A discontinuous function such that $f(x + y) = f(x) + f(y)$
Overview of basic facts about Cauchy functional equation
Is it possible to turn off radio without turning of wifi on Lenovo k900? I am currently having a look at the power consumption of a lenovo k900 phone and on the hardware side it seems Radio and Screen is using most of the power.  I am planning to use wifi but not the radio. Are they disconnected ? If so, can I use just the wifi, but no radio/telephony ? Are the settings or commands/intents to access these settings ?  Thanks! Renaming files in lexicographical order with a numerical pattern that increases sequentially with fixed padding I have files such as  a bb ccc abc emrls   I would like to rename them so that, after sorting them (for example, lexicographically), each file gets the following name:  00001 00002 00003 etc   where the # of digits for the 0 padding is specified a priori, e.g. 6 in the case above (assuming that we know how many digits are needed).  Since the shell I am most comfortable with is zsh, I'm interested in solutions that benefit from features on zsh (maybe using zsh's zmv?). I would also be interested in solutions that are compatible with Bash. Find and remove large files that are open but have been deleted How does one find large files that have been deleted but are still open in an application? How can one remove such a file, even though a process has it open?  The situation is that we are running a process that is filling up a log file at a terrific rate. I know the reason, and I can fix it. Until then, I would like to rm or empty the log file without shutting down the process.   Simply doing rm output.log removes only references to the file, but it continues to occupy space on disk until the process is terminated. Worse: after rming I now have no way to find where the file is or how big it is! Is there any way to find the file, and possibly empty it, even though it is still open in another process?  I specifically refer to Linux-based operating systems such as Debian or RHEL. Is there a symmetry associated to the conservation of information? Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.  We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information. I want to run a script to simply change the current working directory:  #!/bin/bash cd web/www/project   But, after I run it, the current pwd remains unchanged! How can I do that? plist or sqlite The  that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that the mean function $\mu_t = 0 $, since $$E(Y_{t}) = E(e_1+ e_2+ ... +e_t) = E(e_1) + E(e_2) +... +E(e_t) = 0 + 0 + \cdots + 0$$ But why is it that the variance increases linearly with time? Does this have something to do with it's not “pure” random, since the new position is very correlated with the previous one? EDIT: Now I have a much better understanding by visualizing a big sample of random walks, and here we can easily observe that the overall variance does increase over time,  and the mean is as expected around zero. Maybe this was trivial after all, since in the very early stages of the time series (compare time = 10 with time = 100) the random walkers have not had yet the time to explore as much. It is possible to make ls -l output the size field with digits grouped by thousands? If so, how?  For instance:  $ ls -l -rw-rw---- 1 dahl dahl 43,210,052 2012-01-01 21:52 test.py   (Note the commas in the size).  Maybe by modifying the LC_NUMERIC setting inside the locale I'm using (en_US.utf8)?  I'm on Kubuntu 12.04 LTS. Why too many capacitors in parallel for Vdd supply net? Can't we just add all to replace with one big capacitor? Need for a side book for E. Soper's Classical Theory Of Fields I am reading now E. Soper, Classical Theory Of Fields, now and sometimes it is very hard to follow the equations. So I need a side book on  to read it comfortably. Landau &amp; Lifshitz's book is not helping as its content and topics are very much different. Quickly create a large file on a Linux system How can I quickly create a large file on a Linux () system?   will do the job, but reading from /dev/zero and writing to the drive can take a long time when you need a file several hundreds of GBs in size for testing... If you need to do that repeatedly, the time really adds up.  I don't care about the contents of the file, I just want it to be created quickly. How can this be done?  