Recent MathOverflow Questions

Money as a factor in PhD admissions [on hold]

Math Overflow Recent Questions - Fri, 01/05/2018 - 16:36

I am wondering how strong the financial incentive of admitting competitive URG applicants(women and minorities) to PhD math programs (mainly in the U.S) is? From what I've gathered, the main reason why a top 10 institution would accept a URG student who has demonstrated their competency in all the relevant areas but hasn't done anything remarkable, is because of the money the department saves by admitting this candidate and the money gained. The financial incentive I'm referring to is the money that comes from the central administration to cover all the costs of the student and some extra money that goes to the department.

Below are two sources where I discovered that this is what happens in the graduate admissions process

Here the writer says

Even as faculty members on committees expressed philosophical commitment to diversity, Posselt observed financial motivations at play. Some universities offer extra funds for minority graduate students, so that a fellowship might be paid for from general university funds and not departmental funds. Where such incentives exist, they appear to have a strong impact, Posselt writes.

and in What do admission committees look for in a diversity essay?, Paul Garrett says:

being a woman in a STEM field (Science, Tech, Engineering, Math), or of ethnic origin other than northwestern European . . . opens certain money-pots to both the department and to the individual.

If I'm completely off about how the treatment of URG applicants works it would be interesting to get an idea of how the process works for those applicants.

Thanks!

Jordan decomposition of powers of the Shift Matrix

Math Overflow Recent Questions - Fri, 01/05/2018 - 15:55

Given the upper Shift Matrix, which for e.g. dimension $5$ is $$ {\bf E}_{\,{\bf 5}} = \left( {\matrix{ 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right) $$ then its non-negative integral powers are just given by a shift of the non-null diagonal, till ${\bf E}_{\,{\bf 5}} ^{\,{\bf 5}} ={\bf 0}$.

I know that the Jordan decomposition of ${\bf E}_{\,h} ^{\,{\bf n}}$ is given by $$ {\bf E}_{\,h} ^{\,{\bf n}} = {\bf P}_{\,h} (n)\;{\bf C}_{\,h} (n)\;{\bf P}_{\,h} (n)^{\, - \;{\bf 1}} $$ where
- ${\bf P}_{\,h} (n)$ is a permutation matrix;
- ${\bf C}_{\,h} (n)$ is actually a ${\bf E}_{\,h}$ with $n-1$ ones missing in certain positions, i.e. it is also expressible as a permutation.

For instance $$ \eqalign{ & {\bf E}_{\,{\bf 5}} ^{\,{\bf 2}} = \left( {\matrix{ 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right) = {\bf P}_{\,{\bf 5}} (2)\;{\bf C}_{\,{\bf 5}} (2)\;{\bf P}_{\,{\bf 5}} (2)^{\, - \;{\bf 1}} = \cr & = \left( {\matrix{ 1 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 \cr 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 1 & 0 & 0 \cr } } \right)\left( {\matrix{ 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 0 \cr } } \right)\left( {\matrix{ 1 & 0 & 0 & 0 & 0 \cr 0 & 0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 0 & 1 \cr 0 & 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 \cr } } \right) \cr} $$

But, after various trials, I could not yet find an effective way to express the type of P and C permutations wrt $h$ and $n$.
Most probably the subject has been already studied or it is easily assessable with the appropriate approach, and I am asking for hints in this respect.

Liouville measure and Bowen Margulis measure

Math Overflow Recent Questions - Fri, 01/05/2018 - 15:41

There are two natural measures on a Riemannian manifold such that the geodesic flow is measure preserving. One is Liouville measure and the other is Bowen-Margulis measure. Both of them are ergodic with respect to geodesic flow. The later is an analogy of Gibbs measure from discrete version where the maximal entropy(pressure) is attained.

I wonder whether there are some relation between these two measures. Would if be possible that they are the same measure? I would also appreciate for any recommendation of reading about these two measures.

Characterization of "finite" functions $f:\mathbb{R}\rightarrow \{0,1\}$

Math Overflow Recent Questions - Fri, 01/05/2018 - 14:16

Consider the Boolean-valued functions $f:\mathbb{R}\rightarrow \{0,1\}$. There are uncountably many – exactly $2^\mathbb{R}$ – of them.

