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 $g$?

Planar graph permanent can be reduced to determinants and so statistics should be amenable.

Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new additional edge on condition that the new bipartite graph $H$ is planar. Denote $f(G)$ and $f(H)$ to number of perfect matchings of each color respectively. The probability distribution of $G$ is different from $H$ since $H$ is no longer picked from uniform distribution.

What is the probability distribution or at least mean and variance of

number of perfect matchings $f(G)$

number of perfect matchings $f(H)$ (note the probability distribution of $H$ is different from $G$)

number of additional perfect matchings $f(H) - f(G)$?

Note number of additional perfect matchings is not a Markov process (it depends on $f(G)$).

Also since number of edges of planar bipartite graphs are bound we do not have much room to draw edges of $H$ in general and this too depends on starting graph $G$ and is not a Markov process.

Let $i: X\rightarrow Y$ be a cofibration between CW-complexes, more precisely a cellular embedding. Let $A$ be a closed subspace of $Y$ and $Z=i^{-1}(A)$. Let $j: Z\rightarrow A$ be the restriction of $i$ to $Z$, such that $A/Z$ is a retract of a CW-complex. Which of the following statement is true ?

- $j$ is a cofibration.
- The cofiber sequence $Z\rightarrow A\rightarrow A/Z$ induces a long exact sequence in homology.

Obviously if (1) is true (2) follows immediately.

For $x, \lambda > 0$, define $$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$ where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the Hardy-Littlewood prime k-tuples conjecture implies that for fixed $\lambda > 0$, $$S_\lambda(x) \sim e^{-\lambda} \frac{x}{\log x}$$ as $x \to +\infty$.

My question is: If we want only a lower bound of the form $$S_\lambda(x) \gg_\lambda \frac{x}{\log x}, \quad x > 2,$$ for every fixed $\lambda > 0$, has this been proved unconditionally?

Thank you for any reference or suggestion.

[1] Funkhouser,Goldston, Ledoan, Distribution of Large Gaps Between Primes, https://doi.org/10.1007/978-3-319-92777-0_3

Given a K3 surface $X$, the cup product defines a non-degenerate even unimodular structure on the lattice $H^2(X,\mathbb{Z})$. Inside this lattice we have the Neron-Severi group $\text{NS}(X)$, which is also a primitive lattice. The rank of $\text{NS}(X)$, denoted by $\rho(X)$, is called the Picard number of $X$. The orthogonal complement of $\text{NS}(X)$ is by definition the transcendental lattice \begin{equation} T(X):=\text{NS}(X)^\perp \subset H^2(X,\mathbb{Z}). \end{equation}

In the note "Arithmetic of K3 surfaces" by Matthias Schutt, the author says that

"If $X$ is defined over some number field, the lattices of algebraic and transcendental cycles give rise to Galois representations of dimension $\rho(X)$ resp. $22-\rho(X)$."

Does he mean that the Galois representation arise from the etale cohomology $H^2_{et}(X,\mathbb{Q}_\ell)$ splits into the direct sum of two sub-representations with dimension $\rho(X)$ (associated to the algebraic cycles) and $22-\rho(X)$ (associated to the transcendental cycles)?

I guess this statement might be true generally for algebraic surfaces. Could anyone explain it more carefully, and give a reference if possible?

Let $G$ and $H$ be two *undirected* graphs of the same order (i.e., they have the same number of vertices). Denote by $A_G$ and $A_H$ the corresponding adjacency matrices. Furthermore, denote by $\bar G$ and $\bar H$ the *complement* graphs of $G$ and $H$, respectively.

When $G$ and $H$ are **cospectral**, and $\bar G$ and $\bar H$ are **cospectral**, it is known (see e.g., Theorem 3 in Van Dam et al. [1]) that there exists an orthogonal matrix $O$ such that $A_G\cdot O=O\cdot A_H$ and furthermore, $O\cdot \mathbf{1}=\mathbf{1}$, where $\mathbf{1}$ denotes the vector consisting of all ones.

