If $\dfrac{c}{d} &gt; \dfrac{a}{b}$, then $\dfrac{c}{d} &gt; \dfrac{a+c}{b+d} &gt; \dfrac{a}{b}$   This is intuitively clear to me but I don't know how to prove this. I tried rewriting all the fractions to the same for but I still couldn't figure it out because I can't really say anything about the size of $a,b,c,d$ on their own.  What tactic should one use for these types of problems?
We were trying to come up with an easy way to generate a rational number in between two existing rational numbers with a fairly low numerator and denominator (the way we were doing this earlier was to find the average of the two rationals, but that results in a denominator of up to $c * d$. Does this inequality hold for all values of $a, b, c, d$?
I have an ArrayList of objects in Java.  The objects have four fields, two of which I'd use to consider the object equal to another.  I'm looking for the most efficient way, given those two fields, to see if the array contains that object.  The wrench is that these classes are generated based on XSD objects, so I can't modify the classes themselves to overwrite the .equals.    Is there any better way than just looping through and manually comparing the two fields for each object and then breaking when found?  That just seems so messy, looking for a better way.  Edit: the ArrayList comes from a SOAP response that is unmarshalled into objects. Why are the two versions of Chromium and VLC available in Ubuntu software of different sizes? I want to change the default timeout in Chrome. In Firefox I can set the value of network.http.connection-timeout in about:config - but how can I do that in Chrome? Disabling OBSOLETE error in C# Why is it differential equations exist on an interval instead of a domain? I understand a domain is the set of input elements a function is defined for (and can have breaks in it e.g. union of 2 sets) and a interval is a continuous range of real numbers. Why do we speak of the solution to a differential equation over an interval instead of an domain? If the solution to an ODE is just a function, and we normally speak of functions as having a domain, then why wouldn't we use domain? Alternative to itoa() for converting integer to string C++? I was wondering if there was an alternative to itoa() for converting an integer to a string because when I run it in visual Studio I get warnings, and when I try to build my program under Linux, I get a compilation error. I replaced a buggy Windows Vista installation with Ubuntu. All works fine except that the main HD where I had all my files are now inaccessible. Here is the error message I get:   Error mounting: mount exited with exit code 13: ntfs_attr_pread_i: ntfs_pread failed: Input/output error Failed to read NTFS $Bitmap: Input/output error NTFS is either inconsistent, or there is a hardware fault, or it's a SoftRAID/FakeRAID hardware. In the first case run chkdsk /f on Windows then reboot into Windows twice. The usage of the /f parameter is very important! If the device is a SoftRAID/FakeRAID then first activate it and mount a different device under the /dev/mapper/ directory, (e.g. /dev/mapper/nvidia_eahaabcc1). Please see the 'dmraid' documentation for more details   Is it necessarily a hardware problem? If not, is there a way to repair the HD from Ubuntu? How do I go about tracking down IO usage in Linux and other *nix OS'? Occasionally, something on my system starts hammering the disk. I would like to log data that would let me know about I/O usage per process, then I could go back to it later and see what keeps doing this. How should I go about solving this problem? Bonus points for solving this for arbitrary block devices. Could someone explain this shebang line which uses sh and then does exec perl? This is from . It's about executing a Perl script.     Finally, if you are unfortunate enough to be on an ancient Unix system   that doesn’t support the magic #! line, or if the path to your   interpreter is longer than 32 characters (a built-in limit on many   systems), you may be able to work around it like this:   #!/bin/sh -- # perl, to stop looping eval 'exec /usr/bin/perl -S $0 ${1+"$@"}' if 0;   Can you explain step by step what's going on here? I'm trying to make it work or encompass, but to no avail.  At execution of just the above, I'm getting this:  /bin/sh: 0: Illegal option -- In Star Wars: "The Force Awakens," how could this lightsaber have turned up where it did? In The Force Awakens:      Maz Kanata tells Rey that the lightsaber she just picked up and had visions from belonged to Anakin and Luke and now it's hers.   The lightsaber that Luke got from Obi-Wan Kenobi in A New Hope was lost/destroyed when Luke fought Darth Vader in Star Wars: The Empire Strikes Back.  In Return of the Jedi Darth Vader compliments Luke on building a new lightsaber and tells Luke that his skills are now complete.   How could that possibly be the lightsaber that Anakin used originally, and that was then passed on to Luke, when that lightsaber no longer exists?  Writers notes, director's notes, interviews, cast interviews, etc. are welcome. Thanks. Compute $E(X\mid X+Y)$ for independent random variables $X$ and $Y$ with standard exponential distributions How do I fix "Error C1033: cannot open program database"? Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1 The task is as follows:     Given: $$F(x,y,z) = 0$$    Goal:  Show    $\frac{\partial  z}{\partial y}|_x \frac{\partial  y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$   Here is my work so far:  (1) Differentiate with respect to y, I get:  $0 + F_2 + F_3  \frac{\partial z}{\partial y} = 0$  So $ F_3  \frac{\partial z}{\partial y} = - F_2$  (2) Differentiate with respect to x, I get:  $F_1 + F_2 \frac{\partial y}{\partial x}  + 0 = 0$  So $ F_2  \frac{\partial y}{\partial x} = - F_1$  (3) Differentiate with respect to z, I get:  $F_1  \frac{\partial x}{\partial z}    + 0  + F_3 = 0$  So $ F_1  \frac{\partial x}{\partial z} = - F_3$  (4) After some manipulations with the $F_i$, I get to the conclusion that $\frac{\partial z}{\partial y}* \frac{\partial y}{\partial x} * \frac{\partial x}{\partial z}  = -1$, so when evaluated with x, z, y respectively, conclusion is still true  My questions are:  (1) Is my proof correct?  (2) For example, when I differentiate with respect to y, I "let" $F_1$ be 0 and find partials for other coordinates.    I had a hard time trying to explain to my friend on the reason(s) why I can do such "let be 0" thing.  Although I think if I can't do that, then there is no way that I can reach the conclusion, but I somehow feel confused about the fact too.  Since my book is doing it that way, my understanding is that I can do such "let be 0" thing based on the independece of x with respect to y, when I differentiate with respect to y.  But is my thought ok?  Would someone please help me on this question? Thank you very much ^_^ Is there a deb package to install Firefox ESR? I would like to have  on my system but I would like to install it via apt-get. Is there a special package to install this?  Download link would be: Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$  How do I go about proving this? I'm not sure where to begin, I'd be glad for any hint. This code in JS gives me a popup saying "i think null is a number", which I find slightly disturbing. What am I missing?    if (isNaN(null)) {   alert("null is not a number"); } else {   alert("i think null is a number"); }     I'm using Firefox 3. Is that a browser bug?  Other tests:    console.log(null == NaN);   // false console.log(isNaN("text")); // true console.log(NaN == "text"); // false     So, the problem seems not to be an exact comparison with NaN?  Edit: Now the question has been answered, I have cleaned up my post to have a better version for the archive. However, this renders some comments and even some answers a little incomprehensible. Don't blame their authors. Among the things I changed was:   Removed a note saying that I had screwed up the headline in the first place by reverting its meaning Earlier answers showed that I didn't state clearly enough why I thought the behaviour was weird, so I added the examples that check a string and do a manual comparison. How to Add Background Images in 2.8 and later 2.81a here. I am trying to add a "background image" similar to how , so that I see a screen that looks something like this:    However I can't figure out how/where to access this screen in 2.81a as the UI is wildly different. Any ideas?  I am new to Blender so if you can provide a click-by-click list of instructions it would be enormously appreciated! What's an intuitive way to think about the determinant? In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute the determinant of an $n \times n$ matrix by breaking it up into the determinants of smaller matrices. Apparently there is a way by summing over a bunch of permutations. But the notation is really hard for me and I don’t really know what’s going on with them anymore. Can someone help me figure out what a determinant is, intuitively, and how all those definitions of it are related? Cannot update/get any packages. gedit not installed my problem is that my sources.list is out of date for some reason. I already found the fix in another port, but it doesn't work. It says to use gedit to edit sources.list and replace everything with the proper sources. Well, for some reason I don't have it installed (Ubuntu Server 14.04) and I can't install it or else I get the main problem again.  Failed to fetch http:// some ubuntu site Could not resolve hostname "us.archive.ubuntu.com"  Is there any other way to edit sources.list? Or do I need to do a re-install of the OS? Coloring a black and white PDF graphic I have a graphic in PDF format (originally converted from an SVG) that is monochromatic (black on white). Is it possible using TikZ or PSTricks but not (Illustrator or Inkscape) to include this image in a different color?  Example:    While it lasts (there seems to be no long term way to host a pdf), you can .  Edit: Simple was perhaps a bit of a misnomer - the pdfs are cropped and monochromatic, but to intricate to redraw on my own. The above example is a character from a font (which I can recolor), but it gives a good example of what shapes I am talking about.