Distinct subrings of cardinality $p$ of the field $\mathbb F_{p^2} $
Number of subrings of a Field
NullReferenceException in Unity How does Salesforce SSO call just-in-time handler methods Add an unnmbered section to the table of Contents in Lyx Tikz, Tree without overlaps I'm trying to get a nice tree without the overlaps, can anyone help me to improve my code?  \documentclass{standalone} \usepackage{amssymb} \usepackage{amsmath} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{dsfont} \usepackage{tikz} \usetikzlibrary{shapes} \usetikzlibrary{graphs}  \begin{document}       \begin{tikzpicture}[baseline,level/.style={level distance=20mm, sibling distance =35mm/#1},scale=1]         \node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                 child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                     child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         }                     }                 }                    child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             }                         }                     }                     child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                         child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                         child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                             child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             child{ node [ellipse split, draw] {$s_1&lt;s_2$\nodepart{lower}\tiny $(s_1,s_2,s_3,s_4)$}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                                 child{ node [rectangle, draw] {$(s_1,s_2,s_3,s_4)$}}                             }                         }                     }                 }             };     \end{tikzpicture} \end{document} Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$. Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$ for all notnegative value of $x,y,z$.  I think that minimum value is $\frac{3}{4}$ when $x=y=z$ but I have no prove. False proof that $ρe^{iθ} = ρ$ and so complex numbers do not exist? My professor showed the following false proof, which showed that complex numbers do not exist. We were told to find the point where an incorrect step was taken, but I could not find it. Here is the proof: (Complex numbers are of the form $\rho e^{i\theta}$, so the proof begins there) $$\large\rho e^{i\theta} = \rho e^{\frac{2 \pi i \theta}{2\pi}} = \rho (e^{2\pi i})^{\frac{\theta}{2\pi}} = \rho (1)^{\frac{\theta}{2\pi}} = \rho$$ $$Note: e^{i\pi} = -1, e^{2\pi i} = (-1)^2 = 1$$ Since we started with the general form of a complex number and simplified it to a real number (namely, $\rho$), the proof can claim that only real numbers exist and complex numbers do not. My suspicion is that the error occurs in step $4$ to $5$ , but I am not sure if that really is the case. Evaluate  $$\int\frac{e^xdx}{(e^{2x}-1)(e^x+1)}$$  I tried to decompose the integrand into A and B partial fractions.  No success. How does an arcane focus work with an Eldritch Knight? My PC is an Eldritch Knight. I was wondering what is an Arcane Focus? Do I need to have it in my hand? If I get the War Caster Feat does my arcane focus become obsolete? How can I change the local origin of an object, without changing the object location in the world space.  This is the object located at position (0,0,0) in the world space    This is the same object located at position (0,1,0) in the world space, but with its local origin offset so that the object is in the same place    Is there a way to do this in a single step? Can a dual national visit USA under VWP after visa refusal for other nationality? Recently I was refused an E-2 visa under Section 214b. To be honest I don't really know why because I have strong ties with my home country (house, bank account, family) so my lawyer sent an appeal to the legalnet and I'm  waiting for an answer.  I have a second nationality that I used to travel in USA within the last 15 months under ESTA. A few days after my E-2 refusal I checked my ESTA status  and it is still authorized.  Can I travel to the States right now or not?  I have a purchased business over there and have to take care of my assets. Show in linked/related questions sidebar whether a question has answers or not Currently, in the linked/related sidebar, without opening the question the user can only distinguish whether the question has an accepted answer (green icon) or not (grey icon).  I often eagerly open an interesting-sounding question with a grey icon, only to find myself disappointed because it has no answer yet.  To avoid this, I suggest adding a third type of icon for questions that have at least one answer, but no accepted one. Probably a darker/lighter icon or some kind of frame or glow. I've made a quick example with Paint (if relevant, the screenshot I used is from ). I'm in no way satisfied with the look of it, but at least it demonstrates the general idea.    What do you think about this proposal? Would it be a useful addition to the StackExchange interface? Can we have a black hole without a singularity? Assuming we have a sufficiently small and massive object such that it's escape velocity is greater than the speed of light, isn't this a black hole? It has an event horizon that light cannot escape, time freezes at this event horizon, etc. However this object is not a singularity.  If a large star's mass were compressed to the size of, say, a proton, it would certainly have these properties but it would still not be a singularity as a proton does have volume.  The reason "physics breaks down" at singularities is because we cannot divide by zero, but as long as the proton-sized object has volume, physics won't "break down", yet we still have an event horizon and an object that is invisible (but not undetectable) from the outside.  I have read the answers to  related question. I'm not sure if they don't address my specific question or if I don't understand the answers. In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. Equivalence of definitions of prime ideal in commutative ring Converting LiDAR files from LAZ to LAS format I'm trying to convert LiDAR files from the LAZ format (the compressed version) into LAS (the uncompressed version) using  software.   However, I am not really sure how it works. I created a new output folder as I entered in the app interface, but it is totally empty.  I also had tried the following code in the command prompt window:   D:\LiDAR\Laszip\laszip.exe *.laz    and got a message saying 'no input specified'.  In my folder, there are about 100-ish.laz files as well as one imu.laz. Any suggestions? What is the joint distribution of $Z=\min(X,Y)$ and $I_{Z=X}$? Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result:     (A-S 1.6.3) Prove that for every two independent identically distributed real random variables $X$ and $Y$,   $$Pr[|X-Y| \leq 2] \leq 3 Pr[|X-Y| \leq 1].$$ Solve $\int_0^{+\infty} \Big(\frac{\sin t}{t}\Big)^2 \ dt$ 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? Cat 5/6 in same conduit as power: is NM-B required, or is THHN acceptable? Preface: I'd like to avoid the question of whether one should run data and power together; my question is focused on Code compliance.  I'm trying to ensure I interpret Code correctly when running Cat 5/6 cabling and power together in the same conduit.  For starters, I understand that nothing in Code prohibits one from doing this; I'm trying to ensure I understand how to do it properly.  I'm wiring a basement office for power and data, and I'll be running the electrical in EMT.  My plan was to pull THHN for the power rather than pull NM-B, but then 800.133(2) Exception 1 made me scratch my head.  Is my understanding of the Article that if I want to run both data (Cat6) and power in the same conduit, that the power needs to be sheathed (i.e. NM-B).  Is it unacceptable to pull individual THHN conductors and the Cat6 together in the same conduit?     800.133 2) Other Applications. Communications wires and cables shall be separated at least 50 mm (2 in.) from conductors of any electric   light, power, Class 1, non–power-limited fire alarm, or medium-power   network-powered broadband communications circuits.      Exception No. 1: Where either (1) all of the conductors of the   electric light, power, Class 1, non–power-limited fire alarm, and   medium-power network-powered broadband communications circuits are in   a raceway or in metal-sheathed, metal-clad, nonmetallic-sheathed, Type   AC, or Type UF cables, or (2) all of the conductors of communications   circuits are encased in raceway.