Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2015-08-02T22:47:16Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/2128830reference for groupoid cohomologyuser76693http://mathoverflow.net/users/766932015-08-02T21:53:01Z2015-08-02T21:53:01Z
<p>In nLab (groupoid cohomology) says:</p>
<p>"Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types."</p>
<p>Are there references for this?</p>
http://mathoverflow.net/q/2128821Faithul map and (minimal) tensor product of $C^*$-algebrastruebaranhttp://mathoverflow.net/users/240782015-08-02T21:28:11Z2015-08-02T22:01:57Z
<p>Let $f$ be a faithful state on a $C^*$-algebra $A$, i.e. $f(a^*a)=0$ implies $a=0$. in general, call a mapping $T:A \to B$ between $C^*$-algebras faithful if $T(a^*a)=0$ implies $a=0$. How to prove that if $f$ is faithful then the map $f \otimes id: A \otimes A \to A$ is faithful? I think that it should be simple however I don't have an idea how to prove this. </p>
http://mathoverflow.net/q/212879-1How many techniques are there to test colliniarity of n points?anonymoushttp://mathoverflow.net/users/9012015-08-02T20:56:09Z2015-08-02T20:56:09Z
<p>How many techniques are there to test coliniariry of n points?</p>
<p>For example, suppose we have 4 points A, B , C, D. How many ways can it be tested that they are collinear?</p>
http://mathoverflow.net/q/212878-1is graph coloring problem in general np-complete?(solvable)amirhessamhttp://mathoverflow.net/users/766912015-08-02T20:26:53Z2015-08-02T20:26:53Z
<p><strong>graph coloring problem</strong></p>
<blockquote>
<p>Hi,</p>
<p>i tried to find an <strong>algorithm</strong> for this problem and i want to make sure. i found it with this knowledge.</p>
<p>1.is graph coloring problem <em>in general</em> to find the chromatic number?<br>
2.what is the input of this problem? is it just the number of vertex and edge? </p>
<p>good luck .</p>
</blockquote>
http://mathoverflow.net/q/2128760Box counting dimension of the graphs of functions on $\mathbb R \rightarrow \mathbb R$Zachary W. Robertsonhttp://mathoverflow.net/users/711042015-08-02T19:59:27Z2015-08-02T20:21:15Z
<p>Generally speaking box counting techniques are applied to fractals defined by some iterative process, but what about functions? Has the concept of box counting dimension been investigated on the graph of functions?</p>
<p>For instance are there detailed results on formulas to describe the box counting dimension for these functions?</p>
<p>Of course, I'm more interested in results on bounded nowhere differentiable functions. I'm also interested in writing a paper on this subject, so any directions on what regions of this topic to tackle would be appreciated. References to relevant material is also welcomed, I currently only have my own work.</p>
http://mathoverflow.net/q/212875-1Problem about the group theory in dummit [on hold]Mathchohttp://mathoverflow.net/users/766902015-08-02T19:26:31Z2015-08-02T19:44:58Z
<p>I am struggling with Problem 43 of 3.1 of Dummit's algebra book. The problem is:</p>
<blockquote>
<p>Assume $P=\{A_i\}$ is any partition of $G$ with the property that a "quotient operation" is defined as follows: to compute the product of $A_i$ and $A_j$ take any element $a_i$ of $A_i$ and $a_j$ of $A_j$ and define $A_i*A_j$ to be an element of $P$ that contains $a_i*a_j$ (assume this operation is well defined). Prove that an element of $P$ that contains an identity of $G$ is a normal subgroup of $G$ and elements of $P$ are cosets of this normal subgroup.</p>
</blockquote>
<p>I tried to first prove that the element of $P$ that contains an identity is a subgroup and normal...but it seems that I have not enough decks in my hands to play the game. Any help will be greatly appreciated.</p>
http://mathoverflow.net/q/2128700How to write a matrix with some constraints?Turbohttp://mathoverflow.net/users/100352015-08-02T18:28:48Z2015-08-02T21:49:23Z
<p>I want to write down an $m\times n$ $0/1$ matrix such that every row is distinct and every column is distinct.</p>
