I have a few questions. This isn't homework. I just want to understand better.  So if I have  int * b = &amp;k;   Then k must be an integer, and b is a pointer to k's position in memory, correct?  What is the underlying "data type" of b? When I output it, it returns things like 0x22fe4c, which I assume is hexadecimal for memory position 2293324, correct?  Where exactly is memory position '2293324'? The "heap"? How can I output the values at, for example, memory positions 0, 1, 2, etc?  If I output *b, this is the same as outputting k directly, because * somehow means the value pointed to by b. But this seems different than the declaration of b, which was declared int * b = k, so if * means "value of" then doesn't mean this "declare b to the value of k? I know it doesn't but I still want to understand exactly what this means language wise. If I output &amp;b, this is actually returning the address of the pointer itself, and has nothing to do with k, correct? I can also do int &amp; a = k; which seems to be the same as doing int a = k;. Is it generally not necessary to use &amp; in this way?
I know references are syntactic sugar, so code is easier to read and write. But what is the difference between a pointer variable and a reference variable?
The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is surjective (indeed bijective), but I'm afraid that I can't see why this is the case. I wonder whether you might be able to point me in the right direction?  Indeed, the proof begins like this:  "For each $n \in \mathbb{N}$, let $k_n, l_n$ be such that $n = 2^{k_n - 1} \left(2l_n - 1 \right)$; that is, $k_n - 1$ is the power of $2$ in the prime factorisation of $n$, and $2 l_n - 1$ is the (necessarily odd) number $\frac{n}{2^{k_n - 1}}$."  It then states that $n \mapsto \left(k_n , l_n \right)$ is a surjection from $\mathbb{N}$ to $\mathbb{N} \times \mathbb{N}$, and so ends the proof.  I can intuitively see why this should be a bijection, I think, but I'm not sure how to make these feelings rigorous?   I suppose I'd say that the map is surjective since given any $\left(k_n , l_n \right) \in \mathbb{N} \times \mathbb{N}$ we can simply take $n$ indeed to be equal to $2^{k_n - 1} \left(2l_n - 1 \right)$ and note that $k_n - 1 \geq 0$ and thus $2^{k_n - 1}$  is both greater or equal to one so is a natural number (making the obvious inductive argument, noting that multiplication on $\mathbb{N}$ is closed), and similarly that $2 l_n - 1 \geq 2\cdot 1 - 1 = 1$ and is also a natural number, and thus the product of these two, $n$ must also be a natural number. Is it just as simple as this?  I suppose my gut feeling in the proving that the map is injective would be to assume that $2^{k_n - 1} \left(2 l_n - 1 \right) = 2^{k_m - 1} \left(2 l_m - 1 \right)$ and then use the Fundamental Theorem of Arithmetic to conclude that $n = m$. Is this going along the right lines? The 'implicit' definition of the mapping has me a little confused about the approach.    On a related, but separate note, I am indeed aware that if $K$ and $L$ are any countable sets, then so is $K \times L$, so trivially, taking the identity mapping we see trivially that this map is bijective and therefore that $\mathbb{N}$ is certainly countable (!), and thus so is $\mathbb{N} \times \mathbb{N}$. Hence, it's not really the statement that I'm interested in, but rather the exciting excursion into number theory that the above alternative proof provides. Would raising the review limit help with the Close Votes queue? Nesting quotes in JavaScript/HTML Speed of object towards a point not in the object's trajectory? Is there a word to describe someone who tends to disagree with others only to upset them? What's the word to describe someone who acts arrogantly and always disagrees with others unreasonably in order to upset people around him/her?  [I'm not looking for adjectives like unpleasant, annoying, unfriendly, rude, I'm looking for a more specific term like opinionated, didactic, loquacious, gregarious ]  I think I'd come across it before but I've forgotten what it is.. If a set is $F_\sigma$ and is of first category then it is a countable union of closed nowhere dense sets We say a set is $F_\sigma$ if it is a countable union of closed sets. We say a set is of first category if it is a countable union of nowhere dense sets.  The question is, if a set of $F_\sigma$ and of first category, is it true that this set is a countable union of closed nowhere dense sets. How do I increase the render distance? How many solutions for an equation with simple restrictions I'm working on an assignment in which I have to count the number of solutions for this particular equation: $$x_1+x_2+x_3+x_4=20$$for non negative integers with $x_1&lt;8 $ and $x_2&lt;6$   I'm aware that this kind of a task isn't that complicated, but I don't get combinatorics in general that well.  So I've tried two following approaches to get this done.  Firstly I tried to substitute the variable x:  $x_1+x_2+x_3+x_4=20 \Leftrightarrow y_1+y_2+y_3+y_4=34$  in which $y_1=x_1+8$ and $y_2=x_2+6$ (casue $x_1=y_1-8$ and $x_2=y_1-6$) Following this approach the total number of possible solutions would be  $${34+3 \choose 3} $$  But I'm not sure if its the right solution.  