Is a normal covering of the total space of a principal bundle also a principal bundle?

Question: Let $G\to E \to B$ be a principal $G$-bundle over $B$. Take a normal covering $\bar{E}$ of $E$. Does $\bar{E}$ admit a principal bundle structure? Namely, $\bar{G}\to \bar{E}\to \bar{B}$, such that $\bar{G}$ and $\bar{B}$ are normal coverings of $G,B$ respectively.
Comment: Do you mean a principal bundle over the same base, or...?
Comment: @abx I have edited my question just now.
Answer: Without any loss of generality, we may assume that all spaces $B$, $E$ and $\bar{E}$ are connected. Let us first consider that case of $\bar{E}$ being the universal cover $\tilde{E}$ of $E$ (of course I am assuming that spaces are nice enough to admit universal covers). We want to establish the existence of a regular cover $\bar{B}\rightarrow B$ fitting in the commutative diagram
$\require{AMScd}$
\begin{CD}
\tilde{E} @>>> E\\
@V V V @VV  V\\
\bar{B} @>>> B
\end{CD}
in which the columns are principal bundles. Denote the transformation of $E$ which
$g\in G$ induces by $\theta_g:E\rightarrow E$. Let $G'$ be the set of all self-homeomorphisms of $\tilde{E}$ that fit into the a commutative diagram of the form
\begin{CD}
\tilde{E} @>>> \tilde{E}\\
@V V V @VV  V\\
E @>>\theta_g> E
\end{CD}
for some $g\in G$. It is not hard to show that $G'$ is a group of self-homeomorphisms of $\tilde{E}$ acting freely on $\tilde{E}$, and $\tilde{E}/G'$ could be identified with $E/G$; see the related posts:

https://math.stackexchange.com/questions/1667068/on-lifting-an-action-of-g-on-x-to-an-action-of-g-on-tildex
https://math.stackexchange.com/questions/1672763/is-the-quotient-x-g-homeomorphic-to-tildex-g

The group $G'$ is an extension of the discrete group ${\rm{Deck}}(\tilde{E}/E)\cong\pi_1(E)$ by $G$: The group of deck transformations of the universal cover $\tilde{E}\rightarrow E$ is precisely the normal subgroup formed by those elements of $G'$ that lie above $\theta_{{\rm{id}}_G}={\rm{id}}_E$. Any two different lifts of a transformation $\theta_g$ in the previous diagram differ by a deck transformation. Hence the quotient $G'\big/{\rm{Deck}}(\tilde{E}/E)$ may be identified with $G$, and $G'\rightarrow G$ is therefore a normal covering with ${\rm{Deck}}(\tilde{E}/E)$ as its fiber.
In this setting where $\bar{E}=\tilde{E}$, one may take $\bar{B}$ to be $B$ itself, and $\bar{G}$ to be $G'$. In that case, we have a fibration
$$\bar{E}=\tilde{E}\rightarrow \tilde{E}/G'\cong E/G\cong B=\bar{B}$$
which is the quotient map for the free action on $\tilde{E}$ of the normal cover $\bar{G}=G'$ of $G$.
Now let us take an arbitrary normal connected cover $\bar{E}\rightarrow E$. This may be regarded to be an intermediate cover of
$\tilde{E}\rightarrow E=\tilde{E}\big/{\rm{Deck}}(\tilde{E}/E)$; that is, to be in the form of $\bar{E}=\tilde{E}\big/H\rightarrow E=\tilde{E}\big/{\rm{Deck}}(\tilde{E}/E)$ where $H\unlhd{\rm{Deck}}(\tilde{E}/E)$. A lift $\bar{E}\rightarrow\bar{B}$ must be in the form of
\begin{CD}
\bar{E}=\tilde{E}\big/H @>>> E=\tilde{E}\big/{\rm{Deck}}(\tilde{E}/E)\\
@V V V @VV  V\\
\bar{B}=\tilde{E}\big/N @>>> B=\tilde{E}\big/G'
\end{CD}
where $N$ is a normal subgroup of $G'$ containing $H$ as a normal subgroup with $G'/N$ discret. In that situation the columns of the diagram above are principal bundles with structure groups $N/H$ and $G'\big/{\rm{Deck}}(\tilde{E}/E)=G$. So the problem seems to group-theoretic: If $H\unlhd{\rm{Deck}}(\tilde{E}/E)\unlhd G'$, does there exist a subgroup $N$ with $H\unlhd N\unlhd G'$ and $G'/N$ discrete? The key point is that $H\unlhd{\rm{Deck}}(\tilde{E}/E)\unlhd G'$ does not imply $H\unlhd G'$; otherwise, one could take $N$ to be $G'$ itself. Recall a basic fact from group theory: If $H$ happens to be a characteristic subgroup of
${\rm{Deck}}(\tilde{E}/E)$, then we do know that $H\unlhd G'$. So there are definitely cases that the answer is positive; but in general, more information on the extension
$$
1\rightarrow{\rm{Deck}}(\tilde{E}/E)\rightarrow G'\rightarrow G\rightarrow 1
$$
is required.


Syncing two cells in Google sheets

Question: Is there a way to sync two cells on google sheets? So if I have a formula for import data function from another sheet in a column, then I add notes about it in the second column, I would like the second column to auto erase if what is in column 1 changes.
Comment: Welcome to [Web Applications](/tour)  Sheets functions can control what is *displayed*, in a cell but cannot *delete* cell contents.  Consider [editing your question](/posts/175125/edit) to include more details about what you are trying to do and why (see [The XY Problem](https://xyproblem.info/)) and also search this site for other similar questions that may provide the answer you need. **Example search: ["\[google-sheets\] clear"](https://webapps.stackexchange.com/search?tab=&q=%5Bgoogle-sheets%5D+clear "webapps.stackexchange.com/search?tab=&q=[google-sheets]+clear")**
Comment: Agree that it seems like a XY problem. There maybe better alternatives here.
Answer: You might use Google Apps Script; more specifically, you might create an installable trigger. There are other options like using low-code, no-code services like IFTTT, Zapier among many more.
To create a Google Apps Script project bound to your spreadsheet. Start by reading https://developers.google.com/apps-script/guides/sheets. From this, you will learn that you might do the following:

On your spreadsheet, click the menu Extensions > Apps Script.
A new project will be shown if there is still a need for a bounded Google Apps Script project.
Tip: Set a project title to prevent having https://script.google.com full of projects named "Untitled project".
Write a JavaScript function using Class SpreadsheetApp.
Create an installable trigger.
See https://developers.google.com/apps-script/guides/triggers/installable

Ideally, you should start by writing the algorithm—the instructions the Google Apps Script engine should follow to execute the changes you want to be made automatically.
It would help if you decided how the JavaScript function will be run during the algorithm creation. One option is to use an on-change trigger or a time-driven trigger.
Note: Using an AI like ChatGPT might only be effective if you are familiar with Google Sheets, Google Apps Script and JavaScript. It might be possible to use ChatGPT to start from zero, but you should be aware that this might be time-consuming, especially for those who aren't familiar with prompt engineering.

Syncing two cells in Google sheets


First person, third person

Question: So I'm writing a novel myself and the main character is called Jasper.
Whenever he falls asleep, he goes into the dream world type thing. Same dream, same characters, same scenes. When he falls asleep, he wakes up in the real world.
But... but when Jasper's in the real world I write him in third person and when he's in the dream I write him in first person. When he's dreaming, it starts randomly in through a chapter but at the end of the chapter he normally wakes up from the dream.
Any tips writing this?
Comment: Hi Lauren! Welcome to Writing.SE! Please take a look at our [tour] and [help] pages, you might find them helpful. I am unsure I understand what it is you're asking. You're saying you're writing part of your novel in first person, and part in third. What is it you want to know? Whether this is done, or better avoided? It would be helpful if you could [edit] your question to make it clearer.
Answer: Your idea is quite creative. You better stick to third person narrative so that your narration goes smooth and you can even differentiate between dream and real life easily..
Writing in third person also gives you the liberty to explain thoughts of more than one person to the readers while writing in first person restricts you to only Protagonist's PoV. Make sure you create a good and interesting conflict.
HAPPY WRITING!
Answer: There's a narrative style called an epistolary novel that can have multiple 1st-person narrators. The story is presented as a bundle of letters copied verbatim – copying letters word-for-word was like sharing a social media post in 18th Century. No one got confused because (generally) it was explained at the start that the next part is copied from another letter.
Les Liaisons dangereuses and Frankenstein are both epistolary. In Frankenstein it's a letter within a letter (within a letter). Les Liaisons dangereuses works like correspondence in chronological order, with some additional letters from minor characters to fill plot holes.
Your dreamstate POV can be introduced as a dream diary, essentially an epistolary story within the story. The protagonist records everything he can remember from his dreams, starting at a random point already underway and narrating up to when he woke up. His memories of the dream quickly fade to vague impressions soon after he is awake (and sometimes he goes back to sleep), so he started keeping the diary just to confirm the dreams are repeating, and later to piece the longer dream-story together. He's trained himself to write his impressions of the dream as soon as he wakes from it, and doesn't always remember writing in the diary the next day.
Once you have established that the diary is written in 1st-person and "real life" is in 3rd-person (by having the protagonist read from his diary), you can then jump between the two narrative voices without any transition.

First person, third person


How to zoom in and out with opengl?

Question: So I am trying to teach myself opengl and I cant figure out how to zoom in and out. I tried a few different examples and they are not panning out. What I need is the very basics like zooming in and out of a centered cube. i know this is done using some kind of black magic and the glOrtho function but i cant figure in out. Thanks in advance.
Comment: You can either change the field of view (to a narrow FOV centered on the target) or move the object and the camera closer together.  The first will look more like a zoom, the second will make it closer, thus 'zoomed in'.
Answer: If you are using Orthographic projection then your object looks the same size irrespective of your camera position. So here if you want to zoom you should modify the viewing volume by adjusting the parameters to the glOrtho. Increasing the volume will make the object look smaller and decreasing the volume will make the object look bigger. This is more a natural way. I have seen some people use the glScale too but the previous one is more natural in my opinion.
If you want to do zooming in perspective projection then @David M suggestion of modifying the FOV is fine.

How to zoom in and out with opengl?


Is there a generic framework/design pattern/library for real time replication of MySQL database onto other servers?

