Proof by induction of a Fibonacci relation
Inductive proof of the closed formula for the Fibonacci sequence
Edit a truetype font I'm trying to find a free tool (on windows ) that would enable me to change some settings on a TrueType font. Specifically, I'm trying to fix the following errors:  E1311 The macStyle bold bit doesn't match the name table's font subfamily string macStyle bold bit is set, but subfamily is Regular  W2000 The subfamily string is inconsistent with the style of the font OS/2.fsSelection bold bit is set, but subfamily string = 'Regular'  I can do that with fontographer, but would prefer to use something free.  Fontforge doesn't seem to work with this font, when saved it corrupts the cyrillic letters in the font. What is the command to add newline linebreak in \sum \prod? I am sure this question or similar has been asked but I cannot find it by searching so adding it to make it more accessible: what is the command/environment to add newline linebreak in \sum \prod so the expression is on two rows instead of one line hard to read?    MWE  \documentclass{article} \usepackage{amsmath,amsfonts}  \begin{document}  $$\sum^{N_{cut}}_{i=1,\newline e_1\in C_i,e_2\not\in C_i} \prod_{j\in C_i} p_j-\sum^{N_{cut}}_{i=1,e_2\in C_i,e_1\not\in C_i} \prod_{j\in C_i} p_j\geq 0$$  \end{document} MLE of $\theta$ when $X_1,\ldots,X_n $ are i.i.d with $P_\theta(x; \theta)=2x/\theta^2$ for $0\le x \le\theta$ Assume the standard situation, that is, let $X_1, \ldots , X_n$ be independent and identically distributed with $X_k ∼ P_\theta(x; \theta)$ , where $P_\theta(x; \theta) = 2x/\theta^2$  if $0\le x \le\theta$ and $0$ otherwise. It is required to estimate $\theta$. Show that the maximum likelihood estimator for $\theta$ is $\hat{\theta} = \max{[X_1, \ldots , X_n]}$ and then show that the cumulative distribution function of $\hat{\theta}$ is $F_\theta(z) = z^{2n}/\theta^{2n}$. Here's what i did so far: Maximum Likehood estimator: $L_x(θ) = \prod_{i = 1}^{n} P_θ(x_k)$ Here we have $P_θ(x_1,...,x_n;θ) = P_θ(x_1,θ).P_θ(x_2,θ)...P_θ(x_n,θ)$ Likehood = $L_{x,θ}(θ) =P_θ(x_1,θ).P_θ(x_2,θ)...P_θ(x_n,θ)= 2x_1/θ^2. 2x_2/θ^2.. 2x_n/θ^2 = [2^n.\prod_{i = 1}^{n}x_i]/θ^{2n}$ Log-likehood: $\sum_{i = 1}^{n}log(P_θ(x_1,...x_n;θ))= \sum_{i = 1}^{n}log(2x_i/θ^2)$ Is this correct so far? Im still not sure how to get to $\hat{θ} = max{[X_1, . . . , X_n]}$ As for the the cumulative distribution part to show $F_θ(z) = z^{2n}/θ^{2n}$: $F(z) = P(max({x_k})&lt;z) = P(x_1&lt;z).P(x_2&lt;z)..P(x_n&lt;z) = 2x_1/θ^2.2x_2/θ^2...2x_n/θ^n = 2^n.\prod_{i = 1}^{n}x_i/θ^2n$ Not sure if this is correct. Would really appreciate some help. Edit: From the answers below, we can deduce the the estimator is biased. What estimator would be unbiased? How can i find it? I'm from Western Pennsylvania. Until I moved away, I never realized that when I omitted the to be from phrases like needs to be cleaned, my usage was different than what most English speakers are accustomed to. Is it wrong? On Windows 10 Professional, with the November update (Build 10586, Version 1511) , is there any possibility to completely remove Windows Defender ?  I don't want to just disable Windows Defender.  I want to completely uninstall from my system. Get the subdomain from a URL Getting the subdomain from a URL sounds easy at first.  http://www.domain.example   Scan for the first period then return whatever came after the "http://" ...  Then you remember  http://super.duper.domain.example   Oh. So then you think, okay, find the last period, go back a word and get everything before!   Then you remember  http://super.duper.domain.co.uk   And you're back to square one. Anyone have any great ideas besides storing a list of all TLDs? In the movies, the Death Eaters leave a smoky trail behind them as they fly. As the Order arrive in the Ministry of Magic in Order of the Phoenix, they leave a silvery trail behind them. In the books, there’s no mention of how they’re doing this. And J.K. Rowling said in the books that only Voldemort and Snape could fly like this; first seen in the Battle of the Seven Potters.  