Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation $$ (\partial_{t}-\Delta)K(x,y,t)=0, K(x,y,0)=\delta_{x}(y) $$ and its solution $$ f(x)=e^{\Delta}\delta_{P}(x)=\int K(x,y,1)\delta_{P}(y)dy=K(x,P,1) $$ I am wondering if there is any special meaning attached to the case when $t=1$ and $M$ is taken as a homogeneous space. The idea seems natural that one start Brownian motion with a pointed probability distribution and let time flows, then $f(x)$ can be used as a measure of the "distance" between $x$ and $P$. In the case $M=\mathbb{S}^{1}$ via its universal cover $\mathbb{R}^{1}$ we recovered a sum of Gaussians that resembles theta function.

However when I look up online, the explicit formulas we have are rather difficult to interpret. For $g\ge2$ they are quotients of $\mathbb{H}$, and thanks to Mckeans' work we know $$ K(x,y,t)=\frac{\sqrt{2}}{(4\pi t)^{3/2}}e^{-t/4 }\int^{\infty}_{p}\frac{se^{-s^2/4t}}{(\cosh(s)-\cosh(p))^{1/2}}ds,p=d_{\mathbb{H}}(x,y) $$ I do not really know any good interpretation of this when we fix $y=P,t=1$ and let $x$ vary and add up the copies corresponding to $\pi_{1}(M)$. In particular, I am curious what does this say related to the spectrum of the Laplacian in $M$. It should relate to analytic torsion of the Laplacian on the manifold, but so far I have not found any way to relate the two objects.

The question below is the follow-up of this question on MathOverflow.

**Motivation:** As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are related to surfaces with Picard rank 20(see the paper of Elkies and Schuett) in the Dwork family $$x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4.$$ Jesus Guillera found a few Ramanujan-type formulas for $1/\pi^2$(which can be found in W. Zudilin's paper), three of which(formula (92)(93)(94) in Zudilin's paper) are related to the Dwork family $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$$through Picard-Fuchs equation. It is reasonable to conjecture that the Guillera's formulas are related to **"singular"** members in Dwork family.

The end of this paper suggests that the (Hasse-Weil)L-functions of those "singular" members behave differently from those "ordinary" members. It seems that "hypergeometric motive" package(developed by M. Watkins, based on the work of N. Katz et al.) in MAGMA offers a possible way to investigate those L-function numerically(although the L-functions are different from Hasse-Weil L-function).

**Experiment:** M. Watkins and David Roberts tried to find out imprimitive L-function attached to hypergeometric motives in this document(p.29, Table 15), where the L-function attached to the motive can be factorized to the product of two L-functions. One can immediately recognize the numbers corresponding to Guillera formula(formulas (86)(87)(88) in Zudilin's Paper). It is amazing that one can find out that **EVERY** L-function associated to the Guillera formula is **imprimitive.**

**Example:** Guillera conjectured that

$$\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n(\frac{1}{8})_n(\frac{3}{8})_n(\frac{5}{8})_n(\frac{7}{8})_n}{(n!)^5}(1920n^2+304n+15)\frac{1}{7^{4n}}=\frac{56\sqrt{7}}{\pi^2}.$$

The HGM package can evaluate Frobenius trace and Euler factors of the corresponding hypergeometric motive $$H([1/2,1/8,3/8,5/8,7/8],[0,0,0,0,0],\tilde{t}),$$ where $$1/\tilde{t}=t=\frac{1}{7^4}.$$ Almost each Euler factor $L_p(T)$ are quintic polynomials, and the absolute values of roots of Euler factors are $p^2.$ Adapting the primitivity test M. Watkins(p. 25), if one assumes the truth of Selberg's conjecture for Selberg class, then(for a primitive L-function) the second moment of (normalized) Frobenius trace $a_p$ $$\frac{1}{\pi(X)}\sum_{p<X}\left(\frac{a_p}{p^2}\right)^2$$ should have limit 1.

