I have learnt some words can use present tense to express the future.  Like:  Even if it rains, I will go outside. Can I change this one to: Even if you have jobs to do tomorrow, you are still playing!!!  If you booked a ticket, you can enjoy our best service now. Can I change this one to: Even if you booked a ticket, you still can't enjoy the service for some reason. (Here the past tense is about guessing, not the unreality.）  And about Even if he had jobs to do tomorrow, he was still playing.Obviously, he was fired. (Just change the first example to the past, now someone is talking about the reason why he got fired.)  Thanks so much!
I would like to know if there are any other tenses or grammatical structures to use for speaking about the future apart from will (perfect tense included), going to, simple and continuous present.  I mean not the obvious ones. For example, simple present (I could never have imagined it could be used for the future).     My mom arrives tomorrow.   I go to the doctor at 4 o'clock this afternoon.
Proof that Limit of the Log is the Log of the Limit Proof that Limit of the Log is the Log of the Limit. What is the intuition behind this statement? Let $q \in \mathbb C$, $\|q\|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n: n \in \mathbb N\}$ is dense in $S^1$ Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n:  n \in \mathbb N\}$ is dense in $S^1$.  My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ convergent. So given $\epsilon &gt; 0, \exists j_0 \in \mathbb N, \quad |q^{n_j} - q^{n_k}| &lt; \epsilon, \quad \forall j,k \geq j_0$. Supose that $n_j &gt; n_k$, then  $|q^{n_j} - q^{n_k}| = |q^{n_k}||q^{n_j - n_k} - 1| = |q^{n_j - n_k} - 1| &lt; \epsilon, \quad \forall j, k \geq j_0$. So, there is a subsequence of $(q^n)$ that converges to $1$.  Using this fact, how can I ensure that for all $w \in S^1$ there is a subsequence of $(q^n)$ that converges to  $w$? steam fatal error "steam needs to be online to update, but was set to offline movies." I tried re-installing  it but it simply keeps giving me this error in a little window. When I close it, an identical error pops up some 3 seconds later without any new provocation. What does it mean for something to hold "up to isomorphism"? Computing the best constant in classical Hardy's inequality Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327)  If $p&gt;1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then   $$\tag{H} \int_0^\infty \left(\frac{F(x)}{x}\right)^p\, dx &lt; C\int_0^\infty (f(x))^p\, dx $$  unless $f \equiv 0$. The best possibile constant is $C=\left(\frac{p}{p-1}\right)^p$.   I would like to prove the statement in italic regarding the best constant. As already noted by Will Jagy , the book suggests stress-testing the inequality with   $$f(x)=\begin{cases} 0 &amp; 0\le x &lt;1 \\ x^{-\alpha} &amp; 1\le x \end{cases}$$   with $1/p&lt; \alpha &lt; 1$, then have $\alpha \to 1/p$. If I do so I get for $C$ the lower bound  $$\operatorname{lim sup}_{\alpha \to 1/p}\frac{\alpha p -1}{(1-\alpha)^p}\int_1^\infty (x^{-\alpha}-x^{-1})^p\, dx\le C$$  but now I find myself in trouble in computing that lim sup. Can someone lend me a hand, please?    UPDATE: A first attempt, based on an idea by Davide Giraudo, unfortunately failed. Davide pointed out that the claim would easily follow from   $$\tag{!!} \left\lvert \int_1^\infty (x^{-\alpha}-x^{-1})^p\, dx - \int_1^\infty x^{-\alpha p }\, dx\right\rvert \to 0\quad \text{as}\ \alpha \to 1/p. $$  But this is false in general: for example if $p=2$ we get   $$\int_1^\infty (x^{-2\alpha} -x^{-2\alpha} + 2x^{-\alpha-1}-x^{-2})\, dx \to \int_1^\infty(2x^{-3/2}-x^{-2})\, dx \ne 0.$$ I have installed a package using snap and I need to modify one of the files but when I try to change its ownership or permissions, I always get the following message:  sudo chmod +x ./my_file.js chmod: changing permissions of '/snap/my_app/my_file.js': Read-only file system   How can I modify files that are installed via snap? Using the quadratic formula, we have either 0, 1, or 2 solutions. I wonder if we could use it this formula for congruence? Or is there a formula to solve quadratic equation for congruence?  Edit Assume that $ax^2 + bx + c \equiv 0 \pmod{p}$, where $p$ is prime with $(a, p) = 1$, then is there a formula for this equation?    Thanks, How to type \sim above another symbol x? I want to put a "~" above x, just like \hat x but replacing \hat with \sim.   How to do that? Can you enter the Schengen area with a UK visa? Mcrypt extension is missing in 14.04 server for mysql The php mcrypt extension on ubuntu 14.04 shows that it is installed properly, but phpmyadmin says:   The mcrypt extension is missing. Please check your PHP configuration.   My current php config is . If you need anymore information, please contact me through the comments.  $ sudo apt-get install php5-mcrypt     Reading package lists... Done     Building dependency tree            Reading state information... Done     php5-mcrypt is already the newest version.     0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Is there a "win64" identifier in Qmake project files?  documentation does not mention other than unix / macx / win32.  So far I've tried using:  win32:message("using win32") win64:message("using win64") amd64:message("using amd64")   The result is always "using win32".  Must I use a separate project-file for x32 and x64 projects, so they would compile against correct libraries? Is there any other way to identify between 32-bit and 64-bit environments? I pressed F2 to take a few Minecraft screenshots. Later I couldn't find them. Where are they stored on my computer? 100 apples are/is considered as a large number of apples PHP function to build query string from array I'm looking for the name of the PHP function to build a query string from an array of key value pairs. Please note, I am looking for the built in PHP function to do this, not a homebrew one (that's all a google search seems to return). There is one, I just can't remember its name or find it on php.net. IIRC its name isn't that intuitive. How can I prove that $\gcd(a, b) = \gcd (a, b + ma)$?  I have tried this:  let $g = \gcd(a, b)$, then $g \mid a$ and $g \mid b$. This means that $g \mid ax+by$.  I don't know what to do next.  Thanks. Why does Yoda speak the way he does? Yoda's unique speech consisting of reversed grammar is iconic.  Is there any reason he has adopted this speech pattern?  All other alien species encountered seem to have no trouble speaking normally. Why does Yoda speak in this way? For $a,b,c&gt;0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$  how can I find the limit of sequence above?  Provide me a hint or full solution. thanks ^^ how to prevent cell color to cover table's borders? When I change the background color of a cell in a table, the color covers the borders, however I want the horizontal lines to remain.  I have a simple command \cellcolor[gray]{0.9} in my table. Can anybody help me with this problem? Objects not rotating with each other (Rubik's cube) I have a large set of data points of the form (mean, stdev). I wish to reduce this to a single (better) mean, and a (hopefully) smaller standard deviation.  Clearly I could simply compute $\frac{\sum data_{mean}}{N}$, however this does not take in to account the fact that some of the data points are significantly more accurate than others.  To put it simply, I wish to preform a weighted average of these data points, but do not know what the weighting function should be in terms of the standard deviation.