If you have 10 cookies, and your number of cookies increases tenfold, you have 100 cookies, right? That's what I thought at first, but I recalled that if your number of cookies increases 200%, you get N + (200%/100%)N cookies where N was your original number of cookies, so you end up with 30 cookies, not 20. Shouldn't a tenfold (or threefold, or fourfold, or whatever) increase work the same way as a percentage increase?  For what it's worth, I'm interested in whether I should describe an increase from 0.8% to 3.4% as a "fourfold increase" or "threefold increase" (since 3.4%/0.8% = 4.25) in a scientific paper.  (I'm not sure whether this question belongs here or on math.SE.)
and is such interpretation the norm?  When something went from 4 units to 8 units, most authoritative sources seem to agree with the use of "a two-fold increase", even though what was actually increased is more like "one-fold", i.e. the original quantity.  But if the "two-fold increase" is the correct usage, why most people seem to interpret "a 100% increase" the same thing?
If a linear operator has an adjoint operator, it is bounded How do I update the GUI from another thread? ALTER table - adding AUTOINCREMENT in MySQL Square roots -- positive and negative It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.  I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified? How do you get vim to display wrapped lines without inserting newlines? I'm getting back in touch with my inner (g)vim due to an unscheduled MacBook mother(board) of a meltdown (my emergency backup Linux box won't run TextMate). All told I'm happy with vim's efficiency and power, but I'm mortified at how hard it is to get the kind of word wrap that even stupid HTML textareas achieve with no apparent effort.  Consider the text  Etiam ornare mollis tortor. Suspendisse sed neque. Nullam in elit. Cum sociis nullam.  By default, with an 80-character width vim displays this as  Etiam ornare mollis tortor. Suspendisse sed neque. Nullam in elit. Cum sociis nu llam.  This wrapping doesn't care about whitespace, so sometimes it just slices words into pieces (nullam in this case). Of course, you can turn on word wrap, and get this:  Etiam ornare mollis tortor. Suspendisse sed neque. Nullam in elit. Cum sociis nullam.  The problem is that vim inserts a newline at the linebreak, which I most emphatically do not want. In other words, I want the text to display exactly as vim displays it with word wrap turned on, but without inserting a newline. (This way it can be pasted into HTML textareas and email programs, among other places.)  Web searches have yielded nothing of use, despite diligent effort; I hope StackOverflow can succeed where my Google-fu has failed. Second derivative "formula derivation" I've been trying to understand how the second order derivative "formula" works:  $$\lim_{h\to0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$  So, the rate of change of the rate of change for an arbitrary continuous function. It basically feels right, since it samples "the after $x+h$ and the before $x-h$" and the $h^2$ is there (due to the expected /h/h -> /h*h), but I'm having trouble finding the equation on my own.  It's is basically a derivative of a derivative, right? Newtonian notation declares as $f''$ and Leibniz's as $\frac{\partial^2{y}}{\partial{x}^2}$ which dissolves into:  $$(f')'$$ and $$\frac{\partial{}}{\partial{x}}\frac{\partial{f}}{\partial{x}}$$  So, first derivation shows the rate of change of a function's value relative to input. The second derivative shows the rate of change of the actual rate of change, suggesting information relating to how frequenly it changes.  