here $d(n)$ counts the number of positive divisors of $n$.  I've tried 2 things:   Using Bell series. But then again it just showed me that the bell series of the square of a function is not the square of the bell series of the function. Expand it all and use combinatorics. If: $$n = \prod_{i=1}^r p_i^{\alpha_i} $$   then if $d \vert n$:  $$d = \prod_{i=1}^rp_i^{\beta_i}, 0\leq \beta_i \leq \alpha_i$$  so  $$\sum_{t \vert n}d(t) = \sum \prod_{i=1}^r (\beta_i + 1)$$ for all possible combitions of $(\beta_1,\beta_2,...,\beta_r)$ but I couldn't develop it very much
Prove that if $n \in \mathbb{N}$, then $$\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$$ where $d(n)$ is the divisor function. I have know only information is $d(n)=\sum_{d|n}{1}$.
My family PC has Windows Vista and lately it has became unusable, having strange errors and taking hours to do anything at all, so I'm installing Ubuntu 11.10 alongside it.  I went into GParted to resize the Vista partition but found that it had an exclamation point in a red circle next to it:   I tried to resize it but the Resize/Move button was grayed out.  I thought that this might have to do with the exclamation point so I looked at the information and found a load of errors:   What does the exclamation point mean?  How can I resize my partition?  UPDATE 1  I booted into Safe Mode with Command Prompt in Vista as suggested by eug and ranchkdsk c: /F /R (/F Fixed errors on the disk, /R Locates bad sectors and recovers readable information) and got:  Chkdsk cannot run because the volume is in use by another process.  Would you like to schedule this volume to be checked the next time the system restarts? (Y/)   When I reboot it doesn't run chkdsk, I'm selecting Safe Mode with Command Prompt again because I can't boot normally.  How can I stop it being used?  UPDATE 2  I tried doing what Luis suggested.  Here is the terminal output:  ubuntu@ubuntu:~$ sudo ntfsfix /dev/sda3 Mounting volume... OK Processing of $MFT and $MFTMirr completed successfully. NTFS volume version is 3.1. NTFS partition /dev/sda3 was processed successfully. ubuntu@ubuntu:~$ sudo apt-get install ntfsprogs Reading package lists... Done Building dependency tree        Reading state information... Done Note, selecting 'ntfs-3g' instead of 'ntfsprogs' ntfs-3g is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   I still cannot resize the partition.  Ntfsfix doesn't look like it did anything and ntfsprogs is already installed.  I can access the hard drive and do backups with the Ubuntu Live CD which I'm using now.  Any other suggestions?  UPDATE 3  I attempted to use the ntfsresize command but I got an error:  ERROR: Volume is scheduled for check. Run chkdsk /f and please try again, or see option -f.   This is because I ran chkdsk in Vista Safe Mode with Command Prompt and it couldn't do it because it was in use but offered to do it at the next reboot.  I reboot and I get to the menu to the menu to choose whether to Start Windows Normally, Safe Mode, etc instead of checking the disk.  Is there any way I can unschedule?  EDIT  I just found out that I could use the -f option to force it.  It failed and told me to run chkdsk /f in Windows and then reboot twice.  But the chkdsk program is in the partition I am trying to check so it says it is in use. Extraneous solution from substituting in equations I came across this example of how you could end up with an extraneous solution but I was wondering how it arose. We have the equation: $$x^2+x+1=0 $$ Since x=0 does not satisfy the equation, you can divide by x on both sides which yields: $$x+1+\frac{1}{x}=0$$ which is equivalent to our first equation. From our first eqution we can conclude that: $$-x^2=x+1$$ We now substitute that into the second equation to get:$$x^2=\frac{1}{x}$$ which results in $$x^3=1$$ which is equivalent to our previous equation since x cannot be 0. However, one solution from our last equation is x=1, which is not a solution to our original equation. I have a vague idea that it may have to do with the fact that you get a cubic equation and you began with a quadratic, and that steps imply the following and not vice versa, but can you provide a very detailed answer as to why it arises? Can you please provide more examples? How to typeset a file path? Packaging a bike for flying with a bike bag Invert "if" statement to reduce nesting When I ran  on my code, for example:      if (some condition)     {         Some code...                 }   ReSharper gave me the above warning (Invert "if" statement to reduce nesting), and suggested the following correction:     if (!some condition) return;    Some code...   I would like to understand why that's better. I always thought that using "return" in the middle of a method problematic, somewhat like "goto". Firefox running with rare arguments I got this output from ps aux|grep firefox command:  $ ps aux|grep firefox auraham   5443 17.3 11.0 2600628 426484 ?      Sl   18:10   0:54 /usr/lib/firefox/firefox auraham   5496 24.0 17.9 2558108 691700 ?      Sl   18:10   1:15 /usr/lib/firefox/firefox -contentproc -childID 1 -isForBrowser -intPrefs 5:50|6:-1|18:0|28:1000|33:20|34:10|43:128|44:10000|49:0|51:400|52:1|53:0|54:0|59:0|60:120|61:120|91:2|92:1|106:5000|117:0|119:0|130:10000|155:24|156:32768|158:0|159:0|167:5|171:1048576|172:100|173:5000|175:600|176:4|177:1|186:2|200:60000| -boolPrefs 1:0|2:0|4:0|26:1|27:1|30:0|35:1|36:0|37:0|38:0|41:1|42:1|45:0|46:0|47:0|48:0|50:0|55:1|56:1|57:0|58:1|62:1|63:1|64:0|65:1|66:1|67:0|68:1|71:0|72:0|75:1|76:1|80:1|81:1|82:1|83:0|85:0|86:0|87:1|88:0|93:1|94:0|100:0|105:0|108:1|109:0|111:1|112:1|114:1|118:0|120:0|122:0|124:1|125:1|131:0|132:0|133:1|135:0|146:0|153:0|154:0|157:1|160:0|162:1|164:1|165:0|170:0|174:1|179:0|180:0|181:0|182:1|183:0|184:0|185:1|188:1|192:0|193:0|194:1|195:1|196:0|197:1|198:1|199:1|201:0|202:0|204:0|212:1|213:1|214:0|215:0|216:0| -stringPrefs 3:7;release|134:3;1.0|151:332;  ¼½¾ǃː??։֊׃״؉؊٪۔܁܂܃܄ᅟᅠ᜵           ???‐’․‧??????? ‹›⁁⁄⁒ ⅓⅔⅕⅖⅗⅘⅙⅚?⅜⅝⅞⅟∕∶⎮╱⧶⧸⫻⫽⿰⿱⿲⿳⿴⿵⿶⿷⿸⿹⿺⿻　。〔〕〳゠ㅤ㈝㈞㎮㎯㏆㏟꞉︔︕︿﹝﹞?．／｡ﾠ???￼�|152:8;moderate| -greomni /usr/lib/firefox/omni.ja -appomni /usr/lib/firefox/browser/omni.ja -appdir /usr/lib/firefox/browser 5443 true tab auraham   5694  0.0  0.0  15444  1092 pts/2    S+   18:15   0:00 grep --color=auto firefox   According to , this error occurs after firefox crashes. However, every time I run Firefox again, that large line of arguments appears too.  I already try these things:   Run Firefox with addons disabled Reinstall Firefox (sudo apt-get purge firefox; sudo apt-get install firefox)   However, it did not work.  Is it possible to remove such a line of arguments? This is not a duplicate question because I am looking for an explanation directed to a general audience as to the mistakes (if any) in  (reproduced below). (Numberphile is a YouTube channel devoted to pop-math and this particular video has garnered over 3m views.)   By general audience, I mean the same sort of audience as the millions who watch Numberphile. (Which would mean, ideally, making little or no mention of things that a general audience will never have heard of - e.g. Riemann zeta functions, analytic continuations, Casimir forces; and avoiding tactics like appealing to the fact that physicists and other clever people use it in string theory, so therefore it must be correct.)  Numberphile's Proof.  The proof proceeds by evaluating each of the following:  $S_1 = 1 - 1 + 1 - 1 + 1 - 1 + \ldots$   $S_2 = 1 - 2 + 3 - 4 + \ldots $   $S = 1 + 2 + 3 + 4 + \ldots $   "Now the first one is really easy to evaluate ... You stop this at any point. If you stop it at an odd point, you're going to get the answer $1$. If you stop it at an even point, you get the answer $0$. Clearly, that's obvious, right? ... So what number are we going to attach to this infinite sum? Do we stop at an odd or an even point? We don't know, so we take the average of the two. So the answer's a half."   