<p>If I want to write with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns. </p>
<p>Is there a canonical way to write down an $m\times n$ $0/1$ matrix such that every row is distinct and every column is distinct?</p>
<p>If not what are some tricks and strategies?</p>
http://mathoverflow.net/q/2128680A particular interesting elliptic curveuser76689http://mathoverflow.net/users/766892015-08-02T18:14:07Z2015-08-02T19:02:03Z
<p>Given the elliptic curve $E:y^2=x^3-4x+4$.</p>
<p>(a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$.</p>
<p>(b) If we consider the piece of curve on the region $0<x<2, 0<y<2$ with the aid of Magma we find the points
\begin{eqnarray*}
7P &=& \left(\dfrac{10}{9},\dfrac{26}{27}\right),\\
-10P &=& \left(\dfrac{88}{49},\dfrac{554}{343}\right),
\end{eqnarray*}
and $13P, -15P, 18P$, etcetera. What can be said about the sequence $7,-10,13,-15,18,...$?</p>
http://mathoverflow.net/q/2128671Can someone explain some of the steps in this paper clearly?wannabehttp://mathoverflow.net/users/766872015-08-02T18:07:56Z2015-08-02T18:13:23Z
<p>I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$</p>
<blockquote>
<p>Marek Wolf, <em>First occurrence of a given gap between consecutive primes</em>, preprint, April 1997, IFTUWr 911/97 (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.5981" rel="nofollow">Citeseer^x</a>)</p>
</blockquote>
<p>In particular, line 6 where it relates his result on the partial Brun's sums to first occurrence prime gaps. And also the following quadratic. </p>
<p>moreover it seems to be inconsistent with his previous paper in line (4) as it ignores some terms in line 28 of this one.</p>
<blockquote>
<p>Marek Wolf, <em>Generalized Brun's constants</em>, preprint, March 1997, IFTUWr 910/97
(<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.55.8070" rel="nofollow">Citeseer^x</a>)</p>
</blockquote>
http://mathoverflow.net/q/2128661Derived Deformations of associative algebrasMark.Neuhaushttp://mathoverflow.net/users/219652015-08-02T17:55:37Z2015-08-02T18:20:31Z
<p>Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows:</p>
<p>If $A \in \mathfrak{A}rt_k$ is an Artin ring, a deformation of $S$, is a pair $(\mathcal{S},\pi_A)$, where $\mathcal{S}$ is an $A$-algebra and
$\pi_A: \mathcal{S}\otimes_A k\to S$ is an isomorphism of $k$-algebras.</p>
<p>Two such deformations $(\mathcal{S},\pi_A)$ and $(\mathcal{S}',\pi'_A)$ are said to be isomorphic, if there exists an isomorphism of $A$-algebras
$\varphi: \mathcal{S}\to\mathcal{S}'$, such that
$\pi'_A\circ (\varphi\otimes_A id_k)=\pi_A$.</p>
<p>Then the functor </p>
<p>$Def_S: \mathfrak{A}rt_k \to Set\;;\; A \mapsto
\{deformations\; over\; A\}\;/\;isomorphisms$</p>
<p>is a classical (underived) deformation functor.</p>
<p>Now my question is, how exactly can we extend this deformation functor, into
(a model of) the derived setting,if we allow for more general $E_n$ or $E_\infty$ deformations?</p>
<p>Personally, I'm most familiar with Manettis approach to the derived situation, but every explicit extension would be welcome. </p>
http://mathoverflow.net/q/2128651Egyptian fractions similar to Erdos-Straus conjectureasadhttp://mathoverflow.net/users/569472015-08-02T17:44:22Z2015-08-02T17:44:22Z
<p>It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions.
My question is that if it is known that if $a>4$
$$
\frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}
$$
where $k<a$ or it is a conjecture similar to Erdos-Straus one with the same hardness? For instance is it known
$$
\frac 5n=\frac1{x_1}+\frac1{x_2}+\frac1{x_3}+\frac1{x_4}?
$$</p>
http://mathoverflow.net/q/2128600How possibly does a $\alpha$-stable process jump at this stopping time?kennethhttp://mathoverflow.net/users/56562015-08-02T16:07:07Z2015-08-02T16:07:07Z
<p>Lemma 2.3.2 of [Applebum2009] states that,</p>
<p>If $X$ is a Levy process and let $\Delta X(t) = X(t) - X(t-)$,
then $\Delta X(t) = 0$ almost surely for a fixed $t>0$.
