Getting the greatest common divisor of a sequence
Define the sequence $aₙ=20+n²$ and $dₙ=\gcd(aₙ , aₙ₊₁)$ Find with proof the values dₙ takes.
How can something finite become infinite? How can the universe become infinite in spatial extent if it started as a singularity, wouldn't it take infinite time to expand into an infinite universe? I'm using a class I've derived from DocumentPaginator (see below) to print simple (text only) reports from a WPF application. I've got it so that everything prints correctly, But how do I get it to do a print preview before printing? I have a feeling I need to use a DocumentViewer but I can't figure out how.  Here's my Paginator Class:  public class RowPaginator : DocumentPaginator {     private int rows;     private Size pageSize;     private int rowsPerPage;      public RowPaginator(int rows)     {         this.rows = rows;     }      public override DocumentPage GetPage(int pageNumber)     {         int currentRow = rowsPerPage * pageNumber;         int rowsToPrint = Math.Min(rowsPerPage, rows - (rowsPerPage * pageNumber - 1));         var page = new PageElementRenderer(pageNumber + 1, PageCount, currentRow, rowsToPrint)                        {                            Width = PageSize.Width,                            Height = PageSize.Height                        };         page.Measure(PageSize);         page.Arrange(new Rect(new Point(0, 0), PageSize));         return new DocumentPage(page);     }      public override bool IsPageCountValid { get { return true; } }      public override int PageCount { get { return (int)Math.Ceiling(this.rows / (double)this.rowsPerPage); } }      public override Size PageSize     {         get { return this.pageSize; }         set         {             this.pageSize = value;             this.rowsPerPage = PageElementRenderer.RowsPerPage(this.pageSize.Height);             if (rowsPerPage &lt;= 0)                 throw new InvalidOperationException("Page can't fit any rows!");         }     }      public override IDocumentPaginatorSource Source { get { return null; } } }   The PageElementRenderer is just a simple UserControl that displays the data (at the moment just a list of rows).  Here's how I use my Row Paginator  PrintDialog dialog = new PrintDialog(); if (dialog.ShowDialog() == true) {     var paginator = new RowPaginator(rowsToPrint) { PageSize = new Size(dialog.PrintableAreaWidth, dialog.PrintableAreaHeight) };      dialog.PrintDocument(paginator, "Rows Document"); }   Sorry for the code dump, but I didn't want to miss something relevant. How can I apply an effect when a player left-clicks/right-clicks with an item in hand? Identifying records in MC to delete I'm trying to delete a large number of records from SFMC to avoid being charged extra. I've run several different SQLs to help identify the records I don't need, but when I run the Contact Count report, it still shows approx 100K billable contacts and approx 60K distinct email addresses.  Here are the SQLs I've tried:   No Opens in past 5 days Leads/Contacts not in SFDC  SubscriberKey as Email Address Leads/Contacts where SubscriberKey is Empty Unspecified records (DE populated by MC Support)    These efforts reduced the count a good amount, but not as much as I was expecting.   Has anyone else gone through this process before and if so, what queries can you share that might help my case? How to intuitively understand eigenvalue and eigenvector? I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman.  Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to understand.I found that there are many application of eigenvalue and eigenvector in multivariate analysis. For example     In principal components, proportion of total population variance due   to kth principal component equal    $$\frac{\lambda_k}{\lambda_1+\lambda_2+...\lambda_k}$$   I think eigenvalue product corresponding eigenvector has same effect as the matrix product eigenvector geometrically.   I think my former understanding may be too naive so that I cannot find the link between eigenvalue and its application in principal components and others.   I know how to induce almost every step form the assumption to the result mathematically. I'd like to know how to intuitively or geometrically understand eigenvalue and eigenvector in the context of multivariate analysis(in linear algebra is also good).  Thank you! Cannot boot because: Kernel panic - not syncing: Attempted to kill init! This is what I've got. I cannot boot and I see the two of three little lights on the keyboard blinking all the time and the cooling is working on full power, almost.  I saw in other askubuntu topic suggestion for memory test. I did it. I didn't have any errors.  I am 99,9% sure it is not a hardware problem, because I have Windows 7 and I can boot it and it works just fine.  This problem came without any warning or something. I just turned off the laptop one day and at the other day I couldn't boot.   From recovery mode I get this: What is this Disney scifi movie from the late 80s/early 90s? Editing Subject in new Gmail For , the answer given on 7 Feb about how to edit the subject line in the new format of Gmail does not work for me.   There is no box on the left with a down arrow to options to reply, forward or edit subject in my version. There is a book on the right offering options to    Reply Reply to all Forward Filter message Print Delete Report spam Report phising Show original Message text garbled Translate message Mark answered    But, no option to edit subject My first issue was when I tried to apt-get update or apt-get upgrade. On upgrade I get the following error:  You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies: linux-image-server : Depends: linux-image-3.2.0-27-generic but it is not installed E: Unmet dependencies. Try using -f.   I tried running apt-get install -f and this was the output(after saying yes on prompt)  (Reading database ... 