I watched a video featuring Leonard Susskind in which he took a small bowl of water and added three drops of food coloring. He swirled it around. At first you could tell where the drops must have entered the water, but after a while you could not tell anymore. He said that a cornerstone of physics is that information cannot ever be destroyed and that the information is always there -- we just cannot see it.  I see two issues here and I'm hoping for some clarification.   The particles can never be traced back to their origin. The uncertainty principle says that other particles have popped in and out of existence. These may have interfered with the particles in the water and cannot be traced back. So information is in fact lost. Let's take the same type of experiment but narrow the scope. Instead of this, I put four protons into some sort of container that can hold them. Not sure what that is or if it's more of a field, but for this thought experiment, let's just say I have a box with four particles in it. At first you know where they originated by watching them (if you could watch them). But after a long enough time, the particles will occupy the same position and spin state that they did at some point in the past. So the first time they occupied this state they trace back to your initial placement. The second time they hit this state they trace back to the first state. So you cannot actually trace them back to see where they originated in the container.   Perhaps the second thought experiment has some flaws in it but I'm hoping you can see what I'm getting at. That a duplicated state must mean that the originating positions and information is lost, and given enough time you'll always get a duplicated state.
I really can't understand what Leonard Susskind means when he says in the video  that information is indestructible.  Is that information that is lost, through the increase of entropy really recoverable?   He himself said that entropy is hidden information. Then, although the information hidden has measurable effects, I think information lost in an irreversible process cannot be retrieved. However, Susskind's claim is quite the opposite. How does one understand the loss of information by an entropy increasing process, and its connection to the statement “information is indestructible”.  Black hole physics can be used in answers, but, as he proposes a general law of physics, I would prefer an answer not involving black holes.
How to write a good PhD literature review? How do I see the extensions loaded by PHP? Arrays, linked lists and other data structures in cmd.exe (batch) script Given a hotkey, how can I find which program owns it? Using C++ library in C code What can you use Python generator functions for? I'm starting to learn Python and I've come across generator functions, those that have a yield statement in them.  I want to know what types of problems that these functions are really good at solving. Given $a + b = 1$, Prove that $a^ab^b + a^bb^a \le 1$; $a$ and $b$ are positive real numbers. I'm one of very few people at my company who does graphic work, and I do quite a lot of it. Headquarters doesn't quite get that graphics work requires different hardware than, for example, sales work, which is pretty much email/Excel/Powerpoint/web browser and that's it. So they give everybody the same low-end rental PCs to work on.  This was tolerable when we were all on XP.  Recently I got "upgraded" to a new computer, which happens every two years or so as leases expire. The catch: the new PC had the same Core 2 Duo processor that the previous one did, less RAM, it runs Windows 7 instead of XP, and all the Creative Suite applications have been upgraded from CS3 to CS5! I am the first in my office to get 7.  So now I am in a situation where I have the slowest machine in the office, yet I have to do the most demanding work. I got them to OK 2 more gigabytes of RAM, and I hoped that would solve the issues, but it just isn't helping that much at all. I actually have to reboot every day after lunch because the machine stops responding after wake if you leave it asleep for too long.  I am currently talking to Information Systems to see if I can get a capable machine, but it does not bode well for me. Can anyone share any performance optimizations for Win7 that I might be able to use?  