# Math Overflow Recent Questions

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most recent 30 from mathoverflow.net 2018-02-23T10:14:37Z

### Do pseudo-differential operators form a sheaf of algebras?

Mon, 02/19/2018 - 14:14

Let $M$ be a smooth manifold.

I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on $M$). Sadly the best results I could find showed at best that pseudo-differential form a sheaf of left $C^{\infty}$-modules without treating the question of whether the multiplication is well defined or associative. So my question is rather simple:

Do pseudo-differential operators form a sheaf of (associative) $\mathbb{C}$-algebras on $M$? If not, what fails?

I'm pretty sure that if one considers pseudo-differential operators modulo smothing operators than these do form a sheaf of algebras (However a precise reference here would be welcome). As for the entire space of pseudo-differential operators this seems rather non-trivial to me (I'm actually not entirely sure whether this should be true or not). I apologize if this is too elementary for this site.

### Mysterious symmetry - in search for a bijection

Mon, 02/19/2018 - 14:13

I have a mysterious symmetry that I have not managed to prove. First some definitions (see picture below)

Fix a partition that fit in a staircase shape with $n$ rows. There are $Catalan(n)$ such shapes. We can represent this with a diagram $D$, as below, where the gray squares is the partition. The yellow squares are enumerated $1,\dotsc,n$ from top to bottom, and thus any permutation in $S_n$ is seen as a labeling of the yellow squares.

Let $a+b=n$. Given a permutation in $S_n$ seen as a labeling of the yellow squares, the first block is the squares with labels $1,\dotsc,a$ and the second block is the remaining $b$ squares.

A permutation $\sigma \in S_n$ is called $(D,a,b)$-good if the following holds:

• The smallest label in each of the two blocks appear lowest in its block.
• If $i$ and $i+1$ are in the same block, and $i$ below $i+1$, then the square in the same row as $i+1$ and same column as $i$ must be white.

In the diagram, the permutation $342615$ is shown, and one can verify that it is $(D,4,2)$-good. The white squares that has to be white due to the second condition has been marked with bullets.

Let $Good(D,a,b)$ denote the set of $(D,a,b)$-good permutations.

Warmup exercise

Show that $|Good(D,a,b)|=|Good(D,b,a)|$.

Finally, we define the ascent, $asc_D$-statistic on permutations as follows: For every white square $S$, we let $S_1$ be the index of the yellow square in the same row, and $S_2$ be the index of the yellow square in the same column. Then $asc_D(\sigma)$ is the number of white squares $S$, such that $\sigma(S_1)<\sigma(S_2)$. In our diagram, $asc_D(342615) = 1+1+2+0+1 = 5$, where the terms are contributions from each row.

My problem

Show (bijectively) that for every diagram $D$ and choice of $a+b=n$, $$\sum_{\sigma \in Good(D,a,b)} q^{asc_D(\sigma)} =\sum_{\sigma \in Good(D,b,a)} q^{asc_D(\sigma)}.$$

For the diagram $D$ here, we have that $|Good(D,4,2)|=|Good(D,2,4)|=20$, and that both sums above become $$1 + 3 q + 4 q^2 + 4 q^3 + 4 q^4 + 3 q^5 + q^6.$$

Comments: For some diagrams $D$, it is straightforward to produce a bijection, in particular the case when $D$ has no gray squares. One would hope that a bijection would the number ascents 'within blocks', that is, ascents where $S_1$ and $S_2$ belong to the same block. However, this cannot be done for general $D$.

One can generalize the problem to permutations with more than two blocks, but the 2-block case implies the general case.

I am quite confident this result follows (non-bijectively) from a result by C. Athanasiadis, but it requires several messy steps.

Motivation

This is related to the $p_\lambda$-expansion of certain LLT polynomials.

### If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \times \mathbb{R}^{\infty}$ homeomorphic?

Mon, 02/19/2018 - 14:13

Let $X$ and $Y$ be reasonable spaces. Since $\mathbb{R}^{\infty}$ is contractible, $$X \times \mathbb{R}^{\infty} \cong Y \times \mathbb{R}^{\infty} \;\;\; \implies \;\;\; X \simeq Y.$$

Is the converse also true?

My vague intuition: the factors of $\mathbb{R}^{\infty}$ provide so much extra room that there will never be a geometric obstruction to producing a homeomorphism. Evidently, there is no homotopy-theoretic obstruction, so maybe the converse is true.

On the other hand, I really have no idea and could be missing something basic. For example, the plane with two punctures is homotopy equivalent to a wedge of two circles. However, I do not know about a homeomorphism $$(\mathbb{C} - \{0, 1\}) \times \mathbb{R}^{\infty} \overset{?}{\cong} (S^1 \vee S^1) \times \mathbb{R}^{\infty}.$$

### Annihilators of indecomposable representations in the BGG category $\mathcal O$ over semisimple Lie algebra

Mon, 02/19/2018 - 12:21

Let $\mathfrak g$ be a finite-dimensional (complex) semisimple Lie algebra. Then we consider the BGG category $\mathcal O$.

