinite-dimensional vector spacesJake Levinsonhttp://mathoverflow.net/users/455052016-06-25T16:02:10Z2016-06-26T00:43:26Z
<p>Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality.</p>
<blockquote>
<p>Does there exist a bijection $f : B_V \to B_W$ such that, for each
$b_V \in B_V$, the coefficient of $f(b_V)$ in $T(b_V)$ is nonzero?</p>
</blockquote>
<p>If $V,W$ are finite-dimensional, the answer is yes: $\det(T) \ne 0$, hence some monomial term is nonzero. This exhibits a satisfactory $f$. Alternately, build $f$ inductively by Laplace expansion along a row or column. (Some term $a_{ij} \cdot \text{(complementary minor)}$ is nonzero, and so on.)</p>
<p>Does this hold in general? I have tried using Zorn's lemma, but it seems tricky enough that maybe I'm overlooking a straightforward counterexample.</p>
<p>I can state what I've attempted if there's interest.</p>
<p>(I have tagged this as linear and homological algebra since my desired application is in the latter.)</p>
http://mathoverflow.net/q/2430442Cambridge Mathematical Tripos papers from late 19th centuryvvnhttp://mathoverflow.net/users/943252016-06-25T15:55:32Z2016-06-25T21:33:13Z
<p>Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet?</p>
http://mathoverflow.net/q/2430304Unreasonable application of mathematics to the other areas [on hold]Ali Taghavihttp://mathoverflow.net/users/366882016-06-25T10:48:39Z2016-06-26T00:18:07Z
<p>What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science?</p>
<p>I found one paper as follows page 515, the last paragraph(before the discussion)</p>
<p><a href="http://www.sciencedirect.com/science/article/pii/0315086075901135" rel="nofollow">http://www.sciencedirect.com/science/article/pii/0315086075901135</a>.</p>
<p>I search for some more references.Your answer is very appreciated. </p>
<p>Edit: Our question is a particular case of the following post. However our post is motivated by Bishop paper, a paper which we cited it but it is not cited in the following post</p>
<p><a href="http://mathoverflow.net/questions/61408/examples-of-theorems-misapplied-to-non-mathematical-contexts">Examples of theorems misapplied to non-mathematical contexts</a></p>
http://mathoverflow.net/q/2430233Description of connecting maps of Derived functorsShubhodip Mondalhttp://mathoverflow.net/users/239272016-06-25T08:34:12Z2016-06-26T03:48:14Z
<p>Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\delta_i: R^i F(A'') \to R^{i+1}F(A')$. </p>
<p>Now let $0 \to A' \to I^\bullet$ be an injective resolution of $A'$. The resolution $0 \to A' \to A \to A''\to 0$ maps to $0 \to A' \to I^\bullet$ (unique upto homotopy). So we get a map $\phi:A'' \to I^1$ such that $A''$ lands inside $\text{ker} (I^1 \to I^2)$. So this gives a map from $F(A'') \to R^1 F(A')$, which nfdc23 mentioned <a href="http://mathoverflow.net/questions/242894/alternative-construction-of-the-first-chern-class-map">here</a> to be <em>negative</em> the first connecting map.</p>
<p>Now let us choose an injective resolution $0 \to A'' \to K^\bullet$. The map $\phi:A'' \to I^1$ can be extended to obtain a map $K^0 \to I^1$. This extends and gives maps $K^n \to I^{n+1}$ which gives a map of complexes $K^\bullet \to I^{\bullet {+1}}$. Applying $F$ and taking cohomology gives maps $\phi_i:R^i F (A'') \to R^{i+1}F(A')$.</p>
<hr>
<p><strong>Question:</strong> What is the relation between $\delta_i $ and $\phi_i$? Does it follow that $\delta_i = (-1)^{i+1} \phi_i$?</p>
http://mathoverflow.net/q/2430141Relative Leopoldt defectAdel Lillehttp://mathoverflow.net/users/464602016-06-24T23:50:26Z2016-06-25T23:50:27Z
<p>Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$.</p>
<p>Is there a bound of the Leopoldt defect of $M$ ?</p>
http://mathoverflow.net/q/2429196Speed of convergence in Lebesgue's density theoremuser240643http://mathoverflow.net/users/942512016-06-23T17:19:08Z2016-06-26T03:13:31Z
<p>Let $\lambda=\text{unif}([0,1])$ be uniform distribution on $[0,1]$ and $B$ be any Borel set. Lebesgue's density theorem states that for $\lambda$-almost all $x\in[0,1]$ the limit
$$\lim_{\epsilon\downarrow o}\frac{\lambda([x-{\epsilon},x+\epsilon]\cap B)}{2\epsilon}$$
exists and is either $0$ or $1$. Im interested in a 'global' approximation.</p>
