I have some problem with the bevel tool
Why are the longer edges being beveled more?
$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . Using C++ library in C code I have a C++ library that provides various classes for managing data.  I have the source code for the library.  I want to extend the C++ API to support C function calls so that the library can be used with C code and C++ code at the same time.  I'm using GNU tool chain (gcc, glibc, etc), so language and architecture support are not an issue.  Are there any reasons why this is technically not possible?  Are there any gotcha's that I need to watch out for?  Are there resources, example code and/or documentation available regarding this?    Some other things that I have found out:   Use the following to wrap your C++ headers that need to be used by C code.     #ifdef __cplusplus extern "C" {   #endif   //   // Code goes here ...   //   #ifdef __cplusplus   } // extern "C"   #endif    Keep "real" C++ interfaces in separate header files that are not included by C.  Think  here.  Using #ifndef __cplusplus #error stuff helps here to detect any craziness. Careful of C++ identifiers as names in C code Enums varying in size between C and C++ compilers.  Probably not an issue if you're using GNU tool chain, but still, be careful. For structs follow the following form so that C does not get confused.  typedef struct X { ... } X  Then use pointers for passing around C++ objects, they just have to be declared in C as struct X where X is the C++ object.   All of this is courtesy of a friend who's a wizard at C++. While I was reading the book Exploratory examples for real analysis, I came across this:     to show that : $$\lim_{x\rightarrow x_o}\frac{1}{h(x)} = \frac{1}{\lim_{x\rightarrow x_o}{h(x)}} = \frac{1}{M}$$      Discussion:Let $\epsilon \gt 0$.we want to find $\delta \gt 0$ such that       if  $0 \lt |x-x_0| \lt \delta $,then $\Big|\frac{1}{h(x)}-\frac{1}{M}\Big|\lt \epsilon$.      In order to get expression involving $|h(x)-M|$,we first find the common denominator:        $\Big|\frac{1}{h(x)}-\frac{1}{M}\Big|=\Big|\frac{M-h(x)}{M~h(x)}\Big|$      $~=\Big|\frac{h(x)-M}{|M|~|h(x)|}\Big|$.      As we have to deal with variable term $|h(x)|$ in denominator.We can construct bound .Let $\epsilon =\frac{|M|}{2}$ ,since lim$_{x\to x_o}h(x)=M$,there exists $\delta_1\gt 0~~$ s.t.       if $0 \lt |x-x_0| \lt \delta_1 $,then $|h(x)-{M}|\lt \frac{|M|}{2}$.      As you study steps below,identify that step where $h(x)\neq 0$ for all $0 \lt |x-x_0| \lt \delta_1 $ plays a critical role:      $|M|=|M-h(x)+h(x)|$      $~~~~~~\leq |M-h(x)|+|h(x)|$      $~~~~~~= |h(x)-M|+|h(x)|$      $~~~~~~\lt \frac{M}{2}+|h(x)| \implies$      $\frac{|M|}{2} \lt |h(x)| \implies$      $\frac{1}{h(x)}\lt \frac{2}{|M|}$,where $0 \lt |x-x_0| \lt \delta_1 $.      Now that we've bound for $1/h(x)$,we can apply the definition to term $|h(x)-M|$.      Since     lim$_{x\rightarrow x_o}h(x)=M$,we can find $\delta_2 \gt 0$ such that,       $~~~~~~~~~~~~~~~~~~~~~~~~~~~~$if  $0 \lt |x-x_0| \lt \delta_2$, then $|h(x)-M|\lt \frac{|M|^2 \epsilon }{2}$   The thing I can't understand is for what purpose did the author work with such complicated expression involving $\epsilon $ in the end of the discussion? How can I safely upgrade an Ubuntu installation from one release to another? What should I do before upgrading to ensure it goes smoothly, and is there any clean-up I need to do afterwards?  In particular, do I need to do anything special if I am using packages from PPAs, if I have a graphics card that needs drivers, or if I'm running a custom Linux kernel?  This is a general question , but as example we take Ubuntu 12.04 and Ubuntu 12.10 Is there a way to get correctly aligned print export with OpenLayers plugin? I am a beginner using QGIS (1.8.0 Lisboa) on a Mac (OS X 10.6.8 Snow Leopard). I work on a file with two layers in the following sequence (top to bottom):    a vector polygon layer with a few outlines of various buildings in France and Italy (CRS: EPSG:4326 - WGS 84 - "on the fly" re projection is activated ) a Google satellite layer uploaded via open layer plugin.   Sometimes when changing the scale (scrolling in and out) the Google layer updates a little wrong in size or position compared to the polygon layer. But as soon as I "pan" the map everything reconstructs right. Now the problem I have concerns the print composer.  The result of any kind of export - pdf, or image or svg - of my map (vector layer + Google layer in the background) differs from what is shown in the preview of the print composer: the vector layer and the Google layer positions are mismatching and in a different scale. I can send some pictures showing screenshots of the print composer preview and the final results if required.  I tried almost everything:    new files with different CRS settings "cache" update preview "render" update preview different paper sizes,  different scales,   The problem appears only using an open layer map (Google, OpenStreetMap...). If I try the same using two vector layer (shapefile) - one with the buildings, another one with the city in background - everything works out great.  ...exasperating! Since it is a real important research project I would be very grateful for some help and tips. I usually render my animation into image sequence format, It usually takes a lot of time, so I turn off my monitor and come back later to check the rendered result then find that blender crashed before rendering all the frames.  Since rendering can be done from the command prompt I thought about writing a script (running in my OS) to automatically restart rendering from the last successful rendered frame until all frames are finished*.  But I believe this problem is facing a lot of people out there, is there a way to deal with it? instead of reinventing the wheel?     for example if the total render time is 8 hours I don't want come back to check my computer every 20 minutes (some times this is not available I'm asleep or away from my PC) $f(x)$ is a strictly increasing, and continuous, function on $[0,+\infty)$. Consider the integral,  $$\int_0^a f^{-1}(x) dx.$$  Reasoning  The interval $[0,a]$ is "reflected" across the line $y=x$ onto the interval $0 \leq y \leq a.$  $f(x)$ and $f^{-1}(x)$ are symmetric across the line $y=x$. So, the integral is equivalent to the area bounded by $y = 0$, $y=a$, and $f(x)$.  So, "integrating $f$ along an interval on the $y$-axis" is the same as integrating $f^{-1}$ along the same interval on the $x$-axis?  Question  Is my reasoning correct? Note: the eventual outcome of this discussion involved dropping this particular off-topic reason, along with other changes. See:  &nbsp; &mdash;Shog     I was using the reason "demonstrate a minimal understanding" myself quite frequently, but the more I review, the more I have doubts about it and I think this reason can either be dropped entirely or at least rephrased, so I started to use "problems with code you've written must describe the specific problem" instead:     Questions asking for code must demonstrate a minimal understanding of the problem being solved.   So the poster has posted some description, maybe even a snippet of code and in his mind that already proves that he has that minimal understanding required. After all he described the problem and what he tried to solve. So telling him that he lacks that understanding is rather rude, and from the posters point of view doesn't even apply. What both version of the reason not emphasize is that a description of a problem is not various versions of     it doesn't work   but that we rather mean really specific details (see below). I can often see in the reopen review queue that the "improved" version just includes a more verbatim description, while what usually would be expected is some specific error message, a trace or a log, or whatever applies.  Then it continues with:     Include attempted solutions, why they didn't work, and the expected results. See also: Stack Overflow question checklist   comparing it to:     problems with code you've written must describe the specific problem — and include valid code to reproduce it — in the question itself. See SSCCE.org for guidance.   Which says essentially the same, only in more detail.   So the "minimum understanding" reason is IMO a more rude version of the reason telling the user that an SSCCE is missing.  The intent of giving this reason is to tell the user that he needs to give exact details to be able to help him, so IMO a checklist would be much better. Something like:     Your question was closed because it may not provide enough details. Things you can do to improve your question to get an answer include:         Format the code in a way that makes it readable and easily understandable.   Provide the smallest working sample of code demonstrating the problem (link to SSCCE)   Include your input and output.   If applicable provide logs (i.e. Android debugging log, stack trace, etc.).   If applicable provide the exact error messages (compiler, etc.).      And something like this should be best included already in the banner that appears when a question is closed instead of being hidden in some pages you often enough have to actively looking for.  It might be helpful to adjust this, based on the tags or close reason, as I realize that I'm more focused on the programming area, and the above list may not apply to other areas of SO.  Update  Another thing I forgot to mention (but wanted to) is this:     Questions asking for code   Is IMO essentially an invitation of the type of questions     I have a problem, can you provide the solution? (with an optional please)   while     Questions concerning problems with code you've written   puts at least some emphasis on the fact that the poster has to put some effort himself in trying to solve the problem by providing the code which he used to attempt the solution. Lebesgue integral $\int_{(0, \infty)} \frac{\sin x}{x} dm$ doesn't exist but improper Riemann integral exists How to animate multiple lines I wonder what would be the best way of animating multiple 2 dimensional lines in blender.  Lets say I have lots of line, that are parallel and have some sort of rule, kind of like a PCB board with hundreds of circuits.  