For all $-1&lt;x&lt;1$, prove:   $$(1+x)\ln(1+x) + (1-x)\ln(1-x) \leq  2x^2$$   I am sure we should use the Jensen's inequality or the Taylor expansion of $\ln(1+x)$ and $\ln(1-x)$; however, I was not able to do that so far. Any idea?
Find the smallest value of $c$ such that $(1-x)\ln(1-x)+(1+x)\ln(1+x)\leq c x^2$ holds for $0&lt;x&lt;1$.   I saw the curve and realized this is true for $c=2$. How can I prove it? What is the smallest $c$ that still makes the inequality valid?  I think we should use the Taylor expansions of the $\ln$s.
Random weighted choice Consider the class below that represents a Broker:  public class Broker {     public string Name = string.Empty;     public int Weight = 0;      public Broker(string n, int w)     {         this.Name = n;         this.Weight = w;     } }   I'd like to randomly select a Broker from an array, taking into account their weights.  What do you think of the code below?  class Program     {         private static Random _rnd = new Random();          public static Broker GetBroker(List&lt;Broker&gt; brokers, int totalWeight)         {             // totalWeight is the sum of all brokers' weight              int randomNumber = _rnd.Next(0, totalWeight);              Broker selectedBroker = null;             foreach (Broker broker in brokers)             {                 if (randomNumber &lt;= broker.Weight)                 {                     selectedBroker = broker;                     break;                 }                  randomNumber = randomNumber - broker.Weight;             }              return selectedBroker;         }           static void Main(string[] args)         {             List&lt;Broker&gt; brokers = new List&lt;Broker&gt;();             brokers.Add(new Broker("A", 10));             brokers.Add(new Broker("B", 20));             brokers.Add(new Broker("C", 20));             brokers.Add(new Broker("D", 10));              // total the weigth             int totalWeight = 0;             foreach (Broker broker in brokers)             {                 totalWeight += broker.Weight;             }              while (true)             {                 Dictionary&lt;string, int&gt; result = new Dictionary&lt;string, int&gt;();                  Broker selectedBroker = null;                  for (int i = 0; i &lt; 1000; i++)                 {                     selectedBroker = GetBroker(brokers, totalWeight);                     if (selectedBroker != null)                     {                         if (result.ContainsKey(selectedBroker.Name))                         {                             result[selectedBroker.Name] = result[selectedBroker.Name] + 1;                         }                         else                         {                             result.Add(selectedBroker.Name, 1);                         }                     }                 }                   Console.WriteLine("A\t\t" + result["A"]);                 Console.WriteLine("B\t\t" + result["B"]);                 Console.WriteLine("C\t\t" + result["C"]);                 Console.WriteLine("D\t\t" + result["D"]);                  result.Clear();                 Console.WriteLine();                 Console.ReadLine();             }         }     }   I'm not so confident. When I run this, Broker A always gets more hits than Broker D, and they have the same weight.  Is there a more accurate algorithm?  Thanks! MacBook Pro: How to disable discrete GPU permanently from EFI? Are C++ enums signed or unsigned? And by extension is it safe to validate an input by checking that it is &lt;= your max value, and leave out >= your min value (assuming you started at 0 and incremented by 1)? I'm making a mini game (bomb squad) and the armor needs to be enchanted. I know that armor can be enchanted through command blocks and that is what I intend to do. I am wondering if it is possible to enchant armor twice in one command block or should I need multiple command blocks as I have done in the past? Refactoring RadioGroup setter code? $\mathbb R = X^2$ as a Cartesian product I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism? I'm using iPhone 6s 16 GB. My storage isn't full but still I can't download anything. Every time I wanted to start a download it keeps telling:  app name* could not be download at this time Migrating to full Ubuntu after dual installation on WinXP I previously ran 12.04 alongside WinXP to see how it performed on my mom's older laptop.  i also migrated her email to Thunderbird. If I now convert from dual WinXP/Ubuntu to only run Ubuntu will things like her Thunderbird emails be transferred as part of the Ubuntu full install? Or must I do something else? I am fairly tech aware, but not a full IT boffin! Toggle Switch to swap the connections of two buttons I am trying to mod my arcade stick so I can switch two commonly swapped buttons on the controller.  Essentially they are simple 2 terminal momentary switches connected directly to the controller PCB. I would like to switch their responsibilities, so in my case the Back button and the LB button on the XBox pcb I would like to swap so the Back actuates the LB button on the PCB and the LB actuates the Back button.  