Next:  $S_2 \ \ = 1 - 2 + 3 - 4 + \cdots$  $S_2 \ \ = \ \ \ \ \ \ \ 1 - 2 + 3 - 4 + \cdots$  Adding the above two lines, we get:  $2S_2 = 1 - 1 + 1 - 1 + \cdots$  Therefore, $2S_2=S_1=\frac{1}{2}$ and so $S_2=\frac{1}{4}$.  Finally, take  \begin{align} S - S_2 &amp; = 1 + 2 + 3 + 4 + \cdots           \\ &amp; - (1 - 2 + 3 - 4 + \cdots)          \\ &amp; =   0 + 4 + 0 + 8 + \cdots           \\ &amp; = 4 + 8 + 12 + \cdots           \\ &amp; = 4S \end{align}  Hence $-S_2=3S$ or $-\frac{1}{4}=3S$.  And so $S=-\frac{1}{12}$. $\blacksquare$ Banach spaces and their unit sphere Let $X$ be a normed vector space.   Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete if and only if $X$ is Banach.     I have found the first part in a book, but I do not how to prove part 2. Thanx regards How to ensure all data has been physically written to disk? I understand that .NET FileStream's Flush method only writes the current buffer to disk, but dependent on Windows' disk driver and the hard disk firmware this is no guarantee that the data is actually physically written to disk.  Is there a .NET or Win32 method that can give me this guarantee? So if there is power loss one nanosecond after the call to this method comes back, I can still be sure that everything is OK? Can I take clear night sky photographs with my smartphone? is it possible to take clear night sky shots with stars/ planets visible from Redmi Note 4? what settings do I use? when I try to take it just comes black, unless on full moon day with HDR, though stars like Orion's belt won't get captured. Would using a telescope make celestial objects much clear to capture? Can't get armature to move mesh! Edit answer draft is broken The Edit Answer Draft button is broken.    Press Back with text in an answer box. The Edit answer draft bar does not appear.     Seems like an onBackPressed() issue.  Also:   Press ← with text in the answer box The bar appears. Click the bar The answer box is blank.   Seems like the answer isn't properly carried between the Fragments.  Either way, you lost everything you wrote.  Can somebody test this? And especially without a confirm dialog, it is a brush of the finger to lose a long post of code or something.   I hope this report will help because I like the app (I'd say 4/5 stars) but the bugs are preventing a solid 5 stars.   Stack Exchange 1.0.73 Samsung Galaxy S II Skyrocket I727 Android 4.1.2 From where do the permanent magnets get energy from? I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently.  Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy? Determining coordinates of a SHP file i have a SHP file of a building converted from a DXF file. This SHP file is unprojected and have no specific coordinates (origin 0,0). I find WGS84 coordinates (lat, long) of this building from Google Maps. My aim is to replace these coordinates to the SHP file on QGIS.  Could you explain the process step by step?  (I read lots of similar questions, about affine transformation. But i don't know the paramaters.)    What i ask is how to make this transformation in QGIS (step by step)?   SHP file coordinates (not projected)  Scale: 1:142.670.713  X1,Y1: 16.70824, 253.74534  X2,Y2: 1.03521,-6.34918  X3,Y3: 244.00248,-7.49189  Target WGS84 coordinates   X1,Y1: 41.047773, 28.896241  X2,Y2: 41.045452, 28.895929   X3,Y3: 41.045436, 28.899039    I need a second vector layer with coordinates. I have just a SHP file of a building converted from a DXF file. That is not projected. I know where this building is, and can find the WGS84 coordinates from Google Maps. My aim is add WGS84 coordinates to this SHP file and export to replace the building on Google Maps.  My Questions are:  1) I think my SHP file has measurements in meters and unprojected. I have WGS84 lat long coordinates from Google Maps. How can i project this SHP file, for being ready to transformation? Right Click -> Set Layer CRS -> Choose WGS84 EPSG: 4326. Is this enough?  