Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2014-12-22T14:24:45Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/1912801Is it consistent that $\frak{d} < 2^{\aleph_0}$?Dominic van der Zypenhttp://mathoverflow.net/users/86282014-12-22T13:34:44Z2014-12-22T14:23:03Z
<p>Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is said to be <em>dominating</em> if for all $f\in \omega^\omega$ there is $g\in D$ such that $f <^* g$. Set $$\frak{d} = \textrm{min}\{|\mathrm{D}|: \mathrm{D}\subseteq \omega^\omega \textrm{ and } \mathrm{D} \textrm{ is dominating}\}.$$</p>
<p>Is it consistent that $\frak{d} < 2^{\aleph_0}$?</p>
http://mathoverflow.net/q/191278-4Probably some naive question on conditional probability [on hold]Jon D.http://mathoverflow.net/users/642152014-12-22T12:00:25Z2014-12-22T12:00:25Z
<p>As known, three variables x_1, x_2 and y, if x_1 and x_2 are conditional independent given y, we have p(x_1, x_2|y) = p(x_1|y)p(x_2|y).
I was wondering about p(y|x_1, x_2), is that possible to get p(y|x_1, x_2) = p(y|x_1)p(y|x_2) under some condition? </p>
http://mathoverflow.net/q/1912750Powers in compact coset spacesColin Reidhttp://mathoverflow.net/users/40532014-12-22T11:07:04Z2014-12-22T14:07:41Z
<p>Let $G$ be a topological group, let $K$ be a closed cocompact subgroup (i.e. the coset space $G/K$ is compact in the quotient topology) and let $g \in G$. Is there a sequence of positive powers $g^{i_n}$ of $g$ such that $g^{i_n}K$ converges to $K$ in the coset space $G/K$?</p>
<p>If the answer is `no' in general, what if $G$ is totally disconnected and locally compact? (For the application, I'd be happy if I could at least get powers of $g$ to land in $UKV$ for any pair of identity neighbourhoods $U$ and $V$.)</p>
http://mathoverflow.net/q/1912740Two (strictly related) proofs by induction of inequalitiesVincenzo Olivahttp://mathoverflow.net/users/570682014-12-22T10:40:50Z2014-12-22T10:40:50Z
<p>This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the <a href="http://math.stackexchange.com/questions/1056240/two-strictly-related-proofs-by-induction-of-inequalities">original question</a>.</p>
<p>Predictably, I am stuck with the inductive steps.
Let $p_m$ be the largest prime factor of $a_n$ and set $\lim_{n\to \infty}\frac{\log a_n}{p_m}=1$. Suppose also this ratio converges to $1$ faster than $\displaystyle\frac{p_{m+1}}{p_m}$, so that if $n$ is large enough, we always have $\log a_n<p_{m+1}$. </p>
<p>I want to prove that for sufficiently large $n$, with $c$ being a constant and $q<m$, if $$\frac{c}{\log \log a_n}<\frac{\left(1+{\prod_{i=1}^m\left(p_i^{b_i+1}-1\right)^{1/m}}\right)^m}{\prod_{i=1}^m\left(p_i^{b_i+1}-1\right)}, \tag{1}$$ then the following statements are true: </p>
<blockquote>
<p>$$ \frac{c}{\log \left(\log a_n+\log p_q\right)}<\\\frac{\left(1+\prod_{i=1}^{q-1}\left(p_i^{b_i+1}-1\right)^{1/m}\cdot\left(p_q^{b_q+2}-1\right)^{1/m}\cdot\prod_{i=q+1}^{m}\left(p_i^{b_i+1}-1\right)^{1/m}\right)^m}{\prod_{i=1}^{q-1}\left(p_i^{b_i+1}-1\right)\cdot\left(p_q^{b_q+2}-1\right)\cdot\prod_{i=q+1}^{m}\left(p_i^{b_i+1}-1\right)}; \tag{2}$$</p>
<p>$$ \frac{p_{m+1}}{p_m}\frac{c}{\log \left(\log a_n+\log p_{m+1}\right)}<\frac{\left(1+\prod_{i=1}^{m+1}\left(p_i^{b_i+1}-1\right)^{1/(m+1)}\right)^{m+1}}{\prod_{i=1}^{m+1}\left(p_i^{b_i+1}-1\right)}. \tag{3}$$</p>
</blockquote>
<p>To clear it up, in $(2)$ we have $a_n\cdot p_q=a_{n+1}$, in $(3)$ instead $a_n\cdot p_{m+1}=a_{n+1}$.</p>
<p>$(2)$ is fairly intuitive, as the LHS goes to $0$ as $n\to \infty$ while the RHS goes to $1$, but that doesn't tell me so much since if the former is larger than $1$ and slightly smaller than the latter, I cannot say <em>a priori</em> that the LHS is sufficiently fast in its convergence to $0$, to be always less than the RHS.
