## Textbook recommendations: Weakly almost periodic functions

I am currently studying Ergodic Theory from Glasner’s book – in it, weakly almost periodic functions play a large role, as well as general “means” and unitary representations of groups on Hilbert spaces.

I cannot seem to grasp the motivation or intuition behind these notions. What text would be best for me to get a better feel for these objects?

## Keyboard shortcut (almost) never works for Automator service

I have bash script that creates a screenshot in a given directory saved as an Automator service. I’ve then assigned ⇪⌘2 from System Preferences > Keyboard > Shortcuts > Services. The problem is that it doesn’t work much at all. The only place I’ve found were it actually runs when using the macro is (how random) when click on the urlbar of Safari, and even then not without inconveniences, as it popups an error message saying “The “Take Screenshot” service could not be used because the “Take Reference Screenshot” workflow did not provide valid data. – Try running the “Take Reference Screenshot” workflow in Automator.“, even though the process works fine and the screenshot is indeed captured and saved as intended. What’s wrong?

fdate=date screencapture "/Users/username/Unorganized/Studio/$fdate.png"  ## Primary submodules over almost Dedekind domains An integral domain $$R$$ is an almost Dedekind domain if for each maximal ideal $$m$$ of $$R$$, the ring $$R_m$$ is a Dedekind domain, where $$R_m$$ is the localization of $$R$$ at $$m$$. Question: Let $$M$$ be an $$R$$-module, where $$R$$ is an almost Dedekind domain and let $$m$$ be a maximal ideal of $$R$$ and there exists $$x\in M$$ such that $$m$$ is a minimal prime ideal over $$Ann_R(x)$$, where $$Ann_R(x):=\{r\in R\mid rx=0_M\}$$. How can we construct an $$m$$-primary submodule of $$M$$? ## DNS NS record not working after almost 24 hours, but A record changes very quickly I just pointed my domain to a new web server. I use Cloudflare for DNS. I waited a while to get the Name Server to work, but it was taking longer than I have experienced in the past. I then put an A record in Cloudflare to point directly to the server and it worked almost instantaneously. Is having the A record there stopping the name server from working? I removed the A record, and then it could not resolve the domain. I checked through NSLookup and the new nameservers are there, but for some reason doesn’t point to the web server. I had another site on the same server to test using a dot tk domain, using the same nameserver, and it worked within 30 minutes. I have waited a day, and it still isn’t working without the A record. Should I remove the A record and wait? ## can you shrink db when dbo DoscStreams is almost full? I am not an expert in SharePoint server maintenance and i received an notice that one of the database has reached it 93% usage of disk space. i know from research that dbo.DocStreams contains the binary content of documents on your Site Collections within the content database. Now my suggestions is for the users to clean up the documents in the doc library or increase the disk space. but db team is against increasing it. what options do i have? do i shrink the db? would that have any effects on the documents in the site collection? ## Does there exist a continuous nonconstant function$f$that maps almost all irrationals to rationals? [on hold] Let $$f$$ be a continuous nonconstant function on the reals. Could it map almost all irrationals to rationals? This is impossible if $$f$$ maps all irrationals to rationals, by a well known result. This is impossible if $$f$$ preserves measure 0 sets, because then $$\{f(x)| f(x)\textrm{ is irrational}\}$$ has measure $$0$$. ## Invariance under tame almost complex structure of the fibre tangent space of the symplectic normal bundle I am trying to understand the construction of symplectic inflation and I am stuck in the following point. Suppose we have a 4 dimensional symplectic manifold $$(M, \omega)$$. Also suppose that $$N \subset M$$ is a 2 dimensional symplectic submanifold. Let $$J$$ be a tame (with respect to $$\omega$$) almost complex structure. Further we are given that $$N$$ is $$J$$-holomorphic for the above $$J$$. Let $$S(N)$$ denote the symplectic normal bundle of $$N$$. Given a point $$p$$ on the the intersection of a fibre $$F_p$$ of $$S(N)$$ and the zero section are the following statements true? 1)The tangent space $$T_p F_p$$ is invariant under $$J$$. 2)Let $$w=(u,v) \in T_pF_p \oplus T_p N$$ and $$w^\prime= (u^\prime,v^\prime)\in T_pF_p \oplus T_p N$$. Then $$\omega_p((u,v),(u^\prime,v^\prime)) = u^T J_p v + {u^\prime}^T J_p v^\prime$$. I can see that these statements should be true when $$J$$ is compatible with $$\omega$$, but I’m unable to show them for a tame almost complex structure. ## Is there a$2 $dimensional foliation$F$of a 4 dimensional almost complex manifold such that$F$and$JF$have intersecting compact leaves? Before we ask our question we present our motivation for this question; Motivation: Obviously the following situation is impossible: A planar vector fields $$P\partial_x+Q\partial_y$$ possess a closed orbit $$\gamma_1$$ and its rotated vector field $$Q\partial_x-P\partial_y$$ possess a closed orbits $$\gamma_2$$ such that $$\gamma_1$$ and $$\gamma_2$$ have non empty intersection. Question: What is an example of a $$4$$ dimensional almost complex manifold $$(M, J)$$, with 2 dimensional foliations $$F_1, F_2$$, whose tangent spaces are $$J$$-related and $$F_i$$ possess a compact leaf $$L_i$$ such that $$L_1$$ has non trivial intersection with $$L_2$$? Is there an example of such situation with extra condition that each $$L_i$$ has non trivial holonomy? ## Taylor’s theorem for a composition with$\min:\mathbb R^2\to\mathbb R\$ and differentiability Lebesgue almost everywhere