I'm trying to figure out how those

functions with finitely many switches between $0$ and $1$, starting and ending with $0$

might be described in a general and compact way.

First I tried to imagine how such a function may switch its values (viewed arbitrary locally):

  • (1) by a Dirac-like peak from below $0$ to $1$, returning immediately to $0$

  • (2) by a Dirac-like peak from above $1$ to $0$, returning immediately to $1$

  • (3) by a Heaviside-like jump from below $0$ to $1$

  • (4) by a Heaviside-like jump from above $1$ to $0$

I cannot imagine other ways to change values for any function $f:\mathbb{R}\rightarrow \{0,1\}$.

Assuming that these are all the ways there are, each function $f:\mathbb{R}\rightarrow \{0,1\}$ with finitely many switches $\sigma_i$, $i = 0,\dots n$ (starting and ending with $0$, occuring at times $t_i \in \mathbb{R}$) should be uniquely definable by a sequence

$$\sigma_i = \langle \tau_i, \lambda_i\rangle$$

with

  • $\tau_i \in \{(1),(2),(3),(4)\}$,

  • $\lambda_{i} \in \mathbb{R}_{\geq 0}$, $\lambda_{i} = t_{i+1} - t_i$ for $i < n$

  • $\tau_0$ must be (1) or (3),

  • $\tau_n$ must be (1) or (4),

  • $\tau_{i}$ must obey

(And vice versa: Each such sequence defines such a function.)

Is this correct?

(Obviously, there are still uncountably many functions of this kind, as there are uncoutably many parameters $\lambda_i$.)

Addendum: Is there a more general context in which this question might make sense?

Day convolution for prederivators

Math Overflow Recent Questions - Fri, 01/05/2018 - 12:22

Let $\cal J = \bf Cat$ be the strict 2-category of small categories, functors and natural transformations and $\mathbb{V} : {\cal J}°\to \bf MonCAT$ a strict 2-functor taking value on possibly large monoidal categories, i.e. such that $\mathbb V(I)$ is a monoidal category for each $I\in\cal J°$.

This gadget is a monoidal prederivator, as defined in here (caveat: the nLab has a definition too, inspired by a different -higher- $n$POV; I do not consider that!)

Let now $\mathbb X$ be any strict 2-functor ${\cal J}°\to {\bf CAT}$. Consider the functor $$ [\![\mathbb X,\mathbb V]\!] : J\mapsto {\bf PsdNat}[\mathbb X,\mathbb V^J] $$ (the large category of pseudonatural transformations between $\mathbb X$ and the shifted prederivator $\mathbb V^J\colon I\mapsto \mathbb V(I\times J)$.

This works as an internal hom in the 2-category of prederivators, pseudonatural transformations, and modifications.

It defines a new strict 2-functor ${\cal J}\to \bf MonCat$. I would like to call this the pontwise monoidal structure on $[\![\mathbb X,\mathbb V]\!]$: aas a matter of fact, it's easy to prove that each category $[\![\mathbb X,\mathbb V]\!](J)$ is monoidal: given two objects $P,Q$ in it, we define $$ P\otimes Q : J\mapsto P_I\otimes_{I\times J}Q_I$$ to be the composition $$ \mathbb{X}(I)\xrightarrow{\Delta} \mathbb{X}(I)\times\mathbb X(I) \xrightarrow{P_I\times Q_I} \mathbb{V}^J(I)\times \mathbb{V}^J(I) \xrightarrow{\otimes_{I\times J}}\mathbb{V}^J(I)$$ This is a vertical composition of pseudonatural transformations, so it's pseudonatural in $I$, and apparently defines a monoidal structure.

