Given a set of positive integers $A=\{a_1<a_2<...\}\subset \mathbb{N}$ such that $\sum_{n=1}^\infty \frac{1}{a_n}=\infty$. Is it always true that $$\sum_{n=1}^\infty\frac{1}{a_{a_n}}=\infty?$$ At first glance I did not think so but I could not come up with a counter example.

In inverse parabolic problems for pde's (typically heat equation), and especially by the Carleman estimate approach, originaly proposed by Bukhgeim and Klibanov in 1981, when we want to determine the source $f$ from some observations on the solution $y$ we consider
$y(\theta,\cdot)$ as observation with extra observations, where $\theta >0$ is a **non negative** time in the time interval $(0,T)$. For hyperbolic equations we can consider $\theta=0$ leave to extend solutions to $(-T,T)$.

We can not do the same for parabolic equations, and then they mentioned that the case $\theta=0$ still an open problem for **parabolic** problems. Is this only for Carleman estimate method or in general way ?
And can we find any other methods to prove Lipschitz stability in inverse problems ?

How to compute the derivate of $$\frac{\partial{\|XX^T\|_1}}{\partial{X}}$$

where $X$ is matrix and $\|\cdot\|_1$ is the matrix $L_1$ norm.

Can a quiver algebra with acyclic quiver be derived equivalent to a quiver algebra with non-acyclic quiver?

(I moved this question from another thread Derived equivalences of Dyck paths , where the question was originally aksed when I thought the answer was elementary/well known but the question now doesnt fit so good anymore and I started a new thread. For Nakayama algebras the answer should be that acyclic ones can not be derived equivalent to non-acyclic ones as noted by Jeremy Rickard in that thread)

I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate.

Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\mathbb{R}^2$ lies on three or more lines, and no two lines are parallel. The plane will then be separated into $m = {n + 1 \choose 2} + 1$ regions. We can now associate a bipartite graph $G \leq K_{n, m}$ with this arrangement so that vertices of the left half correspond to halfplanes, vertices of the right half correspond to regions, and $xy \in E(G)$ iff the halfplane $x$ contains the region $y$.

Here are my questions:

- How many non-isomorphic graphs $G$ arise from all possible arrangments of $n$ halfplanes (at least asymptotically)? If we denote this quantity as $I(n)$, then one can easily see $I(1) = I(2) = 1$, and $I(3) = 4$. To see the last identity, consider the triangle $T$ formed by the three lines; isomorphism or the arising graphs depends on the number of halfplanes containing $T$ (there could be $0$, $1$, $2$, or $3$).
- Given a graph $G$, how hard is it to recognize it as corresponding to a halfplane arrangement (and, possibly, reconstruct an arrangement giving rise to $G$)?

This questions concerns elliptic complexes and is closely related to Green's operator of elliptic differential operator.

Let $T_f:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator of degree $r$ with coefficients $f \in \Gamma(L(J^r E, F))$, where $J^r E$ is the $r$-th jet bundle of $E$. Given coefficients $f_0$, let $Z^\pm$ be finite-dimensional vector spaces, $T^+: Z^+ \to \Gamma(F)$ and $T^-: \Gamma(E) \to Z^-$ be continuous linear maps such that $$\begin{pmatrix}T_{f_0} & T^+ \\ T^- & 0\end{pmatrix}: \Gamma(E) \times Z^+ \to \Gamma(F) \times Z^- $$ is invertible (for example, we may choose $Z^- = \ker T_{f_0}$ and $Z^+ = \mathrm{coker} \, T_{f_0}$ with $T^\pm$ being the canonical projection/injection). In Theorem II.3.3.3 (p. 157f) of Hamilton's work on the Nash-Moser inverse function theorem it is claimed (and attributed to "standard Fredholm theory") that there exists an open neighborhood $U$ of $f_0$ in $\Gamma(L(J^r E, F))$ such that $$\begin{pmatrix}T_{f} & T^+ \\ T^- & 0\end{pmatrix}$$ is invertible for all $f \in U$.

Side question:

Do you know a reference for the existence of such an open neighborhood $U$?

Real question:

What is the proper generalization of this picture using extended invertible maps to elliptic complexes $\Gamma(E) \overset{T_f}{\to} \Gamma(F) \overset{S_g}{\to} \Gamma(G)$?

