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?

$S$ is the boundary of a genus $n$ handlebody in $S^3$. $\{m_1, m_2,..., m_n\}$ is the collection of the meridian circles of $S$; $\{l_1,l_2,...,l_n\}$ is the collection of the longitude circles on $S$. They are both unlinks.

Suppose there are 2 orientation preserving self-homeomorphisms of $S$, $f$ and $g$, such that $\{f(m_1),f(m_2),\dots,f(m_n)\}$ is isotopic to $\{g(m_1),g(m_2),\dots g(m_n)\}$ as links (that is, there is an isotopy $I_t:\sqcup_{i=1}^n S^1\to S^3$ such that $I_0(S^1_i)=f(m_i)$, $I_1(S^1_i)=g(m_i))$ and they have the same framings ($S$ induces a framing on each $f(m_i), g(m_j)$ such that the framing of $f(m_i)$ is taken by the isotopy to the framing of $g(m_i)$). Similarly for $f(l_i)$ and $g(l_i)$. What can I say about $f$ and $g$?

When $n=1$, I think $f$ has to be isotopic to $g$. Have no idea about $n>1$. In the simple case when $\{f(m_i)\}$ and $\{f(l_i)\}$ are unlinks with framing $0$, what is $f$? (I think uniqueness of Heegaard splitting implies $f$ is identity modulo some simple self-homeomorphisms, right?)

Thanks.

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 $d$?

It's claimed by Granville-Soundararajan that for all large $q$, the portion of Dirichlet L-functions, whose values at point $s=1$ are large, among all these L-functions of conductor $q$, is very small. On the other hand, about $\frac{L'}{L}(1+it, \chi_q)$ for $t \in \mathbb{R}$, we have trivial bound $O(\log q)$ and optimal bound $O(\log\log q)$ assuming GRH.

So I'm wondering if there were any research on the density of Dirichlet characters $\chi$ mod $q$ such that $\frac{L'}{L}(1+it, \chi) = o(\log q)$, or $O(\frac{\log q}{f(q)})$ for some fixed function $f$ going to infinity as $q$ blows up, however slowly? In addition, has anyone secured a positive percentage of $\chi$'s out of all primitive $\chi$ mod $q$, that are known for sure to have bound $O(\log\log q)$?

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?*

Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$.

Define the map $v_k: E^k \rightarrow \mathbb{R}$ that sends a $k$-uple $x_1,\cdots, x_k$ of vectors of E to the norm of their exterior product $\|x_1\wedge\cdots\wedge x_k\|$.

I'm interested in the measure of the set of vectors of bounded norms having their image by $v_k$ bounded by constant, i.e. to the value of the integral: $$ \int_{(\mathcal{B}_0(r))^k} \mathbb{1}[{v_k}<B] d\mu^{\otimes k}, $$ for any $r, B>0$, and where $\mathcal{B}_0(r)$ is the Euclidean ball of radius $r$ centred on $0$.

My try has been to write the value of $\|x_1\wedge\cdots\wedge x_k\|^2$ as a determinant and develop it by rows to find an inductive formula, the base case ($k=1$) being particularly easy, as being the volume of the Euclidean ball of radius $B$... but the computation becomes hideous.

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function

$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$

where $x_0$ is an arbitrary but fixed point in $(0,\infty).$

**The question is: Can we show that always $g_f \neq f?$**

Numerically, it is very easy to set up this eigenvalue problem and it seems to be true that there is no eigenfunction such that $g_f=f,$ but it is not clear how to conclude this analytically.

**Observations:**

The function $$F_f(y):= \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ seems to be real-analytic such that for $(g_f) =f$ to hold, necessarily $F_f(x_0)=0.$

Thus, we see that $(g_f)(y) = \frac{F_f(y)}{y-x_0}$ must also be (real-)analytic. In particular, if $g_f= f$ then $f$ must be (real)-analytic.

Also, $f$ cannot be all strictly positive/negative, as $\frac{1}{y-x_0}$ changes sign and so does $g_f.$ From the formula it is also follows that $(g_f)(0)=0.$

See also this question to see that there are however indeed $f \in L^2$ that get mapped back to $L^2$ under $g_f.$

In case you do not obtain a full result, but only some partial insights about this fixed-point problem, I very much appreciate any further observations.

Thank you for your time.

Let $U_q(\mathfrak{g})$ be the quantized enveloping algebra of a complex semisimple Lie algebra and let $\mathcal{O}_q(G)$ be the quantized coordinate algebra of the corresponding simply-connected group $G$. It is well known that $\mathcal{O}_q(G)$ is Noetherian, by using $R$-matrices to get certain relations between generators (see for example the book "Lectures on Algebraic Quantum Groups" by Brown and Goodearl, chapter 8).

My question is: is it known whether the integral form of $\mathcal{O}_q(G)$ is also Noetherian? Here by integral form I mean the dual Hopf algebra of Lusztig's integral form of $U_q(\mathfrak{g})$. The only thing I found in that direction is in the Inventiones paper "Representations of quantum algebras." by Andersen, Polo, and Kexin. In Polo's appendix (section 12) it is proved that, in type $A$, the integral form of the quantized coordinate algebra is given by the usual generators and relations for quantum $SL_n$. This immediately implies that it is Noetherian. Does anyone know of similar results in other types?

I am trying to find a non-visual proof, that

$\sigma(mx+n)´ = \frac1{1+e^{-mx-n}}´ = \dfrac{m\mathrm{e}^{mx+b}}{\left(\mathrm{e}^{mx+b}+1\right)^2} $.

is increasing for increasing m. I am not sure how to prove it, it seems obvious.

I have typical 2nd order differential equation: $\nabla^2U + \frac{1}{V}\frac{\partial U}{\partial z} = 0$

Let, $z = r(x,t)$ and $r(x,t) = r_0exp(ikx+\omega t)$.

I would like to substitute $r(x,t)$ into the above differential equation.

What I'm doing now, $\frac{\partial^2 U}{\partial x^2} = \frac{\partial }{\partial x}\frac{\partial U}{\partial x} = \frac{\partial }{\partial r}[ \frac{\partial U }{\partial r}\frac{\partial r}{\partial x}]$. Here I calculate $\frac{\partial r}{\partial x}$ w.r.t $r(x,t)$.

But, I get stuck at calculating $\frac{\partial U}{\partial z}$ and its second derivative. Will it be zero ? since $r(x,t)$ is not a function of $z$.

As the substituted equation in an article contains $\frac{1}{V}\frac{\partial U}{\partial r}$ term

Consider the function f(x) = 10 + 6x − x^2, −∞ < x < 4. Find a linear function, defined in the region 4 (more than or equal to) x < ∞, such that together they form a function which is differentiable (and therefore continuous) at the point x = 4.

I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has to do with the remainder I guess... $$|E[\exp(itX_{n,k})|F_{n,k-1}]-1-\frac{1}{2}t^2E[X_{n,k}^2|F_{n,k-1}]|\\ \leq \frac{1}{6}|t|^3E[|X_{n,k}|^3\mathrm{1}_{|X_{n,k}|\leq \epsilon}\big{|}F_{n,k-1}]+t^2E[X_{n,k}^21_{X_{n,k}>\epsilon}|F_{n,k-1}]$$

Where $E[X_{n,k}|F_{n,k-1}]=0$ for all $k,n \in \mathbb{N}$

Especially this is taken from Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES and can be found in the proof of theorem 2.1 equality (4.4). Thanks in advance.