Question: Summary
I need to backup in real time updates/inserts to a MySQL database. Cron/batch jobs appear to be unsuitable for this particular requirement due to the time interval meaning that the data would only be as recent as the last cron job date/time.
Background
I have three LAMP (Linux, Apache, MySQL, PHP/Perl) servers on three locations each issuing small amounts of data regularly requested by client applications on other machines. Each server logs the data that it has issued. Each server holds configuration data that determines the kind of data issued to the clients. Both the logs and the configuration data are to be stored in a mysql database on the server.
Considered solution - servers backup each other in real time, cron jobs not suitable it seems
I am considering backup strategies for each server. Cron job based backups at certain intervals are unsuitable, owing to the necessity for the logs to be current in the backup, containing the very last entry before the server failed. For example, a cron job say every 4 hours would have data that is 4 hours old and not data since then.
So the strategy I am considering is to use the servers to backup themselves, sending their log data to each other, each log entry identifying which server it was from, as well as date stamp. This strategy seems bespoke and specific to this application because the code and data structures are specific. So I wondered:
Is there a design pattern or library or even a framework that does real time MySQL replication?
Answered questions I have already considered on Stack Overflow are:

https://stackoverflow.com/questions/7182791/daemon-software-to-update-mysql-database-in-background (similar question, but their domain - bidding needs a different solution from mine, answers here appear to be based around bidding amount, my domain isn't so much about content of the database entries, just that the latest needs to be backed up)
https://stackoverflow.com/questions/6471879/is-there-a-way-to-listen-for-a-database-event-and-update-a-page-in-real-time (more about keeping the UI up to date to reflect the database (nice question, will use it sometime) but mine is about backup)
https://stackoverflow.com/questions/5221164/need-advice-on-developing-real-time-system-using-php-mysql (nice broad question that would cover my question but theirs is again a slightly different domain concerned with technologies I am not using: OpenID, JQuery, PHP frameworks such as CodeIgnitor)
https://stackoverflow.com/questions/4200515/how-do-i-do-realtime-database-polling-in-mysql-php (for my problem, probably don't want to poll for changes as this has similar limitations to cron in that data is only as up to date as last poll)
Also looked at MySQL documentation, but could not find anything for real time replication, only batch based solutions. (Willing to stand corrected though if others can suggest to me otherwise)
Comment: Have you considered [Replication][1] or [Clustering][2]?

  [1]: http://dev.mysql.com/doc/refman/5.0/en/replication.html
  [2]: http://dev.mysql.com/doc/refman/5.0/en/mysql-cluster.html
Comment: +1 for a very well formatted, presented and researched question.
Comment: Migrating this to our DBA site - it has a better chance of finding the right experts for this particular topic :)
Comment: +1 Thank you @JamWaffles ! and I will continue to research as well as hoping to receive some answers here.
Comment: I think Replication only is solution left for you. Replicating only particular database which will sole this
Comment: (1 of 2):- +1 @Mahesh Patil for the replication and clustering references - thanks for your time in commenting. Looking at Replication (for backup): http://dev.mysql.com/doc/refman/5.0/en/replication-solutions-backups.html - it offers mysqldump for backing up a replicated slave as a solution which to me is non real time and batch based - not suitable for me - updates to the master would continue during the backup. Setting up the slave is here: http://dev.mysql.com/doc/refman/5.0/en/replication-howto-masterbaseconfig.html (documentation quite sparse).
Comment: (2 of 2):- But also consider recovery: In http://dev.mysql.com/doc/refman/5.0/en/replication-solutions-switch.html it says "There is currently no official solution for providing failover between master and slaves in the event of a failure." Oh dear! Looks like a custom bespoke solution might still be worth considering.
Comment: Another option would be to see if there is a solution discussed in: "High Performance MySQL, 2nd Edition Optimization, Backups, Replication, and More" http://shop.oreilly.com/product/9780596101718.do (3rd edition out in March 2012). I have actually bought this book as a PDF download. I will look at it and report back if it offers a solution (without infringing copyright). (I am not associated with O'Reilly). I don't recall it had a solution for my particular situation but I will read again.
Answer: Perhaps I am misreading this, but I am having difficulty understanding how your requirement is any different from built-in MySQL replication.
With MySQL replication, the master database writes changes to a binary log file.  The slaves connect to the master, read the binary logs, write them to their local relay log, and play the statements back.
Please understand that realtime copy and backup are not the same thing.  A real-time slave can help you recover from certain types of problems, but cannot resolve data-corruption issues (oops, I dropped that table!).  For that, you need point-in-time backups.  With a backup and a copy of the binary logs, you can recover from almost any type of failure by replaying the binary logs up to the point where the problem occurred.
Comment: +1 @Aaron Brown thanks, I would agree. I am in the process of researching further into the detail... I will try to report back here when I have more info.


Balance weight beam reading

Question: If there are two balance beam scales with one placed directly on the hard surface of  a floor and the other placed on a thick spongy carpet in between it and the floor, will they both indicate the same weight if you place a 100 pound weight on each, or, would the spongy surface absorb some of the weight?
Answer: If you stand on any surface, be it a piece of steel, concrete, wood, or foam, it gives a little until it pushes back against your weight with the same force. That is still true even if there is a scale between you and the surface. You stand on the scale; you and the scale (under the force of gravity) push down on the surface underneath until that surface pushes back against you and the scale with the same force.
Answer: they will both read the same. the material under the scale has no influence on the weight being measured.

Balance weight beam reading


How do I put colored carpet on my head?

Question: I was messing with command blocks today and I tried to put colored carpet on my head. I don't really know that much about command blocks and that kind of stuff. I already know /replaceitem entity @p slot.armor.head minecraft:carpet 1 0, but I want to know the command for colored carpet.
Answer: The replaceitem syntax for entities is:
/replaceitem entity <selector> <slot> <item> [amount] [data] [dataTag]

The carpet's colour is represented by its data value (sometimes called "damage value"). Currently you have it set to 0, which is white. You can see these values here.
For example, for lime carpet:
/replaceitem entity @p slot.armor.head minecraft:carpet 1 5

Comment: Thanks SirBenet- I haven't been on this website in a while, and I realized that you answered my old question already. I figured the answer out months later.. I'm an idiot.

How do I put colored carpet on my head?


Install app on genymotion device ,meet:"INSTALL_FAILED_CPU_ABI_INCOMPATIBLE"

Question: When I install my app on a genymotion simulater device,it can not be installed well,on console I got "INSTALL_FAILED_CPU_ABI_INCOMPATIBLE"
I trid another app,it installed fine.I do not know why.
Comment: Does your application have any native code (are you using C/C++ and the JNI) or are you only using Java?
Comment: @FuegoFro  No,I didn't...
Comment: any suggestion to solve it for Eclipse editor?
Comment: I followed this and it worked for me:http://forum.xda-developers.com/showthread.php?t=2528952
Answer: The application (certainly a game) must be ARM only.
Genymotion is a x86 platform, so compile the application to target x86.
You may be able to install ARM support manually :
http://forum.xda-developers.com/showthread.php?t=2528952
Comment: Thank you @slvn,I don't know how to compile the app to target x86 but  I followed the instruction of your url to installed ARM support manually ,it worked.
Answer: If you are using IntelliJ it may be related to the project's default configuration.  IntelliJ will assume, unless otherwise specified, that Native libraries will be stored in the Libs folder. This is usually where developer store their JAR libraries.  IntelliJ build process will package up the JAR file into the Native file folder in the APK.
If you experiencing this problem, you can find a good How-to:
INSTALL_FAILED_CPU_ABI_INCOMPATIBLE Intellj
This helped to resolve the issue I had.
Comment: This works for me. The reason is that IntelliJ copies "google-play-services.jar.properties" from google play services to "lib" folder inside the apk. The installer seems to think that "google-play-services.jar.properties" is a CPU architecture name.
Answer: You might want to check out my answer here:
INSTALL_FAILED_CPU_ABI_INCOMPATIBLE on device using intellij
I know it's written for IntelliJ, but you could be running into a similar issue with Eclipse where it thinks that some normal Java files or jar libraries are native code and including it in the compiled APK. A quick way to find out is to open up the final APK (it's just a jar file, so most utilities should be able to decompress it) and make sure that the only things you see are META-INF, res, AndroidManifest.xml, classes.dex, and resources.arsc. It's possible that there are other valid files in a normal Java Android application, but in general if you see anything else in there, particularly a libs folder, it's probably the result of the compilation process thinking that those other things are native libraries (compiled C/C++ code using the JNI).
If this is the case, you'll need to configure Eclipse to not interpret the Java libraries you're using as native code. Now, I don't personally use Eclipse so I don't know where the proper settings would be, but based off of this answer, it looks like Eclipse automatically assumes that the libs folder in the root of your project contains native libraries. The easiest solution may be to move any Java libraries you are using into a different folder (such as libraries).
Answer: http://blog.iteedee.com/2014/02/android-install_failed_cpu_abi_incompatible-intellj/
The INSTALL_FAILED_CPU_ABI_INCOMPATIBLE error is generated when you attempt to install an android application on a device with an unsupported CPU architecture. This error is usually related to the native libraries that are not packaged up for the device's CPU architecture (ie: x86, ARMv6, ARMv7, etc).
Now, if you have an application that doesn't use any native libraries or *.so file you might be scratching your head on this one. If you use IntelliJ IDEA IDE for your android development it might be packaging all of your related .JAR file in to the Native Library file location in your APK.
Answer: The cause of the same problem was different in my case.
I had added some dependency jars in /libs directory of my Android app project. One of these jars had exported library in jar format; i.e. a jar within a jar file. I've noticed that the apk created using jars with such structure are incompatible with x86 based devices (don't know why).

Install app on genymotion device ,meet:“INSTALL_FAILED_CPU_ABI_INCOMPATIBLE”


Phaser3 how to force screen to open in landscape mode

Question: In Phaser3, I want to open the game that will run in webview directly horizontally. In other words, I want the game to open horizontally without the user turning the phone.
The white photo below is the vertical version of the game.
So when I open it vertically, the game itself is landscape, but it does not sit on the screen as landscape, it stands horizontally on the top of the screen, so it does not turn the game.
enter image description here

Here is my game configs :
import Phaser, { Types } from "phaser";

const ratio = window.innerWidth < 600 ? 2 : 1;

const baseGameConfig: Types.Core.GameConfig = {
  type: Phaser.WEBGL,
  scale: {
    mode: Phaser.Scale.FIT,
    autoCenter: Phaser.Scale.CENTER_BOTH,
    width: window.innerWidth * ratio,
    height: window.innerHeight * ratio,
  },
  render: {
    antialias: true,
  },
};

export default baseGameConfig;

import { Types } from "phaser";

import baseGameConfig from "@games/config.base";

import Scenes from "./src/scenes";

export const DEFAULT_WIDTH: number = 1280;
export const DEFAULT_HEIGHT: number = 720;
export const MAX_WIDTH: number = 1920;
export const MAX_HEIGHT: number = 1080;
export let SCALE_MODE: "FIT" | "SMOOTH" = "SMOOTH";

const gameConfig: Types.Core.GameConfig = {
  ...baseGameConfig,
  title: "monkeygo",
  scene: Scenes.BootScene.scene,
  physics: {
    default: "arcade",
    arcade: {
      debug: process.env.NODE_ENV !== "production",
      gravity: {
        y: 800,
      },
      fps: 30,
      fixedStep: true,
      tileBias: 32,
    },
  },
  scale: {
    mode: Phaser.Scale.NONE,
    autoCenter: Phaser.Scale.CENTER_BOTH,
    width: DEFAULT_WIDTH,
    height: DEFAULT_HEIGHT,
  },
  input: {
    activePointers: 3,
  },
};

export default gameConfig;

Answer: within your create function you have to do the following
if(this.scale.orientation.toString() == "landscape-primary"){
///your code here
}else{
//handle your portrait case
}

but if your using a older version of phaser I think this solution might work
this.scale.forceLandscape = true;


