Let $E$ be a measurable set and let $g$ be a function defined on $E$. The product $fg$ belongs to $L^1(E)$ for every function $f \in L^1(E)$ if and only if $g ∈ L^{\infty}(E)$.
Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ for some constant $C$, this follows from the Riesz Representation theorem. But what if we aren't given that such a $C$ exists?
How to extend summations to fractional orders? How does two-weapon fighting work? I'm quite confused about attacking with two weapons in both hands and wondering what is required to be able to do to (in Pathfinder).  1. Base Attack Bonus Looking at the base attack bonus tables for different classes (e.g. the ) it seems like up to +5 you get only one attack per full action but starting from base-attack-bonus of +6 you get a second attack with +1.  2. Feats There is a feat . The existence of the feat sounds like you could not do two-weapon-fighting without it. However, in the description of "normal" it says:     If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. When fighting in this way you suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand. If your off-hand weapon is light, the penalties are reduced by 2 each.   3. Core Rulebook Chapter about combat If I understand the  it says that you need a full-round action do attack with both weapons in two hands.  So how do these things interact? Can I only attack with my off-hand weapon if my base-attack-bonus is high enough that I get second one with +1?  If not, how does the "multiple attacks from high base-attack-bonus" interact with dual-wielding? Assuming I reached a state of base-attack-bonus with +11/+6/+1 and have a full-round action to attack. Does that mean I can either attack with my primary weapon three times or use the primary weapon and off-hand weapon to attack only twice? Why would I then ever choose dual-wielding? Or can I attack three times with both weapons (leading to a total of 6 attacks with heavy malus) Or would I get 3 attacks with the primary weapon + 1 attack with the off-hand weapon? If so, what attack bonus do I have to use for each of the attacks? Remove votes to close question X as a duplicate of Y if question Y is closed as a duplicate of X I just happened across , which has one close vote suggesting that it be closed as an exact duplicate of .  This suggestion is perfectly correct: these are exactly identical questions asked by the same user.  However, there's not much point in closing question A as a duplicate of B, since question B has already been closed as a duplicate of A.  Yet the close vote remains, doing nothing but wasting reviewers' time.  This is not the first time I've seen this happen, so I'd like to suggest that the Stack Exchange software should do something about it.  In particular, I'd like to suggest the following feature:     When a question Y is closed as an exact duplicate of another question X, any votes or flags to close X as a duplicate of Y should be automatically discarded.   Actually, I'm kind of surprised that the SE software doesn't do that already.  If anyone else thinks that this constitutes an actual bug, feel free to retag it. Powering a WS2812B strip Relay: higher voltage allowed at lower current? Is it safe to use a 30VDC/1A relay for a voltage of 60V but a current of &lt;100uA? E.g. this one: Cannot modify files installed by `snap` I have installed a package using snap and I need to modify one of the files but when I try to change its ownership or permissions, I always get the following message:  sudo chmod +x ./my_file.js chmod: changing permissions of '/snap/my_app/my_file.js': Read-only file system   How can I modify files that are installed via snap? Deleted comments are persisting in the global inbox I realise that there is other questions out there about the  (First open one I found is ), but this is marked .  I often get the below issue where it will notify me about comments on my  (in this case what appears to be the user re-writing after deleting their comment) that no longer exist.  Also, occasionally, I see comment snippets with text that doesn't match what is on the page when I view it.  Let me know if I should be trying to edit that one to remove the tag indicating that it has been fixed, and add the above as another answer. Possible to replay repetitive tasks using Python? How to pass arguments to addEventListener listener function? How can I report bugs with or request features for the Stack Exchange mobile apps? I'm using the official Stack Exchange mobile app for iOS or Android, and  My app crashed! It doesn't look good on my new smartphone. It won't run after I updated my device. I noticed another bug. I have an idea for a new feature which is currently not in the app.  