Please can somebody explain how the Death Eaters are able to fly like this? Boot to console (standard method fails) I'm trying to get Ubuntu to boot directly to the console. I don't want to completely remove the GUI because I still kind of need it for training wheels sometimes, but I don't want to deal with it if I don't need it. In the past I have been able to achieve this pretty easily in previous versions by editing /etc/default/grub, but for some reason it's not working on 15.10. Here is my /etc/default/grub file: # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 GRUB_HIDDEN_TIMEOUT=0 GRUB_HIDDEN_TIMEOUT_QUIET=true GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` #GRUB_CMDLINE_LINUX_DEFAULT=&quot;quiet splash&quot; GRUB_CMDLINE_LINUX=&quot;text&quot;  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM=&quot;0x01234567,0xfefefefe,0x89abcdef,0xefefefef&quot;  # Uncomment to disable graphical terminal (grub-pc only) GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass &quot;root=UUID=xxx&quot; parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY=&quot;true&quot;  # Uncomment to get a beep at grub start GRUB_INIT_TUNE=&quot;480 440 1&quot;  As you can see, I have commented out quiet splash, changed the value of GRUB_CMDLINE_LINUX to &quot;text&quot;, and uncommented GRUB_INIT_TUNE. However, the only thing that seems to make a difference is the quiet boot. When I add and remove the hash from that line it affects the behavior when the machine boots, so I know grub is updating when I tell it to. But no matter what I do it still loads the GUI immediately. Let f(x) be a  polynomial satisfying $\lim_{x \to\infty} \frac{x^4f(x)}{x^8+1}=3,$  $f(2)=5 $ and $f(3)=10$, $f(-1)=2$ and $f(-6)=37$ . And then there are some question related to this. In the solution it is given that according to the given information, $f(x)$ is a polynomial of degree 4 with leading coefficient 3. $f(x)=3(x-2)(x-3)(x+1)(x+6)+(x^2+1)$ Now what i think is that he got the coefficient 3 using the limits and the rest of the terms$[(x-2)(x-3)(x+1)(x+6)]$ from the respective $f(x)$ values from what was given and also the $(x^2+1)$ was added for the remainders, as they satisfy all the values that have been given to us in the question. I just wanted to know if my approach is right? or is there another elegant way to do it?. Disallow closing of questions with a certain number of question / answer upvotes With reference to this currently open question   At the time of writing, it has 3 close votes yet has 7 question upvotes and the single answerer (11 upvotes) will have been awarded the enlightened badge (silver).  Therefore in the opinion of many (i) the question adds value and (ii) the answer adds value.  As a staring point for discussion, I suggest that 5 net question upvotes (and perhaps 5 answer net upvotes) should be enough to keep a question permanently open. Let's haggle over the numbers though. Thoughts? How can you combine all lines that end with a backslash character? Using a common command line tool like sed or awk, is it possible to join all lines that end with a given character, like a backslash?  For example, given the file:  foo bar \ bash \ baz dude \ happy   I would like to get this output:  foo bar bash baz dude happy My processor is using a big part of my RAM memory as cache and I want to clean it up because of that; will it prejudice something? App drawer icon gone from Cyanogenmod dock I have a Samsung Galaxy S2, running the latest CM10.1.  I lost the app drawer icon on my Dock; I must have deleted it. The app drawer icon lets you go to the app list, and the widget list.  How do I get the app drawer icon back? The icon looks like a circle with 6 dots inside it. While reading a question here whose answer was Stephen Baxter's &quot;Light of Other Days&quot; book, it reminded me of a short story I read with a similar idea. Probably read at least 30-40 years ago, in English. The fact that the Government has invented a device that can view the past is made public (by a whistleblower, I believe). Historians go nuts besieging the government to look at their favorite time periods to reveal the truth about various historical &quot;facts&quot; and or &quot;mysteries&quot; (Who shot JFK? Who was Jack the Ripper? Was Helen of Troy really that beautiful? Did Jesus exist? etc, etc.) And the government is stonewalls most of them and claims it's expensive to operate, hard to control to get exact time/location, etc. So someone (maybe the whistleblower?), knowing it's possible, does math/construction and builds his own version, and realizes it's easy to make, cheap to operate, and easy to use. So he makes plans to stick it to the government by releasing the info. Maybe