Adjoint action notations $\operatorname{ad}(X)(Y)$, $\operatorname{ad}(X)$ and $\operatorname{ad}_x(Y)$ are equivalent??

Question: As the title says I'm a bit confused with these notations of adjoint action of Lie algebra on itself. Are these notations ($\operatorname{ad}(X)(Y)$, $\operatorname{ad}(X)$ and $\operatorname{ad}_x(Y)$) equivalent?
Answer: No, they are not. The linear map $ad: L\rightarrow \mathfrak{gl}(V)$ is a Lie algebra representation by the Jacobi identity. It is defined by $x\mapsto ad(x)$ with $ad(x)(y)=[x,y]$. Some people write $ad_x(y)$ for $ad(x)(y)$. So $ad(x)$ is an endomorphism of the vector space of $L$, while $ad(x)(y)$ is an element of $L$ itself.
Answer: They are not all quite the same, but the first and last are equivalent.  The key thing here is to keep track of the "data types" involved, i.e. what the inputs and outputs of the functions involved are.
$\DeclareMathOperator{\ad}{ad}$
$\newcommand{\g}{\mathfrak g}$
$\newcommand{\gl}{\mathfrak {gl}}$
We begin with a function $\ad : \g \to \gl(\g)$.  That is, for any $X \in \g$, $\ad(X)$ is an element of $\gl(\g)$, which is to say that $\ad(X)$ is a (not necessarily invertible) linear map from $\g$ to $\g$.  That is, we have $\ad(X):\g \to \g$, so that $\ad(X)$ maps an element $Y \in \g$ to $\ad(X)(Y)$.
To make the notation easier, sometimes we define $\ad_X = \ad(X)$.  Now, we have $\ad_X: \g \to \g$, so that $\ad_X$ maps $Y$ to $\ad_X(Y)$.
So, in particular, given $X,Y \in \g$: $\ad(X) = \ad_X \in \gl(\g)$, while $\ad(X)(Y) = \ad_X(Y) \in \g$


Removed rpm symlink from bin/rpm

Question: I have accidentally removed the symlink to /bin/rpm from my Centos7 installation. Now the rpm command is not working. Does anyone know how to fix this one or how to add this again as symlink ?
Answer: The /bin/rpm is typically not a symlink.
There is a symlink /bin which points to /usr/bin directory.
So if you have removed /bin/rpm - you have removed an actual file, not a symlink.
For restoring it, you will need access to another CentOS 7 machine, running the same architecture (e.g. x86_64).
Copy it from the other machine by uploading it to the server via SFTP.
Ensure matching version afterward, by running:
yum reinstall rpm

Comment: Thanks ,
just "yum reinstall rpm" works for me.
Answer: Just reinstall rpm (yum reinstall rpm, or even yum reinstall /bin/rpm -- if you don't know the package's name). The yum program uses RPM format and databases, but doesn't run rpm itself.
Lucky you. Next time you'll delete something really critical. Be careful when running as root.

Removed rpm symlink from bin/rpm


Folder Action to Convert Image Format after AirDrop?

Question: I often use AirDrop to move photos from my iPhone to my Mac, and I'd like to automatically convert any HEIC photos to JPEG. I created this folder action in Automator, and it works when a photo is dragged into the Downloads folder but not when it has arrived via AirDrop. Is there a trick to getting folder actions to work with AirDropped files?
Answer: This works for me, though the conversion is from jpeg as I have an oldish iPhone :(
I have added lots of notifications to confirm each step - simple debugging!

The pause is a hint I got sometime ago for AirDrop.  Your Get Selected Finder Items is either unneeded or wrong (not sure which).
Comment: It works! Thanks, you're a peach. I had tried a pause but it wasn't long enough. And you're right about the "Get" action—I tried that because I noticed it in a different AirDrop-related folder action but it was not needed.
Answer: I needed to tweak this a little for me on Sonoma. I made the pause 10 seconds just in case I transfer dozens of files in bulk. And then instead of copying the files, I chose to duplicate the finder items because I wanted to convert them and put them in the same folder (in my case, "Downloads". Usually I'm uploading receipts or silly images for fun online so instead of having to send 3 MB per pic, I do a Scale Images down to 30% of the original size and that makes it around 200 kB usually for my images.


How to clear terminal after logging out?

Question: Currently on my laptop when I log out I can still see traces of my last activities (in tty). How can I tell it (Gentoo) to clear the screen before logging out?
Answer: If your shell is bash, add clear_console or reset to ~/.bash_logout. If your shell is zsh, add that command to ~/.zlogout. You might want to run this only when the shell is a login shell on a Linux console, e.g.
if [[ "`tty`" = /dev/tty[1-9] && $(ps -o comm= -p $PPID) = login ]]; then
  clear_console
fi

As the system administrator, you can configure the console to always be cleared before showing the login prompt. The easiest way to do that is to use mingetty instead of getty; you lose support for serial consoles but gain a few features such as screen clearing. First install mingetty. Then, assuming you use SysVinit, in /etc/inittab, change lines like 1:2345:respawn:/sbin/getty 38400 tty1 into
1:2345:respawn:/sbin/mingetty tty1

The Upstart equivalent would to change exec /sbin/getty -8 38400 tty1 to exec /sbin/mingetty tty1 in /etc/init/tty1.conf (these are Ubuntu paths, you may need some adjustments for Gentoo).
Comment: One other way I use (if you have root) is to add `\e[2J\e[f` to the beginning of `/etc/issue`, so the screen gets cleared as part of drawing the login prompt. I like `clear_console` in `.bash_logout` more though
Answer: If you don't have bash or zsh (e.g. you just have ksh), or you want a way that works in all modern shells, this is how I do it:
.profile
test -f "$HOME"/.exitrc && trap ". $HOME/.exitrc" EXIT

.exitrc
type clear >/dev/null 2>&1 && clear

In my case, .bash_profile and .zlogin don't have any commands in them, that way I can just change .profile or .exitrc, rather than having to edit separate files for each shell.
.bash_profile
test -f ~/.bashrc && . ~/.bashrc
test -f ~/.profile && . ~/.profile

.zlogin
test -f ~/.profile && . ~/.profile

Answer: If you don't want to edit anything and you don't need to clear the screen every time, pressing ctrl+c works for clearing the screen back to a login prompt.


Find all triplets $(x,y,p)$ such that $\frac{xy^3}{x+y}=p$

Question: Find all triplets $(x,y,p)$ such that $\frac{xy^3}{x+y}=p$, where $x$ and $y$ are positive integers and $p$ is prime. I don't have much experience with Diophantine equations such as this, and I don't know how to fully solve it. After a bit of fiddling around I found that $(n^4-n,n,n^3-1)$ is a solution if $n^3-1$ is prime, so $(14,2,7)$ is one solution. I suspect this is the only solution since $n^3-1$ is not prime for $n>2$, and I don't see any other obvious ways to generate solutions.
Comment: $n^3-1=(n-1)(n^2+n+1)$
Comment: Yes I see that, but I'm wondering if there are other solutions, and if not how to show that?
Comment: $x(y^3-p)=yp\implies x$ and $y^3-p$ divides $yp$.
Comment: This is (obviuosly) not an answer, but this problem looks like a family of very difficult problems (if you don't know the trick) that can be solved using  infinite descent.
Answer: We have: $y = \dfrac{xy^3}{p} - x$. Thus $p \mid xy^3$. So either $p \mid x$ or $p \mid y^3$ , hene $p \mid y$ since it is a prime number. If $p \mid x \implies x = kp\implies y = ky^3 - kp\implies kp = ky^3-y = y(ky^2-1)$. thus $p \mid y$ or $p \mid ky^2-1$. We have again two cases: if $p \mid y$, then $y = np \implies kp = np(ky^2-1)\implies k = n(ky^2-1)\implies n = \dfrac{k}{ky^2-1}\le \dfrac{k}{k-1}< 2\implies n = 1\implies k = ky^2-1\implies k = \dfrac{1}{y^2-1}\notin \mathbb{N}$. Thus $p \nmid y$, and we must have $p \mid ky^2-1\implies ky^2-1 = pm\implies kp = y(pm)\implies k = my\implies my^3 - 1 = pm\implies m(y^3-p) = 1\implies m = 1 = y^3-p\implies p = y^3-1 = (y-1)(y^2+y+1)\implies y - 1 = p, y^2+y+1 = 1\implies y = 0$ which is absurd, so $y-1 = 1, y^2+y+1 = p\implies p = 2^2+2+1 = 7, k = my = 1(2) = 2, x = kp = 2(7) = 14$. Thus $(x,y,p) = (14,2,7)$. The second case is $p \mid y$, thus $y = rp\implies rp = \dfrac{xr^3p^3}{p} - x\implies rp = xr^3p^2-x\implies x = (xr^3p-r)p\implies p \mid x \implies x = tp\implies r = tr^3p^2 -t=t(r^3p^2-1)\implies t \mid r$. Also, $t = r(tr^2p^2-1)\implies r \mid t\implies r = t\implies 1 = t^3p^2-1\implies 2 = t^3p^2$, and this equation has no solution in $\mathbb{N}$. Thus the only solution to the given equation is: $(x,y,p) = (14,2,7)$.
Comment: nice and **quick** calculus :)
Comment: Thank you for the answer! This suggests a systematic way to solve similar problems so I appreciate it


Getting attacked when I don't have a bounty?