The obvious guess is that one goes over to an extended sequence $$\Gamma(E) \times Z^+ \to \Gamma(F) \times H \to \Gamma(G) \times Z^-,$$ which is exact. Is this worked-out somewhere? (I feel like this is a basic statement in elliptic theory but I couldn't find a reference for both questions.)

Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has *finitely many* orbits, and every point stabiliser $G_x$ has *finite* orbits.

Can we always find a permutation $\tau\in\operatorname{Sym}(X)$ (not necessarily of finite order or finite support) such that $H=\langle G,\tau\rangle$ is transitive, while every point stabiliser $H_x$ still has finite orbits?

As noted in this previous question some obvious choice of $\tau$ will not work in general.

Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch surface $\mathbb{F}_n$.

Thus, $X_\overline{\eta}$ admits a morphism to $\mathbb{P}^1$, and this is defined over some finite extension of $K(Z)$. So, we know that, up to a generically finite base change and birational modification of the main component of the fiber product, we get a morphism $f': X' \rightarrow Z'$ that factors as $g': X' \rightarrow Y'$ and $h': Y' \rightarrow Z'$, where $g'$ and $h'$ are both generically $\mathbb{P}^1$-bundles.

My question is the following. Given the setup above, are there cases when we know that the finite base extension is not needed? My naive hopes rely on two facts. First, the morphism $\mathbb{F}_n \rightarrow \mathbb{P}^1$ is defined over $\mathrm{Spec}(\mathbb{Z})$, and so we can base change it to $K(Z)$. Second, if the base $Z$ is a curve, by the theorem of Graber-Harris-Starr we know that $X_\eta$ has a $K(Z)$-point. For instance, is it reasonable to get something in the direction I want if the base is a curve?

**Question**: Let $\omega_k$ be the number of distinct prime divisors of k.
What is the asymptotic growth of $C_n := \sum_{k=1}^n 2^{\omega_k}$?

Thank you for considering this elementary question. Below I give some motivation for this problem and some of my progress.

**Motivation:**
There are a couple. The first is purely number theoretic. Note that because $2^{\omega_k} = \sum_{d | k} |\mu(d)|$, we have $C_n = \sum_{k=1}^n \sum_{d|k} |\mu(d)|,$ and thus $C_n$ is the asymptotic growth of absolute values of the Möbius function (in other contexts this relates to the Prime Number Theorem, see here).

The second motivation comes from geometric group theory. The *commensurability index* of $A, B \leq G$ (all groups) is $[A : A \cap B][B: A \cap B]$. The *full commensurability growth function* assigned to a pair $A \leq B$ is defined to be
$$
C_n(A,B) = \# \{ \Delta \leq B : c(\Delta, A) \leq n \}.
$$
This is a generalization of the subgroup growth function to pairs and it can be infinite in natural settings. The question above is the case $G(\mathbb{Z}) = \mathbb{Z} \leq \mathbb{R} = G(\mathbb{R})$ of the following

**Problem:** Compute the full commensurability growth function for the pairs $G(\mathbb{Z}) \leq G(\mathbb{R})$ where $G$ is a unipotent linear algebraic group.

**Some progress:**
Write $f \preceq g$ if there exists $C > 0$ such that $f(n) \leq C g(C n)$.
In Proposition .3, it is shown that $n (\log(n))^{\log(2)} \leq C_n \preceq n(\log(n))$.

Schanuel conjecture implies this, so likely it is true.

Let $f(x),g(x)$ be polynomials with coefficient in $\mathbb{Z}[i]$.

Assume that for some complex number $x_0$, both $\exp{f(x_0)}$ and $\exp{g(x_0)}$ are algebraic and $f(x_0) g(x_0) \ne 0$.

Must $\frac{f(x_0)}{g(x_0)}$ be rational?

There are no constraints on $x_0$.

In this famous paper by Borel and Harish-Chandra, Arithmetic Subgroups of Algebraic Groups, it is proved that, in characterisitic zero, arithmetic groups are finitely presented. I have an extremely vague idea of how the argument goes, but I was wondering if anything was known about how this would be done algorithmically. To be specific, (following work by Grunewald and Segal) an arithmetic group $\Gamma \leq \mathrm{GL}(n,\mathbb C)$ is said to be *expicitly given* if we have the following data:

- a system $S$ of polynomial equations defining the algebraic group $V(S)= \mathcal G$ (which contains $\Gamma$),
- a known upper bound $k$ for the index $[\mathrm{GL}(n,\mathbb Z)\cap \mathcal G:\Gamma]$, and
- a procedure which given $g \in \mathrm{GL}(n,\mathbb Z)\cap \mathcal G$ will decide if $g \in \Gamma$.