Virgo Constellation in the Torah

Question: It is known that Elul is Besulah which in its secular name is Virgo where did this constellation come from according to our Torah sources?
Comment: @simchashatorah If you have the answer, why did you post the question here? If you have a more specific question about This Midrash then perhaps you should rewrite your question
Comment: I don't understand the question. Where did *any* of the constellations "come from"?
Comment: Yes, simchashatorah, could you please edit the question so as to clarify what you mean by "where did this constellation come from"? Do you mean where its name _b'sula_ came from? Where the bodies that comprise it came from? Where the idea that those bodies form a constellation came from? Or what?
Comment: Relevant: http://meta.judaism.stackexchange.com/questions/306/if-i-know-the-answer-to-a-question
Comment: I added a paragraph to my answer the above meta question after thinking about this example. The confusion expressed by Dave and msh210 is, I think, a symptom of the fact that this question was much more of a leading, set-up question than a question based on a point of curiosity. The question only really arises if you already know that there's a Midrashic origin tale for Virgo, and if you know that, you probably have the answer already.
Comment: but there could be another answer too i knew the medrash i wanted to know if there was somersetting else
Comment: The Rosh on the Torah:The בני האלהים came to a righteous Virgin  and said live with me she said no unless you give me your wings they gave her their 6 wings and she flew up to Heaven and escaped doing a sin with the בני האלהים and she held on to Hashem's throne. Hashem but clouds around her and accepted her and she is the Consolation of Virgo;Besulah and the Angels where forced to stay on the ground untill the story of Yackovs Ladder for the original see here this is an amazing allegorical Medrash which explains how the constellation of Virgo was formed. for the original See here.
Comment: [1]: http://hebrewbooks.org/pdfpager.aspx?req=14033&st=&pgnum=8&hilite=
Answer: I see what you wrote in the comment section. Fascinating.
I think it possible to trace that legend a bit more. See here.
Thus, (also via this, in Geiger, where he draws connections to the Koran):

"Thus, when Shamhazai noticed a certain maiden whose name was Istahar, he gazed lustfully upon her and pleaded, "Do my bidding." She replied, "I will not do your bidding until you give me your wings and teach me the Explicit Name, which you go up to heaven upon uttering." So he gave her his wings and taught her the Name, whereupon she uttered it, went up to heaven, and was spared from corruption. The Holy One said: Since she shunned transgression, go and set her among the seven stars yonder. Thus, it came about that Istahar was set in the constellation of Draco." - from Jewish Gates

and

When the angels came to earth, and beheld the daughters of men in all their grace and beauty, they could not restrain their passion. Shemhazai saw a maiden named Istehar, and he lost his heart to her. She promised to surrender herself to him, if first he taught her the Ineffable Name, by means of which he raised himself to heaven. He assented to her condition. But once she knew it, she pronounced the Name, and herself ascended to heaven, without fulfilling her promise to the angel. God said, "Because she kept herself aloof from sin, we will place her among the seven stars, that men may never forget her," and she was put in the constellation of the Pleiades.

Finally, there is a connection to a Greek legend:

"Istahar's story is borrowed partly from the Greek writer Aratus (early third century B.C.). He tells how Justice, a daughter of Dawn, ruled mankind virtuously in the Golden Age; but when the Silver and Bronze ages brought greed and slaugter among them, she exclaimed: "Alas, for this evil race!" and mounted into Heaven, where she became the constellation Virgo. The rest of this story is borrowed from Apollodorus's account of Orion's attempt on the seven virgin Pleadies, daugters of Atlas and Pleione, who escaped his embraces transformed into stars. "Istahar," however, is the Babylonian Goddess Ishtar, sometimes identified with Virgo. Popular Egyptian belief identified Orion, the constellation which became Shemhazai, with the sould of Osiris." (The Sons of God and the Daughters of Men, From Hebrew Myths: The Book of Genesis, written by Robert Graves and Raphael Patai)

Read more there. The adaption of a Greek myth and a myth about Ishtar into Jewish midrashim is indeed quite fascinating. Thanks for sharing this riddle.
Comment: *"The adaption of a Greek myth and a myth about Ishtar into Jewish midrashim is indeed quite fascinating."* -- or maybe it was the other way around :) -- although it is interesting that the tale's timeframe predates all the cultures who tell it.
Comment: Well said Menachem
Comment: The flavor (style, genre) of the midrash, though, sounds quite similar to many other Greek myths -- female pursued by deities, achieving a measure of apotheosis, and accounting for natural phenomena. And our standard understanding is that Hashem created all the stars and set the seasons much earlier. And the derivation here is quite slight -- just a chaser spelling. For midrashim entirely of Jewish origin, I usually can find a textual spark for each and every detail of the midrash.
Comment: The story from the Koran has a textual reference in the Chumash from Bereshit.  But otherwise I agree with you.

Virgo Constellation in the Torah


How to integrate $\int_{0}^{2\pi} \frac{1}{\sin^4x + \cos^4 x} \,dx$

Question: So I followed the explanations made in this post and I got that:
$$\int \frac{1}{\sin^4x + \cos^4 x} \,dx = \frac{\sqrt{2}}{2}\arctan\left(\frac{\sqrt{2}}{2}\tan\left(2x\right)\right) + C$$
But when I try to use the Leibniz-Newton formula and evaluate the integral from $0$ to $2\pi$ I get that it's $0$ because $\tan\left(2x\right)$ evaluates to $0$ at both $x=0$ and $x=2\pi$. Is there another way to solve this integral and get the correct answer ($2\pi\sqrt2$)?
Comment: $\tan(x/2)$ is not continuous on $(0,2\pi)$. To use that form, you must split it over intervals where the antiderivative is continuous.
Comment: The core issue is that $\tan(x/2)$ is not monotonic on the interval. You will need to split up into intervals of $\frac{k\pi}{2}$ to $\frac{(k + 1)\pi}{2}$
Comment: Related: https://math.stackexchange.com/questions/1356523/what-are-the-restrictions-on-using-substitution-in-integration (and the linked questions there).
Answer: A correct antiderivative is
$$ \int\frac{1}{\sin^4 x+\cos^4 x}\, \mathrm{d}x = \frac{1}{\sqrt{2}}\arctan\left(\frac{1}{\sqrt{2}}\tan(2x)\right)+\mathsf{C}, $$
which is true on every interval $ \left(\frac{k\pi}{2}-\frac{\pi}{4}, \frac{k\pi}{2}+\frac{\pi}{4} \right) $ with $k \in \mathbb{Z}$. You may check the following plot for the comparison of both sides.