Building on this, I would like to define the analogue of the Day convolution in the setting of (pre)derivators. Let me recall that given a small symmetric monoidal category $(C, \oplus)$ and presheaves $P,Q : C \to \bf Set$ we can define $$ P * Q : c\mapsto \int^{xy}Px\times Qy\times C(c, x\oplus y) $$ (of course you can do something similar in enriched setting); in a beautiful paper (Im-Kelly, "A universal property of the convolution monoidal structure") it is proved that $\widehat{C}^\otimes = ([C,{\bf Set}],*)$ is the free monoidal cocompletion of $C$, and it is the universal category rendering the Yoneda embedding $y : C \to \widehat{C}^\otimes$ a strong monoidal functor (moreover, $\widehat{C}^\otimes$ is monoidal closed; this is less interesting for the moment). So, here's the question:

Let $\mathbb X$ in addition be small and $\mathbb V$-enriched: this means that each $\mathbb X(I)$ is a small $\mathbb V(I)$-category and a bunch of compatibility conditions are satisfied. Is there a way to define a convolution monoidal product rendering the Yoneda morphism $$ y : (\mathbb X^\text{op},\oplus) \xrightarrow{\qquad} ([\![\mathbb X,\mathbb V]\!],*) $$ a "strong monoidal" one (see again Groth for the definition of monoidal morphism)?

Functor of points definition of the Thom space

Math Overflow Recent Questions - Fri, 01/05/2018 - 12:11

Let $X$ be a space (CW complex) and let $E \to X$ be a vector bundle.

Using the language of $\infty$-categories we can can define the Thom space $T(E)$ as the pointed space representing the following $\infty$-pre-cosheaf on the category of pointed spaces:

$$S \mapsto fib(Map(E,S) \to Map(E \setminus X, S))$$

In words, a a pointed map from $T(E)$ to any pointed space $S$ is given by the data of a map $E \to S$ together with a null-homotopy of it's restriction to the complement of the zero section $E \setminus X$.

This is all very nice however most of the interesting occurrences of Thom spaces are when they are mapped into rather than from. The raw imprecise question is therefore the following:

Is there an abstract definition of some manifestly interesting $\infty$-presheaf which $Map(-,T(E))$ happens to represent?

Here's what makes me hopeful:

In Ranicki's book on surgery he proves (what I think is) the following statement about Thom spaces:

Let $X$ be a manifold, $\eta$ a rank $k$ bundle and $N$ an $n$ dimensional smooth manifold.

Definition: The Bordism set $\mathfrak{B}_{m}(N,X, \eta)$ is defined to be the following set of equivalence classes:

Elements are $m$-dimensional submanifolds $j:M \hookrightarrow N$ ($m=n-k$) together with a map $f: M \to X$ s.t. $f^* \eta = \nu_{M \hookrightarrow N}$.

Two such elements $(M_1,f_1)$ and $(M_2,f_2)$ are equivalent iff there exists a submanifold $i: W \hookrightarrow N \times I$ and a map $F: W \to X$ s.t.

  • $\partial W = M \amalg N$

  • $F^* \eta =\nu_{W \hookrightarrow N \times I}$

  • $F |_{\partial W} = f_1 \amalg f_2$

Theorem: Transversal intersection with the zero section induces a natural bijection:

$$\mathfrak{B}_m(N,X,\eta) \cong [N,T(\eta)]$$

This statement seems to hint at the possibility of defining for every vector bundle $E$ on a manifold $X$ the Thom space as a presheaf on the category of smooth manifolds, whose $\pi_0$-presheaf is $\mathfrak{B}_{m}(-,X,E)$. Here's the question (which we have already answered the 0-categorical version of).

Question: Is there a categorical (bordism-flavoured) definition for the $\infty$-presheaf on the full subcategory of spaces generated by spaces homotopy equivalent to smooth manifolds which assigns to every manifold $M$ the mapping space $Map(M,T(E))$? (as we saw the $0$-categorical version is solved by $\pi_0(?)(-)=\mathfrak{B}_m(-,X,E)$)

Hopefully once the above is settled we could define the Thom space as the space representing the left-kan extension of the above presheaf along the inclusion functor to the category of spaces.

Cantelli's inequality: the original source

Math Overflow Recent Questions - Fri, 01/05/2018 - 11:07

Does anyone know where and when Cantelli's inequality was originally published? Strangely enough, I have not been able to find this information online.

