An equivalent condition for Normal spaces Let $X$ be a normal space. By definition it is a $T_1$ space and any pair of disjoint closed sets can be separated by open sets in the sense that they have disjoint neighborhoods. What I want to prove is  $X$ is normal iff each nbd of a closed set contains the closure of some nbd of the set  To prove the first implication I took a closed set $A$ and nbd $U$ of $A$. Since $U'$ is closed and disjoint from $A$ there exist nbd $H_A$ of $A$ and nbd $H_{U'}$ of $U'$ such that they are disjoint. I feel that closure of $H_A$ is subset of $U$ but not able to prove it. For converse part, let $A$ and $B$ be closed disjoint sets in $X$. Then for any nbd $U_A$ and $U_B$ of $A$ and $B$ respectively there exist nbd $V_A$ and $V_B$ of $A$ and $B$ such that $$ A \subseteq V_A \subseteq \bar{V_A} \subseteq U_A $$ and $$ B \subseteq V_B \subseteq \bar{V_B} \subseteq U_B $$ If $U_A$ and $U_A$ are disjoint then we are done. If not how can I proceed further? Please suggest any alternative approach also if possible.
Let X be a T1 space, and show that X is normal if and only if each neighbourhood of a closed set F contains the closure of some neighbourhood of F Let $X$ be a $T_1$-space, and show that $X$ is normal if and only if each neighbourhood of a closed set $F$ contains the closure of some neighbourhood of $F$.  I saw this statement in the book Introduction to topology and modern analysis by Simmons. I tried to prove it but I did not get any where. How may it be proved?
How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$ How would you evaluate the limit inferior of the sequence $n\,|\mathopen{}\sin n|$? That is, $$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$$    Edit. Let $\mu$ be the irrationality measure of $\pi$. Since $\mu$ is not known, I will split the question:   Assuming $\mu &gt; 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$? Assuming $\mu = 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$?     I kind of like its graph... Question Tag problem! Question:     A man like you should not do this, _______?   Which question tag should I use in the blank space?   should he? should you?   Which would be grammatically correct? I think 1 is true. (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . New 16GB of RAM installed yet I see 15.3 on my system. Why? Just switched 4GBs to 16GBs of RAM. However, when I look at how much memory I have, it says 15.3. I'm just wondering why did my memory drop down when I just installed 16 GBs of RAM. I find that the minimum in-call volume on my HTC Desire is still far to loud on some calls.  Since I got the phone, I have flashed several different custom ROMs, radio updates, etc, and have had the issue throughout.  Do you know of any way (setting, hack, app, whatever) to reduce the minimum volume?  My device is rooted and I am currently running one of the CyanogenMod 7 (Gingerbread) nightly builds. What exactly is TLP doing that default system config wouldn't? All the info about TLP on the surface looks more like advertisement and marketing to me, I can't see any detailed comparison to default configurations. What are the precise differences compared to default (K/L)Ubuntu installs under the hood, are they really effective?  Update: And if they really are, why aren't they a part of a default config? Using PTSerif-TLF for Cyrillic with TeX Gyre Pagella Best practice for Rails App to run a long task in the background? I have a Rails application that unfortunately after a request to a controller, has to do some crunching that takes awhile.  What are the best practices in Rails for providing feedback or progress on a long running task or request?  These controller methods usually last 60+ seconds.    I'm not concerned with the client side... 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It must be stable and maintained.  I'm very surprised Rails doesn't have something like this built in.  Edit 3:  I have read this page: .  I would love to read about various experiences with these tools.  Edit 4:  I'm using Passenger Phusion "modrails", if it matters.  Edit 5:  I'm using Windows Vista 64 bit for my development machine; however, my production machine is Ubuntu 8.04 LTS.  Should I consider switching to Linux for my development machine?  Will the solutions presented work on both? Question about the Riemann-Lebesgue Lemma proof Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$  I get all the steps of the proof, except the one which says that if $f$ is an arbitrary integrable function, it may be approximated in the $L^1$ norm by a compactly supported smooth function $g\in L^1$. My question is: Why? How do you know that for all $\epsilon&gt;0$, and for all arbitrary function $f\in L^1$ exists a compactly supported smooth function $g$ so that $||f-g||_{L^1}&lt;\epsilon$? The "proof" can be found in many places, but this step can be found in Wikipedia I have a blender file where a sphere is orbiting a force field. I want it to be more red when it's closer to the center, and more blue when it's further. I already have the math figured out, I just need to know how to get x/y coordinates in nodes.  Here's the file: Open a window on a remote X display (why "Cannot open display")? Once upon a time,  DISPLAY=:0.0 totem /path/to/movie.avi   after ssh 'ing into my desktop from my laptop would cause totem to play movie.avi on my desktop.  Now it gives the error:   No protocol specified Cannot open display:    I reinstalled Debian squeeze when it went stable on both computers, and I guess I broke the config.   I've googled on this, and cannot for the life of me figure out what I'm supposed to be doing.   (VLC has an HTTP interface that works, but it isn't as convenient as ssh.)  The same problem arises when I try to run this from a cron job. Comaximal ideals in a commutative ring Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal? I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space? In how many inequivalent ways can 8 people be seated at a round table? How many different arrangements are there of eight people seated at a round table, where two arrangements are considered the same if one can be obtained from the other by a rotation? Why do electrons not fully fill up all the orbitals sequentially? Understanding complete separation for logistic regression Why does logistic regression not converge for a linearly separable data set?   For linear separable data sets the model parameters go to infinity when mimizing the error function (according to Bishop2006, Pattern recognition and machine learning, section 4.3.2). I don't understand why. It must be connected to setting the score function, i.e. the derivative of the error function, to zero.  The error function is given by $E(\mathbf{w})$ as shown in the figure below. Where $\mathbf{t}$ is the target variable vector of the training set and $\mathbf{w}$ the parameter / coefficient vector. The posterior probabiltiy for class 1 is given by  $$y(n) = p(C_1|\mathbf{x}_n) = \sigma(\mathbf{w}'\mathbf{x}_n).$$ Where sigma is the logistic sigmoid function and $\mathbf{x}_n$ is the feature vector for instance $n$ including the predictors.     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While players wait, it's not uncommon for them to fire off random shots, especially since a lot of the environment is destructable (it's entertaining to watch plush onions sail through the air in the arcade).   Do shots fired before the match starts count towards accuracy statistics at the end of the game? Selecting features with NULL attributes in QGIS How do I identify which features have NULL attributes in the attribute table of a shapefile?  When i try [field]= 'NULL' it selects the features with value = 0 (see picture)