Variability of K SVD components
Relationship between SVD and PCA. How to use SVD to perform PCA?
$\int_0^\infty \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx$ Trying to compute   I was facing: \begin{align}J=\int_0^\infty \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx\end{align}  I want to prove that $\displaystyle J=0$, or equivalently, that,  \begin{align}\int_0^1 \frac{\ln\left(1+x-\sqrt{2x}\right)}{1+x^2}\,dx=-\dfrac{1}{2}\text{G}\end{align}  $\text{G}$ being the Catalan constant.  Read carefully please.   I know, using so-called Feynman's trick, how to prove this. I would like to obtain a proof, using only integration by parts and change of variable in simple integrals (that is, no multiple integrals) I don't know, if, under these restrictions, such computation is possible.  NB: You're probably wondering what is the link between : $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$ and $J$.  The link is, for $x\in\left[0;\frac{\pi}{2}\right]$, $(\sin{x}+\cos{x}+\sqrt{\sin{2x}})(\sin{x}+\cos{x}-\sqrt{\sin{2x}})=1$ and $\sin(2x)=2\sin x\cos x$ Is a URL allowed to contain a space? Is a URI (specifically an HTTP URL) allowed to contain one or more space characters? If a URL must be encoded, is + just a commonly followed convention, or a legitimate alternative?    In particular, can someone point to an RFC that indicates that a URL with a space must be encoded?  Motivation for question: While beta-testing a web site, I noted that some URLs were constructed with spaces in them. Firefox seemed to do the right thing, which surprised me! But I wanted to be able to point the developers to an RFC so that they would feel the need to fix those URLs. I'm looking to schedule 16 players to play a round robin tournament with each other such that there are 4 players at each table.  I'd like for each player to play with each other player exactly once over the course of 5 rounds.  I just took a look at this post:  , and I was wondering if anyone knows if the case n=m=4 has been solved, and if so what that solution is. I'm trying to work with  in Java.   I want to implement arithmetic functions.  For this, I will first require a way to normalize the functions.  I know I can't add 1/6 and 1/2 until I have a common denominator. I will have to add 1/6 and 3/6.  A naive approach would have me add 2/12 and 6/12 and then reduce.  How can I achieve a common denominator with the least performance penalty?  What algorithm is best for this?    Version 8 (thanks to ):     Improvements include:         the equals() method is now consistent with the compareTo() method      final class Fraction extends Number {     private int numerator;     private int denominator;      public Fraction(int numerator, int denominator) {         if(denominator == 0) {             throw new IllegalArgumentException("denominator is zero");         }         if(denominator &lt; 0) {             numerator *= -1;             denominator *= -1;         }         this.numerator = numerator;         this.denominator = denominator;     }      public Fraction(int numerator) {         this.numerator = numerator;         this.denominator = 1;     }      public int getNumerator() {         return this.numerator;     }      public int getDenominator() {         return this.denominator;     }      public byte byteValue() {         return (byte) this.doubleValue();     }      public double doubleValue() {         return ((double) numerator)/((double) denominator);     }      public float floatValue() {         return (float) this.doubleValue();     }      public int intValue() {         return (int) this.doubleValue();     }      public long longValue() {         return (long) this.doubleValue();     }      public short shortValue() {         return (short) this.doubleValue();     }      public boolean equals(Fraction frac) {         return this.compareTo(frac) == 0;     }      public int compareTo(Fraction frac) {         long t = this.getNumerator() * frac.getDenominator();         long f = frac.getNumerator() * this.getDenominator();         int result = 0;         if(t&gt;f) {             result = 1;         }         else if(f&gt;t) {             result = -1;         }         return result;     } }     I have removed all previous versions.  My thanks to: Let $x,y,z\ge 0$, show that $$\dfrac{y}{xy+2y+1}+\dfrac{z}{yz+2z+1}+\dfrac{x}{zx+2x+1}\le\dfrac{3}{4}$$  I had solve $$\sum_{cyc}\dfrac{y}{xy+y+1}\le 1$$ becasuse After some simple computations, it is equivalent to  $$(1-xyz)^2\ge 0$$ How to add and sync folders outside SkyDrive folder? I have folders with documents all over my hard drive, some which I would like to include in SkyDrive without moving them into the SkyDrive folder.  E.g. my SkyDrive folder is in C:\Users\Name\SkyDrive, but I have some files in C:\Users\Name\SyncMePlz that I also would like to be synced with SkyDrive so that I can reach them from other devices.  How can this be achieved? "named" vs "that is named" I want to describe my business and want to refer to it in a sentence.  I am trying to say:      "In 2015 we established a business (that is) named X".   Intuitively, the "that is" part is redundant, and makes the sentence more cumbersome. But I cannot think of a reason why we should be able to drop the verb.  