How do the balls maximizing the maximal function depend on their centers?

Let $ \mu$ be a finite Borel measure on $ \mathbb R^N$ and let $ f\in L^1(\mu)$ be a non-negative function. Let $ M_\mu f$ denote the maximal function of $ f$ relative to $ \mu$ , i.e. $ (M_\mu f)(x)=0$ if $ \mu(B(x,r))=0$ for some $ r>0$ and $ (M_\mu f)(x) = \sup_{0<r<\infty} \frac{1}{\mu(B(x,r))} \int_{B(x,r)}f \, d\mu$ otherwise. (Here $ B(x,r)$ denotes the open ball of radius $ r$ centered at $ x$ .)

Suppose that $ a>0$ and $ K\subset \mathbb R^N$ is a compact such that $ M_\mu f > a$ on $ K$ . Then for each $ x\in K$ there exists $ r_x>0$ such that $ $ \frac{1}{\mu(B(x,r_x))} \int_{B(x,r_x)}f \, d\mu > a. \tag{1}$ $

Question. Is it possible to choose for each $ x\in K$ the radius $ r_x>0$ in such a way that (1) holds and the mapping $ x\mapsto r_x$ is continuous or at least Borel?

This question is inspired by another recent question about Besicovich type covering theorem.

Must this upper bound on mixing time depend on the minimum stationary probability?

It is known fact that for a finite-state, reversible and ergodic Markov chain with transition matrix $ M$ , the following control on the mixing time holds

$ $ \left( \frac{1}{\gamma_\star – 1}\right)\ln{2} \leq t_{mix} \leq \frac{\ln(4/\pi_\star)}{\gamma_\star}$ $

where $ \pi$ is the stationary distribution $ \pi M = \pi$ , $ $ \pi_\star = \min_i \pi(i),$ $ the mixing time is defined by $ $ t_{mix} = \min \{t \geq 1 : \sup_{\mu} \|\mu M^{t-1} – \pi\|_{TV} \leq 1/4\},$ $ and the absolute spectral gap $ $ \gamma_\star = 1 – \max \{\lambda_2(M), |\lambda_d(M)|\}.$ $

Is the upper bound tight on all the parameters ? Specifically, is the dependence on $ \pi_\star$ generally known to be necessary ?

Does software architecture depend on framework being used?

When I was building my backend API I used Django combined with django-rest-framework (DRF) because it was the easiest and fastest way to create what I needed.

And I haven’t thought about an architecture of it.

And if I understand correctly DRF provides some kind of Model-ViewSet-Serializer pattern.

So my question is: Is making decision what architecture would look like based on framework being used wrong?

Magento 2 create admin form selected option depend on another selected option

I have created an module for admin which have admin form and this form have some fields.I have two select option drop down fields.If I choose first dropdown option value then second drop-down display some values and if I choose another option then value should be different.

I have search on forums but not found any solution.Please help me.

screenshot are attached.

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Does the law of a Feller process depend continuously on the initial condition?

Let $ E$ be a locally compact and separable metric space, and suppose $ X$ is a Feller process with transition function $ P_t$ . To be precise, let $ C_0$ denote the space of continuous functions vanishing at infinity, suppose $ P_tf\in C_0$ whenever $ f\in C_0$ , and $ $ \sup_{x\in E}|P_tf(x)- f(x)|\rightarrow 0$ $ as $ t\rightarrow 0$ .

Let $ P_x$ denote the law of the process started at $ x\in E$ . It is stated on page 625 of this paper that the laws $ \{P_x\}_{x\in E}$ are continuous in the space of cadlag paths with the Skorohod topology. A more general question is very briefly discussed here, however I’ve been to unable to find a reference.

Do you know of a book or paper where this is proved?

Does the radius of spirit guardians depend on the size of the caster?

The spirit guardians spell states:

Spirits float around you up to 15 feet away.

Certain other spells like antilife shell state:

A barrier extends from you in a 10-foot radius and moves with you.

Does this mean the range of spirit guardians is affected by the size of the creature who casts it since they are up to 15 feet away from you? Or is the 15 feet based upon your centre of mass so to speak?

Does probability depend on order?

Let’s say I have a playlist of 20 tracks ordered by artist name. The songs’ titles don’t matter.

1) Rihanna 2) Rihanna 3) Rihanna 4) Rihanna  5) Moby 6) Moby 7) Moby 8) Moby  9) Rammstein 10) Rammstein 11) Rammstein 12) Rammstein  13) Eminem 14) Eminem 15) Eminem 16) Eminem  17) Beyonce 18) Beyonce 19) Beyonce 20) Beyonce 

And then I shuffle that playlist. So now it’s like:

1) Moby 2) Rihanna 3) Beyonce 4) Rihanna 5) Eminem 6) Rammstein 7) Beyonce 8) Rihanna 9) Rammstein 10) Rammstein 11) Eminem 12) Moby 13) Moby 14) Eminem 15) Rammstein 16) Eminem 17) Moby 18) Beyonce 19) Rihanna 20) Beyonce 

And the question is: in which case I have more chances to play 2 or 3 Moby’s track in a row if I would randomly pick a track form a list? In other words: does shuffling a shuffled list gives you more chances to get 2 or more items of a same kind?

Check delivery availability depend on customer location

I’m developing a Magento 2 store, My client have several restaurants across the country and he wants to be sure that the customer is within the delivery range for his/her nearest store (5 km).

The customer should have a map to pin his address, if his is on the range he can place the order, otherwise a message will be displayed that he is out of the delivery zone.

I have found several modules that restricted the delivery but uses only zip codes to check the customer location. I need to do the exact same thing but by letting the customer pin his location on a map.

Here is an example of the functionality https://www.mcdelivery.eg/eg/guest.html

Result of Gmail filters depend on the name of the label?

I want to have three types of “multiple inboxes” in Gmail:

  • Waiting for a reply
  • Mails with TODOs
  • Project messages

Project messages should only be shown if they have a “waiting” or “TODO” label. But they should also not show in the first two lists.

I’ve configured my queries as such:

  • Pane 0: is:_TODO AND NOT is:Projects
  • Pane 1: is:_waiting AND NOT is:Projects
  • Pane 2: is:Projects AND (is:_waiting OR is:_TODO)

The issue is the following:

  • Pane 2 correctly shows messages matching the filter, that is, they have the Projects label and one of the other two
  • Pane 1 correctly shows messages with the _waiting label but not the Projects label
  • Pane 0 shows messages with both the _TODO and the Projects label

I don’t understand why the filter would work for Pane 1 but not for Pane 0.

Manually running the search in Gmail, I also get the wrong results, but only if I use the _TODO label. Once I replace _TODO with _waiting, everything works again.

How come?

I have also tired specifying _todo instead of _TODO, but that doesn’t make any difference.