Are there any mathematical discoveries that are considered trivial or useless? And if that's the case, how do you know if they are trivial or useless? I am asking this question, because I am wondering if imaginary numbers were considered useless for so long why we didn't completely ignore it and decided to forget it was ever discovered.
I'm writing an article on  and some people pointed out that this is completely useless.  My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.  I came with some ideas like the Turing machine but I think I'm not grasping the right examples.  Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'?
LWJGL—How to Animate and Render Characters Using a Sprite Sheet I'm making an RPG similar to Super Nintendo RPGs in the early '90s. I'm handling movement much like Squaresoft's Final Fantasy IV.   Notice that the player never actually changes his x and y positions on screen. What I'm asking, to be precise, is how could I, in LWJGL through user input (say, the arrow keys), take the sprite sheet I have of my character's movement animations and render them in a [partially] fluid animation like above?  I've been perusing the internet for hours, and I've not found a surefire way to do this. I don't have any code as of yet; I'm more or less looking for a guideline or some pointers in the right direction concerning how I might tackle this.  And, to note, this'll be a rather large game, so I want to focus partially on speed and efficiency—that meaning that I don't want to dive into Slick2D and libGDX unless absolutely necessary. I saw a lot of times claims that they have to be exhaustive (the examples in such books were always set in such way, that they were indeed), on the other hand I also saw a lot of times books stating they should be exclusive (for example $\mathrm{H}_{0}$ as $\mu_1=\mu_2$ and $\mathrm{H}_{1}$ as $\mu_1&gt;\mu_2$) without clarifying the exhaustive issue. Only before typing in this question I found somewhat stronger statement on the  -- "The alternative need not be the logical negation of the null hypothesis".  Could someone more experienced explain which is true, and I would be grateful for shedding some light on the (historical?) reasons for such difference (the books were written by statisticians after all, i.e. scientists, not philosophers). Is there any quality difference between PNG vs BMP image formats? I have just learned that PNG is a lossless image format while BMP can be both uncompressed or lossless format. Should there be a quality difference between these two? If not, does it mean that other lossless formats like TIFF, JPEG2000 and PNG offer the same quality as BMP? Redundant prepositions? This is a tooltip in an application:     Identifies the site name to which the GL code belongs to.   Can I say that it is incorrect to use "to" twice here and we should remove 1 of them?  Other examples:       Identifies the vendor to which the port speed is associated with.   Provides the vendor to which the milestone is associated with.   (They are both tooltips, too)  The repetitions here are incorrect, aren't they? Please confirm. Let $f:\mathbb{R}_+ \to \mathbb{R}_+$ be a monotone decreasing function defined on the positive real numbers with $$\int_0^\infty f(x)dx &lt;\infty.$$ Show that $$\lim_{x\to\infty} xf(x)=0.$$   This is my proof: Suppose not. Then there is $\varepsilon$  such that for any $M&gt;0$ there exists $x\geq M$ such that $xf(x)\geq \varepsilon$. So we can construct a sequence $(x_n)$ such that $x_n \to \infty $ and $x_n f(x_n ) \geq \varepsilon$. So $$\frac{\varepsilon}{x_n}\leq f(x_n) \implies \sum_{n\in\mathbb{N}}\frac{\varepsilon}{x_n} \leq  \sum_{n\in\mathbb{N}} f(x_n) \leq \int_0^1 f(x)dx.$$ So we get a contradiction.  I feal like I have the correct idea but some details are wrong.  Any help would be appreciated. Let me start this with : Very long "About me" text squeezes both the left side (flair) and the right side (the section with location, GitHub link, etc.)  Now, the reverse thing has happened:         put a long text in their "location", and it causes the right section to be overly extended, effectively squeezing the About Me section.  When will the squeezing game end? Question on groups of order $pq$ I noticed that the locked message still references the FAQ. See :  locked by  yesterday This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. This question and its answers are frozen and cannot be changed. More info: .  Since the FAQ has been replaced by the , this message should provide a link there instead. Currently, the link actually takes you to the ; even if it doesn't link to the help center, the text should probably be something like &quot;More info: &quot;. How to read from two input files using while loop I wanted to know if there is any way of reading from two input files in a nested while loop one line at a time. For example, lets say I have two files FileA and FileB.   FileA:  [jaypal:~/Temp] cat filea this is File A line1 this is File A line2 this is File A line3   FileB:  [jaypal:~/Temp] cat fileb this is File B line1 this is File B line2 this is File B line3   Current Sample Script:  [jaypal:~/Temp] cat read.sh  #!/bin/bash while read lineA     do echo $lineA      while read lineB         do echo $lineB          done &lt; fileb done &lt; filea   Execution:  [jaypal:~/Temp] ./read.sh  this is File A line1 this is File B line1 this is File B line2 this is File B line3 this is File A line2 this is File B line1 this is File B line2 this is File B line3 this is File A line3 this is File B line1 this is File B line2 this is File B line3   Problem and desired output:  This loops over FileB completely for each line in FileA. I tried using continue, break, exit but none of them are meant for achieving the output I am looking for. I would like the script to read just one line from File A and then one line from FileB and exit the loop and continue with second line of File A and second line of File B. Something similar to the following script -  [jaypal:~/Temp] cat read1.sh  #!