"Again, we may put all this together to conclude that our program outputs a boolean value if supplied with an integer input. However, it is important to realise that the presence of $p$ is absolutely essential for the inference to happen. For example, our program might well satisfy $p → q$, but if it doesn’t satisfy $p$ – e.g. if its input is a surname – then we will not be able to derive $q$."   This is text from the book Logic in Computer Science by M. Huth and M. Ryan (page 9).   The second line in the paragraph necessiates the need for p being true, however, the truth table of Implication mentions one case where p is false, but q is true, and the resultant implication is true. How could that be when p is false?
I have just started studying implication in mathematics. So, you'll probably have an idea of where my confusion lies right from the get go.  In the truth table, where $A \implies B$ we obtain this result:  A   B   T/F ___________ T | T | T T | F | F F | T | T F | F | T   Now, my confusion here is regarding why $A\implies B$ given A is false, and B true. The others I understand. The first and last one are obvious, the second one implies, to me anyway, that given A implies B, the truth of B rests upon the truth of A, B is false, A is True, which cannot be, thus not B given A is false.  Now... then, why is B true despite the fact that A is false? Or rather, why is the statement B given not A, True. The truth of B is implied from the truth of A, but A is a false statement. So it's very contradictory that this may be, and in fact, if you go through a real life example, you would find that the third statement seems to be false.  Say $A$ is the statement that it is dark outside, and $B$ the statement that the sun is not on our side of the earth. If it is not dark outside, than stating that the sun is on the other side of the planet must be wrong (in my example i'm going to claim that the sun is the only light source).  Side note:  I just had a thought, if A is false, does that mean you can conclude... mmm, whatever? I mean, B is implied via the truth of A, if A is false, B is NOT implied, but then B could be EITHER true or false, I suppose. Is this correct?  However, would that not mean that A does not imply anything when false. Would that not set the 3rd and 4th statement to be undefined?
How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$? Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for the middle term. I got the limit as $n\to \infty$ of the function inside using Riemann sum but what I cannot get are the lower and upper bound. Compare given date with today How are large amounts of money expected to be made during the early parts of the Waterdeep: Dragon Heist adventure? Black screen after installation of Ubuntu 18.04 I did a fresh complete install of Ubuntu 18.04 on my ASUS M32CD desktop.   Everything during install went fine. At the end it said it was complete complete, and would reboot. Popped the disc out. Rebooted itself back into bios so I changed it back from CD boot as primary to HD.  When it booted up for Ubuntu screen and then everything went black and can’t do anything now! Ugh. Help find command with regex quantifier e.g. {1,2} I have been trying to create a find command string that will find all files that end with a number 1-99 but exclude all others.  e.g. I want to find myfile1 myfile99 but not myfile456 and not myfilebackup  The regex I'm thinking of is myfile[1-9]{1,2} but I can't get this to work with find.  find . -regex '.*myfile[0-9]{1,2}' OR  find . -iname 'myfile[0-9]{1,2}'  From what I can see it's the {1,2} part that is not working.  (by the way can you use -name -regex interchangably?)  Any help appreciated. Finding the null space of symmetric matrix generated by outer product Let $p, q \in \mathbb{R}^n$ such that $\|p\| = \|q\| = 1$ and define $$A = pq^T + qp^T$$ Find the null space of $A$.    I am not having very much luck. I have managed to show that $p + q$ and $p - q$ are eigenvectors, but I can't seem to connect those two. I know that since $A$ is generated by an outer product, it is of rank one. That implies that the rank of the null space is $n - 1$, but I can't seem to get any further. Can anyone help? runtime error: java.lang.ClassNotFoundException: com.mysql.jdbc.Driver Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices. Commenting out large sections To &quot;comment out&quot; a line, I need to insert a % at the beginning of the line (so that the line will not be compiled). Is there a way to comment out a large section without having to manually place % in front of each line? How to get memory usage under Windows in C++ Impulsioned jumping There's one thing that has been puzzling me, and that is how to implement a 'faux-impulsed' jump in a platformer. If you don't know what I'm talking about, then think of the jumps of Mario, Kirby, and Quote from Cave Story. What do they have in common? Well, the height of your jump is determined by how long you keep the jump button pressed.  