I skimmed over this question , and I'm currently under the impression that it's a widespread belief that the empty set contains itself. But a contradiction seems to arise: if the empty set contains itself, then it's NOT empty. After all, if we call a set non-empty just because it contains the empty set, then why should we treat the empty set itself differently?  Then it occurred to me that maybe mathematicians define "empty" differently in set theory. Maybe by "empty set" they mean a "set that contains only itself", instead of a "set that contains absolutely nothing".  Is my surmise correct?
In a  question, an answerer says:  an empty bag is a bag with nothing inside it.  Makes sense, but I'm reading a textbook right now that says:  The empty set has only one subset (namely, itself).  Which seems odd to me. If a set is empty, why would it contain a subset? Regardless of whether or not the subset is also an empty set, the &quot;empty set&quot; still contains a subset, so how is it really &quot;empty&quot;? To go back to the empty bag analogy, it just sounds like my text book is saying:  an empty bag is a bag with nothing but an empty bag inside it.  Which doesn't make sense because then the bag isn't empty. I can't wrap my head around this concept.
Blank white desktop icons on new Windows 7 install I just did a fresh install of Windows 7 professional on my father's Toshiba Satellite laptop and everything went smoothly except for the fact that certain desktop shortcuts appear as a blank white page.    I've tried:    going in and changing the icon to many different things reloading the icon cache by deleting the IconCache.db file uninstalling his Free AVG as he heard the virus scanner might be preventing it (it wasn't and he put it back.)   The one thing these icons have in common is that they are shortcuts to some very old DOS executables.  One of them is Word Star 2000 to give you an idea.  Does anybody have any other suggestions besides what we've tried? I need to find the intersection point of 2 line segments (lines are finite, i.e., they have end points).   e.g. segment 1 from $(x_1, y_1)$ to $(x_2, y_2)$ -- segment 2 from $(x_3, y_3)$ to $(x_4, y_4)$  you can assume $m_1$ and $m_2$ are the gradients of segment 1 and segment 2 respectively  similarly, $c_1$ and $c_2$ being the y-intercepts of segment 1 and segment 2 respectively  Using $y=mx+c$ I can easily find the equations of both lines and then derive the intersection point from those equations and check if that point actually lies on both the line segments (intersection point may not lie between the end points).  My problem is when I calculate the gradients $m_1$ and $m_2$ I do $\frac{y_2-y_1}{x_2-x_1}$ there for if one of the lines is vertical then I am going to have problems. How can I deal with this?  Another problem is that if the two segments intersect at a point which is also the same as one of the end points then I want to assume that they are not intersecting.  e.g. if segment 1 is from $(2, 3)$ to $(3, 7)$ and segment 2 is from $(3, 7)$ to $(7, 3)$ then I want to assume they don't intersect. Colorimetry is not the same In what situation do you put a comma before "because"?   And, when do you not put a comma before "because"?   I've always assumed that you put a comma, because the clause introduced by "because" will always be independent, because it cannot stand on its own, because it wouldn't make sense on its own.   Is this the case? Could you think of an example where it doesn't take a comma? How do I give text or an image a transparent background using CSS? Why are there holes in the icing? I tried to follow the new Blender 2.8 tutorial from Blender Guru.   When entering "Sculpting" with the icing there were suddenly alot of holes in it. Why is this? An entire function whose real part is bounded must be constant. Greets  This is exercise 15.d chapter 3 of Stein &amp; Shakarchi's "Complex Analysis", they hint: "Use the maximum modulus principle", but I didn't see how to do the exercise with this hint rightaway, instead I knew how to do it with the Casorati-Weiestrass Theorem, here is my answer:  Define $g(z)=f(1/z)$ for $z\neq{0}$,then by the hypothesis we must have that for any $\epsilon&gt;0$ $g(D_{\epsilon&gt;0}(0)-\{0\})$ is not dense in $\mathbb{C}$, then the singularity at $0$ of $g$ is not essential, this implies $f$ must be a polynomial, but if $f$ is a non-constant polynomial, it is easy to see that its real part must be unbounded, so $f$ must be constant.  I would like to know an answer with the the maximum modulus principle.  Thanks Suppose we're tying to differentiate the function $f(x)=x^x$. Now the textbook method would be to notice that $f(x)=e^{x \log{x}}$ and use the chain rule to find $$f'(x)=(1+\log{x})\ e^{x \log{x}}=(1+\log{x})\ x^x.