I've recently quit my job as a maths teacher after doing it for 3 years and I have been incredibly lucky to be offered a fully funded doctoral training course at a top university. This is an extraordinary opportunity for me, especially given that I previously withdrew from a PhD, and this time I really want to make it work.  I am quite rusty on some things that many other starting students will not be, not because I can't do them but because it's been a very long time since I needed to apply them. I am currently a bit worried that I may be the weakest in the cohort or that it'll take me a very long time to understand simple things and that I may feel incompetent and like I don't deserve to be there.  What are some good ways to overcome these feelings of idiocy?
I am a third year (starting fourth year in the fall) PhD student in mathematics. I've passed all qualifying exams and am currently doing research. As far as I can tell, I am not doing poorly. I have the good fortune of having a great advisor, being in a very supportive department, and having friends and family who genuinely care about my success.   The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall. I am not complaining about the material being hard, and I am not averse to putting in hard work, but I get frequently discouraged when I realize the vast volumes of mathematics that I yet know nothing about (and probably never will). It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked. I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.   I've talked about this to some number of people; here is some advice I have received:   Take a day off. Putting in hours upon hours of trying things doesn't magically lead to a solution, particularly if the brain is tired and just wants to sleep. Have a hobby. Since math research doesn't exactly provide instant gratification, a hobby might provide some instead.  Talk to other graduate students. Realize that many graduate students go through this.   I'm interested to know how other folks have dealt with being discouraged as a graduate student. Does it get better with time and experience? Is this a sign that research is not for me and that I should seriously consider a life outside academia?
Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $f(f(x) + y) = f(x^2 - y) + 4yf(x)$. Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $$f\big(f(x) + y\big) = f(x^2 - y) + 4yf(x)$$ for all $x,y\in\mathbb{R}$.   $f(x) = 0$ is obvious , but how can we find all other functions?  Thanks in advance! How do I run an application with arguments from the command line without losing the terminal? I use dwm as a window manager on Ubuntu, I like tiled windows and I like them as few as possible. I sometimes need to run an application (let's say it's the pdf viewer mupdf in x11) from the terminal, the application will be launched in a new window but the terminal from which I launched becomes this prompt-less console for that application to log whatever.  I use dmenu to launch applications, but sometimes an application needs arguments (file names) to start. I use uxrvt within x11 managed by dwm. Battle of Hoth: how did the AT-ATs manage to disable the shield generator? (This question is related to , and was originally posted as an answer there, but it's not the same)    In , the Empire attacks Rebel Echo Base located in Hoth. The Rebels are alerted before the attack, and raise some sort of shield to prevent orbital bombardment of their base. The Empire is then forced to deploy land forces to first disable the shield generator, before destroying the base. Presumably, the Imperial forces are deployed using dropships.  My question is about how the Rebel shield generator works, and how the Imperials manage to bypass it.  I think the shield generator is something of a plot hole. At first I incorrectly thought it was a planetary shield (i.e. a shield around all of Hoth) preventing orbital bombardment, and also preventing any dropships (with their AT-AT payload) from landing, but I think : the  states it is an energy field around just an area of Hoth, presumably Echo Base:     VEERS: My lord, the fleet has moved out of light-speed. Com-Scan has   detected an energy field protecting an area around the sixth planet of   the Hoth system. The field is strong enough to deflect any   bombardment.   So my initial objection ("how did the Imperial dropships bypass the shield?") has been answered: they didn't. They landed outside the energy field to deliver the AT-ATs. But now I have another question: does an AT-AT have to walk inside the field to fire at the generator, or can it fire into the field from outside?  Maybe the field simply cannot stop matter, only energy weapons (therefore preventing orbital bombardment), in which case the AT-ATs would be able to just walk inside. However, the script also states:     LEIA: All troop carriers will assemble at the north entrance. The   heavy transport ships will leave as soon as they're loaded. Only two   fighter escorts per ship. The energy shield can only be opened for a   short time, so you'll have to stay very close to your transports.   