Question: I'm in the Dark Brotherhood on the quest recipe for disaster, but when I go to get the guy from the town the guards try to kill me even though I don't have a bounty. I checked if the same happened in other places and they still attack me and I don't have a bounty. How can I fix this?
Comment: Did you ever commit a crime and tell the guards you'd rather die than go to prison?
Comment: You might try hiding in a corner in a different hold for a week (so that the town and its guards get reset). I don't know if that'll fix it, but it sounds suspiciously like a bug I encountered with Faralda in Winterhold the other day.
Answer: You can try to use the command "paycrimegold" - simply use your curser and tap on a guard, then write this command.
If this doesn't work, please say what is attacking you - normal guards, bandits, the emperor men(The Occuloti) because if they attack you you can kill them and the guards might even help you.
Answer: Have you tried going to some far away place and rest for 30 days?  This resets the quest area and NPCs and might fix the problem.
Alternately refer to this extensive answer regarding fixing quests.
Comment: In game time of 30 days.
Comment: When you say 30 days, do you mean in-game or real-life?
Answer: I can't comment yet, but I/we need more info. Is it when you enter Markath to speak with Anton Virane? Having this info, will help us help you.
If that's the case, there's a well known bug, that the guards at Understone Keep will be hostile due to an earlier bounty. Unfortunately it's not fixed by the (Un)official patch yet. Have you tried reloading a save from before you get the quest, and then enter the area before taking the quest? My guess would be to reload an older save.
In general, and especially with mods installed, it's time saving in the long run, to make a hard save every 30 minutes or so. As in Esc > Save > New Save. It takes less than 30 seconds, but can save hours if you run into a bug like this. After a while, when you're sure you don't need the saves, it's simple to delete them to save harddrive space.
Answer: These guards are bugged and glitched, as if they think you're a criminal, but you're not. here are some ways pf fixing this glitch.

Wait and rest in another town or in the wilderness for 30 days or more to reset guards.
Use the cheat on each guard, stopcombat to make them stop combat.
Your bounty might be bugged, so try using payfine cheat.
Cast a Calm spell on the guards which will make them unhostile, but only for a limited time!

If you're still in trouble, then you're doomed.


I am writing some short stories and can't wrap my head around a time travel paradox

Question: I put an object in a time-travel device (say something that looks like a microwave ). I am now going to transport the object in time, but leave it in the same position relative to the Earth.
If it was into the future, that's fine. It could simply disappear and re-appear in 5 mins, BUT, if I transport it into the past, then of course it would have already been in the device when I attempted to put it in...
How the heck can I get around this?
Comment: @gorf wouldn't it be that you always had the object, not that your memory was changed? Also the fun thing to do is to receive an item from the future, then later, send it to the past. Where did it come from?
Comment: In such a way that it makes sense to the story, obviously. It's not as if time travel is a real thing. It's your story, and in your story time travel works the way you want it to work. For example, maybe in your universe a relaxed version of Hawking's [Chronology Protection Conjecture](https://en.wikipedia.org/wiki/Chronology_protection_conjecture) holds, and a person cannot change their own past -- meaning that you cannot send the object back in time where you would have already seen it.
Comment: Hey, let's get REALLY serious about this, and make it a real microwave-time machine. Put a cake in raw, and have it come out 'baked' in the past. THAT would be a real challenge.
Comment: There isn't really an easy solution for the causality problem, but if you're just asking how this could happen: The object would simply have appeared there five minutes ago and you took it out before putting the original in. Or, for extra paradox points, the object is actually the same and was never created :D
Comment: It creates a new timeline, and the microwave will never see his friends or family again. That's the most common excuse
Comment: I don't like the "new timeline" edge that many use.
I would like to somehow expand this.
Let's say I put in the object, send it back. The moment I do my memory is altered to accommodate the existence of the first object, which let's say I take out, lay to the side and proceed.
My memory would then be altered again and so on until I decide not to continue, at which point I never sent anything back in the first place? Or I continue, and produce an infinite amount of duplicate objects. if followed to its impossible conclusion.
Comment: AlexP, Unfortunately my twist in this story necessitates being able to do exactly that.
Comment: @gorf Well, at least you weren't your own father. Wonky things happen when time obeys you.
Comment: @Starpilot. I am not sure (as is absolutely nobody :) ). I envisage that until you do it first time, then it hasn't happened, but once you do it, it has. I don't like the stories where characters say "We need to send this back, because we did".  There needs always (IMO) to be freedom of choice. Something you do in the  past or present affects the future, and something you do in the future affects the past. The only way I see to accommodate this is that all our memories are instantly changed to how it is.
Comment: @gorf the irony about the situation you described is that the reason the time travel happened in the first place is "just because". I guess to me it's just that if someone goes to the past, they will have always been in the past, not just everyone's mind updating when they return to the future. Think of it from the view of someone visiting your present from the future. From your view, future hasn't happened yet, so everything advances according to the changes made by the time traveler. But your version of time travel does allow for more suspense when writing a story.
Comment: Well, you are going to have fused or explosive cakes in some way, assuming the two versions of the timeline are incompatible in the most explosive way possible

Otherwise, this is fundamentally inconsistent, as the entropy of the starting state and the end state are unequal despite sharing the same point in spacetime, and any self consistency principle will prevent it from happening
Answer: Inspired from this genius who made Causality (and perhaps countless other time travel stories I have read in the past 4 years)

Assuming:

The timeline follows the malleable timeline model (the class of models used in movies like Looper, Back to the Future Triology, A Sound of Thunder etc.)
For simplicity, assuming that the ripple is instantaneous, and it does not causes secondary changes to the surrounding environment (you can do all sorts of crazy things with that)
The backward time travel is instantaneous, teleport like (otherwise all sorts of complication can occur, which I have not worked out the dynamics yet for such physics)

Now imagine you set up a camcorder 5 minutes ago to monitor the microwave. Then you decided to place the object (e.g. a box) into the microwave. As soon you turn it on, in your perspective, the box disappears. At the same time, you (unawarely, unless you have ripple proof memory to keep the memory of two versions of a timeline in your brain) remembered that the box was always there 5 minutes ago (you checked the camcorder footage and it is indeed the case, though you do not remember "previously" the object was never there "before" 5 minutes ago "before" you send it back, nor do you find the footage of the camcorder strange, other than a box appearing out of thin air.
Thus to both you and as recorded by the camcorder, the box appeared 5 minutes ago from nowhere, and then just before 5 minutes later, you arrived with the box, place the box into the microwave. Now the trouble here is the time period when you opened the microwave and put the box in (say from 0.5 minutes to 0 minutes), because that's the period where the worldlines of the box and its past self will unavoidably collide. I currently can think of two solutions:

Time travel obey some kind of continuity condition. Thus the original ans new version of the timeline will "smear out", and blend together to accommodate the two boxes. One possible way is that from viewing the camcorders recording, the box that mysteriously appeared out of nowhere disappears in thin air again for no reason at 0.5 minutes, and then you arrived at the empty microwave, opened it, put the box, and then send it back. If the box is a coherent quantum state, then a more perfect blending can occurs, as during those 0.5 minutes where there should be a conflict of positions between the two versions of the boxes (as seen globally in the spacetime diagram), the two boxes as seen in the footage passed into each other, fused with the one you put in (and you do not found that strange memory wise, because you always remembered it is like that, though you will still remember that this box is special that it can somehow passes through each other.). If the box is some electron spin, then it is even easier as from 0.5 to 0 minutes, the two electrons (one which pops out of nowhere) fuses into one violating charge conservation, and then disappears as you turn on the microwave
If such stalemate resolution is not allowed, then the new version take preference, including overriding everything that is involved including the very act you put the box into the microwave. In this new version, the microwave is always full, and then you still decided to press the button anyway to send it back. Here there is no contradiction, because the box that is being sent back at 0 minutes appears at 5 minutes ago, and stays in the microwave until you press it

Now conservation of energy is not conserved, as expected when you warp spacetime to such extent to change its topology. This can be accepted as a property due to phenomenon caused by time travel, or that the microwave have expended some energy to fuel all the relevant rearrangement in that section of the timeline to arrange all matter-energy into the new version. Nobody and nothing will ever remember there is a change in timeline, because everything have been overwritten to accomodate that, through they might remember that probability conservation is violated between 0.5 to 0 minutes without any explanation, as well boxes appearing out of nowhere that is later rationalised to be the same box sent back.
If CPT symmetry holds in your universe, then for Case 1, the mysterious electron that appeared out of nowhere is really a positron (likewise an electron if what you put in is a positron). That way, a more natural annhilation occurs and you have charge conservation as expected. This case is interesting because it will suggest entanglement of charges can be made possible by the microwave such that the two particles to be annihilated in the overlapping region has to have opposite charges, spins etc.
So if time travel does work in the way I laid out, then such a universe will be an interesting one, because any videoing of the time period between the arrival and destination of the object to be sent will reveal a conservation law violation. In addition, the boxes will have an ontological existence, as they are now trapped by your actions (which you do not remember due to memory overwrites) into an ontological loop
There's actually one more solution, which requires the least changes, but also the most physically straightforward

If there is absolutely no compromise or none of the conservation laws are allowed to be violated unless it is from an unexplained process, then some kind of fusion or explosion happens as soon you close the door and press the button. This is because the moment when you do so, the spacetime topology changes and now the box just sent in in the old version now coexists with the new timeline where the box was always there 5 minutes ago (this box will have an ontological existence). We can clearly see that there are two copies of the boxes have overlapped in 0.5 to 0 minutes. This means the atoms will be pushed close enough to trigger fusion, or that pauli exclusion will send the boxes fly apart from the microwave door with great force. Suppose you did survive the ordeal, then you will be lead to think that the microwave converted some electricity into another copy of the box, which is now blow apart. Meanwhile the camcorder will record the box was always there, you then open the microwave, place another box in (acting as if with no clue there is a box there), then as soon the boxes contact, an explosion or fusion happened.
Answer: Time and space are extremely interlinked. You move through space fast, you experience time slower than things that are moving slower and experience time fast. Near a Black hole space is being altered and time also slows, but at the event horizon something magnificently weird happens: time is effectively zero at the event horizon of a Black hole and at that moment space and time switch places, allowing the matter inside the event horizon to instantly be pulled to the center of the hole. Time and space are so interlinked that their physics can actually switch places in certain circumstances!!
Like many others you treat time travel as teleportation, but when pushing something through time its far more likely that it'll behave similar to traveling through space as they are so interlinked: when they meet eachother they'll collide. So you would have to send your object not just through time but to a space where it wont collide with something, or the collision wont be enough to destroy the object.
As an example of a theorized timetravel that might actually be feasible: https://www.google.nl/amp/s/amp.space.com/28000-physicist-kip-thorne-wildest-theories.html
https://www.google.nl/amp/s/www.forbes.com/sites/startswithabang/2017/11/14/how-traveling-back-in-time-could-really-physically-be-possible/amp/
Although the reasoning of "you cant do a grandfather paradox this way" given in the article above is wrong. They assume that the second wormhole is traveling in a straight line and by the time you caught up with it enough time as passed to make it impossible to kill your grandfather. If you were to have it go in a circle you could reach the wormhole multiple times and keep going farther and farther back in time until close to the launch of the wormhole spaceship.
Answer: Time travel is not yet science fact, so all we have are theories.  As a writer, you are free to pick any theory you feel suits your story, and there are at least dozens of them.

