Recent MathOverflow Questions

Convergent net in a quasi-uniform space which is not Cauchy

Math Overflow Recent Questions - Sat, 09/30/2017 - 23:18

The proof of the result that every convergent net in a uniform space is Cauchy, employs symmetry of the uniform space. A quasi-uniform space lacks that symmetry. Is it possible then to find a convergent net in a quasi-uniform space which is not Cauchy?

Coproducts of weak equivalences

Math Overflow Recent Questions - Sat, 09/30/2017 - 21:19

In a model category $\mathcal{C}$ admitting a forgetful functor to simplicial sets, is the coproduct of weak equivalences a weak equivalence? Say even just coproducts indexed by $\mathbf{N}$. A reference?

For example, the model category of simplicial bi-modules over a commutative rings with fibrations and weak equivalences defined to be those that are fibrations/weak equivalences of underlying simplicial sets, satisfies this, because homology of chain complexes commutes with coproducts.

However, just as an example, what about simplicial commutative monoids with analogous model structure, for instance?

Extension of Cauchy Integral

Math Overflow Recent Questions - Sat, 09/30/2017 - 20:15

Let $f$ be holomorphic in the closure of an unbounded Jordan domain. Suppose that $f$ has a limit $f_{\infty}$ as $\left| z \right| \to \infty$. In other words, $$\forall \epsilon > 0 \exists R >0 : \left| z \right| > R \implies \left| f(z) - f_{\infty} \right| < \epsilon.$$ My supervisor made the claim that $f$ can be written as $$f(z) = \frac{1}{2\pi i} \int_{\partial D} \frac{f(\zeta)}{\zeta - z}d\zeta + f_{\infty}.$$

There seems to be no reference online about holomorphic functions and their representations on unbounded domains. Can anyone lend a hand? Cheers.

The Prime impossibility [on hold]

Math Overflow Recent Questions - Sat, 09/30/2017 - 20:09

Im working on a formula to generate prime numbers (i KNOW ITS supossed to be IMPOSSIBLE) and I need to figure out what symbols to use for the following terms:

A natural number that is NOT prime, an specific set of numbers.

Thank you for your help.

I know its not really possible but there's no harm in trying.

Evaluate complex integral [migrated]

Math Overflow Recent Questions - Sat, 09/30/2017 - 19:33

I want to evaluate the integral $$\int_{\left| z \right| =2} \frac{1}{z^{741} +1}dz.$$ It is clear that all singularities of this function are contained in the region of integration. Therefore, the residue theorem would give us that $$\int_{\left| z \right| =2} \frac{1}{z^{741} +1}dz = 2\pi i \sum_{k=1}^{741} \text{Res}_{z_k}.$$ I can't calculate the residues however, can someone assist me?

Pointwise bound on the gradient of solution of poisson equation

Math Overflow Recent Questions - Sat, 09/30/2017 - 19:25

I posted this question on SE .I got stack on this problem for a long time. I hope that someone can help me. This is the problem :

Consider the poisson problem with Dirichlet data :

$-\Delta (u) = f\quad\quad u = u_{d}\quad on\quad\partial \Omega $

Let $u_{h}$ be the finite element solution to this problem relative to a fixed riangulation of a plygonal domain $\Omega\subset R^{2}$

Can i exploit the gradient of $u_{h}$ solution to get informations about the exact solution, more exacly can i have an estimate of the form :

$|\nabla(u-h_{h})(x)|\leq c$

(c must be an explicit computable constant) the following is one try : i write

$-\Delta (u-u_{h}) = f\quad B_{1}\quad\quad u-u_{h} = u_{d}-u_{d,h}\quad on\quad\partial B_{1}$

Note that $f\in H^{-1}$ since $u_{h}$ is only $H^{1}$ and then i used a green representation formula on the unit ball B

Any idea is welcomed

Vladimir Voevodsky's works

Math Overflow Recent Questions - Sat, 09/30/2017 - 19:19

Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and algebraic topology. .

