If Kendall's tau is lower than Spearman, what does it mean?
In my work, we are comparing predicted rankings versus true rankings for some sets of data. Up until recently, we've been using Kendall-Tau alone. A group working on a similar project suggested we try to use the  instead, and that they preferred it. I was wondering what the differences between the different rank correlation algorithms were.  The best I've found was , which claims Spearman is used in place of usual linear correlations, and that Kendall-Tau is less direct and more closely resembles Goodman-Kruskal Gamma. The data I'm working with doesn't seem to have any obvious linear correlations, and the data is heavily skewed and non-normal.  Also, Spearman generally reports higher correlation than Kendall-Tau for our data, and I was wondering what that says about the data specifically. I'm not a statistician, so some of the papers I'm reading on these things just seem like jargon to me, sorry.
Can a smaller sensor's "crop factor" be used to calculate the exact increase in depth of field? If APS-C and similar crop-sensor digital cameras have a focal length multiplying effect such that a 50mm lens has an apparent focal length closer to the field of view of an 80mm on a full frame camera, and yet at the same time the depth of field for the smaller sensor camera is more like the depth of field a 50mm lens would produce on a full frame camera (using the same aperture), then this would seem to suggest the concept of an "aperture dividing effect."  In other words, a 50mm f/1.8 lens on an APS-C camera would act more like a 80mm f/2.8 (approx. 1.8 * 1.6x) lens in 35mm equivalent — for depth of field, not considering exposure.  Can someone with a better understanding of the physics involved clarify this for me.  I've never seen this concept explicitly mentioned anywhere, so I am a bit suspect of it. Could we provide some automatic feedback on (enough) unfriendly/rude comments? If $a_k\geq0$ is a decreasing sequence converging to $0$, do we have $\lim_{n\to\infty}\frac{\sum_{k=1}^n a_k}n = 0$? Let $a_k\geq 0$ be a non-increasing sequence satisfying $\lim_{k\to\infty} a_k = 0$. Is it always true that $$\lim_{n\to\infty} \frac{\sum_{k=1}^n a_k}n=0$$ ?  I tried proving with "brute force" but failed: Let $\varepsilon&gt;0$. Then there exists a $K\in\mathbb N$ such that $a_k\le\varepsilon$ for all $k\geq K$. Let $C=\sum_{k=1}^K a_k$. We have $$\frac{\sum_{k=1}^{\left\lceil\frac C\varepsilon\right\rceil}a_k}{\left\lceil\frac C\varepsilon\right\rceil}=\frac{C}{\left\lceil\frac C\varepsilon\right\rceil}+\sum_{k=K+1}^{\left\lceil\frac C\varepsilon\right\rceil} a_k\le \varepsilon +\varepsilon\cdot\left( \left\lceil\frac C\varepsilon\right\rceil-K\right).$$  However, this is not good, as the right-hand side depends on $C$, which can get large for small $\varepsilon$.   Also, numerical computations for example with $a_k=\frac1{\log\log k}$ seem to confirm the statement though...  (Additionally, note that I didn't use the non-increasing fact, but I am not sure if it is needed.) Question : $$\begin{align}a+b+c &amp;= 0\\ a^3 +b^3 +c^3 &amp;= 12\\ a^5 +b^5 +c^5 &amp;= 40\end{align}$$ Then , $a^4 + b^4 + c^4 = ?$  My try : As a common perspective I just went to find any trick related to it and my first step is as usual as a common man will think $3abc = 12$ and got $abc = 4$ After that I am unable to link it with any other equation I get a message:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Why is there a question limit? 50/month is just not enough :) Is there a conventional wisdom to what gear a bike should be left in, when used infrequently? What is a good general document size for web page mockups and prototypes? I do mockups and prototypes of sites and designs all the time, but I've always wondered if there is a standard size or width I should be using. Generally I'll make my document either 1024, 1200, or 2560px wide, depending on what I'm doing.  So what do you think? What is a good or standard size for this? At least it's my understanding that AES isn't affected by known-plaintext. Is it immune to such an attack, or just resistant? Does this vary for chosen-plaintext? Maximum total distance between points on a sphere What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?  The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points.  All the points are on a single sphere of radius $R$.  Here's a visualization:   Acknowledgements: Based on  Physics S.E. question. Image from . Shortcuts in Objective-C to concatenate NSStrings Are there any shortcuts to (stringByAppendingString:) string concatenation in Objective-C, or shortcuts for working with NSString in general?  For example, I'd like to make:  NSString *myString = @"This"; NSString *test = [myString stringByAppendingString:@" is just a test"];   something more like:  string myString = "This"; string test = myString + " is just a test"; I'm using LyX on Linux Mint 14 ...  Using BibTeX with the apalike style and some modifications in the preamble, I've managed to get the bibliography as it should be except for one thing. When I cite a work with multiple authors it gets printed with the separator "and" which isn't correct for my language (Danish). Curiously it's displayed correctly in the user interface of Lyx - for example "(Wittgenstein og Waismann 2003 - which led me to believe everything was ok but in the pdf the separator "and" is used both in the reference and in the bibliography.  