Recent MathOverflow Questions

Chow Groups of varieties over number fields

Math Overflow Recent Questions - Fri, 12/15/2017 - 17:57

I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely generated. Is it called Beilinson's Conjecture? What is the best reference for this conjecture? Is it know in any non-trivial case?

Why would I get a negative variance?

Math Overflow Recent Questions - Fri, 12/15/2017 - 17:06

I am getting a negative variance for a set of data. I know this should not be the case but I cannot figure out what I am doing wrong.

I keep a tally of the scores and a tally of the scores squared as follows

the tallies are initialized as

tally = 0 -- this is the tally of the scores
tally2 = 0 -- this is the tally of the scores squared

whenever an event that results in a score takes place the new scores are added to the tallies

tally = tally + score
tally2 = tally2 + score*score

at the end I calculate the mean and variance as

mean = tally / number of events
variance = tally2 / number of events - mean*mean

I know that the variance should never be negative, but for some reason I am getting a negative value. Any input on what could cause my variance to be negative would be appreciated. Thanks.

What name can I use for a cocone over the mapping cone diagram?

Math Overflow Recent Questions - Fri, 12/15/2017 - 16:36

In homotopy theory, the mapping cone of a continuous map $f\colon X \to Y$ is the homotopy pushout over the following span:

$$ \require{AMScd} \begin{CD} X @>{f}>> Y\\ @VVV \\ \{*\} \end{CD} $$

I.e., it is universal among all squares of the form $$ \begin{CD} X @>{f}>> Y\\ @VVV @VVV\\ \{*\}@>>> Z \end{CD} $$ where the square commutes up to homotopy.

But what is a good name for such an object $Z$? Normally, I would call it a cocone, but I would rather not use the word cone to mean two different things.

Square and cospan are possibilities, but they seem a bit too general: I want to refer specifically to cocones for the first diagram.

Is there a good alternative word?

Connection between weights in the last eigenvector (corresponding to least eigenvalue) and the corresponding column of a correlation matrix

Math Overflow Recent Questions - Fri, 12/15/2017 - 14:47

This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated and acknowledged.

Consider an $n \times n$ correlation matrix A such that all the off-diagonal entries are between [-1,0]. (Note: A correlation matrix is a positive semi-definite symmetric matrix, with diagonal entries 1 and all off-diagonal entries between [-1,1]).

Let $\alpha_i = \frac{\sum_{j=1,j \neq i}^{n}|A_{ij}|}{n-1}$ denote the mean of magnitudes of off-diagonal entries in $i^{th}$ column.

Let $v_{min} = [v_1,v_2,...,v_n]^T$ be the unit eigenvector corresponding to the least eigenvalue $\lambda_{min}$ of A. Let $v_k$ be the weight with minimum magnitude in $v_{min}$.

Then empirically, I am observing that $\alpha_k$ is also minimum among all $\alpha_i$'s.

I am wondering if this is indeed true and can be proved, or otherwise, if there is any counterexample where this will break?

My attempt so far: I was trying the following approach but that doesn't lead me very far:

$\lambda_{min} = v_{min}^{T}Av_{min}$.
On expanding, we get

$\lambda_{min} = 1 + \sum_{i=1}^{n}\sum_{j=1,j \neq i}^{n} v_i v_j A_{ij}$.

Since $\lambda_{min}$ is the least eigenvalue, we would want $\sum_{i=1}^{n}\sum_{j=1,j \neq i}^{n} v_i v_j A_{ij}$ to be minimized.

It can be shown that for any correlation matrix that has all off-diagonal entries $\leq 0$, all the weights in $v_{min}$ are always of same sign. Therefore, $v_i v_j A_{ij} = - v_i v_j |A_{ij}|\leq 0, \forall i,j \in [1,n], i \neq j$.

Now, for any two columns $A_{:i}$ and $A_{:j}$ such that $|A_{mi}| \geq |A_{mj}| \forall m \in [1,n], m \neq i,j $, $\alpha_i \geq \alpha_j$. Also it is easy to show that $|v_i|$ has to be $\geq |v_j|$ in order to minimize $\lambda_{min}$.

However, the above case is a very special case where column $A_{:j}$ is elementwise smaller than or equal to the other column $A_{:i}$. For other general cases, it doesn't seem to tell us anything more.

Abandoned LCAs on Cantor's Attic : Grand Reflection cardinals, universe cardinals, weak universe cardinals

Math Overflow Recent Questions - Fri, 12/15/2017 - 14:19

Cantor's Attic is a really great website for the various descriptions of large finite numbers, large countable ordinals, and large cardinal axioms.

