Consider the following problem:     Let $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ be real numbers such that   $$a_1^k+\cdots a_n^k= b_1^k+ \cdots+b_n^k$$   for all $1\leq k\leq n$.   Then there is a bijection $\varphi:\{1, 2, \ldots, n\}\to \{1, 2, \ldots, n\}$ such that $a_i=b_{\varphi(i)}$ for all $i$.   This can be proved using , which gives that the polynomials $p(x)=(x-a_1)\cdots(x-a_n)$ and $q(x)=(x-b_1)\cdots(x-b_n)$ are identical.  My question is, can this be said about countable sequences. More precisely, is the following true?     Let $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3,\ldots$ be real numbers such that   $$a_1^k+a_2^k+a_3^k+\cdots = b_1^k+ b_2^k+b_3^k+ \cdots$$   are equal for all $k\geq 1$ (Here we insist that both sums are convergent for each $k$).   Then there is a bijection $\varphi:\mathbb N \to \mathbb N$ such that $a_i=b_{\varphi(i)}$ for all $i$.
Let $(a_n)_{n≥0}$ and $(b_n)_{n≥0}$ be two sequences of a nomed algebra such that $\sum{\| a_n\|}$ and $\sum{\| b_n\|}$ converge, and$ $$$\forall n, \  a_n, b_n \neq 0$$  Show that $(\forall  k \in \mathbb N^*,  \ \sum_{n=0}^{\infty}{a_n}^k = \sum_{n=0}^{\infty}{b_n}^k) \implies \exists\ \sigma \in{\mathfrak S}_{\mathbb N} \ ,\ (a_n) = (b_{\sigma(n)})$  The vector-space should be a finite-dimensional $\mathbb R$-vector-space.   I do not see how to deal with the hypothesis. If you have ideas/hints...  It would be relevant to deal with complex sequences even if it is less general. But I don't think one can generalize using real and complex cases.
I have this graphic. I need to change the color of the graphic to hex color: 1485cc...is there a way to make this graphic match that particular hex color hue...without me having to have to re create the graphic. Install suspension forks on non-suspension bike I have a 1990s Saracen PowerTrax. Is there a way I can install a suspension fork, or will that muck up the bike somehow? I'm actually interested in ways to raise the bars with respect to the seat, so that wouldn't be a problem. My toes rubbing in the wheel, in the other hand, would definitely be a problem. Site blocked by content filter I run a large and high trafficked WordPress powered platform. One of my editors informed me that our site is blocked at his workspace as a "games site". They use WebCat to filter the traffic.  The issue here is that our platform is not a gaming site, although it does have a games category that covers news. And that is a very minor part of the site.  My question is, what can I do to unblock the site? How does WebCat work? Do they use a universal database to filter the traffic (meaning that the site is blocked on all workspace with WebCat) etc. Obviously they need to improve their filtering algorithm. Can I use SPI for asynchronous serial output? I need a second serial async output channel on the MSP430G2553 and would like to use the SPI USCI to do this.  Is the USCI SPI baud rate stable enough to do this at 300 baud, 2400 baud or 9600 baud? I am currently using the 8 MHz clock.  Is there any other option that can save me from bit-banging?  My current thinking for 300 bd Tx is as follows:   set prescaler to 26666 LSB first load byte into Txbuf load 0x7F into Txbuf to generate stop and start bits repeat 2-3 send final stop bits 0xFF   It might also be possible to send ten bits of data on other members of the MSP430 family that contains a SPI.  Thanks for your responses. I'm studying for qualifying exams and ran into this problem.  Show that if $\{a_n\}$ is a nonincreasing sequence of positive real numbers such that $\sum_n a_n$ converges, then $\lim_{n \rightarrow \infty} n a_n = 0$.  Using the definition of the limit, this is equivalent to showing \begin{equation} \forall \varepsilon &gt; 0 \; \exists n_0 \text{ such that } |n a_n| &lt; \varepsilon \; \forall n &gt; n_0 \end{equation} or \begin{equation} \forall \varepsilon &gt; 0 \; \exists n_0 \text{ such that } a_n &lt; \frac{\varepsilon}{n} \; \forall n &gt; n_0 \end{equation} Basically, the terms must be bounded by the harmonic series. Thanks, I'm really stuck on this seemingly simple problem! How do I convert a video to GIF using ffmpeg, with reasonable quality? I'm converting a video to GIF file with ffmpeg:  ffmpeg -i input.flv -ss 00:00:00.000 -pix_fmt rgb24 -r 10 -s 320x240 -t 00:00:10.000 output.gif   It works great, but output gif file has a very low quality.  Any ideas how can I improve quality of converted gif? Does removing the comma before 'which' etc in a non-restrictive clause change the meaning of the sentence? Let $S=\{1,2,3,4,\dotsc,N\}$ and $X=\{ f: S \rightarrow S \mid x < y \Rightarrow f(x)\leqslant f(y)\}$. Then $|X|$ is equal to what? In a question related to introductory Galois theory, I was asked to given an example of a tower of fields $F \subset K \subset E$ such that $E$ is a splitting field for some polynomial $f(x) \in F[x]$, but that $K$ need not be a splitting field. The example that comes to mind for me is $\mathbb Q \subset \mathbb Q(\sqrt[3]{2}) \subset \mathbb Q(\omega_3, \sqrt[3]{2})$.  So I know that the minimal polynomial for $\mathbb Q(\omega_3, \sqrt[3]{2})$ over $\mathbb Q$ is $x^3-2$. This polynomial does not split in the intermediate field $\mathbb Q(\sqrt[3]{2})$.  