This is not surprising, since $\tan(2x)$ has discontinuity at every point of the form $k\pi/2 + \pi/4$ for $k \in \mathbb{Z}$, which naturally restricts the region of validity of the above formula.
If you want to get an expression that is valid on a larger set, you must patch different antiderivatives (with different choices of 'constant of integration') to remove discontinuity. Alternatively, we can exploit the symmetry of the integrand to write
\begin{align*}
\int_{0}^{2\pi} \frac{1}{\sin^4 x+\cos^4 x}\,\mathrm{d}x
&= 8\int_{0}^{\frac{\pi}{4}} \frac{1}{\sin^4 x+\cos^4 x}\,\mathrm{d}x \\
&= 8 \lim_{a \to (\pi/4)^+} \int_{0}^{a} \frac{1}{\sin^4 x+\cos^4 x}\,\mathrm{d}x,
\end{align*}
which then can be computed by the above antiderivative.
Answer: Just an alternate method of evaluation that relies heavily on symmetry.
$$J=\int_0^{2\pi}\frac{dx}{\sin(x)^4+\cos(x)^4}$$
We can do a bunch of trig and notice that this simplifies to
$$J=4\int_0^{2\pi}\frac{dx}{3+\cos(4x)}=16\int_0^{\pi/2}\frac{dx}{3+\cos(4x)}.$$
Preforming $4x\mapsto x$ gives
$$J=4\int_0^{2\pi}\frac{dx}{3+\cos x}=8\int_0^\pi \frac{dx}{3+\cos x}.$$
Then we use my favorite substitution, $t=\tan(x/2)$:
$$J=16\int_0^\infty \frac{1}{3+\frac{1-t^2}{1+t^2}}\frac{dt}{1+t^2}=8\int_0^\infty\frac{dx}{t^2+2}$$
and the rest is rather easy.
Comment: Great solution :-)
Comment: @KevinNivek Thank you very much! I was very proud of myself for finding it! :)
Answer: You cannot directly use the indefinite integral to compute the definite integral because $\tan 2\, x$ is not defined at $\pi/4, 3\pi /4, 5\pi /4, 7\pi /4$. But you can use the  indefinite integral to compute the definite integral from $0$ to $\pi/4 -\epsilon$ , the integral from $\pi/4+\epsilon$ to $3\pi/4 -\epsilon$ etc.  When you take the limit as $\epsilon \to 0$ pay attention to the sign of $\tan $ and $\arctan$. You should now be able to compute the integral.
Note. I have edited the answer based on the comment by Travis.
Comment: The discontinuities of the given antiderivative are at $\frac{\pi}{4}, \frac{3 \pi}{4}, \ldots$, not at $\frac{\pi}{2}$.
Comment: Now I get it. I have to integrate over the intervals where the function is continuous and evaluate the limits at the ends.
Answer: Given integral $$ I  = \int^{2 \pi}_0 \cfrac{1}{\sin^4 x + \cos^4 x} dx = 4 \int^{\pi/2}_0 \cfrac{dx}{\sin^4 x + \cos^4 x} $$
By repeatedly using $ \int^{2a}_0 f(x) dx = \int^a_0 f(x) dx + \int^a_0 f(2a-x) dx$
On multiplying numerator and denominator by $ \sec^4(x)$, followed by substitution of $\tan(x) = u$ $$ \begin{align} I &= 4 \int^{\infty}_0 \cfrac{1 + u^2}{1+ u^4} du \\ &= 4 \int^{\infty}_0 \cfrac{1}{\bigg( u - \frac{1}{u} \bigg)^2 + 2} d \bigg( u - \frac{1}{u} \bigg) \\ &= \cfrac{4}{\sqrt{2}} \arctan \bigg(\cfrac{u-\frac{1}{u}}{\sqrt{2}} \bigg) \bigg|^{\infty}_0 = 2 \sqrt{2} \pi\end{align} $$
Comment: I didn't know about that property of definite integrals.
Comment: Do you need proof? It is quite easy :)
Comment: That would help. I know that to prove $\int^a_{0}f(x)dx=\int^a_{0}f(a-x)dx$ I have to use substitution. Do I use the same method here?
Comment: You have to use $ \int^{2a}_0 f(x) dx = \int^a_0 f(x) dx + \int^{2a}_a f(x) dx$ and then a substitution.
Answer: The given antiderivative, $$\frac{1}{\sqrt 2} \arctan \left( \frac{1}{\sqrt 2} \tan 2 x \right)$$ is not defined everywhere on the interval $[0, 2 \pi]$ of integration---it is undefined at $\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$, so the usual hypotheses of the Fundamental Theorem of Calculus are not satisfied for that interval. In retrospect this is not surprising, since the trigonometric substitution that leads to this formula is not defined at those values.
On the other hand, using the double angle formulas for $\sin$ and $\cos$ (or just their complex representations) shows that the integrand has period $\frac{\pi}{2}$; using this observation and the symmetry of the integrand gives
$$\int_0^{2 \pi} \frac{dx}{\sin^4 x + \cos^4 x} = 8 \int_0^{\frac{\pi}{4}} \frac{dx}{\sin^4 x + \cos^4 x} .$$ The given antiderivative is defined everywhere on $[0, \frac{\pi}{4})$, so we can write
$$\int_0^{2 \pi} \frac{dx}{\sin^4 x + \cos^4 x} = 8 \cdot \lim_{a \nearrow \frac{\pi}{4}} \int_0^a \frac{dx}{\sin^4 x + \cos^4 x}$$
apply the F.T.C., and compute the limit.
Of course, since $\sin$ and $\cos$ do not vanish simultaneously, the original integrand is defined everywhere, and hence it has a continuous antiderivative on all of $\Bbb R$. One can verify that the antiderivative $F$ satisfying $F(0) = 0$ is
$$
F(x) =
\left\{
\begin{array}{cl}
\cdots & \cdots \\
\frac{1}{\sqrt 2} \arctan \left( \frac{1}{\sqrt 2} \tan 2 x \right), & -\frac{\pi}{4} < x < \frac{\pi}{4} \\
\frac{\pi}{2 \sqrt 2}, & x = \frac{\pi}{4} \\
\frac{1}{\sqrt 2} \left(\arctan \left( \frac{1}{\sqrt 2} \tan 2 x \right) + \pi \right) , & \frac{\pi}{2} < x < \frac{3 \pi}{4} \\
\frac{3 \pi}{2 \sqrt 2}, & x = \frac{3 \pi}{4} \\
\frac{1}{\sqrt 2} \left(\arctan \left( \frac{1}{\sqrt 2} \tan 2 x \right) + 2 \pi \right) , & \frac{3 \pi}{2} < x < \frac{5 \pi}{2}\\
\cdots & \cdots
\end{array}
\right. .
$$
It is evidently less practical than appealing to periodicity and symmetry as above, but we can also evaluate the integral by finding this antiderivative---which does satisfy the hypotheses of the F.T.C. on the interval $[0, 2 \pi]$---and then applying the F.T.C.
Answer: You've got many methods here to find the anti-derivative, but hey, why not one more?
Here
\begin{align}
I &= \int \frac{1}{\sin^4(x) + \cos^4(x)}\:dx = \int \frac{1}{\cos^4(x)\left(\tan^4(x) + 1\right)}\:dx \nonumber \\
 &= \int \sec^2(x)\sec^2(x) \cdot\frac{1}{\tan^4(x) + 1}\:dx  = \int \sec^2(x) \cdot\frac{\tan^2(x) + 1}{\tan^4(x) + 1}\:dx
\end{align}
Let $t = \tan(x)$:
\begin{align}
I &= \int \frac{t^2 + 1}{t^4 + 1}\:dt =  \int \frac{t^2 + 1}{\left(t^2 + \sqrt{2}t + 1\right)\left(t^2 - \sqrt{2}t + 1\right)}\:dt \nonumber \\
&= \int \frac{1}{2}\left[\frac{1}{t^2 + \sqrt{2}t + 1} + \frac{1}{t^2 - \sqrt{2}t + 1} \right]\:dt \nonumber \\
&= \frac{1}{2}\left[ \int \frac{1}{t^2 + \sqrt{2}t + 1}\:dt +  \int \frac{1}{t^2 - \sqrt{2}t + 1}\:dt \right] = \frac{1}{2}\left[ J_1 + J_2\right]
\end{align}
We now address $J_1$ and $J_2$ separately. Here we begin with $J_1$:
\begin{equation}
 J_1 = \int \frac{1}{t^2 + \sqrt{2}t + 1}\:dt = \int \frac{1}{\left(t + \frac{1}{\sqrt{2}}\right)^2 + \frac{1}{2}}\:dt = \frac{1}{\sqrt{\frac{1}{2}}}\arctan\left(\frac{t + \frac{1}{\sqrt{2}}}{\sqrt{\frac{1}{2}}}\right)= \sqrt{2}\arctan\left(\sqrt{2}t + 1\right)
\end{equation}
Note I've omitted the constant of integration here. Similarly we find that $J_2$ is
\begin{equation}
 J_2 = \sqrt{2}\arctan\left(\sqrt{2}t - 1\right)
\end{equation}
Thus our integral $I$ becomes:
\begin{align}
I &= \frac{1}{2}\left[ J_1 + J_2\right] = \frac{1}{2}\bigg[\sqrt{2}\arctan\left(\sqrt{2}t + 1\right) + \sqrt{2}\arctan\left(\sqrt{2}t - 1\right)\bigg] + C = \frac{1}{\sqrt{2}}\bigg[ \arctan\left(\sqrt{2}t + 1\right) + \arctan\left(\sqrt{2}t - 1\right)\bigg] + C \nonumber \\
&=\frac{1}{\sqrt{2}}\bigg[ \arctan\left(\sqrt{2}\tan(x) + 1\right) + \arctan\left(\sqrt{2}\tan(x) - 1\right)\bigg] + C
\end{align}
Where $C$ is the constant of integration. Thus,
\begin{equation}
\int \frac{1}{\sin^4(x) + \cos^4(x)}\:dx =\frac{1}{\sqrt{2}}\bigg[ \arctan\left(\sqrt{2}\tan(x) + 1\right) + \arctan\left(\sqrt{2}\tan(x) - 1\right)\bigg] + C
\end{equation}
Comment: Very nice (+1). I did not have the patience to find an antiderivative :)


How to run deluge from CLI?

Question: I want to run deluge in tty3,4,5 . but when I type it there is, No display warning. Then I give top and there is no pid on deluge that is deluge is not running. How do I run it from CLI so that it keeps downloading the torrents?
Answer: Have you started deluged (the deluge daemon) as a service? The command for running the client in CLI is deluge-console.
Here are some guides:

https://wiki.archlinux.org/index.php/Deluge
https://whatbox.ca/wiki/Deluge_Console_Documentation
Comment: Don't install `deluge`!!! just `apt install deluged deluge-console`
Comment: thanx... deluge-console did help/
Answer: This will install it, set up the daemon, and connect to its console:
sudo apt-get install deluge deluged deluge-console
deluged
deluge-console
You can then add torrents like:
add -p /user/Downloads torrent_magnet_link
Comment: everything is great, except you don't need to install deluge, just deluged and deluge-console
Answer: You could also try the local web UI.
Install deluge-web using your package manager (if not already installed). On Ubuntu, you can use
sudo apt-get install deluge-web.
Start the deluge-web service. You can do this by typing deluge-web in your terminal.
Once it's running, open a web browser and navigate to http://localhost:8112. This is the default address for the Deluge web interface.
The default password is deluge. Upon first login, you'll be prompted to change this password.
Once logged in, you can add, manage, and remove torrents just like you would in the desktop client.

How to run deluge from CLI?


Injective von Neumann algebra

Question: Let $G$ be a non-amenable countable discrete group. How can I show that the group von Neumann algebra $L(G)$ has no injective direct summand?
Comment: There is no reason that this subalgebra will be a direct summand
Comment: I must be missing something. Why can't you take an abelian vN subalg of L(G) and then project onto it?
Comment: @Owen: right, for some reason I had "direct summand as a Banach space" in my mind
Comment: Sorry!...I did not know this function.
Answer: Here is an adaptation of the standard proof that $G$ is amenable if $LG$ is injective.  (I believe for instance that it is contained in the book of Brown and Ozawa).
Suppose $p \in LG$ is a non-zero central projection such that $p LG$ is injective.  Thus, there exists a conditional expectation $E: \mathcal B(p \ell^2 G) \to p LG$.  If we view $\ell^\infty G \subset \mathcal B(\ell^2 G)$ as diagonal multiplication operators (for $f \in \ell^\infty G$ and $\xi \in \ell^2 G$ we set $(M_f \xi)(\gamma) = f(\gamma) \xi(\gamma)$), and if we denote by $\tau$ a tracial state on $pLG$ then we can construct a state $\varphi$ on $\ell^\infty G$ by the formula $\varphi(f) = \tau \circ E(p M_f p)$.  If $\gamma \in G$ then we have
$$
\varphi( f \circ \gamma) = \tau \circ E(p M_{f \circ \gamma} p)
$$
$$
= \tau \circ E(p \lambda_{\gamma^{-1}} M_f \lambda_{\gamma} p) = \tau( (p\lambda_{\gamma^{-1}}p) E(p M_f p) (p \lambda_{\gamma}p) ) = \varphi(f).
$$
Thus $\varphi$ is an invariant mean for $G$ and so $G$ is amenable.

Injective von Neumann algebra


Are you allowed to cry because of how beautiful the Quran is?(In Salat)

Question: are you allowed to cry because of how beautiful the Quran is?*(In Salat) because of how beautiful the signs and meanings are?
Comment: Depends on situation, whether or not you lead the prayer and whether or not it was spontaneously and also intensity and loudness play a role.
Answer: السَّلامُ عليكم ورحمةالله وبركاته
Crying in response to recitation does not invalidate Salah, it is actually desirable if due to fear and love for Allah. There is an hadith where the Prophet صلى الله عليه وسلم said "I have established palaces at the highest levels of paradise for those who shed tears and no one will share to with him."
Comment: The hadith reference actually somewhat contradicts your claim. As it means that a person cries in secret so how would this be desirable in case of a congregation?


it's possible to save materials in cycles node editor to reuse them in other project?

Question: instead of copy and paste them from other projects I'd like to store them somehow and load them directly
Answer: You can create a .blend file that is a master collection of your finished materials, and then Link (not Append) this to your various other .blends.

Be cautious with this kind of workflow though, because if you change anything in this master file it means your materials in all other .blends may render differently - so for example, re-rendered frames of animation may look different.
It's generally best to work with Linked files on a project basis, not a global basis.


Any good, undergraduate level introductions to Functional Analysis?