There's the "alternate timeline" approach, where you can't send things to your past.  You instead send them to the past of an alternate timeline, which then plays out differently.
There's the idea of time preventing two objects from occupying the same space-time or from causing an inconsistency.  In such a case, sending it back in time is like trying to push a ball through a concrete wall.  It simply pushes back, and doesn't let it happen.
12 Monkeys used an approach where messing with time sent waves down the line which slowly brought things into consistency, like ripples on a pond slowly becoming muted.
Dr. Who had the idea that if something like this happened, time would tear, and extra-dimensional scavengers would come consume our dying timeline.

As you can see, there's many ways to resolve it.
My personal favorite to work with is the Novikov Self-Consistency Principle.  It simply states that the universe is consistent across all spacetime.  The classic example is done with billiard balls and wormholes.  A billiard ball is sent through a wormhole, back in time.  It is put on a course such that, when it comes out in the past, it is on a collision path with itself such that it knocks itself away from the worm.  This appears to cause a paradox, as now the ball would not go through the wormhole, so it wouldn't knock itself off course, so it would go through the wormhole... oh bother.
Physicists have discovered that there is a solution.  When the ball goes through the wormhole, it comes out at a slightly different angle than it went in.  This change in angle is enough to turn a solid impact with itself into a grazing blow.  This grazing blow changes the trajectory of the past billiard ball just enough to enter the wormhole at an angle consistent with the angle it came out, so that it can graze itself (again?).
They proved mathematically that this works for all possible configurations of billiard ball initial states and wormhole positions.  They postulate that this could also be true for more objects more complicated than billiard balls.
Or if that doesn't work, go make your own rules!  Time travel is not science fact yet (time travel was not science fact yet?  Tenses get bothersome when exploring time travel).  You can do anything you want.  If you want to make up your own rule, I recommend following Sanderson's First Law of Magics:

Sanderson's First Law of Magics: An author's ability to solve conflict with magic is DIRECTLY PROPORTIONAL to how well the reader understands said magic.

While he wrote the law for magic, if you apply Arthur C. Clarke's famous rule, "Any sufficiently advanced technology is indistinguishable from magic," you find that it applies directly in the science fiction domain as well.  If your readers understand what happens when the object is sent back in time, the effects can be as wondrous as you please.
Comment: I am loving the comments. No more clear where I should go, but great food for thought. My missus is gonna get really tired of me. :)
Comment: "If your readers understand what happens when the object is sent back in time, the effects can be as wondrous as you please."
I like this, but had wanted to keep it as scientific as possible.
Comment: @gorf To my own understanding, I believe that the Novikov Self-Consistency Principle is the only approach which is popular within the scientific community, so I do believe that is the only approach they would vote for.  It is the most natural extension of the equations we use to model our universe without creating unnecessary unknowable regions (such as other timelines).
Answer: Now you must realize that from a scientific perspective, this kind of time travel is near impossible. Whenever the object reappears, if it materialises, it will invariably take the space of whatever was there in that volume already. And probably cause some intense action at the molecular level.
The only way such a device can work is if it creates an absolute vaccuum at the place where the object is supposed to teletemport (I just made that word up). For this, the device should know well in hand when an incoming object is expected, and as such, shouldn't allow outside interaction during that period.
If the said device is what is the object of this question, then there are two cases possible:
1. Causality is maintained, and when you open the microwave for the first time, you already find a box inside. You take it out, put the new box in, and resend it for your past self to find. The box undergoes a linear timeline containing a loop. There are no other timelines or world branchings involved.
2. Causality is not maintained, and when you open the microwave for the first time, it is empty. You proceed to send it back in time. This creates a branching timeline, where the box is already in there, and when you open the microwave, you find the box already inside. You take it out, and may or may not decide to put the box in again.
In the second case, the box completely disappeared from the first worldline, and there are two boxes in the second.
If the you from the second world also decides to send the box back, you might end up in a pseudo causal system, where the box is sent to another parallel worldline, where it again comes out from the microwave, and a different you again puts his box back in. So while to each of you it would appear that you are in a causal system and indeed getting your box back, you are in fact getting the box from the previous worldline, and yours is getting sent to the next. It "appears" to be a causal system for everyone but the initial you.
One way to check which one you are in is, to not completely observe the box that came out of the microwave, and make a tiny mark somewhere on the underside of the box you are going to send into the microwave. If the box that came out of the microwave doesn't have this mark, you can be sure of multiple worldlines. If the mark is there, it still mightn't be a causal system, as the previous worldline's you could also have decided to mark the box.
My 2 cents
Answer: I tend to stay away from 'time travel' issues, because they are inherently contradictory. I can not resolve your conundrum, but I can point you in a direction.
I would like to bring to your attention that these paradoxes only exist because of life, particularly human life.
Life implies knowledge and knowing. Knowing implies memory. Memory is particularly uni-directional. As a great mind once asked, 'Why can we remember the past, but we can't remember the future?' Only living things have memory. Only living things experience time. Only living things are concerned with the concept of past, present, and future.
It is the memory problem that you have to work around. Why can't we remember something we did in the future? If you resolve that, you resolve the problem. You have to remember that you put the object in the microwave five minutes from now and sent it back to the past, in order to not be surprised that it is there now. You must be able to say, "Yes, I remember that I will put it there in another five minutes.' instead of 'Yes I remember I put it there five minutes ago.'
Since inanimate objects have no memory, it really makes no difference to them if something 'appears' from the future. There is never any memory of weather they were there a microsecond ago or not. In fact, it makes no difference if they wee there a microsecond ago or not. To an inanimate object, there is only the present. That is, the past or the future never effects the present. Only the present effects the present.
Answer: I prefer the theory that you can not change time in your timeline, and that as you travel back in time, you create or move into a parallel universe where the change you made occurred. Then, by placing the object on the time machine and sending it to the past, it would have gone to another timeline where the other you from that timeline will find it when it opens the device. If you use the time machine on yourself, you would go back in time but would go to the other timeline. This does not break many law of physics as far as I know, but it depends on the possibility of parallel universes and travel between them. In the same way as in transporting matter from one universe to another, that matter which did not exist before only appears from nothing, in "time travel" if this matter would appear in the new universe, but in an earlier or later moment in the temporal line.
Comment: I am personally not keen on the parallel universe / alternate timeline idea. For one,  if you are "creating" the timeline and all the matter that is in it, then where does the energy to do so come from?
Secondly, it doesn't fit with the crux of the story :)
Comment: I am now thinking more along these lines. But maybe an alternate reality that shares the same universe.
In other words, everything that exists, other than the item that is sent back, is as it was. But the object that is sent back becomes misaligned with its surroundings in such a way that the original "me" that puts the object into the "microwave" cannot perceive it, even though it is already there.
But then if the time-travelling object is a "ghost" object, how would it stay still in the "microwave"?


Open port 1433 to use SSMS securely

Question: I have a SQL Server placed in DMZ and I want to give access to internal users (and VPN clients) to use SSMS to remotely connect to it via port 1433 so server will accept the connection only form defined IP sources over this port, is it a safe approach for a production server?
Comment: I would not have any SQL Servers in the DMZ. Period.
Comment: I have a single server that hast to serve as Web Server and also has to manage database task, what is you suggestion then?
Answer: First let's answer your initial question then your follow up question.
I would go further and:
change the SQL Server port to a non standard one, turning off the SQL browser service, renaming SA, turning on the firewall on the local machine not to accept any connections on the SQL Server port outside of your internal IPs, and have all of your traffic hit the web server which serves as a secured abstraction layer for outside connections.
Follow up question:
How about HyperV and setup one server for SQL internally and a second one for the web server in the DMZ?  Why does the web server need to be in the DMZ?  Are you able to just forward all 80/8080/443/etc. ports over to that web server?
BTW, since you have both on 1 machine, have you set 'max memory to use' options in SQL Server and 'lock pages in memory' so IIS doesn't steal precious RAM?
Comment: Port 1433 only open internally and behind 2 layers of security (windows firewall and NAT/Router Firewall) Check.  Other ports and remote mgmt locked down: Check.  I feel you have enough compensating controls assuming you're not bound by PCI/HIPA/SOX or other major compliance requirements.  In those cases, you'd want to document all of your compensating controls and write some policies around it.  I personally would be fine with this implementation with your current constraints.
Comment: Thanks for answer

This is a very basic sever (configuration wise, it is a VM located somewhere (that I don't know where), with very low amount of memory and very low hard drive capacity)  but the load will be very very light  on it and there are no concerns about that (indeed I have no access to change any of the setting you have mentioned)
Why DMZ, because of the IT department decisions which I have no control over. ...
Comment: Why DMZ, because of the IT department decisions which I have no control over.
this server does not have route-able IP, so 80 and 443 ports are NATed to receive HTTP and HTTPS requests from inside/outside of company network, no more ports will be exposed to outside but on the other side of the wire, local network has access to some other ports to do development tasks, one port for deployment already opened and and one for RDP.
I don't think that changing the default port is necessary since eventually this will be moved to Azure which only supports 1433 for SQL Database.
Comment: Indeed I do not have access to open the port 1433 on the server, I need to ask IT to do it but I wanted to be sure that it does not cause any serious issue. Currently we have to RDP to the sever to work with sql sever. note that this port 1433 will only be opened toward internal network not internet.
So according to these details are there any security concerns here?
Comment: Thanks, I'll ask IT to open the port, the rest is on them.

Open port 1433 to use SSMS securely


How to replicate files between two separate apache machines in bidirectional

Question: How to replicate /var/www/html/ between two separate machines which has separate hard disk storage running Apache web servers. I was planning to setup load balancing between these two Apache servers and configured Nginx as front end for those two Apache servers,  the files should sync from server 1 to server 2  and vice-versa
Answer: Multiple options / solutions come to my mind:

NFS:

Network File System (NFS) is a distributed file system protocol originally developed by Sun Microsystems in 1984,[1] allowing a user on a client computer to access files over a computer network much like local storage is accessed.