I wanted to make sure that I'm justifying that this intermediate field is not a splitting field correctly. Now I may be confused about the books terminology. When we say something is not a splitting field, we always have to refer to a specific polynomial correct? I've read a term called a normal extension which I think refers to one in which every polynomial splits, but my book doesn't mention these.   If splitting field means with respect to the polynomial $x^3-2$, then I've already demonstrated that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field. Also, it couldn't be a normal extension either because that polynomial doesn't split.  Does it sound like I'm understanding this correctly or are there some subtleties I might be missing? gnome maps program never runs I have this problem while running GNOME Maps ~$ gnome-maps  (org.gnome.Maps:10414): Gjs-WARNING **: 19:43:34.553: Some code called array.toString() on a Uint8Array instance. Previously this would have interpreted the bytes of the array as a string, but that is nonstandard. In the future this will return the bytes as comma-separated digits. For the time being, the old behavior has been preserved, but please fix your code anyway to explicitly call ByteArray.toString(array). (Note that array.toString() may have been called implicitly.) 0 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmTypes.js&quot;:32] 1 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmEditDialog.js&quot;:35] 2 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/osmEdit.js&quot;:25] 3 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/contextMenu.js&quot;:33] 4 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/mainWindow.js&quot;:33] 5 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/application.js&quot;:35] 6 &lt;TOP LEVEL&gt; [&quot;resource:///org/gnome/Maps/js/main.js&quot;:43] 7 start() [&quot;resource:///org/gnome/gjs/modules/package.js&quot;:209] 8 &lt;TOP LEVEL&gt; [&quot;/usr/bin/gnome-maps&quot;:2]   (org.gnome.Maps:10414): Gdk-WARNING **: 19:43:36.525: Native Windows taller than 65535 pixels are not supported Gdk-Message: 19:43:36.574: Error 71 (Protocol error) dispatching to Wayland display.  19:43:36.574 (This message has changed many times) What's this? OLS is BLUE. But what if I don't care about unbiasedness and linearity? Positive euphemism for "harped on" How can I replace harped on with a more positive expression?  The production manager harped on the new quality assurance regulations for nearly an hour. Unable to Retrieve Metadata How do you match only valid roman numerals with a regular expression? Can the Force be used to fly? In The Empire Strikes Back Yoda uses the Force to lift Luke's X-Wing out of the water and put it on land.  In Attack of the Clones and in many instances in the second series of The Clone Wars, Anakin (and others) use the Force to jump higher than they could otherwise, or to jump from very high locations, but I don't think I've ever seen the Force used for outright self-levitation.  But if Yoda could lift the X-Wing, shouldn't a strong Jedi be able to lift himself and, essentially, fly with the help of the Force? Countable sets have the same cardinality as the natural numbers (or a subset of the natural numbers, depending on who you ask).  Can one make the same claim about uncountable sets and real numbers?  In other words, do all uncountable sets have same cardinality as real numbers? Today I've got 10 upvotes for an answer. Answer was a first answer in the question. I got an Nice Answer badge for this, but no Enlightened badge.  Shouldn't I get both?  is my answer :) dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with different file `/usr/lib/xorg/protocol-precise.txt', not allowed I just installed Ubuntu (12.04.4) and I upgraded to 13.10  I have no idea how to fix this :/  $ sudo apt-get install -f  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be REMOVED:   xserver-common-lts-saucy 0 upgraded, 0 newly installed, 1 to remove and 2 not upgraded. 1 not fully installed or removed. After this operation, 1,718 kB disk space will be freed. Do you want to continue [Y/n]? y (Reading database ... 175174 files and directories currently installed.) Removing xserver-common-lts-saucy ... Removing 'diversion of /usr/lib/xorg/protocol.txt to /usr/lib/xorg/protocol-precise.txt by xserver-common-lts-saucy' dpkg-divert: error: rename involves overwriting `/usr/lib/xorg/protocol.txt' with   different file `/usr/lib/xorg/protocol-precise.txt', not allowed dpkg: error processing xserver-common-lts-saucy (--remove):  subprocess installed post-removal script returned error exit status 2 Errors were encountered while processing:  xserver-common-lts-saucy E: Sub-process /usr/bin/dpkg returned an error code (1) Chat gives me really, really specific time gap notifications How does "virus been detected" on android pop up on chrome when visiting some sites? What is the cause of this chrome bug on android? The last site I went to was www.destructoid.com and all of a sudden it gets redirected to the following url. This happened previously awhile back, but I didn't document everything as I did today. I'm also on android chrome canary if that matters.   I took some screenshots before closing out of that tab. But my question is:   What chrome bug or malware is causing these random redirects? How can one detect if they have this vulnerability?   The url is: