The squirrel's riddle - how many marks on the tree trunk?
What are all the correct answers to the Squirrel's questions, and what is the reward when I get them all correct?
Proof of orthogonal matrix In the following question we regard vectors in $\Bbb R^n$ as columns and define the dot product in the usual way which means that $x · y$ = $x^{t}y$  a) If A is an n × n matrix show that $A_{ij}$ = $e_i \cdot A e_j$ where $e_i$, $i =1, \dots , n$ are the standard basis vectors in $\Bbb R^n$  b) Show that a matrix $P$ is orthogonal if and only if ($Px$) · ($Py$) =$x · y$ for all $x, y$ ∈ $\Bbb R^n$  Working -   I got part (a) alright but have no idea of where to start with part (b) except to use the information that an n x n matrix $P$ is orthogonal if $P^{-1}$= $P^{t}$ and that $P^{t}$$P$=$I$, $P$$P^{t}$=$I$, where $I$ is the identity matrix. Any suggests are gracefully appreciated. Feel free to edit my question for clarity. Thank you Starters on Stack Overflow tend to lose interest as soon as their first question starts getting downvotes. Their first question also sometimes gets closed because it was either not framed properly or it missed some very common things.  There should be way to cope with this for instance give some starter bonuses to new users. Can I skip over releases when upgrading? iPhone, reproduce the magnifier effect 1) "The book 'The Three Musketeers' is a wonderful example of..." Here we have a proper noun, a title that happens to end in a plural, and I have no sense that the verb should be plural. "Musketeers"/is seems proper/natural/correct, and 'are' not at all.   However, such certainty begins to wane in other examples: 2) "'The Heidi Chronicles' is a wonderful example of..." doesn't completely shake my confidence, but it moves the needle, at least a little bit. But, practically speaking, I wouldn't have a second thought about correcting a student who used 'are' instead of 'is'.  Here's the one that really bothers me: 3a) " 'The Basement Tapes' are a wonderful example of..." I don't think this sounds any less proper/natural/correct than 3b) " 'The Basement Tapes' is..." In fact, 3a seems less awkward than 3b.  Why? For only my index pages, I am receiving this warning in Google Search Console:     Duplicate, submitted URL not selected as canonical   I know that this occurs because Google is attempting to track both the default landing "page" (i.e. www.domain.example/) and the page to which to which DirectoryIndex points (i.e. www.domain.example/index.html). They are, in fact, the same page, and so the warning is the result of a false positive.  Two questions for our community:   Why does Google not conflate these and count them as a single page? Alternately, is there something wrong with my set up if this has happened? I can easily force both pages to point to a single canonical URL. But which is preferable, "/" or "index.html"? A generic question across most board games: once a player has taken their turn (taken their finger off the piece they are playing), should they be allowed to change their mind?  Similarly, if a player forgets to take a privilege, such as an occupied building in Puerto Rico giving an extra worker, or plantation, if they remember a few moves later, should they be allowed to take the privilege?  Very few rules explicitly express how these events should be dealt with, so they are generally left up to the group. However, my group's rules will change depending on the personality of the players playing (not all members are available each time we play). I would like to make this consistent, and would appreciate feedback on how others deal with this scenario. How can two dates be compared in a shell?  Here is an example of how I would like to use this, though it does not work as-is:  todate=2013-07-18 cond=2013-07-15  if [ $todate -ge $cond ]; then     break fi                              How can I achieve the desired result? Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $. How does one prove the following limit? $$   \lim_{n \to \infty}   \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$ A surjective endomorphism (of a Noetherian ring) is injective. The problem is stated as follows:  "Let $R$ be a Noetherian ring and $\theta$ be a ring homomorphism from $R$ to $R$. Show that if $\theta$ is surjective then it is also injective."  Regardless of the right solution, I don't understand why is the following wrong:     We have $\theta: R\to R$. By the isomorphism theorem $R/\ker\theta\cong\operatorname{Im}\theta$. Since $\operatorname{Im}\theta = R$, it follows $\ker\theta =0$, so it's injective.   Harsh criticism will be appreciated. Thanks. How do I fix "Error C1033: cannot open program database"? Editing the "Self-Learner" Badge Description Stemming from  the text for "Self-Learner" badge is pretty easy to get confused by.  Current Text     Answered your own question with score of 3 or more    Issue  The person in the above question was confused as they thought they needed to answer their own question, and that the question had to have a score of 3 or more. What it actually means is that they have to answer their own question, and the answer needs to have a score of three or more. The way it is phrased is ambiguous, so why not make it clear?  Suggestion  Edit the text as follows:     Provide an answer with a score of 3 or more to your own question   No more confusion. Using Process.Start() to start a process as a different user from within a Windows Service I'd like to periodically run an arbitrary .NET exe under a specified user account from a Windows Service.   So far I've got my windows service running with logic to decide what the target process is, and when to run it. The target process is started in the following manner:   The Windows Service is started using "administrator" credentials. When the time comes, an intermediate .NET process is executed with arguments detailing which process should be started (filename, username, domain, password). This process creates a new System.Diagnostics.Process, associates a ProcessStartInfo object filled with the arguments passed to it, and then calls Start() on the process object.   