Suppose that, *in addition*, $G$ and $H$ have a **common equitable partition**. That is, there exist partitions ${\cal V}=\{V_1,\ldots,V_\ell\}$ of the vertices in $G$ and ${\cal W}=\{W_1,\ldots,W_\ell\}$ of the vertices in $H$ such that (i) $|V_i|=|W_i|$ for all $i=1,\ldots,\ell$; and (ii) $\text{deg}(v,V_j)=\text{deg}(w,W_j)$ for any $v$ in $V_i$ and $w$ in $W_i$, and this for all $i,j=1,\ldots,\ell$.

**Question:**

- What
*extra*structural conditions on the orthogonal matrix $O$, apart from $A_G\cdot O=O\cdot A_H$ and $O\cdot \mathbf{1}=\mathbf{1}$, can be derived when $G$ and $H$ are cospectral, have cospectral complements,**and**have a common equitable partition?

I am particularly interested in showing that one can assume that **$O$ is block structured according to the partitions** involved. That is, if $\mathbf{1}_{V_i}$ and $\mathbf{1}_{W_i}$ denote the indicator vectors of the (common) partitions ${\cal V}$ and ${\cal W}$, respectively, can $O$ be assumed to satisfy
$$
\text{diag}(\mathbf{1}_{V_i})\cdot O=O\cdot \text{diag}(\mathbf{1}_{W_i}),
$$
for $i=1,\ldots,\ell$? Here, $\text{diag}(v)$ for a vector $v$ denotes the diagonal matrix with $v$ on its diagonal.

**UPDATE**

Since my posting, the following came to my attention:

- Cospectral graphs with a common equitable partition
*necessarily*have cospectral complements. So, the latter assumption can be removed. Indeed, graphs with a common equitable partition are easily seen to have the same number of walks of any length. When, in addition the graphs are cospectral, Theorem 3 in [1] and Theorem 1.3.5 in [2] imply that they must have cospectral complements. - Graphs $G$ and $H$ for which there exists an orthogonal matrix $O$ such that (i) $A_GO=OA_H$; (ii) $O\mathbf{1}=\mathbf{1}$; and (iii) $\mathsf{diag}(\mathbf{1}_{V_i})O=O\mathsf{diag}(\mathbf{1}_{W_i})$ for $i=1,\ldots,\ell$, where characterised [3] as
**graphs that are cospectral wrt to the WL$_1$-closure of the adjacency matrices**. Here, the $\mathsf{WL}_1$-closure can be considered to be an extension of the generalized adjacency matrix. It can be inductively defined by means of symbolic expressions $e$, as follows.- basic expressions $e=X$, $e=I$, $e=J$ with $X$ a matrix variable, $I$ identity matrix, $J$ the all-ones matrix (all of the same dimension) are in $\mathsf{WL}_1(X)$;
- if $e_1$ and $e_2$ are expressions in $\mathsf{WL}_1(X)$, then also $e_1+e_2$, $e_1\cdot e_2$, $e_1^*$, and $a\cdot e_1$ for scalars $a\in\mathbb{C}$, are in $\mathsf{WL}_1(X)$;
- if $e$ is an expression in $\mathsf{WL}_1(X)$, then also $\mathsf{diag}(e(X)\mathbf{1})$ is in $\mathsf{WL}_1(X)$.

Then $G$ and $H$ are cospectral wrt $\mathsf{WL}_1(A_G)$ and $\mathsf{WL}_1(A_H)$ when $e(A_G)$ and $e(A_H)$ are cospectral for any expression $e$ in $\mathsf{WL}_1(X)$. (In particular, they will be cospectral wrt their adjacency matrices, adjacency matrices of their complements, Seidel matrix, Laplacian, normalized Laplacian, ...)

It is shown (Lemma 9 in [3]) that when $G$ and $H$ are cospectral wrt $\mathsf{WL}_1(A_G)$ and $\mathsf{WL}_1(A_H)$ **then** $G$ and $H$ are cospectral (trivial) and $G$ and $H$ have a common equitable partition.

**Revised question**
My original question thus asked whether the converse also holds. That is, are any two cospectral graphs with a common equitable partition necessarily cospectral wrt to the WL$_1$-closure of the adjacency matrices? If not, what is a counter example?