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

Math Overflow Recent Questions - Fri, 01/05/2018 - 10:59

We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.)

Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains only algebraic numbers.]

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

That is, does there exist a curve $X$, a finite etale morphism $X\to \mathbb{A}^1-S$ and a finite etale morphism $X\to \mathbb{A}^1-\{0,1\}$?

For which $S$ is the answer positive?

Techniques for proving relaxed one-wayness of functions

Math Overflow Recent Questions - Fri, 01/05/2018 - 10:01

Existence of one-way functions is a widely accepted conjecture in complexity theory. A function is one-way if it is computable in polynomial-time but not invertible in polynomial-time (this is different from the notion used in cryptography where average-case hardness is required). It seems we don't have any proof techniques that proves one-wayness.

Let us relax the requirement such that one-wayness means the function $f(x)$ is computable in $O(n^{c})$ but $f^{-1}(x)$ is not computable in $O(n^{t \cdot c})$ time for some integer $t \gt 2$.

Is there any known current technique for proving this relaxed notion of one-wayness? Is there a natural function $f$ that was proven to be one-way in this relaxed setting?

I am interested in honest injective functions where $|x|< p(|f(x)|)$ for some polynomial $p$.

Continuous monotone real functions of several real variables

Math Overflow Recent Questions - Fri, 01/05/2018 - 07:44

Let $O$ be an open bounded connected set in $R^n$ and K its boundary. Given a continuous real function $f$ defined on $K$, I would like to extend $f$ to a continuous real function $g$ (i.e. $g$ restricted to $K$ equals $f$) defined in the closure of $O$, in such a way that g is also monotone (e.g. the min and max of $g$ on each closed ball contained in $O$ is attained on the boundary of the ball). In case $K$ has the extra property of being Holder-continuous, there exists a harmonic function $g$ as intended; in particular $g$ is real-analytic in $O$ and is monotonic by the min and max principles. What about if no extra property of $K$ is known, can one still find a monotonic $g$ as intended?

Toda Hierarchy and Quantum Cohomology of $\mathbb{P}^1$ Frobenius manifolds

Math Overflow Recent Questions - Fri, 01/05/2018 - 06:31

People usually say that the quantum cohomology of $\mathbb{P}^1$ Frobenius manifold $QH^*(\mathbb{P}^1)$, corresponds to dispersionless extended Toda hierarchy (e.g. page 6 of https://arxiv.org/pdf/math/0308152.pdf). I'm trying to understand this correspondence in more detail but I'm some having troubles. More specifically how do I recover the $\tau$-function? Should I be able to get the Frobenius potential $F$ of $QH^*(\mathbb{P}^1)$ from the $\tau$-function by \begin{equation} F = \frac{1}{2}(t^1)^2t^2 + e^{t^2} = \lim_{\epsilon \rightarrow 0}\epsilon^2\log \tau |_{t^{\alpha,p > 0} = 0, t^\alpha := t^{\alpha,0}} ? \end{equation} (Where $\epsilon \rightarrow 0$ is the dispersionless limit. I'm sort of getting this from Theorem 6.1 of https://arxiv.org/pdf/hep-th/9407018.pdf, but the result is for KdV). If I didn't set $t^{\alpha,p > 0} = 0$, should I get the Gromov-Witten potential of $\mathbb{P}^1$ with descendences from $\lim_{\epsilon \rightarrow 0}\epsilon^2\log \tau$?