DRBD aka Distributed Replicated Block Device:

DRBD® software is a distributed replicated storage system for the Linux platform. It is implemented as several userspace management applications and some shell scripts and is normally used on high availability (HA) computer clusters.

csync2:

Csync2 is a cluster synchronization tool. It can be used to keep files on multiple hosts in a cluster in sync. Csync2 can handle complex setups with
much more than just 2 hosts, handle file deletions and can detect conflicts.

There are other options available as well, just take this as a starting point.
Answer: Having code and content managed, promoted, and deployed to all servers is a far better solution to setting up a 2-way-sync.  Consider deploying server assets to all servers in your pool, rather than updating on one server, and counting on a sync/replication, or managing a sharing filesystem.
This allows you to keep your content/code managed separately, so you can review/manage/roll-back changes.  An unintenional edit or delete will nor harm you.
This will also allow you to spin up additional nodes and help with deploying via  a repeatable process.
The simplest iteration of this is to have your web servers periodically do a git pull of a particular branch of your repo.
In an interation of this, do the git pull to a staging location, and rsync to your httphome directory.  This will allow you to exclude things like your repo's .git directory and README files.
For example:
rsync -art --delete --delete-excluded --exclude='.git/' --exclude='README*' #{repo_path}/ #{site_path}

As far as using NGINX to front Apache, WHY?  What are you hoping to gain from that?  They're both web servers who can do reverse-proxy and run applications.  Unless you have a really good reason, pick a webserver, and let the loadbalancer balance load.
Comment: Sorry, I had a bit of a cold sweat there...  Ideally, you could get people to stop logging into servers and use a code management process.  (It doesn't have to be git; any source control is better than none.)  Setting the expectation that servers can go away or be redeployed at any time might be beneficial.
Comment: This works, as long as the files don't dynamically change, for example via uploads triggered by random users. +1
Comment: It really depends on the specific setup(s) and what you're trying to achieve. Just one example: I'm operating multiple tileservers, requests from clients get round-robin to these. If a client requests a tile which wasn't rendered before, it's rendered and after this, as fast as possible, should be available on all other tileservers as well (to minimize the risk that it's getting rendered again and to prevent costly database queries, which aren't necessary). `csync2` or `DRBD` are great for this, because the transfer is nearly instantaneous, ...
Comment: ...(continued) doing something like `git add && git commit && git push` and a `git pull` (or using some different `vcs`) afterwards sounds just crazy: How do you notify the servers to do a `git pull`? Doing it regularly? I think, this approach for this use-case doesn't work well, because it doesn't scale: It's to slow, at least `git` isn't fast at something like ~ 250GB of data. All in all, even if this is quite old, but I think it's still valid: _Use the right tool(s) for the job_.


FormView or not?

Question: I have an ASP.NET page with a Wizard control containing several pages of form fields. The data is collected and inserted to a database from the code behind page. I need to set this form up so you can not only insert, but edit a record as well. Since the form is long and complex, I would rather use the existing one and not make a duplicate one for editing, especially since I want to keep both forms exactly the same and any edits would have to be made to both. But it looks like this is what I need to do if I'm going to databind it. But this would also involve putting the Wizard inside of a FormView, and then I'd have to use FindControl to access any of the fields which would mean altering all my already-existing code (which of course would be time-consuming). So should I manually enter all the values from the code behind instead of databinding it? Which is better, to use a FormView and have duplicate forms (plus have to go in and redo the way I access the fields), or to do everything from the code behind?
Answer: I cheat in this circumstance. :)
Create each screen as 2 separate user controls
One for edit, and one for view
Then you get access to all you usual coding
Then embed the controls into the Wizard/FormView
Answer: I would suggest you to Go Ahead using FormView, as using DataBind control you have more control the functionality and layout Insert/Edit/View template. Since you have specified that your form is very complex and long, if you control from code behind you have to do lot of work to handle this in code behind and lot code required.
Since I have personal experience to develope very complex form using FormView and it was easy for me bind the Value in directly in formview instead if you assign/Get Values of each conrol in code behind and sometimes you have to hide.

FormView or not?


Date/Time stored as floating point, which algorithm is used?

Question: I'm have access to a 3rd party application's database, and I see a field called "date" which stores date/time values as floating point numbers, but I'm not sure how this floating point number is mapped to a date/time. There is no documentation for this database.
Here is some sample data:
<code>date-field        actual-date-time
253507382.168744  1/12/09 6:43 PM PST
253507480.136126  1/12/09 6:44 PM PST
253508091.838982  1/12/09 6:54 PM PST
256703604.015055  2/18/09 6:33 PM PST
256704413.484674  2/18/09 6:46 PM PST
</code>
Note: I had to enter these values manually so there's a slight chance they may be off a bit. If you would like to see more data, let me know and I'll add more.
I'm hoping someone is familiar with storing dates in this format and can let me know how to get a date/time given a floating point number.
Answer: If you look at the change in the numbers over the 10 and 13 minute intervals, you'll see that it's about 60. Therefore I conclude that it's a count of the number of seconds from a base date.
I think the base date is 1/1/2000 or 1/1/2001.
Edit: The base date appears to be 1/1/2001, and the time appears to be adjusted as well - it's probably UTC with your local time offset added.
Answer: If you subtract any of the two points you'll see that the values represent the number of seconds, at microsecond accuracy. It should be easy to work out the base date where the clock "started". On Unix and related systems this is January 1st, 1970.
Answer: The timestamps are 'number of seconds elapsed since 00:00 on January 1st 2001'. It's not a common date format but at least it should be easy to work with now you know what it represents!


What's the best way to go around HR?

Question: If, for instance, you find (and contact) a senior office (the president/ CEO, VP, CFO, CMO, etc) of a company, and send them a networking query, is this a legitimate way of seeking if there are openings in a company? Note: I wouldn't directly ASK for a position (that's not what networking is for) but to address what's going on in the industry as a whole and in their company more specifically. Also, what is the best way to go about doing so without scaring away a potential job offer (or client to perform services for)?
Comment: a lot of times I do. Executive assistant to the CEO. Marketing/ writing for a small company, and the person in charge is the owner, and so on.
Comment: A classic joke is "HR" doesn't mean "human resources" it means "Human resistance." Going through the proper channels often doesn't work for a significant portion of the population. That means those it doesn't work for need to improvise to stand out from the crowd, hence skirting the system.
Answer: If you know someone well enough to use them as a networking contact into their company, you can just ask them directly what you should do. If it's possible to bypass part of the HR process, they may shepherd that.
If you don't have a genuine relationship with this person (i.e. they would feel comfortable giving you a personal reference), trying to use them as a means to get around HR or any of the normal hiring processes has a high potential to backfire. Most people don't like to have their name dropped by someone to whom they are not close and upon hearing that happen may even go so far as having a negative impact on your hiring possibilities.  In other words, nobody likes to be used.
Additionally, I would advise against trying to use high level direct contacts of people you don't actually know.  Most executives are busy and on the chance they even see your inquiry, they are not likely to respond positively because they don't actually know you.
Comment: @JoeStrazzere not to directly inquire. Networking is a bit more subtle than that. Industry talk, discussions about company problems, and so on is more the gist of what I'm saying
Comment: Define "don't actually know" I know lots of people through LinkedIn (They're my 2nd and 3rd level contacts, and I know how to contact them through the service by various means), but it makes a difference that I'm a LION - Linked In Open Networker.
Answer: If you know them well enough to consider them a personal friend or acquaintance, I'd say go for it.
But be wary of this if you're not absolutely certain you know them well enough. If you wouldn't go out to coffee with them, don't do this.
Furthermore, you have to be careful in how you approach the conversation. Don't just say "Hey I need work." Instead, say something to the effect of "So what's Company X doing in Y field? Anything interesting or exciting?"
Even so, many larger companies have policies in place to prevent higher ups from influencing hiring decisions too heavily - often the best they can do is offer you a good recommendation (Although that in and off itself is still very valuable)
Answer: If you're fairly high up in the food chain -- if you're someone well enough known that a company might actually make a position in order to bring you on board even if they weren't hiring otherwise -- coming in from the top can make sense. This is one of the somewhat specialized techniques executive outplacement firms train their clients in.
If you're a student from a top school that that company has a history of preferentially hiring from, this may be worth considering. For example, IBM used to screen MIT recruiting through the research division rather than the normal paths.
If you're looking for your first job right out if school, and again have exceptional grades and/or an exceptional school, the sheer chutzpah of contacting an exec directly -- if done respectfully -- might occasionally work in your favor. I found myself unexpectedly in an interview with CDC's director of research -- turned out he answered his own phone and appreciated the fact that I had made an extra effort to find out exactly how to contact him. (I didn't get an invitation for an on-site interview, though, so arguably that wasn't a success.)
But if you are an ordinary schlub rather than something special, they will probably just hand you right back to HR. Whether they are annoyed or not will, again, depend on exactly how respectful you were of the value of their time, how much they enjoy teaching, their mood on that particular day...
It can be made to work, sometimes, for some people in some situations. But it does risk being a double-or-nothing bet. Cue Dirty Harry: "Do ya feel lucky?"

What's the best way to go around HR?


Can I use smart switches for my 3-way?

Question: I have a 3 way setup and one switch has white black and copper wire and the other one has the same but with a red wire. Can I use smart switches for these?
Thanks!
Joe
Comment: I am going to have to get some sleep. I will post again tomorrow with pictures. One switch does have a few common wires tied together. With the black and red and copper wire. The other switch has a black white and ground. Thanks for trying. It's 3am and I am having trouble seeing straight. Good night,
Comment: 3-ways are simple enough once you label them.  But 99.9% are unlabeled, and those are much more complicated.  The info you provided is not sufficient to understand the circuit.  We need to know about all 3 non-grounds per switch, which cables they are grouped into, and the colors of the screws on the switches. Or some photos. [Edit] your question to add the info.
Answer: Yes, a smart 3-way switch can be installed.I have used this 3-way switch in the past. Additionally, with the new smart switches, in general, the use for 3-way IMHO is becoming obsolete. Most link to your smart home hub (i.e. Alexa Echo or Google Home) and you simply have to tell it to turn on or off.
If you don't want a 3-way you could also get this alternative.
Answer: Stop!!!
white black and copper is not a legal 3 way to start with. It is not safe and a code violation.
Yes there some smart switches that can make this right and have some nice features like dimming and 10 seconds fade to off
Lutron majestic? Or maestro has just a switch I have used several times in years past one is a master or controller and the second is a slave or remote.

