help proving an binomial identity about catalan numbers
Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths
I can't get to the bottom of this error, because when the debugger is attached, it does not seem to occur.  Collection was modified; enumeration operation may not execute  Below is the code. This is a WCF server in a Windows service. The method NotifySubscribers() is called by the service whenever there is a data event (at random intervals, but not very often - about 800 times per day). When a Windows Forms client subscribes, the subscriber ID is added to the subscribers dictionary, and when the client unsubscribes, it is deleted from the dictionary. The error happens when (or after) a client unsubscribes. It appears that the next time the NotifySubscribers() method is called, the foreach() loop fails with the error in the subject line. The method writes the error into the application log as shown in the code below. When a debugger is attached and a client unsubscribes, the code executes fine. Do you see a problem with this code? Do I need to make the dictionary thread-safe? [ServiceBehavior(InstanceContextMode=InstanceContextMode.Single)] public class SubscriptionServer : ISubscriptionServer {     private static IDictionary&lt;Guid, Subscriber&gt; subscribers;      public SubscriptionServer()     {                     subscribers = new Dictionary&lt;Guid, Subscriber&gt;();     }      public void NotifySubscribers(DataRecord sr)     {         foreach(Subscriber s in subscribers.Values)         {             try             {                 s.Callback.SignalData(sr);             }             catch (Exception e)             {                 DCS.WriteToApplicationLog(e.Message,                    System.Diagnostics.EventLogEntryType.Error);                  UnsubscribeEvent(s.ClientId);             }         }     }          public Guid SubscribeEvent(string clientDescription)     {         Subscriber subscriber = new Subscriber();         subscriber.Callback = OperationContext.Current.                 GetCallbackChannel&lt;IDCSCallback&gt;();          subscribers.Add(subscriber.ClientId, subscriber);                  return subscriber.ClientId;     }      public void UnsubscribeEvent(Guid clientId)     {         try         {             subscribers.Remove(clientId);         }         catch(Exception e)         {             System.Diagnostics.Debug.WriteLine(&quot;Unsubscribe Error &quot; +                      e.Message);         }     } } Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists? Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists ? If it does then what is the value ?   I don't think even L'Hospital's rule can be applied . Please help . Thanks in advance Our GM won't let us affect the story; the more I play the less fun I have. What do I do? I was wondering if anyone could give me some advice or anything on a GM situation.   My current GM is running a campaign and it is very video game/anime like. It started out super fun and the deeper we get the less fun I have. The reason is, it feels like GM is telling us a story but we cannot interact or change anything during his, "cut scenes". No matter what any player does he just says, "No it didn't work," or, "Yeah you do that. See if you roll a 20." We also can't damage any boss he has until we reach the point in the story that he has planned, so our interactions we have with villains don't matter.  I feel like he is making us sit there and listen to an anime he has come up with in his head. Since the players can't change or react to the key situations. I find my self sitting there thinking, why would I react, it doesn't matter.  It's not just me either. Another player feels the same way. I am going to talk to my other fellow players soon about it and see how they feel.  Side note: the GM as a person is very defensive and I am afraid to confront him about it because I know he will throw a fit. We have also caught him straight lying about abilities and stats on his own PC in other campaigns, and he doesn't take to kindly when you call him out.  He is new to GMing. This is only the second game I have played with him as a GM. The first one had one session the he just never wrote more for.  Is there anything I can do to maybe get him to more lenient with his story? I really enjoy the story for the most part and if he would just let our characters do things and give us a chance at stopping his villians or letting us change the story based on our actions, everything would be cool. Switching from one area of graduate study to another? Is it possible to switch from one area of graduate study to another in US universities? For example, suppose someone has enrolled in a computer science phd program. Can he switch over to math(or physics) phd program in the same school later?(or say from applied mathematics to pure mathematics?)  