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Evaluating $\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx$ Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate   $$  \int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx.  $$   Well, the case $n=1$ is easy and the integral equals $\frac{1}{\alpha+1}$, for every $\mathbb R \ni \alpha \ne - 1$.  I've done also the case $n=2$ and, if I'm not wrong, it's $\displaystyle \frac{2}{(\alpha+1)(\alpha+2)}$.  My big problem is that I cannot understand how to deal with the general case. Any ideas?   Thanks in advance. How did "When" become the customary answer to "say when"? When a waiter at a restaurant comes by with pepper or Parmesan cheese, he says, "say when" and starts putting it on your food.  Many people will say "OK" or "that's enough," but it seems that the customary answer is "when".  How did this become the customary answer?  Did everyone decide to make the exact same joke, until it became commonplace.  I thought it was funny when I was a kid, but even now as an adult, I still say it.  For evidence that this is indeed commonplace, see .  The answer "when" is used and is external to the joke of the strip. But even so, I'm curious as to just how commonplace this really is.  I see it a lot in the US.  Is it used elsewhere in the world?  (by the way, I'm new to asking questions on this site, and I have no idea what tags to put on this question) How do I redirect to another webpage? Is there a systematic way to obtain all conserved quantities of a system? I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's equations. These equations are invariant under many transformations (Lorentz transformation, rotations, spatial and temporal translations, etc. By the way is there a way, maybe from group theory, to find all the possible transformations that leaves the equation(s) invariant?) which imply as many conserved quantities thanks to Noether's theorem. In wikipedia I can see an equation that seems to give all the conserved quantities () but it involves the Lagrangian and I'm not sure whether the formula is valid for all systems whose Lagrangian is possible to obtain. How to negate a specific symbol? I want to negate the symbol whose code is \models (this would be specific to Formal Logic). How can I do that? I tried a number of combination, but didn't find how this could be done. I tried \neqmodels, \nemodels, \notmodels, \nmodels  but all of them are wrong.  Is there a general way to negate symbols? Equivalence between least squares and MLE in Gaussian model I am new to Machine Learning, and am trying to learn it on my own. Recently I was reading through  and had a basic question.   Slide 13 says that "Least Square Estimate is same as Maximum Likelihood Estimate under a Gaussian model". It seems like it is something simple, but I am unable to see this. Can someone please explain what is going on here? I am interested in seeing the Math.   I will later try to see the probabilistic viewpoint of Ridge and Lasso regression also, so if there are any suggestions that will help me, that will be much appreciated also. I have a battery pack of NiMH batteries. It is ten cells with 1.2V, 4000mAh each, put together in series. So rated voltage is 12V. After charging, i.e. when the charging device says that it is finished, I measure 14.3V.   When I now discharge, at what voltage should I consider my battery as "empty"? Is this the lowest voltage that I can discharge it to without damaging it? Which voltage is that? What is the normal operating temperature for the MacBook (2016 generation)? Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$. Let $B$ be a compact subset of a metric space $X$. Let $x \in X \setminus B$. Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.   Is this an existence proof? If yes, do I get to define $U,V$? I am currently working my way through "A Concise Introduction to Pure Mathematics" by Martin Liebeck, and have been stuck on an excersise for a couple of days now. The question reads:  "Show that for an integer n $\geq2$, the period of the decimal expression for the rational number $\frac{1}{n}$ is at most $n-1$."  I've been trying to solve this without making any substantial progress. And yes, I have seen explanations that when performing a long division the decimal expression either terminates to $0$ or eventually repeats. But I haven't seen any proof of this that I've been satisfied with, even less do I know how to formalise this type of proof. I've tried to write out long division with arbitrary digits, but that obviously turns out to be too much of a hassle and basically impossible to draw any conclusions from. I want to be able to lock my translation/scale operations to two axes at a time, say xy, xz or yz. It currently takes two operations (one for each axis).  Example, say for a cylinder, I would like to reduce the radius. I currently have to scale on the x axis first, then on the y axis, or I have to manually input the scale values in the object properties panel.  This option is easily accessible in 3dsmax via the axis constraints command bar. Or I can click the intersecting area between the 3d manipulator of the axes in question and move in that plane. List contents of tarball in tree format Per the answers  I'm able to list the contents of my tarball in ls format. However, I'd like to be able to list them in tree format, i.e. something like the latter instead of the former:  With tar:  $ tar -tf foo.tar  foo/ foo/baz/ foo/baz/qux/ foo/baz/qux/hisfile.txt foo/bar/ foo/bar/myfile.txt foo/bar/yourfile.txt   With tree:  $ tree foo foo ├── bar │   ├── myfile.txt │   └── yourfile.txt └── baz     └── qux         └── hisfile.txt   Is it possible to do this without extracting the tarball? I'd prefer to have to avoid extracting the tarball due to their size. Order of an element in a finite cyclic group Handling circular entity references with Migrate I am using  on a Drupal 6 to Drupal 7 migration right now.  