I am taking a Lie algebras course as a prerequisite to study Lie groups. The idea of a radical of a Lie algebra (maximal solvable ideal) has been defined in class but no other statements or theorems are available.  I have a question that asks me to calculate the radical of $\mathfrak{gl}(2,\mathbb{C})$, which is the general linear Lie algebra of $2 \times 2$ matrices with complex entries.  I just have no idea how to start the question; I know how to check something is an ideal of $\mathfrak{gl}(2,\mathbb{C})$ but I have no idea how to check it's a maximal ideal, or how to check that it's maximal with respect to solvable Lie algebras.  I thought of just guessing a maximal ideal (from a list of common ideals, like $\mathfrak{u}(2,\mathbb{C})$ (upper triangular matrices)) and using the correspondence theorem to prove no bigger ideal exists, but I couldn't get anywhere with it.
I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof:  Proof. Let $\mathfrak s$ denote the scalar matrices. Clearly $\mathfrak s\subset\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$. Suppose that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is generated by more than one element, so that $X\in\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$, but $X\notin\mathfrak s$. We can change basis such that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is upper-triangular. (The change of basis leaves the scalar matrices invariant.) Then $X$ is upper-triangular.  I want to conclude that there exists a $Y\in\mathfrak{gl}_n\mathbb F$ s.t. $[X,Y]$ is not upper-triangular. How can I see that $Y$ always exists, except by waving hands?
I know that there's a difference between relativistic and rest mass. Relativistic mass is "acquired" when an object is moving at speeds comparable to the speed of light. Rest mass is the inherent mass that something has regardless of the speed its moving at.  When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$.   How does the rest mass of an object turn into energy? Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles.  Also, Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy? Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$. Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$.   I can find all the infinite subgroups of the form $n\mathbb{Z}\times m\mathbb{Z}$, where $n,m$ run over $\mathbb{Z}$. But I don't know how to write out all the finite subgroups in a uniform expression.  Any suggestions? Thanks in advance. Generating ordered combination of numbers Find out BIOS Version from Windows Microphone Not Working With FaceTime New install - no network devices available - Realtek Device b822 I just purchased a Lenovo Yoga 730 15" laptop, and installed Ubuntu 16.04 in place of Windows 10. Install seemingly worked fine, but now I have no wifi device available.  Tried a reinstall just to be sure, and the same issue is happening. I don't believe it to be a hardware issue, as wifi was working without issue on Windows before I wiped it out.  When I run rfkill list I see:  0: ideapad_wlan: Wireless LAN Soft blocked: no Hard blocked: no   But when I run lspci I only see one item in the list claiming to be a network device:  3a:00.0 Network controller: Realtek Semiconductor Co., Ltd. Device b822 (rev ff)   I have a USB wifi adapter and that gets me connected without issue, but obviously that is not the desired solution.  Does anyone have any insight on what might be causing the issue?  Output from modinfo -p rtl8822be:  fix_rate: (int) debug_level: (int) swenc:Set to 1 for software crypto (default 0) (bool) ips:Set to 0 to not use link power save (default 1) (bool) swlps:Set to 1 to use SW control power save (default 0) (bool) fwlps:Set to 1 to use FW control power save (default 1) (bool) msi:Set to 1 to use MSI interrupts mode (default 1) (bool) dma64:Set to 1 to use DMA 64 (default 0) (bool) aspm:Set to 1 to enable ASPM (default 1) (int) debug:Set debug level (0-5) (default 0) debug_mask:Set debug mask (default 0) (ullong) disable_watchdog:Set to 1 to disable the watchdog (default 0) (bool) Desktop doesn't remember brightness settings after a reboot Every time I reboot my machine the brightness goes back to 100% in Gnome. I wish it would keep the last setting. Is there anyway? Here's the information :     There are four different pairs of   opening and closing tags which can be   used in PHP. Two of those, &lt;?php ?&gt;   and &lt;script language="php"&gt; &lt;/script&gt;,   are always available. The other two   are short tags and ASP style tags, and   can be turned on and off from the   php.ini configuration file. As such,   while some people find short tags and   ASP style tags convenient, they are   less portable, and generally not   recommended.   In my experience most servers do have short tags enabled. Typing  &lt;?=   is far more convenient than typing  &lt;?php echo    The programmers convenience is an important factor, so why are they not recommended? Average distance between two random points in a square A square with side $a$ is given. What is the average distance between two uniformly-distributed random points inside the square?  For more general "rectangle" case, see . The proof found there is fairly complex, and I am looking for a simpler proof for this special case. I expect it could be significantly simpler.  See also . today I installed Ubuntu 64-bit on a netbook. When the netbook boots this, error report appeared. I think there is a problem with the partitions.  Gave up waiting for root device. Common problems:   — Boot args (cat /proc/cmdline)     — Check rootdelay= (did the system wait long enough?)     — Check root= (did the system wait for the right device?)   — Missing modules (cat /proc/modules; ls /dev) ALERT! /dev/mapper/ubuntu--vg-root does not exist. Dropping to a shell!   BusyBox v.1.21.1 (Ubuntu 1:1.21.1-1ubuntu1) built-in shell (ash)    Enter 'help' for list of built-in commands.    (initramfs)   Output of lsblk as requested:  $ lsblk NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 232.9G  0 disk  ├─sda1   8:1    0   231G  0 part  ├─sda2   8:2    0     1K  0 part  └─sda5   8:5    0     2G  0 part [SWAP] sr0     11:0    1   1.2G  0 rom  /cdrom loop0    7:0    0   1.1G  1 loop /rofs Reflectivity of a glowing-hot metal surface When a polished piece of metal (or steel in particular) is heated to incandescence, how do its reflective properties change?  Given a mirror-like surface, would the object temporarily cease to act like a mirror at a certain temperature?  Would the specular reflection of light turn into diffuse reflection; in other words, would the surface temporarily become matte/dull?  I am not asking about permanent changes to the object such as melting, but rather can there be a temporary change in the reflectivity only while the the object is hot, that disappears when the object is cooled down?  Background of the question: I have a laser scanner that usually has trouble scanning shiny surfaces with high specular/ low diffuse reflectivity; and it seems to work a lot better for hot steel pieces than for cold ones. Glowing steel also looks less shiny to me than cold steel, but of course that is hard to tell with all the glowing I accidentally opened too many menus (or windows) and I want to reset them to default.  It is really annoying, and effecting my animation.  Thanks How can I implement "cut outs" for lighting in OpenGL? So, I'm working with OpenGL (I'm not exactly sure of the version), and I want to do an old-style lighting setup by essentially drawing a black rectangle over the screen, and drawing white circles over the areas that should be lit. I already have the rectangle and the circles, but I can't figure out how to make it so that the white circles are transparent when applied to the black areas. Any ideas? I tried to use blending modes, but it still evades me...  EDIT: Here's the code that I'm using (Python).  view_buf = Buffer(GL_INT, 4) glGetIntegerv(GL_VIEWPORT, view_buf) view = view_buf.to_list() if hasattr(view_buf, "to_list") else view_buf.list  glDisable(GL_DEPTH_TEST)  glDisable(GL_LIGHTING)  glEnable(GL_BLEND)  if logic.lights['smooth']:     glEnable(GL_LINE_SMOOTH)  # # Make sure we're using smooth shading instead of flat # glShadeModel(GL_SMOOTH)  # Setup the matrices  glMatrixMode(GL_PROJECTION) glPushMatrix() glLoadIdentity() gluOrtho2D(0, view[2], 0, view[3]) glMatrixMode(GL_MODELVIEW) glPushMatrix() glLoadIdentity()  #glAlphaFunc(GL_GREATER, 0.5)  glColor4f(0.0, 0, 0, 0.75)  glBegin(GL_QUADS);                              # Draw Black screen glVertex2d(0.0, 0.0); glVertex2d(view[2], 0.0); glVertex2d(view[2], view[3]); glVertex2d(0.0, view[3]); glEnd()  glColor4f(1, 1, 1, 1)  degtorad = 3.14159/180.0   glBlendFunc(GL_ZERO, GL_SRC_ALPHA)  def DrawCircle(x, y, radius, fill = 1, step = 11):      glVertex2f(x, y)      if fill:         glBegin(GL_TRIANGLE_FAN)     else:         glBegin(GL_LINE_LOOP)      for a in range(0, 360, step):         angle = degtorad * a         glVertex2f(x + (sin(angle) * radius), y + (cos(angle) * radius))      glEnd()   #for layer in range(len(logic.lights['layers'])): #   size =  for light in logic.lights['points']:     DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], step = light[2])     if logic.lights['smooth']:         DrawCircle(light[0][0] * view[2], (1 - light[0][1]) * view[3], light[1], 0, light[2])  # Reset the state glPopMatrix() glMatrixMode(GL_PROJECTION) glPopMatrix() Surjectivity of Composition of Surjective Functions Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto? Mouse Pointer Locator Due to a stroke, I have had some vision impairment in that I sometimes have a hard time locating my mouse pointer on the screen.  I have tried making the mouse pointer larger which helps but is still not the solution.  I am new to the Mac world and would appreciate your help.  In Windows, there is an option where pressing the CTRL key will help to identify the mouse pointer using a temporary display of several radiating concentric circles around the mouse pointer.  Is there something like this on OS X? Applying options to already loaded package Is there a way to apply some options to a package which is loaded in the cls file I am using? For instance, suppose that the cls file loads graphicx, and I want to apply the option demo to it from my tex file. I asked a  recently, looking for examples of an online tool. Per the , I answered my own question with a few examples that I had already found. I did this so that these examples would be included in the answers, along with the rest.  The question has received only down votes and insults saying that this shouldn't be done.   Should I not answer my own questions? DRAM Frequency showing 800 instead of 1600 I am running a Core i7 CPU originally at 3Ghz overclocked to 4Ghz using a multiplier of x20 and a Bus Speed of 200 Mhz  My memory is 1600 Mhz OCZ.  I installed CPU-Z to see what is going on - but it shows me (under the "Memory" Tab):  DRAM Frequency 800 Mhz  Channel Triple - NB Frequency 3800.2 Mhz  So what is the actual frequency its running on then? I am slightly confused. The  is the difference between the distances travelled by two waves meeting at a point. Given the path difference, how does one calculate the phase difference? Can we have the table name as "option" in MySQL? I am very, very new to MYSQL.I tried to create a table named "option". My SQL Query is :  create table option(   id int not null primary key auto_increment,  choice varchar(30)  )  While executing this query it shows the following error  Error Code : 1064 You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near 'option( id int not null primary key auto_increment, choice varchar(30) )' at line 1 (0 ms taken)  If I try with the table name as "choice" it is working.  can we have the table name as "option" in mysql?   thanks