Using a sparse file won't work for this. I need the file to be allocated disk space. Extension fields isomorphic to fields of matrices Suppose $K \subset L$ is a finite field extension of degree $m$. Is it true that there exists some natural $n$ such that $L$ is isomorphic to a subfield of $M^{n\times n}(K)$, the ring of $n\times n$ matrices with entries in $K$? What is the smallest dimension (as a function of $m$) for which an isomorphism exists?  The example I had in mind was the 2-dimensional extension $\mathbb{R} \subset \mathbb{C}$ with $\mathbb{C}$ being isomorphic to $\{ \begin{pmatrix}a&amp;-b\\b&amp;a\end{pmatrix} : a, b \in \mathbb{R}\}$. The relation between acceleration and inertial force How should I budget for ally/third party NPCs in 4e combats? I've done a few combats recently where the PCs and their enemies were not the only participants in the fight.  I wasn't sure how to budget XP in this case.  In the case of an NPC ally to the PCs, I see two options.     Draw the NPC up as a PC and increase the number of members in the party.    Draw the NPC up as an NPC and take its XP into account when budgeting.   I'm not sure if either option is correct, or if there are others.  But the ambiguity doesn't end there - what about if there's more than one side in the fight and nobody wants to team up?  I'd rather not treat it as one combat budget divided over two factions.  Unless the factions immediately unite, it's going to be an absurdly easy combat.  Would I be better off budgeting for two encounters than happen at the same time, and hoping that no PC gets ganged up on too much?  In general, how is XP budgeting done in complex combats with NPC allies and/or more than two factions? I typically have two needs:  Scenario A. Change a single PDF page.  In this case I have a PDF but not the original source file used to create the PDF. I don't want to try to recreate the document from scratch. I'd like to open the PDF and change a few things. A good example of this scenario: I was responsible for planning a big event at a campground site, I had a PDF of the site. I wanted to start with that document, highlight some parts, add some labels, remove some parts that weren't relevant.   or   Scenario B. Combine PDFs or extract page(s) from a PDF  This scenario usually arises because I want a single PDF deliverable that is made up of parts that are best created in different programs. In this case I have the source files for all the documents but they don't play well enough together to easily create a single PDF deliverable. For part of it, I may want to use Libre Office Writer. For another page I may want to use Gimp. Still another page I may use Libre Office Calc. I could use Writer as the master document and embed images or the Calc object into that, but for ultimate control, you can't beat separate PDF documents that are then combined.  What are the best tools / processes for editing PDFs in Ubuntu? apt-get remove with wildcard removed way more than expected. why? Why does Windows 7 ask for additional drivers when installing from a USB 3.0 drive? Is there such a thing as a line/vertices symbology? If not a workaround? In some situations, particularly when editing, I would like to see my lines as well as all the vertices similar to the effect I get when using the Edit Vertices tool but on all features in the feature class not just the selected feature.  The solution should be dynamic, without having to create a separate point layer but having one line features class loaded twice in the TOC is acceptable.  Programmatic solution using an example (addin, ArcObjects &amp; VB.Net) is acceptable.  I have the ArcEditor license in case there is a trick to use with representations. PHP regex delimiter, what's the point? Number of solutions of $x_1+2x_2+\cdots+kx_k=n$? Suppose $n$ be a given positive integer. Then the Diophantine equation $x=n$ has only $1$ solution. Just by inspection, I found that the Diophantine equation $x+2y=n$ has $\left\lfloor \dfrac{n}{2}+1\right\rfloor$ non-negative solutions for $(x,y).$ Also, according to  the Diophantine equation $x+2y+3z=n$ has $\left\lfloor \dfrac{n^2}{12}+\dfrac{n}{2}+1 \right\rfloor$ non-negative solutions for $(x,y).$  Is there any closed form for the number of non-negative integer solutions for $(x_1,x_2,\cdots,x_k)$ of $$x_1+2x_2+3x_3+\cdots+kx_k=n$$ for a given $k\in\Bbb{N}$? How can I prove these formulas rigorously?  EDIT After a very tedious calculation I found that the equation $w+2x+3y+4z=n$ has  $\left\lfloor \dfrac{n^3}{144}+\dfrac{5n^2}{48}+\dfrac{(15+(-1)^n)n}{32}+1 \right\rfloor$ solutions. This solution completely agree with the approximation given by Rus May. However still I believe that we can do something more than this.