Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
Question on groups of order $pq$
Proving the geometric sum formula by induction I am designing gamecards in LaTeX  Everything is going pretty ok so far but when i try to insert the %-sign for a question about percentages my file keeps giving errors.  I tried putting the percentage between dollarsigns because i know just % will make it a 'remark' and that the dollarsign makes it 'mathematical data'.   Whats the reason for the error?   P.s. the error says: file ended while scanning use of \kaart. ...    Codesample:   {\kaart{Bekijk statistiek kaart F:   Schat het percentage mensen dat als lievelingseten spaghetti opgaven, je mag er slechts $ 2% $ langs zitten. } { $ 11% $}} msbuild, defining Conditional Compilation Symbols I'm possibly just blind, but is there a command line to specify conditional compilation symbols in MSBUILD?  I currently have this Line in my buildscript:  SET MSBUILD=C:\WINDOWS\Microsoft.NET\Framework\v2.0.50727\MSBuild.exe SET CONFIG=Debug %MSBUILD% /p:Configuration=%CONFIG% /p:OutputPath=..\..\output source\MyProject\MyProject.csproj   And I'd like to add a condition. In Visual Studio, i can just go into Project Properties => Build => Conditional compilation symbols, but I have not seen that option for msbuild?  Bonus Karma if you know if I can completely override all symbols already specified in the .csproj files to make sure that only the conditionals from my Buildscript go in. Can a child inherit both their parents' clans jutsu? Can a child in Naruto inherit both abilities from his parents' clans?  For example, if Ino and Sasuke had a child is there a possibility for it to have both Uchiha's Sharingan and Yamanaka's mind transfer jutsu? Two ethernet ports on motherboard. How can I get double the bandwidth? Direct Question:  My evga classified motherboard has two ethernet ports.  I want to use them simultaneously to improve the speed of a remote connection.  (Yay for programming remotely).  How do I do this?  At present /etc/network/interfaces:  auto lo eth0 eth1  iface lo inet loopback  iface eth0 inet static address 192.168.1.202 netmask 255.255.255.0 gateway 192.168.1.2  ifrace eth1 inet static address 192.168.1.201 netmask 255.255.255.0   What should I change? Is there a package I need to install to bond the two channels.  Background: I do a lot of remote computing both from home to office and office to home.  But (to take my ISP out of the equation), let's frame the question as if I'm really just interested in maximizing the file transfer rate between two computers on the same (local) network.  Both boxes have dual ethernet motherboards.  Using a simple Linksys WRT54GS as a 5 port switch (ignore wireless and isp) -- how do I maximize the transfer rates between the computers?  Please Note: I'm not interested in load balancing -- Correct me if I'm wrong, but I want to bond the two channels together so that the box appears as if it has one connection that's twice as fast. Does a subsequent charm effect override a previous one? In a recent session, I was confused about how multiple charm effects would interact with each other:  A vampire used its 'Charm' ability to charm a PC. It reads (paraphrased):     The target must succeed on a saving throw or be charmed by the vampire. It takes the vampire's requests in the most favourable way it can. It can repeat the saving throw if the vampire or its companions harm it. Otherwise, the effect last 24 hours or until the vampire is destroyed, is on another plane of existence, or takes a bonus action to end the effect.   Another PC then decided to attempt to wrest control back from the vampire by casting crown of madness on their friend. it reads (paraphrased):     The creature must succeed on a saving throw or be charmed. The charmed target must use its action before moving on each of its turn to make a melee attack against a creature other than itself that you mentally choose. The target can act normally on its turn if you choose no creature, or none are within its reach.   At the time, I ruled that both effects would take place simultaneously. This meant that Crown of Madness took effect first, and the vampire's charm afterwards.  Was this the correct ruling? Can a creature be charmed twice? Or does one charm effect override the other? Is the claim that functions of independent random variables are themselves independent, true?   