Question: In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As an undergrad who feels like at times he knows more about math then the actual rigor and computation involved, are there any good recommendations to an introductory functional analysis book that give a reasonable treatment as well as show some connections it may have to other fields of math?
Comment: In my opinion (and Halmos') the best way to study Functional Analysis is by a problem solving approach. Halmos' *A Hilbert space problem book* may not be easy (or appropriate for beginning). My suggestion is "Trenoguin's *Problems and exercises in Functional Analysis*. Mir."
Comment: **Introduction to Topology and Modern Analysis** by George F. Simmons is a possibility.
Comment: So what is your background? How much real analysis? Any measure theory?
Comment: @MichaelGreinecker In regards to measure, my background doesn't go much past the definition of measure and the Baire-Category theorem, which we touched upon in my topology course.
Answer: One of the few books I've seen that is fully self-contained at the undergraduate level is available from Dover: Functional Analysis by George Bachman and Lawrence Narici. The exposition is verbose and unusually well-written; so don't be too put off by the length. You can teach yourself the subject from this book.
http://www.amazon.com/Functional-Analysis-Dover-Books-Mathematics/dp/0486402517/ref=pd_sxp_grid_i_1_0
Comment: Completely agree,one of the few books on the subject suitable for self study.
Answer: A gentle path that brings you relatively far is to work through Introduction to the Analysis of Metric Spaces by John Giles and then through his Introduction to the Analysis of Normed Linear Spaces. In spite of its name, the first book has a lot of material on normed spaces.
A nice book with an applied outlook is Optimization by Vector Space Methods by David Luenberger.
Answer: Griffels's book is not commonly mentioned, perhaps because it is focused on applied math, but it gives good intuition on concepts. I've found it very useful.
https://www.amazon.com/Applied-Functional-Analysis-Dover-Mathematics/dp/0486422585


How are Cepheids used to evaluate their distance?

Question: Cepheids are used to evaluate distances. What is the math and physics behind their use?
Answer: The reason Cepheids can be used to determine distance is because the are variable stars that change brightness according to a regular pattern.  The stars literally change size, swelling larger and smaller and while doing so change in brightness.  It was discovered by Henrietta Leavitt in the early 1900's that the period of the Cepheid variability was directly correlated with it absolute maximum brightness.  The longer the period, the larger the absolute luminosity of the star.
So by measuring the period of a Cepheid variable, we know what it absolute magnitude (brightness at a distance of 10 parsecs) should be.  Given that we can measure is apparent magnitude and the two are related to each other by distance according the the following formula:
$$m - M = 5\log (D/10)$$
where M is the absolute magnitude, m is the apparent magnitude and d is the distance in parsecs.
Comment: There are some Cepheids close enough that we can determine there distance through other methods such as parallax (the apparent mostion of the star relative to more distant ones as the earth moves along its orbit around the sun).  So she had appearant brightness and distance with which to work out the absolute brightness for those particular Cepheids.  Once she had the absolute brightness and the periods she was able to determine the relationship.
Comment: Modern calibrations (eg [Feast & Catchpole](http://adsabs.harvard.edu/abs/1997MNRAS.286L...1F)) use parallax, but I believe [Leavitt measured a bunch in the LMC](http://adsabs.harvard.edu/abs/1912HarCi.173....1L), which are all effectively at the same distance, and noted the correlation between the apparent magnitude and period.
Comment: Every method used to determine astronomical distances generally has to be calibrated by corroborating it with another, already established method. The most confident methods are generally the ones applied to the nearest objects, starting with parallax. This leads to a sequence of growing distance methods called the [cosmic distance ladder](http://en.wikipedia.org/wiki/Cosmic_distance_ladder).
Comment: How did Henrietta Leavitt determine the relationship without knowing the absolute brightness of a cepheid to start with?
Comment: I recently attended a presentation that included this information, but the key piece that wasn't presented there either was in your above comment - starting with a defined distance from some other method.
Answer: Cepheids pulse, with periods on the order of days to months. The longer the period, the brighter the star is intrinsically.
This was first known experimentally with parallax studies in the 1910s. "Stellar pulsation theory," which sought to explain the experimental relationship, was a huge field for a long time; the first complete theoretical justifications came decades later.
The first model to get it pretty much right was the celebrated paper in 1953 by Sergei Zhevakin, which explains things in terms of ionization of helium (from He+ to He++). These models are hard, but the short version is this:
He++ is more opaque than He+ to photons. In stellar structure, this effectively means that a shell of He++ surrounding a the interior of a star will trap more photons inside the star than He+ will. This will accelerate the rate of fusion within the core. As the rate of fusion builds, the pressure in the core increases, pushing the He++ outwards. The He++ thus expands. As it expands, it loses energy, like any other gas.
But the expansion of the He++ causes it to fall in temperature. This fall in temperature means there is less energy available, so the ionization fraction drops - eventually, the shell becomes mostly He+. Now, the shell is optically thin! The photons inside can rush out, and the core temperature drops as fusion slows down. The star deflates.
But once it has deflated, the He+ compresses, increases in temperature, and ionizes to form He++. The process then repeats.
We see the pulses in this process because the brightness of the star is proportional to the square of its radius; when the star expands, it gets brighter, when it contracts, it gets dimmer.
The rate of pulsation depends on the total mass of the star, since the total mass of the star governs the rate of fusion in the core. Intuitively, we might expect that a bigger star (which has faster fusion in the core) will be relatively less affected by the presence of He++ or He+ than a dim star which is struggling to stay alive. The real mathematical answer is something like this, but comes in the form of a huge array of many, many, many differential equations and can only be solved on a computer, not analytically.
Comment: [This reference](http://e-collection.library.ethz.ch/eserv/eth:28283/eth-28283-01.pdf) (pages 11 and 12) that I found states that Zhevakin's discovery was published in 1953 (in Russian).
Answer: Cepheid variables have a strong correlation between their pulsation period and luminosity.  Thus, as we constrain a given Cepheids pulsation period, we can determine it's luminosity.  The distance to any object can be found if we know it's luminosity using the relationship between flux and luminosity.


Is a rock obstacle in a pond good for fish vitality - or is it an urban legend?

Question: There are fables such as this one suggesting that fish do better in a pond whose surface is pierced by one or several obstacles, such as rocks. The rationale is that such "islands" turn the pond into channels, encouraging more movement and therefore well-being of the fish. I wonder - is that true - or just an urban legend?
Answer: First of all yes this is an urban legend but it is true that a pond with objects that pokes out of the water can help increase the quality of life for your fish.
Rocks in a pond or in a stream will slow down the waterflow and create a micro environment for plants and other life in the water,this can improve water quality for your fish.
Rocks that pokes out of the water and rocks on the bottom of your pond will provide a larger surface for bacteria to grow on,the rocks will be hiding places for insects and this will provide some snacks for the fish that they have to hunt for to get.
BE SURE TO ONLY USE ROCKS THAT ARE SAFE FOR YOUR POND,do not use marble or rocks containing dissolvable minerals or metals,river rocks and rocks from streams are in general safe to use in your pond.
Comment: @Allerleirauh putting a ball in the water keeps an area of open water during winter,it is used to let toxic gasses like methane and co2 escape from the water,cold water will be saturated with oxygen  so as long as one get the co2 out the fish will be fine.
Comment: A stream is much more effective improving conditions than some rocks. I have a 30 ft/10M stream for my small pond. Flow is about 200 gal/hour with a low wattage pump ( 30 watt ?).  Danios jump the few 3" drops to lay eggs. Only problem I now need to get rid of 100+ giant danios and chain pet shop won't take them.  The stream also replaces a filter. The referenced article was likely written by a "writer", not someone familiar with ponds.
Comment: I remember also that something poking out of the water hinder the surface to freeze completely, and this would give some more oxygene into the water?
Comment: @blacksmith37 a stream with rocks and plants are better than putting the rocks in the pond but having rocks in the pond will provide a micro climate for insects and other life living off waste in a pond,i focused on the in pond question this time but next time if we get a question about streams i can focus on that :)


How to send an automated email from Win XP Apache

Question: I need to send myself an automated email once a day from my windows XP dev. machine. I've got Apache, PHP, and MySQL running here. I don't mind which email address the email gets sent from since i can add it to my address bar. I'm wondering though, what do i need to enable/install to be able to send emails?
Answer: you can look into scheduled tasks that and commandline php. Commandline php lets you execute php files from the cmd for example php emailsend.php where you can use the mail() function. :)
Comment: Still need an MTA, and a normal LAMP stack on XP won't find a mail server.
Comment: ah didnt see he hadnt sendmail on it, can be downloaded and configured easily from apachefriends.org
Answer: You'll need some sort of mail transport agent - like sendmail.  You could theoretically use a PHP library for sending email and have it relay to an SMTP server hosted somewhere else.
Answer: You can try using the swift mailer library
http://swiftmailer.org/
Answer: Check out BLAT:
You can set it up to run in a batch file and then use the Windows Task scheduler to execute the batch file at the specified time each day.


Update Product Names To Title Case

Question: After exporting my previous database, allot of my product names where upper case. I now want to run a sql query to update the product names to title case.
What is correct way to do this
Answer: A few ideas:
Use CSS! (Courtesy @McNab):
h2 { text-transform: capitalize; }

Echo it from PHP in your View:
You shouldn't have to store it into the database, you can do this with a PHP wrapper around your echo. Instead of:
<?php echo $product->getName(); ?>

Use:
<?php echo ucwords(strtolower( $product->getName() )); ?>

Create a view helper:
Or, instead, register it in a Magento helper. Let's call that helper's module mymodule and use it as such:
class MyCompany_MyModule_Helper_Data extends Mage_Core_Helper_Abstract
{
    public function toUppercase($name)
    {
        return ucwords(strtolower( $name ));
    }
}

And use it like this:
<?php echo Mage::helper('mymodule')->toUppercase($product->getName()); ?>

I don't care, I want to do it in the DB:
Because the product names are stored in EAV your attribute code may differ. So you'll want to avoid direct SQL. Never query the db directly anyhow. It's bad practice.
You can load the collection in a one-off script and update all the product names as such:
$products = Mage::getModel('catalog/product')->getCollection()->addAttributeToSelect('*');

foreach($products as $product){
    $name = $product->getName();
    $product->setName( ucwords(strtolower( $name )) );
    $product->save();
}

If you have flat catalog enabled you'll want to re-index after this operation as the flat tables may not be enabled. You may also want to disable all indexes prior to running this.

Seriously, I just want a direct SQL query:
Good luck! Go read this article on creating your own Title Case function:
https://stackoverflow.com/a/1191683/582138
Comment: +1 for this - I have be honest, I would have just done `text-transform: capitalize` in the css ;)
Comment: Ah, another good point. Forgot about good-ol' CSS.
Comment: @philwinkle Thanks for all of these options :) Really appreciate your time. I'm not sure which method to go with here.

**McNab**, Good point completely over looked the CSS option. Thanks!

Update Product Names To Title Case


Tumour cell injection into a mice

Question: I am going to inject mice tumour cells into mice to create them tumours and I am wondering if I need to have special precautions for that, even they are cells from mice tumour I am afraid what can happen if I inject myself by accident, I guess is not necessary to do it a hood either?
Comment: I hope you understand from the answers and comments that injecting mice (or any other thing) with live tumor cells is not without serious risks.
Answer: You should not be asking random people on the internet about this. Talk to your lab safety people. Get detailed instructions on handling requirements. As for the hood, again, do not ask here, ask your animal care people (IACUC or equivalent).  If you don't have both lab safety people and animal care people immediately available and involved before the experiment begins, you should not be doing this experiment.
Comment: And you will most likely need a permission by an ethics comission.
Comment: +1 This is occupational safety and health for labs 101: If you develop a procedure, it needs to be reviewed to ensure the proper personnel, training, controls and PPE are available. *Before* you perform the procedure!
Comment: IACUC is an ethics commission. If humans are not being experimented on, there probably won't be a need for a second ethics review.
Comment: Ah, I didn't knew this abbreviation. But since I have seen people who wanted to start immediately, I thought I mention it briefly...

