Let $n$ be a natural number such that $n \ge 2$ and given complex number $z_1, z_2, \ldots, z_n$ that is contained in an open unit disk centered at origin. Prove that there exists $\epsilon_l = \pm 1$ with $l = 1, 2, \ldots, n$ such that  \begin{align*} \left|\sum_{l=1}^n \epsilon_l z_l\right|\le\sqrt{3} \end{align*}
Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|&lt;1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that $$\left|\sum_{i=1}^n a_iz_i\right|&lt;\sqrt3.$$
Mob Arena - Mob vs. Mob I want to make an arena where mobs will fight each other and a player can bet on one side or the other. For example: There are 2 skeletons in dyed armor; red and blue. Each color representing a team. When they spawn I want them to fight each and then when one dies I want it to announce which team is the winner and give the player the "money" due. I'm just not sure how to get the skeletons to attack each other without a player instigating it. I am trying to prove that the set of irrational numbers $\mathbb{R} \setminus \mathbb{Q}$ is not an $F_{\sigma}$ set. Here's my attempt:  Assume that indeed $\mathbb{R} \setminus \mathbb{Q}$ is an $F_{\sigma}$ set. Then we may write it as a countable union of closed subsets $C_i$: $$ \mathbb{R} \setminus \mathbb{Q} = \bigcup_{i=1}^{\infty}  \ C_i $$ But $\text{int} ( \mathbb{R} \setminus \mathbb{Q}) = \emptyset$, so in fact each $C_i$ has empty interior as well. But then each $C_i$ is nowhere dense, hence $ \mathbb{R} \setminus \mathbb{Q} = \bigcup_{i=1}^{\infty}  \ C_i$ is thin. But we know $\mathbb{R} \setminus \mathbb{Q}$ is thick, a contradiction.  This seems a bit too simple. I looked this up online, and although I haven't found the solution anywhere, many times there is a hint: Use Baire's Theorem. Have I skipped an important step I should explain further or is Baire's Theorem used implicitly in my proof? Or is my proof wrong? Thanks.  EDIT: Thin and thick might not be the most standard terms so:  Thin = meager = 1st Baire category Comma after introductory words, phrases, clauses: unacceptable, obligatory or optional? I am no native speaker and always confused about the comma in introductory phrases, in particular in prepositional phrases. Is there any hard rule when a comma must be set? If I make a google search for certain phrases, I often find both variants. Typical examples where I am not sure whether a comma must be set are:        First[ly][,] I must phone mother   In this case[,] we must...   For simple problems[,] the algorithm...   From Lemma 1.2[,] we obtain...   For a typical user[,] the algorithm...   In our theory[,] we...   A dark line on the horizon[,] the mountain range we were headed for seemed somehow not as threatening as when we had started out on the trip.   Avidly reading her novel[,] Jane did not notice that her stop was approaching.   To be sure[,] it is not easy to see a solution.   When you get to Rome[,] give me a ring. A company is any form of business whether it is small or large. Generally the term &quot;company&quot; indicates a particular kind of business dealing in a specific product. An organisation is the larger form and generally comprises of a number of companies.  Simply, a company is an organization, but an organization is not just a company. An industry is the combination of companies in same line of business. Firm, corporation and business  are synonyms of &quot;company&quot;. An Agency is a particular kind of company, which serves as an intermediary between clients (other companies or individuals).  Is this correct? IINM my system is failing when bashing  for i in {0..10000000}; #   Seven zeroes. do     false; done #   `bash` exited and its `tmux` pane/window was closed.   or  for i in $(seq 0 10000000); #   Seven zeroes. do     false; done #   `bash` exited and its `tmux` pane/window was closed.   but not when  for i in {0..1000000}; #   Six zeroes. do     false; done #   Finished correctly.   Can you please briefly explain the internals of this behavior and prompt a workaround for getting the task done? What is causing different spacing between a header and content on one page only? I have found that for only one page (so far) the spacing between the header and content is different from the spacing on all the other pages. You can see this behavior in the images below (from pages 2 and 3 of the document):     My .tex file contents are shown below.  \documentclass[a4paper]{report} \usepackage{fullpage} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \setlength{\headsep}{0.5cm}  \begin{document} \renewcommand{\headrulewidth}{0.4pt} \renewcommand{\chaptermark}[1]{\markboth{\thechapter. #1}{}} \fancyhead[L]{\bfseries \leftmark} \fancyhead[R]{\bfseries \rightmark}  \chapter{Version Control} \lipsum \lipsum \lipsum \lipsum  \end{document} Pretty formattings for warning and info blocks I'm looking for pretty formatting for warnings, and others for some info or tips.  For the moment, I use the following formatting but maybe someone has one better idea.    To obtain this, I use the following code.  