Prove that if $a&gt;0\ then\ 1+a^9\le \frac1a+a^{10}$  I started off by moving every term to the left side of the inequality and multiplied every term by a since a>0 but not sure where to go after that. The hint given is to complete the square.  $a+a^{10}-a^{11}\le0$
Prove that if $a&gt;0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $.   Using the fact that $a \gt 0$, multiply by $a$ on both sides and get everything to one side we have; $a^{11}-a^{10}-a+1 \geq 0$. By factoring $(a^{10}-1)(a-1) \geq 0 $.  I am not sure how to proceed any further.
Recommend a statistics fundamentals book To give you some background, I have a grasp on the basics of statistics and probability theory and even remember touching Bayes theorem at the university data mining course. But being a few years away from the university made my math got extremely rusty (so much for last-minute pre-exam cramming). While I remember various random basic concepts, there are a lot of gaps in my understanding of them.  What would be a good material (a book, a site, or otherwise equally accessible medium) to revise the fundamentals and go beyond basics? I'd like a book that can be actually read as a book (most statistics books are really dry and are close to being reference material, rather than a book). The 'ok' on the 'Your Answer' dialog on unregistered sites is a bit too high The 'ok' on the 'Your Answer' dialog on unregistered sites is a bit too high:     You can see the top of the k is outside the actual box, increasing the margin-top by 10px in the containing div fixes it. For the command   sudo apt-get update   I get following error:  W: Failed to fetch http://ch.archive.ubuntu.com/ubuntu/dists/maverick-updates/multiverse/binary-i386/Packages.gz   407  Proxy Authentication Required (The ISA Server requires authorization to fulfill the request. Access to the Web Proxy filter is denied. )   I am running Ubuntu 10.10 installed on Windows XP using VirtualBox. For Internet connections I am using proxy server with an authentication. I tried to use gnome-network-proxy tool to set proxy settings system-wide.   After that /etc/environment has been updated by http_proxy variable with the format http://my_proxy:port/, there were no authentication data.  I checked this with Firefox. Browser asked my for login and password and everything was working fine. It was unfortunately not the case for apt-get. I have also tried to do as described . Unfortunately it does not work.  May it be somehow related to the fact that a proxy is in a Windows domain, any ideas?  My proxy name is http-proxy. Is '-' a special character here? Put numbers in legend using pgf pie I would like to have the numbers that are inside the pie chart to be in the legend. How can I make that happen? Here is the MWE: \documentclass{article} \usepackage{pgf-pie} \usepackage{tikz} \usetikzlibrary{shadows}  \begin{document} \begin{tikzpicture} \pie[text=legend,style=drop shadow,radius=2]{51.3/Protestant, 23.9/Roman Catholic, 3.3/Other Christian, 1.7/Jewish, 0.7/Buddhist, 0.6/Muslim, 0.4/Hindu, 1.2/Other religions, 16.1/No religion } \end{tikzpicture} \end{document}  EDIT: I also have another pie chart, where the text is quite long. How can I set the width of the text so that it wraps around before the page ends? Here is what it looks like:  I forgot that PGF-PIE was not on CTAN. Here is a link to the google source: Proof of "well-founded iff there is no sequence $x_n$ s.t. $(x_{n+1}, x_n) \in R$" I tried to do the following exercise from a book I'm reading, can you tell me if my solution is correct? Thanks!  Theorem 2: Let $R$ be a binary relation on a set $X$. Then $R$ is strictly well-founded iff there is no sequence $(x_n)_{n \in \mathbb N}$ of elements of $X$ such that $(x_{n+1}, x_n) \in R$ for all $n \in \mathbb N$.  Exercise:  (i) Formulate an analogue of Theorem 2 for  well-founded relations.   (ii) Prove Theorem 2.    (i) Theorem 2': Let $R$ be a binary relation on a set $X$. Then $R$ is well-founded iff there is no sequence $(x_n)_{n \in \mathbb N}$ of elements of $X$ such that $(x_{n+1}, x_n) \in R$ and $x_n \neq x_{n+1}$ for all $n \in \mathbb N$.  (ii)   $\implies$: Assume there is a sequence $x_n$ with $(x_{n+1},x_n) \in R$ for all $n \in \mathbb N$. Then the set $Y = \{x_n \mid n \in \mathbb N\}$ and hence $R$ is not well-founded since for every element $x_n$ there is $x_{n+1} \in Y$ with $(x_{n+1},x_n) \in R$.  $\Longleftarrow$: Let $R$ be strictly well-founded. Assume we have a sequence $x_n$ with $(x_{n+1},x_n) \in R$ for all $n \in \mathbb N$. Then by the same argument as above $R$ is not well-founded, hence there cannot be such a sequence.   