How to fix javascript to change image file name from "download" to something else for example "test.png"? HTML
Is there any way to specify a suggested filename when using data: URI?
Calculating $1+\frac13+\frac{1\cdot3}{3\cdot6}+\frac{1\cdot3\cdot5}{3\cdot6\cdot9}+\frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $ Why does an integral change signs when flipping the boundaries? Let us define a very simple integral:   $f(x) = \int_{a}^{b}{x}$    for $a,b\ge 0$.  Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$?  I drew the graphs and thought about it but to me integration, at least in two-dimensions, is just taking the area underneath a curve so why does it matter which direction you take the sum? Integrate $\int_{0}^{2\pi}\log|1-e^{i\theta}| d\theta$ Here is exercise 11, chapter 3 from Stein &amp; Shakarchi's Complex Analysis II:  Show that if $|a|&lt;1$, then:  $$\int_{0}^{2\pi}\log|1-ae^{i\theta}|\,d\theta = 0$$  Then, prove that the above result remains true if we assume only that $|a|\leq 1.$ ($a\in \mathbb{C}$)  I've already proved it for $|a|&lt;1$, but couldn't make it for $|a|=1$, which I thought would be the easy part. But the only thing I could come up with was that, WLOG, it is enough to prove that:  $$\int_{0}^{2\pi}\log|1-e^{i\theta}|\,d\theta = 0$$  Tried to look for some kind of symmetry, but could't find it. Any ideas? Thanks! How to get Microsoft Search to index common file types like .pdf and .pub I'm trying to get Windows Search (4.0) to index PDFs and .pub, .doc, etc. files.  The files are on an XP Pro PC acting as a file server. But I don't have MS Office, etc. installed (no need, trying to keep it "slim and trim") but WS 4.0 is doing the indexing of those files and not indexing .pdf and .pub.  I've found a few  (from Foxit PDF reader) rated 3/5 and only support WS 2.0 or 3.0.  Looking for other solutions and also a way to index .pub files. Extension fields isomorphic to fields of matrices Suppose $K \subset L$ is a finite field extension of degree $m$. Is it true that there exists some natural $n$ such that $L$ is isomorphic to a subfield of $M^{n\times n}(K)$, the ring of $n\times n$ matrices with entries in $K$? What is the smallest dimension (as a function of $m$) for which an isomorphism exists?  The example I had in mind was the 2-dimensional extension $\mathbb{R} \subset \mathbb{C}$ with $\mathbb{C}$ being isomorphic to $\{ \begin{pmatrix}a&amp;-b\\b&amp;a\end{pmatrix} : a, b \in \mathbb{R}\}$. How do I remove Windows but keep Ubuntu? Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$ Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$.  The contest is just ended, so you may freely answer. (I did not attend the contest: it is an Italian contest for schools and the groups may ask other people for hints. This is how I got involved, but of course asking on MSE would have been cheating)  I found three solutions: I am explaining below what I did. I would like however to know if there is a more general approach. Prove $(k,n+k)=1$ iff $(k,n)=1$ (a) Prove $(k,n+k)=1$ iff $(k,n)=1$. (b) Is it true that $(k,n+k)=d$ iff $(k,n)=d$? (all variables are integers, (a,b) means gcd(a,b))  For (a), I can only get so far.  $(k,n+k)=1\ \Rightarrow\ kq_1+(n+k)q_2=k(q_1+q_2)+nq_2=1$.  Note that we don't yet have that theorem (from a different class) that says something like if $d=(a,b)$, then $d$ is the smallest (or is it biggest) number which can be made by $ak_1+bk_2$, (which I'm not quite sure how it would help, anyway). So I'm stuck. Bold italic vectors I am trying to display vectors as bold and italic. The command \mathbf sets the letters non-italic. I also googled a bit and stumbled upon \mathbf{\em y} or \mathbfit{y}, which all didn't work.  I hope someone can help me on this. $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ twice Does there exist a continuous function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ exactly two times? When are "if" and "whether" equivalent? Are if and whether equivalent in sentences like the ones below?     How to determine if my saddle is too high?   How to determine whether my saddle is too high?         We should check if everything is okay now.   We should check whether everything is okay now. Resize multiple layers around their respective anchor points Live USB for installation does not boot, black error screen followed by visual artifacts on a purple screen I'm trying to install Ubuntu 18.04 desktop with Rufus on a computer with the following specs:  CPU: AMD Ryzen 5 2600 MOBO: ASUS Prime B450M-A/CSM RAM: G-Skill Ripjaws DDR4 F4-2400C15D-16GVB GPU: Nvidia GeForce GTX750 PSU: Apexgaming AG-650M SSHD: Seagate Firecuda Compute mATX: Rosewill SCM-01  Basically it boots from the USB (FAT32) and brings up a black error screen with words and numbers. Then it transitions to the purple splash screen with &quot;ubuntu&quot; and 5 dots, however the screen shows various visual glitches depending on which USB port I plug into. When trying to boot the computer, it boots to a black screen with the following error message: ACPI Error: Needed [Integer/String/Buffer], found [Region]    that shows what I mean. I am trying to learn data in higher space into lower space. To have a clue, I'd like to know how to transform the data in the image below into a lower dimension preserving the structure. Hope to hear some explanations and what should I study to learn mapping data to lower dimension? can any one tell me how to write long equation in multline i tried with coding below but some errors of \right and \left appear \begin{multline*} \displaystyle \left \| f  \right \|_{M\dot{K}^{\alpha(\cdot),q}_{p(\cdot), \lambda}(\mathbb{R}^{n})} \approx \max \left\{ \sup_{L\leqslant0,L\in  \mathbb{Z}}2^{-L\lambda}\left ( \sum_{k=-\infty}^{L}2^{k\alpha(0)q}\left \|  f\chi_{k} \right \|^{q}_{L^{p(\cdot)}} \right )^{1/q},\\\sup_{L&gt;0,L\in \mathbb{Z}}\left[ 2^{-L\lambda}\left ( \sum_{k=-\infty}^{-1}2^{k\alpha(0)q} \left \| f\chi_{k} \right \|^{q}_{L^{p(\cdot)}} \right )^{1/q}\\ +2^{-L\lambda}\left ( \sum_{k=0}^{L}2^{k\alpha(\infty)q}\left \|  f\chi_{k} \right \|^{q}_{L^{p(\cdot)}} \right )^{1/q}\right] \right\}  \end{multline*} I've been wondering this for a while, why is there free space at the end of a drive that was formatted to NTFS in windows?  You can only see the unallocated space in Linux. So why is it there? Does it serve a purpose? Does removing an iPhone from my icloud account also erase the data? I lost my iPhone 6S recently. I think it fell off my car and was crushed because it has been offline (according to Find My iPhone) ever since. I want to erase the phone and remove it from my iCloud account since the phone is gone.  However, I can't erase the phone until it comes back online. I do have the option to remove the device from my iCloud account. If I remove the phone from my iCloud account, does that also erase the data? The phone has a password on it, will it continue to be locked in case someone finds it after I have removed it from my iCloud account? Generic htaccess redirect www to non-www I would like to redirect www.example.com to example.com. The following htaccess code makes this happen:  RewriteCond %{HTTP_HOST} ^www\.example\.com [NC] RewriteRule ^(.*)$ http://example.com/$1 [L,R=301]   But, is there a way to do this in a generic fashion without hardcoding the domain name? If A is a finite set and B is an infinite set, then B\A is an infinite set Prove: If $A$ is a finite set and $B$ is an infinite set, then $B \setminus A$ is an infinite set.   I've tried proving the contrapositive, that if $B \setminus A$ is a finite set then $A$ is infinite or $B$ is finite, but I don't think this is the way to go. I am not able to sum all the opportunities under the account ?  How will I do that?  What I have to use in this to run the trigger? Is it possible to create because opp. is child and account is parent ?  trigger sumAllopptyAmtAccount on Account(before insert) {     Set &lt; ID &gt; AccID = New Set &lt; ID &gt; ();     if (Trigger.isInsert)         for (Account oppAc: trigger.New) {             AccID.add(oppAc.AccountID);         }     List &lt; Account &gt; acc = [select id, SumAll_Opprty__c from Account where id in: a.AccID];     List &lt; Opportunity &gt; opp = [selece id from Opportunity where AccountID in: a.AccID];     for (Account a: acc) {         a.SumAll_Opprty__c = con.size();     } }