Here is what I have gathered so far (mainly from https://arxiv.org/pdf/nlin/0306060.pdf), please correct me if I misunderstood something. The extended Toda hierarchy is given by the bihamiltonian structure on the loop space $\mathcal{L}(\mathbb{R}^2)$: \begin{align*} \{u(x),u(y)\}_1 = \{v(x), v(y)\}_1 &= 0, \\ \{u(x), v(y)\}_1 &= \frac{1}{\epsilon}(e^{\epsilon \partial_x} - 1)\delta(x - y),\\ \{u(x), u(y)\}_2 &= \frac{1}{\epsilon}(e^{\epsilon \partial_x}e^{v(x)} - e^{v(x)}e^{-\epsilon\partial_x})\delta(x-y)\\ \{u(x),v(y)\}_2 &= \frac{1}{\epsilon}u(x)(e^{\epsilon\partial_x} - 1)\delta(x-y)\\ \{v(x),v(y)\}_2 &= \frac{1}{\epsilon}(e^{\epsilon\partial_x} - e^{-\epsilon\partial_x})\delta(x-y). \end{align*} From this, I'm guessing some sort of Miura transformation can be used to put $\{.,.\}_2 - \lambda\{.,.\}_1$ into the normal form and from that we get a set of Hamiltonian densitites $\{h_{\alpha,p} = h_{\alpha,p}(x,u,v,u_x,v_x,...)\}$. The Hamiltonians are given by $H_{\alpha, p} := \int_0^{2\pi}h_{\alpha,p}dx$. The $t^{\alpha,p}$-evolution is \begin{equation} \frac{\partial w^\alpha}{\partial t^{\beta,p}} = \{w^\alpha,H_{\beta,p}\}_1, \qquad w = u,v;\ \alpha,\beta = 1,2;\ p \geq 0. \end{equation} All of these came from Theorem 3.1 of https://arxiv.org/pdf/nlin/0306060.pdf. From here we can use the standard result of biharmiltonian theory that $\partial h_{\alpha,p-1}/\partial t^{\beta,q} = \partial h_{\beta,q-1}/\partial t^{\alpha,p} =: \frac{1}{\epsilon}(e^{\epsilon\partial_x} - 1)\Omega_{\alpha,p;\beta,q}$, therefore there is a function $\log \tau$ such that \begin{equation} \Omega_{\alpha,p;\beta,q} = \epsilon^2\frac{\partial^2\log \tau}{\partial t^{\alpha,p}\partial t^{\beta,q}}. \end{equation}

Ideally, I would like to calculate $\log\tau$ from here and directly and answer my question above. However, the explicit form of $h_{\alpha,p}$ as given in Theorem 3.1 is complicated and I'm not sure how to proceed. Also, I have never seen anyone written down the equation $F = \lim_{\epsilon \rightarrow 0}\epsilon^2\log \tau |_{t^{\alpha,p > 0} = 0, t^\alpha := t^{\alpha,0}}$ explicitely anywhere, so maybe it could be wrong. Could anyone give me some advices or correct my understand or provide me with a suitable reference?

On a vanishing integral inner product

Math Overflow Recent Questions - Fri, 01/05/2018 - 02:48

Let $G(z)$ be an $n\times m$ rational matrix-valued function of full column rank on the unit circle. Further, let $P(z)$ be an $m\times m$ rational matrix-valued function positive definite on the unit circle and let $A$, $B$ be $m\times n$ complex (constant) matrices. Consider the following inner-product condition (on $\mathcal{L}_2^{m\times m}[-\pi,\pi]$) $$\tag{1}\label{eq:cond} \langle G^*XG,G^* \Delta G \rangle_2 = \mathrm{tr}\int_{-\pi}^{\pi} G(e^{i\theta})^* X G(e^{i\theta}) G(e^{i\theta})^* \Delta(e^{i\theta}) G(e^{i\theta}) \frac{\mathrm{d}\theta}{2\pi}=0,\ \, \forall X\in\mathcal{H}_n, $$ where $(\cdot)^*$ denotes Hermitian transposition, $\mathcal{H}_n$ denotes the space of Hermitian $n\times n$ matrices, and $$ \Delta(e^{i\theta}) := A^* P(e^{i\theta}) B + B^* P(e^{i\theta}) A. $$

My question. Does condition \eqref{eq:cond} necessarily imply $G(e^{i\theta})^*\Delta(e^{i\theta})G(e^{i\theta})\equiv 0$?

Some comments.

