I am doing M.Sc. in Mathematics from a reputed college in India. But I have a humble background in mathematics. At my undergraduate college most of my batch mates were not interested in mathematics, and the teacher taught in a way that we could sail through exams. But I tried to learn on my own from online courses like NPTEL and OCW lectures and somehow I got a position for doing master's. At that time I used to study for just an hour and manage to understand, and I didn't solve any problems. But here, the teachers are extremely good and they take 3-4 hours of lectures (i.e., one hour for each course). But now I am not able to grasp the concepts (develop intuition, visualization, looking at the big picture, etc.). To understand what my teacher/course instructor taught takes me 2-3 days of work and I am left behind. If I include memorization time then almost a week or so. I am becoming overweight and suffering from lower back problems. If stay late up at night then I don't get sunlight giving vitamin D deficiency. I don't know but I know I want to do research. But how should I deal with this, I have 6 (5courses+seminar under professor). And here we have to write an exam and be in the top 100 percent to get admitted for PhD from a reputed college with proper funding. Please guide me, help me. How can I maintain my health and study?
I'm doing a masters course at a different university than I did my undergrad (both in maths). I did well in my undergrad but now I'm finding it very difficult. The shock of the change is hard to cope with.  I'm not just finding it very difficult, but I get zero feedback: we don't have any tests and I was not able to make friends so I don't know how others are finding it. I'm starting to think that I might be an idiot.   How can I build some confidence in my abilities at the same time as I rush and struggle to keep up with the massive rate of new material I need to learn? How might I get myself some reassurance that I can actually pass, or stop worrying about this so much? (I feel terribly guilty using so much money from my family to be here) It's possible that I don't really have a chance at all but knowing that would be fine.
Entropy and the principle of least action Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ? Drawing text on an image background that has dark and bright parts Often, I want to write text over a background image. If the background image is bright, I use a dark color for the text, and vice versa. But, often the image has both dark and bright parts. So a dark text is unreadable on the dark parts, and a bright text is unreadable on the bright parts.  I currently use one of several solutions, all of them are imperfect:   Brighten the image and use a dark font, or darken the image and use a white font; the problem is that it reduces the beauty of the background image. Use grey text (as in this question:  ). But grey text often looks unappealing, and is still only barely readable. Use dark text, and try to position it on the bright spots of the image (or vice versa). This takes a lot of time because it requires me to position each word separately.   Are there better solutions to this problem?   In particular: is there an automatic tool that takes an image and a text, and positions the text only on the dark/bright parts of the image? I have very old Computer  What's the Best Operating System for Old Computer? show that $$\lim_{n\to \infty}\sin{(n^m)}$$  divergent,where $m\in N^{+}$  I have kown $$\lim_{n\to\infty}\sin{n^2}$$  to be divergent and dense in $[-1,1]$. This is very  famous problem,the problem is first is post $AMM$( The American Mathematical Monthly,1970-1975) problem,and after some years  $AMM$ post $\sin{(n^{14})}$ in $[-1,1]$ was dense? and this is open problem.so I Guess  $$\sin{(n^m)}$$ is dense in $[-1,1]$. Proving that two vector spaces of equal dimension are isomorphic Given the following problem     Let $U$ and $V$ be finite dimensional vector spaces, such that $\dim(U) = \dim(V)$. Prove that $U$ and $V$ are isomorphic.   I was just wondering if somebody could critique and validate my proof.  Proof  Two vector spaces $U$ and $V$ are isomorphic if and only if there exists a bijection $\phi: U \to V$. That is, an invertible linear map between the two. Let $\dim(U) = \dim(V) = n$ and let $\left\{u_1, \dots, u_n\right\}$ and $\left\{v_1, \dots, v_n\right\}$ be bases for $U$ and $V$, respectively. It follows that any vector $u \in U$ can be written as  $$ u = c_1u_1 + \cdots + c_nu_n $$  We define $\phi: U \to V$ as  $$ \phi(c_1u_1 + \cdots + c_nu_n) = c_1v_1 + \cdots + c_nv_n $$  Observe that $\phi$ is invertible. That is, given a vector  $$ v = d_1v_1 + \cdots + d_nv_n $$  in $V$, we define $\phi^{-1}: V \to U$ as  $$ \phi^{-1}(d_1v_1 + \cdots + d_nv_n) = d_1u_1 + \cdots + d_nu_n $$  Since there exists a bijection between $U$ and $V$, they are isomorphic. Setting the hostname: FQDN or short name? I've noticed that the "preferred" method of setting the system hostname is fundamentally different between Red Hat/CentOS and Debian/Ubuntu systems.   and the  say the hostname should be the FQDN:     HOSTNAME=&lt;value&gt;, where &lt;value&gt; should be the Fully Qualified Domain   Name (FQDN), such as hostname.example.com, but can be whatever   hostname is necessary.   The  is slightly more ambiguous:     Setup prompts you to supply a host name for this computer, either as a   fully-qualified domain name (FQDN) in the format hostname.domainname   or as a short host name in the format hostname.    says the hostname should not use the FQDN:     3.5.5. The hostname      The kernel maintains the system hostname. The init script in runlevel   S which is symlinked to "/etc/init.d/hostname.sh" sets the system   hostname at boot time (using the hostname command) to the name stored   in "/etc/hostname". This file should contain only the system hostname,   not a fully qualified domain name.   I haven't seen any specific recommendations from IBM about which to use, but  seems to have a preference.  