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 Apr21 comment How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ I added some details, see if that helps. Apr21 revised How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ add details on specific example and uniform distribution Apr20 comment Am I a toroid or not? Many of us increase our genus with body piercings. Apr20 comment How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ Yes, if you set the output range to be from $0$ to $2^{22} - 1$ or from $1$ to $2^{22}$, the range size will be $2^{22}$ and it's easy to solve. For example, if the input is from $-2^{31}$ to $2^{31} - 1$ and the output is from $0$ to $2^{22} - 1$, $y = (x + 2^{31}) \mod 2^{22}$ works. Apr19 comment How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ So, $0$ to $2^{22}$. The size of that output range is not a power of $2$, so this is not possible (I added a note in my answer about this). On the other hand, if you want $0$ to $2^{22} - 1$, that's easy. Apr19 revised How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ add note on impossibility Apr19 comment How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ Can you give an example of an output range that you're interested in? Apr19 comment what is smooth embedding Yes, $g$ wraps the real line $\mathbf{R}$ around a circle, so its image $g(\mathbf{R})$ is a circle. Now you need to visualize what $f$ does. Apr19 comment How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ But if it's a random sequence of bits, the interpretation of it as a signed or unsigned integer is up to you. For example, if you have a signed integer between -127 and 128, you can easily convert it to an unsigned integer between 0 and 255; they're both simply 8 bits. Apr19 answered How to scale a random integer in $[A,B]$ and produce a random integer in $[C,D]$ Apr19 comment what is smooth embedding OK, I suggest you start with something simpler. Define $g: \mathbf{R} \to \mathbf{R}^2$, $g(t) = (\cos(2\pi t), \sin(2\pi t))$. You can try out values of $t$ to get a feel for this function, or look it up under the name "wrapping function" if that helps. Once you understand $g$, $f$ should be easier. Apr19 comment Weak topology and the closed unit ball It seems to me that the open unit ball is a neighborhood of $0$ which lies inside the closed unit ball, so the statement is false. But maybe I'm missing something. Can you give more details? You're assuming some kind of topological vector space, I think; anything else? Apr19 comment what is smooth embedding I think if you draw what $f(\mathbf{R})$ looks like, you will quickly know whether it's homeomorphic to $\mathbf{R}$. To prove it, you could look for a continuous map in the other direction (from $f(\mathbf{R})$ to $\mathbf{R}$) which would be an inverse to $f$. Apr19 answered Can a sequence whose final term is an axiom, be considered a formal proof? Mar13 answered Abstract/formal interest of rings Mar6 comment Verification that a statement is true or not. I see. Well, if you write $a = px$ (where $p \nmid x$) and $b = py$ (where $p \nmid y$), then it looks pretty clear that $p^2 \mid ab$ but $p^3 \nmid ab$. See if that helps. Mar6 comment Verification that a statement is true or not. I'm still not sure what you're trying to prove, but if $p^3$ is the greatest common divisor of $ab$ and $p^4$, then in particular $p^3$ must be a divisor of $ab$. So it seems clear that the first statement is true. Mar6 answered Verification that a statement is true or not. Mar3 comment Are there variations on definition of locally finite category? "Is there already a name for the locally 2-finite categories?" I'll take that as a no. Mar1 asked Are there variations on definition of locally finite category?