dg.differential geometry – What is the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of generalized Smooth Spaces?

Let $G$ and $H$ be Lie Groupoids. We know that there are two notions of equivalences of Lie Groupoids:

  1. Strongly Equivalent Lie Groupoids: (My terminology)

A homomorphism $phi:G rightarrow H$ of Lie Groupoids is called a strong equivalence if there is a Lie groupoid homomorphism $psi:H rightarrow G$ and natural transformation of Lie Groupoid homomorphism $T: phi circ psi Rightarrow id_H$ and $S: psi circ phi rightarrow id_G$. In this case $G$ and $H$ is said to be Strongly Equivalent Lie Groupoids.

  1. Weakly Equivalent or Morita Equivalent Lie Groupoids :

A homomorphism $phi:G rightarrow H$ of Lie Groupoids is called a weak equivalence if it satisfies the following two conditions

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where $H_0$, $H_1$ are object set and morphism set of Lie Groupoid H respectively. Similar meaning holds for symbols $G_0$ and $G_1$. Here Symbols $s$ and $t$ are source and target maps respectively. The notation $pr_1$ is the projection to the first factor from the fibre product. from t. Here the condition (ES) says about Essential Surjectivity and the condition (FF) says about Fully Faithfulness.

One says that two Lie Groupoids $G$ and $H$ are weakly equivalent or Morita Equivalent if there exist weak equivalences $phi:P rightarrow G$ and $phi’:P rightarrow H$ for a third Lie groupoid $P$.

(According to https://ncatlab.org/nlab/show/Lie+groupoid#2CatOfGrpds one motivation for introducing Morita Equivalence is the failure of Axiom of Choice in the category of smooth manifolds )


Now let we replace $G$ and $H$ by categories $G’$ and $H’$ which are categories internal to a category of Generalized smooth spaces (For example, category of Chen Spaces or Category of Difffeological spaces,..etc). For instance, our categories $G’$ , $H’$ can be Path groupoids.

Analogous to the case of Lie Groupoids I can easily define the notion of Strongly Equivalent Categories internal to a category of Generalized Smooth Spaces.

Now if I assume that the Axiom of choice fails also in the category of generalized smooth spaces then it seems reasonable to introduce a notion of Weakly Equivalent or some sort of Morita Equivalent categories internal to a category of generalized smooth spaces.

But it seems that we cannot directly define the notion of Weak Equivalent or Morita Equivalent Categories internal to a category of Generalized Smooth Spaces in an analogous way as we have done for Lie Groupoids. Precisely in the condition of Essential Surjectiveness (ES) we need a notion of surjective submersion but I don’t know the analogue of surjective submersion for generalized smooth spaces

I heard that Morita Equivalence of Lie Groupoids are actually something called “Anaequivalences” between Lie Groupoids.(Though I don’t have much idea about anafunctors and anaequivalences).

So my guess is that the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of Generalized smooth spaces has something to do with anaequivalence between categories internal to a category of generalized smooth spaces. Is it correct?

My Questions is the following:

What is the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of generalized Smooth Spaces?

unity – Does animation have internal “update”-cycle where I can make calculations between steps?

I am new to Unity and not game developer at all, so sorry if this question might be stupid. I have an animation in my project. Is there some points where Unity calculates/recalculates the next position of animated character? Is there some function like for example “OnAnimationSample()”? I want to make calculations between theese points. I didn’t found anything in the documentation and hope that someone more experienced than me know some way to tackle, although the task is quite peculiar.

I am programming a simulation environment for robots. There is a need to take screenshots at certain events happening by means of raycasting (gathering img data for CNN). The robot has various physical specification like for example speed. I don’t want the environment depends on speed of the robot, so I am planning to make calculations and screenshots constatly for example in between two animation “milestones”. That will slow fps and overall game down, but I don’t mind since I need only right data at right step. For example suppose there are exist such points in the Unity, when before taking any movement the Unity calculates the next state of the object being animated. I want to stuff my calculation and such, that next animation step won’t start until all is done.

I hope that explained not so tricky. Say if the explanation is obscure I will try my best again.

wallet – Bitcoin Core internal Electrum Server

I think the preferred technique to use ‘bitcoin core as a wallet backend’ is through block filters, like the new compact block filters (an improvement over bloom filters). The protocol is known as Neutrino and lowers the CPU and storage requirements for your bitcoin core node (an electrumx database takes around 50 GB right now). To create these filters in bitcoin core, you need to launch add the bitcoind option blockfilterindex=1 or blockfilterindex=basic. With these filters, you can filter for blocks that affect the addresses (scriptPubKeys) that belong to your wallet. Several lightning wallets use this technique

This block filtering technique produces some false-positives, so you will fetch blocks that end up not affecting your addresses. Because you are requesting more than just the relevant transactions to your wallet, you end up consuming much more bandwidth than using an electrum server but it has the advantage of being more private.