Can I use smart switches for my 3-way?


To reconnect Motion Tracker in After Effects

Question: I want to reconnect my Motion Tracker. But I cant "Edit Target".
For example: I track the video and forgot to create a null-object for the connection with the tracker. When I do this, there is no more possibility to connect the tracker with the null-object and the only chance for me is to track the motion again (but the former tracking path is visible, but no more connectable). Anyone an idea?
Answer: Steps:

Add your footage.
Go to the Tracker panel
Select the layer to be tracked
Click Track Motion
Position your track point (or points if you're doing Scale and Rotation as well)
Click the Play icon in the Tracker panel
If it mis-tracks, manually reposition the markers, and continue playing from the Tracker panel.
A load of motion tracker keyframes appear in your footage
Add a null to your comp
Go back to the Tracker panel.
Click 'Edit Target'
Select your null and press OK.
Click 'Apply' and choose whether you want X and/or Y coordinates applied.
Your null now has position keyframes derived from the tracker marker.
If you delete the null and make another one, you can repeat the process.
Select your footage layer
Go to the Tracker panel.
Choose the correct Motion Source (your footage layer).
Choose the correct Current Track (probably Tracker 1).
Choose Track Type - probably 'Stabilize'.
Click Edit Target, choose your Null, and hit Apply as before.

I suspect the 'Track Type' pull-down is working differently, and this is what you had to do, in order to be able to re-apply to a new null.. I don't remember it working like that before?


What are some good ways to earn cash?

Question: Getting a large sum of money in Assassin's Creed 3 is much different then previous version of the game. Gone are they days of buying every shop in town, and now we appear to have to actually do a little bit of labor for our money. What are some of the best ways of quickly earning large sums of cash. Are there some items that are cheap/easy to craft but sell for a boatload?
Comment: Get a job, you hippie!
Comment: There might be a more efficient way, so I'm putting this as a comment. I found playing the pub games and betting the most I can is useful. Also, hunting for about 15 - 20 minutes, skinning, then trading everything I had will get me about 2000 - 3000 every time. This is all based on me playing though, I'm not super far.
Comment: @leety The games thing only works if you're actually *good* at them. Not that I'm bitter at getting schooled by the computer at checkers and X Men's Morris all the time or anything.
Comment: I hear you @AnnaLear :) I figured I'd throw it in there for a comment though. I personally would feel bad saying "play this game" for an answer. It really relies on your skill to actually make money.
Answer: If you're like me and completely suck at the tavern games...
Hunting and selling pelts can be quite profitable, though a bit time-consuming. I like hunting wolves - they travel in packs and can be killed instantly with the hidden blade (unlike bears which take 3 hits) for a pelt in perfect condition. Beaver pelts are good as well. Beavers are slow and easy to track, and their pelts sell for boatloads.
I'm only on sequence 6 so far, and I'm also making decent cash from opening chests around the world, though I suspect this source will dry up shortly. :)
Haven't done much crafting yet since I need to gather more resources and artisans, but it doesn't seem like an efficient money-maker, since making the same item increases its cost for a while.
Sending your assassin recruits on missions is another source of income. It's not very quick, but it's pretty hands-off and passive.
Answer: Go to the Homestead, buy as many bear pelts as you can. You can sell them on convoys and you''ll make a great profit. You should get more and more if you send multiple convoys at once.
Comment: Before you can get Bear, Beaver works really well too.  It has the added bonus of you being able to farm them en-masse by just going up and down the river, the place is lousy with them.
Comment: I feel sorry for beavers. They are so bad at not being killed. :(
Answer: Buy bear pelt for £240 then sell it by naval convoy to saint Augustine for £1258 profit per pelt.The only downside is it takes 32 minutes but with 1 maxed out convoy you can sell 15 bear pelts for £19k
Answer: A good idea for money making is to hunt beaver. They're relatively easy, and a great spot to find a few is around the mine where Norris works. Just around the river, there's raccoons, and beavers. Just use an arrow, or run up to them, they aren't very fast. The hides sell for quite a few pounds if you market them through the use of caravans. Otherwise, they're only 160 pounds. Make sure to sell them to the FIRST general store you unlock, the one from the tutorial. The second one won't pay nearly as much.
Answer: Sell ale and upgrade your homestead as early as possible. Officer's swords 1220, cannon 1400, fine clothing 1230, fine shoes 1002 pelts make good filler whilst stockpiling parts to craft large items. Try to rotate artisans wood buttons, blades, linsywools, send pelts. Next trip back make weapon handles, then make fine clothing and officers swords. Also with 4 major artisans you can rotate stock pretty effectively.
Answer: Elk and bear pelt farming for trading.  There are tricks to hunting bears.  Hit them from an air attack, or jump right before they attack you.  Then you can 1shot them.  Here's a vid of bear and elk pelt farming that can be done over and over:
Assassin's Creed III bear and elk pelt farming
Answer: Also, a good way to make money is by finding chests. Some may have 10k gold and others 7500 or 1250. You can buy chest maps at any general store for 2000 or 2500 gold.
Answer: Do one liberation mission that grants the ability where your friends become guards and take you as a prisoner. (This should be done before Sequence 7, or otherwise before you do any liberation missions.) Go into a fort under British control that hasn't been freed and use the aforementioned ability without entering alerting anyone. Open the chest, leave, and enter again.
Find the fort with the most chest cash to make a ton of money. This makes about 10K every 10 minutes if you find the right fort.
Comment: Hello, welcome to gaming.stackexchange.com, and thank you for your answer. However, as it stands your answer is both difficult to read and hard to make sense of. Please consider editing it to clarify the grammar. Unfortunately I'm not familiar enough with the game to make an accurate guess at what you mean or I'd do it for you.

What are some good ways to earn cash?


Is the intersection of infinitely many recursive sets recursive?

Question: Is the intersection of infinitely many recursive sets $\bigcap_{i}U_{i}$ (where each set is different ) recursive? Recursively enumerable?
I know the union need not be recursive, because deciding if an element is in the set $U_i$ is the same as deciding the Halting Problem since you for each element you'll have to decide if the function that computes if $x \in U_i$ for some $i$  will terminate.
Comment: Should "infinite recursive sets" be "infinitely *many* recursive sets"?
Comment: @AndrejBauer yes
Comment: By that reasoning everything is the same as deciding the halting problem. I want to know if the input number equals 42, to do that I have to decide if the program that only halts if the input number is 42 will halt.
Answer: For each $i\in \mathbb{N}$, take $S_i = \mathbb{N}\setminus \{i\}$. You can now build any set you want as $X = \bigcap_{i\notin X}S_i$.
Similarly, an easier proof that a union of recursive sets can be anything at all is to let $T_i=\{i\}$ and now any set $X$ is $X = \bigcup_{i\in X}T_i$.
Answer: Since you already know about unions, you could figure this out by remembering the de Morgan laws: the complement of a union is the complement of the intersection of complements. With this in mind: let $U_i$ be the set of (indices of) Turing machines that do not halt within $i$ steps of execution. So, the complement $\mathbb{N} \setminus U_i$ is the set of (indices) of those machines that halt within the first $i$ steps.
Now we have:
$$\textstyle\bigcap_i U_i =
  \mathbb{N} \setminus \bigcup_i (\mathbb{N} \setminus U_i) =
  \mathbb{N} \setminus H$$
where $H$ is the halting set. The complement of $H$ is neither recursive nor recursively enumerable.
By the way, you state that "deciding if an element is in the set $U_i$
is the same as deciding the Halting Problem ..." which is not necessarily true. For instance, if I set $U_i = \{42\}$ for all $i$, then all the $U_i$ are decidable. You need to pay attention to what your $U_i$'s are.


Sample Programs for Design Pattern Recover Tool

Question: I'm developing a Design Pattern Recovery tool for my final year project. It is a java based system which can be used in identifying, implemented design patterns in a particular java program. When you input the java program (Source code) to the design pattern recovery tool, it analyzes the static and dynamic features of the java program and identify the implemented design patterns.
So far, i have been using dummy projects to test the functionality of the tool. But, now i need sample programs with proper documentations about the implemented design patterns to  test the tool. I want to know, where i can find such programs (with source code) so that i can test my system properly.
I know this is a bit of broad question. What  i need is sample java programs to test my tool. In order to ensure the accuracy of the results, those programs should have proper documentation about the implementations details specially about the design patterns used.
Comment: Way too long question, **github.com**.
Answer: I would look for some open source java projects. Must be many out there.


Charge and conducting thick slab

Question: I have thought over this problem but I haven't found the solution: There is an electric charge $q$ at a distance $d$ from a conducting slab with thickness $t$, the problem is to find the potential everywhere, assumming that the slab is grounded. So, on the side of the slab where the charge is I think one may use the image method solution to find the potential in this region of space. Inside the slab $\phi =0$, but the question remains how to find the potential  on the other side? The method of images does not seem to work on that side(?), and other solutions seem difficult to implement. Charge conservation seems to imply that the total charge induced on each surface of the slab is $\pm q$.
Is there any easy or semi-easy way to do it? I guess my question would be: what is the most straightforward way to do it?
Any help is greatly appreciated.
Answer: Let the "other side" exist in the lower-half space $z<0$.  The conductor imposes the following boundary condition on the potential $\Phi = \Phi(x,y,z)$:
\begin{align}
  \Phi(x,y,0) = 0.
\end{align}
The charge density vanishes for $z<0$, so we simply need to solve the Laplace equation $\nabla^2\Phi = 0$ for the region $z<0$ with this boundary condition at $z=0$.  Note that $\Phi = 0$ is a solution to this equation satisfying the desired boundary condition, so this is the solution by uniqueness;
\begin{align}
  \Phi(x,y,z) = 0, \qquad \text{for all $z<0$}.
\end{align}
Answer: Considering a spherical conducting shell of radius $R$ and thickness $t$, with a charge $q$ at a distance $d$ outside. So, what is the potential inside the shell? Extending $R\to\infty$ now and you get it.
Comment: I stand corrected. Indeed the method you gave works well as the charge density on the opposite side to the plate will vanish, as joshphysics pointed out. It is remarkable that the thickness plays no role whatsoever.
Comment: I am afraid this will not work. The electric field inside a conducting spherical shell will always be zero (conductors shield electric fields), hence the surface charge densities in the inner shell will be zero. This cannot be the case for the infinite slab, so the solution cannot be obtained in this way.

