Introduction  I was playing around with the zeta function when I thought of this series:  $$\sum_{n=2}^{\infty} \ln(\zeta(n))$$  According to  this sum converges and gives the following approximation:  $$\sum_{n=2}^{\infty} \ln(\zeta(n))\approx 0.83067$$    Questions   Is this a "known" series from before? I find this series quite cool and I thought I would find it somewhere but I didn't.  says it confirmed the convergence by the root test, can anybody confirm this? What's the closed-form of what the series converges to (exact value)?     Extra: My attempts  I tried to solve the 3rd question, but I don't have much ideas. An attempt I made was to note that:  \begin{align} \sum_{n=2}^{\infty} \ln(\zeta(n)) &amp;= \sum_{n=2}^{\infty} \ln(\prod_p \frac{1}{1-p^{-n}}) \\\\ &amp; = \sum_{n=2}^{\infty} \sum_p \ln(\frac{1}{1-p^{-n}})) \\\\ &amp; = -\sum_{n=2}^{\infty} \sum_p \ln(1-p^{-n})) \end{align}  ...but it didn't get me anywhere.  also lists some integrals that are equal to $\ln(\zeta(n))$, but I'm not that familiar with them. Here's an example:  $$\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\,\mathrm {d} x$$ And also: $$\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }J(x)x^{-s-1}\,\mathrm {d} x$$ where $J(x)$ is the Riemann prime-counting function. I doubt that these are going to help, but I'll just put them here.  I'm not experienced in evaluating infinite series so I don't have anymore ideas. It would be highly appreciated if some of you could shed a little bit of light on this question!  Thanks in advance!   EDIT 1:  As  noted,  is quite similar because my series is equal to $\ln(v)$ in the other question.  EDIT 2: Using basically the same method as in , we obtain (using one of the equations above):  \begin{align} \sum_{n=2}^{\infty} \ln(\zeta(n)) &amp;= \sum_{n=2}^{\infty} n\int _{0}^{\infty }J(x)x^{-n-1}\,\mathrm {d} \\\\ &amp; = \int _{0}^{\infty}J(x)(\sum_{n=2}^{\infty}\frac{n}{x^{n+1}})\,\mathrm {d} x \\\\ &amp;= \int_0^{\infty} J(x) \frac{2x-1}{(x-1)^2x^2}\,\mathrm {d} x\end{align}  This is a quite interesting result which I believe is correct, but I'm still not satisfied. We can do better!
Another problem from quora.  What can be said about $v =\prod_{s=2}^{\infty} \zeta(s) $?  Wolfy says that $v \approx 2.294856591673313794183 $.  The Inverse Symbolic Calculator () finds nothing.  Here are the close values:  $2294856391585950 = (0064) sum((7/6*n^3-3*n^2+65/6*n-3)/(Fibo(n)+1),n=1..inf)\\ 2294856397653578 = (0001) GAM(5/6)*BesI(1,2)^{GAM(7/12)}\\ 2294856469349552 = (0006) 1/4357571\\ 2294856473388934 = (0324) 1/10*(10+14^{1/2}*10^{1/4})^{1/2}*10^{3/4}\\ 2294856488490537 = (0001) BesJ(0,1)^{GAM(1/3)/GAM(5/12)}\\ 2294856524215021 = (0314) sin(Pi*8/45)-sin(Pi*14/51)\\ 2294856591673313 \text{ would be here}\\ 2294856740350177 = (0011) sum((2*n^3-5*n^2+22*n-14)/(n!+2),n=1..inf)\\ 2294856991319600 = (0324) 23^{1/2}/(12^{3/4}-3^{2/3})^{1/2}\\ 2294857017919943 = (0001) Feig2*Li4(1/2)/GAM(5/6)\\ 2294857089879260 = (0248) F(11/26;27/50;1) \\ 2294857493303450 = (0404) Psi(1/21)+Psi(19/21)+Psi(13/14)\\ $  I tried using Euler's zeta product and partition sum, but this didn't seem to help. Here is what resulted:  $\zeta(s) =\dfrac1{\prod_p (1-p^{-s})} $.  $\begin{array}\\ v &amp;=\prod_{s=2}^{\infty} \zeta(s)\\ &amp;=\prod_{s=2}^{\infty} \dfrac1{\prod_p (1-p^{-s})}\\ &amp;=\prod_{s=2}^{\infty} \prod_p\dfrac1{ (1-p^{-s})}\\ &amp;= \prod_p\prod_{s=2}^{\infty}\dfrac1{ (1-p^{-s})}\\ \end{array} $  Euler's product identity states that $\prod_{n=1}^{\infty}\dfrac1{1-zx^n} =\sum_{n=1}^{\infty} \dfrac{x^nz^n}{\prod_{k=1}^n(1-x^k)} $. Putting $x=z=\dfrac1{p} $, $\prod_{n=2}^{\infty}\dfrac1{1-p^{-n}} =\sum_{n=1}^{\infty} \dfrac{p^{-2n}}{\prod_{k=1}^n(1-p^{-k})} $.  That's as far as I can go.
How to change an input button image using CSS So, I can create an input button with an image using &lt;INPUT type=&quot;image&quot; src=&quot;/images/Btn.PNG&quot; value=&quot;&quot;&gt;  But, I can't get the same behavior using CSS. For instance, I've tried &lt;INPUT type=&quot;image&quot; class=&quot;myButton&quot; value=&quot;&quot;&gt;  where &quot;myButton&quot; is defined in the CSS file as .myButton {     background:url(/images/Btn.PNG) no-repeat;     cursor:pointer;     width: 200px;     height: 100px;     border: none; }  If that's all I wanted to do, I could use the original style, but I want to change the button's appearance on hover (using a myButton:hover class). I know the links are good, because I've been able to load them for a background image for other parts of the page (just as a check). I found examples on the web of how to do it using JavaScript, but I'm looking for a CSS solution. I'm using Firefox 3.0.3 if that makes a difference. Managing error with GPS routes (theoretical framework?) I'm looking for the appropriate theoretical framework or speciality to help me deal with understanding how to deal with the errors that the GPS system has - especially when dealing with routes.  