I have seen that result often used implicitly in some proofs, for example in the proof of independence between the sample mean and the sample variance of a normal distribution, but I have not been able to find justification for it. It seems that some authors take it as given but I am not certain that this is always the case. "Communicate over" vs. "communicate through" Also schemas can communicate over database links to schemas in other remote databases.    Could one replace the preposition over with through? Will it change the meaning, and if so, when should we use over instead of through? This is from problem $8$, Chapter II of Rudin's Real and Complex Analysis.  The problem asks for a Borel set $M$ on $R$, such that for any interval $I$, $M \cap I$ has measure greater than $0$ and less than $m(I)$.  I was thinking of taking the Cantor approach: taking $R$ to be the union of $[a,b]$ with $a$ and $b$ rationals, and for each $[a,b]$ we construct Cantor sets inside it. During theconstruction of each Cantor set, in order to have positive measure on it, we need to take off smaller and smaller intervals from it, namely the proportion goes to $0$. As a result, these Cantor sets are extremely "dense" on their ends. If for an interval $I$ it intersects with the Cantor set on $[a,b]$ while $b-a&gt;&gt;m(I)$, we shall expect the measure of intersection to be rather close to $m(I)$ and then we lose control on these cases.  Is there any way to fix this or shall I consider other approaches?  Thank you What may happen after a stack of spells resolves? Within the same phase, once a stack of spells has resolved in its entirety, is priority once again given to a player so that more spells may be cast?  For example:  I attack with a creature. Opponent opts not to block. Now, I want to buff (e.g. +3/+3) my creature multiple times, but I'm afraid of my opponent responding with a burn (e.g. Deal 3 damage) spell at the top of the stack. Can I buff once, ask if opponent wants to respond, then allow the stack to resolve. Then, in the same phase, cast more buff? The Start menu no longer shows up when I press the Start button How do you get a timestamp in JavaScript? Something similar to Unix's timestamp, that is a single number that represents the current time and date. Either as a number or a string. Can I see why a package is installed? VirtualBox 5.0.40 crashes the whole host since update to kernel 4.13.0-26-generic Why Doesn't a Bootable USB Boot This question is asked often in Ask Ubuntu, sometimes with few hints about the situation.   Please provide a list of possible reasons to help troubleshoot the problem. Definition of Significant Figures In my textbooks,  are defined as:   “Significant figures by definition are the reliable digits in a number that are known with certainty.”  “A significant figure is the one which is known to be reasonably reliable.”    Reliable means giving the same result on successive trials or reliable information can be trusted to be accurate. However, at the end of each of the two definitions it is also written that ‘the last digit of a number is generally considered uncertain in the absence of qualifying information.' For example if the mass of an object is 12.248 gm, the last digit which is ‘8’ is uncertain by plus or minus 0.001 gm. This uncertainty is unreliability in the information. So, the digit ‘8’ is uncertain, and thus according to the definition 1 it is not a significant figure. However, the rules for significant figures say that ‘non-zero digits are all significant’. Due to the last digit ‘8’, if the object was weighed with careful handling minimizing the chances of error, the figures preceding ‘8’ that are (1, 2, 2, 4) are certain and reliable. If the mass of another object is 2 gm, then it is uncertain by plus or minus 1 gm i.e. its mass could be 3 gm or could be 1 gm.  ‘2’ is therefore not certain! What is the reason that significant figures are defined to be the reliable digits in a number? In what sense they are said to be reliable? "As part of" versus "as a part of" Why won't SparseArray let me store values with the head List? Do I need to license fonts included with my OS? Notepad ignoring linebreaks Most documents I'm opening in notepad.exe under windows 7 lately, there are no linebreaks... everything just runs together.  The same documents in any other program, even previewed directly in Explorer, will show the linebreaks correctly.  this is happening with many .info files, .css, .js.. but not all of them.   I am guessing it's some kind of character encoding unix line endings something or other, but wondering if there's anything I can do or if anyone else has experienced similar and if I can make notepad work correctly. I prefer it to notepad++/other text editors for certain uses.