Charge and conducting thick slab


Is it haram to touch my nipple( I am a male teenager ) and slightly stimulate myself?

Question: When I stay alone like the time of studying I automatically have thoughts like having sex with someone, these moments are very disturbing for me , I can't do any work then .( I am a male teenager).. If I just touch my nipples then I feel slightly excited, and I feel better then , but I don't touch my penis , so I don't musterbate...so is it okay to touch my nipple and slightly stimulate myself ? I don't think about sex ,just touch my nipples and it slightly stimulates me . That's it .
Answer: It is very good to hear you do not masterbate. But Allah, forbids all acts of shamefulness:
Allah commands justice, the doing of good, and liberty to kith and kin, and He forbids all shameful deeds, and injustice and rebellion: He instructs you, that ye may receive admonition. (90) Fulfil the covenant of Allah when ye have entered into it, and break not your oaths after ye have confirmed them; indeed ye have made Allah your surety; for Allah knoweth all that ye do. (16:91)
Remember satan is everywhere and he commmads people to do shameful deeds. The advice in the quran is:
When satan assults you remember Allah, it will help. see verse below:
Indeed, those who fear (Allah) when touches them an evil thought from the Shaitaan, they remember (Allah) and then they (are) those who see (aright).(7:201)
Try to recite surah Baqara in your place of study. It is said, that the house surah baqara is read satan does not enter it for 3 days. It is long but since satan does not enter your house for 3 days you can read it over 3 days.
recite after fajr and magrib prayer:
La ilaha illallahu wahdahu la sharika lahu lahul mulku wa lahul hamdu wa huwa 'ala kulli shay in qadir one hundred times you will be protected from Shaytan all day.
recite after fajr and magrib prayer the 3 Qul's X3 (i.e.112:1-4,113:1-5 and 114:1-6), 3 times spit blow on hand and wipe yourself from head to teo 3 times.
It can be taxing to ask someone to compeletly block unwanted thoughts, you cannot stop them but what you can do is not to prolog them yourself. You know when you are doing this. If you prolong the bad thought yourself, then you can make a solemn oath to Allah to couple this act to another act that will stop you from doing it for example:

read 200 nafl salat without break

2)fast 2 months without break

give 1000 pound in charity

If you do this your bad thoughts will very quickly vanish.
You must be strict in which ever rule you choose otherwise it will not work. These number work, but some people are weak so you can adjust them like 10 days fast instead, but try to get the balance right. And indeed it says in the Quran those who turn off evil with good:
and those who observe patience in order to seek the pleasure of their Lord and establish Salāh and spend secretly and openly from what We have given to them, and repel evil with good. Those are the ones for whom there is the ultimate abode, (13:22)
And there is more good news for those who believe and restain there souls from lowly desires:
And for such as had entertained the fear of standing before their Lord's (tribunal) and had restrained (their) soul from lower Desires, (79:40) Their Abode will be the Garden. (79:41)
stimulating yourself by touching you nipple, is an act of shamfulness. It is not a good idea to curb your bad thoughts with a bad act. But rather throught remeberance of Allah, seeking from Allah protection from shaytan and by turning it off with good deeds. And I also adive you to ask forgiveness for any sin of shamfulness.


Conditional Expectation E[X] = E[X|Y<=a] + E[X|Y>a]

Question: Is it generally true that for random variables $X$ and $Y$, regardless of being dependent or independent, that $E[X] = E[X \mid Y \le a] + E[X \mid Y>a]$ ?
Comment: Consider a homely analogue: is the average height equal to the average height for those below average weight PLUS the average height for those above average weight? Answer: No.
Comment: Assume $X$ and $Y$ are highly correlated standard normal variables. Then $E[X\mid Y\le L] + E[X \mid Y > L] \approx 0 + L = L$ when $L$ is a very large number. If you assume that $L$ is a large negative number you get a contradiction.
Answer: Not quite, if we use the law of total expectation we would have that
$$
E(X) = E(X| Y \le a)P(Y \le a) + E(X|Y > a) P(Y>a)
$$
Answer: Exactly Chappers, and I can just add if $X$ and $Y$ are independent continuous random variables, $E[X|Y]=E[X]$.
So Mauro, your formula would be false because in this case, you'll have
$E[X]=E[X|Y\le a]+E[X|Y>a]=E[X]+E[X]=2*E[X]$
Finally, regardless the dependence between $X$ and $Y$, we have always:
$E[X]=E[E[X|Y]]$
Comment: Ah yeah sorry I feel dumb now ;_; I was trying to prove it with like integrals. Just to verify though is:  f(x, y<=a) +f(x,y>a) = f(x,y) Where f is the joint distribution of X,Y. If so I think I got it from the integrals too.


Markov chain transition matrix with each row the same?

Question: In my linear algebra class, we were learning about Markov chains, and the prof. put a confusing question on the board. Namely, can one have a transition matrix in which each row is the same, i.e., let's say for a $3 \times 3$ matrix, each row is $[a,b,c]$, with the constraint being that $a + b +c  =1$. Is this possible? Further, what if the values of $a,b,c$ are the same as the values of the steady-state vector?
Comment: Why not?  It might be a peculiar  Markov chain, but the matrix fits all the conditions.
Answer: The values $a,b,c$ are going to be the values of the steady state vector - we can think about this as such: in the long run, the probability from going from any state to state 1 is $a$, and correspondingly $b$ and $c$ for states 2 and 3 respectively.
The transition matrix you mentioned is definitely a valid transition matrix. The only requirement is that each row sums to 1.
Answer: A Markov chain $X_1, X_2, \dots$ with a transition matrix in which each row is the same
corresponds to the special case in which each $X_i$ is independent of $X_{i-1}.$
In general, a Markov chain has
$$P(X_i = x_i|X_{i-1}=x_{i-1},\, X_{i-2} = x_{i-2},\, \dots,\, X_{i-k} = x_{i-l}) = P(X_i = x_i|X_{i-1}=x_{i-1}),$$
so colloquially, dependence is only on the most recent specified step. However,
there is no requirement for any dependence at all. An sequence of independent random variables can be Markovian.
Note: The power $P^n$ of a Markov chain shows $n$-step transition probabilities. Some Markov chains have the property that $P^n,$ for sufficiently large $n,$ has rows that are very nearly the same, indicating that
$\lim_{n\rightarrow\infty} P^n$ exists. This means that, with sufficient
passage of time, the state of a Markov chain at step $n$ does not depend
on its initial state at step $1.$ A chain with such limiting distribution
is called ergodic.


Generate sentences from keywords by adding formal word

Question: For example, I have a list of keywords like I, hungry => output: I am hungry or I, author, poem => output: I am the author of this poem.
Can someone please suggest the simplest way to achieve this?
I am a newbie, please tell me which knowledge I must have to solve this problem.
Answer: I am also fairly new to this, but am working on the same type of problem where I have to predict the sentence based on previous data. Like you want giving X value predict y value.
So, basically you need to create a dataset as per your requirement and once your dataset is ready you can train your model with encoder-decoder or LSTM layers.
Please find below links that may be helpful to you to fulfill your requirements. See as X and Y value input output prediction.

Gmail style smart compose
Neural Machine Translation with Attention
Sentence Prediction using LSTMs


How do I prove that a symmetric matrix has a set of $N$ orthonormal real eigenvectors?

Question: Accepting the fact that, like all others normal matrices, a hermitian matrix (with $N$ lines) has a set of $N$ orthonormal complex eigenvectors (with real eigenvalues, degenerate or not), how do I prove that a symmetric matrix (real hermitian) has a set of  $N$ orthonormal real eigenvectors?
Answer: It follows simply from the fact that the eigenvalues of the symmetric matrix are real. Assume that $A$ has a real eigenvalue $\lambda$ and a nonzero eigenvector $u=v+iw$, where $v$ and $w$ are real. Then it follows $Au=\lambda u$ that $Av=\lambda v$ and $Aw=\lambda w$ and hence $v$ and $w$ are in the same eigenspace associated with $\lambda$. Consequently, the eigenvectors can be chosen to be real.
Comment: I see... But, let's say, we have an eigenvalue lamba with L orthonormal eigenvectors (L <= N, of course). It's not obvious to me that from the imaginary and real parts of each vector (2L real vectors) I could get L real and linearly independent vectors.


Chinese visiting England, have Schengen visa

Question: In Sweden on vacation with my Chinese wife. We plan to go to London 3-4 days. I heard, that there is simple way to get a visa to England, when you already have a Schengen visa. Its impossible to get any information from embassy or consulate, and there is nothing written on internet.
Comment: What's your citizenship?
Answer: Yes indeed there is a simple way for Chinese nationals to get a UK visa at the same time they get a Schengen.
This question/answer pair reveals the details Obtaining a UK and Schengen Visa in a Single Application or Single Step in China
Upon reading the article, you will find that the procedure extracts fields from a UK application and puts the information in the corresponding fields on the Schengen application (pretty neat, huh?).
When this is complete, the person can make a single visit to the VAC and submit both applications at the same time.  It's amazing!  :)  The VAC will forward the applications to their respective centres, the analysis and decision-making pipelines are totally separate.
Based upon what you wrote, you already have a Schengen, so the procedures will not be of use to you.  You will need to apply for a UK visa from the beginning and if you're travelling in 4 days, you should use their Super Premium Service.
If you are currently in Sweden, you can apply at the British Embassy in Stockholm.  Fortunately for you, the Stockholm consulate contains the hub for Scandinavia so your application can be processed locally.
It's also likely you may have heard about the UK's Approved Destination Status scheme for Chinese tour operators.  It's not applicable for you because you need to sign up for a tour originating in China with a licensed tour operator.
Finally, Relaxed's answer to your question accurately describes the EU procedures.  These procedures are also available to you and worth checking out as well.
Answer: I am not aware of any "simple" way to get a UK visa once you have a Schengen visa. A good history of travels in the Schengen area (but also the US, Canada, etc.) helps a bit and if you are ready to pay some extra fees, the UK offers premium visa services that might help you get a visa quickly but you still need to fulfill all the usual requirements.
Assuming you are an EU citizen, there are two other rules that might be of some relevance, but I don't think they will be immediately helpful to you in this case:

If your wife would hold a residence card (but not merely a visa) as a "family member of a Union citizen" in a Schengen/EU country, she can visit the UK with you without visa.
Since she would be traveling with you, your wife should be eligible for an EEA family permit. That's relatively "easy" compared to a regular visa in that it's free of charge and the main requirement is proving that she is indeed your wife. But it can still take some time and effort to get one (I am not sure about the current status of all this but in spite of the EU's recommendation that such visas should be issued "quickly", it could in fact take longer than a regular visa because excessive processing time for EEA family permits was a serious problem a while back).