Fundamentally, I'm looking for the requirements on the data and any algorithms to use to be able to establish the length of a trail. The answer needs to be trustworthy.  A friend of mine was the race director of a race which was billed as 160km but the Garmin watches everybody has makes it more like 190km+. It caused quite some grief at the finish line, let me tell you!  So my friend went back to the course with various GPS devices in order to remap it and the results are interesting.  Using a handheld Garmin Oregon 300 she got 33.7km for one leg. For the same leg on a wrist watch Garmin Forerunner 310xt it came out to 38.3km.  When I got the data from the Oregon it was obvious that it was only recording data every 90 seconds or so. The Forerunner does it every couple of seconds.  When I plotted the data from the Oregon I could see that it got confused by some switchbacks and put a line straight through them and a curve was made a little less.  However, I muse that the difference in the recording frequency is much of the explanation. i.e. by recording every couple of seconds the Forerunner is closer to the real route. However, there will be an amount of error because of the way GPS works. If the points recorded are spread around the real route randomly (because of the error) then the total distance will be larger than the real route. (A wiggle line going either side of a straight line is longer than the straight line).  So, my questions: 1. Are there any techniques I can use on a single dataset to reduce the error in a valid way? 2. Does my theory about the difference in recording frequency hold water? 3. If I have multiple recordings of the same routes are there any valid techniques to combine them to get closer to the real route?  As I say, I don't really know what to search for to find any useful science about this. I'm looking for ways to establish just how long a given piece of trail is and it is very important to people. An extra 30km in a race is an extra 5+ hours we weren't expecting.  As requested here is some sample data:       Thanks for any advice you can give. Clothing store access in Pokémon X & Y Once the player obtains access to Lumiose City for the first time, they'll be able to visit Boutique Couture from the city's Vernal Avenue (South fork, connecting from South Boulevard). Upon entering the store without performing any other actions, the player will be turned away and informed they "aren't stylish enough" and that they should visit more places in Lumiose City before visiting again.  This description is dubious, and doesn't inform the player of what they need to accomplish to gain access. Specifically:  Even beating the game (completing the Pokémon League) fails to allow access, as does modifying the player's look, visiting most shops in all areas of Lumiose City, and performing all hair stylist, PR studio, and associated actions. Completing a set of triple battles in Restaurant Le Wow did set the event flag that allowed access in my copy of Y, but it is not clear if this is the only requirement for access.  So other players may benefit, and so I know how to repeat this process if I obtain a copy of X or the future release of Z, what exact requirements yield access to this store for the first time? Given the positive integer numbers $n$,prove that $$\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$$  where $\sigma(n)$ is defined as $$\sigma{(N)}=\sum_{d|N}d$$  This problem I think is very interesting, and have nice methods?  Thank you everyone.  and I find some of this     or can see: Sensitivity to initial conditions and topological transitivity Forgive the possibly naive question, but I'm returning to a basic undergrad-level study of chaotic dynamical systems and can't quite seem to reconcile the notion that in order for a dynamical system to be considered chaotic, it must have positive global Lyapunov exponents. Such exponents, of course, drive the divergence of trajectories. However, topological transitivity implies that trajectories can get arbitrarily close as well over some finite time. It seems that positive Lyapunov exponents would disallow this mixing. Is there a point at which these positive exponents no longer describe the system, and the mixing behavior begins? prove that $24 \mid a(a^2-1)$ If $a$ is an odd integer, prove that $24 \mid a(a^2-1)$.  