The second approach is to sum all of the possible values that $x_1$ and $x_2$ could possibly take, also $x_1=0,1,2,3,4,5,6,7$ and $x_2=0,1,2,3,4,5,6$ And then  count all the possibilities for each of the variables $${20 -x_1-x_2+1\choose 1}$$ and sum them together like this: $${21\choose 1}+{20\choose 1}+{19\choose 1}+{18\choose 1}+... $$ and so on...  I'm sure I'll get the correct number with this one, but I'm not feeling like summing all of this possibilities. There's got to be a better, more elegant way to deal with this.  My professor gave me a hint that I should do it using the complement. Why do we use the simple past but not the present or future in the following expressions:  Don't you think it's time we went a little further Don't you think it's time we ate Don't you think it's time we went home Deriving curvature formula How do you derive the formula for unsigned curvature of a curve $\gamma (t) = (x(t), y(t)$ which is not necessarily parameterised by arc-length.   The formula given is $$|\kappa (t)| = \left|\frac{x'' y' - y'' x'}{((x')^2 + (y')^2)^{3/2}}\right|$$  All the definitions of curvature in my notes give it in terms for an arc-length parameterisation $T'(s) = \kappa(s) N(s)$. This is apparently meant to be 'easily derivable', am I missing something? Why is $(2, 1+\sqrt{-5})$ not principal? Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbb{Z}[\sqrt{-5}]$?   Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb Z[\sqrt{-5}]$. Writing $2=\alpha\beta$ in $\mathbb Z[\sqrt{-5}]$ and taking norms, $4=N(\alpha)N(\beta)$ in $\mathbb Z$, so $N(\alpha)\mid4$ in $\mathbb Z$. Similarly, since $\sqrt{-5}\in(\alpha)$ we get $N(\alpha)\mid5$, thus $N(\alpha)$ is a common divisor of $4$ and $5$, therefore $N(\alpha)=(1)$, so $\alpha$ is a unit. But that means $1\in (2,1+\sqrt{-5})=(\alpha)$, so it must be the whole ring, but it cannot to reach a contradiction, so how can I find an element, which is not in $(2, 1+\sqrt{-5})$.  Or is there a simpler method ? (Maybe not UFD would imply not PID) Best way to remove an event handler in jQuery? I have an input type="image". This acts like the cell notes in Microsoft Excel. If someone enters a number into the text box that this input-image is paired with, I setup an event handler for the input-image. Then when the user clicks the image, they get a little popup to add some notes to the data.  My problem is that when a user enters a zero into the text box, I need to disable the input-image's event handler. I have tried the following, but to no avail.  $('#myimage').click(function { return false; }); Why normal acceleration doesn't bring a change in speed? Suppose There's a particle of mass m moving with speed $v_{0}$, at certain moment a force starts acting on it (centripetal force) and it start uniform circular motion  Now every other reference states that centripetal force only brings about change only in direction  But plugging-in things in the equations paint a different picture  suppose  normal/centripetal acceleration = $A$  then $|v| = \sqrt{v_{0}^2+\frac{(A t)^2}{1}}$  we can certainly see that factor of $t$ will bring change in magnitude of velocity   Secondly, why the velocity is always tangential to the circle, but how can we mathematically show such? Intuitively for me if there's constant acceleration toward a certain then at some point of time, the body should 'slack' and fall toward the direction of acceleration   Here' a picture for a reference: SINGLE_EMAIL_LIMIT_EXCEEDED. What does it mean? Create file and its parent directory I know that touch command creates a file:  touch test1.txt   but how I can create a file and its full path?  for example my desktop contains nothing:  ~/Desktop/$ ls ~/Desktop/$   and I want to create 1.txt in ~/Desktop/a/b/c/d/e/f/g/h/1.txt. Can I do this with a simple command like:  $ touch ~/Desktop/a/b/c/d/e/f/g/h/1.txt   instead of create full path manually and then create the file? What is a natural response to &quot;Good to see you. How you doing?&quot;? Will a natural response include &quot;doing&quot; as in &quot;I'm doing okay&quot; or will &quot;I'm okay&quot; be more natural? Continuous function preserving rational difference. Find all the continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that if $x-y\in \mathbb{Q}$ then $f(x)-f(y)\in \mathbb{Q}$.   Can somebody give me some hint to solve this problem. I think that the functions are $f(x)=mx+b$ where $m\in\mathbb{Q}$.  Thanks! How to blacklist kernel modules? How do I disable loading of unnecessary kernel modules. Kernel 3.2.4 I'm writing a contribution for a double-blind submission system.   Since I would like to supply more data than I could show on a small number of pages, I'm looking for a way to reference supplemental data in my manuscript, such as software scripts, figures or databases.  I could upload on GitHub, RPubs, Figshare and Zenodo, but all of these would expose my real user name (and I don't feel like making a throwaway e-mail account with a new repository for every anonymous publication that I intend to write).  Is there any way to get the above services to do what I want? Or is there a similar service that would allow anonymous hosting of research data?   It would be even better if I could later (e.g. after publication) declassify the information, with the link from the manuscript still being intact. Logical or operator not behaving as expected