Imagine a circle cut in half. Now, are the two angles on it right? I can imagine when going limitedly close that they would be. Got curious about this question after seeing  on the internet.
If you have a line passing through the middle of a circle, does it create a right angle at the intersection of the line and curve? More generally, is it valid to define an angle created between a line and a curve? Is the tangent to the curve at the point of intersection a valid interpretation (I.e a semi circle has 2 right angles) I saw it in a debate thread and it got me curious now.
Is "The empty set is a subset of any set" a convention? Recently I learned that for any set A, we have $\varnothing\subset A$.  I found some explanation of why it holds.     $\varnothing\subset A$ means "for every object $x$, if $x$ belongs to the empty set, then $x$ also belongs to the set A". This is a vacuous truth, because the antecedent ($x$ belongs to the empty set) could never be true, so the conclusion always holds ($x$ also belongs to the set A). So $\varnothing\subset A$ holds.   What confused me was that, the following expression was also a vacuous truth.     For every object  $x$, if $x$ belongs to the empty set, then $x$ doesn't belong to the set A.   According to the definition of the vacuous truth, the conclusion ($x$ doesn't belong to the set A) holds, so $\varnothing\not\subset A$ would be true, too.  Which one is correct? Or is it just a convention to let $\varnothing\subset A$? Solution for A/C Condensate Drain Line that doesn't require emptying a bucket of water every day? Can I get from Gatwick airport to London by Oyster card? I read many contradictions as to whether one can use a Oyster card from Gatwick Airport to central London.  What is the most time-efficient way to go from Gatwick to "Old Billingsgate" (Monument tube station is nearby)? Public transportation is preferred over taxi. There is no set of all sets.. So there are a few posts about this already, but they skip over the problem I have.  The proof of Cantor's theorem elegantly shows that if we consider $f:A\rightarrow \mathcal{P}(A)$, the set $B=\{x\in A : x\not\in f(x) \}$ by its definition cannot lie in the image of $f$, precluding it from being a bijection - hence power sets have strictly greater cardinality.  But now, I thought one (obvious) way to characterise the "set of all sets" would be $S=\mathcal{P}(S)$, as indeed, it contains all possible sets (perhaps here is the problem, but I don't immediately see how using this as a definition for the set of all sets is bad). Now consider simply taking the identity function $i:S\rightarrow S$. Then $B=\{x\in S : x\not\in x \}$ - the set of all sets which do not contain themselves. But this doesn't exist. It's exactly the set of Russell's paradox. If this isn't a set, and so is not in $S$, then the proof no longer offers an obstruction to $i$ being surjective.    Each other post I've seen on the matter uses Cantor's theorem to show such a set doesn't exist. My query is to find that something else that must be happening, since I think the proof has issues for this particular set. I'm expecting other issues to arise with the definition $S=\mathcal{P}(S)$... From this , I now understand the gradient to be the directional derivative that returns the steepest slope at a point. However, reading my textbooks, they all say that the gradient is normal to the tangent plane. In the above SE question, the gradient sounds like it should be parallel to a direction in the $xy$ plane, not actually up on the surface.   My question is therefore do I have a misunderstanding of the tangent plane, gradient or directional derivative?   In addition, I question how the gradient always passes through the origin or the $xy$ plane? If for instance in the graphic, the gradient had been another direction (say horizontal), then it wouldn't pass through the origin! In addition, the gradient of other points in the graphic would satisfy this example. Many thanks! How to fix very slow Ubuntu booting? When I boot up my my newly installed copy of Ubuntu on my custom rig w/4GB DDR3 RAM and a 3 Ghz Intel Core i7, it takes about a minute and a half at the purple loading screen, and then after logging in it takes a while for the icons to show up on the screen. This computer could run and boot Windows 7 faster, so  i have a problem here. How to get back a declined Outlook invitation I accidentally clicked "decline" on an Outlook invitation and now I can't reverse it -- the event has completely disappeared from my calendar!  