/bin/bash count=1 while read lineA     do echo $lineA          lineB=`sed -n "$count"p fileb`         echo $lineB         count=`expr $count + 1` done &lt; filea  [jaypal:~/Temp] ./read1.sh  this is File A line1 this is File B line1 this is File A line2 this is File B line2 this is File A line3 this is File B line3   Is this possible to achieve with while loop? How can I save my hardcore map after I die? My brother jumped into the voice of the doom in my Hardcore Creative map. I didn't press the ΄΄Delete This World΄΄ button yet. Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.  My attempt:  Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$  As $f\mid g, g$ all the roots of $f$ are roots of $g$.  This means I have to prove the statement:  If $c$ is a multiple root of $f \Longrightarrow c$ is multiple a root of $g$.  Equivalently:  If $c$ is not a multiple root of $g \Longrightarrow c$ is not a multiple root of $f$.  I'll try to prove this last statement:  Suppose $\exists c$ such that $c$ is a multiple root of $g$, then   $$g(c)=0 \implies g'(c)=0$$  $$ g=(x-1)(1+x+\cdots+x^n) \implies  g'=nx^n $$ $$ g'(c)=nc^n=0 \implies c=0 $$ But  $$ g(0)\neq 0 $$ And we've found the contradiction: $\not \exists c: g(c)=g'(c)=0$. Then $f$ has no multiple roots.  Is this correct?   Also: Assuming this is correct, is it possible to prove $\sum_{k=0}^n \frac {x^n}{n!} $ has no multiple roots in a similar manner? Formation of supermassive black holes Scientists have found very bright source of light which they call  and the are found to be . So these black holes are so massive that they cannot be formed by a supernova. So how are these formed? Product Notation for Multiplication in Reverse Order Is there a standard notation for multiplication in reverse order? For example consider the problem  $$x_{k+1} = A_k x_k$$  where $x_i \in \mathbb{R}^n$ and $A_i \in M_n(\mathbb{R})$, ($i=0,1,2,\dots$) without any further assumptions on $A_i$. The solution to this problem is  $$x_k = A_{k-1} \dots A_1 A_0 x_0$$  Obviously, writing  $$x_k = \prod_{i=0}^{k-1} A_i x_0$$  is wrong because of the multiplication order. But I'm also unconfortable to use  $$x_k = \prod_{i=k-1}^{0} A_i x_0$$  as the notation suggests that we need to increment $i$, not decrement it. Also, one may need to write $\prod_{i=k}^{m} A_i$. Then, we need to explicitly state $m &lt; k$ to imply ordering.  I can always write the open form but as similar products frequently occur in calculations in my work I want to express them in a more compact way. The question is, is there a standard and nice notation to do so? There are some suggestions in , but I think none of them are "nice enough" to work with.  I came up with the notation  $$x_k = \coprod_{i=k-1}^{0} A_i x_0$$  to imply decrementing $i$ rather than increment it, but I don't know if it is used in somewhere else or there is a standard notation. Conditional distributions of the multivariate normal What permissions should a udev rule have for a USB device? 644? 755?  For example on Ubuntu I'm looking at this file: /etc/udev/rules.d/99-lpcscrypt.rules How do you unlock the Dream Suite? Allow cancelling of stars from message Currently, the only way to cancel stars in a chat transcript is when the chat-message is visible in the chatroom's sidebar.    However, in a fast-moving, multi-star chatroom, the window of opportunity to cancel can be short, and rapid changes in starring can quickly make this impossible to do in a timely way.  We use stars as markers in moderator chat events (where the Community managers disseminate important information to the moderator teams) as well as for town halls during elections.  Feature request: Allow mods/room owners to cancel stars from the chat message menu box, including in the transcript. Reason for the discreteness arising in quantum mechanics? What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the boundary conditions, and the quanta in harmonic oscillator arise due to the commutation relations of the ladder operators, which give energy eigenvalues differing by a multiple of $\hbar$. But what actually is the reason for the discreteness in quantum theory? Which postulate is responsible for that. I tried going backwards, but for me it somehow seems to come magically out of the mathematics. Today I was informed by VFS that my tourist visa for UAE got approved. I'm travelling to UAE on 16th December 2014 and returning on 7th Jan 2015. But when I checked the "valid until" date on the visa it states 30 December 2014.    Does this date mean I should return before December 30th? Should I get it corrected from VFS? Remove impossible badges from per-site metas Per-site metas lists some badges that are impossible to get. For example  lists:         But these are impossibly to get because per-site metas do not support bounties, were not in a beta stage or Area 51 proposals. I therefore suggest that these badges are removed from the per-site metas. They have no function and they are confusing. There are already badges that are unavailable on per-site metas but available on the main sites:    Convention Quorum  This seems to suggest that per-site metas should only contain badges that can be possibly awarded.