**Then my question is:** If $\Gamma$ is an explicitly given arithmetic group, is it possible to compute the presentation of $\Gamma$ or, even better, to get the orbihedron whose fundamental group is $\Gamma$, which is used in the construction of Borel and Harish-Chandra?

If there is a reference for this, or if this is known to be impossible, that would be awesome!

Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we get $$(f^p)_{xx}(t)=p(p-1)f^{p-2}(t)(f_x(t))^2+pf^{p-1}(t)f_{xx}(t).$$ Since for every $t\in[0,T]$,$f(t)$ and $f_x^2(t)$ are in $C(0,L)$, we have $(f^p)_{xx}(t)$ in $H^2(0,L)$ using above formula. Does this prove the statement?

Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space.

What are some general results on the relationship between $H^i_{\mathrm{et}}(\mathscr{X}, \underline{\mathbf{Z}_\ell})$ and $H^i_{\mathrm{et}}(X, \underline{\mathbf{Z}_\ell})$? By the usual spectral sequence argument, I guess I'm asking for what some general results are about the pushforwards $R^i \pi_* \underline{\mathbf{Z}_\ell}$. If I'm not mistaken, proper base change should tell us that these are (constructible? lcc?) $\ell$-adic sheaves with stalks $H^i_{\mathrm{et}}(\mathscr{X}_x, \underline{\mathbf{Z}_\ell})$.

I'm happy for results that work in significantly less generality, or to know what some interesting conditions on $\mathscr{X}$ are which make this question easier. Conversely, I'd love to hear something that works when $\underline{\mathbf{Z}_\ell}$ is replaced with some other (lcc etc.) $\ell$-adic sheaf.

My motivation for this question came from the case where $\mathscr{X} = [Y/G]$ for $Y$ a smooth projective variety (even a hypersurface) over $\mathbf{C}$ and $G$ a finite cyclic group acting with non-discrete fixed points, and I wanted to compute torsion in the singular cohomology with $\mathbf{Z}$-coefficients.

I've been wanting to ask this question, because I have no insights into the way other mathematicians prepare papers (for eventual publication).

How much are editing, revising, updating, adding to, etc., part of the "normal" process of process of drafting math papers? Specifically, papers of moderate (10-15 pages) length. I've noticed I tend to do this for several months, and the thought has occurred to me that perhaps I'm being too "fussy" and that I'm wasting time.

Let $\mathbb{B} = \langle B, \wedge, \vee, \leq, \neg, 0, 1 \rangle$ be an atomless complete Boolean algebra.

We call $f$ a *splitting function on $\mathbb{B}$* iff

- $f : B-\{1\} \longrightarrow B \times B \ \ \ (b \mapsto (b_0, b_1))$,
- $b_i \leq b$ for all $b \in B-\{1\}$ and $i \in \{0, 1\}$,
- $b_0 \wedge b_1 = 0$ for all $b \in B-\{1\}$.

A splitting function on $\mathbb{B}$ is *monotone* iff

- $b \leq b'$ implies $b_i \leq b_{i}'$, for all $\{b, b'\} \subset B-\{1\}$ and $i \in \{0, 1\}$.

Monotone splitting functions trivially exist: consider eg. the map $b \mapsto (0, b)$. However, it is not so clear to me how non-trivial these can be.

In particular, is there always a monotone splitting function on $\mathbb{B}$ satisfying

- $\sup\{b_{i}' : b \leq b'\} = \neg b_{1-i}$ for all $b \in B-\{1\}$ and $i \in \{0, 1\}$?

If not, what additional conditions on $\mathbb{B}$ can guarantee its existence, and in what cases does it fail to exist?

Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number Theorem to prove that $$\frac{1}{x}\sum_{n \in S:\, n \le x} = \frac{\log^{k-1} \log x}{(k-1)!\log x} (1+o(1)),$$ and he obtains some information on the size of the error term.

I am trying to find what is known, conditionally (on RH) and unconditionally, on the asymptotics of $\sum_{n\in S: x\le n \le x+x^{c}} 1$ as $x$ goes to infinity. Landau's result answers this for $c=1$.