A calculation with second moment of normalized Frobenius trace of the example above(up to $p\approx 246500$) is $\approx 2.018$, suggesting the mean value is $2$. The same test is performed for each hypergeometric motive attached to Guillera's formulas(formula (86)-(95), up to $p\approx 40000$), and all mean values are close to $2.$ The same phenomena are also observed for identities discovered by Ramanujan, B. Gourevich and J. Cullen(formulas on p. 33 of this paper). I also tried to find imprimitive L-function other than those associated to Guillera's formulas, but without any success.

**Question:** I am really amazed by the fact that each L-function associated to the Guillera formulas is likely to be able to be factorized(and one of the factors seems to be either Riemann zeta function or Dirichlet L-function(tested for Euler factor for each Guillera formula up to $p\approx 1000$ and $p\approx 100$ for higher order examples of B. Gourevich and J. Cullen)), while the L-function of **a randomly chosen** hypergeometric motive **cannot be further factorized**(which is suggested by calculation with MAGMA).

- Is there any interpretation that how the imprimitivity of L-function attached to hypergeometric motives leads to period relation of hypergeometric functions?
- Is it possible to find new Ramanujan-type formula with this imprimitive test?

For context, consider a sum $X$ of $n$ independent random variables in $[0,1]$. For this situation, some Chernoff bounds bound the probability of a deviation with additive error, while others consider multiplicative error. E.g., letting $\mu = E[X]$, one standard Chernoff bound bounds the probability of deviating by an additive error:

$$\Pr[X \ge \mu + \epsilon n] ~\le~ \exp(-\epsilon^2 n).$$

Another gives a slightly stronger bound on the probability of deviating by a multiplicative error:

$$\Pr[X \ge (1+\epsilon) \mu] ~\le~ \exp(-\epsilon^2 \mu / 2).$$

Now, McDiarmid's inequality as it is usually stated gives a bound on the first kind of deviation (additive error).

Is there a form of McDiarmid's inequality that gives a multiplicative-error bound? (Assuming here we are using it to bound, say, a function $f(x_1,..,x_n)$ that is always non-negative, and changes by at most 1 when any $x_i$ is changed.)

In case the answer to that is no, here is the specific application I have in mind. Does a multiplicative error bound hold for this application?

Fix $n$ arbitrary values $x_1, x_2, ..., x_n$ in $[0,1]$, and an integer $k$. Obtain $k$-set $S$ by drawing $k$ times randomly *without replacement* from $\{1,2,..,n\}$. Define r.v. $X = \sum_{i \in S} x_i$.

Following the hint here, we can use McDiarmid's inequality to give a bound such as

$$\Pr[X \ge \mu + \epsilon k] ~\le~ \exp(- \epsilon^2 k / 3),$$

where $\mu = E[X]$.

In this case, does it also hold that, say,

$$\Pr[X \ge \mu (1+\epsilon)] ~\le~ \exp(- \epsilon^2 \mu / c)$$

for some constant $c > 0$?

It seems plausible that, at least in this specific case, one might be able to prove a multiplicative-error bound using some carefully designed super-martingale.

A related but different question was asked here.

Let E be a vector bundle of rank 2 over a variety X. Is there a counterexample so that $E$ is not isomorphic to $E^*\otimes det$ $E$?

I have asked this question on a different forum. I am asking it here as well in order to increase the number of different people who see it.

Consider a special function defined as:

$$f(a_1,a_2,a_3;b_1,b_2,b_3;c;x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3)=\\ =\sum_{i_1,i_2,i_3,\\j_1,j_2,j_3,=0\\k_1,k_2,k_3}^\infty \frac{(a_1)_{i_1+i_2+i_3}(a_2)_{j_1+j_2+j_3}(a_3)_{k_1+k_2+k_3}(b_1)_{i_1+j_1+k_1}(b_2)_{i_2+j_2+k_2}(b_3)_{i_3+j_3+k_3}}{i_1!i_2!i_3!j_1!j_2!j_3!k_1!k_2!k_3!(c)_{i_1+i_2+i_3+j_1+j_2+j_3+k_1+k_2+k_3}}\prod_{r=1}^3x_r^{i_r}y_r^{j_r}z_r^{k_r}$$

where $(x)_y=\Gamma(x+y)/\Gamma(x)$ is the Pochhammer symbol.