On the other hand, it is only my istinct that says $(3)$ holds, but I might be wrong.</p>
<p>Here is how I tackled both inequalities, hoping to simplify things a bit (and not "too much"). Call $L_t$ and $R_t$ respectively the LHS and RHS of $(1)$, $(2)$ and $(3)$. So $(2)$ is the same as $$ L_1 \frac{L_2}{L_1}<R_1\frac{R_2}{R_1},$$ and since $L_1<R_1$ by hypothesis, $(2)$ is implied by $$ \frac{\log \log a_n}{\log \left(\log a_n+\log p_q\right)}<\\ \frac{p_q^{b_q+1}-1}{p_q^{b_q+2}-1}\left(\frac{1+\prod_{i=1}^{q-1}\left(p_i^{b_i+1}-1\right)^{1/m}\cdot\left(p_q^{b_q+2}-1\right)^{1/m}\cdot\prod_{i=q+1}^{m}\left(p_i^{b_i+1}-1\right)^{1/m}}{1+{\prod_{i=1}^m\left(p_i^{b_i+1}-1\right)^{1/m}}}\right)^m.\tag{4}$$
Similarly, $(3)$ follows from $$ \frac{p_{m+1}}{p_m}\frac{\log \log a_n}{\log \left(\log a_n+\log p_{m+1}\right)}<\\ \left(1+\prod_{i=1}^{m+1}\left(p_i^{b_i+1}-1\right)^{1/(m+1)}\right)\left(\frac{1+\prod_{i=1}^{m+1}\left(p_i^{b_i+1}-1\right)^{1/(m+1)}}{1+{\prod_{i=1}^m\left(p_i^{b_i+1}-1\right)^{1/m}}}\right)^m.\tag{5}$$
This said, I do not know how to prove $(4)$ and $(5)$ either. Any ideas? Thanks in advance.</p>
http://mathoverflow.net/q/191273-3Proving integration techniques [on hold]Anas Ayubihttp://mathoverflow.net/users/642192014-12-22T10:35:34Z2014-12-22T10:35:34Z
<p>I've recently competed my A levels and now that I'm in the university I finally found the time to understand calculus on a intuitive level. So I've been reading up on books such as "Calculus with easy" by Thompson, visiting khan academy and betterexplained.com
However, even now I have some doubts that I'm trying to clear out especially with integration. My main question is: how does one find out the procedure of solving an integral without knowing that it is an anti-derivative, i.e. without having the fundamental theorem to lean on?</p>
<p>If you guys have any other resources that I could possibly use to understand calculus through an intuitive approach, please share!! Would really appreciate it. </p>
http://mathoverflow.net/q/1912720Inequivalent definitions of Cartan subalgebraTim kinsellahttp://mathoverflow.net/users/383472014-12-22T10:34:41Z2014-12-22T10:34:41Z
<p>As far as I can tell, there exists no acknowledgment on the internet of the fact (or maybe it's not a fact) that inequivalent definitions of "Cartan subalgebra" of a real Lie algebra exist in the literature. I was hoping an expert could remedy this by listing the definitions and confirming that they are indeed inequivalent. </p>
<p>My apologies if this is not appropriate for MO. I asked this question here: <a href="http://math.stackexchange.com/questions/1071417/are-there-common-inequivalent-definitions-of-cartan-subalgebra-of-a-real-lie-alg">http://math.stackexchange.com/questions/1071417/are-there-common-inequivalent-definitions-of-cartan-subalgebra-of-a-real-lie-alg</a> </p>
<p>and I'll be happy to delete that post or this post if the other is answered (assuming this one isn't closed). </p>
http://mathoverflow.net/q/1912662Anti-compactnessDominic van der Zypenhttp://mathoverflow.net/users/86282014-12-22T08:11:02Z2014-12-22T09:12:44Z
<p>Let $(X,\tau)$ be a topological space such that $\tau$ is a proper superset of $\{\emptyset, X\}$. We call an open cover $\mathcal{U}$ of $(X,\tau)$ <em>proper</em> if $X\notin \mathcal{U}$. Moreover we say that $(X,\tau)$ is</p>
<ul>
<li><em>anti-compact</em> if it does not have a finite proper cover;</li>
<li><em>anti-paracompact</em> if for every proper cover $\mathcal{U}$ there is $x\in X$ such that every neighborhood intersects infinitely many members of $\mathcal{U}$;</li>
<li><em>anti-metacompact</em> if for every proper cover $\mathcal{U}$ there is $x\in X$ such that $x$ is a member of infinitely many members of $\mathcal{U}$. </li>
</ul>
<p>We have anti-metacompact $\Rightarrow$ anti-paracompact $\Rightarrow$ anti-compact.</p>
<p>Do any of the converse implications hold?</p>
http://mathoverflow.net/q/1912625Sierpinski's construction of a non-measurable setAsaf Karagilahttp://mathoverflow.net/users/72062014-12-22T05:50:51Z2014-12-22T08:52:05Z
<p>In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to the following paper:</p>
<blockquote>
<p>Sierpinski, W. <strong>"L’axiome de M. Zermelo et son rôle dans la théorie des ensembles et l’analyse."</strong> <em>Bulletin de l’Académie des Sciences de Cracovie, Classe des Sciences Math., Sér. A (1918)</em> (1918): 97-152.</p>
</blockquote>
<p>(And even more specifically, to pages 124-125.)</p>