Thank you very much in advance! SS the root of $N$th transpose matrix For $n=3,$ $L$ matrix is as below:  $$ \left[\begin{matrix} 1&amp;0&amp;0\\ -1&amp;1&amp;0\\ -0.5&amp;-0.5&amp;1\\ \end{matrix}\right] $$  for $n=4$ we have:  $$ \left[\begin{matrix} 1&amp;0&amp;0&amp;0\\ -1&amp;1&amp;0&amp;0\\ -0.5&amp;-0.5&amp;1&amp;0\\ -1/3&amp;-1/3&amp;-1/3&amp;1 \end{matrix} \right] $$  and for $N$th we have: $$ \left[ \begin{matrix} 1&amp;0&amp;0&amp;0&amp;\cdots&amp;0&amp;0\\ -1&amp;1&amp;0&amp;0&amp;\cdots&amp;0&amp;0\\ -0.5&amp;-0.5&amp;1&amp;0&amp;\cdots&amp;0&amp;0\\ \vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp; &amp; \vdots&amp;\vdots\\ -1/(N-2)&amp;-1/(N-2)&amp;-1/(N-2)&amp;-1/(N-2)&amp;\cdots&amp;1&amp;0\\ -1/(N-1)&amp;-1/(N-1)&amp;-1/(N-1)&amp;-1/(N-1)&amp;\cdots&amp;-1/(N-1)&amp;1 \end{matrix} \right] $$  How to get the roots (eigenvalues) of the $N$th order of the $L+L^T$ matrix? is there a specific logic between the roots of this matrix? Finding the limit of $(1-\cos x)/x^2$ Kernel of injective group homomorphism If $\phi$ is a homomorphism from the group $G$ with identity $e$ to the group $G'$ with identity $e'$, then $\phi$ is injective if and only if $\ker\phi=$?  I think the $Ker\phi = \{x\in G:\phi(x)=1\}$  because you are looking at an identity then it is all reals $0$ does not work a because if $x=0, e'x=0$  but then i realize it is injective how does that affect the kernel? Universe being flat and why we can't see or access the space "behind" our universe plane? I'm a layman, but i watched some intereting videos about big bang on youtube[michio kaku, hawking this kind of things, not some crackpots :)]   I described everything on my picture:     So is there any possibility to see that inner side, for example, by creating wormhole, or entering new dimension? Or i get something wrong, on the way, and my model is totally wrong?  And another question. I heard that even if you would fly a ship across the universe you would eventually end in the place where you started, because we live in a 3d space that is defined on a surface of big sphere. But why a big sphere? Universe might be a big mobius strip, and the same principle of ending at start will apply. Is it possible?  Thanks in advance, and please don't bash me for silly questions. Im just curious, but i don't have any degree in physics. Im also not an english native speaker. Convolution is uniformly continuous and bounded Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function.  The definition of the convolution is $(f\ast K)(x)=\int_\mathbb{R}f(x-y)K(y)dy$.   There is the inequality $\|f\ast K\|_\infty\leq\|f\|_\infty\|K\|_1$, which yields that $f\ast K$ is bounded. But what about uniform continuity? I'm a big fan of the podcast Astronomy Cast and a while back I was listening to a Q&amp;A episode they did. A listener sent in a question that I found fascinating and have been wondering about ever since.   From the show :      Arunus Gidgowdusk from Lithuania asks: "If you took a one kilogram mass and accelerated it close to the speed of light would it form into a black hole? Would it stay a black hole if you then decreased the speed?"   Dr. Gay, an astrophysicist and one of the hosts, explained that she'd asked a number of her colleagues and that none of them could provide a satisfactory answer. I asked her more recently on Facebook if anyone had come forward with one and she said they had not.  So I thought maybe this would be a good place to ask. Is there an easier way to install Chromium than what is described ?  Furthermore, is there an easier way to stay up to date with current Chromium releases rather than having to re-build Chromium each release? (a script which automatically grabs and installs the latest build possibly?).  Related: No internet connection after ubuntu server 20.04 install, ifconfig not available I recently installed Ubuntu Server 20.04 LTS on my desktop, and found out there was no internet connection even though I had the LAN cable connected. ifconfig and iwconfig failed saying the corresponding packages net-tools and wireless-tools were not installed.   EDIT  Here is the info requested. Please find the image How do you choose Spells Known when creating a new character above 1st level? It's not clear to me how to choose Spells Known when creating a new character at a higher level.  Let's take the Bard as a working example first:  Bard level 2:   Spells Known: 5 Slots per Spell level: 1st &times; 3   Bard level 3:   Spells Known: 6 Slots per Spell level: 1st &times; 4, 2nd &times; 2   I'm interpreting this to mean that, if an existing 2nd-level Bard gains a new level (level 3), he's gaining 1 (one) new spell known. He can then choose the new spell known to be either 1st or 2nd, correct?  