I encountered the ideal $\sum_{i \in I} \mathfrak{a}_i$, where the $\mathfrak{a}_i$'s are ideals of an entire ring. However, no explanation was given about this ideal. Is it the ideal $\lbrace \sum_{i \in I} a_i \mid a_i \in \mathfrak{a}_i \rbrace$? Is it possible to express this ideal as the ideal generated by some set, i.e. the intersection of all ideals containing some set?
I've been trying to prove that given a ring $R$ and a collection of ideals $\{I_{\beta}\}_{\beta \in B}$ in $R$, the set $$\sum_{\beta \in B} I_{\beta}=\{f_1+\cdots+f_r:f_j \in I_{\beta_j}, \mbox{ for some } I_{\beta_j}\}$$ is an ideal and is the smallest ideal containing $\cup_{\beta \in B} I_{\beta}$.  I've already proved when I only have two ideals, and so, using induction, I've seen it for finite sums. But I don't see how to generalise that to an arbitrary family of indexes (maybe infinite) $\beta$.   Could someone please give me a hand? Thank you.
What is the best class or classes in 3.5e to build a character that uses both a sword and either arcane magic or psionics which can be fluffed as arcane magic and focuses on fighting powerful individual creatures?    I have a foundness for those that can sling spell along with blade, like Dak'kon from Planescape: Torment.  So, given that, how do I make the best Gishy character I can?  Specifically, the group is looking for a character focused on delivering high damage or other debilitating effects to an individual target.  There is no need to balance sword and magic, though I would prefer to have the melee abilities remain at least relevant through at least mid-levels.  This question:  is related.  But I am specifically looking to optimize for high targeted damage output and have no need to balance between sword and spell if focusing on one is more effective.  I have considered the duskblade and that may be the best answer, but as I have limited experience with 3.5 I want to find out if I should use a pure duskblade, a duskblade which steps into other classes, or some other class I have not considered.  In this particular case, the characters will start at level 7, and equipment rules are liberal as long as the characters can afford it.  I'm not sure how long this will last so I am focused on the next few levels instead of trying to plan all the way out to 20.  It will be a party of 5.  Not all of the rest of the characters have been fully fleshed out, but we will have a cleric and a wizard.  The other two are more up in the air but they one is looking at various rogue classes and the other is looking at some sort of melee class, with this gishy rounding out the team.  The initial encounter is up in the air, but there is a definite fondness for gryphons.  We do know that we will be working up to dragon in relatively short order, which is a factor in wanting to focus on large damage output to individual creatures as opposed to being able to deal with numerous weaker creatures.  The setting is "modified Forgotten Realms".    I personally own absolutely no 3.5 books, but we have access to a fair bit and just about anything from WoTC is permitted.  I am new to 3.5, but I have a fair bit of roleplaying experience including AD&amp;D.    Thematically, I envision the character as a generally honorable, but working as a monster hunter and mercenary.  I definitely favor swords though specifically which sword is flexible.  Dak'kon from Planescape would be an excellent example of what I would like to build, as would The Witcher.  Based on Kryan's excellent answer, it seems to be leading up to using duskblade and a specialist wizard to lead into Abjurant Champion. So I have a program written in C++.  It can tell me how long it took to do all the calculations and it does a lot of quite heavy, multi-threaded calculation.   I've just noticed that if I run the program on the exact same machine, it takes around 20-21 seconds to do all the calculations if started from the TTY, and only about 0.2 seconds if I start it from GNOME terminal.  What is causing that? It's literally the exact same file on the same machine. Do you think there is a big difference in for...in and for loops? What kind of "for" do you prefer to use and why?  