  1. A simple observation is that the answer is in the affirmative if $P(e^{i\theta})=p(e^{i\theta})M$ where $p(e^{i\theta})$ is scalar and $M$ is a constant $m\times m$ positive definite matrix. Indeed, in this case, the thesis readily follows from the fact that, picking $X=A^* M B + B^* M A$ in \eqref{eq:cond}, yields $$ \left\|p^{1/2} G^*(A^* M B + B^* M A)G\right\|_2=0. $$
  2. Condition \eqref{eq:cond} can be equivalently rewritten as $$\tag{2}\label{eq:cond2} \int_{-\pi}^{\pi} G(e^{i\theta}) G(e^{i\theta})^* \Delta(e^{i\theta}) G(e^{i\theta}) G(e^{i\theta})^* \frac{\mathrm{d}\theta}{2\pi}=0, $$ since $\mathrm{tr}(XY)=0$, $\forall X \in\mathcal{H}_n$ and a fixed $Y\in\mathcal{H}_n$, is equivalent to $Y=0$.

Examples of integer sequences coincidences

Math Overflow Recent Questions - Thu, 01/04/2018 - 22:50

For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the same mark, which suggests a connection between a priori independent mathematical areas. The most famous example like that is perhaps the Catalan numbers sequence: A000108.

Question: What are the examples of pair of integer sequences coinciding on all the known terms, but for which the coincidence for all the terms is unknown?

Cheating is not allowed. By cheating I mean artificial examples like:
$u_n = v_n =n$ for $n \neq 10$, and if RH is true then $u_{10} = v_{10} = 10$, else $u_{10}+1 = v_{10} = 1$.
The existence of an OEIS entry could act as safety.

sequence generated with polynomials

Math Overflow Recent Questions - Thu, 01/04/2018 - 21:06

Below is an old olympiad problem, that turned out to be notoriously hard, that we couldn't solve it. If anyone has a solution for it, I'd be grateful.

Let $P(x)=x+1$, and $Q(x)=x^2+1$. We consider all sequences of pairs, namely, $\{(x_k,y_k)\}_{k=1}^\infty$ such that, we have the following rule to generate: $(x_1,y_1)=(1,3)$, and for every $k$, $(x_{k+1},y_{k+1})$ is, $$ \text{either } (P(x_k),Q(y_k)) \text{ or } (Q(x_k),P(y_k)). $$ Call a positive integer $n$ as cute, if there exists at least one such sequence, on which, $x_n=y_n$. Find all cute $n$'s.

The number of representations of an integer as the inner product of integral lattice points

Math Overflow Recent Questions - Thu, 01/04/2018 - 19:54

I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a question I asked about four years ago. In particular I wrote that for an arbitrary fixed integer $k>1$ we have the following asymptotic expansion:

$$\Psi_k(n)=\sum_{\substack{\mathbf{u}\cdot\mathbf{v}=n\\(\mathbf{u},\mathbf{v})\in \mathbb{N}^k\times \mathbb{N}^k}}1=\left|\{(\mathbf{u},\mathbf{v})\in \mathbb{N}^k\times \mathbb{N}^k:\mathbf{u}\cdot\mathbf{v}=n\}\right|\sim \frac{\sigma_{k-1}(n)\text{log}(n)^{k}}{\zeta(k)(k-1)!}$$

With $\sigma_{k-1}(n)=\sum_{d\mid n}d^{k-1}$ and $\zeta(k)=\sum_{n=1}^{\infty}\frac{1}{n^k}$ the Riemann zeta function. Now alternatively if one expands the inner product of $\mathbf{u}\cdot\mathbf{v}$ we see $\Psi_k$ can be expressed in a number of other ways: $$\Psi_k(n)=\left|\{(u_1,v_1,u_2,v_2,\ldots u_k,v_k)\in \mathbb{N}^{2k}:n=u_1v_1+u_2v_2+\cdots +u_kv_k\}\right|\\=\sum_{\substack{m_1+m_2+m_3+\cdots +m_k=n\\(m_1,m_2,m_3,\ldots ,m_k)\in \mathbb{N}^k}}d(m_1)d(m_2)d(m_3)\cdots d(m_k)$$