Could a genetically modified plant become a house?

Question: I was checking out art made by Hyrotrioskjan on Deviantart and found his Nea project. One art piece features large genetically modified plants that are able to essentially house a community. These large barrel-like plants grow fast and can be hollowed out to make rooms. I was wondering if such a thing was feasible? Especially considering the side of the plants.
Comment: Possible [duplicate](https://worldbuilding.stackexchange.com/q/34513/55743). (Duplicates are judged by whether the answers fulfill the criteria of a question, not the actual question being identical).
Answer: Some trees can already be used as shelter, thanks to the size of their trunk, like baobabs.

To host a community you would need a much more large tree, or better a set of trees with their branches forming a cover, like it has already being done here and there

It doesn't therefore seem too far fetched to have something like you describe.
Comment: The only issue with a modified baobab is that you would need to speed up the growth rate significantly in order for it to be practical.
Answer: People are already working on it using tree weaving as a base. The biggest problem is the time it takes to grow the house which is on the order of decades  under ideal conditions. perhaps with genetic engineering something more practical might be possible.
take a look at the willow palace in Germany or Patrick Dougherty's tree weaving art.

https://inhabitat.com/grow-your-own-treehouse/
Answer: By figuring out how cells assemble themselves
We're only beginning to understand how cells know where to go in order to form a coherent organism. But say a genius botanist and geneticist figures out the secret, he might as well program a plant to grow a complete house that comes ready with all the furniture. The house could even grow continually new floors, eventually become a tree skyscraper... in a million years. You could even add different reactions to chemicals. Say for example you sprinkle the plant with chemical x, the plant responds to the chemical and grows a living room. Use chemical y and it grows a swimming pool on the roof.
I suggest using a strain of bamboo to create fast growing houses. The fastest growing bamboo was recorded to grow 91cm a day. Bamboo grows by inflating its cells with water, which were grown before the shoot came out of the ground. What's interesting is that bamboo are networked by their roots and can 'pump' water into newly growing shoots. This could also be modified into a sewer system. Just remember that bamboo is an invasive grass and that one tree house will quickly become a treehouse city.
Answer: Another potential plant for growing housing could be gourds.  If you could modify them to bear large enough fruit, you could hollow and dry them to make human-sized shelters.  People already use them for birdhouses, so it would just be a question of size.


What professions give the largest stat bonuses for a Cataclysm PvP Warrior?

Question: What professions give the largest stat bonuses for a Cataclysm PvP Warrior?
Answer: Update for Patch 4.0.6!
Leatherworking is now exactly as good as blacksmithing / enchanting / alchemy / inscription. Pick Jewelcrafting if you need that 1 extra stat, but the rest are totally a wash.
Original Answer
Jewelcrafting can give up to +81 to a primary Stat, whereas enchanting / blacksmithing / inscription etc. only give +80. (Exceedingly minor difference.)
Also worth looking into is Leatherworking, which offers unique arms-slot enchants that are unique in that they provide a primary stat bonus (such as strength) otherwise unobtainable as an enchantment in that slot.
In general, the gathering professions provide less useful benefits than the secondary professions, a possible exception being Mining, which provides health. (Strength is a better damage increaser than both crit rating from skinning or an activated haste+heal from herbalism).
So the bleeding edge optimization would indicate Leatherworking and Jewelcrafting as the best, in terms of pure +stats. Mining / Jewelcrafting is easier to level, and the bonus health from Mining is not wasted during the period before (if) you switch it for leatherworking.
Beginning with Wrath, and developed to completion in Cataclysm, class is largely meaningless when looking at profession bonuses. The core primary stats of Intellect / Spirit / Strength / Agility are usually all available within a single profession (if they're present at all), making class choice largely irrelevant.*
Even Tailoring isn't useless for melee characters, as it provides a +AP cloak enchant (though its proc-based benefit compared to simply passive +strength is debatable)
Comment: Engineering is also spectacularly useless for Melee classes at the moment. There is no Agi or Strength equivalent to Synapse Springs, the +Intellect glove tinker.
Comment: Just a side note, in order to keep most (if not all) profession specific augments working, you need to maintain a certain level in the particular prof. So you cannot level it, get the enchants and then drop it to get a new one.
Comment: Of course, the day after I answer this, the PTR gets updated with enchanting recipes for stats to bracers, of which the leatherworking embossments are *surprise* 80 higher.
Comment: Further correction thanks to todays PTR update: "Synapse Springs now increase Agility, Strength, or Intellect (whichever is highest for the character)." -  i.e. Engineering just became *awesome* for everyone.
Comment: (For reference, Synapse Springs provide 480 [Stat] for 12 seconds on a 60 second cooldown, or an average value of 96 [stat] if used on cooldown. This, combined with the increased value of large bursts of [stat] over a lower average value makes Engineering a *very* strong profession.)


show that two vector are perpendicular to other

Question: Let $S : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be the transformation induced by the $3 \times 3$ matrix $\left(I-u\,u^T\right)$. Show that for $x \in \mathbb{R}^3$, $S(x)$ is perpendicular to $u$,
where $u = [b , c, d]^T.$
Comment: Show that $u^\top S(x) = 0$ by plugging in the definition of $S(x)$. I think you also need the condition that $u^\top u = 1$ which isn't mentioned in the problem.
Answer: $S(x) = (I-u\,u^T)\,x = x - u\,u^T\,x$
$\langle u,S(x)\rangle = u^T\,S(x) = u^T\,x - u^T\,u\,u^T\,x = (u^T\,x)\,(1-u^T\,u)$

If we include the condition: $b^2 + c^2 + d^2 = 1 \Leftrightarrow \|u\| = 1$
$1-u^T\,u = 0$, which means that $\langle u,S(x)\rangle = 0$, i.e., $S(x)$ is perpendicular to $u$.

If $u$ is a unit vector, then $(u\,u^T\,x) = (u^T\,x)\,u$ is the projection of $x$ in the $u$-direction. Thus, $S(x)$ subtracts from $x$ its projection in the $u$-direction, and only the perpendicular part remains.


Is a Chinese multiple-entry, 6-month tourist visa (L) available to a Sri Lankan citizen?

Question: I'm a Sri Lankan passport holder planning to travel to Mainland China in May and a few more times in the following six months.
Comment: @pnuts might be worth elaborating WHY you suggest they change their plans.
Comment: What is your premise? Have you had Chinese visa before, and in the same passport?
Comment: @mts Yeah. I visited Shanghai last month
Answer: Simple answer: yes. The visa form from the Chinese embassy in your country has this option.
More complex answer: maybe. Not all embassies want to immediately issue a multiple entry visa, and you need to meet the requirements to be able to get one. Based on my own experience with China, I was only able to get a multiple entry visa if I had visited China at least 3 times in the last 2 years (but I'm no Sri Lankan resident, so who knows different rules apply to you).
Also, it is quite common that you need more documentation as proof than what is listed on the website. Sometimes the website doesn't list any documentation, but you need plenty of proof. The best solution is to call the Chinese embassy and ask them.


Could I list the conference that I didn't attend in the PhD application system even I got my paper accepted?

Question: I am applying for a Ph.D. position in a university. Their system required me to fill in the conferences I attended.
I published my paper on one conference although I didn't attend it personally for some reasons. My advisor went on my behalf. However, if I denied that I had attended that conference, the system doesn't even allow me to upload my paper. I think the paper is an important proof of my research ability.
What should I do? Should I just pretend I attended it. I don't even think they will really check about this thing.
Comment: Can you specify a list of publications, too? Then this is where your paper should go, and then you shouldn't claim double credit if you didn't attend the conference.
Comment: Doesn't the application include a CV?  Just include the publication on that.
Answer: In general, no. If they ask you to specify a list of attended conferences, then a conference that you didn't attend doesn't belong in your answer, even if your paper was accepted. Claiming that you attended the conference  would be incorrect and could lead to problems if someone found out.
Normally, you should be able to get credit for your accepted conference paper in another category, where you're asked for a list of your peer-reviewed articles.
However, the form could be designed in a way which neglects the existence of peer-reviewed conference papers. For example, you might be asked for a list for your journal articles only, because the form designers were unaware of fields in which journal articles are not the only kind of peer-reviewed publications. In this case, you should seek contact to the admission committee and ask how to proceed.
Answer: My reading here is that Conferences Attended and Papers Published are not the same thing.
You could attend a conference purely to watch the other presentations, to make contacts or to participate in workshops. You can also attend non-academic conferences, or conferences held within your university.
Based on the way this has been listed, I would not included the paper in this section. However, hopefully there will be the opportunity to submit your list of Publications elsewhere.


do you sit in or at an openspace

Question: If I was to write:
I will be sitting all day on open space
would that be correct
or should I say
I will be sitting all day at open space
or otherwise. By openspace I mean the typical corporate space where employees sit and work
Comment: You sit *in* the open space. You cannot sit *on* it unless you climb onto the roof, and sitting *at* it would mean you're not actually sitting *in* the space but in some area adjacent to it.
Answer: Because it's not a proper noun, your "open space" likely needs a definite or indefinite article. If the audience of your message understands that there is one specific open space to which you are referring, use the definite article ("the open space"); otherwise, use an indefinite article ("an open space").
As to which preposition to use, you would generally say that you sit in a space.
I will be sitting all day in the open space.
You would say that you sit on a piece of furniture, like a chair or a bench, or you would sit on the floor, the ground, or the lawn, as the case may be.
You would sit at your desk, or at a landmark, perhaps, or any other place that you are physically next to but not precisely "in" or "on."
Comment: Thanks, I forgot about the indefinite article but of course you are right, it should be there

do you sit in or at an openspace


Are there any examples of hidden Markov models that are not mixture models?