There is also a warning that, this lemma may fail for a stopping time.</p>
<p>Consider an $\alpha$-stable 1-dimensional
process $X$ for some $\alpha\in (0,2)$ with
initial $X(0) = 1$ with its Levy symbol
$$\eta(u) = - |u|^{\alpha}.$$
It has its levy measure
$$\nu(dy) = dy/ |y|^{1+\alpha}.$$</p>
<p>[Q.] Let $T = \inf\{t>0: X(t-) \le 0\}$. I want to know, for $\alpha\in (0,1)$,
$$\mathbb P(\Delta X(T) = 0) = 1?$$</p>
<p>Intuitively, since $X$ is finite variation for $\alpha\in (0,1)$, we may think
$X$ has less frequent small jumps as of $\alpha \in [1, 2)$. However,
I am not confident for this statement. Can you either prove or disprove it?
Thanks.</p>
http://mathoverflow.net/q/2128582Can the projective line be provided with a ring structure?Wolfgang Tintemannhttp://mathoverflow.net/users/490562015-08-02T15:37:02Z2015-08-02T18:56:27Z
<p>A definition of multiplication on the projective $1$-points $(a:b)$ of $P_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the multiplication of complex numbers.</p>
<p>Is it known whether there a way to define an addition between projective $1$-points ? </p>
http://mathoverflow.net/q/2128532map between $C^*$-algebras with special properties, why is $f(a^2)=f(a)^2 $ for all $a\in A$?eddy9000http://mathoverflow.net/users/753382015-08-02T12:54:56Z2015-08-02T21:56:04Z
<p>Let $A$ and $B$ unital $C^\ast$-algebras, $f:A\to B$ a linear, bounded map such that $f(a^*)=f(a)^*$ for all $a\in A$, $f(1_A)=1_B$ and $f(a)f(b)=0$ for all $a,b\in A_{sa}$ with $ab=0$. Follows $f(a^2)=f(a)^2$ for all $a\in A$?</p>
<p>I have tried to proof this first for self-adjoint elements $a\in A$ using the continuous functional calculus but I'm stuck. Do you know a proof or a reference? Greetings</p>
http://mathoverflow.net/q/2128488Does "cardinal arithmetic is well-defined" imply axiom of choice?Wojowuhttp://mathoverflow.net/users/301862015-08-02T11:32:42Z2015-08-02T15:40:33Z
<p>Let me quickly explain what I mean with my question.</p>
<p>Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in I}\kappa_i$ as follows: take collection $(A_i)_{i\in I}$ of sets such that $|A_i|=\kappa_i$, and let $A=\{(a,i):i\in I,a\in A_i\}$. Then we say $\sum\limits_{i\in I}\kappa_i=|A|$. We can do similar trick with multiplication, by setting $B=\{f:f\text{ is a function from }I\text{ such that }f(i)\in A_i\}$.</p>
<p>If we assume AC, then these notions are well-defined: First of all, we <em>can</em> choose such a sequence $(A_i)_{i\in I}$, and if we take two collections $(A_i)_{i\in I}$, $(A_i')_{i\in I}$ such that $|A_i|=|A_i'|=\kappa_i$, then we can use choice to get bijections between $A_i$ and $A_i'$ for every $i\in I$ and then use there to create bijection between $A=\{(a,i):i\in I,a\in A_i\}$ and $A'=\{(a',i):i\in I,a'\in A_i'\}$, so this procedure defines only one cardinal. Similar trick for multiplication.</p>
<p>However, if we do <em>not</em> assume choice, then 1. we don't know a priori that we can always choose any sequence $(A_i)_{i\in I}$, and even if we can, we have no guarantee that there is only one possible size of $A$ we can get in this way. I don't know of scenario where we cannot find any sequence of sets with given cardinalities, but I know that uniqueness of size can fail quite spectacularly, even if we add $2$ countably many times (see <a href="http://dml.cz/bitstream/handle/10338.dmlcz/119630/CommentatMathUnivCarolRetro_47-2006-4_15.pdf">here</a>).</p>
<p>I have three questions:</p>
<blockquote>
<ol>
<li>Is it possible in abscence of choice that there is a collection of cardinals $(\kappa_i)_{i\in I}$, but there is no collection of sets $(A_i)_{i\in I}$ having respective cardinalities? (I suspect the answer is "yes")</li>
<li>If the answer to 1. is yes, then is it known that existence of such a collection implies axiom of choice? (I suspect answer to this to be "doesn't imply" or "not known")</li>
<li>Does existence of collections like above and "value of the sum and product of cardinals doesn't depend on choice of collection" imply axiom of choice? (no clue, but I'm hoping for "yes")</li>
</ol>
</blockquote>
<p>These are main questions I'm interested in, but we can ask more by asking which combinations of below three imply choice:</p>
<blockquote>
<p>For any collection of cardinals, there is collection of sets of respective cardinalities.