As a final remark, Theorem 6.2 in [4] seems to imply that cospectrality and having a common equitable partition is equivalent to the existence of an orthogonal matrix $O$ such that $A_GO=OA_H$, $O=S+T$ for a doubly stochastic matrix $S$, $A_GS=SA_H$, and such that for $i=1,\ldots,\ell$,

- $\mathbf{1}_{V_i}=S\mathbf{1}_{W_i}$, $\mathbf{1}^t_{V_i}S=\mathbf{1}^t_{W_i}$
- $\mathbf{0}=T\mathbf{1}_{W_i}$, $\mathbf{0}=\mathbf{1}^t_{V_i}T$.

Hence, the orthogonal matrix $O$ can be assumed to satisfy $\mathbf{1}_{V_i}=O\mathbf{1}_{W_i}$ for $i=1,\ldots,\ell$. This is a weaker condition than $\mathsf{diag}(\mathbf{1}_{V_i})O=O\mathsf{diag}(\mathbf{1}_{W_i})$.

[1] *Cospectral graphs and the generalized adjacency matrix*, E.R. van Dam, W.H. Haemers, J.H. Koolen. Linear Algebra and its Applications 423 (2007) 33–41. https://doi.org/10.1016/j.laa.2006.07.017

[2] Dragoš M. Cvetković, Peter Rowlinson, and Slobodan Simić. *An Introduction to the Theory of Graph Spectra*. London Mathematical Society Student Texts. Cambridge University Press, 2009. https://doi.org/10.1017/CBO9780511801518.

[3] Mario Thüne. *Eigenvalues of matrices and graphs*. PhD thesis, University of Leipzig, 2012. http://ul.qucosa.de/api/qucosa%3A12068/attachment/ATT-0/

[4] Ada Chan and Chris D. Godsil. *Symmetry and eigenvectors*. In Graph symmetry (Montreal, PQ, 1996), volume 497 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 75–106. Kluwer Acad. Publ., Dordrecht, 1997. https://doi.org/10.1007/978- 94- 015- 8937- 6_3.

Suppose E is an ample vector bundle on a curve. Is there a way to manufacture a (slope)stable vector bundle out of it, without destroying its ampleness & retaining as much information about the original bundle as possible?

Sorry if this question is too vague. I am at present not too familiar with the general theory of stable bundles.

I am not a math guy, just curious about the following question:

For any multivariate real polynomial system of equations with degree of two, is there any standard way to show the existence or non-existence of solutions?

PS1: I understand fixed point theory and groebner basis may help to show the existence. I want to know whether there are more standard ways.

Consider a power series $\sum_{n=0}^{\infty}c_{n}z^{n}$ for which $c_{n}\in\left\{ 0,1\right\}$ for all $n$. One can write this as: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ for some set $V$ of positive integers. I call this the “set-series” of $V$. There is a beautiful theorem due to Otto Szëgo which, for the case of set-series, shows that $\varsigma_{V}\left(z\right)$ is either a rational function whose poles are simple and located at roots of unity, or that $\varsigma_{V}\left(z\right)$ is a transcendental function with the unit circle ($\partial\mathbb{D}$) as a natural boundary.

Natural boundaries generally occur as the result of singularities clustering arbitrarily close to one another. My intuition tells me that in the case where $\varsigma_{V}\left(z\right)$ has a natural boundary (example: $V=\left\{ 2^{n}:n\geq0\right\}$, $V=\left\{ n^{2}:n\geq0\right\}$, etc), the clustering singularities in question are *simple poles*.

I figure a good way to try to see this would be via Padé approximants. The “rigorous” statement of my intution would then be something along the lines of: *for an appropriately chosen sequence of Padé approximants $\left\{ P_{n}\left(z\right)\right\} _{n\geq1}$ of $\varsigma_{V}\left(z\right)$ (where $\varsigma_{V}\left(z\right)$ has a natural boundary on $\partial\mathbb{D})$, for every $\epsilon>0$ and every $\xi\in\partial\mathbb{D}$, there is an $N_{\epsilon,\xi}\geq1$ so that, for all $n\geq N_{\epsilon,\xi}$, any pole $s$ of $P_{n}\left(z\right)$ satisfying $\left|s-\xi\right|<\epsilon$ is necessarily simple*.

With the literature on Padé Approximants appears to be quite extensive (while the literature on natural boundaries appears to be comparatively paltry), I was wondering if anyone knew of anything about this question, or something similar. Insight and/or references would be most appreciated.