<p>Moore writes that Sierpinski proved that if $[\Bbb R]^\omega$ has cardinality $2^{\aleph_0}$, then there is a non-measurable set. Lebesgue argued that Sierpinski uses the axiom of choice, but Moore points out that while the assumption requires some amount of choice (as we full well know today), the implication does not.</p>
<p>Being a historical book about the axiom of choice, Moore doesn't provide a sketch of the proof. However, I've been unable to locate the paper. Which brings me to my question.</p>
<p><strong>Question.</strong> Is there any accessible (preferably English) reference to what the argument of Sierpinski was?</p>
http://mathoverflow.net/q/191261-3Two problems in functional analysis [on hold]Hyz Yuzhouhttp://mathoverflow.net/users/626922014-12-22T04:40:59Z2014-12-22T04:58:04Z
<ol>
<li>Let $f$ be linear functional on Banach space $B$ and $ker f$ is closed subspace of $B$, prove that $f$ is a bounded linear functional.</li>
<li>Let $\{e_n\}$ be an orthonormal basis of Hilbert space H. T is a linear operator on H, satisfy $\sum \Vert Te_n\Vert < \infty$, prove that T is bounded linear operator.</li>
</ol>
<p>These two problems appear in my homework. Both them are to show a linear operator is bounded(or continuous), but I have no idea.</p>
http://mathoverflow.net/q/1912605Is there a "good" reason that the universal central extension of $SL(2,\mathbb Z)$ is $Br_3$?Allen Knutsonhttp://mathoverflow.net/users/3912014-12-22T04:08:00Z2014-12-22T06:59:40Z
<p>Let
$$ Br_3 := \langle \tau_1,\tau_2\ :\ \tau_1 \tau_2 \tau_1 = \tau_2 \tau_1 \tau_2 \rangle $$
be the braid group on three strands, and consider the surjection
$$\phi : Br_3 \twoheadrightarrow SL_2(\mathbb Z), \qquad
\tau_1 \mapsto \begin{pmatrix} 1&0\\ 1&1\end{pmatrix}, \quad
\tau_2 \mapsto \begin{pmatrix} 1&-1\\ 0&1\end{pmatrix}.
$$
According to Wikipedia, this is the universal central extension. Moreover,
it arises as the fiber product of the inclusion $SL_2(\mathbb Z) \hookrightarrow SL_2(\mathbb R)$ and the universal cover $\widetilde{SL_2(\mathbb R)} \twoheadrightarrow SL_2(\mathbb R)$.</p>
<blockquote>
<p>Is there a satisfying reason why there <em>should</em> be a map $\phi$?</p>
</blockquote>
<p>Perhaps something Galois-theoretic, based on $Br_3$ being the fundamental group of the configuration space of $3$ points in the plane?</p>
<p>Feel free to retag. If there are many satisfying reasons, I'll make it community wiki!</p>
<p>ADDED: This is, if not the same question, at least really close to <a href="http://mathoverflow.net/questions/20281/details-for-the-action-of-the-braid-group-b-3-on-modular-forms">Details for the action of the braid group B_3 on modular forms</a> , and Dylan Thurston's answer is pretty satisfying (even if he isn't satisfied).</p>
http://mathoverflow.net/q/1912530Bombieri-Vinogradov in short intervalsStijnhttp://mathoverflow.net/users/494382014-12-22T02:05:57Z2014-12-22T09:33:38Z
<p>In 1985 <a href="http://link.springer.com/article/10.1007%2FBF01388653#page-1" rel="nofollow">Perelli, Pintz & Salerno</a> proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the reduction of the lower bound on $\theta$? </p>
<p>I'm not fussed if some of the other variables are changed as, clearly, $\psi$ would have to be altered if we could get $\theta < 1/2$, but I just need a good $\theta$ whilst keeping (roughly) the same R.H.S..</p>
http://mathoverflow.net/q/191248-1Maximum connected components $0-1$ matrixTurbohttp://mathoverflow.net/users/100352014-12-21T22:53:19Z2014-12-22T04:22:18Z
<p>Let the notion of connected matrix be as in here <a href="http://mathoverflow.net/questions/190981/connected-components-0-1-matrices">Connected components $0-1$ matrices</a></p>
<p>Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where each step you take you step on another $1$.</p>
<p>The answer to the question "Can every $0-1$ be converted to a matrix of one component by permutations of rows and columns?" was posted there. Not every matrix has one component.</p>
<p>Define a $k$ component matrix as a matrix that cannot be reduced to $k-1$ components by permutations of rows and columns.</p>
<p>Infact, the approach in the answer in post <a href="http://mathoverflow.net/questions/190981/connected-components-0-1-matrices">Connected components $0-1$ matrices</a>
can be directly used to get $k$ component matrices for any constant number of components $k$.</p>
<p>It would be nice to know how big $k$ (upper bound) can be for a given $n$. Is there any sharp estimate for $k$ as a function of $n$?</p>
