Use strong induction to show that every intger can be expressed as the sum and/or difference of 3's
Show that every nonzero integer has balanced ternary expansion?
I was wondering, why there are passport checks at the check-in counter, Immigration and even at the gate?    For example, I was at Istanbul flying to either UK or US. There's a security check at the check-in counter asking a bunch of questions about my immigration status - even more than the Immigration officer at the border. Also, at the gate, there's also Security officer checking my immigration status and passport asking the same questions.    I thought that would be the responsibility of an Immigration officer rather than some third party company?   The thing I'm curious the most is they don't even know which countries this passport can go to with or without visa. They have to go back to the counter and check with the system which takes a long time. I was wondering why they hire these kind of companies to slow down the process? Formal writing: "…for my colleagues and {I/me/myself}."? I'm currently using Cambridge English Advanced 1. It's a book that contains past examination papers, and includes numerous samples of authentic writing. This material helps, candidates and teachers, understand what the examiners are ‘testing’ and how these papers are marked. The assessment covers four categories: content, communicative achievement, organisation and language. Each category is awarded a mark between 1 and 5, so the maximum score is 20, and each mark has a brief note attached by the examiner. Any errors of punctuation, orthography, grammar, appropriacy, and vocabulary are left intact. In fact, there are no corrections because the  ‘examiner’ does not specify where the errors lie. This can be frustrating, even though spelling mistakes are rare at the advanced level, and errors in style, collocation or register are still relatively easy to identify, sometimes I'll read a phrase that forces me to ponder.  Dear Director, [ TEXT ] To conclude, this letter is a polite request to cover the costs of a 2 month language course for my colleagues and me. We would be very pleased if the company would do us this favour. Yours sincerely John Smith  The following marks were awarded Content                    5    blah, blah, … Communicative Achievement  2    blah, blah, …     Organisation               3    blah, blah, … Language                   3    blah, blah, …  I am able to pick out six minor errors in that brief extract, maybe some users will identify more, maybe some will identify fewer, and maybe some will say that the language used is perfectly acceptable. But if I can help a candidate attain that elusive B,  I would be delighted. I am interested in (what could be) the 7th error, emphasised in bold. Because the letter of proposal is formal, I feel the phrase, for my colleagues and me, is jarring. I want to change it to for my colleagues and I, but the antecedent requires an object. You would not say: “This is a request to cover the cost […] for I”. So, why use the subject pronoun I in the expression “my colleagues and…”?  Could I use instead, myself?  Which of the following is preferable in a formal written proposal?    …for my colleagues and me …for my colleagues and I …for my colleagues and myself  EDITED: I found a   of the writing sample (11/11) if anyone is interested.  I've looked at the following question,  Some answers appear to be contradictory, the accepted answer says using  I and me are both grammatical, which in my example is not true. Moreover, there's no mention of myself, as a possible solution, in the question. Possible (but not serious) glitch in close/reopen I just had one of my questions closed. (Hey - it happens to everyone) When I clicked on reopen - lo and behold - it worked. I cast a reopen vote!  Is this a glitch? I know we need 4 more votes anyway but still...  And no  tag please :) The locus of the intersection point of two perpendicular tangents to a given ellipse For a given ellipse, find the locus of all points P for which the two tangents are perpendicular.   I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof. How do I install an application from a .dmg disk image? I've downloaded Firefox and it's now sitting on my desktop as Firefox.dmg. When I double click on Firefox.dmg it opens up a window with Firefox in it so I can use it - does this mean the application has been installed?    Community Wiki'd the question because I think it's a worthwhile question to have on the site for pointing people to, it's not exactly intuitive to do and not a question I think I deserve any rep from. Why is 8 so special? I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having infinitely many different dimensions of numbers, the only composition algebras are of 1, 2, 4, and 8 dimensions. What's so special about 8? I was going through fluid dynamics and I came upon this section on the shape of a liquid surface in a rotating cylindrical vessel. The working in the book is something like this:   It considers a mass $m$ of water on the surface of the rotating liquid at a distance $x$ from its axis of rotation .It takes into account a pseudo force ($m \cdot x \cdot \omega^2$) radially outwards and its weight force ($mg$).It then proceeds to calculate the net force and the angle it makes with the vertical.From there through basic integration it gets the equation of the surface of liquid  i.e. $y= \dfrac{(x^2 \cdot \omega^2)}{2g}$ .  I get this working.  But if I were to view the same rotation through an inertial frame,how would I proceed? I mean, there wouldn't be any pseudo force, rather a centripetal acceleration of same magnitude radially inwards.This is where I am facing certain doubts.  