I have three types: type1, type2, type3.  Each of the three types has a entityreference field on it, which is configured to be able to point to all three types.  This means that there can be circular references, and when a particular node is being migrated, the destination node may not exist yet and could potentially be any of the three types.  My migration class looks like this:  class MyNodeMigration extends DrupalNode6Migration {   public function __construct(array $arguments) {     parent::__construct($arguments);      $this-&gt;addFieldMapping('field_legacy_nid', 'nid');      $this-&gt;addFieldMapping('field_the_reference', 'field_the_reference')       -&gt;sourceMigration(array('type1', 'type2', 'type3');   } }  $node_arguments = array(   array(     'class_name' =&gt; 'MyNodeMigration',     'description' =&gt; 'type1',     'machine_name' =&gt; 'type1',     'source_type' =&gt; 'type1',     'destination_type' =&gt; 'type1',     'dependencies' =&gt; array('roles', 'users'),   ),    array(     'class_name' =&gt; 'MyNodeMigration',     'description' =&gt; 'type2',     'machine_name' =&gt; 'type2',     'source_type' =&gt; 'type2',     'destination_type' =&gt; 'type2',     'dependencies' =&gt; array('roles', 'users'),   ),    array(     'class_name' =&gt; 'MyNodeMigration',     'description' =&gt; 'type3',     'machine_name' =&gt; 'type3',     'source_type' =&gt; 'type3',     'destination_type' =&gt; 'type3',     'dependencies' =&gt; array('roles', 'users'),   ), );  $common_node_arguments = $common_arguments + array(   'user_migration' =&gt; 'users', );  foreach ($node_arguments as $arguments) {   $arguments = array_merge_recursive($arguments, $common_node_arguments);    Migration::registerMigration($arguments['class_name'], $arguments['machine_name'], $arguments); }   When I run the migrations, most nodes are fine, but I get a decent amount of Stubs (*) created.  Some of the destination nodes have the entityreference pointing to these Stub nodes.  The source nodes are fine.  What I have uncovered is that the bundle on the destination nodes that are being made as stubs are incorrect, and look like they are all type1.  A node of the proper type does end up being created, but it never gets set as the entityreference target.  Another way to explain this is that I end up with   node: nid=123, type=type1 node: nid=234, type=type1, field_the_reference=123 node: nid=345, type=type2   where it should be   node: nid=234, type=type1, field_the_reference=345 node: nid=345, type=type2   How do I handle this situation?  (*): a "Stub" in the context of this question here refers to a node which is created for the purpose of being a reference target, but does not get its data initially. The term is used throughout the Migrate documentation and source code. Is there a grammatical term for moving a word to the front or back of a sentence? There is a function in Arabic grammar where you may bring forward in sentence order a word - as well as deferring it.  For example: if the sentence order is Subject - Verb - Object, you can bring the object forward to be before the subject,  yielding Object - Verb - Subject or Object - Subject - Verb.  Similarly, if you have Noun Phrase - Verb Phrase, you can move the verb phrase forward so you get Verb Phrase - Noun Phrase.  This is used to add emphasis to the item brought forward. The same applies to moving the item to the end of the sentence.  Is there a grammatical term in English that represents this function or functions? com.canonical.unity.webapps   In order to work properly, Unity Tweak Tool recommends you install the necessary packages Can I partition a HDD for playing Xbox 360 games? I am planning to buy an xbox 360 console. my budget permits me acess to only the 4 GB one which needs extra space for many games' install. I have an external HDD of size 40GB. My question is can I divide it into partitions such thet I can use one for gaming on xbox and the other for data storage. There are now errors when updating and I cannot install most software due to a corrupted  /etc/apt/sources.list   file. Is there a copy I can download to replace it?  The file would be for Ubuntu 12.04 (Final Beta) in the United States. QGIS 3.4 : Add points to exact locations? I would like to add several coordinates at specific British National Grid Easting and Northing values. I know this can be done by bringing in a .csv, but I want to know is there a way to manually enter the numbers in QGIS 3.4, and then the point moves to that location ? I've read that in QGIS 2.x there was a plugin called Numerical Vertex Editor that doesn't seem to exist for 3.x ? Center of a free product Let $A$ and $B$ be two groups and let $G=A*B$ be their free product. Suppose both $A$ and $B$ are nontrivial. I want now to prove that $G$ is infinite and that $Z(G)$ is trivial.  I think I can prove that $G$ is infinite using this argument: two reduced words of $A*B$ are equal if and only if they have the same length and the ith letter of the first word is the same as the ith letter of the second one. Hence one can find a reduced word of $G$ of length arbitrarily big (since we are sure it won't be equal to a shorter word). Do you think it works? Has anyone another suggestion?  On the other hand, I don't really have a clue for the second part except trying to analyze a long list of cases of reduced words and prove for each one that the only element with which they commute is the identity. Does anyone have a smarter idea?  Thanks in advance!