Tumour cell injection into a mice


Prime divisors of $\prod_{i=1}^n (i^2+1)$

Question: Is it true that for every positive integer $n$ there is a prime $p>n,$ which divides   $\prod_{i=1}^n (i^2+1)$ ?
Comment: How about the following partial solution? The case of $n=1$ is trivial. So suppose $n>1$. By Bertrand's postulate there exists an odd prime $p$ with $n<p<2n$. Suppose $p\equiv 1 \pmod{4}$. In this case the equation $x^2\equiv -1 \pmod{p}$ has two solutions in $\{1,\dots,p-1\}$ whose sum is $p$. Hence one of them must be less than $\frac{p}{2}<n$. We conclude that there exists $1\leq i\leq n$ with $p\mid i^2+1$.
Comment: See [OEIS sequence A219586](https://oeis.org/A219586).  The paper of Hooley linked there seems relevant.
Comment: @KhashF: Great observation. See my response about how to complete this partial solution.
Comment: If you like my answer, please accept it officially (so that it turns green). Thanks in advance!
Answer: For $n\geq 7$, Erdős proved in 1932 that there is a prime $n<p<2n$ of the form $p=4k+1$. From this it follows, by the comment of KhashF, that $p$ divides $\prod_{i=1}^n (i^2+1)$. The remaining cases $1\leq n\leq 6$ are easy to check by hand, so the answer to the OP's question is affirmative.
Answer: A great question studied by many of the masters.
Let $P_n$ denote the largest prime factor of this product. Cebotarev proved that $P_n/n \rightarrow \infty$, which gives a complete answer to your question asymptotically. There were various improvements by  Nagell and Erdős, and Hooley proved that $P_n \gg n^{1 + 1/10}$, see:

Hooley, Christopher, On the greatest prime factor of a quadratic polynomial, Acta Math. 117 (1967) pp 281–299, doi:10.1007/BF02395047.

This answers the question (and more) in the asymptotic range. For small $n$, you can use the following trick relevant to the arguments of these papers (also I see KhashF says this in the comments): take a prime $n < p \le 2n$ of the form $1 \mod 4$, and then note that $-1$ is a quadratic residue mod $p$ so there is an $x < p/2 \ge n$ so that $x^2+1$ is divisible by $p$. So now you win assuming that $\pi(2n;4,1) > \pi(n;4,1)$ where $\pi_1(x;4,1)$ is the number of primes less than or equal to $x$ which are $1 \pmod 4$. Now you dust off some explicit estimates for these quantities.  More than good enough for this purpose is:

Michael A. Bennett, Greg Martin, Kevin O'Bryant, Andrew Rechnitzer, Explicit bounds for primes in arithmetic progressions, Illinois J. Math. 62 Number 1–4 (2018) pp 427–532, doi:10.1215/ijm/1552442669, arXiv:1802.00085.

which proves that
$$\left|\pi(x;4,1) -  \frac{Li(x)}{2} \right| \le (0.0005028) \frac{x}{(\log x)^2}$$
for $x \ge 5438260589$. This is good enough to win then for $n \ge 5438260589$ because then
$$\pi(2x;4,1) - \pi(x;4,1)
\ge \frac{Li(2x)}{2} - \frac{Li(x)}{2} - (0.0005028) \frac{2x}{(\log 2x)^2} - (0.0005028) \frac{x}{(\log x)^2} > 0.
$$
For smaller $n$ you can do the usual ladder trick, e.g.:
$$1^2 + 1 = 2,$$
$$2^2 + 2 = 5,$$
$$4^2 + 1 = 17,$$
$$14^2 + 1 = 197,$$
$$184^2 + 1 = 33857,$$
$$33794^2 + 1 =  1142034437,$$
$$1142034424^2 + 1 = 1304242625601011777,$$
$$1304242625601011736^2 + 1 = 1701048826434620873794109106809733697,$$
$$1701048826434620873794109106809733360^2 + 1 = q_0,$$
where $q_0$ is prime,
$$(q_0-131)^2 + 1 = q_1,$$
where $q_1$ is prime,
$$(q_1 - 185)^2 + 1 = q_2,$$
where $q_2$ is prime,
$$(q_2 - 483)^2 + 1 = q_3,$$
where $q_3$ is prime, and... the PARI stack overflows!
so it is at least true for
$$n \le q_3 = 491437705438594121865352195975\ldots6899550122553886277,$$
where $q_3 > 10^{579}$, but this is better than the bound we already established, so done. (This latter trick goes far enough that you can use crappier estimates at the previous step if you prefer.)


How can I see my follower's health?

Question: Simply put, I don't see any way to gauge how much health a follower has so I know when to heal them. In Oblivion, a health bar showed up when you placed the crosshair on either an enemy or an ally, why would they leave this out?
Comment: Step 1. Shoot follower with arrow Step 2. Look at enemy health bar... You now know your followers Health Level!
Answer: Seems like this only works in combat. (or maybe only when they are damaged?)
I have seen the health bar when I put my cursor over them, but it only works when we're in the middle of a fight.   Not sure if it is triggered by the combat, or by the fact that they have less than full health.
Comment: I can't seem to figure this out either, I'm assuming they are still missing health after the fight and the game doesn't automatically heal them.  So I'm thinking it only shows during battle.
Answer: Their health bar will show up only when you are in combat.  Specifically, you and your follower need at least to hit an enemy or receive some damage for it to work.
Comment: Thanks for the answer, I guess I have just been too busy during battles to notice.
Answer: Given the usual caveats regarding using computing systems and variances of results...
In my experience so far, from release through the 220.127.116.11 update, I have not noticed any consistent pattern at all (sometimes it appears but normally does not).  This includes:

the bar appearing when in combat or not
when their health is not 100% or not
when they were recently damaged or not
whether the player's char is in combat or not (I thought for a while I saw a pattern here... sadly it was not repeatable)
when I am healing them, buffing, etc... or not, lol

Note:
My rig is fairly high end and the settings are almost all set at the highest levels.  As mentioned above, this has not changed at anytime though the different versions from final release which also includes before and after the "evisceration of larger address space, including use of more than 2 gigs of memory" as well as using the 4gb launcher.
Answer: If you're playing on the PC, the Floating Healthbars mod can add floating healthbars to any actor engaged in combat.
Answer: Their health bar shows up when they're damaged. My follower (Lydia) seems to regain her health really quickly so you won't notice it unless they're really hurt.

How can I see my follower's health?


Can I use M1 straight away and get rid of IV in Merkle–Damgård?

Question: In Merkle–Damgård is there any reason why we use a fixed $IV$ at the beginning? Can we use the first block ($M_1$) right away instead of $IV$ and feed it through the compression function with $M_2$.
Comment: I'm pretty sure you can just copy and paste "Merkle–Damgård" from the edits in the previous question, please do so.
Answer: TL;DR: A simple reason why we use the IV is to mitigate second preimage attacks.
This is the typical Merkle–Damgård construction: you split your messages in blocs of $k$ bits (input size of the compression function).
<code>    M1         M2 ...     Mn
    |          |          |
    +--|\      +--|\      +--|\
       | \        | \        | \
 IV ---|  |-------|  |-------|  |---  Hash
       +--+       +--+       +--+
</code>
your proposal is the following :
<code>    M2         M3 ...     Mn
    |          |          |
    +--|\      +--|\      +--|\
       | \        | \        | \
 M1 ---|  |-------|  |-------|  |---  Hash
       +--+       +--+       +--+
</code>
This assumes that the message is longer than $k$ bits! What if the message is shorter ? Well you get the following because you don't have an IV to compress with.
<code> M1 ---  Hash
</code>
Also this idea of using $M_1$ as an $IV$ assumes that the inputs of your compression function have the same size e.g. if you take $f(x,y) \to h$ where:

$\texttt{sizeof}(x) = 128 \texttt{bits}$ and
$\texttt{sizeof}(h) = 160 \texttt{bits}$

This implies that  $\texttt{sizeof}(y) = 160 \texttt{bits}$. Therefore the use of an IV is encouraged unless you want to manage a padding for the first input...
What if we have a fixed last bloc compression ?
Such that:
<code>    M2         M3 ...     Mn ...     LB
    |          |          |          |
    +--|\      +--|\      +--|\      +--|\
       | \        | \        | \        | \
 M1 ---|  |-------|  |-------|  |-------|  |---  Hash
       +--+       +--+       +--+       +--+
</code>
And in the case of very short messages:
<code>    LB
    |
    +--|\
       | \
 M1 ---|  |---  Hash
       +--+
</code>
In the end don't you agree that this is the same has having a fixed first block, also called an IV ? You still need to save this value. Whether it is at the end or at the beginning, it is needed.
A Second Pre-image attack
Quick reminder:

Given an input $m_1$ it should be difficult to find another input $m_2$ such that $m_1 \neq m_2$ and $hash(m_1)=hash(m_2)$. Functions that lack this property are vulnerable to second-preimage attacks.

Feeding directly $M_1$ instead of the IV whatever the construction above (with or without a fixed last block) leads to a second-preimage attack:
Let $M_1$ such that $M_1 = a || b || c || d$ where $||$ is the concatenation and $a,b,c,d$ are of the right size. We pose $H(a || b || c || d) = h$.
<code>    b          c          d
    |          |          |
    +--|\      +--|\      +--|\
       | \        | \        | \
  a ---|  |---h'--|  |---h"--|  |---  h
       +--+       +--+       +--+
</code>
Notice that we have $h'$ and $h''$ such that $h = H(a||b)$ and $h'' = H(a||b||c)$.
What happen if we compute $H(h'||c||d)$ or $H(h''||d)$ ?
<code>   c         d
   |         |
   +--|\     +--|\
      | \       | \
  h'--|  |------|  |---  h
      +--+      +--+

    d
    |
    +--|\
       | \
 h'' --|  |---  h
       +--+
</code>
In both case you will get $h$ which leads to:
$$\begin{align*}
H(a || b || c || d) &= h\\
&= H(h' || c || d)\\
&= H(h''||d)
\end{align*}$$
Thus we have a collision through a second preimage attack (note that the compression function may be perfectly secure!).
How to solve this?
Multiple solutions exists:

Use an IV.
Encode the length of the input in the last block.
add a domain separation encoding at the end of each block.

