NASA's "Impossible" Space Engine
Is the EmDrive, or "Relativity Drive" possible?
I am trying to solve the following:  Let $W_t$ be a brownian motion and $a,b &gt;0.$  Define $\tau$ to be the exit time of $W_t$ from $[-a,b]$, that is $$\tau = \inf\{t \ge 0\ :\ W_{t}(\omega) \notin [-a,b]\}.$$  Show that $\tau$ is integrable and compute $\mathbb{E}(\tau)$.  I have seen this stated without proof multiple places that $\mathbb{E}(\tau)=ab.$  What is the logical argument that backs up this statement?  How are we guaranteed $\tau$ is integrable? Consider the following table:  id  title   genre1  genre2  genre3  genre4  genre5 1   Movie1  Thriller    Crime   NULL    NULL    NULL 2   Movie2  Fantasy NULL    NULL    NULL    NULL 3   Movie3  Political   Philosophical   Historical  NULL    NULL 4   Movie4  Science Fiction NULL    NULL    NULL    NULL   When I unpivot genre1, genre2, ... the NULL values are removed, so that, for example, the rows for Movie1 would be:  id title  genre     original_column 1  Movie1 Thriller  genre1 1  Movie1 Crime     genre2   Is there a way to include NULL values, for example:  id title  genre     original_column 1  Movie1 Thriller  genre1 1  Movie1 Crime     genre2 1  Movie1 NULL      genre3 1  Movie1 NULL      genre4 1  Movie1 NULL      genre5   I can SELECT ISNULL(genre1, '*'), ISNULL(genre2, '*') ..., but suppose I want the values to appear as NULL? Is there a better way than:  WITH  -- SELECT that unpivots table AS unpivoted SELECT unpivoted.id,    unpivoted.title,    ,CASE unpivoted.genre WHEN '*' THEN NULL ELSE unpivoted.genre END    ,unpivoted.original_transaction FROM unpivoted   Also, I notice that the query that returns the source data to be unpivoted MUST have an alias. For example:  SELECT ... FROM   -- *must* give it an alias or it's a syntax error   (SELECT ... FROM) AS source    UNPIVOT(genre for original_column IN (genre1,genre2, genre3,genre4,genre5))   Does anyone know why? I have not found a way to refer to the alias anywhere in the query. Windows Starter Edition, Home and Home Premium do not include gpedit, how do I install it? Why is it considered a bad practice to omit curly braces? Why does everyone tell me writing code like this is a bad practice?  if (foo)     Bar();  //or  for(int i = 0 i &lt; count; i++)     Bar(i);   My biggest argument for omitting the curly braces is that it can sometimes be twice as many lines with them. For example, here is some code to paint a glow effect for a label in C#.  using (Brush br = new SolidBrush(Color.FromArgb(15, GlowColor))) {     for (int x = 0; x &lt;= GlowAmount; x++)     {         for (int y = 0; y &lt;= GlowAmount; y++)         {             g.DrawString(Text, this.Font, br, new Point(IconOffset + x, y));         }      }  }  //versus using (Brush br = new SolidBrush(Color.FromArgb(15, GlowColor)))     for (int x = 0; x &lt;= GlowAmount; x++)         for (int y = 0; y &lt;= GlowAmount; y++)             g.DrawString(Text, this.Font, br, new Point(IconOffset + x, y));   You can also get the added benefit of chaining usings together without having to indent a million times.  using (Graphics g = Graphics.FromImage(bmp)) {     using (Brush brush = new SolidBrush(backgroundColor))     {         using (Pen pen = new Pen(Color.FromArgb(penColor)))         {             //do lots of work         }     }  } //versus using (Graphics g = Graphics.FromImage(bmp)) using (Brush brush = new SolidBrush(backgroundColor)) using (Pen pen = new Pen(Color.FromArgb(penColor))) {     //do lots of work }     The most common argument for curly braces revolves around maintance programming, and the problems that would ensue by inserting code between the original if statement and its intended result:  if (foo)     Bar();     Biz();     Questions:   Is it wrong to want to use the more compact syntax which the language offers? The people that design these languages are smart, I can't imagine they would put a feature which is always bad to use. Should we or Shouldn't we write code so the lowest common denominator can understand and have no problems working with it?  