Trying to get idea of reduction
Which of those functions are PRFs?
I am calculating the Intraclass Correlation coefficient (ICC) as per the method of Shrout-Fleiss (1979), using the (3, 1) model in which the judges are fixed as are the targets.  Some of my ICC values are negative.  Is this to be expected?  How to interpret this?  I understand that the maximum ICC is 1.0, but is there a minimum value?  How are negative values interpreted? Why does the co-efficent of kinetic friction of a block on an inclined plane not change when you increase the angle of the plane? Show that $\aleph_{\alpha}^{\aleph_{1}} = \aleph_{\alpha}^{\aleph_{0}}\cdot 2^{\aleph_1}$ So basically I want to show:  $\aleph_{\alpha}^{\aleph_{1}} = \aleph_{\alpha}^{\aleph_{0}}\cdot2^{\aleph_1}$ , Where $\omega\leq\alpha&lt;\omega_1$  There is a proof that I found;    But I don't understand his proof for the limit case of $\alpha$.  To be more specific, I dont understand why $ \left( \sup_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1} \le \left( \prod_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1}$  Any insights or help is deeply appreciated.  Cheers difference fn(String... args) vs fn(String[] args) Whats this syntax useful for :       function(String... args)   Is this same as writing       function(String[] args)    with difference only while invoking this method or is there any other feature involved with it ? Does a first generation iPod touch have the ability to download apps and use emojis? I am getting a 1st gen iPod touch for a friend and I was wondering if you can download like apps and games on it and if it has emojis? It would help to get an answer before I buy it! Possible Duplicate:          If I am using the word model in the context of financial models and the UK, then which of these words should I use? Also, are there any key difference in the meanings of the two? Do I choose the target or the ally for the Help action? I'm new to D&amp;D and am making my first character for a home game.  What should I discuss with my DM prior to my first game? And what do I need their permission for? Classes, races and sub-races I intend to choose? There is a lot of info in multiple sources; any word on what I have to to discuss with the DM? Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$ $x,y,z &gt;0$, prove   $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$   Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem for National TST of an Asian country a few years back. However, upon receiving the official solution, the committee decided to drop this problem immediately. They don't believe that any students can solve this problem in 3 hour time frame.   Update 1: In this forum, somebody said  that BW is the only solution for this problem, which to the best of my knowledge is wrong. This problem is listed as "coffin problems" in my country. The official solution is very elementary and elegant.   Update 2: Although there are some solutions (or partial solution) based on numerical method, I am more interested in the approach with "pencil and papers." I think the approach by  in  may help.    Update 3: Michael has tried to apply Peter Scholze's method but not found the solution yet.  Update 4: Symbolic expanding with computer is employed and verify the inequality. However, detail solution that not involved computer has not been found. Whoever can solve this inequality using high school math knowledge will be considered as the "King of Inequality". Combinatorics - binary - very interesting question I tried to solve this problem without any success. I really hope that you will know the answer. $f(n,m)$ presents the number of binary strings (empty string included) that include at most n times '1', and at most m times '0'.      Prove: $$f(n,m) = \binom{n+m+2}{n+1}-1$$ In QGIS, I want to make polygons from a set of intersecting lines. Looking at the screenshot here, I want chop the purple line (it's a closed line, not a polygon) where it crosses the green and blue lines. The result hopefully would be a polygon where I have hatched in red. I have quite a few to do.    I have tried the QGIS tool previously called Polygonizer (see ), but now called Polygonize. This is a tool that, given a layer containing a set of crossing lines arranged essentially like a # symbol, should create a polygon from the enclosed space in the middle.  (In QGIS 2.01, Polygonize is in the top menu bar: go to Processing -> Commander, then type "Polygonize" on the command line, select "Processing algorithm: Polygonize" from the list that appears, and press Enter.)   At the momemt Polygonize doesn't work properly. Is there another way to do what I want to do?   