Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "white"). If any cell, lies in the Von Neumann neighbourhood of a certain cell and also has the same colour as of *that* central cell, it is said to belong to the same cluster as that central cell. Moreover, if any cell belongs to the Moore neighbourhood of a certain cell, but not its Von Neumann neighbourhood, and is of the same as of that cell, it is considered to belong the cluster as of that central cell with a probability $q$.

I wrote a program to plot the "Euler number" graphs, that is, the $\chi(p) \ [=N_B(p)-N_W(p)]$ vs. $p$ graphs, for different values of $q$, where $N_B(p)$ is the number of black clusters and $N_W(p)$ is the number of white clusters, at a probability $p$.

For a $1000\times 1000$ matrix (averaged over $100$ iterations) the Euler number graph's variation with $q$ is as follows:

When $q=0.5$ the middle root of the curve is extremely close to $0.5$.

I plotted the middle roots ($p_0$'s) in another graph:

For $1000\times 1000$ the middle root $p_0$'s variation with $q$ seems to be almost linear. Also, I plotted the same graph for a few more sizes: $125\times 125$, $250\times 250$ and $500\times 500$. I noticed that as system size increases the "middle root" graph gets more and more smooth and linear.

For what it's worth, I also noticed a similar trend (i.e. "site percolation threshold vs. $q$" graphs getting linear and smoother with increasing size) for the (approximate) site percolation thresholds for these finite size lattices.

**Is there any mathematical justification for this trend?**

P.S: Answers addressing *only* the site percolation threshold trend or *only* the $p_0$ trend are also welcome.

What English translations are there of work done by the Italian school of algebraic geometry?

Perhaps I'm being too spoiled here, given that mathematical French, German, Italian are much easier to pick up on the fly than say, mathematical Russian or Japanese, for a native English speaker.

This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?

In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of size $1$ clusters in a random matrix are always greater than the number of sizes $2$ clusters in large $N\times N$ matrices (for any $p\in (0,1)$). From my data files, it also seems that the number of sizes $2$ clusters will be greater than the number of size $3$ clusters for all $p\in (0,1)$. That is, at least for the first few natural numbers $n$, the number of clusters of size $n$ is greater than the number of clusters of size $n+1$. Around the site percolation threshold $p=0.407$ there seem to be some fluctuations, however, still, for the first few natural numbers, the cluster sizes continue showing the above trend.

So, my question basically is: Is it possible to generalize the above trend? If yes, up to which natural number $n$ can it be generalized, and why?

**P.S:**

@SylvainJULIEN made an interesting comment:

This sounds a bit like some kind of graphic Benford's law.

I'm not sure if Benford's law is somehow applicable in this situation. However, I'd be interested to hear if someone has any idea regarding this.

Let $k$ be an infinite perfect field (e.g. I'm happy to assume that $k$ has characteristic $0$. On the other hand, the algebraically closed case is not interesting for this question). The question is happening inside the vector space $k^{21}$.

Let $f$ be a homogeneous polynomial of degree 6 in 21 variables with coefficients in $k$. For $\lambda \in k$, let $S_{f=\lambda}$ be the corresponding level set of $f$ in $k^{21}$, i.e. $S_{f=\lambda}:=\lbrace x\in k^{21}~\vert~f(x)=\lambda \rbrace$.

QUESTION: What kind of conditions can one give on $f$ to ensure that for any $8$-dimensional vector subspace $W<k^{21}$, the set $W\cap \big( \bigcup \limits_{\lambda \in (k^{\times})^2} S_{f=\lambda} \big)$ is non-empty?

I hope that the given parameters $21,6$ and $8$ are not really relevant, but this is what I get in my specific situation. Any comment on how to think about this question is welcome!

When in differential geometry one shows , on a riemannian manifold, that a (unique) connection exists, (Levi Civita connection), is it possible to "lift" that notion to the principal bundle of frames over that manifold?In that case, is that lifted connection unique? (if so, the Yang Mills field would precisely be the LV connection)

Does the bundle of frames always admit a one form, even if the manifold has no metric on it?In that case that would provide a lifting to the LC connection...

Quoting from this blog of Prof Terry

“…as the very useful Sobolev embedding theorem, which allows one to trade regularity for integrability…”

Thats one use of Sobolev spaces. In this context, Fourier and Plancheral methods come very handy, when the corresponding Sobolev space is also a Hilbert space. But that is not always the case…. Only $L^2$ based Sobolev spaces are Hilbert spaces. According to Sobolev emebdding, if the function in $\mathbb{R}^d$ need to be holder continuous, then its gardient needs to be $L^p$ integrable with $p >= d+1$. So for $d >1$, we need $p > 2$, so the associated Sobolev space cannot have a Hilbert space structure. So in this context, we cannot use Fourier Plancheral techniques. “But if” (stress If)… I say, that I can always find a Hilbert space, for any d, (even for cases when d>1), how useful a tool that it would be, in the context of PDE. What would the impact be? Any examples of PDE, on which there would be impact? Appreciate your valuable comment.