Sum of cardinals, if defined, is well-defined. (i.e. sum doesn't depend on choice of collection, but we allow possibility that the collection doesn't exist)
Product of cardinals, if defined, is well-defined.</p>
</blockquote>
<p>Thanks in advance for all feedback.</p>
http://mathoverflow.net/q/2128460Connectivity of weighted graph and zero Laplacian eigenvaluesMichelehttp://mathoverflow.net/users/267782015-08-02T09:47:26Z2015-08-02T16:58:44Z
<p>Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any pair of vertices. </p>
<p>The adjacency matrix is
\begin{equation}
a_{ij} =
\begin{cases}
1 & If \, (i,j) \in E \\
0 & \textrm{Otherwise}
\end{cases},
\end{equation}
and the Laplacian is $L = \delta_{ij} \sum_{k} a_{ik} - a_{ij}$. </p>
<p>It is known that the number of connected components of $G$ is equal to the multiplicity of the zero eigenvalue of $L$. </p>
<p>Is this result true also for weighted undirected graphs, where
\begin{equation}
a_{ij} =
\begin{cases}
w_{ij} & If \, (i,j) \in E \\
0 & \textrm{Otherwise}
\end{cases},
\end{equation}
with $w_{ij}$ some positive weight? If so, may you please give me a reference where I can find the proof for weighted graph?</p>
<p>Thank you</p>
http://mathoverflow.net/q/2128433Proof of a Fourier pair with Bessel functions?Pavelhttp://mathoverflow.net/users/766802015-08-02T09:24:01Z2015-08-02T15:47:31Z
<p>How can we prove that the Fourier transform of the function
$$
f(x)
=
\begin{cases}
(a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\
0 & \text{otherwise}
\end{cases}
$$
is
$$
\hat f(y)
=
\sqrt{2\pi a} \times (a^c) \times (b^c) \times (b^2+y^2)^{-c/2-1/4} \times BesselJ[c+1/2,a\sqrt{b^2+y^2}]
?
$$</p>
<p>This Fourier transform pair is given in the book
<em>Formeln und Satze fur die speziellen Funktionen der mathematischer Physik</em>
(Julius Springer, Berlin, 1943) p. 119.</p>
<p><a href="http://www.pokyrek.cz/Nijboer/Magnus_Hettinger.pdf" rel="nofollow">http://www.pokyrek.cz/Nijboer/Magnus_Hettinger.pdf</a></p>
<p>Numerically computation suggests this is correct.</p>
<p>I need this formula for $c=1$.</p>
http://mathoverflow.net/q/2128405Random suborbits of a rotationStéphane Laurenthttp://mathoverflow.net/users/213392015-08-02T08:35:22Z2015-08-02T21:10:24Z
<p>Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely many times).</p>
<p>Now, let $(\epsilon_n)_{n \geq 0}$ be a sequence of independent uniform Bernoulli random variables on $\{0,1\}$, and define the random integers $K_n=\sum_{i = 0}^n \epsilon_i2^i$. Is it true that the random sequence $(u_{K_n})_{n \geq 0}$ almost surely visits every open interval infinitely many times ? </p>
<p>As a conditional and secondary question, if it is true, I would also like to know:</p>
<ul>
<li><p>is it more generally true for random integers $K_n$ defined in the same way but using another representation of the integer numbers instead of the binary one ? </p></li>
<li><p>is it more generally true for the orbits of a minimal homeomorphism ? (after replacing intervals with open sets)</p></li>
</ul>
http://mathoverflow.net/q/2128304Deceptive linear algebra problemuserhttp://mathoverflow.net/users/170872015-08-02T00:56:03Z2015-08-02T14:56:25Z
<p>Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before:</p>
<p>Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots a_p$ and $n_1, n_2, \ldots n_p$:
$$ \left[ \begin{array}{ccc}
d_1\\
d_2\\
d_3\\
\vdots\\
d_{2p}
\end{array} \right]
= \left[ \begin{array}{ccc}
a_1, ~a_2, \cdots ~a_p\\
a_1^2, ~a_2^2, \cdots ~a_p^2\\
a_1^3, ~a_2^3, \cdots ~a_p^3\\
\vdots\\
a_1^{2p}, ~a_2^{2p}, \cdots ~a_p^{2p}
\end{array} \right] \cdot
\left[ \begin{array}{c}
n_1\\
n_2\\
\vdots\\
n_p
\end{array} \right] $$</p>
<p>(and where $a \succeq 0$ and $n \succeq 0$ and only real values are considered, so wherever there can be positive and negative roots for the values of $a_i$, only the positive roots need be considered). </p>
