I have just started reading real analysis. The topics include sequences, series, continuity, differentiability, integration, series of functions , calculus in higher dimensions etc.  I am in need of a text and a reference book.  Can someone please tell me about it?  The text should not be very broad as I need something to finish before my exam.But the reference should be broad, with full of exercise and examples. Hint to exercise will help me a lot.  Waiting for your response.
Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.   While it's a good book, I'm not sure it's suited for self study by itself.  I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway.
I'm setting up a new server and want to support UTF-8 fully in my web application. I have tried this in the past on existing servers and always seem to end up having to fall back to ISO-8859-1.  Where exactly do I need to set the encoding/charsets? I'm aware that I need to configure Apache, MySQL, and PHP to do this — is there some standard checklist I can follow, or perhaps troubleshoot where the mismatches occur?  This is for a new Linux server, running MySQL 5, PHP, 5 and Apache 2. What is relationship of widower to late wife's family (i.e. in-laws)? I am a widower.  I still have a strong personal relationship with my late wife's family, and I regard them as part of my own family.  They are blood of my blood in the sense that my children are related by blood to both me and them,  but if we had not had children, that link would not exist.    My only direct relation to them was by marriage, and with my wife's death, it would seem that that bond no longer exists; at least not formally.     Is there any enduring relationship by law or tradition for ex relations by marriage, and what would it be called?   Also, what if instead of death, we had been separated by divorce?    And what if I were to remarry,  would that change my relationship to my first in-laws? How can I disable Ctrl+Q for all applications (system wide)? A lot of applications like Firefox, Chrome etc, get closed when I accidentally press Ctrl+Q instead of Ctrl+W because of the proximity of the Q and W keys on the keyboard. Is there a way this shortcut can be removed or reset on a system wide basis? How do I count attendance in Excel? Prove that you can make change for any amount greater than 7 cents using only 3 and 5 cent coins. I am pretty sure this question has something to do with the Least Common Multiple.    I was thinking that the proof was that every number either is or isn't a multiple of $3, 5$, and $8\left(3 + 5\right)$. If it isn't a multiple of $3,5$, or $8$, great. You have nothing to prove. But if it is divisible by one of them, I couldn't find a general proof that showed that it wouldn't be divisible by another one. Say $15$, it is divisible by $3$ and $5$, but not $8$. Prospective PhD contacts potential supervisor but receives no answer after 2-3 emails, what to do? Say a prospective PhD has a high reply rate (90-95%) to first contact emails with potential superviors. He/She has a well thought out e-mail, but a particular professor (here: USA) that is in the top preferences does not reply after two to three e-mails that were sent within a time frame of three months. I also followed the instructions on the professor personal website in the section "prospective PhD students". The student has already contacted some of the advisor students and has received positive feedback about the professor.  Should the student just give up and move on? Should the student keep trying?  Aside from the obvious "I have too many incoming e-mails to answer" or "I am out in the field", what could create a no response behavior from the professor?  I have seen the simplistic answer "move on, bad advisor, doesn't have time, etc". But this seems a contradiction in this case. The PhD students like the professor and say he/she is personable and no e-mail response after a few well written emails. divisibility of binomial coefficients by primes If $p$ is a prime number and $r = 1,2,3 \dots ,p -1$, how can I prove that the binomial coefficients $_nC_r(p,r)$ are divisible by $p$? By divisible I mean $_nC_r(p,r)/p$ leaves remainder $0$.  oivind How to change the date format? Presently the date is showing as mmm dd hh:mm:ss. Is there any way that I can change it to as per my liking. For example : dd mmm, hh:mm:ss. I am new to the telecom simulations system specially multisim, which I am having a project that I want to simulate.  My project in (Analog Communications) is consisting to have FM Transmitter (Modulator) that accepts a generated signal (Sin, Cos) and get me the modulated output as a Graph.  I've found this FM transmitter circuit (3 km range):   ![two muppets][1]  and I am building that circuit on Multisim, and I failed. Because I faced many problems, first of, the variable capacitor symbol is weird here in the multisim, and the regular capacitor doesn't have variable options. Secondly, when I Use signal generator as (Sin, Cos), the output gets weird after 5 second of the simulation.  