Does this function have a name, and if so what is it?

EDIT:

Note for instance a curiosity - for e.g. $a_2,a_3=0$ the function reduces to a Lauricella function:

$$f(a,0,0,b_1,b_2,b_3,c;x_1,x_2,x_3,...) = \sum_{i_1,i_2,i_3=0}^{\infty} \frac{(a)_{i_1+i_2+i_3} (b_1)_{i_1} (b_2)_{i_2} (b_3)_{i_3}} {(c)_{i_1+i_2+i_3} \,i_1! \,i_2! \,i_3!} \,x_1^{i_1}x_2^{i_2}x_3^{i_3}$$

Given a 3-manifold $M$ with a triangulation $T$, will every essential surface in $M$ be a fundamental one? If not, then what are the conditions on $T$ so that these essential surfaces become fundamental?

It seems to be standard that connective spectra are "the same" as infinite loop space. However, I do not understand the reason why the associated spectrum is connective.

For me, an infinite loop space is a space $Y_0$ together with a collection of pointed spaces $Y_1, Y_2, \dots$ and homotopy equivalences (or homeomorphisms) $f_j: Y_{j-1} \rightarrow \Omega Y_j$, $j=1, 2, \dots$. Now there is an obvious way to associate an $\Omega$-spectrum $X$ to this data: Just set $$ X_j = \begin{cases} Y_j & j \geq 0 \\ \Omega^j Y_0 & j < 0 \end{cases}$$ for the underlying spaces with the map $g_j: X_{j-1} \rightarrow \Omega X_j$ being given by $f_j$ for $j \geq 0$ and the identity for $j < 0$.

Clearly, there is no reason why $X$ should be connective: Just take $Y_0$ the zero space of a non-connective $\Omega$-spectrum.

So what is the precise meaning of this statement that infinite loop spaces are the same as connective spectra?

I am trying to understand the integer points in the rowspace of the $2t\times (2t+1)$ matrix of form $$\underbrace{\begin{bmatrix} -\frac{b^t}{a^t}&0&0&0&\dots&0&0&0&0&\dots&0&0&1\\ -\frac{b^{t-1}}{a^{t-1}}&0&0&0&\dots&0&0&0&0&\dots&0&1&0\\ -\frac{b^{t-2}}{a^{t-2}}&0&0&0&\dots&0&0&0&0&\dots&1&0&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ -\frac{b^{2}}{a^{2}}&0&0&0&\dots&0&0&0&1&\dots&0&0&0\\ -\frac{b^{1}}{a^{1}}&0&0&0&\dots&0&0&1&0&\dots&0&0&0\\ -\frac{a^{1}}{b^{1}}&0&0&0&\dots&0&1&0&0&\dots&0&0&0\\ -\frac{a^2}{b^2}&0&0&0&\dots&1&0&0&0&\dots&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ -\frac{a^{t-2}}{b^{t-2}}&0&0&1&\dots&0&0&0&0&\dots&0&0&0\\ -\frac{a^{t-1}}{b^{t-1}}&0&1&0&\dots&0&0&0&0&\dots&0&0&0\\ -\underbrace{\frac{a^{t-2}}{b^{t-2}}}_{\text{ col } 1}&\underbrace{1}_{\text{ col } 2}&0&0&\dots&0&\underbrace{0}_{\text{ col } t+1}&\underbrace{0}_{\text{ col } t+2}&0&\dots&0&0&\underbrace{0}_{\text{ col } 2t+1}\\ \end{bmatrix}}_{2t+1}$$

where $a,b$ are very large coprime and $$0<a<(1+\delta)b<(1+\delta)^2a$$ holds for a small non-negative $\delta$.