My assumption is that I would need a switch with 8 terminals, 4 in 4 out.  What type of switch am I looking for? How to show $f'(0)$ exist and is equal to $1$? Is it possible to determine the exact date and time (if possible) of creation of an Apple ID?   Is there any official or unofficial way to establish this if you don't have the initial sign up information recorded? Would time freeze if you could travel at the speed of light? I'm looking to build a collection of fantasy miniatures, preferably pewter - but I don't want to buy them one at a time from my local hobby store... Much too expensive, and they don't often carry anything I'm interested in buying, which makes it a long term venture as well. So I'm looking for ideas on how to obtain many miniatures at once, preferably for cheap. D&amp;D Miniatures aren't a good source for me as they're collectibles, and the prices on those are also prohibitive.  By the way - they don't have to be pewter, that's just my preference. =) SQL JOIN - WHERE clause vs. ON clause After reading it, this is not a duplicate of . The answer may be related (or even the same) but the question is different.    What is the difference and what should go in each?  If I understand the theory correctly, the query optimizer should be able to use both interchangeably. How could I select one vertex from my mesh if I know ID of that vertex? I enabled bpy.app.debug = True and I can see all IDs but cant find a specific one. I have a 1-year residence permit from one Schengen country that is now expiring. Upon its expiration, I would like to get my 90/180 days validity as a tourist to stay, or return here as soon as possible.  I understand that the tourist short-stay scheme is based on the 90/180 days "counter". But what happens if my previous stay was not tourism but residence (as searching year). Wouldn't that mean that I can quickly exit and re-enter the Schengen area to activate my tourist stay?   Also, is there a way to change this purpose of stay without exiting the Schengen area? Can't move files to trash without any permission issues This is not like one of the cases addressed before - I've checked them all.  I've suddenly begun getting that message "... can't be put in the trash. Do you want to delete it immediately?" this morning in my Home directory only.  I have a Home partition (ext4, mounted properly) with symlinks for user folders like Downloads, Documents, etc. to folders with the same names in a Data partition (ntfs, mounted properly).  For example, I cannot move a file in Downloads folder to Trash when I try via Home, but no problem if I try via Data partition, although the file in question has the same ownership and permissions at both locations.  Also no problem at all with files in my Home folder, including sub-folders like Public that are not symlinked to my Data partition.  Interesting puzzle for me...  PS: Just upgraded to 15.04 but I don't think it matters... On second thought, it might perhaps be something to do with upgrading from Nautilus 3.10 to 3.14?  PPS: This bug has been finally resolved some time later in version 16.10, and no longer exists under Ubuntu 17.04. What is color temperature and how does it affect my photography? What is a word that means "someone who pretends to be your friend but is actually your enemy?" What do you call someone who pretends to be your friend but is actually your enemy?  A friend suggested spy for me, but that does not nearly describe the word I need for an English project. The character is very good at manipulating how people see him. He's a good liar. And pretends to be your friend but actually is your enemy. What is the best place for storing binary files that are related to data in your database? Should you:   Store in the database with a blob Store on the filesystem with a link in the database Store in the filesystem but rename to a hash of the contents and store the hash on the database Something I've not thought of   The advantages of (1) are (among others) that atomicity of transactions is preserved. The cost is that you might dramatically increase storage (and associated streaming/backup) requirements  The goal of (3) is to preserve atomicity to some degree - if you can enforce that the filesystem you are writing to does not allow files to be changed or deleted, and always has the correct hash as filename. The idea would be to write the file to the filesystem before permitting the insert/update referencing the hash - if this transaction fails after the filesystem write but before the database DML, that is fine because the filesystem is 'faking' being the repository of all possible files and hashes - it doesn't matter if there are some files in there that are not being pointed to (and you could clean them up periodically if you are careful)  EDIT:  It looks like some RDBMSs have this covered in their individual ways - I'd be interested to know how others do it - and particularly in a solution for postgres