http://mathoverflow.net/q/19122220Cantor's theorem for presheaves?Todd Trimblehttp://mathoverflow.net/users/29262014-12-21T15:13:29Z2014-12-22T13:00:40Z
<p>Some years back (before MathOverflow was born), Tom Leinster <a href="https://golem.ph.utexas.edu/category/2008/01/2toposes.html#c014438">asked</a> an interesting question at the $n$-Category Café which I don't recall ever seeing an answer for: </p>
<blockquote>
<p>Does there exist a category $C$ that admits an essentially surjective functor $F: C \to Set^{C^{op}}$? </p>
</blockquote>
<p>Terminology: we say that a functor $F: C \to D$ is <a href="http://ncatlab.org:8080/nlab/show/essentially+surjective+functor">essentially surjective</a> if every object $d$ of $D$ is isomorphic to some value $F(c)$. This is the good notion of surjectivity for the 2-category $Cat$, or at least one good notion. </p>
<p>As is well-known from categorical circles, the presheaf category $Set^{C^{op}}$ here plays a role of "power object" $P(C)$ that is usefully regarded as analogous to power sets in set theory or more generally in toposes. (For example, the Yoneda embedding $y_C: C \to Set^{C^{op}}$ plays a role analogous to the singleton mapping $\{-\}: S \to P(S)$ from set theory.) In fact Tom's question is embedded in a larger discussion of what one should mean by a '2-topos' -- see that discussion for more on the analogy. </p>
<p>So the question above is analogous to one that Cantor's theorem answers: can one have a set $S$ that maps onto its power set $S$? So the expected answer to the question is '<strong>no</strong>'. Note however that the standard diagonalization technique behind Cantor's theorem, as explained for example <a href="http://arxiv.org/abs/math/0305282">here</a>, doesn't apply in any obvious way since there is no general decent notion of diagonal map $C \to C \times C^{op}$. </p>
<p>Regarding foundational issues: I'll leave that up to you. :-) If you want me to impose a constraint, we might add the condition that $C$ is locally small, but note that we'll soon be leaving the land of local smallness anyway, since there is a <a href="http://www.tac.mta.ca/tac/volumes/1995/n9/1-09abs.html">result</a> due to Freyd and Street that if also $Set^{C^{op}}$ is locally small, then $C$ is (equivalent to) a small category, and that would be a <em>huge</em> constraint that makes the question not so interesting. </p>
http://mathoverflow.net/q/1912014Do geodesics in SL2R map to geodesics in the hyperbolic plane?HenrikRüpinghttp://mathoverflow.net/users/39692014-12-21T08:15:13Z2014-12-22T07:04:50Z
<p>I am looking for a reference/proof/disproof of the following statement.</p>
<p>Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle A,B\rangle_e :=tr(AB^*)$. Let $pr:SL_2(\mathbb{R})\rightarrow SL_2(\mathbb{R})/SO_2(\mathbb{R})=\mathbb{H}^2$ be the canonical projection.</p>
<p>Is is true that $pr\circ \gamma$ is a (constant speed) geodesic in $\mathbb{H}^2$ for any geodesic $\gamma$ in $SL_2(\mathbb{R})$? </p>
<p>The speed might even be $0$. Another similar example, where geodesics map to constant speed geodesics would just be the orthogonal projection $\mathbb{R^2}\rightarrow \mathbb{R}\times \{0\}$.</p>
http://mathoverflow.net/q/1911822Prime labelling of graphsMarcAndresonhttp://mathoverflow.net/users/641212014-12-20T19:51:22Z2014-12-22T13:16:57Z
<p>A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two adjacent vertices are relatively prime).</p>
<p>What can <em>prime labeling</em> be useful for? I was thinking of any applications like scheduling etc. but I'm not sure.</p>
http://mathoverflow.net/q/1911667Application of the Riemann hypothesis and the ABC conjecture to independence resultsMohammad Golshanihttp://mathoverflow.net/users/111152014-12-20T13:45:39Z2014-12-22T05:18:40Z
<p>In <a href="http://wwwmath.uni-muenster.de/u/weiermann/" rel="nofollow">Old Home Page of Andreas Weiermann</a> Andreas Weiermann has stated the following:</p>
<blockquote>
<p>Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to independence results for publication.</p>
</blockquote>
<p><strong>Question 1.</strong> can someone give a short description of the stated work(s)?</p>
<p><strong>Question 2.</strong> Do such independence results say anything about the independence of Riemann hypothesis or ABC conjecture in $PA$ or some of its weaker sub-theories?</p>
<p>The paper <a href="https://biblio.ugent.be/publication/861950" rel="nofollow">Unprovability, phase transitions and the Riemann zeta-function</a> by Bovykin-Wiermann may be related.