Consider the row space of the matrix $$\mathcal R=\{u_1R_1+\dots+u_{2t}R_{2t}:u_i\in\Bbb R\}$$ where $R_i$ is row $i$ and its integral subspace $$\mathcal R_\Bbb Z=\{v\in\mathcal R\cap \Bbb Z^{2t+1}:v\neq(0,\dots,0)\}.$$

Fix $j\in\{1,\dots,t\}$.

- Is there a $c_j>0$ such that if $$(v_0,v_1,\dots,v_{2t})\in\mathcal R_\Bbb Z$$ satisfies $$\max_i|v_i|<c_j\cdot \min(a^{2j},b^{2j})$$ then $v_i=0$ if $i\not\in\{0,t-j+1,\dots,t,t+1,\dots,t+j\}$?

This trivially is true at $t=1$ or more generally if $j=t$ holds and the problem is whether it holds if $1\leq j<t$ holds.

To show 1. I think it suffices to show that at every $j\in\{1,\dots,t\}$ there is a $c_j>0$ such that $$2t+1-(2j+1)=codim(\mathcal R_{\Bbb Z,j})$$ holds where $$\mathcal R_{\Bbb Z,j}=\{v\in\mathcal R_\Bbb Z:\max_i|v_i|<c_j\cdot \min(a^{2j},b^{2j})\}$$ is defined. This looks true since the space of solutions in 1. is a valid one and has codimension exactly $2t+1-(2j+1)$ and one expects the codimension to decrease uniformly as $j$ increases.

- Does this hold at even the
**simplest**non-trivial case of $j=1$ at any $t>1$?

Can any one give a reference on the number of periodic solutions of the linear fractional ODE $(-\frac{d^2}{dx^2} )^s u= u$ on $x\in (0, T)$ with $u(0)= u(T)$ and $s\in (0, 1)$.

An example of a particular case of $s$ being a fraction will help.

Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.

*Question 1*: What class of lattices the following property characterizes? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ It follows that $\mu(\hat{0},\hat{1}) = \pm1$.

*Remark*: It is satisfied by any boolean lattice, more generally by the face lattice of any convex polytope (as suggested by John Shareshian), and more generally by any Eulerian lattice.

*Proof*: An Eulerian lattice is a graded lattice $L$ such that for any $a,b \in L$ with $a \le b$, we have $\mu(a,b)= (-1)^{|b|-|a|} $, with $a \to |a|$ the rank function. The result is immediate. $\square$

*Question 2*: Is there a non-Eulerian lattice with the above property on the Möbius function?

**Yes**, see the answer of John Machacek.

As suggested by Sam Hopkins, *for starters*:

*Question 3*: Is there a non-Eulerian *atomistic* lattice with the above property on the Möbius function?

In my research I have stumbled across the following 1st order complex differential equation for smooth functions $\eta:\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{C}-\lbrace0\rbrace$ defined on the circle,
$$i\frac{\partial\eta}{\partial t}+(re^{it}+\varepsilon i)\bar\eta=0$$
where $\varepsilon\in\mathbb{R}_+$ is sufficiently small and *fixed* (so choose $0<\varepsilon<<1$ but maybe not $\varepsilon=\frac{1}{2}$) and $r\in\mathbb{R}_+$ is a nonzero positive real *parameter*. Motivated by some perturbation theory and functional analysis, I believe there must exist a nontrivial solution $\eta$ to this ODE for *at least one* choice of $r$. **What is such an explicit pair $(r,\eta)$?**

This problem arises when studying the asymptotics of punctured $J$-holomorphic curves, and I need the solutions to do what I want to do (when perturbing $J$). Ultimately I desire the set of all such distinct $r$ and the dimension of the kernel of the corresponding differential operators.

