How to show that $P_n(x)$ have n distinct roots
Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros
Unmarshal a single element list fails Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$ For the $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0,$ my text book asks the following questions.  (b) Why is $f$ uniformly continuous on any bounded subset of $\mathbb{R}$?   (c) Is $f$ uniformly continuous on $\mathbb{R}$??  The graph for the function is this.    For the question (b), if I take subset between $[0.2,0.6]$ or the subset where the slope is steep, I don't think the function is uniformly continuous because I think for a given $\epsilon&gt;0$, there is no unique $\delta &gt;0$ for the bounded subset. Therefore, it also cannot be uniformly continuous on $\mathbb{R}.$ However, the questions sounds like the function is uniformly continuous and the book says that it is uniformly continuous. The answer on the book says something but I need more explanation. Thanks. Q) Prove that if $0\rightarrow V_1\rightarrow V_2\rightarrow...\rightarrow V_k\rightarrow 0$ is an exact sequence of finite dimensional vector spaces, then $\sum_{i=1}^{k}(-1)^idim (V_i) = 0$.  For $k=3$, I see that the statement is true. For $k&gt;3$, am I correct in understanding that $V_2 = V_1\oplus V_3 (\implies dim(V_2)=dim(V_1)+dim(V_3))$ and $V_3 = V_2\oplus V_4(\implies dim(V_3)=dim(V_2)+dim(V_4))$ and so on?   If it is true, then for $k=4$, $\sum_{i=1}^{4}(-1)^idim (V_i) = -m_1+m_1+m_3-m_2-m_4+m_4$ and am not sure why $m_2=m_3$ for the sum to be $0$? where $m_i = dim(V_i)$. Thanks. Did English ever have a word for 'yes' for negative questions? The Germans have doch and the French have si as a word that means "yes" in response to a negative question, such as:     Don't you want some ice-cream?   Yes [I do]!   In English, we only have yes (as far as I know) and further clarification is required in order to be unambiguous.  Did we ever have such a word in English and, if so, what happened to it? If not, given the French and Germanic influences on English, any idea why not? $a$ and $b$ are natural numbers and $p$ is a prime number. Prove that if $a^{p}-b^{p}$ is divisible by $p$, then it is also divisible by $p^{2}$.  My attempt: Based on Fermat's theorem $(a^{p}-a)$ and $(b^{p}-b)$ are divisible by p, therefore their difference is divisible by $p$. I.e. $(a^{p}-a)-(b^{p}-b)=(a^{p}-b^{p})-(a-b)$ is divisible by $p$. Since the left hand side of this equation is divisible by $p$, the right hand side also should be divisible by $p$. Based on the problem's assumption $(a^{p}-b^{p})$ is divisible by $p$ therefore we conclude $(a-b)$ must be divisible by $p$. Now We can factorize $(a^{p}-b^{p})$ $$(a^{p}-b^{p})=(a-b)(a^{p-1}+a^{p-2}b+....+b^{p-1})$$ we have to prove $(a^{p}-b^{p})$ is multiple of $p^2$ which means we have to show the right hand side should be multiple of $p^{2}$. But $(a-b)$ is multiple of p therefore we have to show  $(a^{p-1}+a^{p-2}b+\cdots+b^{p-1})$ is multiple of $p$ and I am stuck here. Any help would be appreciated. How can I get crash data (stack traces at least) from my Android application? At least when working on my own device being retrieved by cable, but ideally from any instance of my application running on the wild so that I can improve it and make it more solid. Plotting heat equation in a Manipulate expression Is there ever a case where you would want to accept an answer, but not upvote it?  I can't think of a single situation where this would be the case.. So would it not make sense to have the "Accept" button upvote the answer too? What is the emergency phone number in different countries? msiexec is command prompt software that installs an  program. But I have found that you can install an MSI file from the command line by just typing in the name of the MSI file on the command line.  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I believe, at this point the title became misleading, so let me make it clear:  I am not seeking an advice on data manipulation. I will not manipulate my data. However, I ask to myself "how the hell this can be detected?"  And now, I don't only ask to myself, but to whole community. How is this detected? For editors, referees out there, have you ever detected something like this? How to create tagged PDF such that they will be:   good enough for PDF/UA "reflowable" on smaller screens and ebook readers   Any syntax &amp; any engine are of interest, but I'm most interested in using LaTeX with [Xe|pdf]latex engines. How do I log into my Linux laptop if I have forgotten both the username and password? I see that the code in many packages and examples contains percent signs % at the end of (many) lines. What are they used for? Do they affect the parsing of those lines? I have a Nexus 7 (2013) and a Moto X (2014) running Lollipop. On both of these, if I get a heads-up notification (i.e., those that show up on top of the screen when the device is unlocked) and swipe down on it with 2 fingers, it 'breaks' further heads-up notifications:   I can't tap on them to open the responsible app For, e.g., an incoming phone call with 'answer' and 'dismiss' buttons, tapping the buttons does nothing. I can't answer phone calls anymore!   I can still swipe to the side to dismiss the notifications, and they still disappear on their own after a few seconds. I haven't been able to find a way to get the correct behavior back aside from a restart.  Yeah, the obvious workaround is to not two-finger swipe on them, but it's second nature as that same gesture expands notifications in other contexts.  I'm guessing the problem is generic, but I've triggered it on Hangouts and Phone notifications.  Do others experience the same thing? Any known 'fixes' aside from restarting? How good is it for SEO if you have a widget that lives on other sites? I made a widget that would ideally live on other sites.  Here is an example:   I guess since the widget would link back to me, it would be an SEO boost for my site.  Is that correct?  Or would it be just an SEO boost for that particular URL?  If it is just an SEO boost for the particular URL, it does me little good since that page does not link to any of my other pages.  Am I thinking about this correctly?  How are these things typically handled so that there is  a benefit to my site's SEO? Regex exec only returning first match I have an ArrayList&lt;String&gt; that I'd like to return a copy of.  ArrayList has a clone method which has the following signature:  public Object clone()   After I call this method, how do I cast the returned Object back to ArrayList&lt;String&gt;? Are airlines in the EU obligated to provide water for passengers? How to change 'et al' to italic font in citation call-outs, with 'agsm' bibliography style