What are the steps for doing for doing that? How can I get an underbrace and an overbrace to partially overlap in an equation? Sometimes I want to describe portions of an equation using \underbrace and \overbrace in such a way that the portions are not completely distinct, for example:          /-------\ x = A + B + C + D + E     \-------/   \---/      What is the best way to do this? If asking a question or giving instruction towards a group of objects where the number is unknown (and possibly could be one) what is the correct way to phrase the sentence? Which of the following sentences are more correct:     Has there been any studies regarding this and if so what has been the   conclusion?   OR     Has there been any studies regarding this and if so what have been the   conclusions? Console freezes on auto-complete of /mnt/folder (CIFS) share I have a Ubuntu 14.04 server that will lock up after an undetermined amount of time when you attempt to tab-auto-complete the path of a CIFS share in the /mnt directory. I mount the share and it seems to work fine for a uncertain amount of time. I can do ls or run du on folders within the mounted share. Then I will go to tab out a folder path and my console locks up. I can start a new session but it will lock as soon as I try to mess with the mounted share; this even happens to other users if I have them try as well. I have no idea why this is happening? I've never seen this issue. latex : Exclude draftwatermark from specific pages I am using draftwatermark package. Is there any option to exclude watermark from some specific pages? Say no watermark from pg 3-4. I can not use xwatermark package. It conflicts with the geometry package of my document.  MWE:  \documentclass[12pt,a4paper]{exam} \usepackage{draftwatermark}  \usepackage{lipsum}   \SetWatermarkAngle{45}  \SetWatermarkLightness{.8}  \SetWatermarkFontSize{10cm}  \SetWatermarkScale{5}  \SetWatermarkText{\tt{12345}}  \begin{document}  \lipsum[1-20]  \end{document} Is a Schengen visa/transit visa required for a transit flight through two Schengen airports? How to get back to a site when viewing one of its questions When viewing a question on a site which I don't use (I never registered on it), I can't figure out how to get back to the list of questions of the site.  See this example  from the Stack Exchange Android app. Where should I click to see the list of questions? (I chose Mathematics Educators because it's the most recent site, it is very likely that most users don't have an account on this website)  Here is a random example:    When viewing a question, I don't know how to view the list of questions. This site can't be found in the sidebar since it's not in my Your Sites list. The top menu containing the site's title only opens the sidebar. I have this MySQL query:  SELECT DAYOFYEAR(`date`)  AS d, COUNT(*)  FROM  `orders`  WHERE  `hasPaid` &gt; 0 GROUP  BY d ORDER  BY d   Which returns something like this:  d  | COUNT(*) | 20 |  5       | 21 |  7       | 22 | 12       | 23 |  4       |   What I'd really like is another column on the end to show the running total:  d  | COUNT(*) | ??? | 20 |  5       |   5 | 21 |  7       |  12 | 22 | 12       |  24 | 23 |  4       |  28 |   Is this possible? Ubuntu 20.04 on DualBoot fail 90% of tries - Initramfs unpacking failed: Decoding failed Help + Noun + Gerund or Infinitive P.d.f of the absolute value of a normally distributed variable I came across this question as an exercise, had a brief idea, but didn't know how to proceed.     Let $X \sim N(0, 1)$. What is the p.d.f of $|X|$ ?   I know the final p.d.f looks just like the right half of the original pdf, but extended vertically for a factor of 2.  Could someone please show mathematically what the p.d.f of variable |X| is going to be, and explain briefly? Thank you. Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$ Look at the profile page of , who has just "lost" 100k reputation and all their bronze badges, because of this broken design.  Shouldn't the primary usercard be kept intact no matter how other elements grow?  P.S. So many design bugs these days, did SE do a large CSS refactory?  P.P.S  that squeezes the right pane ( to reproduce) create mesh then add vertices to it in python Cron Job not working in plugin I have written a plugin that will convert posts to excel format and then email the .xls to an email id. I can make it work with when function is declared in functions.php but does not work with function defined in plugin file.   if ( ! wp_next_scheduled( 'xls_func_hook1' ) ) {   wp_schedule_event( time(), 'hourly', 'xls_func_hook1' ); }  add_action( 'xls_func_hook1', 'sendxls1' ); //senxls1 is a function in functions.php if ( ! wp_next_scheduled( 'xls_func_hook2' ) ) {    wp_schedule_event( time(), 'hourly', 'xls_func_hook2' ); } add_action( 'xls_func_hook2', 'export2excel' );   The full code is at MacBook Pro 2018 Random Restart > Previous shutdown cause: -20 As far as I can tell, it's something like this:  damage reduction = armor / (armor + 100)   Is this accurate?