2) qgsAffine plugin just need transformation matrix paramaters. But the answer of @Jochen Schwarze is a bit complex for me. Is there a online calculator? (Just give the source XYs and target XYs and calculate the parameters.)    I think, I have a projection problem before affine transformation. I have a DXF file of a building from AutoCAD. I saved this DXF file as a SHP file in QGIS. But this SHP file is converted from a plain DXF file and is not referenced/projected, it is drawn from coordinates 0,0 in AutoCAD in meters. And the drawing has meters, WGS84 has decimal degrees.  When I set Layer CRS as WGS84 in QGIS, the coordinates don't change.  How can I project this SHP file as WGS84 in QGIS? SQL Server 2016 memory optimised table: Precreated files constantly growing I have a memory optimised table in SQL Server 2016, with file durability enabled. The table is only 100MB. SQL Server is constantly creating new files on disk, which start off at a few hundred MB, but then grow constantly until the drive runs out of space then they clear down and the process starts again. The files are growing at about 900MB an hour. Running the SQL below shows that all the space is in Precreated Data - so it's not even actually storing large amounts of data, just allocating space to use later.  SELECT state_desc, file_type_desc, COUNT(*) AS [count],    SUM(file_size_in_bytes) / 1024 / 1024 AS [on-disk size MB]    FROM sys.dm_db_xtp_checkpoint_files   GROUP BY state, state_desc, file_type, file_type_desc   ORDER BY state, file_type     File growth on the file group is set to unlimited, so I'm assuming I can set a value on this and stop it growing, but I can't find anything to say why it's allocating so much space in the first place. The data in the table does change very rapidly (a lot of inserts and deletes as the data is constantly processed, but it the data that isn't Precreated never gets over 4GB.  Is there a setting in SQL Server that controls this file precreation? Is it safe to set a limit on the autogrowth for the file group? Can we make it more plain to recruiters that we're not open to relocation? The preference for being open to relocation is a checkbox, and if you don't check it, nothing about relocation shows up in your profile.  Evidently recruiters can't understand that a lack of "I am open to relocation" is the same thing as "I am not open to relocation", and so every single contact I've gotten through Careers is for a job that I would have to relocate for.  Can we make it more obvious to the recruiters (especially ones from Amazon) that we have not checked that box on purpose? How do I find this limit: $\lim_{n\to\infty} \left[ n-{n\over e}\left(1+{1\over n}\right)^n \right]$? wrappers and JavaScript pop-up? Use Case:  What i want is a list of records, and allow this user to choose 2 of those records, then click a button to go to a new VF page, if the user selects more than 2, i want him to get a notification of some kind saying you need to select 2 records, I do not want the page refeshed though.  Strategy:  I'm creating a wrapper class for the records like this:  and i have that part going by creating a custonm controller... I need the notification part working.   Problem:  I don't know how to call methods from the customController in javascript Also, i do not want to refresh the entire page. I just want them to go back to where they were. I was thinking a pop-up notification but I'm flexible.. What is "offset null" in IC 741? What is Offset null in 1st and 5th pin in IC 741 (Op-Amp)? Why it is used, though it is not used in many circuits? Give me explanation regarding the offset null! Why offset voltage was formed in IC 741? Does one need to adjust for multiple comparisons when using MANOVA? There are 10 dependent variables and 2 groups. When I wanted to compare 10 dependent variables between two groups, I performed MANOVA.  I wonder if there is a multiple comparison problem in that case. Should I have performed Bonferroni correction because of 10 dependent variables? Or is Bonferroni useless because MANOVA just does analysis once?