The original one is rather straightforward:  $$\frac{\Delta y}{\Delta x} = \lim_{h\to0} \frac{f(x+h) - f(x)}{x + h - x} = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$  And can easily be shown that $f'(x) = nx^{n-1} + \dots$ is correct for the more forthcoming of polynomial functions. So, my logic suggests that to get the derivative of a derivative, one only needs to send the derivative function as input to finding the new derivative. I'll drop the $\lim_{h\to0}$ for simplicity:  $$f'(x) = \frac{f(x+h) - f(x)}{h}$$  So, the derivative of the derivative should be:  $$f''(x) = \lim_{h\to0} \frac{f'(x+h) - f'(x)}{h}$$  $$f''(x) = \lim_{h\to0} \frac{  \frac{ f(x+2h) - f(x+h)}{h} - \frac{ f(x+h) - f(x)}{h}  }{h}$$  $$f''(x) = \lim_{h\to0} \frac{  \frac{ f(x+2h) - f(x+h) - f(x+h) + f(x)}{h}  }{h}$$  $$f''(x) = \lim_{h\to0} \frac{  f(x+2h) - f(x+h) - f(x+h) + f(x)  }{h^2}$$  $$f''(x) = \lim_{h\to0} \frac{  f(x+2h) - 2f(x+h) + f(x)  }{h^2}$$  What am I doing wrong? Perhaps it is the mess of it all, but I just can't see it. Please help. I'm robotic engineer. I'm using OpenSCAD to model robotic components (gears, pulleys, parts, etc). But I need application to model physics and interaction of the components (for i.e. how will robot move if I will rotate current gear).  So, is it possible to import OpenSCAD models into Blender and model interactions in it?  I have models in STL format (for i.e. gears). I need to make reductor from it and see how it moves (if possible - measure rotation speed and torque). How do I enable automatic updates? Update Manager is constantly offering me updates (e.g. security fixes, updates from PPAs).  How can I tell my Ubuntu installation to automatically download and install updates whenever they become available? AttributeError: MapDocObject: Unable to save on mxd.save/saveACopy? I'm trying to use arcpy to save an .mxd and am having an issue.  It doesn't matter if I use mxd.save or mxd.saveACopy I get the same error.  I know it doesn't look like I am doing anything w/ it here, but it's just a portion of my code:  import arcpy import arcpy.mapping as map arcpy.env.overwriteOutput = True  env.workspace = r"C:\MGIS\FinalProject2.gdb     mxd = map.MapDocument(r"C:\MGIS\Finalproject2.mxd") df = map.ListDataFrames(mxd)[0]  for df in map.ListDataFrames(mxd): df.scale = 2000000  #Add layers into map document addLayer = map.Layer(OKXcounties) addLayer1 = map.Layer(contourLabelText) addLayer2 = map.Layer(contoursClipLyr) addLayer3 = map.Layer(clipOKXLyr) map.AddLayer(df, addLayer, "AUTO_ARRANGE") map.AddLayer(df, addLayer1, "AUTO_ARRANGE") map.AddLayer(df, addLayer2, "AUTO_ARRANGE") map.AddLayer(df, addLayer3, "AUTO_ARRANGE")  # Add contour Labels for lyr in map.ListLayers(mxd):     if lyr.name == "contourLabelText":         lyr.showLabels = True     if lyr.name == "Contour Features":         lyr.transparency = 100  mxd.saveACopy(mxd)   I get the following error:   Traceback (most recent call last):   File "C:\Python26\ArcGIS10.0\Lib\site-packages\pythonwin\pywin\framework\scriptutils.py", line 312, in RunScript     exec codeObject in __main__.__dict__   File "C:\MGIS\geog485\FinalProject2\mxdtest.py", line 7, in &lt;module&gt;     mxd.saveACopy(mxd)   File "C:\Program Files\ArcGIS\Desktop10.0\arcpy\arcpy\utils.py", line 181, in fn_     return fn(*args, **kw)   File "C:\Program Files\ArcGIS\Desktop10.0\arcpy\arcpy\_mapping.py", line 668, in saveACopy     self._arc_object.saveACopy(file_name) AttributeError: MapDocObject: Unable to save.  Check to make sure you have write access to the specified file and that there is enough space on the storage device to hold your document.   Space is not the issue, I have 60 GB of free space and I've checked the windows properties of the .mxd file.  Read-only box is unchecked and I've "allowed" all permissions for all groups:    Does anyone have any idea how if this is a bug or how I can fix this?  I'm using ArcInfo 10.0 SP4. How to use manual partitioning during installation? I'd like to see the full How-To on how to use manual partitioning during Ubuntu installation. The existing guides (at least those I found here) cover only automatic part and leave untouched the manual part (or extremely short and contain no pictures).  I'd like to cover such situations:     If your disk contains other systems: Why is jumping into water from high altitude fatal? If I jump from an airplane straight positioned upright into the ocean, why is it the same as jumping straight on the ground?  Water is a liquid as opposed to the ground, so I would expect that by plunging straight in the water, I would enter it aerodynamically and then be slowed in the water. All Stack Exchange data dumps The original ClearBits link to the September 2011 data dump has broken. How can I access it and other previous data dumps? The Internet Archive only has . For which values of $a$ does $a-2 \mid a^3+4$ Was Severus Snape made a Death Eater before or after he told Voldemort about the Prophecy? Based on discussion in comments on , was Severus Snape a Death Eater (real one, with Dark Mark) before he overheard Sybil give the Prophecy to Dumbledore and relayed it to Voldemort? Or was he made a DE after that?  