Find all the functions for which $f(x^{2})-f(y^{2})=(x+y)(f(x)-f(y)):\forall x, y\in\mathbb{R}$ Hey guys I need help on solving the following problem Find all the functions for which : $$f(x^{2})-f(y^{2})=(x+y)(f(x)-f(y)):\forall x, y\in\mathbb{R}$$ I started with taking $x=y$, and there you see that the both sides are equal to $0$, therefore the solution is the function $y=x$, but after that I can't prove that that is the only solution (if it is the only). Would appreciate some help, thanks.
Find all functions $f$ such that $(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$ Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$  I've tried subbing in heaps of values but I keep getting things like $f(0)=f(0)$ and other such useless results.  Any help would be hugely appreciated.
-Bash command not found appearing every time I start terminal every time I start my terminal, it always says "-bash:  .: command not found" twice.  Does anyone know how I can get rid of that? Cannot mount NTFS partition in Ubuntu 13.04 After I installed Ubuntu 13.04, I can't mount my NTFS partition.  Error mounting /dev/sda5 at /media/mrxfly/xFLyDiskData: Command-line `mount -t "ntfs" -o "uhelper=udisks2,nodev,nosuid,uid=1000,gid=1000,dmask=0077,fmask=0177" "/dev/sda5" "/media/mrxfly/xFLyDiskData"' exited with non-zero exit status 14: The disk contains an unclean file system (0, 0). Metadata kept in Windows cache, refused to mount. Failed to mount '/dev/sda5': Operation not permitted The NTFS partition is in an unsafe state. Please resume and shutdown Windows fully (no hibernation or fast restarting), or mount the volume read-only with the 'ro' mount option. finite and infinite product of discrete space Creating Ellipse in WGS84, but with metric parameters I have the function  --Ellipse(x,y,rx,ry,rotation,#of segments in 1/4 of ellipse) SELECT ST_Translate( ST_Rotate( ST_Scale( ST_Buffer(ST_Point(0,0), 0.5, $6), $3, $4), $5), $1, $2)   I have points in WGS84 (x,y), radiuses in meters (rx, ry), rotation angle in degrees. And I need a new WGS84 geometry as a result.  As you can see I can't call this function without conversions between WGS84 and some metric SRID. But my points aren't in a definite region. So a SRID which is optimal for one point, throws an error for another point during conversion.  Is there a common way to get an optimal metric projection (SRID) for a WGS84 point? Or at least to use some rough, but world-wide universal metric projection? Or maybe there's a trick for this case? Is it possible to create a geocoding and reverse geocoding service based on the shapefiles data Though tangentially related to another post on MathOverflow (), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...    Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given topology, $\tau$, on $X$ denote by $\mathfrak{B}_X(\tau)$ the Borel algebra over $X$ generated by $\tau$.     Question 1. Does there exist a topology, $\tau$, on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$?   If the answer to the previous question is affirmative, it makes sense to ask for the following too:     Question 2. Denote by ${\frak{T}}_X(\Sigma)$ the family of all topologies $\tau$ on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$ and let $\tau_X(\Sigma) := \bigcap_{\tau \in {\frak{T}}_X(\Sigma)} \tau$. Is $\Sigma = \mathfrak{B}_X({\frak{T}}_X(\Sigma))$?   Updates. Q2 was answered in the negative by Mike (). Ethernet Without Magnetics? How / can you connect Ethernet devices together without magnetics? How do you wire it up? What sorts of practical limitations are there on doing this?  To clarify, I'm not trying to connect two computers here. I'm thinking specifically of creating a small network of microcontrollers on a single board. Lets say the total length of a network segment is no longer than 6 inches. This would be for 10BaseT or at most 100BaseT. How to adjust the breaking in the bibliography? Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function? If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples? Are old (>35) faculty candidates discriminated against all over the world? I think that in India research institutes will never take a faculty candidate who is over 35. Is it so all over the world?   I am asking this question from the point of view of if someone starts their PhD. at the age of 27 then its unlikely they will be faculty candidates till they are 40. I just enabled the Blenderkit addon but cannot seem to figure out how to edit the objects. I looked in this reddit thread and amazingly none of the solutions worked for me.  