Proof that the order of $D_n$ is $2n$
How to show that $|D_{2n}| = 2n$ via the presentation?
Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$? Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:  $$ \nabla_A \operatorname{trace}( ABA^{T}C )  = CAB + C^T AB^T $$  There is a proof . However, I was not able to understand the second line of the proof. it says something like this:  $$ \nabla_A \operatorname{trace}( ABA^T C )= \nabla_\bullet \operatorname{trace}(f(\bullet)A^{T}C) + \nabla_\bullet \operatorname{trace}(f(A) \bullet^T C) $$  Where  $f(A) = AB$  I tried searching the proof or any hint using google, but I couldn't fine anything. Could anyone please help me with this second line. I cannot sleep. I am obsessed with this proof. Mysql: Select rows from a table that are not in another How to vertically center content with variable height within a div? Java Object Reuse Should Java Objects be reused as often as it can be reused ? Or should we reuse it only when they are "heavyweight", ie have OS resources associated with it ?  All old articles on the internet talk about object reuse and object pooling as much as possible, but I have read recent articles that say new Object() is highly optimized now ( 10 instructions ) and Object reuse is not as big a deal as it used to be.  What is the current best practice and how are you people doing it ? Which characters are valid in CSS class names/selectors? What characters/symbols are allowed within the CSS class selectors? I know that the following characters are invalid, but what characters are valid?  ~ ! @ $ % ^ &amp; * ( ) + = , . / ' ; : " ? &gt; &lt; [ ] \ { } | ` # no wifi in Ubuntu 15.04 Using a java library from python I have a python app and java app. The python app generates input for the java app and invokes it on the command line.  I'm sure there must be a more elegant solution to this; just like using JNI to invoke C code from Java.  Any pointers? (FYI I'm v. new to Python)    Clarification (at the cost of a long question: apologies) The py app (which I don't own) takes user input in the form of a number of configuration files. It then interprits these and farms work off to a number of (hidden) tools via a plugin mechanism. I'm looking to add support for the functionality provided by the legacy Java app.  So it doesn't make sense to call the python app from the java app and I can't run the py app in a jython environment (on the JVM).  Since there is no obvious mechanism for this I think the simple CL invocation is the best solution. Can there be true conclusions without assumptions? I was thinking of the sentence  "I think therefore I am",  which I had for a long time considered indisputable because it's self-evident.   Then I considered the hypothetical situation where my thoughts are illogical – and of course I can't be aware that they are – and so to a logical outsider I am not saying something that is actually indisputable and self-evident, but in fact may be total nonsense. Perhaps there is a hypothetical evil demon altering my ability to reason. However convincing it seems to me that I MUST exist since, in whatever since "I" or "exist" is, there is the ability to question whether I do, the conclusion can't actually be trusted without doubt since I cannot know if I am in fact reasoning logically.  Even what I consider to be trivial tautologies require this assumption of sanity, and therefore there is no true conclusion without assumption – an assumption about the nature of my own reasoning. The only absolute truth I have is unspeakable subjective experience, and any articulation (and I realize the apparent circular reasoning I've cornered myself into here...) is necessarily uncertain.  I realize that perhaps this doesn't even make sense, but was just curious if anyone had entertained this line of thought and come to some satisfying resolution. How can I safely upgrade an Ubuntu installation from one release to another? What should I do before upgrading to ensure it goes smoothly, and is there any clean-up I need to do afterwards?  In particular, do I need to do anything special if I am using packages from PPAs, if I have a graphics card that needs drivers, or if I'm running a custom Linux kernel?  This is a general question , but as example we take Ubuntu 12.04 and Ubuntu 12.10 Is normal extension of normal extension always normal? Calling future method from batch class? How do I quickly rename a MySQL database (change schema name)? The MySQL manual at  covers this.  Usually I just dump the database and reimport it with a new name. This is not an option for very big databases. Apparently RENAME {DATABASE | SCHEMA} db_name TO new_db_name; .  This needs to work with , which stores things very differently than . How to Understand the Relationships Among Random Variables, Samples, and Populations? Different ways to prove there are infinitely many primes? This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so many different ways.  Proof 1 : Euclid's. If there are finitely many primes then $p_1 p_2 ... p_n + 1$ is coprime to all of these guys. This is the basic idea in most proofs : generate a number coprime to all previous primes.  Proof 2 : Consider the sequence $a_n = 2^{2^n} + 1$. We have that  $$ 2^{2^n}-1 = (2^{2^1} - 1) \prod_{m=1}^{n-1} (2^{2^m}+1), $$  so that for $m &lt; n$, $(2^{2^m} + 1, 2^{2^n} + 1) \, | \, (2^{2^n}-1, 2^{2^n} +1) = 1$. Since we have an infinite sequence of numbers coprime in pairs, at least one prime number must divide each one of them and they are all distinct primes, thus giving an infinity of them.  