$$  But suppose that I didn't make this observation and instead tried to apply the following differentiation rules:  $$\frac{d}{dx}x^c=cx^{c-1} \qquad (1)\\ \frac{d}{dx}c^x = \log{c}\ \cdot c^x \quad (2)$$  which are valid for any constant $c$. Obviously neither rules are applicable to the form $x^x$ because in this case neither the base nor the exponent are constant. But if I pretend that the exponent is constant and apply rule $(1)$, I would get $f'(x)\stackrel{?}{=}x\cdot x^{x-1}=x^x.$ Likewise, if I pretend that the base is constant and apply rule $(2)$ I obtain $f'(x)\stackrel{?}{=}\log{x}\cdot x^x$.  It isn't hard to see that neither of the derivatives are correct. But here's where the magic happens: if we sum the two “derivatives” we end up with $$x^x+ \log{x}\cdot x^x=(1+\log{x})\ x^x$$ which is the correct expression for $f'(x)$.  This same trick yields correct results in other contexts as well. In fact, in some cases it turns out to be a more efficient way of taking derivatives. For example, consider $$g(x)=x^2 = \color{blue} x\cdot \color{red} x.$$ If we pretend the blue $\color{blue} x$ is a constant we would get $g'(x)\stackrel{?}{=}\color{blue}x\cdot 1=x$. Now if we pretend the red $\color{red}x$ is constant we get $g'(x)\stackrel{?}{=}1\cdot \color{red} x=x$. Summing both expressions we end up with $2x$ which is of course a correct expression for the derivative.  These observations have led me to the following conjecture:     Let $f(x,y)$ be a differentiable function mapping $\mathbb{R}^2$ to $\mathbb{R}.$ Let $f'_1 (x,y)=\frac{\partial}{\partial x} f(x,y)$ and $f'_2 (x,y)=\frac{\partial}{\partial y} f(x,y)$. Then for any $t$ we have: $$\frac{d}{dt}f(t,t)=f'_1 (t,t) + f'_2 (t,t).$$   (I apologise for the somewhat awkward notation which I could not seem to get around without causing undue ambiguity.)  This formulation also seems to lend itself to following generalisation:     Let $f:\mathbb{R}^N \to \mathbb{R}$ be a function differentiable in each of its variables $x_1,x_2,\ldots,x_N$. For $n=1,2,\ldots,N$ define $f'_n(x_1,x_2,\ldots,x_N)=\frac{\partial}{\partial x_n}f(x_1,x_2,\ldots,x_N)$. Let $t$ be any real number and define the $N$-tuple $T=(t,t,\ldots,t)$. Then one has: $$\frac{d}{dt} f(T)=\sum_n f'_n(T).$$   Thus my question is:   Is this true? How can it be proven? (Specifically in the case $N=2$ but also in the general case.) What is the volatile keyword useful for? At work today, I came across the volatile keyword in Java. Not being very familiar with it, I found . Given the detail in which that article explains the keyword in question, do you ever use it or could you ever see a case in which you could use this keyword in the correct manner? "Activity" tab for the Community user's network profile sometimes leads to a server error I'm trying to write a basic program in C using gedit or sublime text (which I prefer), but no matter what I do when I try to compile or run, or even while writing, I have no option to use 'stdio.h' or any similar package. Even when I start typing #include &lt;st the only word completion sublime suggests is 'struct'.  I've read dozens of threads, but all managed to fix this by installing gcc, which I've tried to reinstall multiple times, with no luck.  This is the output from gcc -v:  Using built-in specs. COLLECT_GCC=gcc COLLECT_LTO_WRAPPER=/usr/lib/gcc/x86_64-linux-gnu/7/lto-wrapper OFFLOAD_TARGET_NAMES=nvptx-none OFFLOAD_TARGET_DEFAULT=1 Target: x86_64-linux-gnu Configured with: ../src/configure -v --with-pkgversion='Ubuntu 7.2.0-8ubuntu3' --with-bugurl=file:///usr/share/doc/gcc-7/README.Bugs --enable-languages=c,ada,c++,go,brig,d,fortran,objc,obj-c++ --prefix=/usr --with-gcc-major-version-only --program-suffix=-7 --program-prefix=x86_64-linux-gnu- --enable-shared --enable-linker-build-id --libexecdir=/usr/lib --without-included-gettext --enable-threads=posix --libdir=/usr/lib --enable-nls --with-sysroot=/ --enable-clocale=gnu --enable-libstdcxx-debug --enable-libstdcxx-time=yes --with-default-libstdcxx-abi=new --enable-gnu-unique-object --disable-vtable-verify --enable-libmpx --enable-plugin --enable-default-pie --with-system-zlib --with-target-system-zlib --enable-objc-gc=auto --enable-multiarch --disable-werror --with-arch-32=i686 --with-abi=m64 --with-multilib-list=m32,m64,mx32 --enable-multilib --with-tune=generic --enable-offload-targets=nvptx-none --without-cuda-driver --enable-checking=release --build=x86_64-linux-gnu --host=x86_64-linux-gnu --target=x86_64-linux-gnu Thread model: posix gcc version 7.2.0 (Ubuntu 7.2.0-8ubuntu3) `   This is the code:   #include &lt;stdio.h&gt;  int main(void) { printf("hello, world\n"); }   When I compile using gcc, I get nothing, no error at least. but when I try to run using just ./hello.c this is what I get. How can I remove duplicate values from a multi-dimensional array in PHP?  