For a given indecomposable module $M$ in $\mathcal O$. And assume that we know all composition factors $L(\lambda_i)$ (simple height weight module of weight $\lambda_i$) of $M$, for $i =1,\ldots, m$.

${\bf My ~question}$: Is there any method to detect the annihilator of $\text{Ann}M$? Can we say something about $\text{Ann} M$ in terms of annihilators $\text{Ann}L(\lambda_i)$, $i=1,\ldots ,m$. (In general I think $\text{Ann}M \neq \bigcap_{i=1}^m\text{Ann}L(\lambda_i)$, is this correct?) Thanks!

*$\text{Ann} N = \{x\in \mathfrak g|~xN=0\}$, for all $N \in \mathcal O$.

### Could the sequence A287326 be generalized in order to receive expansion of natural power n>3?

Mon, 02/19/2018 - 11:55

The sequence https://oeis.org/A287326 - is Binomial distributed triangular array, that shows us necessary items to expand perfect cube $n^3$. Summation of $n$-th row of Triangle A287326 from $0$ to $n-1$ returns $n^3$. But is it exist simillar patterns in order to receive expansion of power $n>3$, where $n$ - positive integer?

$$\begin{matrix} & & & & & 1\\ & & & & 1 & & 1\\ & & & 1 & & 7& & 1\\ & & 1 & & 13& & 13& & 1\\ & 1 & & 19& & 25& & 19& & 1\\ \end{matrix}$$ Figure 1. Triangle A287326.

It derived by means of identity $$x^3=\sum\limits_{m=0}^{x-1}3!\cdot mx-3!\cdot m^2+1$$

For detailed info on derivation, please, reffer to links below. Thank you !

### Tate-Shafarevich group over number fields

Mon, 02/19/2018 - 10:55

Let $A$ be an abelian variety over a number field $K$, $\text{Sha}(A/K)$ its Tate-Shafarevich group, $\ell$ a prime.

Is it known that the $\ell$-primary torsion subgroup $\text{Sha}(A/K)\{\ell\}$ is trivial for almost all primes $\ell$?

### Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?

Mon, 02/19/2018 - 01:33

Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$.

Question (?) : Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \mathcal{O}_D$ which is not a (possibly infinite order) differential operator. I.e. not of the form:

$$\phi=\Sigma_{n=0}^{\infty} b_n(z) \partial^n$$

Where $\partial =\frac{d}{d z}$ and $b_n \in \mathcal{O}_D$ .

EDIT: Suppose I require that $\phi$ be continuous w.r.t. to the natural frechet topology on $\mathcal{O}_D$ coming from uniform convergence on compact subsets, does the answer change?

### Haar measure on $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$

Mon, 02/19/2018 - 00:05

The bi-invariant Haar measure on the quotient $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}_2(\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. An element of the quotient can be written (using the Iwasawa decomposition) as: $\begin{pmatrix} y^{1/2} & xy^{-1/2} \\ 0 & y^{-1/2} \end{pmatrix}$ with $z=x+iy$ in a fundamental domain of the action of $\mathrm{SL}_2(\mathbb{Z})$ over the upper-half plane, such as $D = \{(x,y) : x^2+y^2 \geq 1,|x| \leq 1/2,y > 0\}$. In these coordinates, the invariant measure is a scale of the hyperbolic measure, namely $\frac{3}{\pi} \frac{dx \, dy}{y^2}.$ Is there a similar nice parametrisation for the rank 3 case (namely explicating the bi-invariant Haar measure of $\mathrm{SL}_3(\mathbb{Z}) \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})$?

### What is this quotient of the triangle 2-3-7 group?

Sun, 02/18/2018 - 23:51

I have been working with Hurwitz groups, and I came across the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, ([a,b]^2ab)^6 \rangle$. I'm trying to figure out exactly what this group is. I know it contains two copies of ${\rm PSL}(2,13)$, and there seems to be a very large 2-group in there as well, of order $2^{28}$ at least. Is this group even finite, and if so, what is the order? Does it contain any other simple groups?

### $K[[X_1,...]]$ is a UFD (Nishimura's Theorem)

Sun, 02/18/2018 - 23:18

Let us define the infinitely-many-variable formal power series ring

$$K[[X_1,\ldots]] \colon= \underset{m \geq 1}{\varprojlim}\,K[[X_1,\ldots,X_m]].$$

$K[[X_1,\ldots]]$ is known to be a UFD by a theorem of Nishimura (c.f. On the unique factorisation theorem for formal power series, Journal Math. Kyoto. Univ. Vol 7. No 2. 1967, 151-160).