However, after looking through the archives of the website, I have found that originally, the following cardinals were included and never given a definition:

  • Grand reflection cardinals
  • Universe cardinals
  • Weak universe cardinals

The universe cardinals and weak universe cardinals were replaced by the worldly cardinals in the same spot, so it makes sense that the term "universe" was renamed to worldly. However, that doesn't explain what "weak universe" cardinals are.

The grand reflection cardinals were created and never replaced. They still remain on the upper attic today, although hidden by code. You can see the link there, but it links to nothing.

So what are these cardinals? Does anybody know? The best person I could think of to answer this would be @JoelDavidHamkins himself, who was the one to put these on Cantor's Attic.

Comparison of the absolute value of an operator with its positive parts, II

Math Overflow Recent Questions - Fri, 12/15/2017 - 14:09

Suppose $A,B\in M_n(\mathbb C)$ are self-adjoint. Does there exist a constant $C>0$ dependent only on $n$ such that $$ |A+iB| \leq C(|A| + |B|)? $$

If $A$ and $B$ are positive then this was answered in the affirmative in this question. The argument given is not obviously extendable to this situation.

Efficient characterisation of the generalized symmetric groups

Math Overflow Recent Questions - Fri, 12/15/2017 - 14:05

This is a follow-up to normal form for some finite groups, extending the small groups library.

Not being familiar with groups, I wonder whether it is possible to check efficiently whether a group (given as a permutation group) is isomorphic to a generalized symmetric group.

Initial computer experiments indicate that the parameter $m$ in $\mathbb Z_m\wr\mathfrak S_n$ might be twice the index of the derived subgroup in the group.

From a practical point of view, I am trying to do this with GAP.

Invariance of domain for smooth functions

Math Overflow Recent Questions - Fri, 12/15/2017 - 13:54

Let $f \colon U \to \mathbb R^n$ ($U$ open in $\mathbb R^n$) be of class $C^1$ and assume furthermore that $f$ is injective. The theorem of the invariance of domain tells us that $f(U)$ is open.

Is it possible to show that $f(U)$ is open by not using the invariance of domain theorem. But instead using that $f$ is smooth?

Cutting a Gaussian in two pieces that are maximally separated in the Wasserstein metric

Math Overflow Recent Questions - Fri, 12/15/2017 - 12:11

Denote the standard Gaussian probability measure on $\mathbb R^n$ by $\gamma$. We partition $\mathbb R^n$ into two sets $A$ and $A^c$ such that $\gamma(A) = \gamma(A^c) = 1/2$.

Denote by $\gamma_{A}$ to Gaussian measure restricted to $A$, and normalized so that it is a probability measure. Similarly, define $\gamma_{A^c}$ to be the Gaussian measure restricted to $A^c$ and normalized.

My question is the following:

What is the optimal $A$ such that $\gamma_A$ and $\gamma_{A^c}$ are the farthest apart; i.e., solving

$$\arg\max_{A} W_2(\gamma_A, \gamma_{A^c}),$$

where $W_2$ is the 2-Wasserstein distance?

Possible generalization: Instead of constructing $\gamma_A$ and $\gamma_{A^c}$ as above, we could start with any two probability measures $\gamma_1$ and $\gamma_2$ such that $\gamma = \frac{\gamma_1 + \gamma_2}{2}$ and find $\arg \max_{\gamma_1, \gamma_2} W_2(\gamma_1, \gamma_2)$.

Finding upper bounds on the $W_2$ distance is also of interest. A natural conjecture, inspired by the Gaussian isoperimetric inequality, would be that $A$ should be a half-space. Counterexamples to this are also welcome!

Irreducibility of fixed point scheme

Math Overflow Recent Questions - Fri, 12/15/2017 - 10:08

Let $X$ be an integral affine scheme $X = Spec(A)$ endowed with a finite groupe action by $G$, which is of order $n$.

Consider the fixed points scheme $X^G$

Assume $n$ is invertible in $A$. Is there a result out there describing the irreducible components of $X^G$ ?

I tried several easy examples (permutations of coordinates in affine $d$-space) and every time I found that $X^G$ was irreducible.

Original version of Thurston's Notes

Math Overflow Recent Questions - Fri, 12/15/2017 - 10:00

I'm looking for a scanned version of the famous Thurston's notes (as it were in ~1980).

I have true difficulties to find the original version ... since the electronic (TeX) version is now everywhere on the web.

Any help appreciated !

Computing Hausdorff-Gromov distance between these objects

Math Overflow Recent Questions - Fri, 12/15/2017 - 09:42

As a bonus question in an exam we were asked to find compact metrix spaces $X,Y$ and $Z$ such that $d_{GH}(X,Y)=d_{GH}(X,Z)=d_{GH}(Y,Z)>0$.

