I am to prove that the variation of Dirichlet's function is infinite. I would really appreciate any help, because I do not know how to start. $$f(x)= \begin{cases} 1  &amp; \text{if $x\in\mathbb{Q}$}\\ 0 &amp; \text{if $ x \notin\mathbb{Q}$} \\ \end{cases}$$
I was trying to do some of these questions to check my understanding about the topic, but I'm not sure if they're correct. Here are my answers.   1) Suppose $f$ is continuous on $[0,1]$. Must there be a nondegenerate closed subinterval $[a,b]$ of $[0,1]$ for which the restriction of $f$ to $[a,b]$ is of bounded variation?   No, because $f(x)$ might be a constant function. In this case, $f(x)$ will not be the difference of two increasing functions...and so will not be of bounded variation.  2) Let $f$ be the Dirichlet function, the characteristic function of the rationals in $[0,1]$. Is $f$ of bounded variation on $[0,1]$?   No, because we cannot express it as the difference of two increasing functions.  3) Let $f$ be a step function on $[a,b]$. Find a formula for its total variation.  If we partition $[a,b]$ in such a way that for each $[x_{i-1}, x_i]$ (which is part of the partition), $x_{i-1}$ and $x_i$ each correspond to the midpoint of a step. I know this is not a formal answer, but I'm just trying to check my understanding.
How would I go about detecting monopoles? A question needed for a "solid" sci-fi author: How to detect a strong magnetic monopole? (yes, I know no such thing is to be found on Earth).  Think of basic construction details, principles of operation and necessary components of a device capable of detecting/recognizing a macroscopic object emitting magnetic field of equivalent of order ~0.1-10 Tesla near its surface, but with only one pole, reliably distinguishing it from normal (2-pole) magnets, preferably at a distance.  Preferably a robust method, not involving extremely advanced technology. Detect the presence, possibly distance (or field strength) and direction.  I know of SQUIDs, but these concentrate on extreme sensitivity. I'm thinking of something less sensitive but more robust (like, no need for the monopole to fall through the loop) and still able to recognize a monopole against a magnet.  Also, how would such a macroscopic object behave practically? Such a "one-pole magnet" about the size and strength of a refrigerator magnets - how would it behave around ferromagnetics, normal magnets and so on? Black artifacts when looking through many transparent objects? Using v2.73, cycles and a very simple scene: an array of cubes, no big deal.   If I set the shader's material to transparent, the cubes in the foreground do become transparent but those far away in the back do not disappear completely and this strange dark things appear (visible in rendered mode and in final renders)    I've played with the clipping distance on the viewport and camera, but I can't figure out how to make those ghosts go away... Since the Denoise feature was added to Blender I only use the default setting with it and it really does the work. But sometimes the denoise is noticable on some of my renders. How can I overcome the blurred effect? Denoising for me is a big help because my system is quite aged. Some people have a diamond after their username (e.g. )    What special privileges do diamond moderators have? How can I become a diamond moderator? Who are the diamond moderators?  How many are there?      For more information, see  in the .     Note: not to confuse with certain Stack Exchange employees, for example  members or "community managers", who often have diamonds as well, but are not ordinary moderators. Labels disappear while zooming What file sizes and formats are most appropriate for enlarging and projecting an image onto a large wallspace for painting? Package system is broken. How to fix it? I just recently bought an arduino and requires gcc-avr/avrdude to compile the software. I installed avr for another microprocessor component a while ago but is obviously an outdated version (gcc version 3.3 20030512 (prerelease)) so I went ahead to update these but it didn't work.   (Please note that before hand I also broke my aptdaemon through an incomplete wine installation (couldn't get passed font installation agreement) but I fixed that via a re-installation then accepting the agreement.)   