Example array:  Array (     [0] =&gt; Array     (         [0] =&gt; abc         [1] =&gt; def     )      [1] =&gt; Array     (         [0] =&gt; ghi         [1] =&gt; jkl     )      [2] =&gt; Array     (         [0] =&gt; mno         [1] =&gt; pql     )      [3] =&gt; Array     (         [0] =&gt; abc         [1] =&gt; def     )      [4] =&gt; Array     (         [0] =&gt; ghi         [1] =&gt; jkl     )      [5] =&gt; Array     (         [0] =&gt; mno         [1] =&gt; pql     )  ) Are there any simple rules for choosing the definite vs. indefinite (vs. none) article? Selecting the size of a System.Drawing.Icon? At this moment there are 4 meta-site questions on the HNQ list:    (World Building Meta)  (Meta StackOverflow)  (Workplace Meta)  (Mathematica Meta)   When I first saw it, I only saw the SO Moderator question, so I assumed that this was a one-off special thing for the election; but now we've got four questions crept in. Was this an intentional change, or a bug? Can you explain the sentence structure 'In a hole in the ground there lived a Hobbit'? Why put the verb before the subject? The opening sentence to The Hobbit by J.R.R. Tolkien reads,     In a hole in the ground there lived [verb] a hobbit [subject].   I wonder if there are accepted stylistic purposes for such a structure.  When is it natural, and when is it unnatural? Can I set max_retries for requests.request? Escape velocity from a rotating black hole Under Newton, the escape velocity is  $$v_{esc} = \rm c \ \sqrt{r_s/r}$$  where $\rm r_s=2 \ GM/c^2$. In the nonrotating relativistic case (the Schwarzschild case) the radial escape velocity is the same:  $$v^{\parallel}_{esc} = \rm c \ \sqrt{r_s/r}$$  which is the speed of light, $v={\rm c}$, at the event horizon at $\rm r=r_s$. Since the photon sphere, where an object with $v={\rm c}$ would not escape, but be captured into orbit, is at $r=3 \ {\rm GM/c^2}$, the transverse escape velocity is  $$v^{\perp}_{esc} = \rm c \ \sqrt{r_s/r}\div \sqrt{1-r_s/r}$$  and for the observed (shapiro delayed) velocity ${\rm v}$ the transformation is  $${\rm v}^{\perp}=v^{\perp} \ \sqrt{\rm 1-r_s/r} \ , \ \ {\rm v}^{\parallel}=v^{\parallel} \ (\rm 1-r_s/r)$$  so the observed angular motion is the same as with Newton, yet locally higher, while the observed radial motion is slowed down because of the radial length contraction and time dilation.  But what about the Kerr case where the rotation is significant and the event horizon is at $\rm r=(1+\sqrt{1-a^2}) \ {\rm GM/c^2}$ instead of $\rm r=r_s$?  A testparticle which is locally at rest and therefore corotating with the angular velocity of the frame dragging  $${\Omega = \rm d \phi/ {\rm d t} = {\rm 2 \ a \ r/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  should experience the time dilation factor  $$\rm \dot t = {\rm d t}/ {\rm d}\tau = \sqrt{{\rm ((a^2+(r-2) \ r) (a^2 \cos ^2(\theta )+r^2))/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  relatively to the far away observer, so multiplying $\Omega$ with $\rm \dot t$ should give the local frame dragging angular velocity, which, multiplied with the $\rm \sqrt{x^2+y^2}$ value of the Boyer-Lindquist to cartesian transformation should give the transverse velocity - this is slightly below $\rm c$ for a maximaly rotating black hole (with $\rm a = 1$ - at the edge of the outer equatorial ergosphere where $\rm r=1$ and $\rm R=\sqrt{r^2+a^2}$).  But how do I find the radial and total escape velocity, and what are the rules to convert the shapiro-delayed coordinate velocities $\rm d r/d t \ , \ d \phi/d t \ , d \theta/d t$ to the local 3-velocity components $v^{\parallel}$ and $v^{\perp}$ of a test particle (local meaning relatively to a frame dragged probe with contant $\rm r$)? The radial length contraction seems a bit different than in the Schwarzschild-case, where it is the same as the gravitational time dilation factor... Ohm's law states that  electric current is directly proportional to voltage provided that physical conditions like temperature remain constant i.e. $$V = IR$$  On the other hand,  $$\text{Power = Voltage} \times \text{Current}$$  So here it seems that the higher the voltage, the lower the current, provided that the power remains constant (i.e. current is inversely proportional to the voltage here which is against Ohm's Law.).   Now my question is how do physicists explain this apparent contradiction? Or maybe this not a contradiction because I am analysing things incorrectly?   P.S.: I am a tenth grade student so please refrain from the usage of highly complicated terminologies in your answers. Using C# and WPF under .NET (rather than  or console), what is the correct way to create an application that can only be run as a single instance?  I know it has something to do with some mythical thing called a mutex, rarely can I find someone that bothers to stop and explain what one of these are.  The code needs to also inform the already-running instance that the user tried to start a second one, and maybe also pass any command-line arguments if any existed.