Let

• $$f\in C^3(\mathbb R)$$ with $$f>0$$
• $$g:=\ln f$$ (and assume $$g’$$ is Lipschitz continuous)
• $$n\in\mathbb N$$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(x,y):=\min\left(1,e^{s(x,\:y)}\right)$$ for $$x,y\in\mathbb R^n$$
• $$x\in\mathbb R^n$$ and $$Y$$ be a $$\mathbb R^n$$-valued normally distributed random variable on a probability space $$(\Omega,\mathcal A,\operatorname P)$$ with mean vector $$x$$ and covariance matrix $$\sigma I_n$$ for some $$\sigma>0$$ ($$I_n$$ denoting the $$n\times n$$ identity matrix)

I want to make the following argumentation rigorous: By Taylor’s theorem, $$$$\begin{split}h(x,Y)-h(x,(x_1,Y_2,\ldots,Y_n))&=\frac{\partial h}{\partial y_1}(x,(x_1,Y_2,\ldots,Y_n))(Y_1-x_1)\&=\frac12\frac{\partial^2h}{\partial y_1^2}(x,(Z_1,Y_2,\ldots,Y_n))(Y_1-X_1)^2\end{split}\tag1$$$$ for some real-valued random variable $$Z_1$$ with $$Z_1\in[\min(x_1,Y_1),\max(x_1,Y_1)]$$. Thus, $$$$\begin{split}\left.\operatorname E\left[h(x,(y_1,Y_2,\ldots,Y_n))\right]\right|_{y_1\:=\:Y_1}&=\operatorname E\left[\min\left(1,e^A\right)\right]+g'(x_1)\operatorname E\left[1_{\left\{\:A\:<\:0\:\right\}}e^A\right](Y_1-x_1)\&+\frac12(g”(Z_1)+\left|g'(Z_1)\right|^2)\left.\operatorname E\left[1_{\left\{\:B\:<\:0\:\right\}}e^B\right]\right|_{z_1\:=\:Z_1}(Y_1-x_1)^2.\end{split}\tag2$$$$ Above, I wrote $$A:=\sum_{i=2}^n(g(Y_i)-g(x_i))$$ and $$B:=g(z_1)-g(x_1)+\sum_{i=2}^n(g(Y_i)-g(x_i))$$ in order to make the equation more readable (you need to replace them where they occur).

Question 1: There are two issues: The first one is that $$(x,y)\mapsto\min(x,y)$$ is partially differentiable in both arguments except on the diagonal $$\Delta_2:=\left\{(x,y)\in\mathbb R^2:x=y\right\}$$. Are we able to conclude the existence of $$Z_1$$ anyway? Note that $$\frac{\partial h}{\partial y_1}(x,y)=\begin{cases}\displaystyle g'(y_1)e^{s(x,\:y)}&\text{, if }s(x,y)<0\0&\text{, if }s(x,y)>0\end{cases}\tag3$$ and $$\frac{\partial^2h}{\partial y_1^2}(x,y)=\begin{cases}\displaystyle(g”(y_1)+|g'(y_1)|^2)e^{s(x,\:y)}&\text{, if }s(x,y)<0\0&\text{, if }s(x,y)>0\end{cases}\tag4$$ for all $$y\in\mathbb R^n$$.

Question 2: The second issue is the case $$s(x,y)=0$$. In order for $$(3)$$ to hold, we need to show that the probability of the corresponding event is $$0$$ (this seems to be related to the question whether the set on which the occurring function is not differentiable has Lebesgue measure $$0$$; and it’s clear that $$\Delta$$ has Lebesgue measure $$0$$). How can we do that?

While it’s clear that $$h$$ is partially differentiable with respect to the second variable except on a countable set, it is not clear to me why $$h$$ is even twice differentiable with respect to the second variable except on a set (at least) of Lebesgue measure $$0$$ (see this related question).

## When is an almost Hermitian manifold is almost Kähler?

Let $$(M, J,h)$$ be an almost Hermitian manifold, where $$J$$ is an almost complex structure and $$h$$ is a Hermitian metric. Let $$D$$ be the unique $$h$$-connection compatible with $$J$$, i.e. $$Dh=0=DJ$$. Let $$\tau$$ be the torsion of $$D$$.If we decompose our connection $$D$$ into (1,0) and (0,1) part: $$D=D’+D”$$. Then the tortion $$\tau$$ of $$D$$ will also be decomposed $$\tau=\tau’+\tau”$$. It is not hard to see that $$\tau’=N$$, the Nijenhuis tensor for $$J$$, which is eactly the obstruction for an integral complex structure. What about the other part $$\tau”$$? Is there any geometric meaning?

So far, I find that it will be an obstruction for being alomst Kähler, i.e. $$d\omega=0$$, where $$\Im h:=\omega$$. I mean, the following holds: $$d\omega=0 \Rightarrow \tau”=0$$.

My question is about the converse. Is that true $$\tau”=0 \Rightarrow d\omega=0$$?

BTW, if $$M$$ itself a Hemrmitian manifold. It is well-known that $$\tau”=\tau$$, which will be the exact obstruction for being Kähler.