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[x11names,svgnames]{xcolor} \usepackage{graphicx} \usepackage{fourier-orns} \usepackage[framemethod=tikz]{mdframed} \usetikzlibrary{calc}  \makeatletter % Default settings for warnings     \newcommand{\warningSymbol}{\raisebox{0.8\depth}{\danger}}      \definecolor{warningColorText}{named}{Red3}     \definecolor{warningColorLine}{named}{Red3}     \definecolor{warningColorBack}{named}{LemonChiffon1}     \definecolor{warningColorBackSymbol}{named}{white}  % Inline mode     \newcommand{\warning}[1]{%         \textcolor{warningColorText}{\warningSymbol{}\,#1}%     }  % Block mode     \tikzset{         warningsymbol/.style={             rectangle,             draw  = warningColorText,             fill  = warningColorBackSymbol,             scale = 1,             overlay         }     }      \mdfdefinestyle{warning}{%         hidealllines      = true,         leftline          = true,         skipabove         = 12,         skipbelow         = 12pt,         innertopmargin    = 0.4em,%         innerbottommargin = 0.4em,%         innerrightmargin  = 0.7em,%         rightmargin       = 0.85em,%         innerleftmargin   = 1.1em,%         leftmargin        = 0.85em,%         middlelinewidth   = .2em,%         linecolor         = warningColorLine,%         backgroundcolor   = warningColorBack,%         fontcolor         = warningColorText,%         firstextra        = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)             node[warningsymbol] {\warningSymbol};         },%         secondextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         middleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         singleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%     }      \newmdenv[style=warning]{@Warning}     \newenvironment{Warning}{\let\warning\relax\begin{@Warning}}{\end{@Warning}} \makeatother  \usepackage{lipsum}  \begin{document}  \warning{IMPORTANT !} \lipsum[1]  \begin{Warning}      \lipsum[1] \end{Warning}  \end{document} Setting print layout to portrait using QGIS 3? Add high-resolution-graphics and / or .svg-icons Linked server returned message "Query timeout expired." We have two separate SQL Servers. On one server we have a data warehouse (DWH), on the other we have sales information database.  Now on the DWH server there is an ETL job that collects the information from the sales server. The job runs daily after midnight. The DWH collects the information via linked server from the sales database.  Now, the most time the ETL job runs without any problems. But sometimes it fails because of query timeout. We have found out, that there is a specific pattern: The failure happens every 11th day. So on the 11th day the ETL job fails to collect information.   The following error occurs:     SQLNCLI11 for linked server “my linked server” returned message “Query timeout expired.”   Note: The job fails usually 10 minutes after start.  We have searched everything and could not found out what this issue causes. Also we know that the amount of data is every time, almost the same. Also there is no any scheduled job that runs all 11 days or something.  The remote query timeout on the linked server is set to 0.  Our next step will be, to turn off the antivirus programm on the sales server, to check if this causes the problem.  Does anyone have any clue or idea, where i can search further to find the problem? Minion Pro, TL2016 and LuaLaTeX Let $V$ be an n-dimensional vector space over $\mathbb{R}$ and $T:V \rightarrow V$ be linear. Call a vector $v \in V$ cyclic if $V$ is spanned by $\{v, \ Tv, \ T^2v,..\}$.  Question:  Show that $v$ is cyclic for $T$ then there is some $k \geq 1$ such that   $$B=\{v, \ Tv,..T^{k-1}v \}$$ is linearly independent in $V$ and that $T^kv$ lies in  the span of $B$.  Thoughts:  If we suppose that no such $k$ exists, then every subset of the form $\{v, \ Tv,..T^{l-1}v \}$ is linearly dependent, or it is linearly dependent with $T^lv$ not lying in it span.  There are in effect then two cases to consider.  Suppose every subset of the form $\{v, \ Tv,..T^{l-1}v \}$ is linearly dependent. Then $T^{l-1}$ can be expressed as a linear combination of $v,...,T^{l-2}v$. Now, $\{v, \ Tv,..T^{l-2}v \}$ is also linearly dependent by assumption and so we can express $T^{l-2}$ as a linear combination of  $v,...,T^{l-3}v$ etc... Continuing in this way, we show that $T^{l-1}v= \lambda v$ for some $\lambda \in \mathbb{R}$. Thus $$B=\{ \mu v \ | \ \mu \ \text{is an eigenvalue of} \ T^mv \ \text{for some} \ m \}$$ This is 'at best' countably infinite, and therefore could not span $V$ even if it were 1-dimensional, wince our base field is $\mathbb{R}$, which is uncountable.  Suppose then that $B$ is linearly independent for some $k$ but that $T^kv$ does not lie in its span. I can't see how to progress this argument.  Note: This question is for second year Undergraduates, and I think there must be some more simple approach; even the first part surprises me if we need to argue using the uncountability of $\mathbb{R}$, so any alternative approaches are also appreciated. Linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T.$ Let $T$ be a linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T$ for some $\lambda\in\mathbb{R}.$ Then which of the following is/are true?  $1.\|T(x)\|=|\lambda|\|x\|$ for all $x\in\mathbb{R}^n.$  $2.$ If $\|T(x)\|=\|x\|$ for some non-zero $x\in\mathbb{R}^{n}$, then $\lambda=\pm 1.$  $3.T=\lambda I$ where $I$ is the identity transformation on $\mathbb{R}^{n}.$  $4.