$\underline{\text{Attempt}}$ I originally posted this on StackExchange a month ago with many edits but not much luck. My attempt was to Fourier expand $\eta(t)=\sum_{k\in\mathbb{Z}}a_ke^{ikt}$ and obtain the recurrence relation $$-ka_k+r\bar a_{1-k}+\varepsilon i\bar a_{-k}=0$$ I need a specific collection $\lbrace a_k\rbrace\subset\mathbb{C}$ which solves this (for some $r>0$). What I get at the least is $r=\varepsilon i\frac{a_0}{a_1}$ (and subsequently $\frac{a_0}{a_1}$ must be purely complex and nonzero). But there is still a good chance that the coefficients $a_k$ will "blow up" as $k\to\infty$ if not chosen carefully. I've done some manipulations and I cannot parse whether they are helpful or harmful.

Also, this *complex* ODE is equivalent to two coupled *real* ODEs, but I don't think it helps. Decompose $\eta=x+iy$ and attempt to solve the equivalent system:
$$\dot y(t) -r\cdot\cos(t)\cdot x(t) -[\varepsilon +r\cdot\sin(t)]\cdot y(t) = 0$$
$$\dot x(t) -r\cdot\cos(t)\cdot y(t) +[\varepsilon +r\cdot\sin(t)]\cdot x(t) = 0$$
Perhaps there are numerical methods to find "approximate" periodic solutions, or some software to plot $(x,y)$ for various values of $r\in\mathbb{R}_+$?

I ran into Hua's identity without intending to, meaning that I do not have a concrete reference available, and my background is not in Ring Theory.

It is apparent to me that the identity is something of a big deal, but I couldn't find any explanation why. Authors pretty much assume that if you are reading about it, you already have a feeling for how important it is. Can you give me a hint?

Let $I$ be a set and $\mathcal{U}$ be an ultrafilter on $I$. Suppose that $(X_{i}, d_{i})_{i\in I}$ is a family of pointed metric spaces with a distinguished point $e_{i}$ for each $i\in I$. We set $$(X_{i})_{\mathcal{U}}:=\{(x_{i})_{i}\in \prod X_{i}:\sup_{i\in I}d_{i}(x_{i},e_{i})<\infty\}/\sim, $$ where the equivalent equation $\sim$ is defined as: $(x_{i})_{i}\sim (x'_{i})_{i}$ if and only if $\lim_{\mathcal{U}}d_{i}(x_{i},x'_{i})=0$.

A natural metric $d$ is defined on $(X_{i})_{\mathcal{U}}$ as: $d((x_{i})_{\mathcal{U}},(x'_{i})_{\mathcal{U}})=\lim_{\mathcal{U}}d_{i}(x_{i},x'_{i}).$

A canonical embedding $J: (X^{\#}_{i})_{\mathcal{U}}\rightarrow ((X_{i})_{\mathcal{U}})^{\#}$ is defined by $$\langle J((f_{i})_{\mathcal{U}}), (x_{i})_{\mathcal{U}}\rangle=\lim_{\mathcal{U}}f_{i}(x_{i}).$$

Where, for a pointed metric space $X$, $X^{\#}$ denotes the space of all real-valued Lipschitz functions defined on $X$ which vanish at a distinguished point $0$, with Lipschitz norm.

Question: Is the canonical embedding $J$ an into isometry?

Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ are irreducible and of equal degree what is the best technique to factor such polynomials?

Assume $GCD$ of coefficients is $1$ after removing denominator.

Is there a deterministic or a randomized algorithm that is polynomial time in complexity in total degree ($d$), variables ($n$) and number of bits in coefficients ($L$)?

Is there an algorithm that is at least in $O(2^{c\cdot n}(d\cdot L)^c)$ for a fixed real $c$?

Recently I'm reading Donaldson's Geometry of four manifolds. It seems to me that the book requires a lot for background. Additionally, the proof in the book is too sketchy without too much detail. I had a really hard time to digest the content in the book. Do we have other textbook demonstrating the same topic with more detail and background?

I have checked everything "homology of loop spaces"-like, but was not able to find what is $H_*(\Omega^2S^3, \mathbb{Z})$. Therefore I ask you how to compute that?

Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that \begin{equation} UV^\top :=\arg\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{F}, \end{equation} where $USV^\top$ is the SVD of $A^\top B $ and $\mathcal{O}^{r\times r}$ means the set of $r\times r$ orthonormal matrices.

However, if I change the metric from Frobenius norm to operator norm, what is the best orthonormal matrix?

In other words, what's $R$ that attains the minimum of the following? \begin{equation} \min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{op}. \end{equation}

It seems that the two rotation matrices are not the same (for Frobenius and for operator norm). If this is true, what can we say about \begin{equation} \|AUV^\top-B\|_\mathrm{op} \end{equation} and how worse is it compared with the optimal one?

Assuming that x,y,z are positive real numbers and x<=y<=z and x * y * z = 1.

How can I prove that (x+1)*(z+1) > 3?

I know that x has to be less than 1 and z greater than 1 and I have already tried to rearrange that term but this didn't help me.

So do you have any ideas?

Thanks in advance

Assume that $M$ is a complex manifold.

Let $G$ be the group of all (real) smooth diffeomorphisms $\phi$ of $M$ such that $\phi^* (X)$ is a holomorphic vector field for all holomorphic vector fields $X$ on $M$. Is $G$ a finite dimensional Lie group?(With respect to a natural smooth structure on $G$).

In 1963 Grothendieck introduced the algebraic de Rham cohomolog in a letter to Atiyah, later published in the Publications Mathématiques de l'IHES, N°29.

If $X$ is an algebraic scheme over $\mathbb C$ the definition is via the hypercohomology of the complex of Kähler differntial forms: $$H^i_{dR}(X)=\mathbb H^i(X,\Omega_{X/\mathbb C}^{\bullet})$$ If in particular$X$ is affine: $$H^i_{dR}(X)=h^i(\Omega_{X/\mathbb C}^\bullet (X))$$ If moreover $X$ is affine and smooth: $$H^i_{dR}(X)=H^i_{sing}(X(\mathbb C),\mathbb C)$$ so that we get a purely algebraic means of computing the cohomology of the set of closed points $X(\mathbb C)$ of our scheme provided with its transcendental topology (derived from the topology of the metric space $\mathbb C$).

In order to understand this algebraic de Rham cohomology I tried to compute some examples in the non smooth case, in particular for the cusp $V\subset \mathbb A^2_\mathbb C$ given by $y^2=x^3$.

The slightly surprising result is that $H^1_{dR}(V)$ is the one-dimensional complex vector space with basis the class of the differential form $x^2ydx$.

This is in stark contrast with the fact that $H^1_{sing}(V(\mathbb C),\mathbb C)=0$, since $V(\mathbb C)$ is homeomorphic to $\mathbb C$.
I made some very stupid preliminary mistakes: in particular I hadn't realised at the start of my calculation that $d(x^2ydx)=x^2dy\wedge dx=0$, as follows from the equality $2xydy=3x^2dx$ (and wedge-multiplying by $dy$).

In other words $\Omega^2(V)$ was more complicated than I thought: although I knew the formula for, say, the differentials of order one for a hypersurface $X=V(f)\subset \mathbb A^n_\mathbb C $, namely $\Omega^1(X)=\frac { \sum \mathcal O(X)dx_j}{\langle \sum \frac {\partial f}{\partial x_j} dx_j\rangle }$, I realized that I didn't know the algorithm for computing higher order global differentials.

This is the motivation for my

**ACTUAL QUESTION**

Given the subscheme $$X=V(f_1(x),f_2(x),\cdots, f_r(x))\subset \mathbb A^n_\mathbb C \quad (f_j(x)\in \mathbb C[x_1,\cdots, x_n]),$$ how does one compute the space of algebraic global differential forms $\Omega^i_{X/\mathbb C}(X)$ ?

(Ideally I'd love to learn an algorithm which for example would calculate $\Omega^2(V)$ for the cusp above in a picosecond)

**Edit**

The software showed me a related question that my computation for the cusp solves.

I have used this opportunity to write down the details of that computation as an answer to the related question.