Making it a single user object, data, and materials did not work. I do not see any option to append when adding a model form the Blenderkit asset bar (I even tried saving it and appending the object in another instance of Blender and it still would not let me edit it). And when I right click I do not see an option to make local on outliner collection. I am using Blender 2.83 if that makes a difference but can use 2.9 if need be. Any other solutions? What is "it" in the following sentence: It is clear that Bob likes doughnuts I used WolframAlpha to calculate a sum but it didn't show me the way :( Anybody has a hint or a solution for proving this sum? $$\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$$ How to change language using command line on CentOS 7? I use the following command line to set my system in English (installed in French) :  localectl set-locale LANG=en_US.utf8   However, when I boot into graphical mode, the desktop menu, directories names etc. are still in French.  The only way I can set everything in english is to use the graphical gnome menu "Applications > System tools > Settings > Region and Language".  Is there a way to change system language using command line as if I used the graphical menu?  EDIT :  In fact, my problem was coming from the fact that I was modifying through SSH with user root the system language but then loging through graphical console (workstation VM) under a standard user who has its language already defined. localectl works well but to define default language for a specific user, I must define $LANG in my .bashrc How to distinguish between American Indians and Indian Indians in native English (language) parlance? How to distinguish between American Indians and Indian Indians in native English (language) parlance?  Can I say Indian Indian to say Indian from Asia compared to the Native Americans? How to append multiple lines of text to a file? This is my first question in /photography. I'm a very amateur photographer, doing things wrong and by eye with lots of trial-and-error all the time.  I currently own a reflex Nikon D70 my uncle gave us after he bought a better one. It has some nice settings, and he bought a a Nikkor 28-80mm F/3.5-5.6 zoom lens for it. It behaves nicely, but it's getting old (it has those old cartridges before SD cards existed) and every new charge takes longer than the last.  I'm planning to upgrade to a better camera, and there are some neat offers on a nearby superstore (MediaMarkt), this one catching my eye:  Sony DSC-H400 63x, Sensor Super HAD 1/2/3 CCD 20.1MP -> for less than 300€ taxes included  I don't have any type of extra information aside from that, all I can get from the official site is clinical data from other cameras that I suspect it's not 100% accurate, and there's no photography expert on the store to ask.  The common things I use my D70 for are: taking pictures of scenery (castles, towns, forests, waves hitting rocks, etc...), nearby (&lt;5m) persons and objects, and well, things where I want to save a memory in digital format, not for professional purposes at all.  Being that my objective is to have a decent, good-zooming (superzoom looks awesome), amateur (but with manual settings) camera, is this Sony a good replacement? Will I miss any features? Is there any better, similarly-priced alternative you know of?  Basically, I want to know whether I will get better quality photos than I currently do, given the examples, with any averagely-priced bridge camera similar to that Sony, and if it's worth it.  examples of current camera results  Full zoom-out of town at night, bad quality I know; notice the pink blur at the left.    Maximum zoom on that pink blur from almost the same spot:    Daylight, cloudy, modified with Adobe Lightroom, scenery:    Same day, same beach, different settings:    Same day, same beach, rocks: How to manually parse a floating point number from a string I started reading "The Mathematics of nonlinear Programming - AL peressini" recently.  Theorem : Suppose that $f(x) , f'(x) ,f''(x)$ exist on the closed interval $[a,b]$. If $x^* ,x $ are two different points of $[a,b]$ , then there exists a point $z$ strictly between $x^*$ and $x$ such that   $$f(x)=f(x^*)+f'(x^*)(x-x^*)+\frac{f''(z)}{2}(x-x^*)^2$$  I tried to prove this theorem using The Fundamental theorem of calculus. But I can not stop it at the second derivative . Do I have to use mean value theorem ? or is there a simple proof for this ? Emoji Characters I'm trying to display Emoji Characters inside my XeLaTeX Document. It seems to work for some glyphs like the numbers and pound sign but not for all. Here is a MWE:  \documentclass{scrartcl}  \usepackage{fontspec}  \setmainfont{Apple Color Emoji} % The emoji Font   \begin{document} \char"0030{}  %Works \char"1F60D{} % Doesent Work should out put SMILEY FACE WITH HEART SHAPED EYES \end{document}   What is the reason for this strange behavior and is there a way to include Emojis in XeLaTeX?  Edit:   It seems the Character is there. It is selectable but not visible. I can copy&amp;paste it.