he tries to force the government to release the info, or they realize he has made it, so they grab him and try to convince him to stop. But he is not convinced until they tell him, it can't see back very far (maybe just a few days/weeks, because it needs exponentially more power the further you look back.) Finally the agents tell him: &quot;Don't you see, you can use it to see what's happening just a few minutes ago, or even now. Your company can't make widgets? Zoom in on your competitors and see how they make them. Think your wife is fooling around on you? Zoom in on her and track her to see if she is. Your pretty neighbor is taking a bath now? Zoom in and watch her... These devices mean no one will have any privacy now.&quot; At which point he tells them it's already too late. They kept him there so long that the letters he mailed will already be delivered to various newspapers, scientists, historians, etc. What Happened to The Latex Symbols In the Libreoffice-Texmaths Equation Editor? I recently purged Libreoffice because it was buggy.  I then reinstalled the latest version from apt and now it's missing the symbols above the equation editor.  Here is the current Texmaths equation editor:   There used to be shortcuts for latex including frac{}{} and various set theory operations, as well as the binomial distribution on the top of the window.  What happened to them and how do I get them back? I just figured that they were in some package and installed all packages useing sudo apt install libreoffice*, but this didn't help after I refreshed the program.  EDIT 0: I'm using Kubuntu 19.04  sarah@LesserArk:~$ sudo apt-cache policy libreoffice-texmaths libreoffice-writer texlive libreoffice-texmaths:   Installed: 0.43-2   Candidate: 0.43-2   Version table:  *** 0.43-2 500         500 http://us.archive.ubuntu.com/ubuntu disco/universe amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu disco/universe i386 Packages         100 /var/lib/dpkg/status libreoffice-writer:   Installed: 1:6.2.2-0ubuntu2   Candidate: 1:6.2.2-0ubuntu2   Version table:  *** 1:6.2.2-0ubuntu2 500         500 http://us.archive.ubuntu.com/ubuntu disco/main amd64 Packages         100 /var/lib/dpkg/status texlive:   Installed: 2018.20190227-1   Candidate: 2018.20190227-1   Version table:  *** 2018.20190227-1 500         500 http://us.archive.ubuntu.com/ubuntu disco/universe amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu disco/universe i386 Packages         100 /var/lib/dpkg/status How to check if installing a package will require a reboot? Travelling in Schengen area while waiting for permit of stay renewal in Italy Understanding of the theorem that all norms are equivalent in finite dimensional vector spaces The following is a well-known result in functional analysis:     If the vector space $X$ is finite dimensional, all norms are equivalent.     Here is the standard proof in one textbook. First, pick a norm for $X$, say  $$\|x\|_1=\sum_{i=1}^n|\alpha_i|$$ where $x=\sum_{i=1}^n\alpha_ix_i$, and $(x_i)_{i=1}^n$ is a basis for $X$. Then show that every norm for $X$ is equivalent to $\|\cdot\|_1$, i.e.,  $$c\|x\|\leq\|x\|_1\leq C\|x\|.$$ For the first inequality, one can easily get $c$ by triangle inequality for the norm. For the second inequality, instead of constructing $C$, the  is applied to construct a contradiction.   The strategies for proving these two inequalities are so different. Here is my question,      Can one prove this theorem without Bolzano-Weierstrass theorem?   UPDATE:     Is the converse of the theorem true? In other words, if all norms for a vector space $X$ are equivalent, then can one conclude that $X$ is of finite dimension? Logical XOR operator in C++? Polynomial irreducible - maximal ideal I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal.  $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$  $\mathbf 1)$ Is $I_1$ a maximal ideal in $\mathbb{Q}[x]$?  Yes, since $I_1$ is irreducible with $p=3$ using Eisenstein's criterion, thus maximal ideal.  $\mathbf 2)$ Is $I_2$ a prime ideal in $\mathbb{Q}[x]$?  Yes, since $I_2$ is obviously irreducible, and thus a maximal ideal, and every maximal ideal is a prime ideal.  $\mathbf 3)$ Is $I_2$ a maximal ideal in $\mathbb{Z}[x]$?  Yes, $I_2$ is obviously irreducible, and thus a maximal ideal.  $\mathbf{Edit:}$ No, as it is not a field. $$ $$  Am I right in my conclusions?  Appreciate any help.