Question: Mixture models lend themselves nicely to HMM (hidden Markov model) treatment. Obviously, HMM can be nonmixture or not resolve mixtures when there is only one resolvable input or the superposition of self-similar inputs. What I am asking for is whether or not there are any not so trivial examples of nonmixture HMM data, with the impact relating to an attempt to define HMM as non-trivially different than mixture model analysis, e.g., see https://stats.stackexchange.com/a/249712/99274 $\leftarrow$Let me ask, for example, if the example I gave is exactly correct or not.
Be mindful please that a hidden Markov model can be considered a generalization of a mixture model where the hidden variables (or latent variables), which control the mixture component to be selected for each observation, are related through a Markov process rather than independent of each other..
And moreover that In the standard type of hidden Markov model considered here, the state space of the hidden variables is discrete, while the observations themselves can either be discrete (typically generated from a categorical distribution) or continuous (typically from a Gaussian distribution).
Comment: I am not sure of your semantics, but do *[particle filters](https://en.wikipedia.org/wiki/Particle_filter)* count? (The Wikipedia article for that one also appears a bit particular ... but many would include Kalman/particle filters as a type of HMM)
Comment: @Taylor that is fine. I was using "particle filter" as shorthand for something like "continuous-state HMM with no pre-defined parametric PDFs built in". Really just the idea that the class of state-space models/applications where particle filters are used must typically deal with multimodal state PDFs in practice (e.g. [robotics](https://scholar.google.com/scholar?cluster=15009205118336839875)).
Comment: Rabiner has a 1988 tutorial on HMMs in speech recognition that is widely read and cited to this day. He reviews several different types of HMMs. http://www.cs.cornell.edu/courses/cs481/2004fa/rabiner.pdf
Comment: @GeoMatt22 a particle filter is the algorithm, not the model. But I agree with the rest
Comment: @GeoMatt22 Tried to fix semantics, let me know please.
Comment: @DJohnson That's the stuff I wanted. +1, but make an answer, so I can vote. Also what about convolution? That is not mixture except as a subset.
Comment: Thanks. Actually, I don't have enough to say about this to motivate an  answer worth a vote, especially wrt convolution.
Comment: @DJohnson I know (or knew) zip about HMM. I learn a lot by putting stuff forward, although, admittedly there is a downside to that. For example, I have learned several foreign languages by opening my mouth and speaking, and even when I have gotten stupid looks as a reward, those stupid looks have dissipated in frequency of occurrence as I have gotten better at expressing myself. Go for it....
Comment: Absolutely. That's the right attitude towards lots of things. In fact, it delineates one of the divides between entrepreneurial vs corporate mindsets. Reminds me of Tom Lehman, co-founder of RapGenius.com, who said "The first version of Rap Genius was really bad — it sucked, and I'm glad it sucked," he says. "Only in its sucking could it have taught me the secret of how to build things on the internet, and that secret is 'worse is better.'" So it goes with many questions, comments and answers on CV. I've had a long-standing interest in HMMs. Let me think about writing a "response."
Answer: This might deserve to be a comment:
One of the reasons I don't understand your question, or in particular your second sentence, is that I'm not sure unimodal is equivalent with non-mixture. I guess it would be nice to think of those as equivalent if the observed random variables were continuous and in particular Gaussian, but also even in that case, I bet if the locations of each of those Gaussians are close enough, or if the weights are in such a configuration, that one of the conditional distributions of the observations could be unimodal, still. Are we talking about the predictive distribution, or some other conditional distribution?
And it IS very different because the proportions are changing (this might be more of the answer you're looking for). Assuming a HMM is assuming a model for how the mixing proportions change (the probabilities, the locations, the scales, etc.). You're assuming they are a Markov Chain. A mixture model is a static mixture. A HMM is a dynamic mixture.
Comment: I frankly do not know if it helps because I do entirely understand some of what you are saying.  Did you ask me to consider that, for example, the HMM model $f[N(X;\mu_1,\sigma_1),t]+g[N(X;\mu_2,\sigma_2),t]$, where $\int_{-\infty}^{\infty}f+gdt=1$ is somehow not a mixture model? Are you saying that a parametric generalization of a mixture model can be an HMM? Unimodal versus nonmixture and Gaussian, yup they seem used together in that context and I do not know about other contexts, any clarification appreciated.
Comment: @Carl no I don't recognize that. One thing I am saying is that there exist mixtures that are unimodal.
Comment: Dumped the word unimodal, see if my changes make sense, please, BTW +1 for effort.
Answer: HMMs model ordered sequences of observations or time-series $X = \{x_1, x_2,..., x_T\}$, where each $x_i$ can be a discrete or continuous, uni-dimensional or multi-dimensional observation/data sample. The order of the samples, from $t = 1$ to $t=T$ is modeled by the HMM as well (via the transition matrix).
Mixture models cannot model any dynamics in the data. So HMM can be seen as a Dynamic Mixture Model as noted in this research report for instance that defines the HMMs in the context of Dirichlet mixture models (as their dynamic counterpart). The multimodality does not change anything to this, even when mixtures are not used, HMMs embed data dynamics while mixture models "see" data as unordered. (By the way, as someone commented, multimodal is a very confusing term I find in the context of HMMs...).
The only cases HMMs are similar to mixture models are if the number of states is 1 or, as said in the comments, if the transition matrix is the identity matrix added to the fact the initial state probability mass function is a vector of 0 with a unique 1, and that simple distributions and not mixtures are assigned to the sates. Otherwise, it is more a "collection" of mixtures. However, knowing that the transition matrix is the identity adds an information about the dynamism of the data, and I would see it as a "weak-equivalence". In the case of a mixture-based HMM, the mixing matrix also impacts making this "equivalence" even more challenged...
In general, a good introduction reading to HMMs is this paper by Rabiner and Juang who popularized their use in the speech processing community.
Comment: "So HMMs cannot be similar to mixture models except if the number of states is 1." This isn't true. You can have more than $1$ states along with an identity transition matrix
Comment: @Eskapp +1 for effort and effect, look over semantic changes, see if I got that right or not, please. What about convolution?
Comment: Do not understand as the observations can be either discrete or continuous. Yes, you can help. Put quotes in question to help clarify.
Comment: Very true! I forgot this case :) Thank you for reminding me. I edit my answer
Comment: @Carl I added details and references. Hope it helps :)
Comment: @Taylor However, knowing that the transition matrix is the identity, adds some information about the data you are working with. So it is a "weak equivalence". Also, the initial state pmf impacts. If it is not a vector of zeros with a unique one, then the HMM is like a "collection" of mixture models.
Comment: @Carl Always a pleasure to discuss these topics :) I cannot see the changes yet, I'll make sure to check and discuss them if needed. I am not at all an expert in convolution so I cannot say much.   I know the existence of Convolutional HMMs but that's pretty much it...
Answer: With regrets, I did not want to answer my own question. What I was asking about was something other than mixture modelling. Here is one; Convolution HMM. @GeoMatt22, I had originally thought that "continuous-state HMM with no pre-defined parametric PDFs built in" was the most natural mixture model HMM definition, and did not even know about discrete modelling until it was mentioned here and I started looking into it a bit deeper. The general filter problem may be even more general than convolution as I suspect that its solution may be a general inverse solution, which is very general, indeed. But, the point here is that most people would probably agree that mixture models are distinct from convolution, and from the link, it would seem that there is such a thing as a convolution HMM. Indeed, a highly cited one, 600+. Having said that, I will award +1 to anyone who can (and has) helped with the semantics. What I am trying to do is come up with a short, good definition of HMM. And as @Geomatt22 says, the Wikipedia entry may be a bit 'particular' not to say wonky.
Comment: As far as I know, the most general definition of a hidden Markov model is a state-space model for an evolving state $x[t]$ (an arbitrary vector; can combine continuous and categorical components), whose dynamics satisfy **1)** the Markov property, such that given the current state $x_t$, the (possibly stochastic) dynamics is independent of the history (i.e. $p[x_{t+1}|x_{0:t}]=p[x_{t+1}|x_t]$), and **2)** whose state is at least partially hidden (i.e. some components are not observed). (Typically an HMM will be associated with a measurement function, cf. [tag:state-space-models] wiki).
Comment: @BrianCannard No. Convolution of random variates adds realizations of paired events. See my Q/A [here]{https://stats.stackexchange.com/q/331973/99274}. That is not a mixture model in which a random variate tends to weighted sum of two or more probability densities.
Comment: Isn't convolution to add probabilities, isn't a mixture such a sum? https://en.m.wikipedia.org/wiki/Convolution_of_probability_distributions#:~:text=The%20convolution%2Fsum%20of%20probability,linear%20combinations%20of%20random%20variables.


How to reach the planets core in Astroneer?

Question: I started playing Astroneer again but I'm unable to find the last teleportation point somewhere near the planets core. I searched like dozens of caves without success. How do I find it? Is it entombed?
Answer: In my experience, you have to dig downward to reach the center, which is called a Gateway Engine (so yes, they are entombed).  You can't find them by just exploring caves. In theory, you should be able to dig straight down and discover it but I wouldn't recommend this since you will almost certainly hit a cavern along the way and fall to death (not to mention likely suffocate).
I've notice that once you get closer to the center of the planet, gravity will lessen, allowing you to jump higher. All audio effects will also sound a lot more echoey.  When you begin to notice these signs, you should be getting fairly close, so just keep digging downward.
I've found the best strategy is to dig at roughly a 45 degree angle downward. You could go steeper, but it may be hard to climb your way back up eventually.
Answer: get a tractor and start digging down you will use the tractor for o2 and if you hit a cave get in in and you will not need to worry about fall damage i suggest placeing a medium wind turbine on the tractor for power and if you fail to hop in the tractor if you hit a cave when falling run at a wall of the cave for no fall damage.