What I don't understand then, is the case of creating a new character starting at a higher level; say, creating a new Bard at level 3.  In that case you're forced to choose 6 spells, which must be from the list of 1st- and 2nd-level spells, correct?  When making this 3rd-level Bard from scratch, though, can I select 2nd level spells for all 6 spells known?  As another example, say you're creating a very high level Bard from scratch, starting at level 15. The book says your Spells Known is 19, and your spell slots are 1st × 4, 2nd × 3, 3rd × 3, 4th × 3, 5th × 2, 6th × 1, 7th × 1, and 8th × 1.  So, for those 19 Spells Known, you can choose any spells from 1st through 8th spell level. Am I free then to select as many spells as I can at 8th spell level (5 in number), working backwards, and in the end ignoring the 1st, 2nd, 3rd level spells entirely?  As another example, the Ranger doesn't know any spells at 1st level, nor have any spell slots, but the book says “You know two 1st-level spells of your choice from the ranger spell list.” Does it mean at 2nd level and above?  Long story short: When choosing Spells Known for a spellcasting character created above 1st level, am I allowed to choose my Spells Known at whatever spell level I like, just so long as I have at least one spell slot of that level?  I'm working on a D&amp;D 5e character generator and need to clarify this. I have filed my 2012 tax return jointly with my wife, along with my wife's ITIN application, back in late March.  In April, I got a letter for IRS ITIN department saying that we should send my wife's original passport to them. So we sent my wife's passport on May 15th. USPS tracking number confirmed that IRS received it on May 20th.  However, last week (end of June) we have received another letter from IRS ITIN department saying that our ITIN application has been rejected, rejected because we have not sent the original identification document that they requested!  Okay, let's put our tax return and ITIN aside. Where is my wife's passport? Right now my wife does not have her personal identification, nor proof of legal status in U.S. Could IRS have lost it?  I tried to call the IRS several times. One time I waited for 1 hour and 20 minutes to talk to a person, and that person said he has only 10 minutes to talk before he has to leave. He said they'll investigate and someone(?) will call us back soon, but never called back. I wrote them a letter, but no response. I tried calling several times more, but machine is saying 'sorry too many calls right now, we can't accept your call'.  What can I do? With proper nouns, we don't use the except for river names, newspaper names, etc.  I want to know why we use the with White House. I mean, under which rule can we categorize it? What could other similar examples for that rule be? Show that $(L^{p},\|\|_{p})$ is a Banach space.  My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So,  $(L^{1},\|\|_{1})$ is vector space. To show that is a Banach space, note that  Prop. 1: Let $(f_{n})_{n\in\mathbb{N}}\subset\mathcal{L}^{1}$ (where $\mathcal{L}^{1}=\mathcal{L}^{1}(X,\tau,\mu)$ is the space of all integrals function), such that $\sum_{i}{\|f_{n}\|}_{1}&lt;\infty$, then the sequences $\left(\sum_{n=1}^{N}{f_{n}}\right)_{n\in\mathbb{N}}$ converges almost everywhere to the integral function, that we called $\sum_{n\in\mathbb{N}}{f_{n}}$. Furthermore $$\sum_{n\in\mathbb{N}}{\int{f_{n}d\mu}}=\int{\sum_{n\in\mathbb{N}}{f_{n}d\mu}}\quad\lim_{N\to\infty}{\|\sum_{n=1}^{N}{f_{n}}-\sum_{n\in\mathbb{N}}{f_{n}}\|_{1}}=0$$ Prop. 2: Let $(E,\|\|)$ a normed vector space. Then $E$  is a Banach space if and only if for all $(e_{k})_{k\in\mathbb{N}}\subset E$ such that $\sum_{k\in\mathbb{N}}{\|e_{k}\|}&lt;\infty$, $\left(\sum_{k=1}^{n}{e_{k}}\right)_{n\in\mathbb{N}}$ converges in $E$  We take the proposition 2. Let $(f_{n})_{n\in\mathbb{N}}\subset L^{1}$ such that $\sum_{n}{\|\hat{f}_{n}\|_{1}}&lt;\infty$. Let $f_{n}\in\hat{f}_{n}$, then $\|f_{n}\|_{1}=\|\hat{f}_{n}\|_{1}$ and then $\sum_{n}{\|f_{n}\|_{1}}&lt;\infty$, by prop. 1, there exist $F\in\mathcal{L}^{1}$ such that $\|\sum_{n=1}^{N}{f_{n}-F\|_{1}}\to 0$, and then $\|\sum_{n=1}^{N}{\hat{f}_{n}-F\|_{1}}\to 0$. Finally note that $\hat{F}\in L^{1}$ is a Banach space. This was my answer for  $L^{1}$ space, but how I prove the general statement for $L^{p}$ (I want to do a similar response). Edit: I'm stuck in the problem, any idea or hint is appreciated.Thanks!!