I was wondering if after starting the session, session_id automatically stores a value? or does session_id will ever have a value if some user successfully logged on the page?
How unique is the php session id? I got the impression from various things that I've read that I should not rely on two users never getting the same sessionid. Isn't it a GUID?
Available space not changing when resizing a partition In my computer I have several partitions. I have one partition with Fedora that had just below 40 GB of space, and which I wanted to make bigger. Therefore, using GParted, I moved 10 GB from another partition to the Fedora partition. Now, according to GParted, I should have just below 50 GB (see sda12):  However, from Fedora itself the available space looks different:  This was the same space that was available before I resized the partition. How do I make that space actually available to Fedora? I'm new to field extensions and I can neither see why or how to prove this statement.  If I'm not mistaken, to construct a splitting field, you construct rupture fields one after the other, using that the $8$ element field is the splitting field of $X^8-X$ over $\mathbb F_2$, since $2$ is the only prime divisor of 8.  Over $\mathbb F_2$ $X^8-X$ has two roots : $0$ and $1$. Therefore $X^8-X=X(X-1)(1+X+...+X^6)$. The last factor, call it $P$, has no roots. So let $\alpha$ be a root in some extension $\mathbb F_2(\alpha)$ of $\mathbb F_2$.  We now have $P=(X-\alpha)(X^5+\alpha X^4+...+\alpha^4X+\alpha^5)$.  I suspect $\alpha$ might not be the way to go and if so, my second plan would be to reduce $P$. I don't know how to do this but assuming it to be done, hence having $P=P_1...P_n$, I could start using $P_1$'s rupture field, and then my best guess is that if $\mathbb F_2[X]/(P_1)$ is not sufficient, I re-reduce $P$ over this new field and consider the rupture field of one of the factors, and so on.  Now assuming I wasn't given the statement itself, is it elementary to construct the $8$ element field using $\mathbb F_2$ ? This question is from [Number Theory George E. Andrews 1-1 #3].     Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$   This problem is driving me crazy. $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$  $(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can't figure it out    This is a similar problem in the book and I tried this method but it wasn't working out      $\quad$Thereom $\bf1$-$\bf2$: $\,\,\,\,$ If $\,x$ is any real number other than $1$, then $$\sum_{j=0}^{n-1}x^j=1+x+x^2+\ldots+x^{n-1}=\dfrac{x^n-1}{x-1}.$$    $\quad$Remark: $\displaystyle\sum_{j=0}^{n-1}A_j$ is shorthand for $A_0+A_1+A_2+\ldots+A_{n-1}.$     $\quad$Proof: Again we proceed by mathematical induction. If $n=1$ then $\displaystyle\sum_{j=0}^{1-1}x^j=x^0=1$ and $(x-1)/(x-1)=1$. Thus the theorem is true for $n=1$.  $\quad$ Assuming that $\displaystyle\sum_{j=0}^{k-1}x^j=(x^k-1)/(x-1)$, we find that $$ \eqalign{ \sum^{(k+1)-1}_ {j=0}x^j &amp; = \sum^{k-1}_ {j=0}x^j+x^k=\dfrac{x^k-1}{x-1}+x^k \\ &amp;= \dfrac{x^k-1+x^{k+1}-x^k}{x-1}\\ &amp;= \dfrac{x^{k+1}-1}{x-1}. }$$   Hence condition $(\rm ii)$ is fulfilled, and we have established the theorem.    $\quad$Corollary $\bf1$-$\bf1$: $\,\,$ If $\,m$ and $n$ are positive integers and if $m&gt;1$, then $n&lt;m^n.$ Manually installing a program on Debian Best Fake Proofs? (A M.SE April Fools Day collection) I'm seeking for a proof of the following:     Let $A$ be an invertible matrix. Then the determinant of $A^{-1}$ equals:   $$\left|A^{-1}\right|=|A|^{-1} $$   I don't know where to begin the proof. Any suggestions? Alignment issue for the last modified information in the "Top Questions" page In the  (homepage), the asked/modified X mins ago text was left-aligned next to the tags. Previously the text was right-aligned. Seems some alignment issue introduced on that page recently. Screenshot for reference:      Configuration Version     Operating System Windows 10 Pro   Mozilla Firefox 89.0 (64-bit)   Google Chrome 91.0.4472.101 (Official Build) (64-bit)     Update: I see the same alignment issue in all other Stack Exchange site's &quot;Top Questions&quot; page too. why Booleans slow my computer more than expected "Is the same" or "are the same"? Which is the correct one?     On average, the polyunsaturated fat content in one teaspoon of the olive oil and canola oil is the same        On average, the polyunsaturated fat content in one teaspoon of the olive oil and canola oil are the same Explaining Horizontal Shifting and Scaling I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.  For example, $f(x+1)$ is a horizontal shift to the left (a shift toward the negative side of the $x$-axis), whereas a cursory glance would cause one to suspect that adding a positive amount should shift in the positive direction. Similarly, $f(2x)$ causes the graph to shrink horizontally, not expand.  I generally explain this by saying $x$ is getting a "head start". For example, suppose $f(x)$ has a root at $x = 5$. The graph of $f(x+1)$ is getting a unit for free, and so we only need $x = 4$ to get the same output before as before (i.e. a root). Thus, the root that used to be at $x=5$ is now at $x=4$, which is a shift to the left.  