The quotient manifold theorem says that

If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$ is a submersion.

I was wondering what happens when the action is not free. My intuition suggests that we get corners, I have in mind this example: The action of $\frac{\mathbb{Z}}{2\mathbb{Z}}$ over $\mathbb{S}^2$ induced by the reflection wrt the $zy$-plane. The quotient manifold obtained is $\mathbb{D}^2$ and the boundary $\partial\mathbb{D}^2 $ can be identified with the fixed points of the action i.e. $\mathbb{S}^2\cap zy \text{-plane}$.

Does anyone know a theorem that covers the non-free case? Where can I read about it?

Suppose $M$ is a smooth connected complete Riemannian manifold of dimension $n\geq 2$. Let $d:M\times M\rightarrow \mathbb{R}^+$ be the distance induced by the Riemannian metric on $M$. For $p\in M$ we set $d_p:=d(p,\cdot)$. We know that $d_p$ is smooth on $M\setminus (C_p\cup\{p\})$, where $C_p$ is the cut locus of $p$, which is a null set according to the Riemannian measure on $M$. Moreover, $d_p$ is regular in any point $q\in M\setminus (C_p\cup\{p\})$, since its gradient at $q$ is the derivative of the unique minimal geodesic at $d_p(q)$ joining $p$ and $q$.

For $R>0$ consider the level set $d_p^{-1}(R)$. Since $R$ does not need to be a regular value of $d_p$, we may not be able to define a normal vector field globally on $d_p^{-1}(R)$. Is there some characterisation of the intersection $C_p\cap d_p^{-1}(R)$, which states that set is "small" maybe in the sense of some $N-2$-dimensional Hausdorff measure or in some topological sense? Or is there some other way to define a unit normal vector field "almost everywhere" on $d_p^{-1}(R)$?

Assume that $f$ is smooth function defined in the unit disk $D: x^2+y^2\le 1$, and consider the integral $$I=\int_D f dxdy=\int_0^1r \int_0^{2\pi} f(re^{it})dt.$$

Then it is clear that for $r\in[0,1]$ there is $t_r\in [0,2\pi]$ so that $$I=2\pi \int_0^1 r f(re^{it_r})dr.$$ My question is, can we choose $t_r$ to depend smoothly on $r$.

Can you solve this problem for me?

*Premise*

Let $K$ be a field of characteristic zero and $f\in K[X_1,\dots,X_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N_i,\nu_i)\}_i$ be the numerical data of a fixed log resolution. The quantity $$ lct_K(f):=\min_{i}\frac{\nu_i}{N_i} $$ does not depend on the choice of the log resolution and it is called the log canonical threshold of $f$ over $K$.

*Questions*

Let $f\in \mathbb{Q}[X_1,\dots,X_m]$. By definition, we have $$ lct_{\mathbb{Q}}(f)\ge lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) \ge lct_{\mathbb{C}}(f_{\mathbb{C}}). $$ On the other hand, from Denef's formula for the motivic Igusa zeta function it follows that for all but finitely many $p$ one has $$ lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) \ge lct_{\mathbb{Q}}(f_{\mathbb{Q}}). $$ This shows that $$ lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) = lct_{\mathbb{Q}}(f_{\mathbb{Q}}) \quad \forall\forall p. $$

**1. Is this equality actually true for all $p$?**

In all the counterexamples I have found in the literature for the validity of Denef's formula for the "bad" primes (in the sense of Denef) one still has $lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) = lct_{\mathbb{Q}}(f_{\mathbb{Q}})$ also for bad primes $p$. Were this not always the case, has anybody a counterexample at hand?

**2. What can we say about the comparison with $lct_{\mathbb{C}}(f_{\mathbb{C}})$?**

The equation $x_1x_2+y_1y_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an *additive divisor problem.* The number of solutions in positive integers is $n(c_2\log^2n+c_1\log n+c_0)+O(n^{1-\delta})$ where $c_i$'s are some explicit arithmetic functions of $n$.

The same tools allow to prove similar formula for the number of solutions of the equation $ax_1x_2+by_1y_2=n$. In this case $c_i$'s will depend on $n,a$ and $b$.

Is it possible to find this result in a literature?

I wonder if the following Kunneth formula for semidirect product is valid $$ H^n(N\rtimes_\phi G;\mathbb{Z}) = \sum_{i+j=n} H^i(G; H^j(N;\mathbb{Z})),$$ where $H^*$ is the group cohomology and $G$ has a proper action on $H^j(N;\mathbb{Z})$ as induced by $\phi$. (For direct product, $G$ has no action on $H^j(N;\mathbb{Z})$ and the above reduces to the standard Kunneth formula.)

https://arxiv.org/abs/math/0406130 only showed above when $N$ has a form $\mathbb{Z}^k$.

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "white"). Cells in the Moore neighbourhood of any central cell and having the same colour as the central cell, are considered to belong to the same ("black" or "white") *cluster* as that of the central cell.