<p>It seems straightforward to solve with elimination: From the first row, let $a_1 = \frac{d_1-\sum_{i>1} a_i n_i}{n_1}$. Then from the second row, solve for $a_2$ (which will now involve a quadratic, etc.). Because the polynomials get larger, solving it with elimination in this manner can be quite slow (I crashed sage doing this on a fairly small problem).</p>
<p>Has this family of problems been characterized well in the literature and can it be solved efficiently in practice? Even a numerical solution would suffice, as long as it will converge certainly (no multi-start Levenberg-Marquardt, etc.)...</p>
http://mathoverflow.net/q/2127995soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEsCSAhttp://mathoverflow.net/users/368862015-08-01T12:43:53Z2015-08-02T18:02:41Z
<p><a href="https://en.wikipedia.org/wiki/Pseudogroup" rel="nofollow">I was reading this article</a> on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics? </p>
http://mathoverflow.net/q/2127450Proofs needed for observations regarding prime-partitionable numbersChristopher Hunt Gribblehttp://mathoverflow.net/users/766352015-07-31T16:24:48Z2015-08-02T21:29:30Z
<p>Below is the definition of a <strong>prime-partitionable</strong> integer taken from W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1987) 263-272 and is apparently the same in W. T. Trotter, Jr, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142:</p>
<p>An integer $n>=2$ is said to be prime-partitionable if there is a partition {$P_1,P_2$} of the set $P$ of all primes less than $n$ such that, for all natural numbers $n_1$ and $n_2$ satisfying $n_1+n_2=n$ we have that either $gcd(n_1,p_1) \ne 1$ or $gcd(n_2,p_2) \ne 1$ or both, for some pair $(p_1,p_2) = P_1 \times P_2$.</p>
<p>Conjectures:
If $P_1 =$ {$p_{1a}, p_{1b}$}, $p_{1a}$ and $p_{1b}$ are odd primes and $p_{1a}< p_{1b}$, it appears that, if $pp = kp_{1a} + 1 = p_{1b} + p_{1a}$ for $k$ odd and $1 < k ≤ p_{1a}-2 $ then:</p>
<ol>
<li><p>$pp$ is prime-partitionable,</p></li>
<li><p>no two values of $p_{1b}$ are the same and</p></li>
<li><p>the number of values of $k ≥ 1$ for each $p_{1a} ≥ 5$.</p></li>
</ol>
<p>Can proofs be provided for these observations please.</p>
http://mathoverflow.net/q/2126818"Most Similar Vector Problem" on an Integer Lattice?Berk U.http://mathoverflow.net/users/496732015-07-31T01:18:58Z2015-08-02T22:19:27Z
<p>I am currently working on problem that I think could be expressed as an integer lattice problem.</p>
<p>Given $u \in \mathbb{R}^n$ and a <em>bounded</em> integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like to find an integer vector $v \in L$ that minimizes the <em>angle</em> between $u$ and $v$. That is, I would like $$v \in \text{argmax}_{w \in L} \frac{u.w}{\|u\|\|w\|}$$</p>
<p>Here, the objective is maximizing the cosine of the angle between $u$ and $w$ (i.e. minimizing the angle between $u$ and $w$). The vectors $u$ and $w$ are said to be "similar" if this quantity is close to 1.</p>
<p>I am wondering:</p>
<ul>
<li><p>Is this problem related to a well-known integer lattice problem (e.g. a closest vector problem)?</p></li>
<li><p>Could this problem be solved using existing lattice algorithms (e.g. the LLL algorithm?)</p></li>
</ul>
http://mathoverflow.net/q/2126253Terminology for polygonsAnton Petruninhttp://mathoverflow.net/users/14412015-07-30T09:57:22Z2015-08-02T17:39:13Z
<p>As you may know term "polygon" might mean few different things
and its meaning has to guessed from context.
By some reason I have to use few of these meaning in one place.</p>
<p>So I converge to the following convention:</p>
<p>Polygon is a cyclically ordered set of points.
Then I could define its <em>sides</em>;
I can say then if the polygon is <em>simple</em>;
for simple polygons I can define its <em>interior</em>.