I want to show the input signal (before modulation) and the output signal (transmitted or modulated).  I guess my output result where not the as expected, as the signal (Sin with phase 90&deg;) is giving weird result of few seconds of simulating.  Below are screenshots of the simulation results within time.          Any ideas how to edit or fix the above circuit to have this output is similar to what it should be? As stated above, I can't connect to my home wi-fi with natty but can with Windows 7. The strange thing is I that I have gotten it to connect to other password protected networks with natty, it's just mine that it dosn't like.   I am using a WPA2 pass-code with AES Encryption and all other devises work fine on my network. Laptop used is a Dell Latitude D430. How do I check if an integer is even or odd? Here’s something that sucks:   Plug in external monitor to laptop (Mac OS X Leopard in my case). Arrange your windows to have the IDE on one screen and browser on another. (Etc etc.  Resize, rearrange, fuss, fuss, fuss.) Unplug monitor to run to a meeting, or whatever. Goto 1.   I’d like to replace step 2 with “computer remembers exactly how I had it and just puts it back that way”.  (I personally only need a solution for Mac but collecting solutions for other systems here might be useful to others.  I’m sure it’s a common problem.) Copying files from command line to clipboard Did she judge him "wrong" or "wrongly"? Which one is the correct use?  She judged him wrong. She judged him wrongly.  Or, are both correct, but have slightly different meanings? In my Master thesis, I used this:  \begin{itemize}[label=\ding{167}] \item Access Windows XM from Ubuntu Server \item List the directories contents of Windows XP \end{itemize}   The output is:  My question: must I avoid the use of  such beautiful things when writing my thesis or is it a normal thing to do ? Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$ There is a famous Theorem telling that:  For $n≥5$, $A_n$ is the only proper nontrivial normal subgroup of $S_n$.  For the proof, we firstly start with assuming a subgroup of $S_n$ which $1≠N⊲S_n$. We proceed until at the last part of proof's body, we assume $N∩A_n=\{1\}$. This assumption should be meet a contradiction with normality of $N$ in $S_n$. There; we get $N=\{1,\pi $} in which $\pi$ is an odd permutation of order $2$. Now for meeting desire inconsistency, I have two approaches:  (a) Since every normal subgroup, having two elements, lies in the center of $G$ so, our $N⊆ Z(S_n)=\{1\}$ for $n≥5$ and then $N=\{1\}$.  (b) Clearly, $1≠N$ acting on set $\Omega=\{1,2,...,n\}$ is intransitive wherein $|\Omega|≥5$ and according to the following Proposition $S_n$ would be imprimitive.  Proposition 7.1: If the transitive group $G$ contains an intransitive normal subgroup different from $1$, then $G$ is imprimitive (Finite Permutation Groups by H.Wielandt).  May I ask if the second approach is valid? I am fond of knowing new approach if exists. Thanks. Evaluate forecasting ability of GARCH models with RMSE and MAE I am evaluating different forecasting models and their ability to forecast index volatility during period of market turmoil, using two measurements, Root Mean Square Error and Mean Absolute Error. For the previous evaluated models, ARMA (1,1) as an example, I was able to obtain the residuals and calculate the RMSE quite easily in Stata.   When estimating the GARCH (1,1) model in Stata I am however not able to correctly obtain the residuals in the post estimation procedure, no option to directly obtain the RMSE is available. Perhaps I have misunderstood how one should evaluate the forecasting ability of GARCH models, since the models specifies the conditional variance unlike ARMA which specifies the conditional mean.   Does anyone have a suggestion on how to obtain these evaluation measurements after estimating a GARCH model? And preferably how to do it in Stata. What is the best symbol to use for the d'Alembertian? Normally, most people use the symbol $\Box$ to represent the  (including the linked to Wikipedia page).  Recently I wanted to use \hat\Box and \tilde\Box, but they do not render very well.    What is the "best" way to create such symbols?    Aside: The above image is rendered using my local install of pdflatex: pdfTeX 3.1415926-1.40.10-2.2 (TeX Live 2009/Debian). The symbols seem to render fine on some online services: e.g. "" produces    from   \begin{align*} \Box &amp;= \sum_a \partial^a\partial_a \,, &amp; \hat\Box &amp;= \sum_a \mathcal{D}^a\mathcal{D}_a \end{align*} How to get back Windows 10 [UEFI] after Legacy installation of Fedora 24? Restore SQL Server 2012 backup to a SQL Server 2008 database?