An algebraic foliation chart for a foliated manifold is a foliation chart for which the transition maps are polynomial maps.

What is an example of an analytic foliation of the Euclidean space $\mathbb{R}^n$ which does not admit an algebraic foliation chart? In particular is there an algebraic foliation chart for foliation of the plane tangent to $cos y \partial_x +sin y \partial _y$

Is there a PL version of Urysohn lemma (something that works for PL manifolds) ? What is the precise statement and where can I find a reference ? Thanks

Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space $$X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},$$ endowed with the compact–open topology. I would expect $X$ to be contractible because $U$ is contractible, but I simply know too little about these kind of spaces.

I am looking for a textbook to cover the following areas:

- Tensor products of Hilbert spaces
- Tensor products of operators on Hilbert spaces (i.e., something like $A\otimes B$ is the operator with $(A\otimes B)(\psi\otimes\phi)=A\psi\otimes B\phi$).
- Tensor products of superoperators (i.e., linear functions from operators to operators, e.g., $\mathcal E\otimes\mathcal F$ defined by $(\mathcal E\otimes\mathcal F)(\rho\otimes\sigma)=\mathcal E(\rho)\otimes\mathcal F(\sigma)$ for trace-class operators $\rho,\sigma$).

Background: tensor products of (completely positive trace-preserving) superoperators would correspond to composing quantum channels.

The text should be a mathematically rigorous textbook, and not limited only to finite or separable Hilbert spaces. I have found textbooks covering the first two points to varying degrees, but nothing for the third one.

I need it as a reference for citing in a research paper.

In their paper recently published in the PNAS, Zagier et al demonstrated that

- The Jensen polynomials $J_{\alpha}^{d,n}(X)$ of the Riemann zeta function of degree $d$ and shift $n$ are hyperbolic for each $d\geq 1$ and every sufficiently large positive integer $n$.

On page 1 of the paper, they state that

- if $J_{\alpha}^{d,n}(X)$ is hyperbolic then so is $J_{\alpha}^{d, 0}(X)$,
*since the hyperbolicity property is preserved under differentiation."*

On combining (1) and (2), does it follow that $J_{\alpha}^{d,n}(X)$ is hyperbolic for all integers $d\geq 1$ and $n\geq 0$ ?

Following Positive proportion of logarithmic gaps between consecutive primes let for given $\lambda$, $\alpha$ and for any $x$ all positive the quantities $S^{-}_{\lambda,\alpha}(x):=\#\{p_{n+1}\leqslant x\colon p_{n+1}-p_{n}\leqslant\lambda\log^{\alpha}x\}$ and $S^{+}_{\lambda,\alpha}(x):=\#\{p_{n+1}\leqslant x\colon p_{n+1}-p_{n}\geqslant\lambda\log^{\alpha}x\}$.

Is it presentely known, 6 years after Yitang Zhang's 2013 breakthrough, whether $S_{1,1-\alpha}^{-}(x)\sim S_{1,1+\alpha}^{+}(x)$? If not, is it a consequence of some widely believed conjecture such as Hardy-Littlewood $k$-tuple conjecture?

I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function?

I have been trying for days to find the limit of this sequence, I'm desperately hoping that someone could help find it. Any help is very much appreciated!

Let a and b be natural numbers, with $1\leq a< b$

We define $U_{n}$ by:

$U_{n} = \frac{1}{b+n} * \left [ \frac{1}{a+n} + \sum_{i=0}^{n-1}\left ( U_{i} * \sum_{k=n-i}^{b+n-1} \frac{1}{k} \right ) \right ]$ for $n\geq 1$

$U_{0} = \frac{1}{a*b}$

We want to find $\lim_{n \to +\infty}U_{n}$

Set $\phi (x) = u(x)+iv(x)$, $x=(x_1,...,x_N)$, a $T$-periodic function in $H^1_{loc}(\mathbb{R}^N)$, that is $\phi (x) = \phi (x_1 + T ,..., X_N + T)$ for all $x$ and where $u = Re\ \phi$ and $v = Im \ \phi$. I wonder if $$ \int_{[0,T]^N}\frac{u_{x_1}^2}{2} + \frac{v_{x_1}}{\sqrt{2}}(u-1)^2\ dx > 0$$ for functions satisfying $$ \int_{[0,T]^N} v_{x_1}(u-1)\ dx = \varepsilon > 0$$ for some small and fixed $\varepsilon > 0$. Any idea or help to show this would be appreciate! I am stucked. Thank you in advance.

