Lets assume an ordered finite sequence of integers A = {0,1,2, ... k}. Im looking for a (seeded) function f: A -&gt; A where the image of f seems to be ramdom, but every element is hit exactly once (bijective).  It's importatant that someone who has access to some results of f should neither be able to estimate if he's looking at direkt neighbours f(n), f(n+1) nor should he be able to guess the next result.  One idea would be to map a range(1, k) on a shuffle(range(1, k)). But this seems very low in performance (for large k) and does miss the basic idea of a generator. The first element in the sequece should be as expensive to calculate as the n-th element.  Difference to 'normal' randomgenerators  Read carefully before marking as dublicated. I'm not looking for randomly picking from a range where collisions are possible. Also I want to calculate the n-th element without calculating all n-1elements before, like I would have to using the MT or Xorshift.
I'd like to generate unique random numbers between 0 and 1000 that never repeat (i.e. 6 doesn't show up twice), but that doesn't resort to something like an O(N) search of previous values to do it. Is this possible?
Suppose that $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$,  where $X_1,X_2,\ldots$ and $Y_i,Y_2, \ldots$ are i.i.d. sequences of random variables with mean $\mu$ and variance $\sigma^2$ and the sequences $\{X_i\}$ and $\{Y_i\}$ are independent.  Then, for all real numbers $x$, show that, as $n\to \infty$,  $$\mathbb{P}(U_n\leq x) \to  \int_{-\infty}^{x} \frac{1}{(\sqrt(2\pi))} \:e^{-\frac 12{u^2}}\mathrm du $$ Approach  I tried to carry out the integration on the right, but this does not integrate nicely. It ends up involving an error function, so I figured this was the wrong was to go.  Should I be looking at convergence in probability to answer this question? Is Chebyshev's inequality relevant here? If so, how and how does that relate to the integral? What is dependency injection? I am looking for a preferably introductory text on Yang-Mills theory with mathematical basis and language. (Differential forms) I have a background in QCD and QED, however, I want to go more in details in geometrical interpretations of Yang-Mills theory. I am sorry if this question has asked before. Thanks in advance. What is EV, when used as an absolute measurement? Inequality $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$ Hi I'm traveling to Uzbekistan and have 12 hours layover in Seoul via Korean Airlines. Do I need a visa to go out of the airport for sightseeing? I'm a Philippine passport holder. How do I shuffle cards for a card game? I am trying to develop an card game for Android Can anyone suggest me how to write code for effectively shuffling the playing cards? Can you concentrate on a special ability while also concentrating on a spell? There exist several abilities that have the text similar to, this effect lasts while you maintain concentration (as if you were concentrating on a spell). One such example is the Bard College of Glamour: Mantle of Majesty ability.     At 6th level, you gain the ability to cloak yourself in a fey magic   that makes others want to serve you. As a bonus action, you cast   command, without expending a spell slot, and you take on an appearance   of unearthly beauty for 1 minute or until your concentration ends (as   if you were concentrating on a spell). During this time, you can cast   command as a bonus action on each of your turns, without expending a   spell slot.   Another example is the Cleric Trickery Domain: Channel Divinity: Invoke Duplicity ability.     Starting at 2nd level, you can use your Channel Divinity to create an   illusory duplicate of yourself.      As an action, you create a perfect illusion of yourself that lasts for   1 minute, or until you lose your concentration (as if you were   concentrating on a spell).   While you are concentrating on maintaining such an ability would you also be able to maintain concentration on a spell that requires concentration?  There does not seem to be any RAW that limits this however the limitation around concentrating on a second spell exists which may suggest that the intent is that a character cannot concentrate on 2 effects at the same time unless explicitly stated. How to use GROUP BY to concatenate strings in MySQL? Basically the question is how to get from this:   foo_id   foo_name 1        A 1        B 2        C   to this:   foo_id   foo_name 1        A B 2        C How do shock waves form in space? Today my aerodynamics professor mentioned that the equations we were learning for atmospheric shock waves can also be applied by astrophysicists to study black hole-related shock waves in space.  How is this possible? Isn't space too rarefied for shock waves? What is an example application? He mentioned something about the density limit before/after a relativistic shock approaching 6. Removal of statistically significant intercept term increases $R^2$ in linear model existence of sequence of polynomial Is there a sequence of polynomials $P_n$ such that   $\displaystyle\lim_{n \to \infty} P_n(z) = \begin{cases} 1, &amp; \text{Im } z &gt; 0\\ 0, &amp; z \text{ is real,} \\ -1, &amp; \text{Im } z &lt; 0 \end{cases}$  I have no clue where to start from. Please provide some hints. Thanks in advance. What should a physics undergrad aspiring to be a string theorist learn before grad school? The question I guess is pretty clear. I am a physics undergrad wishing to pursue research in quantum gravity(string theory?). What are the subjects I should learn other than the usual compulsory undergrad courses such as quantum mechanics, and GR, etc? Things that come to my mind are QFT, lie algebras, differential geometry, algebraic geometry etc? Other than this, could you suggest any topic that are not entirely essential but interesting in mathematical physics, gauge theory etc (e.g. solitons, monopoles etc). Actually I am looking for a mini project of sorts which would give me an idea of what these fields really are. I have done some lie algebras, classical field theory and am currently studying QM. Old post apocalyptic cartoon where the main character has red hair and lives with a nomadic group Has the wall been repaired? Create WiFi hotspot on ubuntu PHP Session data not being saved I have one of those "I swear I didn't touch the server" situations. I honestly didn't touch any of the php scripts. The problem I am having is that php data is not being saved across different pages or page refreshes. I know a new session is being created correctly because I can set a session variable (e.g. $_SESSION['foo'] = "foo" and print it back out on the same page just fine. But when I try to use that same variable on another page it is not set! Is there any php functions or information I can use on my hosts server to see what is going on?  Here is an example script that does not work on my hosts' server as of right now:  &lt;?php session_start(); if(isset($_SESSION['views']))     $_SESSION['views'] = $_SESSION['views']+ 1; else     $_SESSION['views'] = 1;  echo "views = ". $_SESSION['views']; echo '&lt;p&gt;&lt;a href="page1.php"&gt;Refresh&lt;/a&gt;&lt;/p&gt;'; ?&gt;   The 'views' variable never gets incremented after doing a page refresh. I'm thinking this is a  problem on their side, but I wanted to make sure I'm not a complete idiot first.  Here is the phpinfo() for my hosts' server (PHP Version 4.4.7): I have looked in several different Fate games' rulebooks, including Fate Core, DFRPG, Atomic Robo, FAE, and Jadepunk, which all describe Boosts as temporary Aspects you get to invoke for free, one time, and which disappear after being invoked.  In the other "invoke for free" situation I'm aware of (on Aspects newly-created via the Create Advantage action), they aren't singled out as "temporary."  I looked in many books but I'm interested in Fate Core.  So I haven't been able to figure out whether Boosts need to be invoked immediately or they disappear, without having been invoked at all? Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$ It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?  I got:  $2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\mathrm d\alpha \,\mathrm dr$,  where $x = r \cos(\alpha)$ and $y = r \sin(\alpha)$.  but cannot understand and get the right side. The problem emerged when trying to integrate $\displaystyle \int_0^\infty e^{\frac{-x^2}{n^2}}\,\mathrm dx$ where I tried to change the problem knowing $r^2=x^2+y^2$ but stuck to this part. What is the change in the title called and why is it so? According to , the SE chat sites may have individual Easter eggs, depending on the site. If you find them, please post. Because it's so easy to make this stuff up, a screenshot as proof would be nice. (Although Photoshop isn't that hard either...)