Is the version without the "that is" part grammatically correct? How can it be explained? Where can I get Blender test builds? I know that  provides downloads for official releases. I see on forums that people use newer versions of Blender. Where can I get newer version of Blender (nightly builds, alpha versions... etc). For example, in this text:     Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nunc eu tellus vel nunc pretium lacinia. Proin sed lorem. Cras sed ipsum. Nunc a libero quis risus sollicitudin imperdiet.   I want to match the word after 'ipsum'. Is there any intuition or meaning regarding Cauchy-Riemann equations? Holomorphic functions are one of the most beautiful objects in mathematics. However, Cauchy-Riemann equations are a little bit of a mystery for me. What does that mean intuitively when a function satisfyes those equations? Sponsor Gallery, Page, or Something Like That? Can an integrity check be run against a USB boot disk? Meta's tag pages inaccessible due to self-redirect loop Tag pages (route /questions/tagged/tagname) redirect to themselves indefinitely:  $ curl -D - http://meta.stackexchange.com/questions/tagged/featured HTTP/1.1 302 Found Cache-Control: private Content-Type: text/html; charset=utf-8 Location: /questions/tagged/featured X-Frame-Options: SAMEORIGIN Date: Fri, 05 Jul 2013 16:47:42 GMT Content-Length: 143  &lt;html&gt;&lt;head&gt;&lt;title&gt;Object moved&lt;/title&gt;&lt;/head&gt;&lt;body&gt; &lt;h2&gt;Object moved to &lt;a href="/questions/tagged/featured"&gt;here&lt;/a&gt;.&lt;/h2&gt; &lt;/body&gt;&lt;/html&gt;   This only happens on Meta, so far, but only applies to any meta-tag that doesn't have a dash in it; those with a dash seem to be exempt, as do invalid tags:    redirects.  does not.  does not. Combine two columns into a list of all possible combinations of entries I have two columns such as:  a   1  b   2  c    and I need to combine them like:  a   1 a   2 b   1 b   2 c   1 c   2   Is it possible with a formula? I am trying to make a pirate ship and am trying to attach rope to the sails. my rope is made using a path, and soft body physics. I can not find a way to attach my rope to the sail. and I can not use pinned vertices because this is not a cloth modifier. Difference between live and neutral wires In domestic electrical circuits, there are 3 wires - live, earth and neutral. What is the difference between the live and neutral wires?  As there is AC supply, it means that there are no fixed positive and negative terminals. Current rapidly switches direction. Does that not mean that both the live and neutral wires carry the same amount of current, alternatively? Why is the neutral wire called 'neutral'? Using a naked black hole as a mirror? The YouTube video is a good example how the gravity of this merging binary black holes bend light around themselves. Is there a difference here:        It happened that he did not go to work today.   As it happened, he did not go to work today.      Do "it happened that" and "as it happened" have different meanings? sliding a tangent line along a curve Computer unstable at lower temperatures My computer is playing some strange joke at me, and I'm at a loss.  Everybody knows that hot temperatures causes many components to become unstable and even cause damage. When I built this machine, I was careful to do a bunch of experiments with different airflow configurations, with CPU-and-GPU intensive benchmarks running, and measuring the temperature at different points to find the best configuration.  Generally, the internal temperature of this machine is very close to ambient temperature (with 2 exceptions: the GPU always runs ~10°C above ambient, and the CPU goes to at most ~12°C above ambient when running CPU-intensive applications).  Now comes the problem: at really cold mornings. At cold mornings (~10°C (50°F) or below), the system shows really strange general instability. Beeps from the BIOS, the operating system displays strange messages about memory corruption and PCI devices that failed to initialize, and then, everything hangs. The computer hangs to the point that all USB devices turn of. Hitting the Reset button blanks the screen but the mainboard fails to proceed to the boot.  I never thought that low temperatures could cause computer instability. And 10°C is not that cold, it is still within operating temperatures of all components as far as I checked.  I'm quite sure it is related with the temperature. There were too many cold and warm mornings that demonstrated that the problem only showed in the cold ones. I'm working around this problem by letting the computer sit at the GRUB menu for some 5-10 minutes before letting the OS start, or blowing hot air inside the case if I'm in a hurry.  I don't know what should I approach next to find the source of this problem. Since it is difficult to reproduce (I need a cold morning, and the computer must be cooled down (powered off for many hours)), I need to plan ahead the experiment, since it stabilizes after a few minutes on.  Machine is an ASUS P5VDC-X equipped with Intel P4 64-bit 3GHz, 2GB of DDR2 RAM, a 500W PSU and an nVidia GeForce 9600GT. Everything else was already disconnected/removed before and the problem persisted, so the problem should be in one of these remaining parts.  Any help is appreciated!