Why is the main stackexchange.com site still using the old style top-bar? It seems odd for that particular page not to match everything else at this point.  BAD:   GOOD:
,  and 1 still show the old bar. 1 See
Why do elementary matrix operations not affect the row space of a given matrix? Predicting the number of decimal digits needed to express a rational number The number $1/6$ can be expressed with only two digits (and a repeat sign denoted as $^\overline{}$), $$ \frac{1}{6} = \,.1\overline{6}$$  Meanwhile, it takes 49 digits to express the number $1/221$, since a string of 49 digits repeats: $$\frac{1}{221} = .\overline{004524886877828054298642533936651583710407239819}$$ Yet for $1/223$, 222 digits repeat, giving a total of 224 digits needed to express the number.  If $f:\mathbb{Q}\rightarrow\mathbb{N}$ is a function that gives the smallest number of digits needed to express a rational number in decimal notation, what can we say about $f$?  For example, if we do not consider the negative sign to be a digit, then $f$ is an odd function.  Other than that, is there any pattern to it at all? How to change this matrix with respect to standard basis? Given the basis $\beta = \{ (1, 1, 0),\ (1, 0, -1),\ (2, 1, 0)\}$ and the matrix:  $$ A =         \begin{bmatrix}         1 &amp; 1 &amp; 0 \\         1 &amp; 1 &amp; 0 \\         0 &amp; 0 &amp; 1 \\         \end{bmatrix} $$  with respect to  $\beta$.  How to change this matrix with respect to standard basis? Show $f(x)=x\ln x$ is not uniformly continuous Show $f:(0,\infty)\rightarrow \mathbb{R}, f(x)=x\ln x$ on $(0,\infty)$ is not uniformly continuous.    I think that the general way to prove that something is not continuous in a metric space is to let $\epsilon=...$ and show that $\forall\delta&gt;0$, $d&#39;(f(x)-f(y))&gt;\epsilon$. I can't use the Mean Value Theorem because we haven't gone over it yet.    Here's an attempt: Let $\epsilon=1$. Without loss of generality, let $x&gt;y&gt;0$. $|x\ln x-y\ln y|&lt;|(x-y)(\ln x)|\leq|x-y||\ln x|$...$\delta=\frac{\epsilon}{\ln x}$ and therefore doesn't work for all $x$? Hypernym for "audiobook", "podcast", "speech", etc.? I am looking for a short and concise term (preferably one word) that can be used as a hypernym for essentially everything that contains (audio) speech (e.g. audiobooks, podcasts, recorded news bulletins, recorded performances of comedians,..).  Does such a term exist? I've thought about 'speech', but that seems to be more adequate for e.g. political speeches, and not so much for audio books and the like. But maybe I'm wrong. Adjusting font size with TikZ picture Taking from the PGF manual,  \begin{tikzpicture}   \draw [help lines] (0,0) grid (3,3);   \coordinate (a) at (rnd,rnd);   \coordinate (b) at (3-rnd,3-rnd);   \draw (a) -- (b);   \node (c) at (1,2) {x};   \draw let \p1 = ($ (a)!(c)!(b) - (c) $),             \n1 = {veclen(\x1,\y1)}         in circle [at=(c), radius=\n1]; \end{tikzpicture}   I'd like to increase the size of x. The problem is that I have multiple circles in the same diagram. Some of them require a larger sized font, while others are okay at default. How do I do this? How can I parse integers passed to an application as command line arguments if the app is unicode?  Unicode apps have a main like this:  int _tmain(int argc, _TCHAR* argv[])   argv[?] is a wchar_t*. That means i can't use atoi. How can I convert it to an integer? Is stringstream the best option? Alternating sum of binomial coefficients multiplied by (1/k+1) I'm trying to prove that  $$\sum_{k=0}^n {n \choose k} (-1)^k \frac{1}{k+1} = \frac{1}{n+1}$$  So far I've tried induction (which doesn't really work at all), using well known facts such as  $$\sum_{k=0}^n {n \choose k} (-1)^k = 0$$  and trying to apply identities like  $${n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}$$  Would anyone be able to point me towards the right method? Should I be looking to apply an identity or is there a method I'm missing? Unsubscribe anonymous method in C# Is it possible to unsubscribe an anonymous method from an event?  If I subscribe to an event like this:  void MyMethod() {     Console.WriteLine("I did it!"); }  MyEvent += MyMethod;   I can un-subscribe like this:  MyEvent -= MyMethod;   But if I subscribe using an anonymous method:  MyEvent += delegate(){Console.WriteLine("I did it!");};   is it possible to unsubscribe this anonymous method?  If so, how? Verify is tautology by using logical equivalence Verify is tautology by using logical equivalence:  $$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q$$  by showing what laws are using, step by step explanation needed.  Logical equivalence: de morgan's laws, commutativity, associativity, double negative, distribution, idempotent, absorption, identity, universal bound, negation  Rules of inference: modus ponens, modus tollens, generalisation, specialisation, conjunction, ellimination, transivity, proof by division into cases, contradiction  What I got so far:  I need to prove   $$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q ≡ \text{tautology}$$ Since $p → r$ same as $∼ p ∨ r$: $$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q ≡ ((p ∨ q) ∧ (∼ p ∨ r ) ∧ (∼ r)) → q$$  I am stuck here Holomorphic self map on the open unit disc How to make existing public files private? I want to prove the  over a field are prime ideals. To be concrete:      For simplicity, let $I=(x_{11}x_{22}-x_{12}x_{21},x_{11}x_{23}-x_{13}x_{21},x_{12}x_{23}-x_{13}x_{22})$ be an ideal of the polynomial ring $k[x_{11},\ldots,x_{23}]$. I have no idea how to prove that $I$ is a radical ideal (i.e. $I=\sqrt{I}$). Could anyone give some hints?   Generally, let $K$ be an algebraically closed field, then $\{A\mid\mathrm{Rank}(A)\leq r\}\subseteq K^{m\times n}$ is an irreducible algebraic set (I first saw this result from ). And I tried to prove this by myself, then I have proved it (when I see the "Segre embedding").      But I have no idea how to show that the "determinantal ideals" are radical ideals (I hope this is true). BTW, is the statement that the determinantal ideals over a field are prime ideals true ?   Thanks. Seeking QGIS tutorials and web resources? I wish to familiarize myself with the capabilities of QGIS up to and beyond QGIS version 3.0.   Can you recommend some good web resources and tutorials that would show me QGIS' capabilities?   I am interested in both using QGIS through its GUI and in developing applications using QGIS. How can I check if share certificates I found have value? I have been clearing my parents flat and found share certificates from    P and O  European Ferries   Can I find out if they are still valid? If not, is there a market for them as collectibles or the like? How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$? For my proof I distinguished the two possible cases which derive from $7 \mid a^2+b^2$:   Case one: $7\mid a^2$ and $7 \mid b^2$ Case two (which (I think) is not possible): $7$ does not divide $a^2$ and $7$ does not divide $b^2$, but their sum.  I've shown that case $1$ implies $7\mid a$ and $7\mid b$, so I just have show that case 2 isn't possible - I'll be happy for any input.  MfG, Karl Are submodules of finitely generated free modules always free? I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free? Long story short: since custom flag responses are messages to users like any other (answer, comment, suggested edit, chat message etc.), users should be notified about them!!    A problem I have with flagging, as it currently functions, is that flaggers are not notified about custom responses to flags, when they are given. I really don't care if I don't find out about every last flag of mine, whether it was marked "helpful" or not, but if a moderator went to the effort of typing out a custom response to my flag, explaining why it was or was not helpful, I'd like to see that response. The logical place for this response would be in the messages inbox (the red number thingy in the top left corner), where other responses to user actions (like answers, comments, edits, and chat messages) go.  This may also help to alleviate  of users not realizing that their flags go somewhere -- while I agree with her that "We shouldn't send notifications for [all] flag responses; that would be way too noisy for a lot of people," I think that in the event of a custom response, the flagger should be notified (because some mod thought that this information was relevant to the flagger), and it would not be too noisy because (in my experience1) most flags do not receive a custom response. This could provide one way that flaggers can be made aware of their flagging history.2    1 I don't know who can see this, but  on Mi Yodeya, to date, is 55 flags total (49 helpful), and only one custom response (....that I didn't see for a while, till I checked my flagging history for a different reason).  2 While I don't think that this will replace Monica's suggestion entirely, it would certainly help it a little.    To give credit where credit is due, while I came up with this idea myself, I did notice that someone else had posted about a similar idea earlier in a .  Edit: Just spotted  from the related questions along the side of the page; note that while that proposal is similar to this one, it was about sending all declined flags to the inbox, while I suggested that only the ones with custom responses should make it to the inbox. Assigning Multiple Cores to a Python Program I'm trying to put together a comprehensive regex to validate phone numbers. Ideally it would handle international formats, but it must handle US formats, including the following:   1-234-567-8901 1-234-567-8901 x1234 1-234-567-8901 ext1234 1 (234) 567-8901 1.234.567.8901 1/234/567/8901 12345678901   I'll answer with my current attempt, but I'm hoping somebody has something better and/or more elegant.