$ If $\|T(x)\|&gt;\|x\|$ for a nonzero vector $x\in\mathbb{R}^{n},$ then $T$ is singular.  According to me since $T^{2}=\lambda T$ we have $T$ satisfies the polynomial $x^{2}-\lambda x.$ So choices for its minimal polynomial are $x,x-\lambda,x(x-\lambda).$ So $3$rd option is clearly incorrect and $4$th option is not satisfied if $T=I$ and hence correct one. I am confused in first two options . Please help me. Thanks in advance. How do I add the kernel PPA? just wondering if anyone has ever tried embedding and actually integrating any js engine into the .net environment. I could find and actually use (after a LOT of pain and effort, since it's pretty outdated and not quite finished) spidermonkey-dotnet project. Anyone with experience in this area? Engines like SquirrelFish, V8..   Not that I'm not satisfied with Mozilla's Spidermonkey (using it for Rails-like miniframework for custom components inside the core ASP.NET application), but I'd still love to explore a bit further with the options. The command-line solutions are not what I'd need, I cannot rely on anything else than CLR, I need to call methods from/to JavaScript/C# objects.  // c# class public class A {     public string Hello(string msg)     {         return msg + " whatewer";     } }  // js snippet var a = new A(); console.log(a.Hello('Call me')); // i have a console.log implemented, don't worry, it's not a client-side code :)   Just to clarify - I'm not trying to actually program the application itself in server-side javascript. It's used solely for writing custom user subapplications (can be seen as some sort of DSL). It's much easier (and safer) to allow normal people programming in js than C#. With powershell get exactly the same application list as in Add/Remove programms There are various methods to find installed applications on the system but to my knowledge none returns exactly the same list as control panel item Add Remove Programs. Some of the ways I tried are:   WMI via Win32_Product and Win32Reg_AddRemovePrograms Registry, both x32 and x64 branches   For instance, I have 217 entries in AR and   gwmi  Win32_Product   returns 261 items.  gp HKLM:\Software\Microsoft\Windows\CurrentVersion\Uninstall\*, HKLM:\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Uninstall\* | ? {![string]::IsNullOrWhiteSpace($_.DisplayName) } | select DisplayName   returns 119 items.   (gwmi Win32Reg_AddRemovePrograms) + (gwmi Win32Reg_AddRemovePrograms64) | sort DisplayName -Unique    returns 107 items.  I tried to nail it down with more complex filter that included InstallLocation, InstallSource, InstallDate etc without success. Ubuntu MATE freeze on logout/shutdown/reboot when running on Intel graphics card Should I sign, watermark or otherwise label a framed print? And if so how? I have an opportunity to show some of my work in a small local coffee shop.  Are there any guidelines/practices to signing, labeling or watermarking a photo which is to be on display?  Or should a photo be signed, labelled or marked at all? Is signing the back of the photo and showing my name on a separate card below the frame itself the "better" way to do this?   I noticed many people on the web put a very noticeable watermark with their logo, copyright and name right on the photo.  Personally, I find this distracting as it often ruins the photograph itself often turning what could be art to kitsch. Painters most often sign the painting itself but this to me is not distracting because it is done with the painter's own hand and it often "works" with the painting. Why the galaxies form 2D planes (or spiral-like) instead of 3D balls (or spherical-like)? Question: As we know, (1) the macroscopic spatial dimension of our universe is 3 dimension, and (2) gravity attracts massive objects together and the gravitational force is isotropic without directional preferences. Why do we have the spiral 2D plane-like Galaxy(galaxies), instead of spherical or elliptic-like galaxies?   Input: Gravity is (at least, seems to be) isotropic from its force law (Newtonian gravity). It should show no directional preferences from the form of force vector $\vec{F}=\frac{GM(r_1)m(r_2)}{(\vec{r_1}-\vec{r_2})^2} \hat{r_{12}}$. The Einstein gravity also does not show directional dependence at least microscopically.  If the gravity attracts massive objects together isotropically, and the macroscopic space dimension is 3-dimensional, it seems to be natural to have a spherical shape of massive objects gather together. Such as the globular clusters, or GC, are roughly spherical groupings , as shown in the Wiki picture:     However, my impression is that, even if we have observed some more spherical or more-ball-like , it is more common to find more-planar  such as our ? (Is this statement correct? Let me know if I am wrong.)  Also, such have a look at this more-planar Spiral "galaxy" as this NGC 4414 galaxy:    Is there some physics or math theory explains why the  turns out to be planar-like (or spiral-like) instead of spherical-like?  See also a   p.s. Other than the classical stability of a 2D plane perpendicular to a classical angular momentum, is there an interpretation in terms of the quantum theory of vortices in a macroscopic manner (just my personal speculation)?  Thank you for your comments/answers! Guidelines for customizing biblatex styles