Also as Jaso Rute has suggested, the paper <a href="http://cage.ugent.be/~weierman/MSJ.pdf" rel="nofollow">Phase transitions in logic and combinatorics</a> might be also helpful.</p>
http://mathoverflow.net/q/1911572Where do Set Theory and Number Theory meet together?Rahman. Mhttp://mathoverflow.net/users/388662014-12-20T08:13:23Z2014-12-22T06:41:35Z
<p>As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of natural numbers. But on one hand we have Forcing Methods and Theory of Core Model to investigate about reals and the real line, and on the other hand for some statements in Number Theory we have, equivalent statements expressed by real or complex numbers, using Analytic Number Theory. Therefore, it seems it's possible to reconcile two hands!!</p>
<p>Now my question is:</p>
<h3>Is there any theorem in Number Theory that can be proved by tools of Set Theory, especially by methods of consistency results?</h3>
<p>Any reference is appreciated. </p>
http://mathoverflow.net/q/19113911Complete resolutions of GCHJesse Elliotthttp://mathoverflow.net/users/172182014-12-19T21:38:19Z2014-12-22T05:13:30Z
<p>Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ordinals $\alpha$. It is known of course that $F(\alpha) = \alpha+1$ is a complete resolution of GCH (in the positive) that is relatively consistent with ZFC. I read that it's an unpublished theorem of Woodin that $F(\alpha) = \alpha+2$ is a complete resolution of GCH that is relatively consistent with ZFC plus some large cardinal hypothesis. My questions are: (1) What's the weakest known complete resolution of GCH in consistency strength other than $F(\alpha) = \alpha+1$ and what large cardinal axiom is required for it? (2) What are some other complete resolutions of GCH that are known to be consistent relative to specific large cardinal hypotheses, what are their respective large cardinal hypotheses, and how do these consistency strengths relate to one another?</p>
http://mathoverflow.net/q/191118-3Linear algebra over principal rings 1 [on hold]user64148http://mathoverflow.net/users/641482014-12-19T13:39:57Z2014-12-22T10:37:43Z
<p>If N is a left-idea of ring R and R is a left R-module, then submodule N is a direct sum of R if and only if N has a right unit.</p>
http://mathoverflow.net/q/1911021Dual connections for Information GeometryMathearthttp://mathoverflow.net/users/641422014-12-19T08:36:44Z2014-12-22T09:34:00Z
<p>In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied $$X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla^*_XZ)$$
It seems that this definition depends on the metric $g$. My question is could we find another way to define the dual connections above, without the metric involved? </p>
http://mathoverflow.net/q/1908703Question on a proof by Solonnikov,Ladyzhenskaya,Ural'tsevafoo90http://mathoverflow.net/users/585412014-12-16T14:58:03Z2014-12-22T09:25:28Z
<p>I have already asked this question on Mathematics SE, because I suppose that it is not research level. But I haven't got an answer, possibly here someone can answer.</p>
<p>Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$.</p>
<p>In the book "Linear and Quasilinear Equations of Parabolic Type" by O.Ladyzhenskaya, V.Solonnikov, N.Ural'tseva, in Chapter 4.2 there is the proof of the hölder regularity of the simple layer heat potential in a semispace. I explain the step that I haven't understood. We are at page 283 of the book.</p>
<p>Let $\alpha\in]0,1[$. Let $x,z\in\mathbb{R}_+^{n}:=\{(y_1,\ldots,y_n)\in\mathbb{R}^n:y_n>0\}$ and let $K$ be the intersection of the disc in $\mathbb{R}^n$ with center $x$ and radius $2|x-z|$ with the hyperplane $\mathbb{R}^{n-1}\times\{0\}$.Note that $K$ could be empty. If it in not empty we have that $K$ is a disc in $\mathbb{R}^{n-1}$ with center $x'$ (we use this notation: if $x\in\mathbb{R}^n$ then $x'$ is the point in $\mathbb{R}^{n-1}$ made by the first $n-1$ component of $x$ ). In the book is said that
\begin{equation*}
\int_{\mathbb{R}^{n-1}\setminus K} |y'-z'|^\alpha\int_{0}^{\infty}\Big|\frac{\partial G(\tau,x'-y',x_n)}{\partial x_i}-\frac{\partial G(\tau,z'-y',z_n)}{\partial z_i}\Big|d\tau dy'\leq C|x-z|^\alpha,
\end{equation*}
but why?</p>
<p>EDIT: As it is said by @fedja, this problem is probably only a matter of computations and estimates (I think not too long as in the book it is given without any explication). Obviously I don't want you to do the computation for me, but I need to understand the idea behind. For example it is crucial the set K? How it enters in the estimate?</p>
http://mathoverflow.net/q/1891834Which universities teach true infinitesimal calculus?katzhttp://mathoverflow.net/users/281282014-12-08T10:18:51Z2014-12-22T09:44:00Z
<p>My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by Keisler <a href="https://www.math.wisc.edu/~keisler/calc.html" rel="nofollow">https://www.math.wisc.edu/~keisler/calc.html</a>. Two of my colleagues in Belgium are similarly teaching TIC at two universities there. I am also aware of such teaching going on in France in the Strasbourg area, based on Edward Nelson's approach, though I don't have any details on that. </p>