Knowing that these character's 'impulses' are built not before their jump, as in actual physics, but rather while in mid-air - that is, you can very well lift your finger midway of the max height and it will stop, even if with desacceleration between it and the full stop; which is why you can simply tap for a hop and hold it for a long jump -, I am mesmerized by how they keep their trajetories as arcs.  My current implementation works as following:    While the jump button is pressed, gravity is turned off and the avatar's Y coordenate is decremented by the constant value of the gravity. For example, if things fall at Z units per tick, it will rise Z units per tick.  Once the button is released or the limit is reached, the avatar desaccelerates in an amount that would make it cover X units until its speed reaches 0; once it does, it accelerates up until its speed matches gravity - sticking to the example, I could say it accelerates from 0 to Z units/tick while still covering X units.  This implementation, however, makes jumps too diagonal, and unless the avatar's speed is faster than the gravity, which would make it way too fast in my current project (it moves at about 4 pixels per tick and gravity is 10 pixels per tick, at a framerate of 40FPS), it also makes it more vertical than horizontal. Those familiar with platformers would notice that the character's arc'd jump almost always allows them to jump further even if they aren't as fast as the game's gravity, and when it doesn't, if not played right, would prove itself to be very counter-intuitive. I know this because I could attest that my implementation is very annoying.    Has anyone ever attempted at similar mechanics, and maybe even succeeded? I'd like to know what's behind this kind of platformer jumping. If you haven't ever had any experience with this beforehand and want to give it a go, then please, don't try to correct or enhance my explained implementation, unless I was on the right way - try to make up your solution from scratch. I don't care if you use gravity, physics or whatnot, as long as it shows how these pseudo-impulses work, it does the job.  Also, I'd like its presentation to avoid a language-specific coding; like, sharing us a C++ example, or Delphi... As much as I'm using the XNA framework for my project and wouldn't mind C# stuff, I don't have much patience to read other's code, and I'm certain game developers of other languages would be interested in what we achieve here, so don't mind sticking to pseudo-code.  Thank you beforehand. I'm trying to wrap my head around this for a while now. It's a simple question for what I think will be a not so simple answer. How would you go about modeling one leg of this stool? I would like to do it without using sculpt mode because I can't get things quite clean enough. IndentationError: unindent does not match any outer indentation level Does this polynomial evaluate to prime number whenever $x$ is a natural number? Reference template variable within Jinja expression Can you have multiple Dropbox accounts on one computer? How to install specific Ubuntu packages, with exact version? Curved mesh topology 2 I am trying to model a violin belly and I am intrigued, how is it possible to make faces with soft / rounded edges?    The edges I do are totally straight (see the blue amoeba).  Also, do you have some idea on how to avoid the edge loops getting too "compressed" in the middle of the model (see the red amoeba) ?    I know the Subdivision surface modifier, but I want curved edges after applying the modifier... like in this picture :  Any help will be very appreciated. Thank you in advance!  Marcos  2nd part  In fact, I've tryied to model using nurbs and bezier curves, but I wasn't able to find any tool similar to loft that works with curves, then I was forced to convert it in meshes, the results were not terrific and I've started to move the vertices manually, a real pain... Unlike a guitar, each edge loop is on a diferente elevation on z axis and I have to keep the original dimensions, So I have to be extremely careful when using the subdivision suface modifier. I've already modeled this think on three different CAD softwares, but the meshes are far from good... I want to improve the edge loops, but it will be a long path... Thank you again. Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$ It's not difficult to show that  $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$  On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$.  Squaring the first power series and comparing terms gives us  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}2^{-2n}=1$$  that is,  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$$  My question: is there a more direct, combinatorial proof of this identity?  I've been racking my brains trying to come up with one but I'm not having much success. What is the function of the RNA primer in DNA replication?