Are all or any hash functions evenly distributed?  For example: if you had every combination of 256 bits and ran that through SHA256, would that produce approximately 1 of each combination, or would there be peaks?  Are distributions of the hashes calculable without going through the permutations in lower bit sizes?  Is evenness of the distribution of a hash function impacted by the length of the input vs the resultant hash length?  Which hashing functions are most evenly distributed?  (e.g. as close to 1 of each equal-bit-length combination possible)  Edit: The linked "duplicate" doesn't come close to addressing all these questions.
In reading about the  protocol I noticed how much its proofs-of-work apparently depend on  of the SHA-256 hash function. And so  presumably do many other applications.  How do researchers go about proving uniformity of proposed hash functions? Is it just an empirical matter of running many inputs through the function and observing the distribution of outputs, or can one formally prove the property? Where could one (non-crypto-expert) learn about how such proofs are constructed and look one up e.g. for the case of SHA-256?
Add a linebreak in an HTML text area How can i add a line break to the text area in a html page? i use VB.net for server side coding. Intuitive explanation for how could there be "more" irrational numbers than rational? Is there any software or hardware which lets you stop, slow down, speed up or even reverse time? Obviously I'm talking about time in terms of the PC clock rather than real time.  We were testing an application we've developed at work by setting the clock forward and back to simulate different scenarios and I started thinking how useful it would be if you could adjust the rate(?) of the system clock with finer control. So you could make a minute pass in a second or a day pass in 30 seconds and watch how the program you're developing copes with changes in date and time.  Is there any such software available?   I'd be interested to hear if anyone knows of any software or hardware which can let you do some or all of the above. Search window not opening in ArcMap? Google Chrome: Is there a keyboard shortcut to open a link in a new tab? Is is possible to open a link in a new background tab in Google Chrome with a keyboard shortcut?  (I know it's possible with Ctrl+click and middle click, and I know that Shift+Enter opens the link in a new browser window.) For my own understanding, I am interested in manually replicating the calculation of the standard errors of estimated coefficients as, for example, come with the output of the lm() function in R, but haven't been able to pin it down. What is the formula / implementation used? Low wifi signal in 14.04 LTS Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then must there exist a homeomorphism from $A$ to $B$ ? What equipment and settings do I use for panning shots of moving objects? I'm using Windows 7 as the host machine with VMWare Workstation 9. I created a virtual machine Ubuntu Desktop 12.04 LTS 64 bit version. There are two options for log in, Ubuntu and Ubuntu 2D. Whenever I log in to Ubuntu, the task bar will not show up, is like having a black margin all around the desktop background and only the center is visible.  I wish I could attached a pic, but I need at least 10 reputation.  When I move my mouse over the left side of the desktop I can see that something is there but it is shown only as a black square. I tried deleting the virtual machine I created for Ubuntu and re-installed it, but it keeps appearing like that.  When I switched to log in with the Ubuntu 2D version everything worked alright. I could see the task bar, the icons, the clock etc. What might be causing this problem? what is the total number of distinct arrangements of $n$ oranges and $n$ apples around a round table? I have no idea how to go about. How can I get distribution name and version number in a simple shell script? What are the practical applications of the Taylor Series? I started learning about the  in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series.  Question: What are the practical applications of the Taylor Series? Whether it's in a mathematical context, or in real world examples. Why is this simple Excel calculation not netting Zero? It's supposed to be zero but for some reason the result produces a peculiar series of digits at the end.  Also, here's the .xlsx of it: http://83.212.111.36/Book1.xlsx Complete list of Lollipop system icons shown in notification bar? Is there a complete list of these? I've seen lots of Q&amp;A about "What does x icon mean" but it would be really useful if there were a complete list for reference! I saw a video where the person copied a material from one object to another (Cycles).  How can I accomplish this? Proof that the irrational numbers are uncountable Generating random points uniformly on a disk I have to randomly generate 1000 points over a unit disk such that are uniformly distributed on this disk. Now, for that, I select a radius $r$ and angular orientation $\alpha$ such that the radius $r$ is a uniformly distributed variate from $r \in [0,1]$ while $\alpha$ is a uniformly distributed variate from $\alpha \in [0, 2\pi]$ using the following code r &lt;- runif(1000, min=0, max=1)  alpha &lt;- runif(1000, min=0, max=2*pi) x &lt;- r*cos(alpha) y &lt;- r*sin(alpha) plot(x,y, pch=19, col=rgb(0,0,0,0.05), asp=1)  Then I look at my sample space and it looks like this:  This obviously doesn't look like a sample with uniform distribution over the disk. Hence, I guessed that the problem might be occurring as a result of a lack of independence between the variables $r$ and $\alpha$ in contingency to how they've been linked computationally. To take care of that I wrote a new code. rm(list=ls()) r &lt;- runif(32, min=0, max=1) df_res &lt;- data.frame(matrix(c(-Inf, Inf), byrow = T, nrow = 1)) for (i in 1:32) {   for (j in 1:32) {     alpha &lt;- runif(32, min=0, max=2*pi)     r &lt;- runif(32, min=0, max=1)     df &lt;- data.frame(matrix(c(r[i],alpha[j]), byrow = T, nrow = 1))     df_res &lt;- rbind(df_res,df)   } } df_res &lt;- subset(df_res, df_res$X1 != -Inf) x&lt;- df_res$X1 *cos(df_res$X2) y &lt;- df_res$X1 *sin(df_res$X2) plot(x,y, pch=19, col=rgb(0,0,0,0.05), asp=1)  And, yet again the sample looks non-uniformly distributed over the disk  I'm starting to suspect that there is a deeper mathematical problem going on in the vicinity. Could someone help me write code that would create a sample space uniformly distributed over the disk, or explain the mathematical fallacy if any in my reasoning? Why or how did Agen Kolar and Saesee Tiin seem to die so easily without any resistance? Getting started with neural networks for forecasting