And then I need a term for the <em>set formed by the union
of the interior and all the sides of the polygon</em>.
(Maybe "body of the polygon" or "solid polygon"?)</p>
<p>Did you see a term for this used somewhere (I need a term different from "polygon")? </p>
http://mathoverflow.net/q/2097346On combinatorial and cellular model categories and infinity categoriesDavid Whitehttp://mathoverflow.net/users/115402015-06-19T22:36:18Z2015-08-02T19:04:34Z
<p>I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition in Hovey's or Hirschhorn's book, i.e. there are sets of maps I and J who detect the trivial fibrations and fibrations by lifting and whose domains are small relative to I-cell (resp. J-cell). There is a weaker notion used by Emily Riehl, which I will ask about at the very end.</p>
<p>When you want to do Bousfield localization it's common to assume $M$ is in addition either combinatorial or cellular. Combinatorial means cofibrantly generated and locally presentable as a category. Cellular is more mysterious. Morally, it's asking for good control over cell complexes (i.e. things built from the maps in I via gluing). Formally, it's asking for</p>
<ol>
<li>Domains of maps in J are small relative to cofibrations</li>
<li>Cofibrations are effective monomorphisms, i.e. any cofibration $f:X\to Y$ is the equalizer of the two obvious maps $Y\to Y \coprod_X Y$</li>
<li>Domains and codomains of I are compact (in the sense of Hirschhorn, not Hovey) relative to I-cell.</li>
</ol>
<p>Condition 1 is standard and usually easy to verify (for example, it comes for free if $M$ is locally presentable). Condition 2 is basically asking that the intersection of the two copies of $Y$ in $Y\coprod_X Y$ is equal to $X$, so it's also not that hard to check in categories where cofibrations are inclusions of some kind. Condition 3 is a pain to check, because it's really asking for a cardinal $\kappa$ such that for any presentation of a map $f:A\to B$ as a transfinite composition of pushouts along a chosen set of cells then any map from a (co)domain $X$ of a map in $I$ to $B$ factors through some subcomplex of size $\leq \kappa$. Again, if $M$ is locally presentable then this should be true automatically, at least if the collection of subcomplexes is filtered (since you know that all objects are small relative to filtered colimits).</p>
<p>It's clear that not every cellular model category is combinatorial. For example, Top is not (Hovey proves on page 49 of his book that the two-point Sierpinski space is not small). It's clear that not every combinatorial model category is cellular, because condition 2 can fail. For example, Hirschhorn provides such an example as 12.1.7. Another example is in Finnur Larusson's paper "The Homotopy Theory of Equivalence Relations." Another is the category of small categories Cat, as I learned today from a preprint of Amrani Ilias called "Stabilization of the category of simplicial objects in Cat." Similarly, if you take spectra valued in Cat it's not cellular (but it is combinatorial) as Deb Vicinsky has shown in her thesis. There are also examples of model categories which are not cofibrantly generated. My favorite is the Strom model structure on Top, and the proof is in Raptis's paper on Homotopy Theory for Posets. It seems to me that all the other conditions of cellularity are true here (suitably interpreted) except for cofibrantly generated, but perhaps I am wrong. Lastly, there is a model structure on a locally presentable category which is provably not cofibrantly generated, and it's given by Boris Chorny's example from the paper "The Model Category of Maps of Spaces is not Cofibrantly Generated." The category in question is simply the arrow category valued in sSet. When I tell these examples to people in the infinity category community I always get the same question, so now I'm putting it out to the MathOverflow world:</p>
<blockquote>
<p>(1) Is there an example of a complete and cocomplete infinity category which does not admit any cofibrantly generated model?</p>
</blockquote>
<p>This is probably a hard question. Chorny's example doesn't work, because every presentable infinity category gives rise to a combinatorial model category (which in this case will have the same weak equivalences but different cofibrations), obtained by embedding into simplicial presheaves then taking a Bousfield localization. A related question is</p>