For example, wouldn't it be great if the QGIS 'Split Feature' command was able to work like MapInfo's 'Polyline Split', where you select the polygons to split, then select an already-drawn polyline to split by, and voila. But Split Feature seems not to work that way.    UPDATE: Detail of problems with running Polygonize.  I have 64-bit QGIS 2.01 installed to Windows 7 from the installer package, QGIS-OSGeo4W-2.0.1-3-Setup-x86_64.exe.  The routine processed an output from my shapefile, but did not create a polygon from the region enclosed by intersections between the blue, green and purple lines. (Note: in the actual input file, all these lines are on the same layer; the screenshot just colours them differently for illustrative purposes). What I got instead was a simple conversion of the purple outline to a polygon, as though the program completely ignored the blue and green lines.  When I node-edited the closed purple line, to open it so that I had a set of open crossing lines like in the # symbol, and then ran Polygonize again, nothing happened. No crash, but no visible polygon in the output file.    When I used Nick Hopton's own  data, I got a crash; see screenshot: Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct? Is $\frac{x^2+x}{x+1}$ a polynomial?  Fist question can be: on which field/ring or etc? In basic, let's take over $\mathbb R$.  Actually, it is $x$ if $x \ne -1$.  Can we say again it is a polynomial, or a near-polynomial or what?   Can we say it is a polynomial over $\mathbb R - \{-1\}$? But there is structure here for $\mathbb R-\{-1\}$.  This question can be academically, not sure, but I want to ask it in the level of primary school, then academically. Trying to understand what happens with a brokerage firm's customers when it goes bankrupt (case of study: Lehman Brothers) I'm trying to understand how secure it is to own mutual funds/ETFs, not in terms of market volatility etc., but in terms of judicial security (i.e. what happens to your holdings if the firm goes bankrupt). To understand it, I have read this article about the bankruptcy of Lehman Brothers:  Note: From now on, I'm always referring to investors who own funds/ETFs managed by Lehman Brothers, not by investors who invested in Lehman Brothers or subprime mortgages (e.g. S&amp;P 500 ETF investors). From the above article, what I have understood is that the funds/ETFs managed by Lehman Bros. were shifted to other brokerage firms to be managed by them. Is this what actually happened? Still, what happens with the investors if, hypothetically, no brokerage firm in the world wants to manage these financial products? Let $F$ be a field and $F^\times$ be its group of units. If $F^\times$ is cyclic, then show that $F$ is finite.  I'm a bit stuck. I know that I can represent $F^\times = \langle u \rangle$ for some $u \in F^\times$ and that we must have that $|F^\times| = o(u)$, where $o(u)$ denotes the order of $u$ in $F^\times$.  I tried assuming $o(u) = \infty$, but I'm not sure exactly where to go from there. I was wondering if I could get a hint. How to output multiple image sizes from one design? I am designing an app for Android and as you may know that Android uses 6 different density buckets from LDPI to XXXHDPI.   Is there a way to save multiple outputs of different sizes from one design using either Photoshop, Illustrator, or another software?  If not will I have to resize it 6 times or is there an easy way? Render looks "brighter" than in viewport preview Somehow my viewport preview shows darker result than what comes out in render (check out in corner of the pic)  World strength is at zero and ambient occlusion is also disabled.  Floor is using bump and roughness from picture. Wall has adaptive subdivision surface  modifier.  All lit using planes with emission shaders.  I wanted to play around with render layers and maybe I screw some setting there, hope this helps. I always thought with "any" I should use the plural, but on the internet I can find both:   It can be found in any book. It can be found in any books Do you have any books? It can be said in any language.  This can be understood by anyone.  It has been used in any form.   So, what's correct?  Is there any rule? How to make the taskbar blink my application like Messenger does when a new message arrive? a, b are integers. p is prime. I want to prove: $(a+b)^{p} \equiv a^p + b^p \pmod p$  I know about Fermat's little theorem, but I still can't get it I know this is valid: $(a+b)^{p} \equiv a+b \pmod p$ but from there I don't know what to do.  Also I thought about $(a+b)^{p} = \sum_{k=0}^{p}\binom{p}{k}a^{k}b^{p-k}=\binom{p}{0}b^{p}+\sum_{k=1}^{p-1} \binom{p}{k} a^{k}b^{p-k}+\binom{p}{p}a^{p}=b^{p}+\sum_{k=1}^{p-1}\binom{p}{k}a^{k}b^{p-k}+a^{p}$ Any ideas?  Thanks!