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How to fix "Headers already sent" error in PHP
What are the benefits of nickel-plated chains? Bicycle chains with higher model numbers (and slightly higher prices) are often nickel-plated or half-nickel-plated. What are the benefits of nickel-plated chains over non-nickel-plated chains?  How durable is the plating--does it wear off, or do its benefits last for the life of the chain? I am trying to do SVD by hand:  m&lt;-matrix(c(1,0,1,2,1,1,1,0,0),byrow=TRUE,nrow=3)  U=eigen(m%*%t(m))$vector V=eigen(t(m)%*%m)$vector D=sqrt(diag(eigen(m%*%t(m))$values))  U1=svd(m)$u V1=svd(m)$v D1=diag(svd(m)$d)  U1%*%D1%*%t(V1) U%*%D%*%t(V)   But the last line does not return m back. Why? It seems to has something to do with signs of these eigenvectors... Or did I misunderstand the procedure? What is the difference between public, protected, package-private and private in Java? In Java, are there clear rules on when to use each of access modifiers, namely the default (package private), public, protected and private, while making class and interface and dealing with inheritance? Estimate on the derivative at fixed point Let $D$ be a bounded domain and let $f$ be analytic function from $D$ into $D$. Show that if $z_{0}$ $\in D$ is a fixed point for $f$ , then $|f' (z_{0} )| \leq 1$.  WHAT I WAS THINKING:   To have a conformal mapping from $D$ to the unit disc &amp; after composing it with $f$ ; I was thinking to use Pick's Lemma. So, can anyone tell me any conformal map from any bounded domain to the unit disc?? OR Is there any other way out to solve this problem?? What would you call someone who makes no lasting impression? Example of quasi-compact, non-quasi seperated scheme where qcqs fails? How to transfer between International T2 to Domestic 1B terminals in Mumbai? I will be arriving in Mumbai Terminal T2 on a Turkish Airlines international flight. I have a separately-purchased IndiGo ticket 5 hours later from Terminal 1B (Domestic). In the past I heard of a free bus to transfer.    Is that still the case and if not how do I get to Terminal 1B from Terminal 2? When is a cellular automaton "bidirectional"? Gandalf (as with all the Istari) did not have a corporeal form before coming to Middle-earth and was essentially ageless. We also know that Gandalf did age, albeit very slowly, while on Middle Earth. By the end of the Third Age (before battling Durin's Bane), he had the appearance of an old human male (non-Númenórean, etc) in his 70's or 80's. This leads me to wonder:  Is there any indication in Tolkien's writings that describes Gandalf's appearance (in human-looking years) when he arrived in Middle Earth? In other words, was Gandalf the Grey originally "Gandalf the Tall Dark &amp; Handsome in his 20's"?  I'm not talking about his actual age, but rather his appearance in human years. When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The question is:     When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?   The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?  Edit  I should add that $f:\mathbb R\to\mathbb R$ to the question. I've recently installed Ubuntu on Oracle's Virtualbox.  Now I want to install the guest additions.  I started up my Ubuntu guest OS, clicked "Devices" and chose "Install Guest Additions". Nothing happened, and I don't see the CD icon on my desktop.  How can I make this work? I'm close to maxing out my character and I want to beat Ganon solely for the TRUE ending, but I'm too scared of losing some of my personal favorites from the Guardian Spear++ to fighting those damned Guardians (it was so fun).  So, after beating Ganon    Do monsters disappear? - Including the corrupted Guardians, but also the blood moons, I don't want to just face off against the Yiga Clan... Am I gonna run out of resources? - As a survival game Zelda BOTW has some emphasis on the survival genre but to me it is a survival-esque action-adventure game. Are special weapons still available when you max out? - Like the Savage Lynel Spears or Swords) PHP: Static and non Static functions and Objects $ {L}_{1} $ Regularized Unconstrained Optimization Problem Does $x^2 - y^2 = 12345678$ (x,y are integers) have any solutions? This question appeared in KVPY online exam held on Nov 5. This equation is the equation of a hyperbola so any solution must be of the form $a\sec(k),a\tan(k)$. So no solutions. I'm making an app with a custom list in multiple languages.  