The proposed answer is to take $\{0\},\{-1,1\}$ and $\{-1,0,1\}$. And the distances can be easily calculated by trying all appropriate correspondances and calculating distortions.

However I proposed the following three sets ( there are only two radii and they are $R>r$)

set $X$ is the big closed ball of radius $R$.

Set $Y$ is the small closed ball of radius $r$.

Set $Z$ is a closed ball of radius $r$ union two perpendicular line segments of length $2R$ that intersect in the center of the ball.

The conjecture is that all distances are $R-r$. To achieve this distance simply place the figures concentrictly. (I think we may need that $\frac{r}{R}$ is big)

In order to get lower bounds for the distances involving $Y$ one simply uses the bound $d_{GH}(A,B)\geq |\text{Diam}(A)-\text{Diam}(B)|/2$.

But I am stuck calculating the distance between $X$ and $Z$. One approach is to use contradiction and try to use distortions. If we take $x_1$ and $x_2$ diametrally opposite then if $z_1\sim x_1$ and $x_2\sim z_2$ we must have $d(z_1,z_2)>r$ and so at least one of $z_1$ and $z_2$ is outside the small ball, but Im stuck after this.

Inequality on permutation polytope

Math Overflow Recent Questions - Fri, 12/15/2017 - 09:23

Let $a = (a_1, \cdots, a_n), b = (b_1, \cdots, b_n), c = (c_1, \cdots, c_n) \in \mathbb{R}^n$ with $a_1 \geq \cdots \geq a_n, b_1 \geq \cdots \geq b_n, 0 < c_1 \leq \cdots \leq c_n$.

In addition, assume that $\sum_{i=1}^k b_i \leq \sum_{i=1}^k a_i$ for all $k \in \{1, \cdots, n-1\}$ and $\sum_{i=1}^n b_i = \sum_{i=1}^n a_i$.

Let $A := \{(a_{\sigma(1)}, \cdots, a_{\sigma(n)}) \,|\, \sigma \in S_n\}$ and $K_A$ be the unique minimal convex set containing $A$.

For $b \in K_A$, does $$\sum_{i=1}^n c_i b_i \geq \sum_{i=1}^n c_i a_i$$ hold?

Continuity under various topologies for positive linear functionals

Math Overflow Recent Questions - Fri, 12/15/2017 - 08:22

It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is continuous and $\|f\| = f(1)$.

Can similar statements be produced for a larger class of topological algebras? I am particularly interested in the case when $\mathcal A$ is the algebra $C_b (X)$ of bounded continuous functions on some Hausdorff topological space $X$, endowed with some of the the usual interesting topologies given by modes of convergence (compact convergence, strict topology etc.).

Dodecahedral rolling distance

Math Overflow Recent Questions - Fri, 12/15/2017 - 07:23

Let a dodecahedron sit on the plane, with one face's vertices on an origin-centered unit circle. Fix the orientation so that the edge whose indices are $(1,2)$ is horizontal. For any $p \in \mathbb{R}^2$, define the dodecahedral distance $dd(p)$ from $o=(0,0)$ to $p$ to be the fewest number of edge-rolls that will result in a face of the dodecahedron landing on top of $p$. Equivalently, imagine reflecting a regular pentagon over edges, as illustrated below: It takes $4$ rolls/reflections to cover $p=(5,\pi)$:          
          $p=(5,\pi)$, $dd(p)=4$, $s=(3,1,4,2)$. My main question is:

Q. Given $p$, how can one calculate $dd(p)$?

Greedily choosing, at each step, the roll that is best aligned with the vector $p-o$ does not always succeed.

Could one characterize the sequences of roll indices $s$, where rolling over edge $(i,i+1)$ of the pentagon is index $i\,$? What do all the points $p$ of $\mathbb{R}^2$ with $dd(p)=k$ look like, i.e., what is the shape of a $dd$-circle?

         
          $p=(18.3,-1.4)$, $dd(p) \le 12$, $s=(2, 4, 1, 3, 5, 2, 4, 1, 3, 5, 2, 5)$.

That the dodecahedral distance is well-defined follows, e.g., from "Thinnest covering of the plane by regular pentagons."

A question on Voevodsky´s categories

Math Overflow Recent Questions - Thu, 12/14/2017 - 21:59

I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:

1.- What is the idea behind his construction and what is one possible motivation for this?

2.- What are the basic features of this categories, besides of being triangulated?

3.- Could you suggest me a good reference where I can find a detailed discussion of the construction of this categories?