Neutrino is used by several lightning network wallets:

domain name system – Chrome “Secure DNS” setting in v83 breaks access to internal websites

I have found that if I attempt to access a web based service internal to my home network then Chrome will fail to properly resolve the URL. An example would be if I were to try to browse to my Plex server using its fqdn; plex..com. This is only if the “Secure DNS” setting is enabled. It appears to be directing my web requests to a DNS server outside my network. Rather than taking me to my Plex server web frontend I’ll actually be redirected to a page on my pfsense router with an error warning me of a potential DNS Rebinding attack.

I run my own internal DNS server. It’s a simply Windows 2012r2 server. Everything on my internal network is pointed to this server, which is configured to forward requests to Google’s DNS server for records that it doesn’t know about.

I like the idea of using Secure DNS (or DNS over HTTPS). But I can’t seem to use it for the above reason. I’m wondering though, if I were to configure my internal DNS server to support DNSSEC then that might fix the problem. I’m thinking I’d also need to adjust my Chrome settings to use my internal DNS server for it’s Secure DNS service. Does that all sound correct?

Has anyone else observed this problem?

applications – I Used SD CARD internal storage Having a Problem

I Used SD CARD internal storage Having a Problem

i used sd card internal storage but now i having a problem
i can move app sd card to internal storage and move app from Original storage to sd card internal storage but i cannot run app and manager file from SD Card internal storage
How to fix it?

this photo of problem

enter image description here

help me thanks

security – Proxy local webserver blindly to external port, so internal resources don’t have to be exposed

I have a dedicated webserver running CentOS 7 minimal, I am using HaProxy as my edge server and NginX as my internal application server.

What I would like to do, is set up a small admin area for myself, mostly to administer docker containers. I have several browser based tools running on various local ports which are closed by my firewall. I would like to run httpd on such a blocked port, and proxy it somehow blindly to an exposed port, so that Apache may communicate with internal resources on localhost, but responses are proxied to the exposed port just as requests are proxied to Apache’s port.

Is this possible?

I tried Kali Linux’s httptunnel – but I have realized since I cant just use the server to proxy a single port without a client configured to consume – I guess? In any case when I try:

hts -F localhost:9050 8080

to proxy local 9050 to 8080 (currently open in firewall), I get no response in a browser, just endless loading…

Is there a more obvious way to do this?

P.S. – I would like to not use NginX in any way for this as I want to turn Apache off when I’m not using it, but if that’s the only sensible answer then I accept it.

interaction design – How can I conduct Aha moment analysis through survey result without internal data (retention rate, behavior over time,…)

I’m running my UX pet project to define growth opportunities for a product but I don’t have the company data so I’ve started to conduct a survey to find out Aha-moment.

Now I’m confused because I don’t know how I can define “forever users” and “other users”.

I tried to rate 132 participants on 10 point scale based on their time of use, frequency using, and draw line chart to find out segmentation but I got something like this and I don’t know how to deal with it.

If you have any suggestions, please let me know, I appreciate it.

I've rate 123 users on 10 point scale

What is the upper bound on the number of nodes in a tree with n leaves where each internal node has at least two children?

Is there a way to find the upper bound on the total number of nodes in the tree?

Yes. There is actually a formula for the exact number: see e.g. these notes:

A full $m$-ary tree with $l$ leaves has $n = frac{ml – 1}{m – 1}$ vertices and $i = frac{l-1}{m-1}$ internal vertices.

Here, $m$-ary means that every internal node has between $1$ and $m$ children, and full means that every internal node has exactly $m$ children (the maximum). So in your case, for a full binary tree, just plug in $m = 2$ to these formulas.

By the way, to derive such formulas, you can count the number of vertices in the tree in two ways. First, they are either internal or leaves, so
n = i + l

Second, we can obtain the total number of vertices by adding up the number of children over all nodes, plus the root, so
n = 1 + sum_{v in V} text{children}(v) = 1 + mi,

because each internal node has $m$ children and leaves have no children.
Now set the two equal and we have
i + l = 1 + mi,

from which we can derive the above results.