Proof 3 : (Note : I particularly like this one.) Define a topology on $\mathbb Z$ in the following way : a set $\mathscr N$ of integers is said to be open if for every $n \in \mathscr N$ there is an arithmetic progression $\mathscr A$ such that $n \in \mathscr A \subseteq \mathscr N$. This can easily be proven to define a topology on $\mathbb Z$. Note that under this topology arithmetic progressions are open and closed. Supposing there are finitely many primes, notice that this means that the set $$ \mathscr U \,\,\,\, \overset{def}{=} \,\,\, \bigcup_{p} \,\, p \mathbb Z $$ should be open and closed, but by the fundamental theorem of arithmetic, its complement in $\mathbb Z$ is the set $\{ -1, 1 \}$, which is not open, thus giving a contradiction.   Proof 4 : Let $a,b$ be coprime integers and $c &gt; 0$. There exists $x$ such that $(a+bx, c) = 1$. To see this, choose $x$ such that $a+bx \not\equiv 0 \, \mathrm{mod}$ $p_i$ for all primes $p_i$ dividing $c$. If $a \equiv 0 \, \mathrm{mod}$ $p_i$, since $a$ and $b$ are coprime, $b$ has an inverse mod $p_i$, call it $\overline{b}$. Choosing $x \equiv \overline{b} \, \mathrm{mod}$ $p_i$, you are done. If $a \not\equiv 0 \, \mathrm{mod}$ $p_i$, then choosing $x \equiv 0 \, \mathrm{mod}$ $p_i$ works fine. Find $x$ using the Chinese Remainder Theorem.   Now assuming there are finitely many primes, let $c$ be the product of all of them. Our construction generates an integer coprime to $c$, giving a contradiction to the fundamental theorem of arithmetic.  Proof 5 : Dirichlet's theorem on arithmetic progressions (just so that you not bring it up as an example...)  Do you have any other nice proofs? How to export variable for use with sudo? On Slackware, using sbopkg permits one to build a package from source. Repos is not big as Debian, but it's nice.  Some software can use environment variables, for example on the VICE c64 emulator, if the variable FFMPEG is set to yes, it will enable ffmpeg recording the emulator.  I tried to use  $ export FFMPEG=yes; sudo sbopkg -B -i vice   but ffmpeg is disabled.  Instead I had to use  $ su - $ export FFMPEG=yes $ sbopkg -B -i vice   which works.  How to use environment variables with sudo? Want to cancel a 'Pending Erase' on iphone... Any suggestions? I lost my phone on vacation and activated 'Lost Mode' for a couple of days. The night before heading home I decided to do the Remote Erase ´cause I thought there's no way I'd get it back, I also called my carrier and canceled the data service on my number. The phone has been offline ever since but the hotel just called to say they found my phone at the spa and they are mailing it back to me. So I want to cancel the 'Pending Erase', all I care about is recovering all the pictures from that trip that I hadn't backed up on iCloud yet... Will removing the device from my iCloud account work?  And if I don't have service on the phone, will the 'Erase iPhone' command still go through when I turn it on?  Please help! I have two days to figure this out I am looking at  (dual low-voltage power amplifier). On the fourth page, there is a sample circuit; right at the V+ input there is a symbol that looks like a capacitor. However, it looks like it is shaded in. What is that? I am aware that, to use a Spell Scroll, you must have the related spell on (at least one of) your spell list(s). Then, if you have the spell slot to cast it, you succeed automatically; else, you must make an ability check in order to cast it (failure resulting in the scroll being wasted).  Consider this example : A Cleric 5 / Wizard 4 finds a Raise Dead Spell Scroll. The spell is on the Cleric List, but is a fifth level spell. Considering that the cleric-wizard has fifth level slots, but can only pick up to third level cleric spells, would he have to make the ability check to successfully cast the Raise Dead in the Spell Scroll, or would he succeed automatically ? My prof gave us this definition of an Echelon system:     A system of m linear equations in n variables is called an echelon system if         m ≤ n.   Every variable is the leading variable of at most one equation.   Every leading variable is to the left of the leading variables of all lower equations.   Every equation has a leading variable.      I don't understand this definition at all.  Q1)If you have this system of equations:  $ax_1 + bx_2 + cx_3 = z$  $dx_1 + ex_2 + fx_3 = z$  its not an echelon system because of multiple equations having the same variable as the leading variable and therefore violating rule #2.  But if you switch around variables and rewrite the equation as:  $ax_1 + bx_2 + cx_3 = z$  $ex_2 + dx_1 + fx_3 = z$  It becomes an echelon system! Why would the way you write a system of equations affect its definition?  Q2)Why would point number three matter? Can't you just move around the equations so that they are in order of variables?  Q3)Why would point number four matter? If you have a equation in the system that doesn't have a leading variable can't you just disregard it? For example if you have the equation $0x = 0$ in the equation can't you just ignore it to have a echelon system since it doesn't affect the solution of the system? Optimization paramters read-only mysql 5.7 database? What's the best way to configure a MySQL database that is only read-only, and will never have any modifications made to it?  1) I have increased my innodb_buffer_pool_size to 70% of available RAM  2) Increase query cache size to 40mb  Running against a load generator, after some time the database starts bottlenecking to only 1000 statements per second.  The configuration is 2-cores, 8GB ram 50GB SSD. The database contains a single InnoDB 13 GB table with the same exact query running on two indexed columns.  We are using a vertx server to hit the database. The symptom we see is that the web service performance degrades after 15-20 minutes of heavy load.  Where can the bottleneck be? What parameters should we optimize?