What is charge actually?
Definition of electric charge and proper explanation
"viewed 1 times" When a post is first created, it says "viewed 1 times" in the sidebar. This should say "viewed 1 time". Open Mapping Theorem: counterexample The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to have some counterexamples. After all, how can you appreciate it's meaning without a nice counterexample showing how the conclusion could fail and why the conclusion is not obvious at all.  Let $\ell^1 \subset \mathbb{R}^\infty$ be the set of sequences $(a_1, a_2, \dotsc)$, such that $\sum |a_j| &lt; \infty$. If we consider the $\ell^1$ norm $\|\cdot\|_1$ and the supremum norm $\|\cdot\|_s$, then, $(\ell^1, \|\cdot\|_1)$ is complete, while $(\ell^1, \|\cdot\|_s)$ is not complete.  In this case, the identity $$   \begin{array}{rrcl}     \mathrm{id}:&amp; (\ell^1, \|\cdot\|_1)&amp; \to &amp;(\ell^1, \|\cdot\|_s)     \\                 &amp; x &amp; \mapsto &amp; x   \end{array} $$ is a continuous bijection but it is not open.  I want a counterexample in the opposite direction. That is, I want a linear continuous bijection $T: E \to F$ between normed spaces $E$ and $F$ such that $F$ is Banach but $T$ is not open. This is equivalent to finding a vector space $E$ with non-equivalent norms $\|\cdot\|_c$ and $\|\cdot\|_n$, such that $E$ is complete when considered the norm $\|\cdot\|_c$, and such that $$   \|\cdot\|_c   \leq   \|\cdot\|_n. $$ The Open Mapping Theorem implies that $\|\cdot\|_n$ is not complete.  So, is anyone aware of such a counterexample? One of the biggest issue with cluster analysis is that we may happen to have to derive different conclusion when base on different clustering methods used  (including different linkage methods in hierarchical clustering).  I would like to know your opinion on this - which method will you select, and how. One might say "the best method of clustering is which gives you the right answer"; but I may question in response that cluster analysis is supposed to be an unsupervised technique - so how do I know which method or linkage is the right answer?  In general: is a clustering alone robust enough to rely on? Or we need a second method and get a shared result to be based on both?  My question is not only about possible ways to validate / evaluate clustering performance, but is broader - on what basis do we select/prefer one clustering method/algorithm over another one. Also, is there common warnings that we should look around when we are selecting a method to cluster our data?  I know that it is very general question and very difficult to answer. I only would like to know if you have any comment or any advise or any suggestion for me to learn more about this. The Cartesian product of a finite number of countable sets is countable The Cartesian product of a family $(A_i\mid i\in I)$ is defined as $$\prod\limits_{i\in I}A_i=\{f:I\to\bigcup A_i\mid f(i)\in A_i\}$$ Let $I_n=\{i \in \mathbb N \mid i \le n\}$ for $n\in \Bbb N$ and $A_i$ be countable for all $i\in I_n$. Then $\prod\limits_{i\in I_n}A_i$ is countable.   My attempt: Lemma 1: $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ and $\prod\limits_{i\in I_{n+1}}A_i$ are equinumerous for all $n \in \mathbb N$. (I presented a proof ) Lemma 2: If $A$ and $B$ are countable, then $A \times B$ is countable. (I presented a proof ) I will prove the theorem by induction on $n$. The theorem is trivially true for $n=0$. Assume that the theorem is true for $n=k$, then $\prod\limits_{i\in I_k}A_i$ is countable. Since $\prod\limits_{i\in I_k}A_i$ is countable (by inductive hypothesis) and $A_{k+1}$ is countable, then by Lemma 2 $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ is countable. Furthermore, by Lemma 1 $(\prod\limits_{i\in I_k}A_i) \times A_{k+1}$ and $\prod\limits_{i\in I_{k+1}}A_i$ are equinumerous, then $\prod\limits_{i\in I_{k+1}}A_i$ is countable too. Thus the theorem is true for $n=k+1$. This completes the proof.   Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help! Does this recursive sequence have a closed-form? How to mount a NTFS partition in /etc/fstab? I have two partitions that I want to mount on startup: /dev/sda3   /mnt/devel  ext4    defaults    0   2 /dev/sda2   /mnt/excess ntfs    defaults    0   2  The ext4 partition mounts fine (owned by me, writable only by me), but the NTFS mounts owned by root with R\W permission for all. How to fix this? drwxr-xr-x  7 amanda amanda 4096 2012-03-14 19:07 devel drwxrwxrwx  1 root   root   4096 2012-03-14 22:38 excess Nignx - worker_connections are not enough In Einstein's General Relativity, do the space-time dimensions curve? In Einstein's General Relativity, do the space-time dimensions curve according to the positions of stars, planets, and masses? Is the tabu package obsolete? A limit ordinal $\gamma&gt;0$ is said to be indecomposable iff $\nexists\alpha,\beta&lt;\gamma$ such that $\alpha+\beta=\gamma$.  In view of this definition, I’m trying to prove the equivalence of the following statements for limit ordinals $\gamma$:  (1) $\gamma$ is indecomposable.  (2) $\alpha+\gamma=\gamma$ $\forall\alpha&lt;\gamma$.  (3) $\gamma=\omega^\alpha$ for some $\alpha\in\mathrm{ON}$.  I’ve been able to get $(1)\implies(2)$ and $(3)\implies(1)$, but I have nothing more than scratch work for $(2)\implies(3)$.  Any suggestions would be appreciated. Where did the world go? I was playing on my main world when my laptop ran out of power.  