Let's say we have an array of associative arrays:  var myArray = [{'key': 'value'}, {'key': 'value1'}];   So we can iterate:  for (var i = 0; i &lt; myArray.length; i++)   And:  for (var i in myArray)   I don't see a big difference. Are there any performance issues? Any way to add CSS for a single page/node? I'm cleaning up my big crazy style sheets (possibly pertinent to a future question) and I'm wondering the best way to add custom CSS to a specific node or page. In particular, my work site's home page is a Panel Page and it has a bunch of different styling. Right now the CSS is just included with the main theme style sheet. Is there a way to say, &quot;if this is node Foo, then add foo.css&quot;? Is  what I'm looking for? I might be interested in generalizing this to other nodes/sections/etc, but at the moment I just want to handle this one item. How do I fill an area with faces lined up in a grid over a not flat surface? Hello to everyone again,  I'm looking for an addon or a short cut to fill fast and simmetrically a NOT FLAT surface that needs to be closed with vertices. Doing it manually it takes a lot of time. I tried to do CTRL F and then Grid fill,but it didn't work. Neither F key works. Check the attached screenshot. thanks. Invalid signature for Kali Linux repositories : "The following signatures were invalid: EXPKEYSIG ED444FF07D8D0BF6 Kali Linux Repository" Best methods to fix broken tracks I've made a couple of PCBs which need the odd few 10 mil tracks, but are mostly 20/30 mil. Unfortunately the etching process isn't fantastic and there are numerous breaks in the traces, most of them small.  I've managed to convince solder lumps to go over the top of the smaller gaps, and have used components legs to bridge some of the larger ones. The solder lumps don't feel particularly reliable though, and there are some gaps which are too small and awkward to get a piece of wire across but too big to be bridged with just the iron.  Are there any standard ways to reliably fix up broken traces?  Update: Thanks for the suggestions everyone!  I found and tried out a silver conductive pen. As a first time user I found it difficult to dispense, and obviously I didn't use enough of it or mix it properly because it didn't even end up being conductive.  It seems like the main problem was that I was trying to use pieces of wire which are too short. I found some small gauge tinned wire and used longer lengths, running it over significant lengths of good track on either side of the fault. That makes it much easier to position, and you can tack them down one end at a time. I have a site on my webhotel I would like to run some scheduled tasks on. What methods of achieving this would you recommend?  What I’ve thought out so far is having a script included in the top of every page and then let this script check whether it’s time to run this job or not.  This is just a quick example of what I was thinking about:  if ($alreadyDone == 0 &amp;&amp; time() &gt; $timeToRunMaintainance) {    runTask();    $timeToRunMaintainance = time() + $interval; }    Anything else I should take into consideration or is there a better method than this? Intellisense in IntelliJ no longer working In what order should the Star Wars movies be watched? Please don't lynch me, but I've never sat through an entire Star Wars film. I'm aware that the 3 (new) films are technically prequels, but I feel strange watching 3 new films then 3 old films?  Do I need to watch the new ones then the old ones? How to loop through all the properties of a class? I have a class.  Public Class Foo     Private _Name As String     Public Property Name() As String         Get             Return _Name         End Get         Set(ByVal value As String)             _Name = value         End Set     End Property      Private _Age As String     Public Property Age() As String         Get             Return _Age         End Get         Set(ByVal value As String)             _Age = value         End Set     End Property      Private _ContactNumber As String     Public Property ContactNumber() As String         Get             Return _ContactNumber         End Get         Set(ByVal value As String)             _ContactNumber = value         End Set     End Property   End Class   I want to loop through the properties of the above class. eg;  Public Sub DisplayAll(ByVal Someobject As Foo)     For Each _Property As something In Someobject.Properties         Console.WriteLine(_Property.Name &amp; "=" &amp; _Property.value)     Next End Sub How do I get my PCs to not be a bunch of murderous cretins? Most RPGs teach you that casual violence is the best solution to all your in-game problems. This is so well established a part of the vast majority of RPGs that there are entire satire RPGs like Greg Costikyan's  and John Tynes'  dedicated to showcasing the issue. Even now that the '80's anti-RPG hysteria is past, you see more sober critiques of endemic violence in RPGs in places like . In most RPGs, PCs become inured to murder and other antisocial activities very quickly, and enter the depths of depravity that wouldn't be appropriate in the worst parts of the world. Armed robbery, mass murder, and genocide become routine parts of an adventurer's day, something only stick-in-the-mud characters with the most extremely stated ethics object to. , even though it is not properly applied to just any conflict, is a PC's friend and they generally escalate any conflict in that direction. The sophistication of the gamer mindset towards introspection on this issue can be demonstrated in that the most meaningful question usually debated about in-game violence is some variant on &quot;but should we kill the noncombatant children&quot; or &quot;can we just murder people out of hand as long as they're from a typically evil race?