My questions:   In a heterogeneous environment, is it better to use the vendor recommendation, or choose one and be consistent across all hosts? What software have you encountered which is sensitive to whether the hostname is set to the FQDN or short name? Will a bullet dropped and a bullet fired from a gun horizontally REALLY hit the ground at the same time when air drag is taken into account? Young adult book about a boy(?) who gets sent through a portal to another world which is really the distant future Trying to remember a book about (I think) a boy who goes through a portal (possibly struck by lightning?).   Ends up in a strange, primitive bush/forest land. Alien but somewhat Earth-like plantation everywhere - I believe the grass was purple. There are very primitive humans (or humanoids?) and the main character was possibly made a slave by a tribe? Thinks is on alien planet. It is revealed they are still on earth when someone has a glass bottle which reads "Coca-cola" on it. There was something to do with quicksand which seemed important to the plot.   Probably read sometime around the late 90s, but could have been written earlier.  Was either a short novel or possibly even a short story.  Would be curious if anyone can remember this. Can I use Facebook's fb:friend-selector in an iframe? How to construct semi direct products between Q and H There is a semi direct product if you go $\theta: Q \rightarrow Aut(H)$.  $H = C_{17}, Q = C_2$.  $Aut(H)\cong C_{16}$.  From here, how do I construct the semi direct products?   I said, in $C_{16}$, there are two elements of order that divide $2$, which are $1$ and $2$. Therefore there are two semi direct products. Also $17 \times 2 = 34$ tells me that in both semi direct products there will be $34$ elements.  How do I construct the semi direct products from here though? sed substitution with Bash variables I was trying to make python version 3 as a default version of Ubuntu,  I used nano ~/.bashrc and added the first line of this answer.    Then It just messed up my terminal.  Now when I am clicking on my Terminal or trying to access by ctrl+alt+t, the terminal does open/run. I can only talk to ubuntu using my terminal how could I fix this?  Please advise!  Pls, note that the terminal icon is present, but the terminal does not open when clicked on. How do I get the terminal to open and work?  Update: I deleted that line from .bashrc. However, I still have a problem opening my terminal. I also see this sign on my ubuntu as well.     saying: a problem occurred when checking for the updates ubuntu 18.04 I know that I can make certain blocks only placeable on a specified block, but what is the command to give an item that can only BREAK certain blocks? I have some auto-generated T-SQL, which is probably valid, but I don't really understand.  ALTER TABLE [dbo].[MyTable] WITH CHECK CHECK CONSTRAINT [My_FORIEGN_KEY];   I know what a foreign key constraint is, but what's the CHECK CHECK? Configure Postfix to use external MX servers for delivery of local mail if user is unknown How to prove that the set $A = \left\{ {p:{p^2} < 2,p \in {\Bbb Q^+}} \right\}$ has no greatest element? More specifically, I have to find a $q$ in $A$ such that $q$ is larger than any $p$ in $A$.  The only thing I can think of is using (${2 - {p^2}}$) somehow, or ${(p+x)^2}&lt;2$, but other than that I don't know how to proceed. This exercise is taken from the very first chapter of Walter Rudin's Principles of Mathematical Analysis. Quoting the exercise:     To do this, we associate with each rational $p &gt; 0$ the number:     $$q = p - \frac{{{p^2} - 2}}{{p + 2}} = \frac{{2p + 2}}{{p + 2}}$$   But he didn't explain how he came up with that number.   The only thing I've thought of is the following procedure:  $$\eqalign{   &amp; {p^2} + (2 - {p^2}) = 2 \Rightarrow {p^2} + (\frac{{2 - {p^2}}}{2}) &lt; 2,{\text{ and since }}2 - {p^2}{\text{ is always positive,}}  \cr    &amp; {p^2} &lt; \left[ {{p^2} + (\frac{{2 - {p^2}}}{2})} \right] &lt; 2  \cr    &amp; {\text{Therefore, let }}q = \sqrt {{p^2} + (\frac{{2 - {p^2}}}{2})} ,{\text{ then:}}  \cr    &amp; p &lt; q &lt; \sqrt 2  \cr} $$  However, my result doesn't necessarily belong to ${\Bbb Q}$. I wonder how did he obtain the value of $q$ in the book.  EDIT: Feel free to point out if there are more appropriate tags for this question. Why is super.super.method(); not allowed in Java? I read  and thought that would easily be solved (not that it isn't solvable without) if one could write:   @Override public String toString() {     return super.super.toString(); }   I'm not sure if it is useful in many cases, but I wonder why it isn't and if something like this exists in other languages.  What do you guys think?  EDIT: To clarify: yes I know, that's impossible in Java and I don't really miss it. This is nothing I expected to work and was surprised getting a compiler error. I just had the idea and like to discuss it. Evolution of the Redundancy of the Genetic Code How to tell a lambda function to capture a copy instead of a reference in C#? I've been learning C#, and I'm trying to understand lambdas.  In this sample below, it prints out 10 ten times.  class Program {     delegate void Action();     static void Main(string[] args)     {         List&lt;Action&gt; actions = new List&lt;Action&gt;();          for (int i = 0; i &lt; 10; ++i )             actions.Add(()=&gt;Console.WriteLine(i));          foreach (Action a in actions)             a();     } }   Obviously, the generated class behind the lambda is storing a reference or pointer to the int i variable, and is assigning a new value to the same reference every time the loop iterates.  Is there a way to force the lamda to grab a copy instead, like the C++0x syntax  [&amp;](){ ... } // Capture by reference   vs.  [=](){ ... } // Capture copies Expressing hourly temporal recurrence What is the most idiomatic way to put into words a situation where a recurring event happens every hour at a certain minute. For example, consider that some event takes place at the tenth minute of each hour: 0:10AM, 1:10AM, 2:10AM, ...  My first ideas were "every hour at the tenth minute" or "every hour at minute 10". Are these options idiomatic and if not, what would be an idiomatic way of expressing this matter?