**Problem:** Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c,2^c,3^c,\ldots,n^c$ are integers. Prove that $c$ is an integer.

**My progress:** At first we will deal with the case when $c$ is a rational number. Suppose $c=\frac{a}{b}$. It indeed suffices to prove the statement for rationals of the form $\frac{1}{a}$. Observe that there are $\lfloor{M^{\frac{1}{a}}}\rfloor$ integers of the form $n^{\frac{1}{a}}$ between $1$ and $M$. So the percentage of integers of the form $n^{\frac{1}{a}}$ among the first $M$ integers is
$$\frac{\lfloor{M^{\frac{1}{a}}}\rfloor}{M}\times 100\le \frac{M^{\frac{1}{a}}}{M}\times 100=\frac{100}{M^{1-\frac{1}{a}}}$$
which will be less than 1 for sufficiently large $M$.

But I am unable to prove the problem for any real $c$. I tried approximating reals with a sequence of rational numbers, but it didn't work well.

I was recently working on an open problem of similar kind, and I stumbled upon this sub-problem. How to solve this one(preferably not requiring too much heavy tool)? Thanks.

Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite dominant dimension ($\geq 2$) equal to the Goreinstein dimension in https://www.sciencedirect.com/science/article/pii/S0001870816315055 . Such algebras with finite global dimension are exactly the higher Auslander algebras.

Questions:

In case a minimal Auslander-Gorenstein algebra has infinite global dimension, does it have Cartan determinant not equal to one?

In case a minimal Auslander-Gorenstein algebra has finite global dimension, does it have Cartan determinant equal to one?

I would think that the first question is false, but I was not able to find a counterexample yet.

Let $E$ be an elliptic curve with good and ordinary reduction at an odd prime $p$. Suppose $E[p]$ denotes the $p$-torsion points of $E$ and $G_{\mathbb{Q}_p} := \text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$.

In the article `Selmer group and congruences (page 6)', Greenberg says that one can characterize $\widetilde{E}[p]$ as the maximal unramified quotient of $E[p]$ for the action of $G_{\mathbb{Q}_p}$ where $\widetilde{E}$ denotes the reduction of $E$ in $\mathbb{F}_p$.

This is so because $p$ is assumed to be odd and therefore the action of the inertia subgroup of $G_{\mathbb{Q}_p}$ on the kernel of the reduction map $\pi: E[p] \longrightarrow \widetilde{E}[p]$ is nontrivial.

It will be every helpful if someone can explain how `$p$ being odd' is playing a role in proving the non trivial action of the inertia subgroup on the kernel of the reduction map $\pi$ ?

Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{R}$ satisfying: $$f(M)=g(\sigma_1(M),...,\sigma_n(M))$$ where $M\in\mathbb{R}^{n\times n}$ and $\sigma_i(M)\geq 0$ are singular values of $M$ and $f(0)=g(0)=0$.

Now assume $g$ is Lipschitz in every coordinate, i.e. there exists constant $L>0$ such that for any $i=1,...,n$: $$|g(x_1,...,x_i,...,x_n)-g(x_1,...,x_i^\prime,...,x_n)|\leq L|x_i-x_i^\prime|$$

My question is that does $f$ have some sort of Lipschitz property? For example, for any $M,M^\prime\in\mathbb{R}^{n\times n}$, does there exist a constant $C>0$, such that $$|f(M)-f(M^\prime)|\leq C\|M-M^\prime\|_F$$

On the other direction, does the Lipschitz of $f$ imply the Lipschitz of $g$?