I proved in my class that $a^2=8k+1$ for some $k$, provided $a$ was odd.  So $$a(a^2-1)=(2k-1)(8k+1-1)=8k(2k-1)$$ but I can't pull a $24$ out of it. Inspired by :  In my answer, I had stated that Patronuses appear to not be physical beings because they have no effect on anything other than Dementors.     They don't cause leaves to stir in the movies. They passed straight through the tent. They don't appear to be bound by normal laws pf physics and can travel great distances almost immediately (Shacklebolt was presumably in London when his Patronus issued the warning). They can apparently walk across water without sinking (as they did in The Prisoner of Azkaban). They can jump, walk, and float in mid-air (as seen in the Room of Requirement in the movie version of The Order of the Phoenix). They make no footprints in lightly powdered snow (The Deathly Hallows Chapter 19) and make no sound of crunching snow as Harry does (further down the same page in Chapter 19).   I stated incorrectly that the only place in canon where there is an exception is with Dementors, where the Patronus physically fends them off with its horns.  A Dementor is clearly a physical being, if it prised Dudley's mouth open and (in the movie anyway) physically slid open the train door.)   Based on comments and further thought, I can think of at least two other examples when a Patronus interacts physically with a physical being.   As pointed out by Slytherincess, a Patronus can fend off a Lethifold (Living Shroud). When training, the Patonus took form against a Boggart, which had taken the form of a Dementor.   All three of these things have, AFAIK, only three things in common (but I may be off).   They are evil They are magical They are, indeed, physical beings.   So the question is, why can a Patronus interact with all three of these, but seems to have no physical effect on everyday things? Help with packages in java - import does not work I'm a C++ developer - not a java developer, but have to get this code working...  I have 2 public classes that will be used by another product.  I used the package directive in each of the java files.    package com.company.thing;  class MyClass ...   When I try to compile a test app that uses that I add   import com.company.thing.*;   The javac compiler fails with errors about com.company does not exist.  (even if I compile it in the same directory as the class files I just made a package of)  I am sure I am doing something bone-headed and silly.  I've read the  pages and tried to set up a directory structure like /com/company/thing  etc, but either I have totally screwed it all up or am missing something else.  EDIT thanks for the suggestions - I had tried the classpath previously.  It does not help.  I tried compiling   javac -classpath &lt;parent&gt; client.java    and the result is:  package com.company does not exist   I have the code I want to import (the two java files) in \com\company\product.  I compile those fine.  (they contain MyClass) I even made a jar file for them.  I copied the jar file up to the parent directory.  I then did (in the parent directory with the client java file)   javac -cp &lt;jarfile&gt; *.java   the result is:  cannot access MyClass bad class file: MyClass.class(:MyClass.class) class file contains wrong class: com.company.product.MyClass Please remove or make sure it appears in the correct subdirectory of the classpath.   EDIT  I got the client code to compile and run if I used the fully qualified name for MyClass and compiled it in the parent directory.  I am totally confused now.    compiled with no classpath set - just   javac *.java    in the parent directory - and it worked fine.  I can get a test app to compile, but that is not going to cut it when i have to integrate it into the production code.  Still looking for help.  EDIT:  Finally - not sure why it didn't work before - but I cleaned up all the files all over the directory structure and now it works.    