<p>For $n\geq 1$ let
$$D_n:=\big\{\frac{0}{2^n},\frac{1}{2^n},\frac{2}{2^n},\dots,\frac{2^n-1}{2^n}\big\}.$$</p>
<p>Now i look at the expression </p>
<p>$$a_n:=(1/2)^n\cdot\sum_{x\in D_n}\frac{\lambda\big([x,x+(1/2)^n]\cap B\big)}{(1/2)^n}\cdot\bigg[1-\frac{\lambda\big([x,x+(1/2)^n]\cap B\big)}{(1/2)^n}\bigg].$$
So for large $n$ every summand should be close to Zero. Additionally every summand gets weighted by $(1/2)^n$.</p>
<p>The number $a_n$ tells you something about how 'intervall-like' $B$ is. Since you can approximate every $B$ by finite unions of intervalls you can show that $a_n\rightarrow 0$ for each $B$. I want to know: How fast does $a_n$ tend to Zero? Is it true that $\sum_{n\geq 1}a_n<\infty$ for any measurable $B$? If this is not the case, can you provide an example of a set $B$ with $\sum_n a_n=\infty$? </p>
<p>Remark: This is a special instance of a more general question I posted on Stackexchange (before I knew that Overflow exists, sorry about that): <a href="http://math.stackexchange.com/questions/1835620/switching-independent-experiments-does-i-ax-1-dots-x-n-y-n1-y-n2-dots">http://math.stackexchange.com/questions/1835620/switching-independent-experiments-does-i-ax-1-dots-x-n-y-n1-y-n2-dots</a></p>
http://mathoverflow.net/q/2428664Hilbert modular forms twist-equivalent to their conjugatesDavid Loefflerhttp://mathoverflow.net/users/24812016-06-23T08:31:31Z2016-06-26T04:02:55Z
<p>Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.</p>
<p>Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate newform $f_\sigma$ is twist-equivalent to $f$, i.e. there exists a Hecke character $\chi_\sigma$ of $L$ such that $a_{\sigma(\mathfrak{p})}(f) = \chi_{\sigma}(\mathfrak{p}) a_\mathfrak{p}(f)$ for all but finitely many primes $\mathfrak{p}$ of $L$, where $a_{\mathfrak{p}}(f)$ is the Hecke eigenvalue at $\mathfrak{p}$. </p>
<blockquote>
<p>Does it follow that $f$ is twist-equivalent to the base-change to $L$ of a Hilbert modular form over $K$? </p>
</blockquote>
<p>I'm happy to assume that $f$ is non-CM, and that all weights of $f$ are $\ge 2$, if that helps.</p>
<p>(Note that Galois is acting on $L$ here, not on the coefficients of $f$ -- this is <strong>not</strong> the same setup as the Ribet--Momose theory of "inner twists".)</p>
http://mathoverflow.net/q/2428535Intuitive descriptions of some large cardinalsAnindyahttp://mathoverflow.net/users/942322016-06-23T03:50:34Z2016-06-26T01:45:38Z
<p>I was trying to formulate intuitive descriptions of some large cardinals.<br>
Roughly something equivalent to "A manifold is an object which looks like patches of $R^n$ glued together". Not perfectly rigorous, but hopefully conveys the basic picture.<br>
Here are three descriptions I have in mind:<br>
1) An inaccessible cardinal is a set so large that it can't be reached from smaller infinite sets using unions and power set operations.<br>
2) A measurable cardinal is a set so large that it can't be reached from smaller infinite sets using any set theoretic formula. (I am going for V $\neq$ L)<br>
3) A Reinhardt cardinal is a set so large that all possible properties of the entire set theoretic universe are also true of this set. (If I am correct, this would capture the intuition of why Reinhardt cardinals don't exist. They are simply too ambitious).<br>
I am pretty confident that description 1 is correct in essentials, but are 2) and 3) hopelessly off ? If so, how would one appropriately modify the descriptions ? Or are object like measurable cardinals just too abstract to be expressed in anything but technical definitions ?<br>
Apologies in advance if this question is too elementary for MathOverflow.</p>
http://mathoverflow.net/q/2428526Lexicographic distribution of irreducible polynomialsMartyhttp://mathoverflow.net/users/35452016-06-23T03:50:11Z2016-06-26T00:09:49Z
<p>Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its coefficients as binary digits $\mathbf{b}10011$ and find that it is the $19$th polynomial. Let $f$ be the resulting bijection from $A$ to $\mathbb N$, so $f(P) = 19$.</p>