Solutions has been discussed in the following paper (see section 8:  Implications for sequential hashing).
Comment: @cryptoman updated with the second preimage.
Comment: So you are saying it is not collision resistant? can you give an example of an attack? because to my understanding it looks collision resistant
Answer: If we used this idea of performing $C(M_1,M_2)$ rather than $C(\text{IV},M_1)$ as the first compression step, then for some reasonable choice of the compression function $C$, there would be devastating attacks. That's including with explicit length of the input in the last block, contrary to the second-preimage attack in the other answer.
An example is when the compression function $C$ is an (assumed ideal) block cipher, with $C(S,M)=E_M(S)$ (that is, encryption of the hash state using a message block as key). For a message $M$ leading to a 2-block padded message $M_1\|M_2$, with the proposed shortcut, the hash is $h=H(M)=E_{M_1}(M_2)$ rather than the usual $h=H(M)=E_{M_2}(E_{M_1}(\text{IV}))$.
To reach any desired hash $h$, we chose any valid $M_2$ (the choice is constrained only by the length padding), and compute $M_1=E^{-1}_{M_2}(h)$ (that is, we decipher $h$ under key $M_2$). We then form $M$ from $M_1$ and $M_2$, and $H(M)=h$ holds. This first-preimage attack is trivially extended to second-preimage and collision.
Sure, common hashes tend to use $C(S,M)=E_M(S)\oplus S$ per the Davies-Meyer construction (or a small variation), and it it is unusual to directly use a block cipher as the compression function. But assuming we do, with the usual choice of IV (an arbitrary public random-like value clearly independent of the block cipher), we still get a reasonably secure construct (collision-resistance to usual levels can be proven with standard arguments; preimage resistance could be weaker than usual due to greater freedom in multi-collision attacks).
This example illustrates that the shortcut weakens our security insurance; and that making a security argument for the construction would require stronger hypothesis on $C$, or/and more complex reasoning.

Note: a comparatively minor difficulty with the proposed shortcut is that we'd need a special case for messages short enough to fit 1 block after length padding; like $h=H(M)=C(M_1,M_1)$.
Answer: Actually, the result will be collision resistant, Im afraid the answer that Biv gave is wrong since he didn't pad the message with the length
Comment: Reread the last part of my answer. I do mention the length encoding.
Comment: I read it twice when you wrote it, and now I read it again for fariness, Im sorry but what I said is still applied, the scheme that cryptoman suggested is collision resistant and Im afraid your answer is wrong.
Comment: the reason IV is used is to make the analysis more convenient, with or without IV, it doesn't matter, which means that if you do use IV, and not adding the length, the scheme would not be collision resistant


Mysql general query log tool to play back queries in time?

Question: I am trying to reproduce production activity in time for benchmarking purposes from the mysql general query log. I'd like to be able to use the date data from the log to reprduce the time intervals between queries. So if query 1 executes at 9AM and query 2 executes at 10AM then I want the benchmark to execute these queries an hour apart.
Is there are a tool that does this? I'm assuming there is. The maatkit tool mk-log-player parses out the queries but replaying them with no delay.
Answer: I don't think there is a particular tool to do this. I'd write a simple script to parse log and extract queries to be run.


How safe is RSA generated public-private key pair?

Question: If one has been using the same set of public-private key pair for quite sometime now, given his public key(which is public), can't an attacker find the corresponding private key for that public key by bruteforce?
If not bruteforce then what else methods are there to break the rsa encryption?
Comment: For 128bit, at 1 attempt per nanosecond, it will take 11,342,745,564,031,282,115,445 years to try all keys.  You're bruteforce safe.
Comment: @ChuckFricano, 128 bits tend to be for symmetric keys, not RSA (asymmetric) keys. That's not quite the same problem.
Comment: What key size are you using?
Comment: @mikeazo : 2048
Comment: @user1139023 - So your key size is larger then the number of stars in the galaxy and your asking if its safe?  Its currently safe, eventually change it to a larger key size, always good to do that.
Answer: They can, but bruteforceing these keys takes too much time to actually be a plausible attack. This depends on things like key length, but in general we are talking about on the order of thousands of years (or more!) to bruteforce. And as far as I know, RSA keys aren't subject to key wear, so using a public-private key for a while won't help attackers discover the private key.
Comment: Thanks for answering. But if not bruteforce then what other method is there to break rsa.
Comment: If RSA is implemented correctly, brute-force is the best you can do in attempting to break RSA. However, if RSA is implemented poorly, there are several attacks you can execute. For example, if the algorithm chooses primes too close to each other, or the primes are too small, you can break RSA more easily. This is why you should never implement things like this yourself, and use a trusted library.
Answer: If the attacker has discovered the solution to the RSA problem, they are probably too busy heading to the bank to worry about whats on the other end of the line.
http://en.wikipedia.org/wiki/RSA_problem
Answer: Like all credential schemes, RSA is easily cracked by not being cracked but by doing an end run around it. If you realtime your malware to boost the entered key out to an IRC channel or other type of IM, you have about 30 seconds to steal both factors and use them to break an account. The major defense becomes keeping the computer used to do the login operation clean of all malware.
Comment: This isn't "cracking RSA" its just observing the (unencrypted) private key.  IRC/IM/30sec limit has nothing to do with it.  If you have malware running on the same system (with appropriate access controls) as the private key at any point you can copy the private key (and transfer it via any method IM/IRC/SCP/FTP/HTTP/Email/blog post/etc.), and use it to authenticate in to the remote system.  Yes, if its password/passphrase protected secret key the malware will either have to keylog the password/passphrase or find the decrypted private key from memory when its used.
Comment: Yep, people hate it when their fancy, expensive playtoys get shown to be avoidable. The OP asked if there was a way, and this kind of exploit is out there.


Arduino clone voltage regulator

Question: I am sourcing some parts for an Arduino clone. My question is really simple and I am quite sure that I know the answer, but I wanted to clarify my suspicion. Can I swap out the LM7805 with this part ?? 78M05F
Comment: 2nd link fixed.
Comment: 2nd link is broken.
Answer: Yes, you can use that regulator. It is a 5 volt output linear regulator. You could check to see if the max output currents were similar, but for arduino applications, they do not draw much current.
Comment: The maximum output current is of limited relevance anyway, as the actual maximum may be much, much less depending on the input voltage and heat sinking.
Answer: LM7805 has output voltage 5V and output current 1.5A
And 78M05F has output voltage 5V and output current < 1A
If the load is connected to the LDO requires currents below 1A, you can replace it.
Comment: A linear voltage regulator is not a "power supply" but rather only one component thereof.  The supply rating is the least of what any of its components can handle; in your typical no heatsink except the PCB Arduino style design this is thermal, rather than what the chip itself could achieve in some more ideal installation.  Accessories designed to plug into an Arduino style board should not assume availability of supply current in the realm which would challenge this part's ultimate rating; if they need that, they should have their own supply inputs, **especially** stepper motors.
Answer: If you are powering your clone using batteries, I'd recommend using a Low dropout regulator, depending on you current requirement choose any one below
1.LM2940 this can withstand 1A
2.LM2937 this can withstand 0.5A
3.LT1118-5 this can withstand 0.8A
You can also run it at 3v3
1.LT1763 this can withstand 0.5A
2.LTC3525 this can withstand 0.4A

Arduino clone voltage regulator


Code for deck footers in sugar-sand

Question: What is the Code for "deck footers" that are in a sugar-sandy area?
Answer: If you're in the US, then IRC R507.3 is a pretty safe bet for deck footings.
"Sugar-sandy" sounds like a description of SP soil under the USCS (poorly graded sand, where well graded sand has a wide distribution of particle sizes and better bearing strength), but that's assuming that you're talking about the native soil below the top 12" or so of upper soil. The table under IRC R401.4.1 converts the SP soil classification to a 2000 psf bearing capacity, where you'll need that bearing capacity value for sizing deck footings.
And "tributary area" is a concept that goes along with uniform distributed loading, e.g. "40 psf," to find loads on structural members. For the typical grid pattern of posts, the tributary areas are a bunch of rectangles. For a heuristic, imagine the deflected shape of the flat deck surface under a uniformly distributed 40 psf and now flip it upside down. Under a slow, uniform rain, each post is the drain position for its tributary area. So a post gets half of any beam going to another post. It gets half the joists plus deck spanning from the half beam to any adjacent beam (or ledger). The post gets the full load of any joists cantilevering from one of its tributary beam halves. Etc. Etc.

Code for deck footers in sugar-sand


Deriving Einstein-Rosen Bridge

Question: I know that an einstein rosen bridge is derived by a coordinate transformation on the schwarzschild metric, but I can't find much on it online, could someone please show how to change the metric into a "wormhole" one and explain it a bit?
Answer: The usual Schwarzchild metric is given by
$$
ds^{2}=(1-2 M / r) d t^{2}-\frac{d r^{2}}{1-2 M / r}-r^{2} d \Omega^{2}.
$$
We can make a coordinate transformation of the form $u^2 = r - 2M$.
This gives $1-2M/r = \frac{u^2}{u^2 + 2M}$ and $dr = 2u du$ such that the line element becomes
$$
ds^{2}=\frac{u^2}{u^2 + 2M} d t^{2}-4(u^2 + 2M)du^2-(u^2 + 2M)^2 d \Omega^{2}.
$$
This is the line element for the Einstein-Rosen bridge. This coordinate transformation gives the spacetime two distinct asymptotically flat regions, one defined by $u \rightarrow \infty$ and the other by $u \rightarrow -\infty$. Where a singularity once was, at $r = 2M$ (which was only a coordinate singularity anyway), we now see that there is a 'bridge' (now located at $u = 0$) connecting each asymptotically flat region $u = \pm \infty$. This is the so-called wormhole.
Comment: Thank you for your help. One thing that I'm confused about is if we set r= 2m, then u=0, making no time pass. Is this correct?
Comment: It seems like it might be, if only because the Schwarzchild black hole is entirely hypothetical:  Astrophysical black holes (for which there's much observational evidence) are described by the Kerr or (if electrically charged) Kerr-Newman metrics, reportedly much more complex.
Comment: ah i see. I wonder if you could or if some source can derive the wormhole mathematics from  a kerr metric, for a spinning black hole I guess. And I also had some trouble understanding the -infinity to positive infinity, could you show how they found that from the line element?

Deriving Einstein-Rosen Bridge


How to transmit computer audio to a wireless microphone receiver?

Question: I'm currently trying to broadcast my computer's audio output to a wireless microphone system receiver so that I can accompany my school's coming presentation with music. The frequency range is 480-598 MHz, and I can't find a device capable of emitting a signal that can be picked up by this system. Please, if you have any ideas on how I may do this, or have come across a product with said capabilities, please let me know. Thanks in advance!
Comment: Welcome to stackexchange! You should state your country, since 480-598MHz lies in the UHF broadcast band and is subject to licensing in many countries.  It would also be useful to state your microphone system.
Comment: In the US, the FCC allows unlicensed low power transmissions in certain broadcast bands under Part 15 rules.
Answer: If you have an extra microphone, it may be as simple as placing it near the computer's speaker and adjusting the volume levels. This requires the mic and computer to be in a quiet area where it wont pick up outside noise.
If the computer is in a noisy area, you could tap into the microphone circuit at the input and run a headphone cord direct between the two. You may need to use a transformer to get the impedances right.
Check the wireless microphone system's receiver and amplifier for auxilliary inputs. You may not need wireless at all!
Answer: An educational SDR kit, such as the ADI ADALM-Pluto or a Lime SDR Mini, can be used as a UHF exciter, with any modulation that your receiver requires generated by software synthesis (GNU radio , et. al.).  I might use a Raspberry Pi to run the SDR modulator/synthesizer code.
If legal within your jurisdiction, you may need to use a suitable filter before any antenna, and way to estimate or measure the RF output power and purity, to make sure your RF emissions stay within regulations (FCC Part 15 in the US).
The harmonics of a toggling Raspberry Pi GPIO pin can also generate modulated UHF signals (see librpitx/RPiTx on GitHub), but a GPIO will definitely need filtering to produce a clean enough legal signal for any antenna.
Answer: I have indeed done this simply by disassembling the wireless microphone and wiring in an auxiliary input jack in place of the mic element along with an attenuator so the AUX IN signal from the computer's earphone jack will not overload the transmitter inside the mic- as suggested above by R Pau.
It is also possible to signal-trace the (tiny) PC board inside the mic to find a point in the signal path where the signal level is close to that which your computer earphone jack furnishes so you don't need the attenuator, and then wire up the AUX IN to that point.
It is also possible to install a microswitch so you can turn off the AUX IN function and use the mic for its original purpose instead.