<p>Which schools, colleges, or universities teach true infinitesimal calculus? Anyone with any additional information in this direction is requested to provide it.</p>
<p>A colleague in Italy has recently told me about a conference on using infinitesimals in teaching in Italian highschools. This NSA (nonstandard analysis) conference was apparently well attended (over 100 teachers showed up). Anybody with more information about this (who to contact, what the current status of the proposal is, etc.) is hereby requested to provide such information here.</p>
<p>A new piece of information that just came in is that in Geneva there are two highschools that have been teaching calculus using ultrasmall numbers for the past 10 years. Anybody with more information about this is requested to provide it here.</p>
http://mathoverflow.net/q/1890201Different definitions of spin structuresJjmhttp://mathoverflow.net/users/623672014-12-06T09:22:39Z2014-12-22T13:12:12Z
<p>This is the definition of spin structure according to Wikipedia:</p>
<p><img src="http://i.stack.imgur.com/6M9jN.png" alt="enter image description here"></p>
<p>which is supposed to be the standard definition. But in the book <em>The Geometry of Four-Manifolds</em> (Donaldson-Kronheimer, page 76) one finds a rather different definition, at least for a 4-dimensional vector space ($S$ is supposed to be a general two-dimensional complex vector space with Hermitian metric and compatible complex symplectic form):</p>
<p><img src="http://i.stack.imgur.com/ElHdK.png" alt="enter image description here"></p>
<p><strong>What is the meaning of the second definition?</strong> Everything seems quite involved and unrelated. Any idea will be helpful and welcomed.</p>
http://mathoverflow.net/q/1845950Poisson approximation of random sub-graphsOlivierhttp://mathoverflow.net/users/582712014-10-16T13:52:42Z2014-12-22T04:56:22Z
<p>I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the process, $0 \leq m \leq {n \choose 2}$, the fraction $\rho^n_k(m)$ of vertices in components with size $k$ is given by:
$$\rho^n_k(m)= \sum_{v \in V(G_n)} \frac{1_{\{|\mathcal C_m(v)|=k\}}}{n}, k \in \{1, \ldots,n\},$$
with $|\mathcal C_m(v)|$the size of the component that contains the vertex $v \in V(G_n)$ at step $m$. As $n$ go to $\infty$, the (random) sequence of functions $(\rho^n(\lfloor t.n \rfloor ), t \geq 0)$ is known to converge to $(\rho(t), t \geq 0)$ the unique (deterministic) solution of the (discrete) Smoluchowski (coagulation) equation:
$$\rho'_k(t)= k \bigg[ (\rho \star \rho)_k(t) - 2 \rho_k(t) \sum_{k \in \mathbb N} \rho_k(t) \bigg]$$
started at $\rho_k(0)=1_{\{k =1\}}$. I used $(\rho \star \rho)_k = \sum_{m+n=k} \rho_m \rho_n$ to denote the convolution.
(The solution of this equation is explicit and involves the Borel Tanner distribution, that describes the size of a Poisson Galton Watson tree).</p>
<p><strong>Question:</strong> </p>
<p>I now replace $G(n)$ by a sequence of (deterministic or random) graphs $H(n)$ with $n$ vertices. These $H(n)$ are supposed to have enough edges so that the problem is meaningful (that is $\omega(n)$ edges). Also, in case $H(n)$ is random, the alea should be independent of the construction of the process.</p>
<p>I wonder what are the relevant (geometric, probabilistic, ...) conditions on the sequence of graphs $H(n)$ that ensure the convergence of $(\rho^n(\lfloor t.n \rfloor), t \geq 0)$ to $(\rho(t), t \geq 0)$ the solution of the Smoluchowski equation?</p>
<p><strong>Intuition</strong>: </p>
<p>An example where convergence does not hold is the discrete torus $H(n)=(\mathbb Z/m \mathbb Z)^d$, for fixed $d$, with $n= m^d$. If $d=d(n) \to \infty$ however this should work.</p>
<p>A condition like "all degrees in $H(n)$ are equal, and diverge with $n$" is relevant as a sufficient condition. It is however in no way a necessary condition. For instance we may take $H(n)=G(n,m)$ defined above, for $m=\omega(n)$. </p>
<p>All comments are welcomed!</p>
http://mathoverflow.net/q/1816041Coloring algorithm maximising color difference between neighborsLin Chenhttp://mathoverflow.net/users/528712014-09-23T08:09:24Z2014-12-22T11:58:54Z
<p>Consider a graph and a set of <strong>ordered</strong> colors ${\cal C} = \{1,2,\cdots,C\}$. I want to color each node $i$ with a color $c_i\in{\cal C}$ so as to maximize the minimum color difference between two neighbor nodes. That is, let two neighbor nodes $i$ and $j$ colored $c_i$ and $c_j$, let $\delta_{ij}\triangleq |c_i-c_j|$, I want to find a distributed coloring algorithm that maximises the minimum of $\delta_{ij}$ of the graph. The difference between this problem with the classical distributed coloring problem is the ordered color set.</p>
http://mathoverflow.net/q/1724992All solutions to a set of integral equationsuser54297http://mathoverflow.net/users/542972014-06-23T16:59:47Z2014-12-22T13:58:55Z
<p>I would like a better understanding of the set of pairs $(f_1,f_2)$ of functions $[0,1] \times [0,1] \to [0,1]$ which satisfy the following conditions:</p>
<ol>
<li><p>For all $y \in [0,1]$, $f_1(x,y) \geq f_1(x',y) \Leftrightarrow x>x'$. For all $x \in [0,1]$, $f_2(x,y) \geq f_2(x,y') \Leftrightarrow y>y'$.</p></li>