Is there another argument that I'm missing? Intersection of three vector subspaces We say that two vector subspaces $U,W\subset V$ intersect transversally (written $U\pitchfork W$) if $U+W=V$.  Let $V_1,V_2,V_3$ be vector subspaces of $V$ where $\dim V=n$. We say they intersect normally if $V_i\pitchfork(V_k\cap V_l)$  for $i\ne k,l$. Prove that $V_1,V_2,V_3$ intersect normally if  the following condition holds:  $\operatorname{codim}(V_1\cap V_2\cap V_3)=\operatorname{codim} V_1+\operatorname{codim} V_2+\operatorname{codim} V_3,$   where $\operatorname{codim} V_i=n-\dim V_i$.  How to approach this problem? I tried to order dimensions of $V_i$ but I got that the LHS is $\ge n-\min (\dim V_i,i=1,2,3)$ and the RHS is $\le 3( n-\min (\dim V_i,i=1,2,3))$ which does not lead anywhere. What is the right way to do this? Layover at Narita, Japan. Do I need a transit visa? Prove $\sup(f+g) \le \sup f + \sup g$ Suppose $D$ is a nonempty bounded subset of reals. Let $f:D \to \mathbb R$ and $g:D \to \mathbb R$. Define $(f+g)(x)=f(x)+g(x)$. Prove $\sup(f+g)(D) \le \sup f(D) + \sup g(D)$ (also prove that $\sup (f+g)$ exists).   I understand why this is the case, just not how to prove it. Left side is pretty much $\sup  (f(x)+g(x))$ and right side is $\sup (f(x) + g(y))$ for $x,\,y \in D$. Basically $f+g$ has to use the same variable and $f(D)+g(D)$ use different ones. But I don't know how to go about proving this.     The second part of the question is to find a specific example where strict inequality holds.    Let $D=\{a,b\}$ and $f: a \to 1,\, b\to 0, \,g: a \to 0,\, b\to 1$. $\sup f(D) = 1,\, \sup g(D) = 1,\, \sup f(D) + \sup g(D) = 2.$ $\sup (f+g)(D) = 1$ (if we choose a, $f+g = 1+0,\, b,\, f+g=0+1$). How can I create Light Streaks in Cycles? $P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$ Let $\mathfrak{F} = (A_n)_{n \in \mathbb{N}}$. Prove that $P(\limsup A_n)=1$ if $\forall A \in \mathfrak{F}$ s.t. $P(A) &gt; 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$. (Side question 1 Is second Borel-Cantelli out because we don't know if the $A_n$'s are independent?)  Suppose $\forall A \in \mathfrak{F}$ s.t. $P(A) &gt; 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$, but $P(\limsup A_n)&lt;1$. My prof gave us this convenient hint: Show $\exists M &gt; 0$ s.t. $P(\bigcap_{j=M}^{\infty} A_j^{c}) &gt; 0$.  $1 &gt; P(\limsup A_n)$  $1 - P(\limsup A_n) &gt; 0$  $P(\liminf A_n^{c}) &gt; 0$  By definition of $\liminf A_n^{c}$, $\exists M &gt; 0 $ s.t. $P(\bigcap_{j=M}^{\infty} A_j^{c}) &gt; 0$. (Side questions 2 and 3 Isn't this =1? ...Wait, is that what we're trying to prove? Haha)  $\to 1 - P(\bigcup_{j=M}^{\infty} A_j) &gt; 0$  $\to 1 &gt; P(\bigcup_{j=M}^{\infty} A_j)$  Main question: Now I guess we have to come up with some A s.t.  $\to 1 &gt; P(\bigcup_{j=M}^{\infty} A_j) \geq \sum_{n=1}^{\infty} P(A \cap A_n)$, or is there something else to do? Can my commander ever not be my commander? Once and for all: Is there anything that can possibly be done to my commander that would cause it to cease being my commander?    All of these questions ask the same thing, but aren't general enough to be marked as a duplicate.      Is a commander still a commander if it gets turned into a forest?     Is a commander still a commander if it gets turned face down? Easy way to model spiral engraving (knurling) that is usually used on metal for better grip? What is the easiest way to model this diamond grip pattern into a cylinder? Automatically run commands over SSH on many servers There is a list of IP addresses in a .txt file, ex.:  1.1.1.1 2.2.2.2 3.3.3.3   Behind every IP address there is a server, and on every server there is an sshd running on port 22. Not every server is in the known_hosts list (on my PC, Ubuntu 10.04 LTS/bash).  How can I run commands on these servers, and collect the output?  Ideally, I'd like to run the commands in parallel on all the servers.  I'll be using public key authentication on all the servers.  