My explanation seems to help some students and mystify others. I was hoping someone else in the community had an enlightening way to explain these phenomena. Again, I emphasize that the purpose is to strengthen the student's intuition; a rigorous algebraic approach is not what I'm looking for. How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$. Prove that there is no continuous functions $f:[0,1]\to R$, such that   $$ f(x)+f(x^2)=x. $$   My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all $x\in [0,1]$, \begin{align*} f(x)&amp;=x-f(x^2)=x-\big(x^2-f(x^4)\big)=x-x^2+\big(x^4-f(x^8)\big)=\cdots\\ &amp;=x-x^2+x^4-\cdots+(-1)^n\left(x^{2^n}-f\big(x^{2^{n+1}}\big)\right) \end{align*}  since $f(0)=0$ and $\displaystyle\lim_{n\to \infty}x^{2^{n+1}}=0$ for any $x\in(0,1)$,it follows by the continuity of $f$ that $\displaystyle\lim_{n\to \infty}f\big(x^{2^{n+1}}\big)=0$, hence $$f(x)=x-x^2+x^4-x^8+\cdots+(-1)^nx^{2^n}+\cdots$$ for any $x\in (0,1)$  Why do I have prove that there is exists such a continuous functions $f$? Maybe  my example is wrong? Why?  if my   method is wrong,then How prove this problem ? Thank you. Does an increasing sequence of reals converge if the difference of consecutive terms approaches zero? If $a_n$ is a sequence such that   $$a_1 \leq a_2 \leq a_3 \leq \dotsb$$   and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent?   I know that without the condition that the sequence is increasing, this is not true, as we could consider the sequence given in  to a similar question that does not require the sequence to be increasing.  $$0, 1, \frac12, 0, \frac13, \frac23, 1, \frac34, \frac12, \frac14, 0, \frac15, \frac25, \frac35, \frac45, 1, \dotsc$$  This oscillates between $0$ and $1$, while the difference of consecutive terms approaches $0$ since the difference is always of the form $\pm\frac1m$ and $m$ increases the further we go in this sequence.    So how can we use the condition that $a_n$ is increasing to show that $a_n$ must converge?  Or is this still not sufficient? Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future. iPhone 6s with iOS 12.4 not connecting to MacBook Pro mid2015 with macOS 10.14.6 I am using an iPhone 6s with iOS 12.4 and a MacBook Pro mid2015 with macOS 10.14.6. Since recently (I am not exactly sure when it started), I cannot charge the phone with my computer anymore. As soon as I plug the phone in, it will play the charging sound repeatedly. The right-hand side port sometimes works, the lefthand side port never works. When I connect my old iPhone 5 (using the same cable), it charges without any issues. My iPad mini 4 shows the same issue.  Sometimes the "Do you trust this computer" setting flashes on the phone and disappears before I can click anything.   How can I fix this?  Update 1  Sometimes I get the Software Update dialogue in macOS. After approx. 2 minutes it shows the following:     Update 2  For some reason the issue disappeared for a while but is back now. Disabling "Settings" > "Touch ID &amp; Passcode" > "USB Accessoires" seems like it makes the problem a bit less bad, but not go away.  Update 3  This is driving me crazy. I upgraded to Mojave 10.14.3 mostly to get rid of the issue. It persists. Some more random information: Before logging in (Filevault pre-boot authentication), the phone will charge fine. Only when a user account is loaded, the issue appears. The likelihood of a successful connection increases with the battery state of the iPhone. Workaround 2 in my answer below suggest that this a "Trust issue" from the iPhone's side. I suppose the solution will have to be hidden somewhere there...    What I tried so far (will update):   Reset Privacy settings on iPhone Different (original, almost new Apple) lightning cables   Remove and add "iPhone USB" in Network Settings  SMC reset (), PRAM reset Upgrade to latest iOS 12.4 and MacOS 10.14.6 Declare global variables in Visual Studio 2010 and VB.NET Can anyone explain why following code won't compile? At least on g++ 4.2.4.  And more interesting, why it will compile when I cast MEMBER to int?  #include &lt;vector&gt;  class Foo {   public:       static const int MEMBER = 1;   };  int main(){       vector&lt;int&gt; v;       v.push_back( Foo::MEMBER );       // undefined reference to `Foo::MEMBER'     v.push_back( (int) Foo::MEMBER ); // OK       return 0; } Can thebibliography be sorted by citation order? Or is there a solution with an inlined bibliography that can be sorted?  Not having an additional .bib file would be preferable.  My current bilbiography is inluded in the rest of the article inside a thebibliography block.  \begin{document} \section{Example} The citations in\cite{ne} the bibliography should\cite{flowers} be ordered  according which order they appear in the text.   \begin{thebibliography}{9}  \bibitem{fetebok} \emph{Internal Combustion Engine Fundamentals}  \bibitem{ne} \emph{Förbränningsmotorer}  \bibitem{flowers} \emph{HCCI Research Towards Development for Stationary Power  Applications}  \end{thebibliography}   Should output:  The citations in[1] the bibliography should[2] be ordered  according which order they appear in the text.   Bilbiography [1] Förbränningsmotorer [2] HCCI Research Towards Development for Stationary Power Applications [3] Internal Combustion Engine Fundamentals Does the Sun produce audible sound? Theoretically if I were able to build some sort of device that let me sit 1 foot away from the surface of the Sun (or any star for that matter) without being vaporized, would a star produce any sort of audible sound at that distance?   From my understanding, sound waves need a medium to propagate through (air, water, etc).   Does this mean that all stars are completely silent (in the audible spectrum of frequencies at least) as they fuse  hydrogen into helium?   I know stars emit radio, xray, infrared, ultraviolet electromagnetic waves, but how about audible (20 Hz to 20Khz) frequencies?   For some reason it seems like this process would produce some sort of sound. How do I change player name in Minecraft smp in offline mode on a Mac? Which tool for fixing wobbly crank? Any idea what kind of tool should I but to fix the loose left crank on my bike? I found this  but not sure if it's gonna suit. Or is it enough to buy only nut driver?