186183 files and directories currently installed.) Unpacking linux-image-3.2.0-27-generic (from .../linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb) ... Done. dpkg: error processing /var/cache/apt/archives/linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb (--unpack):  failed in write on buffer copy for backend dpkg-deb during `./boot/System.map-3.2.0-27-generic': No space left on device  No apport report written because the error message indicates a disk full error                                                                           dpkg-deb:    error: subprocess paste was killed by signal (Broken pipe) Examining /etc/kernel/postrm.d . run-parts: executing /etc/kernel/postrm.d/initramfs-tools 3.2.0-27-generic   /boot/vmlinuz-3.2.0-27-generic run-parts: executing /etc/kernel/postrm.d/zz-update-grub 3.2.0-27-generic /boot/vmlinuz-3.2.0-27-generic Errors were encountered while processing: /var/cache/apt/archives/linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   I tried running apt-get autoremove and it gives me the same error as apt-get upgrade.  When I run df, I get this for /boot:  /dev/sda1                    233191     230297         0 100% /boot   So, I read elsewhere that I should try to purge old kernels. I checked for what kernels I had with:  $ dpkg -l linux-image-\* | grep ^ii ii  linux-image-2.6.38-13-server  2.6.38-13.52  Linux kernel image for version 2.6.38 on x86_64 ii  linux-image-3.0.0-13-server   3.0.0-13.22   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-14-server   3.0.0-14.23   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-15-server   3.0.0-15.26   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-16-server   3.0.0-16.29   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-17-server   3.0.0-17.30   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.2.0-24-generic  3.2.0-24.39   Linux kernel image for version 3.2.0  on 64 bit x86 SMP ii  linux-image-3.2.0-25-generic  3.2.0-25.40   Linux kernel image for version 3.2.0  on 64 bit x86 SMP ii  linux-image-3.2.0-26-generic  3.2.0-26.41   Linux kernel image for version 3.2.0  on 64 bit x86 SMP   When I try to remove the oldest with this:  $ sudo apt-get purge linux-image-2.6.38-13-server Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: linux-image-server : Depends: linux-image-3.2.0-27-generic but it is not going to be     installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).   How can I free up or extend boot without messing up my installation? I'm very new to the GIS world, so my problem could be very stupid, but I'm going to try anyway.  Objective  Given a list of coordinates in longitude/latitude, stored as a .csv file, I want to create a buffer with distance of x km around those coordinates. (In case this is relevant, these coordinates are locations of the societies in the . These societies scatter all around the world.)  Problem  I'm only able to create buffers with distance in radial degrees, but I want to do them in (kilo)meters. I'm aware of the following Q&amp;A's, but following what's suggested therein doesn't seem to solve my problem:          What I have tried...   I start QGIS (v2.4), and Add delimited text layer  choose my .csv file and the x/y fields, click OK select WGS 84 as my CRS (also tried NAD83 but didn't make a difference), click OK right-click on layer and select Save As..., in the dialog box, choose the following and click OK:  add sccs_meter.shp as layer to the current project Select Vector > Geoprocessing Tools > Buffers > set buffer distance as 10   The result is a buffer with radius of approximately 1068km, which suggests that buffer distance is 10 degrees rather than 10 meters.  What did I do wrong here? Recursively delete empty directories in Windows How to draw tangent vectors and component vectors on a curve I am searching for a way to have a figure drawn using LaTeX. I am aware of some packages TikZ and PSTricks that could accomplish this, and have been tinkering around with them recently but do not have enough skill set at the moment to accomplish this drawing. Is there a way someone can accomplish the drawing, preferably with TikZ but could be any package your most comfortable with in a reduced amount of coding (non-advanced) to where I could understand what is happening from the code to the picture. If this is not possible, then I will have to study the package more and work with it until I understand the method (code) to create the graphic. The drawing is below.  Note: This is a rough sketch. The optimal output would have smooth quadratic curves and all quantities are vectors. I'm taking an intro course in Probability theory, and we have just defined expectation for a random variable as $E(X) = E(X^+) - E(X^-)$ if either of them is finite (extending the definition first from simple positive rv to positive rv, and then to all rvs, positive or not).  Then I came across this result for the expectation of a positive random variable. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X \ge 0$ be a positive rv:  $$ E(X) = \int_{\Omega} X dP = \int_\Omega \int_0^\infty \mathbf{1}_{\{X &gt; t\}} dt  \, d P = \int_{0}^\infty P(X &gt; t) dt. $$ Actually, I'm familiar with this result, and the proof I've seen before involves converting $\int^\infty_0 P(X &gt; t) \, dt$ to a double integral. What I'm struggling with is the step $ \int_{\Omega} X dP = \int_\Omega \int_0^\infty \mathbf{1}_{\{X &gt; t\}} dt  \, d P $, and also the formal definition of an expectation itself.   I have no previous with integration or a "hard analysis" background, and the material covered so far in the course I'm taking has only used $\int_\Omega X dP$ as a notational device for $E(X)$. In general, any pointers to literature or "intuitive" explanations are more than welcome.  Thanks in advance! I have 3987MB of RAM on my laptop. Task (System) Manager says that I am using 700MB of this RAM, whilst the command "free -m" says I am using nearly 1500MB. Which one is correct and why do they show two different values?   Thanks for any help. Why can't we initialize members inside a structure? Why can't we initialize members inside a structure ?  example:  struct s {    int i = 10; }; Hyperlinks are not visible in the generated PDF CV How can one change a coordinate system for a .lasd (Lidar) dataset?  The file is currently in UTM_Zone_15N and I am trying to convert it to IL_East. I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is? Multiple file upload in php How to programmatically login/authenticate a user?