A vector field over a complex riemannian manifold

Math Overflow Recent Questions - Thu, 12/14/2017 - 13:45

Let be a complex riemannian manifold $(M,g,J)$. Is the following canonical vector field studied ? $$ X_J = \sum_{i=1}^{2n} \nabla^{LC}_{e_i}e_i +\nabla^{LC}_{Je_i}Je_i+ J[e_i,Je_i], $$ with the $(e_i)$, an orthonormal basis and $\nabla^{LC}$, the Levi-Civita connection. It seems indeed that $\nabla_e^{LC}f + \nabla_{Jf}^{LC} Je + J[e,Jf]$ is a tensor in $(e,f)$. If the manifold is Kaehler, it can be reduced to the torsion, so zero. We can also introduce a tensor with a quaternionic structure : $$ T_H (X,Y)=\nabla_{IX}IY + \nabla_{JY}JX + K[IX,JY] $$ with $IJK=-1$, the quaternionic structure. If it is hyperkaehler, it reduces also to zero torsion.

Conjecture on Odd Perfect Numbers

Math Overflow Recent Questions - Wed, 12/13/2017 - 16:24

I'm a grad student in mathematics and I've been working with a very gifted high school student (likely the smartest high school student I've ever met) on problems he's brought up and some competition math problems. This student has developed an interest in perfect numbers and the question regarding existence of odd perfect numbers. He has come up with a conjecture about odd perfect numbers, but I have not studied number theory and hence am not necessarily aware of well-known results of the field. So, here we are.

His idea:

Suppose $N \in \mathbb{N}$, with prime decomposition $N = p_1^{q_1}\cdots p_n^{q_n}$ Define $\tilde{N} = p_1\cdots p_n$.

Conjecture: If $N$ is an odd perfect number, then the sum of reciprocals of all factors of $\tilde{N}$ (excluding 1, including $\tilde{N}$) is less than 1.

Q. Does this conjecture appear to be equivalent to something that has already been established? If this conjecture is true, does it appear to have any obvious implications?

Invariants of exterior powers

Math Overflow Recent Questions - Wed, 12/13/2017 - 04:20

Let $\mathfrak{g}$ be the Lie algebra of $GL(n,\mathbb{R})$. Let $\theta(X) = - X^T$ be the Cartan involution on $\mathfrak{g}$; it induces decomposition as $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ of $+1$-eigenspace $\mathfrak{k}$ and $-1$ eigenspace $\mathfrak{p}$. Then $\mathfrak{k}$ is the Lie algebra of $K = O(n)$. I want to compute the dimensions of $K$-invariants $\textrm{Hom}_K(\wedge^q \mathfrak{p}, \mathbb{C})$, which, I suppose, is equal to $(W/ \mathfrak{k} W)^*$ where $W = \wedge^q \mathfrak{p}$, where $\mathfrak{p}$ is viewed as $\mathfrak{k}$-module by adjoint-representation. Could you someone point a way further?

Topological similarity of solutions to Dirichlet problem

Math Overflow Recent Questions - Tue, 12/12/2017 - 10:11

Let $\varphi_{1},\varphi_{2}:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be two smooth general position (Morse) functions having the same set of critical points $\left\{ p_{1},...,p_{n}\right\} \subset\mathbb{S}^{1}$ ($n$ is even) and both $\varphi_{1}$ and $\varphi_{2}$ have a local maximum at $p_{1}$. Suppose that $\varphi_{1}$ and $\varphi_{2}$ are similar in the following sense:

$\left( \varphi_{1}(p_{i})-\varphi_{1}(p_{j})\right) \left( \varphi _{2}(p_{i})-\varphi_{2}(p_{j})\right) >0$ for any $i\neq j$,

i.e. the critical level sets of $\varphi_{1}$ and $\varphi_{2}$ are in some sense similar.

Consider the corresponding two Dirichlet problems:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Delta u=0$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u|_{\mathbb{S}^{1}}=\varphi_{i}$ , $i=1,2$,

getting in such a way two harmonic solutions $u_{1},u_{2}:\mathbb{B}% ^{2}\rightarrow\mathbb{R}$.

Then is it true that the level lines portraits of $u_{1}$ and $u_{2}$ are the same up to topological equivalence, i.e. there is a homeomorphism $h:\mathbb{B}^{2}\rightarrow\mathbb{B}^{2}$ fixing all $p_{i}$ and sending the level lines of $u_{1}$ onto the level lines of $u_{2}$? Then, of course, $h$ is sending the critical set of $u_{1}$ onto the critical set of $u_{2}$.

In brief: does the similarity of the boundary conditions implies similarity between the solutions of the corresponding Dirichlet problems?

Note that we don't assume $\varphi_{1}$ and $\varphi_{2}$ to be close in any sense.

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