Thanks Fetching all (javascript) global variables in a page An image alt="" text should contain information about that image in the context of the page, so for instance if you have an article about cats on skateboards and then a photograph of the cat on his skateboard you'd probably give it the alt text of alt="Fluffy pulling some mad tricks on his pimped-out board".          &ndash; Wireframes created with   However, what should happen when that image is a link to a page (such as the Wikipedia cat page)?  I don't think there is any point using the title attribute here because screenreaders just ignore that when an alt attribute exists - the only people who would see the title attribute are mouse users who hover over the image, so that excludes keyboard only users too. It's not really any benefit to accessibility to include a title in this situation (Overall the title attribute is a bit useless really).  So, what should the image ALT text say in this case?   states:     Provide text alternatives for any non-text content so that it can be changed into other forms people need, such as large print, braille, speech, symbols or simpler language.    Therefore giving it the alt text of cat article page on Wikipedia is incorrect because that doesn't describe the image, neither does skateboarding cat image that links to the wikipedia article because the alt text should refer to the image itself, not the link associated with it. Identifying a thin, black, worm-like creature with a hammer-shaped head I came across the creature shown in the photo below yesterday evening while walking the dogs. At first, I thought it was an earthworm, but a flashlight revealed it to be something I had never seen before.    It is approximately 12 inches (300 mm) in length and about 1/3 of an inch (10 mm) in diameter at its maximum. It is pitch black and very shiny—like piano black. It moves very slowly, using a snake-like motion. It had a very strange head, like a hammerhead shark.  I live in Durban, South Africa, if that helps to narrow down the possibilities.  Can anyone help in identifying this creature? Any Video formats that render with Alpha Channel? Does anyone know if there are any video formats (besides the HUGE uncompressed avis) that you can render in that maintain the alpha channel? (i.e. for after FX etc.)   I used to use quicktime, but that's no longer an option.  I know you can render in png. or other image formats, but video is significantly smaller, and I often work on machines with less space &amp; resources. In the following snippet:  public class a {     public void otherMethod(){}     public void doStuff(String str, InnerClass b){}     public void method(a){         doStuff("asd",             new InnerClass(){                 public void innerMethod(){                     otherMethod();                 }             }         );     } }   Is there a keyword to refer to the outer class from the inner class?  Basically what I want to do is outer.otherMethod(), or something of the like, but can't seem to find anything. What does \s! do? 1st example:  \starttypescript [\s!serif] [paratype]   \definefontsynonym [ParatypeSerif-Regular] [\s!name:PTSerif-Regular] [\s!features=\s!default]   \definefontsynonym [ParatypeSerif-Italic] [\s!name:PTSerif-Italic] [\s!features=\s!default]   \definefontsynonym [ParatypeSerif-Bold] [\s!name:PTSerif-Bold] [\s!features=\s!default]   \definefontsynonym [ParatypeSerif-BoldItalic] [\s!name:PTSerif-BoldItalic] [\s!features=\s!default] \stoptypescript   2nd example:  \definefont[SerifS][\s!file:GenBkBasR at \smallfontsize]   Is \s! only related to fonts? Should I use it in my typescript files? Why use a certain measure of forecast error (e.g. MAD) as opposed to another (e.g. MSE)? MAD = Mean Absolute Deviation MSE = Mean Squared Error  I've seen suggestions from various places that MSE is used despite some undesirable qualities (e.g. , which states on p8 "It is commonly believed that MAD is a better criterion than MSE. However, mathematically MSE is more convenient than MAD.")  Is there more to it than that? Is there a paper that thoroughly analyzes the situations in which various methods of measuring forecast error are more/less appropriate? My google searches haven't revealed anything.   A similar question to this was asked at , and the user was asked to post on stats.stackexchange.com, but I don't think they ever did. Looking for a short story about a man who is visiting a planet where you get your position/job through playing a game I read this story 8-15 years ago in German. I'm not sure but I think the book had 2 short, totally different stories in it.  