<li><p>$f_1(x,y)+f_2(x,y) = 1$, for all $(x,y) \in [0,1] \times [0,1]$</p></li>
<li><p>$f_1(x,y)=f_2(y,x)$ for all $(x,y) \in [0,1] \times [0,1]$</p></li>
<li><p>Due to monotonicity (Condition 1), the functions $f_1$ and $f_2$ are integrable. Now define
$$p_1(x,0) = xf_1(x,0) - \int_0^x f_1(t,0) dt,$$
$$p_2(0,y) = yf_2(0,y) - \int_0^y f_2(0,t) dt,$$
$$p_1(x,y) = p_1(x,0) + xf_1(x,y) - \int_0^x f_1(t,y) dt,$$
$$p_2(x,y) = p_2(0,y) + yf_2(x,y) - \int_0^y f_2(x,t) dt.$$</p></li>
</ol>
<p>Then for all $(x,y) \in [0,1] \times [0,1]$, $p_1(x,y)+p_2(x,y)=0$.</p>
<blockquote>
<p>Example: Let $\alpha \geq 0$. For $x \geq y$, let
$$f_1(x,y) = (1/2)+(1/2)(x^\alpha-y^\alpha)$$
$$f_2(x,y) = (1/2)-(1/2)(x^\alpha-y^\alpha)$$
For $x < y$, let
$$f_1(x,y) = (1/2)-(1/2)(y^\alpha-x^\alpha)$$
$$f_2(x,y) = (1/2)+(1/2)(y^\alpha-x^\alpha)$$
Then $(f_1,f_2)$ satisfy the conditions.</p>
</blockquote>
http://mathoverflow.net/q/17198819The error in Petrovski and Landis' proof of the 16th Hilbert problemAli Taghavihttp://mathoverflow.net/users/366882014-06-17T02:05:30Z2014-12-22T08:26:53Z
<blockquote>
<p><strong>Edit:For a recent progress on the Hilbert 16th problem see <a href="http://arxiv.org/abs/1411.6814v1" rel="nofollow">the following note</a>. Best wishes for the authors of this paper and their final success. I thank <a href="http://mathoverflow.net/users/24309/lo%c3%afc-teyssier">Loic Teyssier</a> who informed me of existence of this paper.</strong></p>
</blockquote>
<p>What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?</p>
<p>Please see this related <a href="http://mathoverflow.net/questions/879/most-interesting-mathematics-mistake/967#967">post</a> and also the <a href="http://mathoverflow.net/questions/77425/failures-that-lead-eventually-to-new-mathematics/77503#77503">following post.</a></p>
<p><strong>Added :</strong> According to their method, what of the following two statements are true?:</p>
<p>There are uniform numbers $\tilde{H}(n)$ such that every polynomial vector field $X$ of degree $n$ satisfies:</p>
<p><strong>Statement 1)</strong> There are at most $\tilde{H}(n)$ real limit cycles of $X$ which lie on the same leaf.</p>
<p><strong>statement 2)</strong> There are at most $\tilde{H}(n)$ distinct complex leaves which contains real limit cycles.</p>
<p>By "Leaf" I mean the leaf of the corresponding complex singular foliation of $\mathbb{C}P^{2}$. Some technical and historical aspects of these foliations are explained <a href="http://qcpages.qc.cuny.edu/~zakeri/papers/SHFC_zak.pdf" rel="nofollow">here</a>. However in this linked paper there is no an explicit explianation about the "error".</p>
<p>According to the video of lecture of Ilyashenko, provided in the answer to this question by Andrey Gogolev, we ask: </p>
<blockquote>
<p>What is the fate of the "persistence problem" which is mentioned by Ilyashenko? How it can be revised to become a true statement?</p>
</blockquote>
<p>According to the first page of the english version of the paper of Petrovski_Landis we ask </p>
<blockquote>
<p>"How they assume that a solution of the equation can be considered as an <strong>entire</strong> map from $\mathbb{C}$ to $\mathbb{C}P^{2}$? Can every leaf be parametrized by an entire map?</p>
</blockquote>
<p>According to comments and answers to this question, there is no a written paper which explain the <strong>error</strong>, explicitly.<strong>Why really this is the case?</strong></p>
http://mathoverflow.net/q/886117Decomposition of an integral operator into a compositionVictor Liuhttp://mathoverflow.net/users/10742012-02-16T09:04:51Z2014-12-22T05:56:22Z
<p>I've been musing about the following question for a while now. Given an integral operator $G$ defined by
$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$
Is it possible to decompose this into two separate "one-sided" integral operators $L$ and $R$ such that
$$ (Gf)(x) = \int_x^1 L(x,u) \int_0^x R(u,v) f(v)\,dv\,du $$
If so, under what assumptions on $G$? And are there expressions for the left and right operators $L$ and $R$ in terms of $G$?</p>
http://mathoverflow.net/q/4008252Why do we teach calculus students the derivative as a limit?Steven Samhttp://mathoverflow.net/users/3212010-09-27T05:29:56Z2014-12-22T12:18:30Z
<p>I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?</p>
<p>Something a teacher might do is ask students to calculate the derivative of a function like $3x^2$ using this definition on an exam, but it makes me wonder what the point of doing something like that is. Once one sees the definition and learns the basic rules, you can basically calculate the derivative of a lot of reasonable functions quickly. I tried to turn that around and ask myself if there are good examples of a function (that calculus students would understand) where there isn't already a well-established rule for taking the derivative. The best I could come up with is a piecewise defined function, but that's no good at all.</p>
<p>More practically, this question came up because when trying to get students to do this, they seemed rather impatient (and maybe angry?) at why they couldn't use the "shortcut" (that they learned from friends or whatever). </p>