The forces on the liquid would be  the centripetal force inwards and the weight force downwards.The net force would therefore be in a different direction,and consequently the slope of the surface at that point would be in a different direction.How is it possible for the same surface to have two different shapes in two different frames?   I would be obliged if someone could help me out by pointing out the flaw in my understanding. Finding the perimeter of a right triangle with area given I'm going to encrypt the characters Zhu, and decrypt them using RSA. I'm using the public key $\{e, n\}$ and private key $\{d, n\}$. The values of $e$, $d$ and $p$ I get from my textbook: $e = 17$, $d = 53$, $n = 77$.  It works fine when the number is smaller than 76, but in my example below, it doesn't work:  z = 90  h = 104  u = 107  e = 17 d = 53 n = 77  print z,h,u  c_z = z ** e % n c_h = h ** e % n c_u = u ** e % n  print c_z, c_h, c_u  p_z = c_z ** d % n p_h = c_h ** d % n p_u = c_u ** d % n  print p_z, p_h, p_u   The outputs are:  90 104 107 62 69 46 13 27 30     Do you have any idea what's going wrong? Search a specific word and copy line to another Sheet Let $a\ge 2$ be an integer. Show that for positive integers $m,n$, we have $a^n - 1$ divides $a^m - 1$ if and only if $n$ divides $m$.   I am having trouble showing this. I've seen a similar problem on here but it only shows one direction for the proof. I would like to see both directions.  This is what I have so far: Let $m = nq + r$, then $a^m - 1 = a^{nq + r}- 1 = a^{nq}a^r - 1 = a^{nq}a^r - a^r + a^r - 1 = a^r(a^{nq} - 1) + a^r - 1$. Since $0\le r\lt n$, $r = 0$ then $a^m - 1$ = $a^{nq} - 1$ so $a^m$ = $a^{nq}$ . Thus $m = nq$ hence $n\mid m$. I get a long list of such arrays  {     "id": "byu6g6c4cjys5mdkg5znh8ju8c",     "create_at": 1511875272294,     "update_at": 1511875272294,     "delete_at": 0,     "display_name": "BMA",     "name": "BMA",     "description": "",     "email": "aleksandar.ivanov@random.com",     "type": "O",     "company_name": "",     "allowed_domains": "",     "invite_id": "gdgz1tbxuinntx1fbr1ax7kehy",     "allow_open_invite": false,     "scheme_id": null   }   I want to get by JQ only the ID where the name is BMA. At the moment I parse " jq -r '.[]["name"]" and I can filter the output from curl by name and I will get "BMA" and also 100 other names, but I need to filter only the ID where name is =BMA. Any ideas? Is there a geometric reason why two merging black holes never “decay” into two separate black holes All black hole mergers seem to result in one Kerr black hole after ring-down.   Could it be possible, that two initial black holes with two separate event horizons form an “intermediate compound state” with one event horizon, which - at a later time - “decays” in two final black holes?  I know that the answer to this question is “no” when applying Bekenstein-Hawking entropy where the decay would violate the second law which postulates $dA/dt &gt; 0$.   And the answer seems to be “no” when relying on numerical relativity for black hole mergers.  My question is if there is a more elegant, geometrical proof following entirely from general relativity, which rules out such a decay.  Thanks I use Kali Linux and discovered the following "problem":  root@kali:/proc/self# ls -l cmdline -r--r--r-- l root root 0 May 27 13:43 cmdline root@kali:/proc/self# chmod 744 cmdline chmod: changing permissions of `cmdline': Operation not permitted   I am not allowed to change the permissions of that file... but I am root? Why is that? Are there ways to work around this? I allready tried some things like moving the file (what is obviously not possible), change the runlevel etc. What exactly is weakly informative prior? Is there a precise definition of weakly informative prior?  How is it different from a subjective prior with broad support? I have just updated Windows 7 to Windows 10.  My touchpad scolling (right-side) is not working anymore.    I have no idea how to fix this in Windows 10.  I have hit the start button and typed "synaptic", but have found nothing useful.  I used to have an icon on the taskbar that showed where I was touching the touchpad.  I used to be able to right-click that icon to change settings, but Windows 10 has not brought that icon over.  Also, the asus website has no synaptic touchpad drivers available for Windows 10.  I have tried the enaltech drivers, but upon the installation atempt I am told that my hardware is not correct. Why doesn't \makeatletter work inside \newcommand? After hours of banging my brain against my skull, I narrowed down an issue I'm having to the fact that \makeatletter doesn't work inside \newcommand. Commands that I call after \makeatletter are registered as undefined, as if \makeatletter didn't do anything.  Why is this the case?  In the example below, I define two commands. Calling one of them from the document body causes the undefined error, while the other doesn't.  \documentclass[11pt]{book} \usepackage{kvsetkeys}  \newcommand{\commandThatFails}{     \makeatletter \comma@parse{}{} \makeatother }  \makeatletter     \newcommand{\commandThatWorks}{         \comma@parse{}{}     } \makeatother  \begin{document}  Hi there.  \commandThatFails \commandThatWorks  \end{document} I want to change the default timeout in Chrome. In Firefox I can set the value of network.http.connection-timeout in about:config - but how can I do that in Chrome? Why do the measurements of this object seem erroneous? delegate keyword vs. lambda notation Once it is compiled, is there a difference between:  delegate { x = 0; }   and  () =&gt; { x = 0 }   ?