How do I report bugs with or request features for the iOS and Android apps? Also:  I've used the built-in functionality of the app to report a bug or submit a feature request. Did I do the right thing? The app prompted me to post on Meta after an error occurred. Should I post about it here? I'm wondering why the latest version is from 2017. I've heard rumors that the apps are no longer supported; is that true? I can't find the app on my device's app store anymore! Where did it go?     We get a lot of bug reports from users about the mobile apps, which are unsupported, and at the moment, there is no centralized Q&amp;A pair for those to be closed as duplicates of (the relevant answers are spread across multiple unrelated questions). This question is an FAQ proposal, to serve as a directly-related duplicate target for those questions. Reference for general-topology Bibliography per chapter with continuous numbering I would like to create a document with two chapters. After the first chapter there should be a bibliography. The next chapter should be able to reference items of the former chapter, but only show the new items in its bibliography, starting with the next number from the first bibliography.  Example:     Chapter 1 - The Fu   Fu has been done by Bar [1]. Early versions ...   Bibliography   [1] Bar et. al., Doning the Fu      Chapter 2 - Compining Fu with Herp   Doning Fu as described in [1]   neglects the advantages of Herp pointed out by Derp [2].   Consequently ....   Bibliography   [2] Derp et. al., The Advantages of Using Herp   Please not too much discussion about if that is a friendly-to-use idea, it is not my own demand :) How many ways can the average of n dice be a? Variance of sum of $k$ randomly drawn numbers from $1,...,n$. Is there a limit of $\cos(n!)$? I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks If user input is inserted without modification into an SQL query, then the application becomes vulnerable to , like in the following example:  $unsafe_variable = $_POST['user_input'];   mysql_query("INSERT INTO `table` (`column`) VALUES ('$unsafe_variable')");   That's because the user can input something like value'); DROP TABLE table;--, and the query becomes:  INSERT INTO `table` (`column`) VALUES('value'); DROP TABLE table;--')   What can be done to prevent this from happening? Merging dictionaries in C# What's the best way to merge 2 or more dictionaries (Dictionary&lt;T1,T2&gt;) in C#? (3.0 features like LINQ are fine).  I'm thinking of a method signature along the lines of:  public static Dictionary&lt;TKey,TValue&gt;                  Merge&lt;TKey,TValue&gt;(Dictionary&lt;TKey,TValue&gt;[] dictionaries);   or  public static Dictionary&lt;TKey,TValue&gt;                  Merge&lt;TKey,TValue&gt;(IEnumerable&lt;Dictionary&lt;TKey,TValue&gt;&gt; dictionaries);   EDIT: Got a cool solution from JaredPar and Jon Skeet, but I was thinking of something that handles duplicate keys. In case of collision, it doesn't matter which value is saved to the dict as long as it's consistent. Everywhere I get to read statements like EM radiations and EM fields carry "energy as well as momentum " . I wonder if energy and momentum are inalienable . Is it possible for a field to carry only energy without having any momentum term ? I mean isn't it sufficient to say that fields or EM waves carry energy or Doesn't the ability to transfer energy, of the fields, automatically imply that they can change the momentum of another particle while transferring energy to them? How to colorize an image as in the example? I have seen such an effect being used in many places however I haven't figured out how to replicate it via Photoshop.  I have tried desaturating the image, placing it under a layer of solid colour and then lowering the opacity of the colour. I have also tried a couple of blend modes, to no avail. The effect is nowhere near the same.  Can someone please tell me the effect used on these images? lcm versus phi in RSA In textbook RSA, the Euler $\varphi$ function $$\varphi(pq) = (p-1)(q-1)$$ is used to define the private exponent $d$. On the other hand, real-world cryptographic specifications require the Carmichael lcm function $$\lambda(pq) = \operatorname{lcm}(p-1,q-1)$$ to define $d´$. It is clear that the $d´$ divides $d$, and therefore using $d´$ may be more efficient than using $d$. My question is: Are there any further reasons, e.g. regarding security, why one should not use $d$?