So it's clear that the energy field is also capable of stopping matter, in this case troop transports and fighters. It follows that it should also prevent AT-ATs from walking through it. So the only way I can see the walkers disabling the generator is by firing into the energy field!  This would mean the field is strong enough to stop matter such as ships and fighters, and energy weapons used for orbital bombardment (and also the Ion Cannon the Rebels use as an anti-orbit battery: Leia also states they must lower the shield to use it), but not able to stop whatever energy weapons the AT-ATs use!  So how on Earth did Imperial Walkers manage to disable the shield generator? I am trying to parse url-encoded strings that are made up of key=value pairs separated by either &amp; or &amp;amp;.   The following will only match the first occurrence, breaking apart the keys and values into separate result elements:  var result = mystring.match(/(?:&amp;|&amp;amp;)?([^=]+)=([^&amp;]+)/)   The results for the string '1111342=Adam%20Franco&amp;348572=Bob%20Jones' would be:  ['1111342', 'Adam%20Franco']   Using the global flag, 'g', will match all occurrences, but only return the fully matched sub-strings, not the separated keys and values:  var result = mystring.match(/(?:&amp;|&amp;amp;)?([^=]+)=([^&amp;]+)/g)   The results for the string '1111342=Adam%20Franco&amp;348572=Bob%20Jones' would be:  ['1111342=Adam%20Franco', '&amp;348572=Bob%20Jones']   While I could split the string on &amp; and break apart each key/value pair individually, is there any way using JavaScript's regular expression support to match multiple occurrences of the pattern /(?:&amp;|&amp;amp;)?([^=]+)=([^&amp;]+)/ similar to PHP's preg_match_all() function?  I'm aiming for some way to get results with the sub-matches separated like:  [['1111342', '348572'], ['Adam%20Franco', 'Bob%20Jones']]   or   [['1111342', 'Adam%20Franco'], ['348572', 'Bob%20Jones']] How to hide a folder in sdcard in android Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ having the following properties?   $f(x) = 0$ for all $x \le 0$. $f(x) = 1$ for all $x \ge 1$. For $0 &lt; x &lt; 1$, $f$ is strictly increasing. $f$ is everywhere $C^\infty$. The sequence of $L^\infty$ norms $\langle \left\lVert f \right\rVert_\infty, \left\lVert f' \right\rVert_\infty, \left\lVert f'' \right\rVert_\infty, \dots \rangle$ is bounded.   If we impose only the first four conditions, there is : for $0 &lt; x &lt; 1$, define $f(x)$ by $$ f(x) = \frac{e^{-1/x}}{e^{-1/x} + e^{-1/(1-x)}} = \frac{1}{1 + e^{1/x - 1/(1-x)}} $$ However, the derivatives of this function appear to grow quite rapidly. (I'm not sure how to verify this, but it seems at least exponential to me.)  If such a function does not exist, what is the smallest order of asymptotic growth that the sequence $\langle \left\lVert f \right\rVert_\infty, \left\lVert f' \right\rVert_\infty, \left\lVert f'' \right\rVert_\infty, \dots \rangle$ can have? Using Mosfet with seperate Logic Converter and Arduino Replacement character being used question URL Some questions, for example    Instead of using a correct URL, such as  http://meta.stackexchange.com/questions/79543/٩●̮̮̃-̃۶-forbid-unicode   are using  https://superuser.com/questions/516363/what-is-the-��-character   What is causing this behavior? I have a Canon XTI which I've been using for a few years now. Most of the time I'm been shooting in one of the "semi-manual" modes like aperture priority or shutter priority, but now I'd like to try using full manual mode. I should have a new EF-S 17-55 f/2.8 delivered when I get home today. Before I was using a variable aperture Sigma 17-70 which made manual mode more difficult with the aperture changing arbitrarily.  I understand the effect of changing aperture and shutter speed, but right now I'm accustomed  to only controlling one at a time. The camera is usually setting the other to get the right exposure automatically for me. So if I'm controlling both how do I know I have the correct exposure? I understand I need to use the metering in my camera but I'm not sure exactly how do do that.   First when should I be using each of the metering modes (Evaluative, Partial, Center weighted, or Spot)? These just change the area of the scene which is metered correct?  How do I know what the meter should read off a specific object? Obviously if I just happen to be shooting a 18% gray wall then it should be at zero, but that can't apply to everything? If I'm shooting a black cat and take a spot meter reading off of the cat then the meter should read less then zero right? But how do I know how much less then zero? Are there rules of thumb I can follow? I've ordered a couple gray cards with my new lens. So experimenting with those should help.  How often should I be changing these settings? Say I'm at a party indoors which is consistently lit. I should be able to set the aperture, shutter speed, and ISO and just forget about it unless the lighting changes correct? Does the same apply to outdoor scenes? I've been hesitant to use manual mode in the past because I felt like I was spending a lot of times messing with settings before each shot. But not I'm realizing that this is really only the case when the lighting changes, or I want a different style of photo such as a different depth of field.  