I found a charger and woke it up, but Minecraft had crashed.  Now, it doesn't appear in the world selection screen!  I have verified that it still exists in the Minecraft folder, and the directory contents look to be in order.  Does Minecraft save an internal flag somewhere when it crashes hard?  If so, how do I reset it? What happens with lethargy when Haste is cast again on a creature before first cast runs out? Haste states     When the spell ends, the target can't move or take actions until after its next turn, as a wave of lethargy sweeps over it.   However, let's say Character A cast Haste on Character B. In the 10th round (final round of haste if concentration not broken), Character B casts Haste on himself.  What happens during the next round?    Do they lose a round of the "new" haste while they are lethargic and lose movement and actions? Does the new haste override the lethargy until it ends? Is it some combination? Retain AC/Dex modifications but lose actions/movement? None of the above? Debian root partition resizing Prove that $\sqrt{10} - \sqrt6 - \sqrt5 + \sqrt3$ is irrational I tried the methods shown in  but I cannot extend them well into 4 numbers. Durrett Example 1.9 - Pairwise independence does not imply mutual independence? The example in question is from Rick Durrett's "Elementrary Probability for Applications", and the setup is something like this:     Let $A$ be the event "Alice and Betty have the same birthday", $B$ be the event "Betty and Carol have the same birthday", and $C$ be the event "Carol and Alice have the same birthday".   Durrett goes on to demonstrate that each pair is independent, since for example,   $P(A \cap B) = P(A)P(B)$.   However, he concludes that $A, B$, and $C$ are not independent, since   $P(A \cap B \cap C) = \frac{1}{365^2} \neq \frac{1}{365^3} = P(A)P(B)P(C)$.  I understand the reasoning here, and that one can generally show that arbitrary events $X$ and $Y$ are not independent by showing that $P(X\cap Y) \neq P(X)P(Y)$.   I am a little new to probability, though, and don't understand why exactly $P(A \cap B \cap C) = \frac{1}{365^2}$.   My progress so far:   I do see why $P(A) = P(B) =P(C) = \frac{1}{365}$, and thus why $P(A)P(B)P(C) = \frac{1}{365^3}$.  It seems like the sample space $\Omega = \{ (a, b, c) \mid a,b,c \in [365] \}$ -- i.e., all of the possible triples of numbers from 1 to 365, where 1 denotes January 1st, 2 denotes January 2nd, etc. From that, I can conclude $|\Omega| = 365^3$, but I'm not sure where to go from here.   It seems like once a single birthday is chosen, the rest are completely determined if they're all equal to each other - is this a good direction to go in? How can I get Firefox to open torrent files with Transmission? Every time I download a torrent file (.torrent), Firefox thinks I should open it with a text editor. How can I get it to open with Transmission? In Edit -> Preferences -> Applications, there's no setting for torrent files, and no apparent way to add a new filetype. Join CSV file to shapefile Concurrent requests limit exceeded - how to find the problematic code I am a developer for a Salesforce org that serves as a community portal for thousands of users. We have been consulting with a large team of developers for years and everything we have is custom coded. Our pages are built using AngularJS to remotely call Apex code. We are on an Unlimited Edition org and we are nearing 90% of our total allowed code. We have a lot of callouts to external services and many times these callouts can run long.  We are now running into issues where users are seeing the "Unable to Process Request - Concurrent requests limit exceeded" error. The issue is there is so much code written by so many different developers it is hard to identify what/where the problematic code is. Sometimes users see this error when they are trying to perform a callout. Sometimes it's when they are inserting a record and triggers are firing. The long-running code issue can be anywhere in our org. Another issue is error handling has been done very poorly, so if any process does run long, we most likely are not looking for it or logging it.  Other than enhancing the logging on our callouts, I do not know where to even begin to find other problematic code. Every developer before me has written code with no regards to limits, so inefficient code can be anywhere. Is there any way or best practice to being able to analyze such a complex org and find where the problems may be? All I can think of is enhanced logging EVERYWHERE, but that may not be the best approach and would be very time consuming, especially considering how many developers we have constantly writing and pushing new code.  I have tried to replicate the issue in a sandbox however I cannot seem to get it working. I wrote Javascript to remotely call an Apex method that waits for 6 seconds. I called this remotely 11 times at once, causing 11 synchronous transactions running concurrently each longer than 5 seconds. However instead of getting the concurrent limit error, each Apex transaction just errors out with a timeout.  Any help would be greatly appreciated. I am not familiar with the majority of the code/functionality of our org, so I do not know how to really solve this issue considering I don't even know what could be problematic aside from long running callouts. Thanks! Can I SSH into a specific directory? "Me and my wife" or "my wife and me"