Is there an easy way so that I only have to create the site columns, content type and list once? So I don't have to create them for each language?  My app will be in English, French and Dutch.  My lists elements.xml looks like this  &lt;?xml version="1.0" encoding="utf-8"?&gt; &lt;Elements xmlns="http://schemas.microsoft.com/sharepoint/"&gt;     &lt;!-- Do not change the value of the Name attribute below. If it does not match the folder name of the List project item, an error will occur when the project is run. --&gt;     &lt;ListTemplate         Name="List1"         Type="10000"         BaseType="0"         OnQuickLaunch="TRUE"         SecurityBits="11"         Sequence="410"         DisplayName="List1"         Description="My List Definition"         Image="/_layouts/15/images/itgen.png"/&gt;   &lt;Field       Name="Adres"       ID="{2a32d9eb-ad3b-4dbb-abba-66dd01ca7c6e}"       DisplayName="$Resources:Adres"       Type="Text"       Required="TRUE" /&gt;   &lt;Field       Name="Naampje"       ID="{ac47be45-2f43-4d16-9a06-162c85945ec9}"       DisplayName="$Resources:Naampje"       Type="Text"       Required="FALSE" /&gt;   &lt;Field     Name="Nickname"     ID="{6B0A2CD7-A7F9-41ca-B932-F3BEBB603793}"     DisplayName="$Resources:Bijnaam"     Type="Text"     Required="FALSE" /&gt;  &lt;/Elements&gt;   Well if i change my language in Sharepoint (Office 365) it's not changing the titles from dutch to the other languages i have added. Isomorphism of a product $C_n \times C_m$ of cyclic groups with the cyclic group $C_{mn}$ Given that $C_n$ is a cyclic group of order $n$, what conditions must integers $n$ and $m$ satisfy such that the group $C_n \times C_m$ is isomorphic to C$_{mn}$?   So I attempted to investigate a few lower-order cyclic groups to find a pattern. In particular, I looked at $C_2 \times C_3$ and compared it to $C_6$. Here are a couple of diagrams:  $C_2 \times C_3$ -   $C_6$ -   They Cayley tables look pretty different (identity elements are all over the place for $C_2 \times C_3$) so I concluded that they are not isomorphic.  I did the same for $C_2 \times C_4$ and $C_8$:  $C_2 \times C_4$ -   $C_8$ -   Again, I arrived at the conclusion that the 2 groups are not isomorphic due to their Cayley table structure.  It seems I'm stuck now. Are my conclusions wrong for the examples I examined? If so, how do I find the conditions for the general case of comparing $C_m \times C_n$ with $C_{mn}$? In Episode 1: The Phantom Menace, Anakin's mother, Shmi, said that he does not have a father, although it seemed to me that she doubted that.  Is there a canonical answer as to who Anakin's father is?  If so, who? Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$. Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$.  I'm not really sure of how I should proceed to show this, and it's pretty un-intuitive as well.  I've managed to manipulate the RHS into  $$\frac{e^{4\beta}}{\sqrt{\pi}} \int_0^\infty \frac{1}{\sqrt{u}} e^{-(\beta+2u)^2/4u} du$$ which now vaguely resembles a Gaussian. However, substituting $v = \beta + 2u$ in in order to to get something like $e^{-x^2}$ doesn't particularly help. I got $$ \frac{e^{4\beta}}{\sqrt{2\pi}} \int_0^\infty \frac{1}{\sqrt{v - \beta}} e^{-v^2/2(v-\beta)} dv. $$  I'm really unsure of how I should proceed; any help would be appreciated. Expected number of rolling a pair of dice to generate all possible sums A pair of dice is rolled repeatedly until each outcome (2 through 12) has occurred at least once. What is the expected number of rolls necessary for this to occur?  Notes: This is not very deep conceptually, but because of the unequal probabilities for the outcomes, it seems that the calculations involved are terribly messy.  It must have been done already (dice have been studied for centuries!) but I can't find a discussion in any book, or on line. Can anybody give a reference?