&quot; In the real world, we generally a priori regard anyone having to have this discussion as a monster already. (P.S. The &quot;wipe out a given evil race&quot; thing isn't the point of this question; it is only mentioned to indicate that it's only &quot;that far&quot; that usually causes ethical handwringing from players, when that is a really quite extreme case and we should be uncomfortable with casual violence a lot sooner. ) What I'd like to have is a more realistic in-game treatment of conflict.  People getting wounded and giving up, taking and ransoming of prisoners, not always escalating a fistfight to weapons, not always escalating weapon combat to killing, etc.  Heck, as I write this, I'm watching an episode of Adventure Time with my daughter and the protagonists snuck in and rescued a princess kidnapped by the Ice King and then exit right next to his sleeping form. I thought, &quot;If this were D&amp;D they'd all be carefully coordinating a coup de grace to kill him in his sleep on the grounds that he inconvenienced them.&quot;  The problem isn't limited to D&amp;D of course, sci-fi PCs are happy to neutron bomb planets for convenience too, for example. I'm not part of the &quot;D&amp;D is Satanic&quot; crowd obviously, but I frankly do have compunctions about continually playing in games where the taught behavior is uncomfortably equivalent to the worst examples of human behavior we see on the nightly news. How can I give my PCs a newfound respect for human life? I've updated my Ubuntu to 19.10 couple of weeks ago and on installation system suggests to change or add shortcut for input source selection despite one was already defined previously. I've added ctrl+shift shortcut and just realized that it somehow overlapped any other ctrl+shift+something shortcuts globally, i.e. when I want to open new window in vscode, it opens new tab and it looks like system doesn't see the shift button press.   I tried to search this combination in shortcuts settings, but it's just not there. Azure DNS does not lookup SPF policies for certain domains How to configure @SkipValidation by XML configuration in Struts 2 Riemann integrabilty of an indicator function Consider the following function on $[0,1]$:   $$f(x)=\begin{cases}1&amp;x\in\{a_1,a_2,\dots,a_n\}\\0 &amp;\text{otherwise}\end{cases}$$   The $a_i$ are fixed and all in $[0,1]$. Determine whether $f$ is Riemintegrable or not.   By the Riemann&ndash;Lebesgue criterion $f$ is Riemann integrable. How to prove directly? The difference between GPU and CPU I know what a CPU is(I think). It's the thing who's speed is measured in GigaHertz(these days).  However, you hear a lot about a GPU, and letting the GPU take over, not letting the CPU but the GPU do it, GPU-based rendering, etc...  What is this GPU anyway? How can I access it and use it to my advantage? What am I missing out on here? I often need to view on the canvas the (selected or not) features from the current active vector layer.  In this situation, I only need to view something on the canvas, without doing anything on the canvas itself (so, the active window is always the Attribute Table).  This is what I usually do:   Open the Attribute Table for the current active layer; Select the first feature; Press Ctrl+P button (i.e. the "Pan map to the selected row" button); Minimize the Attribute Table; View something on the canvas for the current feature; Go to the first step until needed.   Is there a way (a method, a shortcut) for directly browsing every feature on the canvas without the needing of going backwards and forwards with the Attribute Table? I mean, something like a "next" button?  (I may use the workaround of splitting the windows of the Attribute Table and of the canvas within the monitor, or also docking the Atrribute Table, but I would like to work on the full QGIS window).    EDIT I saw that my question was casted as a duplicate of  question. The two questions are very similiar, but I think it should be left as open for two reasons:   the related duplicated question doesn't have an accepted answer; my question is slightly different because I'm looking for the possibility of displaying only the selected features (and the @Joseph's , that I marked as accepted, answers well to my request). What does the comma operator , do? I've a Skype interview for PhD application (it will make me miss  today.) I've been asked, in advance, to prepare a short presentation and then there would be some discussion with the panel (I guess.) I expect that they will ask if I have some questions so, I'm wondering what are the best (type of) questions one might consider?