I must be missing something but I can't seem to figure out what. Probability of the union of $3$ events? I need some clarification for why the probability of the union of three events is equal to the right side in the following:  $$P(E\cup F\cup G)=P(E)+P(F)+P(G)-P(E\cap F)-P(E\cap G)-P(F\cap G)+P(E\cap F\cap G)$$  What I don't understand is, why is the last term(intersection of all) added back just once, when it was subtracted three times as it appears from a Venn Diagram?  Here on page 3, this is explained but not in enough details that I can understand it: How do you test non-live sites before switching DNS? I have a few websites that I need to move to a new server - how would I test that they work before switching DNS? What URL would I navigate to? Why the Principle of Least Action? I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$.  Okay, fine.  You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations.  Great.  But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be equivalent to the Principle of Least Action.  After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all:  it's the "Principle of Stationary Action".  Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional.  Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique.  But the problems don't stop there.  Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$.  As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.  From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way.  Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory.  Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean period), and in fact, that it is so natural that we would not even dare abandon these ideas? Nginx installation error in Ubuntu 16.04 :~$ sudo apt-get remove nginx* && sudo apt-get install nginx-full*  Reading package lists... Done Building dependency tree        Reading state information... Done  After this operation, 5,579 kB of additional disk space will be used. Do you want to continue? [Y/n] y Get:1 http://in.archive.ubuntu.com/ubuntu xenial-updates/universe amd64 nginx-full-dbg amd64 1.10.0-0ubuntu0.16.04.1 [3,789 kB] Fetched 3,789 kB in 33s (112 kB/s)                                              Selecting previously unselected package nginx-common. (Reading database ... 189767 files and directories currently installed.) Preparing to unpack .../nginx-common_1.10.0-0ubuntu0.16.04.1_all.deb ... Unpacking nginx-common (1.10.0-0ubuntu0.16.04.1) ... Selecting previously unselected package nginx-full. Preparing to unpack .../nginx-full_1.10.0-0ubuntu0.16.04.1_amd64.deb ... Unpacking nginx-full (1.10.0-0ubuntu0.16.04.1) ... Selecting previously unselected package nginx-full-dbg. Preparing to unpack .../nginx-full-dbg_1.10.0-0ubuntu0.16.04.1_amd64.deb ... Unpacking nginx-full-dbg (1.10.0-0ubuntu0.16.04.1) ... Processing triggers for ufw (0.35-0ubuntu1) ... Rules updated for profile 'Apache Full' Rules updated for profile 'Nginx HTTP'  Processing triggers for systemd (229-4ubuntu4) ... Processing triggers for ureadahead (0.100.0-19) ... Processing triggers for man-db (2.7.5-1) ... Setting up nginx-common (1.10.0-0ubuntu0.16.04.1) ... Setting up nginx-full (1.10.0-0ubuntu0.16.04.1) ... Job for nginx.service failed because the control process exited with error code. See "systemctl status nginx.service" and "journalctl -xe" for details.  invoke-rc.d: initscript nginx, action "start" failed.  dpkg: error processing package nginx-full (--configure):  subprocess installed post-installation script returned error exit status 1 dpkg: dependency problems prevent configuration of nginx-full-dbg:  nginx-full-dbg depends on nginx-full (= 1.10.0-0ubuntu0.16.04.1); however:   Package nginx-full is not configured yet.  dpkg: error processing package nginx-full-dbg (--configure):   dependency problems - leaving unconfigured No apport report written because the error message indicates its a followup error from a previous failure.                            Errors were encountered while processing:  nginx-full   nginx-full-dbg  E: Sub-process /usr/bin/dpkg returned an error code (1) out file with multiple criteria in python? Am I required to pay UK/ USA Visa fees if I have a German residence permit? Here is the situation. I am non-european but staying in Germany with a Residence permit. I can visit all Schengen listed countries with no issue. Assuming that I would want to travel to a non-Schengen country e.g. UK or USA, can I use my residence permit to travel there or am I required to apply as a citizen of my country? Assuming Germans are not required to pay Visa fees to that country, will I have to pay since I only have a residence permit? Third case, Europeans can fly directly to the airport of some countries and apply for a visa from there. In some other countries, you are required to book a Visa appointment at the embassy, which may be granted or not. Does that apply for one with a residence permit? Force pagebreak after every sentence/paragraph Is there a way to force a pagebreak after every sentence or paragraph?  