<blockquote>
<p>(2) Is there an example of a complete and cocomplete infinity category which does not admit a cellular model, but does admit a cofibrantly generated one?</p>
</blockquote>
<p>This seems much more likely to be true to me, since from a model category perspective there really is a difference between cellularity and combinatoriality. But since all statements required for cellularity are about cofibrations, perhaps there's a sneaky way to always choose a nice set of cofibrations when you pass from a presentable infinity category to a model category. Perhaps you could shrink the cofibrations via Bousfield colocalizations, for example.</p>
<blockquote>
<p>(3) Are there examples which have arisen in practice of non presentable infinity categories?</p>
</blockquote>
<p>Lastly, I want to better understand where Riehl's definition of cofibrantly generated fits into this story. In a combinatorial model category the smallness hypotheses are automatic, so it seems Riehl's definition matches the usual one. Similarly, I suppose (1) and (3) in the definition of cellularity force Riehl's definition and the usual one to agree, though perhaps there is a way to weaken cellularity to algebraic cellularity in the sense of Riehl's algebraic wfs. Riehl also has notions of enriched weak factorization systems which allow you to do things like saying the Strom model structure <em>is</em> cofibrantly generated in her enriched sense. I am less certain about Chorny's examples. Anyway, for now my main question regarding Riehl's definition is:</p>
<blockquote>
<p>(4) Is there an infinity category which admits the required colimits to form a factorization system but which does not admit any model that has a factorization system in the sense of Riehl (either simply an algebraic wfs or an enriched wfs)?</p>
</blockquote>
http://mathoverflow.net/q/20693811No limit shape for random Young diagrams under z-measure?Austenhttp://mathoverflow.net/users/251452015-05-18T15:47:10Z2015-08-02T16:28:30Z
<p>In their paper <em><a href="http://arxiv.org/abs/math-ph/0305043" rel="nofollow">Random partitions and the Gamma kernel</a></em> (Advances in Mathematics 194 (2005) 141–202), Borodin and Olshanski state that:</p>
<blockquote>
<p>An important difference between the Plancherel measures and the z-measures is that the random Plancherel diagrams have a limit form... while no such form exists for the z-measures.</p>
</blockquote>
<p>I am not sure if this is a straightforward comment (because the z-measure is in general not positive), or more subtle. That is, <strong>is there still no limit shape for those values where the z-measure is positive?</strong>. If not, why?</p>
http://mathoverflow.net/q/1849786Green's function of the Ornstein-Uhlenbeck operatorSanderhttp://mathoverflow.net/users/607262014-10-21T14:39:49Z2015-08-02T20:29:32Z
<p>The Ornstein-Uhlenbeck operator $L$ is given by
$$
Lu = \Delta u- \frac{1}{2}x\cdot \nabla u.
$$
Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ or at least for $d=2$) ?
Any references?</p>
<p>Thanks a lot!</p>
http://mathoverflow.net/q/1826264Hankel matrix commuting with a Jacobi matrixTwihttp://mathoverflow.net/users/565532014-10-05T15:30:44Z2015-08-02T22:29:33Z
<p>Assume the semi-infinite Hankel matrix $H$ with entries
$$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$
where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a semi-infinite Jacobi matrix $T$ commuting (formally) with $H$ can be determined explicitly. For example, if $H$ is the Hilbert matrix, i.e. $\alpha_{n}=1/n$, then
one can set
$$T_{n,n}=2n(n-1), \quad T_{n,n+1}=T_{n+1,n}=-n^{2} \quad (T_{m,n}=0 \mbox{ otherwise})$$
and $HT=TH$, indeed.</p>
<p>My question is twofold. Is there any systematic way how to determine the diagonal and off-diagonal sequence of the Jacobi matrix $T$ which commutes with a given Hankel matrix $H$ (perhaps a computer based method determining several first entries of the sequences)?</p>
<p>More precisely, the question is: how to find sequences $b_{n}=T_{n,n}$ and $a_{n}=T_{n,n+1}$, such that
$$(a_{j-1}-a_{i-1})\alpha_{i+j-1}+(b_{j}-b_{i})\alpha_{i+j}+(a_{j}-a_{i})\alpha_{i+j+1}=0, \quad \forall i,j\geq1,$$
where one sets $a_{0}=0$ and $\alpha_{n}\in\mathbb{R}$ is given. Of course, only non-trivial solution is of interest.</p>
<p>Second, is any other example of commuting Hankel and Jacobi matrix known?