Where $d(m)=\sum_{d\mid m}1=\sigma_{0}(m)$ counts the divisors of any natural number $m$. Which unearths a $q$-series esque representation for the ordinary generating function of $\Psi_k$ as follows:

$$\sum_{n=1}^{\infty}\Psi_k(n)q^n=\left(\sum_{n=1}^{\infty}q^{n^2}\frac{1+q^n}{1-q^n}\right)^k=\frac{\text{log}(1-q)^k}{\text{log}(q)^k}\sum_{j=0}^k\binom{k}{j}\left(\frac{\psi_{q}(1)}{\text{log}(1-q)}\right)^j$$

Where $\psi_{q}(z)=\frac{1}{\Gamma_{q}(z)}\frac{d}{dz}\Gamma_{q}(z)$ is the $q$-analog of the digamma function defined analogously in terms of the $q$-gamma function, expressible as $\Gamma_{q}(z)=(1-q)^{1-z}\prod_{n=0}^{\infty}\frac{1-q^{n+1}}{1-q^{n+z}}$ for $|q|<1$.

Now working with some of these alternate representations, as well as fiddling with the order of the summands involved I was able to prove by induction on the integer $k>1$ that we have both:

$$\sum_{n\leq N}\Psi_k(n)=\frac{N^k\text{log}(N)^{k}}{k!}+\mathcal{O}(N^k\text{log}(N)^{k-1})$$ $$\sum_{n\leq N}\frac{\sigma_{k-1}(n)\text{log}(n)^{k}}{\zeta(k)(k-1)!}=\frac{N^k\text{log}(N)^{k}}{k!}+\mathcal{O}(N^k\text{log}(N)^{k-1})\\$$

So using the same heuristics as before it seems reasonable that $\Psi_k(n)\sim \frac{\sigma_{k-1}(n)\text{log}(n)^{k}}{\zeta(k)(k-1)!}$ which at least for the case at $k=2$ would coincide with the answer to my previous question. However I'm unable to find a concrete proof of this result and would therefore appreciate any help in the matter.

The von Neumann algebra generated by a non-closable operator

Math Overflow Recent Questions - Tue, 01/02/2018 - 19:55

Let $H$ be a Hilbert space and let $M$ be a densely defined operator $D(M) \subset H \to H$. It is closable iff its adjoint $M^{\star}$ is densely defined, and then its closure $\overline{M}$ is $M^{\star \star}$. Let $\mathcal{M}$ be the smallest von Neumann algebra that $\overline{M}$ is affiliated with; it is called the von Neumann algebra generated by $M$.

Question 1: Is there a bounded operator $X \in B(H)$ such that $W^{\star}(X) = \mathcal{M}$?

In other words: Can a von Neumann algebra generated by a densely defined closable operator, be also generated by a bounded operator?

Question 2: Is there a way to generalize the generation of a von Neumann algebra to any densely defined operator (i.e. non necessarily closable)?

If an answer to Question 1 gives a process defining $X$ from $M$ and if this process works for any densely defined operator, that would also answer Question 2.

The motivation comes from the densely defined operator associated to an integer map $m: \mathbb{N} \to \mathbb{N}$ (i.e. $M: \mathbb{C}[\mathbb{N}] \subset H \to H$ with $Me_n = e_{m(n)}$) such that $\exists n \in \mathbb{N}$ with $m^{-1}(\{ n\})$ infinite.

As a non-obvious example, consider Conway's game of life and let $S$ be the set of states of the grid with only finitely many alive cells. Then the application of the rules produces a map $r:S \to S$, which can be reformulated into a map $m: \mathbb{N} \to \mathbb{N}$ because $S$ is countable infinite. A problem is that the vacuum state (i.e. all cells dead) has an infinite pre-image (and so is any state of $r(S)$). A positive answer to Question 2 would give a way to generate a von Neumann algebra $\mathcal{M}$ from Conway's game of life (or any other cellular automaton); if so, what is $\mathcal{M}$?

The 1-step vanishing polyplets on Conway's game of life

Math Overflow Recent Questions - Tue, 01/02/2018 - 19:33

A $n$-polyplet is a collection of $n$ cells on a grid which are orthogonally or diagonally connected.
The number of $n$-polyplets is given by the OEIS sequence A030222: $1, 2, 5, 22, 94, 524, 3031, \dots$

See below the five $3$-polyplets:

A polyplet will be called $1$-step vanishing on Conway's game of life, if every cell dies after one step.
We observed that for $n\le 4$, a $n$-polyplet is $1$-step vanishing iff $n \le 2$.

