Change periods on x-axis in time series?
xticks steps using Pgfplots / dateplot
Is true that $\sum_{k=1}^n\frac{[kx]}{k}\leq[nx]$, for every $x\in\mathbb{R}$ and for every $n\in\mathbb{Z}^+$? Is true that $\displaystyle\sum_{k=1}^n\dfrac{[kx]}{k}\leq[nx]$, for every $x\in\mathbb{R}$ and for every $n\in\mathbb{Z}^+$?  My work:  Let $x\in\mathbb{R}$ be given. For every $k=1,\dots,n$, define $i_k\in\{0,1,\dots,k-1\}$ such that  $$ [x]+\frac{i_k}{k}\leq x&lt;[x]+\frac{i_k+1}{k}\hspace{20mm} (*). $$  On one hand, we have that $[kx]=k[x]+i_k$ for every $k=1,\dots,n$.  So  $\displaystyle\sum_{k=1}^n\dfrac{[kx]}{k}=n[x]+\displaystyle\sum_{k=1}^n\dfrac{i_k}{k}=[nx]+\displaystyle\sum_{k=1}^n\dfrac{i_k}{k}-i_n$  Thus  $$\displaystyle\sum_{k=1}^n\dfrac{[kx]}{k}\leq[nx]\Leftrightarrow\displaystyle\sum_{k=1}^n\dfrac{i_k}{k}\leq i_n\,.$$  On the other hand, from $(*)$ is direct that $i_k=[k(x-[x])]$. Thus  $$\displaystyle\sum_{k=1}^n\dfrac{[kx]}{k}\leq[nx]\Leftrightarrow\displaystyle\sum_{k=1}^n\dfrac{[k(x-[x])]}{k}\leq [n(x-[x])]\,.$$  In other words, we can assume that $x\in[0,1)$.   Now the questions is:  Is true that $\displaystyle\sum_{k=1}^n\dfrac{[kx]}{k}\leq[nx]$, for every $x\in[0,1)$ and for every $n\in\mathbb{Z}^+$?  Any simpler approach? Any thoughts? Even when I start a brand new file in Blender, I only get four video output file options: AVI raw, AVI JPEG, Frame Server, FFMpeg Video.  I am running version 2.78.5 (I'm using the beta so that I can use denoising.  How can I get access to all the other video outputs other people have? Like this: Can Recovery Mode install OS X onto a blank SSD? I want to install a new Solid State Drive (SSD) into the Mac. Can I just build in the SSD, start the MacBook in Recovery Mode and install OS X from there on?  After reading a lot on the internet I still have some doubts…  My HDD crashed, I can't repair it using disk utility or the fsck methods as mentioned on the internet.  According to  it should be possible.  I know how to replace the HDD with the SSD. I would go for the Samsung 850 EVO SSD - 500GB. My Mac is a MacBook Pro Mid 2012, model A1278. How do I build a house for my NPCs? I read on the wiki that people will move in under the following conditions:   You do something to please them An empty house is available with a light source, table, and chair (also with walls).   I have a house which is about 7x8, then a roof space above that; about half the area. I could put furniture up there too, but will anyone move in? Can anyone provide some pseudo code for a roulette selection function? How would I implement this:    I don't really understand how to read this math notation. I never took any probability or statistics. How to insert Bengali text in IEEE conference paper? I am writing a paper for an IEEE conference using IEEEtran.cls. The primary language of the paper is English but my work deals with Bengali language. I want to know how do I insert some Bengali words such as পারে in the paper. Hi is it possible to exclude traffic from a certain country in google analytics?  If yes, how do I do it? Is it possible and if so how can you pass flags to the ShareLaTeX compiler which one would have passed as parameters to the LaTeX compiler?  I am using the minted package and need to pass the -shell-escape flag. Sum of Cosine/Exponential What are the differences between "=" and "<-" assignment operators in R? When I read about , I see that they should appear as solution to prevent  and  but when I actually try to write a code which would use Lightning promises it still doesn't look like synchronous code, it still resembles both pyramid of doom and callback hell.  main: function(component, helper) {     helper.promise1(component, helper).then(         $A.getCallback(function(record) {             return helper.promise2(component, helper, record).then(                 $A.getCallback(function(){                     console.log("Success");                     helper.showToastMessage(component, "Great Success!!!!")                 }),                  $A.getCallback(function(errorMessage){                     console.log("Rejected by second promise2");                     helper.showToastMessage(component, "Failure " + errorMessage);                 })              )         }),          $A.getCallback(function(errors){             console.log("Rejected by first promise1");             let message = errors &amp;&amp; Array.isArray(errors) &amp;&amp; errors.length &gt; 0 ? errors[0].message :'Unknown error';             helper.showToastMessage(component, "Failure " + message);         })      ) }   Is that possible to refactor this code to look more like synchronous code and without callback hell or pyramid of doom? How to get Total Pupulation From Tables In Sqlserver? selectively make contact groups silent MISSISSIPPI problem How many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's?  I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and the number of arrangements will be $10!/(2!2!4!)$, this is not the correct answer, but what is wrong with my reasoning.  (the correct answer is $11!/(4!4!2!)-7!/(4!2!)\binom{8}{4}$, I understand the solution) Draw integers independently & uniformly at random from 1 to $N$ using fair d6? I wish to draw integers from 1 to some specific $N$ by rolling some number of fair six-sided dice (d6). A good answer will explain why its method produces uniform and independent integers.  As an illustrative example, it would be helpful to explain how a solution works for the case of $N=150$.  Furthermore, I wish for the procedure to be as efficient as possible: roll the least number of d6 on average for each number generated.  Conversions from  to decimal are permissible.    . Is there a way to view Google Chrome browsing history past three months ago? I'm trying to find a page I looked at months ago but it looks like it's disappeared off the face of the planet. I thought I would have better luck looking through my Google Chrome browsing history, but I can't see history past September. Can I force Google Chrome to display history further into the past? Can I retrieve the history manually from a directory? Is the history data gone? I noticed that my American friends tend to use the past perfect form when they use the verb mention in its past form, almost despite of the context of the sentence. I hear them say  "you had mentioned" a lot more often even in situations where the use of simple past would have been perfectly fine. Have you noticed it too and is it one of those unwritten rules, or is there a more formal explanation to it?  It seems like there was more verbs like that, but I am having a hard time remembering it now. If I do, I will edit the post and likely change the title to accommodate other verbs. How do I serialize a C# anonymous type to a JSON string? I'm attempting to use the following code to serialize an anonymous type to JSON:  var serializer = new DataContractJsonSerializer(thing.GetType()); var ms = new MemoryStream(); serializer.WriteObject(ms, thing); var json = Encoding.Default.GetString(ms.ToArray());    However, I get the following exception when this is executed:     Type   '&lt;>f__AnonymousType1`3[System.Int32,System.Int32,System.Object[]]'   cannot be serialized. Consider marking   it with the DataContractAttribute   attribute, and marking all of its   members you want serialized with the   DataMemberAttribute attribute.  See   the Microsoft .NET Framework   documentation for other supported   types.   I can't apply attributes to an anonymous type (as far as I know).  Is there another way to do this serialization or am I missing something? I would like to create a md5 checksum list for all files in a directory.  I want to cat filename | md5sum &gt; ouptput.txt. I want to do this in 1 step for all files in my directory.   Any assistance would be great. I want to print the floating point number with exactly two significant digits in bash (maybe using a common tool like awk, bc, dc, perl etc.).  Examples:   76543 should be printed as 76000 0.0076543 should be printed as 0.0076   In both cases the significant digits are 7 and 6. I have read some answers for similar problems like:      but the answers focus on limiting the number of decimal places (eg. bc command with scale=2 or printf command with %.2f) instead of significant digits.  Is there an easy way to format the number with exactly 2 significant digits or do I have to write my own function?