Canon sources only please (books/interviews/Pottermore) Z-axis issue with the floating review info panel, permits various elements to flow over it I was having a look at an old question of mine when I browsed the  from the timeline. I scrolled downwards and realized the Review completed panel comes along for the ride. While that might be , tags flowing over top of it as pictured below probably isn't.  Here's a .gif of the Question and Duplicate tabs flowing over it as well. Note that the timeline button also flows over. If vote buttons are present, they will also flow over it.  Reproduced on Chrome and Microsoft Edge. Update Per  by Tiago Martins Peres, this issue also happens in other Review types, namely Late Answers, First Posts and Low Quality Posts. In addition to that, it is causing the action buttons on reviews to become unclickable if you scroll low enough.  As a temporary workaround, you can use 's , which  this issue. Restarting with new account on iOS/GameCenter libpng problems for QGIS I'm using ubuntu 18.04 and if I download QGIS 2.18 and I try to run it, I get this:  libpng warning: Application jmp_buf size changed Segmentation fault (core dumped)   If I try with QGIS 3.2, the error is:   libpng warning: Application was compiled with png.h from libpng-1.6.34 libpng warning: Application  is  running with png.c from libpng-1.2.56 libpng error: Incompatible libpng version in application and library   So it's clear that the problem is with libpng.   I purged libpng-dev and reinstall again but nothing changed.   I tried to purge libpng16-16 but I get:  sudo apt purge libpng16-16 Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  libgdk-pixbuf2.0-0 : Depends: libpng16-16 (&gt;= 1.6.2-1) but it is not going to be installed                       Recommends: libgdk-pixbuf2.0-bin  libgtk-3-0 : Depends: libcairo-gobject2 (&gt;= 1.10.0) but it is not going to be installed               Depends: libcairo2 (&gt;= 1.14.0) but it is not going to be installed               Depends: libfontconfig1 (&gt;= 2.12) but it is not going to be installed               Depends: libpango-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: libpangocairo-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: libpangoft2-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: adwaita-icon-theme               Recommends: libgtk-3-bin E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.   and then:  ~$ sudo apt install libgdk-pixbuf2.0-bin libgtk-3-bin Reading package lists... Done Building dependency tree        Reading state information... Done libgdk-pixbuf2.0-bin is already the newest version (2.36.11-2). libgtk-3-bin is already the newest version (3.22.30-1ubuntu1). 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. ~$ sudo apt install libgtk-3-0 libgdk-pixbuf2.0-0 Reading package lists... Done Building dependency tree        Reading state information... Done libgdk-pixbuf2.0-0 is already the newest version (2.36.11-2). libgtk-3-0 is already the newest version (3.22.30-1ubuntu1). 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   At this point I'm afraid of breaking something if I continue trying things, so I come here for advice before.  I deleted and reinstalled ubuntu. I installed first thing qgis and it was working. Afterwards I compiled gdal, from source to have it from python as stand alone. It might be this, but I had done this previously without any problems. What's the difference between $\equiv$ and $\leftrightarrow$ in a formal proof? This question has troubled me for ages despite my several attempts of looking it up in dictionaries or usage books. Do we say, "Do you have any ideas" or "Do you have any idea"? I do see an example where "any" means "it doesn't matter who/which/what", therefore "You can borrow any book you like." Also, does it matter between using it in questions and negatives? Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$