The story was about a man who was sent (I think kinda forced to go) to a planet where he should play a game everyone on that planet plays. I think the goal was to beat their leader and conquer the world like this?  The better you are at the game, the higher is your position in the society. I remember that at the end he was playing against the leader and they had a long back and forth game, but I'm not sure who won.  They played the game on a different planet that had only one stripe of forest around the whole planet and a fire that goes around it. But thats only the location they were playing.  The man was chosen because he was the best player in every game if I remember correctly.  The game was a board like game where you have troops or armies that you move around. It was not an easy game everyone just could play. Should we add more badges for asking questions? This proposal was superseded , which improves on the core concept.   Early on, there was a question about whether Stack Overflow should . At the time,  was:  Getting answers is reward enough for asking questions.  Even then, Stack Overflow was getting more than enough questions, so it didn't make much sense to incentivize more questions.  lists the badges that can be earned by asking quality questions. (There are at least 8). But looking at the list of recent , it looks more like a list of . These badges are difficult to come by in the first place, and they're much harder to earn on sites like  even though there are some  on that site. It turns out that . We , but we still need plenty of oysters. The Student bronze badge is awarded for the first upvoted1 question. There are no equivalent gold or silver badges. Therefore, I'd like to propose the following:  Student—Asked first question with score of 1 or more Sophomore—Asked 10 questions with score of 1 or more Socrates—Asked 50 questions with score of 1 or more  Do you have any suggestions of better names/levels?  To give you an idea of how many of these badges would be awarded, I wrote . The results2 for the original trilogy: users   gold silver bronze none    Site Name  ------- ---- ------ ------ ------- ---------  3000344 7127 67713  805778 2194566 Stack Overflow 235053  115  1698   53073  181980  Super User 156773  53   1191   40063  116710  Server Fault  The gold version on SO would be slightly more rare than  and more common than . Obviously, there would be fewer badges for smaller sites: 5545    1    18     917    4628    Philosophy 4032    4    41     802    3230    History  5663    3    28     917    4746    Parenting   Several older sites would get none at all: 5778    null 29     1063   4715    Quantitative Finance 7884    null 20     927    6957    Project Management 4333    null 17     1630   2703    Motor Vehicle Maintenance and Repair  These smaller sites get at most . I suspect that a gold or silver badge would encourage site regulars to increase that number without sacrificing quality. Certainly the requirement for those questions to stay in positive territory would prevent a person from flooding the site3 with terrible questions in order to get the badge.   1. The [original spec](https://blog.stackoverflow.com/2008/07/stack-overflow-badge-feedbac/) didn't consider voting at all for Student and Teacher.   2. Your results will vary from mine since I'm looking at slightly newer data. The bronze badge count also differs from the number of [Student](https://stackoverflow.com/help/badges/2/student) badges because the badge isn't revoked when a user's question is later downvoted or deleted.   3. There is a potential problem with cookie-cutter or variation-on-a-theme questions. However, I think the typical tools for closing and deleting questions will prevent people from gaming the system for these badges. How do I theme a webform? Mars needs water and women I read this book in high school too many years ago for me to think about. Martian colonist are raiding earth's oceans for water and because of the conditions on Mars where only male children are conceived, they are also on the look out for young women as potential mates. Our main character while on a raid, bags not only a young woman but her brother as well. Returning to Mars they find that Earth has attacked Mars can't remember much more. As the title suggests, I have some kit which is 200m from our current infrastructure (outdoors).  What is the cheapest way to establish a reliable connection across a 200m stretch (Cat 5e ethernet only works up to about 100m from our tests)?