How can I get back the event invitation and change my "Decline" to "Accept" or "tentative"? Is there a word for capitalizing in the middle of a word, as in "eBay"? Usually this sort of word is a trademarked name, such as eBay or BlackBerry.  Is there a word that describes capitalization mid-word? How to parse hex values into a uint? uint color;  bool parsedhex = uint.TryParse(TextBox1.Text, out color);  //where Text is of the form 0xFF0000 if(parsedhex)    //...   doesn't work. What am i doing wrong? How to handle UNABLE_LOCK_ROW errors when this is caused by a batch APEX job we are facing a serious issue. We have a batch APEX job running daily which makes updates on records fulfilling certain conditions. There is another job running parallel to this, in some cases updating or accessing to the same records than the batch job and hence, causing the error:     Update failed. First exception on row 400 with id XXX; first error:    UNABLE_TO_LOCK_ROW, unable to obtain exclusive access to this record or 200    records:   So, I had then read that in order to fix this kind of error you should use FOR_UPDATE in your select statement (records that have to be updated). The problem is, that you can not use FOR UPDATE for batch jobs. We tested this and effectively, we get the error:     System.QueryException: Locking is implied for each batch execution and   therefore FOR UPDATE should not be specified      It will however lock the records within each execution of execute. See   QueryLocator.getQuery()usage:      You cannot use the FOR UPDATE keywords with a getQueryLocator query to   lock  a set of records. The start method automatically locks the set   of records in  the batch.   Could you find a solution to this?. We would like to handle this in the code itself, because we don't have the permission to schedule the batch job for another time as the another job. Beamer: PDF presentation viewer to play embedded videos and show elapsed time Send data from Lightning Web Component (LWC) to a Visualforce page I am adding an LWC component in visualforce and need to add a handler for an event fired in the LWC component on the visualforce page. When googling all I can find is examples of how to do this with aura component using application events. LWC controller: handleSearchClick() {              const searchinfo = {          shipper: this.shipper,         system: this.system,         product: this.product,         status: this.template.querySelector('c-lwc-multi-select').value(),         period: this.period       };     this.dispatchEvent(new CustomEvent(         'dosearch',          {             detail: { data:  searchinfo},             bubbles: true,             composed: true,         }     ));     console.log({ 'lwcNominationSearch searchinfo': searchinfo }); }  AURA app: &lt;aura:application extends=&quot;ltng:outApp&quot; access=&quot;GLOBAL&quot;&gt; &lt;aura:dependency resource=&quot;c:lwcNominationSearch&quot; /&gt; &lt;aura:dependency resource=&quot;markup://force:*&quot; type=&quot;EVENT&quot;/&gt; &lt;/aura:application&gt;  Visualforce: &lt;div id=&quot;rcontainer&quot;&gt; &lt;/div&gt;  &lt;script&gt;     $Lightning.use(&quot;c:NotificationSearch_ltngoutApp&quot;, function()     {         $Lightning.createComponent(         &quot;c:lwcNominationSearch&quot;,         {},         &quot;rcontainer&quot;,         function(cmp)         {             console.log('component created');              //$A.eventService.addHandler({ &quot;event&quot;: &quot;c:myEvent&quot;, &quot;handler&quot; : visualForceFunction})               $A.eventService.addHandler({                 event: 'dosearch',                 handler: function (event) {                          console.log(&quot;vf event success&quot;);                 }             });          });     }); &lt;/script&gt; Bounding box is larger than expected when drawing a curved path I have been working on logo for my department and I have the following code. The problem is that whenever I compile the diagram seems to be larger than the actual logo. I do not know what is the problem. I know the numbers I have selected are not the best but any insights into the matter will be highly appreciated.   \documentclass[letterpaper]{article}  \usepackage[top=2cm,bottom=2cm,left=2cm,right=2cm]{geometry}  %\usepackage{amsmath,amssymb,units}  %\usepackage{enumitem,multicol}  \usepackage{tikz}  %\usetikzlibrary{arrows}  \usepackage{lipsum}   \begin{document}   \lipsum[1-2]  \begin{tikzpicture}[y=0.80pt, x=0.8pt,yscale=-1]  \path[draw=black,fill=white]    (258.9527,290.5199) .. controls (173.9885,538.4766) and (568.5860,261.2969) ..    (306.5098,257.1141) .. controls (44.4337,252.9312) and (429.9845,542.5624) ..    (352.9767,292.0206) .. controls (275.9689,41.4788) and (119.6549,497.6604) ..    (334.1376,346.9999) .. controls (548.6203,196.3394) and (66.4622,188.6439) ..    (276.0276,346.0724) .. controls (485.5930,503.5010) and (343.9169,42.5633) ..    (258.9527,290.5199) -- cycle;  \end{tikzpicture}  \lipsum[1-2]  \end{document}    See image below: I have seen that the new  shows off H.265/HEVC playback, the successor of H.264. I wonder how I may get this support in Linux. Is there a need to install a codec specific for this format? How? Embed LaTeX math equations into Microsoft Word Let's say I have a (comparatively) lovely-looking document in LaTeX, full of lovingly typeset, (relatively) complex equations. Now, let's say some barbarians come along one day and decide that the document has to be put into Microsoft Word (2007). ...after the usual mourning period associated with such events, let's say I value my job (more specifically, the bread it provides) enough to get all the text and tables formatted and references organised into the Word document. –...related questions  and – Now I'm looking at the equations with fear and dread. One option of course is to just lift screenshots from the original document, but this is painstaking if I need to refer to parts of the equation in the text. Also, I might need to edit equations on the fly. Anyone know of a free application which allows embedding LaTeX math into MS Word? I've looked at  and  which do roughly what I want... they build LaTeX images from source and embed them into the Word document, allowing to edit the source later... but both are commerical. ...any help in these troubling times will be greatly appreciated. EDIT: Just a note that  offers a 30-day free trial and is working out really nicely... but still, it's not free. Might be a good solution for those with short-term needs, or money. Why shouldn't root be allowed to login via ssh? why should root access be disabled for ssh? I always hear that's for security reasons. But I don't get it. What's different to login as non-root and then sudo su -? What's the preferred way to handle tasks that need privileged permissions?  Thanks  Thomas Is "I am sat" bad English? I believe it is incorrect and instead either the present continuous     I am sitting   or the predicate adjective     I am seated   should be used.  I hear this quite often, however, and ultimately usage overrules formality.  Does anybody know where this originated or how to describe it grammatically? I'm playing on a Survival world with cheats on, "easy" difficulty, and at night in-game. I didn't have a sword or anything to make a sword with, and I was starving, so I switched to Creative to give myself a stack of food and a bed to set my spawn point. When I switched back to Survival, I couldn't destroy or place blocks. I'm on single player, and my Internet is fine, and I can destroy blocks still if I switch to creative. I've tried resetting the button one, and reloading the map, and it still doesn't work. I've even tried deleting everything I gave myself in Creative. This happened before on a different world and I ended up deleting it.  This only happens on this world. How do I fix it?  If I know I'm in the correct game mode, what else can cause this?  Are there different causes in multiplayer? If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$ $A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$    If $\lambda$ is an eigenvalue of $A^2,$ we have $\lambda\alpha=A^2\alpha$ for some $\alpha.$ Then how can we find a $\beta$ s.t. $\sqrt{\lambda}\beta=A\beta?$ Is there a slowest rate of divergence of a series? $$f(n)=\sum_{i=1}^n\frac{1}{i}$$ diverges slower than $$g(n)=\sum_{i=1}^n\frac{1}{\sqrt{i}}$$ , by which I mean $\lim_{n\rightarrow \infty}(g(n)-f(n))=\infty$. Similarly, $\ln(n)$ diverges as fast as $f(n)$, as $\lim_{n \rightarrow \infty}(f(n)-\ln(n))=\gamma$, so they 'diverge at the same speed'.   I think there are an infinite number of 'speeds of divergence' (for example, $\sum_{i=1}^n\frac{1}{i^k}$ diverge at different rates for different $k&lt;1$). However, is there a slowest speed of divergence?  That is, does there exist a divergent series, $s(n)$, such that for any other divergent series $S(n)$, the limit $\lim_{n \rightarrow \infty}(S(n)-s(n))=\infty$ or $=k$? If so, are there an infinite number of these slowest series?