Are you able to use your insurance policy when you're involved in an innocent vehicle purchase situation?

Question: I purchased a vehicle for $70,000 with what could be a false title. Unfortunately the California DMV was able to register the vehicle, so they too did not know this title or name on the title was possibly fake. I received my registration and eventually vehicle title in the mail. The vehicle was also fully insured and remained this way for aproximately 3-4 months, until the vehicle was finally reported "stolen". The vehicle was taken by law enforcement for further investigation of the parties involved in the sale. At this point if the vehicle is returned to the "rightful" owner, am I able to file a claim through my insurance for full compensation?
Comment: This surely depends on what type of insurance you have.
Comment: Hello Hector, and welcome to law.SE! Unfortunately, your question is missing important detail. Please edit it to explain what type of insurance you have (liability, comprehensive coverage etc.) you have, and the jurisdiction you are in.
Comment: Just as a note: What you need (would have needed ) is probably "title insurance" for the car purchase.
Comment: I'm in the Monterey County (California) and I currently still have full coverage on the vehicle through Allstate Insurance even tough the vehicle is no longer in my possession and is still with law enforcement for further investigation. As to further information to the transaction, the vehicle was previously a luxury rental and an organized crime of individuals were the ones who sold me the car. The "previous owner" and independent vehicle renter did not report the car stolen until 3-4 months after the steal and sold it to me. This is why I was able to register and insured the vehicle.
Comment: @Hector, please add details to the question, not in a comment
Comment: The seller of the vehicle might be liable for fraud - they got money for a good they didn't deliver.
Answer: No
Auto insurance covers loss or damage to the vehicle - the vehicle is neither lost nor damaged.
The risk you describe is called title insurance and is normally bought for real estate.


Best way to implement this box / table design in Adobe Indesign

Question: I am sorry about asking such a newbie question but I am wondering if Adobe Indesign has any easy method of making the box that is on the top of the page that I posted below (With the numbers on top and bottom and text inside of it). I have to make a page similar to this.
Answer: Yes.
Set the text using tabs and paragraph returns between where the table cells are. Then highlight the text and choose Table > Convert Text to Table. Click okay.
From this point it's a matter of adjusting the table. There are many options. You may want to review the Indesign help files regarding tables.
Comment: I decided to do that but create a table in the textbox first and then entering the text.


Why are there no discrete OLEDs?

Question: Why does nobody make a discrete OLED in an SMT or thru-hole package, like we have with LEDs? Is it a cost thing or a limitation of the process?
Comment: @BrendanSimpson Organic LEDs are a healthier alternative to regular arsenic-laden factory LEDs.
Comment: Why would you want one?
Comment: They do http://www.konicaminolta.com/oled/products/
Comment: Ok, let me clarify: Why would you want an OLED in an SMT or thru-hole form factor? What benefit would an OLED provide that a conventional LED doesn't?
Comment: advantage of OLED is form factor, so no SMD or TH, but large plates for example: https://www.lgoledlight.com
Answer: Making an "OLED" and then putting it in a package, like you'd do with a Chip LED is commercially nonsensical from most points of view:

It loses its microscopic-application-advantage
It adds no flexibility advantage
It costs several orders of magnitude more
The optical efficiency is lower (due to less ideal materials for semiconduction)
Decided to edit in based on Spehro's comment above:
Buyers of individual LEDs (in most 'western' areas) don't care as much about non-Arsenic/Non-Indium, etc

So you'd end up being back at paying >=$1 for a single LED, which offers the same amount of light as a begin of the century White LED, in a package in which LEDs get made with tens to hundreds of times the output at $0.05 a piece.
That's not really a great deal to make, is it?
OLEDs have great advantages when you use them for displays, because it's possible to grow them in a usable formation and the materials used offer some manner of flexibility (both in management/bio-factors and mechanically), but they are far from the efficiency of just making a wafer of Aluminium/Gallium/Arsenic/Phosphide/Indium/etc in whatever combination gives the best amount of light.
And "we" in "the west" still want to have the entire world for a dime, so nobody is about to go ahead and offer that deal.

Why are there no discrete OLEDs?


Shortcut to command prompt pointing to specific folder

Question: How can I create a desktop shortcut that opens up Command Prompt (cmd) in a specific folder?  I'm getting tired of navigating from my home directory to my development directories.
Comment: Typing `cmd` in the address bar of any folder in Windows Explorer opens it in the folder too (I'm on Windows 10, not sure if it works in older versions!)
Answer: Create your shortcut and then right click to get the properties dialog and set the "Start in:" property to be your folder.

If you are running the shortcut as an administrator you should add your desired path to the target (instead of "start in"), for example:
%windir%\System32\cmd.exe /k cd c:\crp

or
%windir%\System32\cmd.exe /k pushd c:\crp

because the "start in" value is ignored when running as administrator (all credits go to @barlop for their answer and @T_D for their comment)
Comment: When `Properties > Shortcut Tab > Advanced > Run as administrator` is enabled, this solution does not work.
Comment: @JosephDykstra in that case you should add your desired path to the target (instead of "start in"), for example: `%windir%\System32\cmd.exe /k cd c:\crp` or `%windir%\System32\cmd.exe /k pushd c:\crp` because the "start in" value is ignored when running as administrator (all credits go to @pacoverflow his answer https://superuser.com/a/1067902/550943 here). Cheers!
Comment: This does not seem to work for any drive other than where your system is installed.
Comment: @T_D - it's https://superuser.com/users/42672/barlop - who wrote that answer, but thanks for finding it!
Comment: @ChrisF ah yep, the answer got edited
Comment: If your target is a different drive than C: add the `/d` flag like so `%windir%\System32\cmd.exe /k cd /d f:\crp`
Answer: Or you could just use a target string similar to this:
C:\Windows\system32\cmd.exe /k "c: & cd c:\drv\bat"

Obviously C:\Windows\System32\cmd.exe is to run cmd. The /k carries out the command by string and continues. The string is the "c: & cd c:\drv\bat". Within the string, the c: just changes the drive letter. The & ties both commands together. And the cd c:\drv\bat is basically a change directory. Also one thing to note, the cd c:\drv\bat should also change drives.
Comment: Isn't that "cd /d " for paths on another drive?
Answer: There's more than one way to skin this cat, for sure.

A batch file could also do it for you.  This will start you off in the Program Files directory on the C drive:
@ECHO OFF
CD /D "C:\Program Files\"
CMD

Especially make sure to use the /D switch, if you're changing to a different drive than where CMD normally starts you.

The Open Command Window Here PowerToy for Windows XP is also rather handy for going straight from an Explorer session, into a CMD console at a certain directory.

You can use the AutoRun string value in one of the following Registry keys, to essentially automate the above for all CMD sessions.
For your account only:
HKCU\SOFTWARE\Microsoft\Command Processor\

For all users on this machine:
HKLM\SOFTWARE\Microsoft\Command Processor\

If the AutoRun value doesn't exist, create it as a REG_SZ type (also known as String Value).  For Data, you can either put the path to a batch file like the one above (remove the ECHO and CMD lines) or just insert the CD command as shown in the above example.
The up-side of pointing the Registry to a batch file, instead of just adding the command, is that you can add in as many AutoRun commands as you like to the batch file.  This way you can create custom prompt formats or script other tasks you want done at every launch of CMD.
One thing to keep in mind if you make these Registry changes though, is that they may affect the behavior of other batch files.  Particularly, batch files may find themselves not starting in the directory they were originally written for.
As always, backup your Registry before making any untested changes.
Answer: Answering 7 years after the question asked :-)
It might help someone anyway!
To have the shortcut work with Run as Administrator option enabled:
Insert your commands separated with an & In the Target text box of the Properties dialog after /k switch. Don't forget that the commands containing folder names with spaces (e.g., Program Files), should be enclosed in double quotation marks.
The following example

opens command prompt
runs Node.js batch file
switches to drive D
goes to code directory

C:\Windows\System32\cmd.exe /k "D:\Program Files\nodejs\nodevars.bat" & d: & cd d:\code
Comment: This worked great, should have more upvotes for Administrative shortcuts *(better explanation with examples imo)*.
Comment: I looked into the nodevars.bat and could not find anything that justified a difference (see: https://github.com/nodejs/node/blob/master/tools/msvs/nodevars.bat) Can someone explain it?
Answer: Insurgence's answer didn't work for me, but this did.
For me, I wanted to launch a cmd shell, run the node vars batch file, and then change dir to my repo root (c:\source\myproject).  Here's the command, note the double quotes:
cmd /k ""C:\Program Files\nodejs\nodevars.bat" && cd \source\overlay"

Additionally, to get really fancy and launch angular, I use:
cmd /k ""C:\Program Files\nodejs\nodevars.bat" && cd \source\overlay && ng serve -o"

Answer: You can go to the directory where the cmd.exe shortcut is located by default (the one that comes up when you type "cmd" in the start menu) and change the start directory there.
For me, the shortcut was located in:
C:\Users\<username>\AppData\Roaming\Microsoft\Windows\Start Menu\Programs\System Tools

Right click the cmd shortcut, select properties, and modify the start in directory to be whatever you like. Now, when you start a command prompt from the start menu (as I often do) it will have the directory you want.
Answer: I may be few years too late but Why to make things far too complicated just to run cmd as admin in specific folder.

Create a shortcut.
copy this to target folder
%windir%\system32\cmd.exe /k " cd\ & cd c:\YOURdirectory
if located in different drive other than c: then do this
%windir%\system32\cmd.exe /k " cd\ & YOURdrive: & cd YOURdrive:\YOURdirectory
copy this to start in
%homedrive%%homepath%

on advanced properties choose to run shortcut as admin.
Answer: As a better and more flexible option you can use Autohotkey to create keyboard shortcuts to switch between directories in cmd.
E.g. using the script shown below you can switch to any directory (bound to Ctrl+F12 combo).
It just sends the same commands which you would type in the cmd prompt.
#If winActive("ahk_exe cmd.exe")

^F12::
    send {text}cd /d d:\mydir\mysubdir\
    send {enter}
return

With this approach you don't need to create multiple shortcuts for the executable and
can add other useful commands for navigating, etc.