<p>The prime number theorem and Riemann hypothesis are known for the ring $A$. In fact, I learned from Paul Pollack's thesis (<a href="http://alpha.math.uga.edu/~pollack/thesis/thesis-final.pdf" rel="nofollow">http://alpha.math.uga.edu/~pollack/thesis/thesis-final.pdf</a>) that it is essentially contained in Gauss's unpublished 8th chapter of the Disquisitiones. But what about in this not-so-natural lexicographic order? In other words, let $\pi(x)$ be the number of irreducible polynomials $P$ in $A$ such that $0 \leq f(P) \leq x$.</p>
<p>Is it known or expected that the prime number theorem or Riemann hypothesis is true for $\pi(x)$? Is $\pi(x) \approx li(x)$ with error $O(\sqrt{x} \cdot \log(x))$?</p>
http://mathoverflow.net/q/2427296Counting limit cycles via curvature in Riemannian geometryAli Taghavihttp://mathoverflow.net/users/366882016-06-21T11:56:05Z2016-06-25T21:41:58Z
<p>In this post we would like to give a possible new approach to the second part of the <a href="https://en.wikipedia.org/wiki/Hilbert%27s_sixteenth_problem" rel="nofollow">Hilbert 16th problem</a></p>
<p>First we give a short introduction:</p>
<p>A quadratic system is a polynomial vector field on the plane in the form $$(X_{\alpha})\;\;\;\;\;\;\;\;\;\;\begin{cases}\dot x=P_{\alpha}(x,y)\\ \dot y=Q_{\alpha}(x,y) \end{cases}$$</p>
<p>where $P_{\alpha},Q_{\alpha}$ are polynomials of degree $2$ which are parametrized by a $10$ tuple parameter $\alpha$ of coefficients. </p>
<p>A <strong>center</strong> is a singularity of this vector field which is surrounded by a band of closed orbits.</p>
<p>All quadratic vector fields with center are classified as follows: </p>
<p>They correspond to a finite number of algebraic conditions in $\alpha$,
see <a href="https://www.researchgate.net/publication/246484433_Integrability_of_plane_quadratic_vector_fields" rel="nofollow">"Integrability of plane quadratic vector fields" Expos. Math(1990)3-25.</a>.</p>
<p>We denote these algebraic conditions by $Cent(\alpha)=0$</p>
<p><strong>Question:</strong></p>
<blockquote>
<p>Are there a family of (polynomial) Riemannian metrics $g_{\alpha}$ on the punctured plane(after removing singularities) with Gaussian curvature $\kappa_{\alpha}$ with the following properties:</p>
<p>The solutions of $X_{\alpha}$ are geodesics of $g_{\alpha}$.Moreover $\kappa_{\alpha}$, is not zero except at a finite number of algebraic curves transverse to $X_{\alpha}$. Moreover $\kappa_{\alpha}$ is identically zero if $Cent(\alpha)=0$? The question, in particular its last part is well behaved and consistent since a quadratic system can not have a center and a limit cycle, simultaneously. </p>
</blockquote>
<p>If the answer would be yes, <a href="https://en.wikipedia.org/wiki/Hilbert%27s_sixteenth_problem" rel="nofollow">then $H(2)$, the maximum number of limit cycles of a quadratic system,</a> would be finite.</p>
<p>This question is already discussed at the comment-conversation of the following post:</p>
<p><a href="http://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-curved-space">Limit cycles as closed geodesics(in negatively curved space)</a></p>
http://mathoverflow.net/q/2397292No cuspidal character sheaves on GL(n)Ramihttp://mathoverflow.net/users/46902016-05-25T12:22:43Z2016-06-25T21:09:46Z
<p>We need a reference for the fact that there are no cuspidal character sheaves on $GL_n$ unless $n=1$.</p>
<p>See page 11 of <a href="http://www.kurims.kyoto-u.ac.jp/~arakawa/Henderson_mgsctalk2.pdf" rel="nofollow">http://www.kurims.kyoto-u.ac.jp/~arakawa/Henderson_mgsctalk2.pdf</a>.</p>
http://mathoverflow.net/q/305915Simplicial Model of Hopf Map?Chris Schommer-Prieshttp://mathoverflow.net/users/1842009-10-28T14:24:33Z2016-06-25T23:33:55Z
<p>The Hopf fibration is a famous map S<sup>3</sup> --> S<sup>2</sup> with fiber S<sup>1</sup>, which is the generator in pi_3(S<sup>2</sup>). We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces and the induced map of simplcial sets. But this model is HUGE and isn't really useful for doing calculations. Does anyone know a nice small model for this map in terms of simplicial sets? Something suitable for computations? This map is also the attaching map used to build CP<sup>2</sup> out of S<sup>2</sup>, so I would equivalently be interested in a small combinatorial model for CP<sup>2</sup>.</p>