Voevodsky was awarded the Fields Medal in 2002. Sadly, he died September 30, 2017.

Can one draw a general picture of him works?

(Any expository reference will be appreciated).

Are there universities who take your mathoverflow score seriously [on hold]

Math Overflow Recent Questions - Sat, 09/30/2017 - 19:19

When being considered for a position at a university as a post doc or phd or professor, does your math overflow account mean anything. Are there any universities that take this seriously when looking at your application?

Internal category in an endofunctor category (possible examples)

Math Overflow Recent Questions - Sat, 09/30/2017 - 18:54

A while ago, I asked this question about internal categories in an endofunctor category. The comment that was posted stated that it was not possible because the particular functors don't preserve equalizers. I am not sure if he meant any functors, or specifically frobenius monads. I am currently interested in bimonads (monad/comonad with a mixed distributive law), and especially those bimonads on SET. I am wondering if we can find a bimonad that preserves equalizers and thus becomes an internal category for the reasons I have suggested (See the link).

$2$ dimensional foliations of space whose leaves contain the trajectories of a given vector field

Math Overflow Recent Questions - Sat, 09/30/2017 - 17:12

Assume that $X$ is a non-vanishing vector field on $\mathbb{R}^3$.

Is there a $2$-dimensional foliation of space such that every trajectory of $X$ is contained in a leaf of the $2$-dimensional foliation?

As a related question:

Is there a classification of all $1$-dimensional foliations of space tangent to the unit speed vector field $t$ for which the distributions $\{t,n\}$ and $\{t,b\}$ are integrable? Here $\{t,n,b\}$ is the associated Frenet frame.

Is there a name for this set of vectors?

Math Overflow Recent Questions - Sat, 09/30/2017 - 14:10

Given full rank system of vectors $v_1,\dots,v_n\in\Bbb Z^n$, vector $w\in\Bbb Z^n$ and integers $a,b\in\Bbb Z^n$ such that $$\langle v_i,w\rangle=a$$ at every $1\leq i\leq n$ is there a name for the set of vectors $v\in\Bbb Z^n$ such that $$\langle v',w\rangle=b$$ where $v'=(\langle v_1,v\rangle,\langle v_2,v\rangle,\dots,\langle v_n,v\rangle)\in\Bbb Z^n$?

The category of elements corresponding to a coend as a higher colimit

Math Overflow Recent Questions - Sat, 09/30/2017 - 13:56

Let $D: \mathcal{C} \to \mathbf{Set}$ be a diagram of sets, then we can obtain the colimit of $D$ as the set of connected components of the category of elements of $F$, which we denote by $\mathrm{el}(F)$. The category $\mathrm{el}(F)$ in turn is the oplax colimit of the composition of $F$ and the inclusion $\mathbf{Set} \hookrightarrow \mathbf{Cat}$.
Now let us consider a functor $F: \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$, then the coend of $F$ can again be constructed as the set of connected components of a category $\mathrm{el}^\wedge(F)$ which we now describe:

  • The objects are pairs $(X,x)$ with $X \in \mathcal{C}$ and $x \in F(X,X)$.
  • A morphism $(X,x) \to (Y,y)$ is given by a pair $(f,a)$, where $f: X \to Y$ is morphism in $\mathcal{C}$, and $a \in F(X,Y)$ such that $f_*(x) = f^*(y) = a$.
  • Finally composition of two morphisms $(X,x) \xrightarrow{(f,a)} (Y,y) \xrightarrow{(g,b)} (Z,z)$ is given by $(g \circ f, g_*(a)) = (g \circ f, f^*(b))$.

Does the category $\mathrm{el}^\wedge F$ satisfy a similar universal property as $\mathrm{el}F$?

The above description of the colimit of $D$ feels very natural to me, and I was hoping that this perspective may shed some light on the nature of coends*.