I am trying to update these by running  but the dependencies it requires cannot be installed due to avr... terminal reports   Reading package lists... Done Building dependency tree        Reading state information... Done build-essential is already the newest version. libncurses5-dev is already the newest version. libncurses5-dev set to manually installed. libusb-0.1-4 is already the newest version. libx11-dev is already the newest version. You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: arduino-core : Depends: gcc-avr but it is not going to be installed avr-libc : Depends: gcc-avr (&gt;= 1:4.3.4) but it is not going to be installed Depends: binutils-avr (&gt;= 2.20) but it is not going to be installed binutils-dev : Depends: binutils (= 2.21.0.20110327-2ubuntu3) but 2.21.0.20110327-2ubuntu2 is to be installed libcwidget-dev : Depends: libcwidget3 (= 0.5.16-3ubuntu2) but it is not going to be installed Depends: libsigc++-2.0-dev but it is not going to be installed Depends: libncursesw5-dev but it is not going to be installed libmpfr-dev : Depends: libgmp3-dev (&gt;= 4.2.dfsg-1) but it is not going to be installed python-dev : Depends: python (= 2.7.1-0ubuntu5.1) but 2.7.1-0ubuntu5 is to be installed Depends: python2.7-dev (&gt;= 2.7.1-1~) but it is not going to be installed tk8.4-dev : Depends: tk8.4 (= 8.4.19-4) but it is not going to be installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).   But running   apt-get -f install   results in   (Reading database ... 163021 files and directories currently installed.) Unpacking binutils-avr (from .../binutils-avr_2.20.1-1ubuntu2_i386.deb) ... dpkg: error processing /var/cache/apt/archives/binutils-avr_2.20.1-1ubuntu2_i386.deb (--unpack): trying to overwrite '/usr/bin/avr-size', which is also in package avr-binutils 2.13.90.030512-2 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Unpacking gcc-avr (from .../gcc-avr_1%3a4.3.5-1_i386.deb) ... dpkg: error processing /var/cache/apt/archives/gcc-avr_1%3a4.3.5-1_i386.deb (--unpack): trying to overwrite '/usr/bin/avr-g++', which is also in package avr-gcc-c++ 3.2.90.20030512-2 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing: /var/cache/apt/archives/binutils-avr_2.20.1-1ubuntu2_i386.deb /var/cache/apt/archives/gcc-avr_1%3a4.3.5-1_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   I have tried running update manager and updating my system via it but all I get is an error message  The following packages have unmet dependencies: arduino-core: Depends: gcc-avr but it is not installed avr-libc: Depends: gcc-avr (&gt;= 1:4.3.4) but it is not installed Depends: binutils-avr (&gt;= 2.20) but it is not installed   which then tells me to try running apt-get -f install which just results the same as last time.  So how can I fix my system, I really need the new avr, please :) BTW, my system is Ubuntu 11.04 The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group, I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question. My Question is :   Why is the name Quotient Group kept,normally in the case of division, let us take the example of $\large \frac{16}{4}$ the Quotient of the Division is '$4$' which means that there are four '$4$'s in $16$, I mean we can find only $4$ elements with value $4$     So how can we apply the same logic in the case of Quotient Groups,like consider the Group $A$ and normal subgroup $B$ of $A$, So if $A/B$ refers to "Quotient group", then does it mean:           Are we finding how many copies of $B$ are present in $A$??, like in the case of normal division, or is it something different ??       I understood the Notion of Cosets and Quotient Groups,b ut I want a different Perspective to add Color to the concept. Can anyone tell me the necessity and background for the invention of Quotient Groups?  