This may seem like an odd thing to do, but for catching typos and just general editing I find it much easier to look at a sentence or paragraph in isolation so that I am not tempted to skip ahead.  Currently, I just zoom in so I can only see a few word at a time, but this is not a perfect solution. Making a (simple) directory tree I'm trying to make a simple directory tree representation in LaTeX. Something like:  spam  |  +-- ham  +-- eggs      |      +-- more spam      +-- dead parrots   The tree won't be that long (say, 10--15 entries or so), with three or so levels, and is a one-off thing. So, quick-and-dirty solutions are welcome. When recommending tikz, please note that I have almost no experience with it, so I'd appreciate a clean simple solution, even if it doesn't look quite perfect.   On Tex.SE tikz is almost like the jquery recurring joke on StackOverflow..    Edit: As per Thorsten's suggestion,  is a nice and easy-to-use solution. The example from above could be written as:  \usepackage{dirtree} ... \dirtree{% .1 spam. .2 ham. .2 eggs. .3 more spam. .3 dead parrots. } Convert .sdf database to .mdf database Recovering files from harddrive My laptop got broken (by what it seems the graphics card burned) and it is so old that it is not worth fixing it.   I took the harddrive from it and tried to use it as external hard drive with other computer, however I cannot access all the data, ie anything that is inside my user folder (it shows it to be empty). Couldn't find anything to solve it. I have applied recently for a general visitor visa to the UK to spend 10 days during Christmas with my British unmarried partner and it has been turned down.   The Entry Clearance Officer stated on his refusal notice the following:     I note that you have never previously travelled to the UK or any country with similar immigration requirements and you therefore do not have a history of travel and compliance to consider.      You have submitted an employment letter to confirm your occupation in KSA but I am required to also assess your personal, domestic and economic circumstances in determining whether you are genuine visitor travelling to the UK for the period and purpose you state.      You claim an income of SAR ........ per month which equal ........ GBP and from the evidence submitted this appears to be your sole source of income.   I note that this monthly income is reflected on your bank statements. However, the bank statement you have presented show a recent large deposit that are in excess of your stated income and inconsistent with the account history. You haven't provided evidence toward the origin of this recent deposit and in the absence of reasonable evidence I am not satisfied that these bank statements accurately reflect your personal circumstances.      In view of the above, I am not satisfied you have demonstrated that your circumstances are as you claim and therefore I am not satisfied that you're travelling to the UK for the period and purpose you stated. 41(i)and(ii).   Its true that I have deposited that sum of money right before I apply and it was a commission I received from work earlier this month.   So, could you please give me your advice on the above as I am planning to apply again soon. and how to avoid my application being refused again? Help identifying an error type "tried to help me learning" I have a friend from Russia who is trying to learn English and recently used the sentence "He tried to help me learning..." (implied: the English language)  It is obviously wrong and I corrected it telling her that she was supposed to say: "He tried to help me learn.", but I have a hard time explaining to her why it is incorrect since I'm not an expert on grammar either.  Is it a case of confusing the present simple and present continuous or does it have something to do with gerunds and infinitives? Any help would be highly appreciated! If $n\ne 4$ is composite, then $n$ divides $(n-1)!$. (In, On or At) GitHub? What's the difference between say:     "The project will be on GitHub",      "The project will be in GitHub" and      "The project will be at GitHub"? Adding DC voltages of PSU I need to add two voltages together that share a common ground (power supply from computer). I tried using bridge rectifiers and Op-Amps.