Finally I would like to note that it may be that there are other categories $\mathcal{E}$ than $\mathrm{el}^\wedge F$ such that $\pi_0(\mathcal{E})$ corresponds to the coend of $F$, which are more natural than my construction. If this is the case, I would be grateful for these to be pointed out.

*It is like first taking the quotient of a set by a discrete group in the $(\infty,1)$-category of spaces (or in this case equivalently in the $(2,1)$-category of groupoids) and then applying the left adjoint to $\mathbf{Set} \hookrightarrow \mathbf{Spaces}$ (resp. $\mathbf{Set} \hookrightarrow \mathbf{Groupoids}$).

The supplement of an angle is 40 more than the supplement of the complement of the angle. Find the measure of the angle [on hold]

Math Overflow Recent Questions - Sat, 09/30/2017 - 13:25

Angle = x Comp = 90-x Supp = 180-x

I tried 180-x = 40+180-x(90-x), but did not get the right answer. How would you solve this equation?

Bruhat decomposition over algebraically nonclosed fields

Math Overflow Recent Questions - Sat, 09/30/2017 - 10:15

Let $G$ be the group over algebraically nonclosed field $k$ of characteristic $0$. And let $P$ be a minimal parabolic subgroup defined over $k$. Let $S$ be its maximal split torus and $T$ be its maximal torus defined over $k$ containing $S$. Let $W(T)=N_G(T)/Z_G(T)$ and $W(S)=N_G(S)/Z_G(S)$ be the corresponding Weyl groups. Then we have Bruhat decomposition over $\overline{k}$ $G/P=\bigcup_{n\in N_G(T)(\overline{k})}PnP/P$ and by Borel-Tits we also have the decomposition for $k$ points i.e. $G(k)/P(k)=\bigcup_{w\in N_G(S)(k)}P(k)wP(k)/P(k)$ (which is different from above but each class $P(k)wP(k)/P(k)$ is Zarisski dense in the corresponding class $PwP/P$ of the first decomposition). There is a remark in the paper of Borel and Tits that it can happen that $PnP/P$ is defined over $k$ but does not contain $k$-points. I have the following questions about this issue:

0) What is the easiest example of this phenomenon?

1) Is there a reasonable condition on the field when this cannot happen?

2) Is there a reasonable condition on the field (Galois group) when this does not happen for the given group $P$ and all possible overgroups $G\supset P$ of split rank $1$?

Proper model category of simplicial rings revisited

Math Overflow Recent Questions - Sat, 09/30/2017 - 05:13

Let $s\text{Ring}$ denote the category of simplicial commutative rings. We endow it with the model structure defined by declaring that fibrations, trivial fibrations and weak equivalences are, respectively, those maps inducing fibrations, trivial fibrations and weak equivalences on underlying simplicial sets.

I'm interested in finding a proof of left properness of $s\text{Ring}$ that does not use the Dold-Kan equivalence, either directly or indirectly. In particular, without using the fact that simplicial rings admit a forgetful functor to simplicial abelian groups.

[The proof that I can think of using the fact that every simplicial commutative ring is fibrant (hence, implicitly, the Dold-Kan equivalence) goes this way: let $A\to B$ be a cofibration in $s\text{Ring}$, and $A\to C$ a weak equivalence. The pushout $B\otimes_AC$ (degree-wise tensor product) is equivalent to the derived tensor product, as $B$ is cofibrant as an $A$-module (Corollary on page 6.10, Ch. 2 of Quillen's "Homotopical Algebra"). Now by Thm. 6(b) in loc. cit., we deduce the map $B\to B\otimes_AC$ induces an isomorphism on $\pi_i$ for all $i\ge 0$, and hence is a weak equivalence. QED]

However, the above proof makes use of the fact that every simplicial ring is, in particular, a simplicial abelian group, in a crucial way, e.g.. in applying Thm. 6(b) quoted above, or the corollary quoted right before.

I would like to find a proof that, rather, shows that every free morphism is an $h$-cofibration (in the sense that pushout along it preserves weak equivalences). For the definition of free morphisms in simplicial categories, see Goerss' notes on simplicial methods, Def. 4.19 and the discussion around it.