<p>So here's an actual question:</p>
<p>What benefit is there in emphasizing (or even introducing) to calculus students the $h \to 0$ definition of a derivative (presuming there is a better way to do this?) and secondly, does anyone out there actually use this definition to calculate a derivative that couldn't be obtained by a known symbolic rule? I'd prefer a function whose definition could be understood by a student studying first-year calculus. </p>
<p>I'm not trying to say that this is bad (or good), I just couldn't come up with any good reasons one way or the other myself.</p>
<p><strong>EDIT</strong>: I appreciate all of the responses, but I think my question as posed is too vague. I was worried about being too specific, so let me just tell you the context and apologize for misleading the discussion. This is about teaching first-semester calculus to students straight out of high school in the US, most of whom have already taken a calculus course in high school (and didn't do well or retake it for whatever reason). These are mostly students who have no interest in mathematics (the cause for this is a different discussion I guess) and usually are only taking calculus to fulfill some university requirement. So their view of the instructor trying to get them to learn how to calculate derivatives from the definition on an assignment or on an exam is that they are just making them learn some long, arbitrary way of something that they already have better tools for. </p>
<p>I apologize but I don't really accept the answer of "we teach the limit definition because we need a definition and that's how we do mathematics". I know I am being unfair in my paraphrasing, and I am NOT trying to say that we should not teach definitions. I was trying to understand how one answers the students' common question: "Why can't we just do this the easy way?" (and this was an overwhelming response on a recent mini-evaluation given to them). I like the answer of $\exp(-1/x^2)$ for the purpose of this question though. </p>
<p>It's hard to get students to take you seriously when they think that you're only interested in making them jump through hoops. As a more extreme example, I recall that as an undergraduate, some of my friends who took first year calculus (depending on the instructor) were given an oral exam at the end of the semester in which they would have to give a proof of one of 10 preselected theorems from the class. This seemed completely pointless to me and would only further isolate students from being interested in math, so why are things like this done?</p>
<p>Anyway, sorry for wasting a lot of your time with my poorly-phrased question. I know MathOverflow is not a place for discussions, and I don't want this to degenerate into one, so sorry again and I'll accept an answer (though there were many good ones addressing different points).</p>
http://mathoverflow.net/q/2028115Details for the action of the braid group B_3 on modular formsQiaochu Yuanhttp://mathoverflow.net/users/2902010-04-04T04:17:00Z2014-12-22T06:57:38Z
<p>I'm reading Terry Gannon's <a href="http://books.google.com/books?id=ehrUt21SnsoC&printsec=frontcover&dq=terry+gannon+moonshine&source=bl&ots=7m8tyu7_0n&sig=yOYPV3kAm_eEimKFIGWcidHyR6M&hl=en&ei=3wm4S9HYHMH88Aa7_OXhBw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CA4Q6AEwAg#v=onepage&q=&f=false">Moonshine Beyond the Monster</a>, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an action of the braid group $B_3$. Here is what he says about this action: </p>
<p>First, we lift modular forms $f : \mathbb{H} \to \mathbb{C}$ to functions $\phi_f : SL_2(\mathbb{R}) \to \mathbb{C}$ as follows: let</p>
<p>$$\phi_f \left( \left[ \begin{array}{cc} a & b \\\ c & d \end{array} \right] \right) = f \left( \frac{ai + b}{ci + d} \right) (ci + d)^{-k}.$$</p>
<p>Thinking of $f$ as a function on $SL_2(\mathbb{R})$ invariant under $SO_2(\mathbb{R})$, we have now exchanged invariance under $SO_2(\mathbb{R})$ for invariance under $SL_2(\mathbb{Z})$. ($SO_2(\mathbb{R})$ now acts by the character corresponding to $k$.) In moduli space terms, an element $g \in SL_2(\mathbb{R})$ can be identified with the elliptic curve $\mathbb{C}/\Lambda$ where $\Lambda$ has basis the first and second columns (say) of $g$, and $\phi_f$ is a function on this space invariant under change of basis but covariant under rotation.</p>
<p>Second, $SL_2(\mathbb{R})$ admits a universal cover $\widetilde{SL_2(\mathbb{R})}$ in which the universal central extension $B_3$ of $SL_2(\mathbb{Z})$ sits as a discrete subgroup. Unfortunately, Gannon doesn't give an explicit description of this universal cover (presumably because it's somewhat complicated).</p>
<p><strong>Question:</strong> What is a good explicit description of this universal cover and of how $B_3$ sits in it (hence of how it acts on modular forms)? In particular, does it have a moduli-theoretic interpretation related to the description of $B_3$ as the fundamental group of the space $C_3$ of unordered triplets of distinct points in $\mathbb{C}$? (These triplets $(a, b, c$) can, of course, be identified with elliptic curves $y^2 = 4(x - a)(x - b)(x - c)$.)</p>