How small must an object be in order for it to be subject to the laws of quantum mechanics?
I apologize if this question is posted on the wrong portion of the website, but I knew no better place to ask.  I've come to see that some effects, such as super-conduction may happen at either extremely low temperatures, or at extreme temperatures (the recent discovery of superconducting iron in the bowels of the earth, for one), and this is fascinating.  My question is: can (does?) the same principle be applied to quantum effects? Would it be applicable to, for example, say that although quantum effects happen at nano scales and below they may happen on a grand scale of say, the entire Universe (as a solitary unit)?  Can said quantum effects happen at moments of say, picoseconds after the Big Bang due to the turbulence of forces involves in such an event? Or even within the core of a dying star?  Thank you for any answers or clarification of my misinformation.
Creating your own Tinyurl style uid I'm writing a small article on humanly readable alternatives to Guids/UIDs, for example those used on TinyURL for the url hashes (which are often printed in magazines, so need to be short).  The simple uid I'm generating is - 6 characters: either a lowercase letter (a-z) or 0-9.   "According to my calculations captain", that's 6 mutually exclusive events, although calculating the probability of a clash gets a little harder than P(A or B) = P(A) + P(B), as obviously it includes numbers and from the code below, you can see it works out whether to use a number or letter using 50/50.  I'm interested in the clash rate and if the code below is a realistic simulation of anticipated clash rate you'd get from generating a hash. On average I get 40-50 clashes per million, however bare in mind the uid wouldn't be generated a million times at once, but probably only around 10-1000 times a minute.  What is the probability of a clash each time, and can anyone suggest a better way of doing it?  static Random _random = new Random();  public static void main() {     // Size of the key, 6     HashSet&lt;string&gt; set = new HashSet&lt;string&gt;();     int clashes = 0;     for (int n=0;n &lt; 1000000;n++)     {         StringBuilder builder = new StringBuilder();          for (int i =0;i &lt; 7;i++)         {             if (_random.NextDouble() &gt; 0.5)             {                 builder.Append((char)_random.Next(97,123));             }             else             {                 builder.Append(_random.Next(0,9).ToString());             }         }          if (set.Contains(builder.ToString()))         {             clashes++;             Console.WriteLine("clash: (" +n+ ")" +builder.ToString());         }          set.Add(builder.ToString());         _random.Next();         //Console.Write(builder.ToString());     }      Console.WriteLine("Clashes: " +clashes);     Console.ReadLine(); }   UPDATE:   from this question  I really asked two questions here so I was cheating. The answer I was after was rcar's, however Sklivvz's is also the answer to the 2nd part (an alternative). Is it possible to make a custom unique id generator in a database, or would it be client side (which would be 2 possible reads first)?  The general idea I was after was using Ids in databases or other stores that can be used by phone or printed material, not a giant 16 byte guid.  UPDATE 2: I put the formula for two mutually exclusive events above instead of 2 independent ones (as getting an 'a' the first time doesn't mean you can't get an 'a' the second time). Should've been P(A and B) = P(A) x P(B) If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$. How to restore a system after accidentally removing all kernels? I was trying to delete old kernels, but I must have deleted all of the kernels on my Ubuntu 11.04 laptop. Is there any way to fix this via USB boot or mounting hard drive on another system? Why does second pre-image resistance imply pre-image resistance I am studying hash functions. I can understand why collision resistance implies second preimage resistance, but I don't get why second preimage resistance should imply first preimage resistance. Could anybody be help me with this argument from Introduction to Modern Cryptography by Katz &amp; Lindell, please?   Collision resistance: This is the strongest notion and the one we have considered so far.  Second pre-image resistance: Informally speaking, a hash function is second pre-image resistant if given $s$ and $x$ it is infeasible for a probabilistic polynomial-time adversary to find $x' \ne x$ such that $H^s(x') = H^s(s)$.  Pre-image resistance: Informally, a hash function is pre-image resistant if given $s$ and $y = H^s(x)$ (but not $x$ itself) for a randomly chosen $x$, it is infeasible for a probabilistic polynomial-time adversary to find a value $x'$ such that $H^s(x') = y$. (Looking ahead to later chapters in the book, this essentially means that $H^s$ is one-way.)   [...] Likewise, any hash function that is second pre-image resistant is also pre-image resistant. This is due to the fact that if it is possible to invert $y$ and find an $x^\prime$ such that $H^s (x^\prime ) = y$ then it is possible to take $x$, compute $y = H^s (x)$ and invert it again obtaining $x^\prime$. Since the domain of $H$ is infinite, it follows that with good probability $x \neq x^\prime$. We conclude that the above three security requirements form a hierarchy with each definition implying the one below it. I have the following code and I expect the resulting angle to be 135 degrees but whatevet I try, I get 90.  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \begin{tikzpicture}  \coordinate (A) at (1,0); \coordinate (B) at (-1,1);  \pgfmathanglebetweenpoints{A}{B} \edef\angleAB{\pgfmathresult} \node at (0,0) {\angleAB};  \end{tikzpicture} \end{document}   I also tried the following command without knowing what it means, it also prints 90 except one case. The only working case is when A=(1,0) and B=(-1,0), which it prints 180.  \pgfmathanglebetweenpoints{\pgfpointanchor{A}{center}}{\pgfpointanchor{B}{center}}   Why is that? Couldn't find any reference document for "pgfmathanglebetweenpoints" by the way, that's why I am asking here.  EDIT: B is corrected from (-1,-1) to (-1,1) Citations with no bibliography at the end How do you produce a document with citations (\citep, \citet) working properly in the paper but without a bibliography at the end?  I've come up with one solution that is bad because it breaks my makefile.  latex file.tex bibtex file # comment out \bibliography{} command latex file.tex   Similarly, is there a way to produce the bibliography with no accompanying text?  Again, I've come up with a solution: fill an otherwise blank .tex file with \nocite{} commands.  However, this is a pain because I have to keep two files synced by hand (and regular expressions to search for \citep{author1, author2} can be a huge pain when there are pagebreaks...). How do I remove 'Places' entries from the Nautilus sidebar? Repeating “will” for two verbs with “and” Evaluate $\sum_{k=1}^\infty\frac{x^k}{k(k+1)}$ My partner is tutoring a Civil Engineering student in maths. Conveniently, I am a civil engineer so when any maths that's confusing comes up, I can usually help out. However, we are having a problem with the summing of power series, one area where I am sorely lacking.  After spending two hours on a question, we still cannot solve it and are always left with a $k$. Admittedly, I don't fully know what I am doing.     Evaluate $\displaystyle \sum_{k=1}^\infty\frac{x^k}{k(k+1)}$.   We do know that the answer is $1 + \dfrac{1-x}{x}\ln (1-x)$.  Any hints or help in the direction to go would be very much appreciated. Is there a word for happiness made possible by a tragic situation? Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$. The difference between an analogy and a metaphor? Many a time I've asked what the difference is between an analogy and a metaphor. I've asked it to my teacher, on internet sites, to my parents, so on and so forth. I got a different answer every time, and I never fully grasped what the difference is, so what is the difference? How to find historical airfares? Is there a way to find out how much it would have cost to buy a particular airplane ticket at some point in the past?    For example, how could I find out how much it would have cost to buy a one-way itinerary from airport A to airport B on October 1, 2011 for travel on November 1, 2011? Is the set of all possible binary strings countable? Determine whether the given sets are countable or uncountable. Justify your answers either with bijections or using results/methods. The set of all possible binary strings. Note a binary string is a sequence of 0s and 1s.  If a binary string consists only of 0s and 1s, then is it even possible to find a bijection for it? I start with knowing that two numbers are coprime if:  $n*k + m*j = 1$  So, setting $k = a$ and j = $a+1$ I can solve as follows:  $n*a + m*a + a$  Then,  $a(n+m) + m = 1$  Where can I go from here? I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism? Suppose that $(U,V)$ has the following joint pdf: $$f_{U,V}(u,v)=\exp(-\theta u-\theta^{-1}v)$$ , where $u\geq0$, $v\geq0$, $\theta&gt;0$. Define $X=UV$ and $Y=U/V$. Find the marginal pdf of $X$ and $Y$.  With transformation, I got $U=\sqrt{XY}$ and $V=\sqrt{X/Y}$ and the absolute value of Jacob is $1/2Y$. Then, the joint pdf of $(X,Y)$ is  $$f_{X,Y}(x,y)=\exp(-\theta \sqrt{XY}-\theta^{-1}\sqrt{X/Y})1/2Y, X,Y&gt;0.$$  But I am not sure how to get the marginal pdf?  Thank you! I am trying very hard to install Canon LBP2900b Printer in Ubuntu 13.10.  I have searched and googled a lot for the solution over fortnight but none of the site / link gave me the simple solution for me. How can accomplish my goal? When does Bleeding Attack effect apply? I've designed for a campaign a duo of rogues. My character is as , and his follower is a Knife Mistress. Couple of question about them. Our party is about to ambush several . My Sniper will hide near their camp, and shoot the weakest one when fight begins. His follower will then rush the troll and slice him in several pieces. My Sniper has the Bleeding attack feat:  Bleeding Attack (Ex) Benefit: A rogue with this ability can cause living opponents to  by hitting them with a sneak attack. This attack causes the target to take 1 additional point of damage each round for each die of the rogue's sneak attack (e.g., 4d6 equals 4 points of bleed). Bleeding creatures take that amount of damage every round at the start of each of their turns. The bleeding can be stopped by a DC 15 Heal check or the application of any effect that heals hit point damage.  Emphasis mine. Thus, a normal enemy would bleed every turn, until he get cleaned of the bleeding effect. However, troll have a special ability, which is  5 (acid/fire)  Regeneration (Ex) A creature with this ability is difficult to kill. Creatures with regeneration heal damage at a fixed rate, as with , but they cannot die as long as their regeneration is still functioning (although creatures with regeneration still fall unconscious when their hit points are below 0). Certain attack forms, typically fire and acid, cause a creature’s regeneration to stop functioning on the round following the attack. During this round, the creature does not heal any damage and can die normally. The creature's descriptive text describes the types of damage that cause the regeneration to cease functioning.  So, the troll will automatically get rid of the bleeding effect during his first turn. My question is: Does the bleeding effect apply before the regeneration, or get neglected, because the troll will heal during his turn? Am I right to assume that, Bleeding attack effect getting triggered at the beginning of the troll turn, it will deal him the damages before the wounds heals? $2$ out of $3$ property of the unitary group