We found $1$-step vanishing polyplets with $n=9, 12$, see below:

Question: Is there an other $1$-step vanishing $n$-polyplet with $n \le 12$?
If yes, what are they? (note that there are exactly $37963911$ $n$-polyplets with $n \le 12$).

We can build infinitely many $1$-step vanishing polyplets with such pattern, see below with $n=142$:

Bonus question: Are there $1$-step vanishing polyplets of an other kind?

a question about mapping class groups

Math Overflow Recent Questions - Tue, 01/02/2018 - 19:32

According to Thurston's construction, which can be found for instance in Farb-Margalit's A Primer on Mapping Class Groups, theorem 14.1 (here is a link to the version I am using: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf), If there are two curves $a$ and $b$ on a surface $S_{g,n}$ (a surface of genus $g$ with $n$ boundary components) that fill the surface, then we will be able to figure out the type of the element (periodic, reducible, Anasov) $\phi\in MCG(S_{g,n},\partial S_{g,n})$, $\phi=t_at_b$, based on the intersection number of $a$ and $b$.

I want to know if anything can be said about the type of an element in the mapping class group that consists of three Dehn twists, say $\psi=t_a t_b t_c$, that $a$ and $b$ fill the surface and $t_c\notin <t_a,t_b>$. Is there any general way we can determine whether such an element $\psi$ is period, reducible or Anasov (perhaps based on mutual intersections of these three curves)?

PS: I am actually working on a concrete example, so that $\psi=t_at_bt_c$, $a$ and $b$ fill the surface and $t_at_b$ is pseudo-Anasov, but $t_c\notin <t_a,t_b>$. Apparently since I have not earned enough points on MO, I am not allowed to insert an image. So any general comment about my genral question will be appreciated.

Reduced scheme structure on locally complete intersection

Math Overflow Recent Questions - Tue, 01/02/2018 - 18:35

This question concerns reduced scheme structure on locally complete intersection, and I guess the answer is related to the number of generators of a radical ideal.

I am confused about the following proposition in Hartshorne's Algebraic Geometry, Chapter 2:

Proposition 8.23 Let $Y$ be a locally complete intersection subscheme of a nonsingular variety $X$ over $k$. Then:

(a) $Y$ is Cohen-Macaulay;

(b) $Y$ is normal if and only if it is regular in codimension 1.

In the proof of (b), the author says that "regular in codimension 1" is just (R1) condition and then uses his Theorem 8.22A, but I think he forgets to treat the case of codimension 0. In order to use (8.22A), one needs the fact that any local ring $\mathcal{O}_{Y,y}$ of $Y$ of dimension 0 is regular.

It is trivially true that a dimension 0 reduced local ring is regular (and hence is a field). In contrast, if $\mathcal{O}_{Y,y}$ is not reduced, then it might not be regular. For example, $k[x]/(x^2)$ is a dimension 0 non-reduced non-regular local ring.

So I guess that in (8.23), the author assumed $Y$ to be reduced.

Now my question comes: if $Y$ is not reduced at first, and let $Y^\prime$ be $Y$ with the reduced closed subscheme structure from $X$, is it true that $Y^\prime$ is still a locally complete intersection?

Locally, the question is: Let $A$ be a finitely generated local ring over a field $k$, and $I=\langle x_1,\dots,x_r\rangle$ be an ideal generated by $x_1,\dots,x_r$ in $A$ (the $x_i$'s in fact form a regular sequence in the situation of locally complete intersection). Then is $\sqrt{I}$ also generated by $r$ elements?

I am sorry if this question turns out to be easy.

Harmonic one-form on non-compact manifold

Math Overflow Recent Questions - Tue, 01/02/2018 - 17:43

Let $M$ is a non-compact Riemannian manifold endowed with a metric $g$ of non-negative Ricci curvature. If $\alpha$ denotes a harmonic 1-form on $M$, the function $x\to |\alpha|_{g,x}^2$ is constant?

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