The first time this happens, the target process executes fine and then closes normally. Every subsequent time however, as soon as the target process is started it throws the error "Application failed to initalize properly (0xc0000142)". Restarting the Windows Service will allow the process to run successfully once again (for the first execution).  Naturally, the goal is to have target process execute successfully every time.  Regarding step 2 above: To run a process as a different user .NET calls the win32 function CreateProcessWithLogonW. This function requires a window handle to log the specified user in. Since the Windows Service isn't running in Interactive Mode it has no window handle. This intermediate process solves the issue, as it has a window handle which can be passed to the target process.  Please, no suggestions of using psexec or the windows task planner. I've accepted my lot in life, and that includes solving the problem in the manner stated above. If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \frac{n-1}{n+1}$, find $a_n$ and the sum. If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \dfrac{n-1}{n+1}$, find $a_n$ and the sum.  By definition,  the sum of the series is the $\lim n\to\infty$ of it's $n^{th}$ partial sum.  $$\text{ Sum = } \lim_{n\to\infty} = \dfrac{n-1}{n+1} = 1$$  I am asked to find $a_n$. How do I do this? What is the procedure? Build a basic Python iterator I use the babel package in order to write in greek with font "kerkis".  I want to add code in english ("courier new" font) to my latex document but I get some difficulties.  Can anybody help me, please. Thank you in advance.  \documentclass[12pt]{article} \usepackage[english,greek]{babel} \usepackage[utf8x]{inputenc} \usepackage{graphicx} \usepackage{fancybox} \usepackage{verbatim} \usepackage{courier} \usepackage{kerkis} \usepackage{float} \usepackage{geometry} \usepackage{fancyhdr} \usepackage{titlesec} \usepackage{tocloft} \usepackage{hyperref} \usepackage{afterpage} \usepackage{color}    \usepackage{listings}  \renewcommand{\lstlistingname}{Κώδικας}  \lstset{  %matlab   backgroundcolor=\color{white},     basicstyle=\normalsize,    % COURIER NEW?        commentstyle=\color{green},    % comment style   escapeinside={\%*}{*)},           extendedchars=true,               frame=single,                              keywordstyle=\color{blue},       % keyword style   numbers=left,                     numbersep=5pt,                     numberstyle=\normalsize\color{black},    rulecolor=\color{black},           stepnumber=1,                      stringstyle=\color{magenta},     % string style }   \begin{document}  \selectlanguage{greek} Ελληνικά: \textlatin{this word means greek}  \selectlanguage{english} Now I write in english  \selectlanguage{greek} Ελληνικά  %\selectlanguage{english}???  \begin{lstlisting}[caption = \textlatin{example1.m}] { % Entropy E = 0; for i = 1:J     if p(i) ~= 0         E = E + p(i)*log2(1/p(i));     end end } \end{lstlisting}  \end{document} I would like to know how to upgrade from my current version of Ubuntu to the next stable from the Command line interface? I have a valid US passport. I will fly Jet Airways to Delhi airport with 9 hours of layover to change to another Jet Airways Flight to Abu Dhabi. Do I need a transit visa? Permanently mounting loop device I created a file   touch /datastore/virtual/pseudoblock   created a loop device with it  sudo losetup /dev/loop0 /datastore/virtual/pseudoblock   (although I got a warning about its size)  and then initialized it with random data  dd if=/dev/urandom of=/dev/loop0 bs=1M count=100   My questions are  a) why is it not displayed yet in my block devices' list?  NAME                     MAJ:MIN RM   SIZE RO TYPE  MOUNTPOINT sdb                        8:16   0 489.1G  0 disk   ├─sdb2                     8:18   0   488M  0 part  /boot ├─sdb3                     8:19   0 488.1G  0 part   │ └─sdb3_crypt           253:0    0 488.1G  0 crypt  │   ├─ubuntu--vg-root    253:1    0 472.2G  0 lvm   / │   └─ubuntu--vg-swap_1  253:2    0  15.9G  0 lvm   [SWAP] └─sdb1                     8:17   0   512M  0 part  /boot/efi sda                        8:0    0 931.5G  0 disk   └─sda1                     8:1    0 931.5G  0 part     └─sda1_crypt           253:3    0 931.5G  0 crypt      └─store--vg-store1tb 253:4    0 931.5G  0 lvm   /datastore   b) how can I permanently mount it via /etc/fstab since I cannot get its UUID?  $ blkid /dev/mapper/sdb3_crypt: UUID="fSHjbj-51K2-QfhI-G0Wb-o3lN-4pE1-BB4wCP" TYPE="LVM2_member" /dev/mapper/ubuntu--vg-root: UUID="9e452886-4d8g-fj54-854c-80bc171826e8" TYPE="ext4"   working on   $ lsb_release -a No LSB modules are available. Distributor ID: Ubuntu Description:    Ubuntu 16.04.5 LTS Release:    16.04 Codename:   xenial   with  $ uname -a Linux pkara-pc01 4.15.0-42-generic #45~16.04.1-Ubuntu SMP Mon Nov 19 13:02:27 UTC 2018 x86_64 x86_64 x86_64 GNU/Linux Philosophy is the mother of all science Prologue Running google search:  at the time of this post yields about 114,000 results, and while this Quora post:  puts forth the question to determine the truth of the matter, it does not address the historical, etymological, and philosophical import of the matter. This PhilSE Post:  clearly addresses theory, but does not address the historical aspect. The related question in Researchgate post:  seems to not provide a substantial answer either. Inquiry What is the  context in which the statement 'Philosophy is the mother of all science' or related ones began to recognize as a  the  relationship between philosophy and the advent of modern science under thinkers like ? That is to say, what historical and social consequences were responsible for seeing  somehow superior as a theory to the ? Epilogue The question is intended to determine the metaphilosophical insights of philosophers. For example, if Karl Popper used the phrase or one similar, did he appeal to previous philosophers for insight on that same relationship? See Also