Every cofibration is a retract of a free morphism (and the converse is also true), and $h$-cofibrations are stable under retracts, hence it's enough to show free morphisms are $h$-cofibrations.

Can one reduce to the case the free morphism in question is a generating cofibration, ie. of the form $\mathbf{Z}[\partial\Delta[n]]\to\mathbf{Z}[\Delta[n]]$, $n\ge 0$?

Can one reduce to showing that for every cofibrant simplicial ring $A$, the functor $(\cdot)\otimes_{\mathbf{Z}}A$ preserves weak equivalences?

Is anybody aware of such proof, or of general left properness criteria that might just apply?

The maximal number of copies of a graph $T$ in an $H$-free graph

Math Overflow Recent Questions - Sat, 09/30/2017 - 02:59

Problem. Let $T,H$ be fixed graphs with $H$ being a tree, and let $ex(n,T,H)$ be the maximal number of copies of $T$ in an $H$-free graph on $n$ vertices. Is it always true that $ex(n,T,H)=\Theta(n^k)$ for $k=k(T,H)\in\mathbb N$?

(The problem was posed on 13.10.2016 by Clara Shikhelman. The promised prize for solution is "a bottle of wine in Tel-Aviv", see page 20 at http://www.math.lviv.ua/szkocka/view.php).

Generalized partitions and eta functions

Math Overflow Recent Questions - Sat, 09/30/2017 - 00:23

Let $\sigma$ be an element of $SL_{24}(\mathbb{Z})$ with its Jordan normal form is diagonal and the eigen values are $\epsilon_j$ for $1 \le j \le 24$ are n th root of unity where $n|N$ and $N$ is the finite order of $\sigma$. Equivalently we are describing $\sigma$ through its cycle shape $(a_1)^{b_1}\cdots(a_s)^{b_s}$.

We associate $\sigma$ to the following modular form:

$$\eta_{\sigma}(q) := \eta(\epsilon_1q)\cdots\eta (\epsilon_{24} q)=\eta(q^{a_1})^{b_1}\cdots \eta(q^{a_s})^{b_s}$$

Here $\eta$ stands for the Dedekind eta-function. Using the above defined $\eta_{\sigma}$ we define: $$\sum_{j>0}p_{\sigma}(1+j)q^{1+j} = \frac{q}{\eta_{\sigma}(q)}$$ This is a generalized partition function.

We assume the cycle type of $\sigma$ is $1^{1}23^{1}$ and hence $N = 23$. In this case I have the following questions :

  1. What is this generalized partition function $p_{\sigma}$ and how to find $p_{\sigma}(n)$ for some natural number $n$?

  2. There are many generalisations of partitions functions and hence what is the reference for this particular type of generalized partition function?

Thanks for your time.

Have a good day.

wide sense stationary process

Math Overflow Recent Questions - Thu, 09/28/2017 - 15:51

Let $y_n$ be a wide sense stationary process (wss), i.e., where the mean is independent of $n$ and the correlation depends solely upon $|n_1 - n_2|$.

Let us define:

$u_n = a_1 y_{n−1} + \ldots + a_p y_{n-p}$

with $a_k \in \mathbb{R}$ and $k = 1, \ldots p$.

Is $u_n$ also a wide sense stationary process? Why or why not?

Database of integer edge lengths that can form tetrahedrons

Math Overflow Recent Questions - Thu, 09/28/2017 - 12:32

Is there a collection of lists of six integer edge lengths that form a tetrahedron? Is there a computer program for generating such lists? I need to find approximately thirty such tetrahedral combinations.

Is the series $\sum_n|\sin n|^n/n$ convergent?

Math Overflow Recent Questions - Thu, 09/28/2017 - 12:24

Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?

(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for solution is "butelka miodu pitnego", see page 36 in http://www.math.lviv.ua/szkocka/view.php. To get the prize, write to the e-mail: hsc@pwr.edu.pl).

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