For instance, interesting Hankel matrices correspond to the choice
$\alpha_{n}=1/n^{2}$ (or more general powers of $n$) or $\alpha_{n}=1/n!$.</p>
<p>Any information related to the post would be useful. Thanks. </p>
http://mathoverflow.net/q/1119665flat metrics on the 2-sphere with conical singularitiesAxelhttp://mathoverflow.net/users/262502012-11-10T06:07:14Z2015-08-02T19:40:46Z
<p>Consider some flat metric on $S^2$ with a fixed finite number of conical singularities $p_1,\ldots,p_n$.</p>
<p>What is the moduli space of such metrics up to isometry? In particular what is its dimension?</p>
http://mathoverflow.net/q/191493Linear Regression Coefficients W/ X, Y swappeddsimchahttp://mathoverflow.net/users/48332010-03-23T21:53:07Z2015-08-02T19:45:48Z
<p>Let's say I have a linear regression model of the form $ y = B_x x + I_x + \epsilon $, where $B_x$ is the beta coefficient of the $x$ term, $I_x$ is the intercept term and $\epsilon$ is additive, normally distributed noise. If I have a dataset and perform linear regression, I get a value for $B_x$, which indicates the slope of the relationship. </p>
<p>If I swap the roles of the $x$ and $y$ data, and try to fit a model of $x = B_y y + I_y + \epsilon$, I would expect intuitively that $B_y = \frac{1}{B_x}$. A simple geometric argument can be made to show that swapping the roles of $x$ and $y$ shouldn't change the position of the regression line w.r.t. any data point, and from here it seems like simple algebra that if $y = Bx + I$ then $x = \frac{1}{B} y + \frac{I}{B}$.</p>
<p>Where is this reasoning wrong? Can someone explain to me why $B_x \neq \frac{1}{B_y}$, preferably without resorting to tons of linear algebra or direct derivation from the normal equation?</p>
http://mathoverflow.net/q/1673544Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?Qiaochu Yuanhttp://mathoverflow.net/users/2902010-03-01T02:02:18Z2015-08-02T18:47:12Z
<p>It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} \}$ must lie in certain arithmetic progressions, with a finite number of exceptions. This is because any nonconstant polynomial must have infinitely many distinct prime divisors, which one can prove in a manner imitating Euclid's proof of the infinitude of the primes. For example, taking $p(x) = \Phi_n(x)$, we can prove Dirichlet's theorem for primes congruent to $1 \bmod n$. It is known (see, for example, <a href="http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/dirichleteuclid.pdf">this paper</a> of K. Conrad) that this is possible precisely for the primes congruent to $a \bmod n$ where $a^2 \equiv 1 \bmod n$. </p>
<p>However, the result about polynomials having infinitely many prime divisors has the following generalization: <strong>any</strong> sequence $a_n$ of integers which is eventually monotonically increasing and which grows slower than $O(2^{\sqrt[k]{n}})$ for every positive integer $k$ has infinitely many distinct prime divisors. In particular, any sequence of polynomial growth (not necessarily a polynomial itself) has this property. </p>
<p><strong>Question 1:</strong> Given an arithmetic progression $a \bmod n, (a, n) = 1$ such that $a^2 \not \equiv 1 \bmod n$, is it ever still possible to <em>efficiently</em> construct a monotonically increasing sequence of positive integers satisfying the above growth condition such that, with finitely many exceptions, the prime divisors of any element of the sequence are congruent to $a \bmod n$? ("Efficiently" rules out answers like "the positive integers divisible by primes congruent to $a \bmod n$," since I do not think it is possible to write down this sequence efficiently. On the other hand, evaluating a polynomial is very efficient.) The idea is that such a sequence immediately gives a proof of Dirichlet's theorem for primes congruent to $a \bmod n$ generalizing the Euclid-style proofs. </p>
<p><strong>Question 2:</strong> If the above is not possible, are there any known techniques for proving Dirichlet's theorem or at least some of the special cases not covered above without resorting to the usual analytic machinery? For example, Selberg published an "elementary" proof in 1949, but it relies on the "elementary" proof of the prime number theorem, which to me is "finitary analytic machinery." What is the absolute minimum amount of analysis necessary to produce a proof? (Edit: In response to a suggestion in the comments, one way to describe the kind of answer I'm looking for is that it would generalize to a proof of Chebotarev's density theorem that shows very clearly where the distinction between the number field and function field cases is; aside from some "essential" analytic argument there should be no difference between the two.)</p>
<p>This question is inspired at least in part by the following observation: Dirichlet's theorem is equivalent to the seemingly weaker statement that for every progression $a \bmod n, (a, n) = 1$ there exists <strong>at least one</strong> prime congruent to $a \bmod n$. The reason is that if there exists some such prime $a_1$, then letting $n_1$ be the smallest multiple of $n$ greater than $a_1$, there exists a prime congruent to $a_1 + n \bmod n_1$, and so forth. </p>