Finally, does exposure change with distance to the subject? For example say I want to take a picture of a statue with correct exposure using a gray card. I'd place the card on the statue, and while being careful not the block the light on the card find the correct exposure by spot metering the card. If I were then to step back 20' to get the whole statue in frame would I still have the correct exposure? Show that there are exactly 16 pairs of integers $(x,y)$ such that $11x+8y+17=xy$. Get latest Flash player on Ubuntu "named" vs "that is named" Sci-fi book about a male entering a competition that he wins and gets recruited by millitary Articles: When do I use "a", "the", or "__"? How do I know whether to use the definite (the) or indefinite (a, an) article, and when to omit it altogether? Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$ Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,   $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$   when $X$ has : a) a discrete distribution, b) a continuous distribution.   I assumed that for the case of a continuous distribution, since $F_X (t) = \mathbb{P}(X\leq t)$, then $1-F_X (t) = 1- \mathbb{P}(X\leq t) = \mathbb{P}(X&gt;  t)$. Although how useful integrating that is, I really have no idea. It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ($n^{\underline{m}}$, denoting the falling factorial). But I am thinking about how to calculate the total number of surjective functions $f\colon X \twoheadrightarrow Y $.  The way I thought of doing this is as follows: firstly, since all $n$ elements of the codomain $Y$ need to be mapped to, you choose any $n$ elements from the $m$ elements of the set $X$ to be mapped one-to-one with the $n$ elements of $Y$. This results in $n!$ possible pairings. But the number of ways of choosing $n$ elements from $m$ elements is $\frac{m!}{(m-n)!\,n!}$, so the total number of ways of matching $n$ elements in $X$ to be one-to-one with the $n$ elements of $Y$ is $\frac{m!}{(m-n)!\,n!} \times n! = \frac{m!}{(m-n)!}$.  Now we have 'covered' the codomain $Y$ with $n$ elements from $X$, the remaining unpaired $m-n$ elements from $X$ can be mapped to any of the elements of $Y$, so there are $n^{m-n}$ ways of doing this. Therefore I think that the total number of surjective functions should be $\frac{m!}{(m-n)!} \, n^{m-n}$.  Is this anything like correct or have I made a major mistake here? Regular expression to match a line that doesn't contain a word I know it's possible to match a word and then reverse the matches using other tools (e.g. grep -v). However, is it possible to match lines that do not contain a specific word, e.g. hede, using a regular expression?   Input:  hoho hihi haha hede   Code:  grep "&lt;Regex for 'doesn't contain hede'&gt;" input   Desired output:  hoho hihi haha We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not.  My question is why.  As background there are plenty of mathematical schemes that can be used to define these operators. One of these is   where $H_0(a,b) = b+1$   (successor op) $H_1(a,b) = a+b$   (addition op) $H_2(a,b) = ab $   (multiplication op) $H_3(a,b) = a^b$   (exponentiation op) $H_4(a,b) = a\uparrow \uparrow b$   (tetration op: $a^{(a^{(...a)})}$ nested $b$ times ) etc.    Here it is not obvious to me why $H_1(a,b)=H_1(b,a)$ and $H_2(a,b)=H_2(b,a)$ but not $H_3(a,b)=H_3(b,a)$  Can anyone explain why this symmetry breaks, in a reasonably intuitive fashion?  Thanks. Question list on profile page not vertically aligned There's something wrong with the vertical alignment on the :   (click to enlarge)  On the , which uses a similar list but without the Favorite column, the vertical alignment is correct.  (Tested in the latest Firefox, Safari and Chrome on macOS Catalina.) Linux kernel update stuck I just installed a new kernel on Ubuntu GNOME 17.04 that came in through the normal updates. However, it is now stuck for a while on the following line:  Setting up linux-headers-4.10.0-20-generic (4.10.0-20.22) ... Examining /etc/kernel/header_postinst.d. run-parts: executing /etc/kernel/header_postinst.d/dkms 4.10.0-20-generic /boot/vmlinuz-4.10.0-20-generic   Something seems to hang the process. Does someone know what could cause this?  EDIT: This is the content of /boot as requested:  abi-4.10.0-19-generic     grub                          System.map-4.10.0-19-generic abi-4.10.0-20-generic     initrd.img-4.10.0-19-generic  System.map-4.10.0-20-generic config-4.10.0-19-generic  memtest86+.bin                vmlinuz-4.10.0-19-generic config-4.10.0-20-generic  memtest86+.elf                vmlinuz-4.10.0-19-generic.efi.signed efi                       memtest86+_multiboot.bin      vmlinuz-4.10.0-20-generic   EDIT 2: Just tried purging the new kernel but got this:  E: dpkg was interrupted, you must manually run 'sudo dpkg --configure -a' to correct the problem.   Using that command returns me to the original problem. I am stuck now :(