Now let us choose an irreducible element $f \in K[[X_1,\ldots]]$ and consider the image $f_m \in K[[X_1,\ldots,X_m]]$ of $f$ by the natural quotient ring homomorphism $K[[X_1,\ldots]] \twoheadrightarrow K[[X_1,\ldots,X_m]]$.

Q. Is $f_m$ also irreducible for $m \gg 0$?

### Is every true statement independent of $PA$ equivalent to some consistency statement?

Sat, 02/17/2018 - 17:15

Most true statements independent of PA that I know of is equivalent to some consistency statement. For example

• Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$
• Goodstein's theorem is equivalent to Con(PA)
• Any conjunction or disjunction of the above.

Is every true statement independent of PA equivalent to some consistency statement?

By "equivalent to some consistency statement", I mean that $PA \vdash S \iff Con(T)$, for some theory $T$. Also, $T$ should be either finite, or specified by a Turing machine that outputs its axioms (and such that PA proves that the Turing machine never stops outputting statements), so that the description of $T$ doesn't throw PA off.

EDIT: In particular, are there are $\Pi^0_1$ examples?

### Fast Comparing of the Volume of Simplices Defined by Sidelengths

Sat, 02/17/2018 - 01:06

I have a problem, that requires sorting a set of simplices, that are defined via their sidelengths, according to volume; the value of the individual volumes isn't relevant in my problem.

Question:

are there faster methods of comparing the volumes of two simplices (that are defined via their sidelengths), than comparing the absolute values of their Cayley Menger determinants?

• if no, where can I find details about the initial proof?
• if yes, what are relevant algorithms for that problem?

### Estimating the critical probabilities $\mathrm{P_{c1}}$ and $\mathrm{P_{c2}}$ mathematically for the infinite system case

Sat, 02/17/2018 - 00:47

Suppose I have an $\mathrm{N\times N}$ square matrix consisting of only $0$'s and $1$'s. The probability of a certain element being $1$ is $\mathrm{p}$. At that certain probability $\mathrm{p}$, we count the number the number of white clusters and number of black clusters in the matrix. Any two elements who share a side (i.e. any one of them lies to the left, right, top or bottom of the other) or share a vertex are considered to belong to the same cluster. BUT, there's one special case i.e. if anywhere in the matrix there occurs a situation like this:

0 | 1 1 | 0

OR

1 | 0 0 | 1

That is two $1$'s share a vertex (diagonally connected) and two $0$'s also share a vertex as shown, the $50\%$ of the times, one should consider the $1$'s to belong to same cluster and other $50\%$ of the times one should consider the $0$'s to belong to the same cluster.

Suppose I am gradually increasing $\mathrm{p}$ from $0$ to $100$ in steps of $0.001$ and counting the number of black and white clusters for each $\mathrm{p}$. My aim is to find that critical probability $\mathrm{P_{c1}}$ at which number of white clusters increases from $1$ to a number greater than $1$ as $\mathrm{N}\to\infty$. Also I need to find $\mathrm{P_{c2}}$ at which number of black clusters decreases from a number greater than $1$ to $1$.

I wrote a program for the $1000\times 1000$ case, and averaged the critical probability over $10$ iterations. (The program basically had a loop which ran from $\mathrm{p}=0$ to $\mathrm{p}=100$ and I outputted those $\mathrm{p}$'s at which the number of white clusters increased from $1$ and number of black clusters decreased to $1$) The value I got for $\mathrm{P_{c1}}$ was $0.05$ and for $\mathrm{P_{c2}}$ it was $0.96$. However, I don't know if any purely mathematical technique exists to find the convergence value of these two critical probabilities for the infinite matrix case (i.e. $\mathrm{N}\to \infty$). Around $3$ decimal places of accuracy will be enough for my purpose. Any help will be appreciated.

Edit:

I realize that my question wasn't very clear. I'm adding an example.

$$\begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{bmatrix}$$ Here, $f$ and $k$ share a vertex and so do $g$ and $j$.

If $f$ and $k$ were both $1$'s and $g$ and $j$ were both $0$'s, then you should either consider the two $1$'s ($f$ and $k$) to belong to the same "black" cluster (i.e. connected along the vertex) and $0$'s ($g$ and $j$) to belong to the same "white" cluster. BUT, suppose if you did consider the two $1$'s to belong to the same cluster (i.e. connected along the vertex), then you should not consider the two $0$'s to be connected along the vertex.

### Decomposition of injective modules over Noetherian rings

Sat, 02/17/2018 - 00:42

Let $A=\mathbb{C}[x_1,\ldots,x_n]$ be a polynomial algebra over the complex numbers. I am interested in injective modules over $A$.