Note: I tried my level best in formatting and typing with proper protocol,if in case, any errors still persist, I beg everyone to explain the reason of their downvote (if any), so that I can rectify myself, Thank you. What are some good books for learning the concepts of Kinematics, Newton laws, 2D Motion of Object etc.? I recently made the switch to Ubuntu. Everything on my laptop works fine except my hybrid graphics card. It is Intel graphics combined with Nvidia GeForce GT650M. I tried Bumblebee, but primusrun doesn't work (segmentation fault, core dumped) and optirun works with glxspheres64, but steam games look worse with optirun than with Intel graphics (e.g. Surgeon Simulator 2013 runs fine with Intel and hangs a lot with optirun)   I tried Nvidia Prime, but it gives me a black screen on startup after install, so I can't do anything and have to power off my PC with the power button. I also tried nvidia-prime 0.5 on Ubuntu 14.04, and then I get 'low-graphics mode' and I can't get to Unity. Let $(X,\mathcal{M},\mu)$ be a measure space and $\{f\}$ be a sequence of functions on $X$, each of which is integrable over $X$. Show that $\{f_n\}$ is uniformly integrable if and only if for each $\varepsilon \gt 0$, there is a $\delta \gt 0$ such that for any natural number $n$ and measurable subset $E$ of $X$, if $\mu(E) \lt \delta$, $$ \left|\int_E f_n~d\mu\right| \lt \epsilon.$$     I think one direction $(\Rightarrow)$ is clear. Since $f_n$ being uniformly integrable imply that for every $\varepsilon\gt 0,~\exists \delta \gt 0$ such that for any $E\in \mathcal{M},~\mu(E)\lt \delta$, $\int_E |f_n|~d\mu$ for every natural $n$. But then   $$ \left|\int_E f_n~d\mu\right|\le \int_E|f_n|~d\mu \lt \varepsilon.$$    Any suggestions for the other direction?   Edit:  Following Davide's suggestions, I have   $$ \begin{align*} \int_E |f_n|~d\mu &amp; \le \int_{E\cap [f_n \ge 0]} f_n~d\mu + \int_{E\cap [f_n \lt 0]} (-f_n)~d\mu\\ &amp; = \left| \int_{E\cap [f_n \ge 0]} f_n~d\mu \right| + \left| \int_{E\cap [f_n \lt 0]}f_n~d\mu \right| \\ &amp; \lt \varepsilon + \varepsilon = 2\varepsilon. \end{align*} $$ Use nmcli to modify VPN password? With nmcli, I can modify VPN data like this:  nmcli con modify myvpn vpn.data 'refuse-pap = yes, user = xxx'   So I'm wondering if I can set password this way as well? I was naively validating my binomial logit models by testing on a test dataset. I had randomly divided the available data (~2000 rows) into training (~1500) and validation (~500) datasets.   I now read a post in another thread ( Frank Harrell) that causes me to question my approach:     Data splitting is not very reliable unless you have more than 15,000   observations. In other words, if you split the data again, accuracy   indexes will vary too much from what you obtained with the first   split.   How serious is this worry and what are ways around it? The OP speaks of "resampling" but not sure how that works here for validation.   Edit: Adding context as per @Bernhard's comment below: Drawing users' attention to unrecoverable actions? Star vs Heart icon to represent save as a favourite I am working on a mobile app that involves the user saving a favourite location. I wanted to use an icon to represent that feature, but I'm not sure which one is better – a star or a heart icon.  Some say heart icon tends to increase the rate of productivity, and some say they are used to a star icon to 'save' or 'bookmark' something.  What do go you guys think? Retrieve Voice Memos from device? iOS seems to refuse to allow me to email "long" voice memos, and I'm unaware of any way to extract them from iTunes backups of my phone. How can I retrieve entire voice memos that aren't really short? Find all integer solutions to Diophantine equation $x^3+y^3+z^3=w^3$ Compute all integer solutions to the equation  $$x^3+y^3+z^3=w^3$$ Conky does not work properly on Ubuntu 16.04 GNOME I have this strange problem with conky. It doesn't show properly on the desktop:    I've tried everything from editing conky widget file to reinstalling conky and conky manager (v.2.4). Who knows how to fix it? Problems with pie charts There seems to be in increasing discussion about pie charts.   The main arguments against it seem to be:   Area is perceived with less power than length. Pie charts have very low data-point-to-pixel ratio   However, I think they can be somehow useful when portraying proportions. I agree to use a table in most cases but when you are writing a business report and you've just included hundreds of tables why not having a pie chart?  I'm curious about what the community thinks about this topic. Further references are welcome.  I include a couple of links:         In order to conclude this question I decided to build an example of pie-chart vs waffle-chart. Dedicated mysql server slow response