**To be more formal**:

Define a *cluster* of "black" cells as a maximal connected component in the
graph of cells with the colour "black", where edges connect cells
whose rows and columns both differ by at most $1$ (so up to eight
neighbours for each cell). Define a *cluster* of "white" cells in a similar

fashion.

I wrote a program for this situation (for a $1000\times 1000$ matrix) and found the cluster size distributions, that is, like (say) at $p=0.40$, the number of "black" clusters of size $1$ is $a_1$, the number of "black" clusters of size $2$ is $a_2$, and so on (averaged over $100$ iterations).

Now, interestingly, I found that $\forall p\in (0,1)$, for a matrix of size $1000\times 1000$ the number of clusters of size $1$ is always the greatest (when averaged over $100$ iterations). Is this by fluke or is there a mathematical proof for why this is true? Also, will the result that "number of black clusters of size $1$ is always the greatest for any $p\in (0,1)$", even in the limit $N\to \infty$?

**P.S:** By "a cluster of size $1$" I mean a cluster having a single cell; by "a cluster of size $2$" I mean a cluster having two cells, and so on.

**N.B:** All the data files and plots can be found here.

Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$ with initial data $u(0,\cdot) = u_0$. Fix $\alpha >0$. How can I estimate (without using explicitly the heat kernel) $$\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2 \ dx,$$ in terms of the initial data? Could you point out a reference where such an estimate is obtained?

Is it fair to call what we obtain a decay estimate?

In the textbook https://www.springer.com/gp/book/9783034851688 (Klassische elementare Analysis, by M. Koecher) the following elegant recurrence relation is proved for $\zeta(2n)$ (on p. 157):
$$\left(n+\frac{1}{2}\right)\zeta(2n)=\sum\limits_{m=1}^{n-1}\zeta(2m)\,\zeta(2n-2m). \tag{1}$$

In fact (1) is equivalent to Euler's recurrence relation for Bernoulli numbers (independently found by Ramanujan)
$$(2n+1)B_{2n}=-\sum\limits_{m=1}^{n-1}\binom{2n}{2m}B_{2m}\,B_{2n-2m}. \tag{2}$$
Why, In contrast to (2), (1) can seldom be found in the literature (I was able to find only https://link.springer.com/article/10.1007/s00591-007-0022-2 that mentions (1))? Are there any other references that discuss (1)?

**P.S. In addition to juan's answer.**
G.T. Williams was not the first to state the result in this form. It can be found at least in N. Nielsen, Handbuch der theorie der gammafunktion, Leipzig:
Druck und Verlag von B.G. Teubner, 1906, p. 49. I found this reference thanks to the paper "Some identities involving the Riemann zeta function. II." by R. Sitaramachandrarao and B. Davis, Indian J. Pure Appl. Math. 17(10):1175–1186, 1986. https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a50_1175.pdf This reference also has (1) and proves (among others) an interesting generalization of (1):
$$4\sum\limits_{i+j+k=n}\zeta(2i)\zeta(2j)\zeta(2k)=(n+1)(2n+1)\zeta(2n)-6\zeta(2)\zeta(2n-2),$$ where $n\ge 3$ and the sum extends over all ordered triples $(i,j,k)$ of positive integers satisfying $i+j+k=n$.

Let

$$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$

where $A$ is a given $n \times m$ matrix (where $m \gg n$),

$$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$

and $I_n$ is the $n \times n$ identity matrix. $\lambda_1, \lambda_2, \dots, \lambda_m$ and $\gamma$ are independent random variables with a gamma distribution. I would like to compute the expected value $\mathbb E(R^{-1})$.

Any idea on how to approximate this? I am not able to find any analytical solution for this. Thanks.

In equation 6 of Computing Persistent Homology (page 8), the authors put forward the following identity: $$\deg \hat{e_i}+\deg M_k (i,j)=\deg e_j$$ Where $\hat{e_i}$ and $e_j$ are elements of homogeneous bases for the space of (k-1)-chains and k-chains, respectively, and $M_k$ is a matrix representation of $\partial_k$ with respect to these bases.

This identity is intuitive and makes it easy to come up with matrix representations for the boundary operators. However, it's the only part of the paper that I've never felt fully comfortable with. The authors treat this identity as obvious, saying that the reader may verify it "using this example as intuition" (referring to an example introduced at the beginning of the paper). I'm not sure if this means verify that the identity holds in *the case* of the example or verify that it holds in general. I'm not sure how to approach the latter.

Is there an example of a metric space $(X,d)$ whose corresponding path metric, $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$?

Let $M$ be the Banach algebra of measures on the circle with $L_1$ naturally sitting as a closed ideal of $M$. Then $M$ carries the strict topology implemented by the family of seminorms $\|\mu\|_f = \|f\ast \mu\|$ where $f\in L_1$.

*Let $F\colon M\to C(X_M)$ be the Gelfand (Fourier-Stieltjes) transform, where $X_M$ is the maximal ideal space of $M$. Is $F$ strictly-to-norm continous?*

Edit: Of course, the answer is negative as explained in the comments. I would be then interested in the following follow-up question.

*Is there a measure that cannot be approximated by invertible measures in the strict topology?*