Since $A$ is projective over itself, the $\mathbb{C}$-dual module $A^\ast=\mathbb{C}[[x_1,\cdots,x_n]]$ is known to be injective and the generatings $x_i$ of $A$ act on $A^\ast$ by partial derivatives $\partial/\partial x^i$. As the ring $A$ is Noetherian, $A^\ast$ splits in the direct sum of indecomposable injective modules. My question is about the structure of this decomposition.

As I can see, for any $y\in \mathbb{C}^n$ the module $A^\ast$ has the indecomposible $A$-submodules $$M_y=\{p(x)e^{yx}| p(x)\in A\}\,,$$ where $yx$ stands for the inner product of vectors in $\mathbb{C}^n$. Furthermore, it seems that $M_y$ is isomorphic to the injective hull $E_A(\mathbb{C})$ of the trivial $A$-module $\mathbb{C}$ on which $p(x)\in A$ acts by multiplication by $p(y)\in \mathbb{C}$.

Is it true that

$$A^\ast\simeq \bigoplus_{y\in \mathbb{C}^n} M_y \quad?$$ This would be rather strange, as any finite sum of functions of the form $p(x)e^{yx}$ for some $p$'s and $y$'s can't produce a divergent power series from $A^\ast$.

What is a good reference on the subject?

### Log-concavity of the maximum of gaussians

Sat, 02/17/2018 - 00:23

Let $Z_1,\ldots, Z_n$ be independent gaussian random variables. Is it true that $X=\max\{Z_1,\ldots,Z_n\}$ has a log-concave distribution function?

### Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic?

Sat, 02/17/2018 - 00:18

This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along a smooth plane cubic or not. I believe that the answer shoud be "no" by Matsushima's criterion saying that if the automorphism group of the variety is not reductive then there is no such metric.

I was thinking that the following might be a (sketch of) proof: The automorphisms of $\mathbb{P}^3$ fixing a plane (say $x_3=0$) are of the form $$\begin{pmatrix} * & * & * & * \\ * & * & * & * \\ * & * & * & * \\ 0 & 0 & 0 & * \end{pmatrix}$$
This group has projective dimension 12 and since 9 points determines a cubic in $\mathbb{P}^2$ we have that the automorphism group of the blow-up is 3-dimensional. On the other hand, the 3-dimensional unipotent subgroup $$\begin{pmatrix} 1 & 0 & 0 & * \\ 0 & 1 & 0 & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ acts trivially on the plane $x_3=0$, so in particular fixes the 9 points determining the cubic and hence lifts to the automorphism group of the blow-up. Since they have the same dimension (but the latter might be non connected), the automorphism group if a finite extension of a unipotent group, hence not reductive.

Am I right? Sorry for being sketchy! Thanks a lot in advance for any comment, I just started to introduce myself to the subject.

PS: By the way, does the fact that $\operatorname{Aut}(X)$ is reductive implies that the connected component of the identity $\operatorname{Aut}^0(X)$ is reductive as well?

### Graphs of groups with homomorphisms not necessarily injective

Fri, 02/16/2018 - 23:44

I'm wondering if there is any literature on graphs of groups where the maps $G\to H$ from an edge group $G$ to its endpoint group $H$ are not necessarily $\pi_1$-injective. Or is this just too general to actually say anything meaningful? Are there any results on this subject?

### Numerical Methods

Fri, 02/16/2018 - 22:26

While using the adaptive step size Runge Kutta method to approximate the solution of ordinary differential equations, what must generally be the tolerance to compare the error with?

### Specialization map étale cohomology

Fri, 02/16/2018 - 21:59

Let $R$ be a henselian dvr, $s,\eta\in\text{Spec}(R)$ the closed and generic points, and $f : X\to \text{Spec}(R)$ a proper smooth scheme.

For a prime $\ell$ invertible on $R$, is there a specialization map

$$sp^i_{\eta,s} : R^if_*(\mu_{\ell^n})_{\eta}\to R^if_*(\mu_{\ell^n})_{s}$$

(with $\eta$ and $s$, not the geometric points over them)?

Is it an isomorphism, injective, surjective?

### Amortization differences [on hold]

Fri, 02/16/2018 - 19:44

not a math guru but work in the mortgage field and have a question I am hoping someone can answer.

Scenario: one loan for $200,000 at 5pct over 30 years has a monthly principle and interest payment of$1074.

If broken up into two loans, each for $100,000, one loan with a 4% interest rate and the other with a 6pct interest rate, the payments are$477 and $600 respectively. Totaling$1077.

Considering that the combination scenario has an average rate equal to 5% and the amortization schedules each end at 360 months...... why the difference in total payments?

This payment difference increases with larger loan amounts and bigger swings in interest rates (e.g. 2% and 8%) for the same average rate.