Is there any way to fix this using BibTeX, or can it only be done with biber 1.5, as suggested in the last comment here:   I really hope their is because getting biber to work using the following guide has thus far proven to be impossible:   Help much appreciated ... I really hope I won't have to relapse into setting my master thesis up in LibreOffice. Is there any way to tell Google to use a particular account for all Google services by default? I typically use 2 Google accounts simultaneously in my browser. So whenever I fire up any Google service like Google Maps or YouTube or Google Search, it uses one of the accounts. The problem is that I don't want that account to be used and no matter how many times I switch the account for these services, they seem to default to same account.  Is there any way to tell Google to use a particular account for all Google services by default?    I am not talking about using Google account to sign in to third party websites. I am talking only about Google's own services.  I am using Chrome if anyone is going to suggest client-side solution. Taking the second derivative of a parametric curve Solving Keplerian two-body problem numerically? I'm trying to solve the two-body problem numerically, setting up G, m1 and m2 =1.0.  The masses are placed at the positions -10 and 10 respectively along the x-axis and gave them both 0 on the y-axis. I am having some real issue with the initial conditions fitting for a keplerian orbit with a given eccentricity (e=0.9)! Also how would the initial condition change for different mass ratios? Prove that the product of four consecutive positive integers plus one is a perfect square I need to prove the following, but I am not able to do it. This is not homework, nor something related to research, but rather something that came up in preparation for an exam.     If $n = 1 + m$, where $m$ is the product of four consecutive positive   integers, prove that $n$ is a perfect square.   Now since $m = p(p+1)(p+2)(p+3)$;     $p = 0, n = 1$ - Perfect Square      $p = 1, n = 25$ - Perfect Square      $p = 2, n = 121$ - Perfect Square   Is there any way to prove the above without induction? My approach was to expand $m = p(p+1)(p+2)(p+3)$ into a 4th degree equation, and then try proving that $n = m + 1$ is a perfect square, but I wasn't able to do it. Any idea if it is possible? Async Await Handler Deadlock How to see an embedded image's dpi (ppi) in a pdf in Acrobat Pro? The following situation baffled me slightly: after preparing a pdf in a different program I wanted to make sure that an embeded image in the pdf file had the right dpi, yet I found no way in adobe acrobat pro to inspect the file/image dpi.   Can that be, or am I missing where this feature is hidden? Quantum Hall effect for dummies In the past few days I've become increasingly intrigued by the QHE, mainly thanks to very interesting questions and answers that have appeared here. Unfortunately, I am as of yet very confused by all the (seemingly disparate) stuff I learned.  First, here are some random points that I've been able to gather   I(nteger)QHE occurs due to the presence of Landau levels IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone IQHE requires negligible electron-electron interations and so is dependent on the presence of impurities that shield from Coulomb force F(ractional)QHE occurs because of formation of anyons. In this case Coulomb interaction can't be neglected but it turns out an effective non-interacting description emerges with particles obeying parastatistics and having fractional charge FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff FQHE has something to do with hierarchy states   So, here are the questions        Most importantly, do these points make sense? Please correct any mistakes I made and/or fill in other important observations   How do explanations 1. and 2. of IQHE come together? Landau quantization only talks about electron states while topological picture doesn't mention them at all (they should be replaced by global topological states that are stable w.r.t. perturbations)   How do explanations 4., 5. and 6. relate together   Is there any accessible introductory literature into these matters?   Do IQHE and FQHE have anything (besides last three letters) in common so that e.g. IQHE can be treated as a special case? My understanding (based on 3.) is that this is not the case but several points hint into opposite direction. That's also why I ask about both QHE in a single question Does ReLU layer work well for a shallow network? Is it touch(1) from coreutils-8.4-37.el6.x86_64 or my brain which is broken?  $ touch abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:06.189751847 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:06.189751847 +0800 $ touch -a abc $ LANG=C stat abc   File: `abc'   Size: 0               Blocks: 0          IO Block: 4096   regular empty file Device: fd04h/64772d    Inode: 10485773    Links: 1 Access: (0664/-rw-rw-r--)  Uid: (3060399/ nodakai)   Gid: (  418/   quant) Access: 2016-10-14 18:42:17.374235446 +0800 Modify: 2016-10-14 18:42:06.189751847 +0800 Change: 2016-10-14 18:42:17.374235446 +0800 $ touch --help | grep 'access time'   -a                     change only the access time   As you see, it was not only atime but also ctime which was updated by touch -a !?!?  The filesystem is ext4 over LVM if that makes any differences.