Here are some potential pitfalls:   The ssh prompts me to put the given servers ssh key to my known_hosts file. The given commands might return a nonzero exit code, indicating that the output is potentially invalid. I need to recognize that. A connection might fail to be established to a given server, for example because of a network error. There should be a timeout, in case the command runs for longer than expected or the server goes down while running the command.   The servers are AIX/ksh (but I think that doesn't really matter. Run Ubuntu ISO live image from Windows BIOS - no USB or DVD What provokes an opportunity attack?  In 3.X moving out of a threatened square, standing up from prone, casting a spell (unless cast defensively), or making a ranged attack provoked, while in 4e moving out of a threatened square or making a ranged/area attack provoked. in my mariadb i have replication, i tried to change the location of the binlogs by changing the configuration file as so: log-bin = /new/path/mysql-bin I also tried to change the index location as so: log_bin_index = /new/path/mysql-bin.index but every time when I restart I see that the location of the binlog files is /var/lib/mysql... i tried to check with mysqld --verbos --help | grep bin and I saw that the lob-bin parameter is mysql-bin and not what I wrote in the cnf... how to I make the MySQL to take the value of the log-bin from the cnf file? this is the cnf file of the MySQL for sure... I'm currently building a town in Minecraft Pocket Edition and I'm wondering how many blocks long and how many blocks wide the world is. Software Tester Skill Matrix with Levels I would like to know if there's a standard Skill set for Skill Matrix for a Software Tester of different levels, like for example, what are the skills needed for an Entry level tester as well as what technologies and responsibilities he needs. And for the Mid Level tester and Senior Level as well.  Note that I am working in a company who designs and develops websites. How do I debug Node.js applications? Get three highest values from the array How do I install drivers for the RTL8821CE Network Controller on Ubuntu 20.04? My Laptop (HP Laptop 14-ck1xxx) came with the RTL8821CE Network Controller. When I run lspci | grep Network I get: 03:00.0 Network controller: Realtek Semiconductor Co., Ltd. RTL8821CE 802.11ac PCIe Wireless Network Adapter  I‘m unable to get the Wi-Fi to work. What I did so far:  Download the repo at . Unzip the file and rename the folder from rtl8821ce-master to rtl8821ce. Run the following commands in the terminal:  cd Downloads/rtl8821ce/ sudo apt install bc module-assistant build-essential dkms sudo m-a prepare sudo ./dkms-install.sh    Everything seemed to install and execute successfully. Then, I rebooted my laptop and made sure secure boot is disabled. However, Ubuntu still displays: No Wi-Fi Adapter Found - make sure you have a Wi-Fi adapter plugged in and turned on  The command uname -r returns: 5.4.0-7634-generic  The command sudo lshw -C network returns: *-network                         description: Ethernet interface        product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:02:00.0        logical name: eno1        version: 15        serial: c4:65:16:08:52:bf        size: 1Gbit/s        capacity: 1Gbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 duplex=full firmware=rtl8168h-2_0.0.2 02/26/15 ip=192.168.178.37 latency=0 link=yes multicast=yes port=MII speed=1Gbit/s        resources: irq:16 ioport:4000(size=256) memory:a1204000-a1204fff memory:a1200000-a1203fff   *-network UNCLAIMED        description: Network controller        product: RTL8821CE 802.11ac PCIe Wireless Network Adapter        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        version: 00        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress cap_list        configuration: latency=0        resources: ioport:3000(size=256) memory:a1100000-a110ffff  What am I doing wrong? How can this be solved?