For $k=1$, I am just asking about actual primes in short intervals. RH gives the asymptotics when $c>1/2$, and the work of Huxley gives the asymptotics unconditionally when $c>7/12$.
For $k>1$ all I can find are 'almost everywhere' results. What results are known for *all* intervals when $k>1$?

Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)$.

I do not get the motivation behind the construction given in Kobayashi and Nomizu which I will write down below.

Idea is to construct a fibre bundle with fibre $F$ i.e., we need to construct a smooth manifold $E$ and a smooth map $\pi_E:E\rightarrow M$ that gives a fiber bundle with fiber $F$.

Kobayashi's proof goes as follows :

They consider the product manifold $P\times F$ with an action of $G$ as $g.(u,\xi)=(ug,g^{-1}\xi)$. Then they consider the quotient space $(P\times F)/G$ and call this $E$.

Consider the map projection map $P\times F\rightarrow M$ defined as $(p,\xi)\mapsto \pi(p)$.

This induces a map $\pi_E:E=(P\times F)/G\rightarrow M$. As $P\rightarrow M$ is a principal $G$ bundle given $x\in M$ there exists an open set $U$ containing $x$ and a local trivialization $\pi^{-1}(U)\rightarrow U\times G$. They then give a bijection $\pi_E^{-1}(U)\rightarrow U\times F$ and give a smooth structure on $E$ so that these bijections are difeomorphisms. Then, they cal $(E,\pi_E,M,P,F)$ the fiber bundle associated to principal $G$ bundle.

I am trying to understand the motivation for the above construction.

Suppose $F=H$, a Lie group and the action of $G$ on $H$ is given by a morphism of Lie grroup $\phi:G\rightarrow H$ with $G\times H\rightarrow H$ given by $(g,h)\mapsto \phi(g)^{-1}h$ do get a principal $H$ bundle in above construction?

Edit : I thinnk above content looks like it is asking why do we need the construction of associated fiber bundles. No, what I am asking is, suppose I have a Principal $G$ bundle $P(M,G)$ with an action of $G$ on a manifold $F$ and I want to associate some fiber bundle on $M$ with fiber $F$. Then, **what suggests you to think of above construction**. How does it occur naturally? Are there any other properties of Fiber bundle I should have in mind which suggest this way of construction.

Suppose we have a simultaneous game, that has a strong Nash equilibrium (SNA), i.e. a weak Pareto efficient Nash equilibrium (no deviation of any subset of player brings a benefit to them).

Now suppose we play this game repeatedly. Does the repeated game has a strong Nash equilibrium, too?

I would keep the question about the payoff function of the repeated game as open. Choose whatever adopted payoff works for the answer.

The idea behind the question is as follows: Games that don't have SNA, like prisoner's dilemma, might have those in the repeated scenario, since additional effects like long term strategies come into play.

Based on this, my guess is, that playing the SNA in every game, would also give a SNA in the repeated game.

Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is the quotient of a smooth affine variety to the algebraic action of a finite group LCI? If no, is there any condition on the action that guarantees such a property?

Let $I=[0,1]$ be the unit interval. Suppose we are given a sequence of nonnegative scalars $\{p_i\}_{i=1}^k$ such that $\sum_{i=1}^k p_i=1$, and denote by $p$ the vector that consists of these scalars. Let ${\cal B}(p)$ denote the set of disjoint measurable partitions of $I$ consistent with $p$, i.e., $\{B_i\}_{i=1}^k \in {\cal B} (p)$ if and only if $\cup_{i=1}^k B_i = I$, $B_i\cap B_j=\emptyset$ for $i\neq j$, and for all $i$ we have that $B_i$ is Lebesgue measurable and satisfies $\int_{B_i} dx =p_i$. Consider the set $Z=\left\{z ~|~ z_k = \int_{B_k} x ~ dx ~~\forall k, \mbox{ and } \{B_i\}_{i=1}^k\in {\cal B}(p)\right\}$.

I'm trying to understand the structure of this set $Z$. In particular,

- Is it polyhedral? If so, do the extreme points admit a simple characterization?
- Is it possible to characterize $Z$ by only focusing on $\{B_k\}$ that constitute intervals?
- Is there any reference which studies objects similar to those defined above?