Question on the Prime Number Theorem (the Tchebychev Function)
Estimating the integrated Tchebychev function and calculating its error
How to remove local (untracked) files from the current Git working tree Is it true that $||a|^p - |b|^p| \le C_p|a - b|^p$ for $1 < p < \infty$? If $a, b \in \mathbb{C}$, then we have the standard triangle inequality for the difference:  $$||a| - |b|| \le |a - b|.$$  I am wondering if this inequality generalizes to exponents greater that one.     My question is, for $1 &lt; p &lt; \infty$, does there exists a constant $C_p$ such that $||a|^p - |b|^p| \le C_p|a - b|^p$ for all $a, b \in \mathbb{C}$?   I am aware of the "standard" triangle inequality for $p &gt; 1$:  $$|a +b|^p \le 2^{p}(|a|^p + |b|^p),$$  and that $2^p$ may not be sharpest constant possible. If I try to use this estimate to prove, say, that $|a|^p - |b|^p \le C_p |a - b|^p$, I get stuck with an extra term that I'm not sure what to do with.  $$|a|^p - |b|^p \le 2^p|a -b|^p + (2^p -1)|b|^p.$$  A resolution on this matter is greatly appreciated. PlotLegends disappearing with list of functions Is it possible to make abstract classes in Python? Encopresis, I'm at my wits end? Does anybody have suggestions? Android: How to open a specific folder via Intent and show its content in a file browser? See code:   var file1 = "50.xsl"; var file2 = "30.doc"; getFileExtension(file1); //returns xsl getFileExtension(file2); //returns doc  function getFileExtension(filename) {     /*TODO*/ } Post Feedback link is broken I first noticed this a while back. But it has yet to resolve itself.  When I click on the  in the 10k tools, it hangs for a while.Then it gives me an error message: How to change from German to English in the File Manager in Ubuntu 15.10? My months and days in my File Manager are in German, how do I change that? On the photo you see 'Dez' : It means Dezember (German for December). I also saw one day Mittwoch (German for Wednesday). How do I change that German issue? This is only my File Manager and not completly but partially. Weird. How to find and replace a string by increasing its numerical part? My input file is,  ami21 ami65 ami67 ami66 ami88 ami76 ami29 ami55  ami54 ami32   Using a single command line I need the output as,  ami22 ami66 ami68 ami67 ami89 ami77 ami30 ami56 ami55 ami33   I used the command  awk -vRS=ami '{$0=$0+1;ORS=RT}++n' inputfile &gt; outputfile   but I got the outputfile as ami21ami65ami67ami66ami88ami76ami29ami55ami54ami32  i.e. all strings are written in same line and without space. Could anyone suggext me some better command line. Is a citizen of Canada required to apply for a US visitor visa after visiting Iran? Under new rules, non-US citizens from Visa Waiver Program countries who have visited Iraq, Syria, Iran, Sudan, Libya, Somalia, or Yemen are required to apply for a proper visa because they are no longer eligible for the VWP.   Does the above apply to citizens of Canada also? Canada is not a Visa Waiver Program country, but instead Canadian citizens can enter the US visa-free. If a Canadian citizen has visited Iran for example, is the entry procedure to the US any different?  I have looked through  but that page does not mention Canada. Corrupt Backupt GPT Table Is it possible to configure Windows 7 to use YYYY-MM-DD instead of DD/MM/YYYY as the date format? How? Where? Do save bonuses stack from multiple magic items in AL? I play in Adventurers League and have a character that has a +1 Ring of Protection, which gives +1 to AC and saves. I also have a Stone of Good Luck which gives a +1 to saves and ability checks.  If I equip both items and my base Dex save is +4, would it become +6 due to receiving both bonuses, or +5 with just one? I can't find much in the PHB or Google to answer this. Silencing "Your disk is almost full" notification After upgrading to macOS Sierra, I'm getting the notification that "Your disk is almost full. Save space by optimizing storage.":    The options appear to be to store my files in iCloud, automatically delete files, or manually delete files:    My problem/irritation is that I have 80GB free of my 440GB volume.  The question: is there a way to silence this notification in a (semi-)permanent fashion, or to change the threshold it uses for the notification? Floating point calculations in LaTeX? Weighting features prior to SVM I'm building an object detector using HOG features and linear SVM. Some of the regions of the object are more "distinctive" so I would like to give more weight to the features extracted from those regions. (e.g. imagine we want to detect between mugs and glasses, then the most "distinctive" part is the mug's handle)  How could I do this? For sure there should be a more intelligent way than just replicating the features of those "distinctive" regions. What are the pros and cons of engineered hardwood? Are there any downsides to engineered hardwood? I'm looking to install hardwood in dining room and kitchen areas. How do you properly use namespaces in C++? I come from a Java background, where packages are used, not namespaces. I'm used to putting classes that work together to form a complete object into packages, and then reusing them later from that package. But now I'm working in C++.  How do you use namespaces in C++? Do you create a single namespace for the entire application, or do you create namespaces for the major components? If so, how do you create objects from classes in other namespaces? When I have a query that checks if a column of type uniqueidentifer does not exist in a table that has a null value then I get no results back. If the subquery does not return a null it works fine and it only happens when using not in.   I know I can just do a not null check in my subquery, but I am curious why this does not work.  Query Example:  select a.guid from tableA a where a.guid not in (select b.guid from tableB AS b)   Working Test:  select 1 where newid() not in (select newid())   Broken Test:  select 1 where newid() not in (select null)