The starting frame would be when nothing is visible and during the animation the lines would slowly build up from nothing all the way to the end forming the finished lines.  I have a couple of ideas that might work, but all include heavy hand animation, like:   grease pencil animation, drawing them one by one, converting to curve, then build up modifier some dynamic paint hack   There maybe other more efficient methods that I don't know of and if so, please let me know. In PHP, the HEREDOC string declarations are really useful for outputting a block of html.  You can have it parse in variables just by prefixing them with $, but for more complicated syntax (like $var[2][3]), you have to put your expression inside {} braces.  In PHP 5, it is possible to actually make function calls within {} braces inside a HEREDOC string, but you have to go through a bit of work.  The function name itself has to be stored in a variable, and you have to call it like it is a dynamically-named function.  For example:  $fn = 'testfunction'; function testfunction() { return 'ok'; } $string = &lt;&lt;&lt; heredoc plain text and now a function: {$fn()} heredoc;   As you can see, this is a bit more messy than just:  $string = &lt;&lt;&lt; heredoc plain text and now a function: {testfunction()} heredoc;   There are other ways besides the first code example, such as breaking out of the HEREDOC to call the function, or reversing the issue and doing something like:  ?&gt; &lt;!-- directly output html and only breaking into php for the function --&gt; plain text and now a function: &lt;?PHP print testfunction(); ?&gt;   The latter has the disadvantage that the output is directly put into the output stream (unless I'm using output buffering), which might not be what I want.  So, the essence of my question is: is there a more elegant way to approach this?  Edit based on responses: It certainly does seem like some kind of template engine would make my life much easier, but it would require me basically invert my usual PHP style.  Not that that's a bad thing, but it explains my inertia..  I'm up for figuring out ways to make life easier though, so I'm looking into templates now. Is there a way of installing Ubuntu (Windows Subsystem for Linux) on Win10 (v1709) without using the Store? The Microsoft Store app is disabled by Group Policy by my university. I have administrator rights on my computer and I was wondering if there was a way of installing Windows Subsystem for Linux (WSL) directly? Or maybe via Chocolatey or some other command line method? I would like to download and install the Ubuntu  on windows without using the Microsoft store.  Note that since version 1709 (Fall Creator's Upgrade), only enabling Developer Mode is no longer sufficient. Does anyone know any example that invalidates the following affirmation:     If a morphism $f:A\to A$ induces the identity $\hat f:\operatorname{Spec} \left( A \right) \to \operatorname{Spec} \left( A \right)$ then $f = \operatorname{id} _A $. Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$  I have tried to fiddle with it as follows:   $$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$ $$ 3 \cdot 2^{3x+1}+ 2 \cdot 3^{3x+1}=7 \cdot 6^x(2^x+3^x)$$ Dividing both sides by $6$ gives us $$2^{3x}+3^{3x}=7 \cdot 6^{x-1}(2^x+3^x)$$  Is this helpful? If so, how should I proceed form here? If not any hints would be greatly appreciated. Why is this CW post saying 36% unknown? Looking at         it says "36% unknown". No user by the name of 'unknown' appears in the  ()  What madness is this? Who is this 'unknown' you speak of? In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show this?  Ideas: The sum of the diagonal entries of $M$ is the trace, which is also the sum of the eigenvalues. The product of the eigenvalues is the determinant. All the diagonal entries of $A$ must be real and strictly positive. Also we have $|m_{ij}|\leq \sqrt{m_{ii}m_{jj}}\leq \frac{m_{ii}+m_{jj}}{2}$. So far nothing is working...  We might try to write the Gram matrix $m_{ij} = \langle v_i, v_j \rangle$, and then $\prod_i m_{ii} = \prod_i \|v_i\|^2$. Now we'd need to show that the determinant, which is a polynomial in the entries of $v_i$, is less than $\prod_i \|v_i\|^2$ somehow. "Same as I do" or "same as I have" She has the same book as I have.  Isn't it supposed to be She has the same book as I do?  But I often hear the former.  Why? $n$-th digit in the sequence of natural numbers By accident, I used rm on a file I didn't want to delete. Is there a way that I can get it back under Linux? So I am running on Windows 7 (64-bit) Home Premium with 12 GB RAM (Intel Core i7 920).  I have restricted the Page File to 400MB since it was eating up a lot of space om my SSD (that is 80GB).   After that I sometimes get the "low memory"-warning, like this, except for the behold-comment:   When I get those warning I have usually a lot of RAM available; when I check Resource Manager I have over 2GB "Free" and over 2GB "Available" - usually more than that.  The diskspace on all my drives have over 10GB free.  So the question is - why does Windows complain? It actually restarted by computer (hard boot) when it happened, and as I said - lots or RAM available.