Gravity propagation speed
"Speed" of Gravity and Speed of Light
Unexpected behavior of rule matching a pattern Why is the upper Riemann integral the infimum of all upper sums? I was reading the theory of Riemann integration when I cam across the following ,   If $f$ is bounded on $[a,b]$, and $P = \{x_0,x_1,x_2.......x_n\}$ is a partition of $[a,b]$, let $$M_j = \sup_{x_{j-1}\leq x\leq x_j}f(x)$$ The upper sum of f over P is $$S(P) = \sum_{j=1}^{n} M_j(x_j-x_{j-1})$$ and the upper integral of $f$ over $[a,b]$, denoted by $$\int_{a}^{b^-} f(x)dx$$ is the infimum of all upper sums.  The theorem similarly goes on to state the result for lower sums.  My doubt is : I do not understand how is $$\int_{a}^{b^-} f(x)dx$$ the infimum of all upper sums. I understand that if we refine the partition P, then the upper sum would decrease, so it may be a lower limit for all the upper sums computed on the refinements of P (but still being the lower limit does not prove that it is the infimum) and what about those partitions for which P itself is the refinement of?  How do I know that it will be a lower limit for those, let alone a infimum? Solutions to the equation $ze^{a-z}=1$ Let $a \in \mathbb{R}, a&gt;1,$ and consider the equation $$ze^{a-z}=1$$ How many solutions does this equation have in the unit disk, and how many of those solutions are real?   My attempt to solve was to use the Argument Principle on the function $f(z)=ze^{a-z}-1$, in hopes of recovering the number of zeroes that way. The integral I attempted to compute for this was $$\int_{|z|=1} \frac{e^{a-z}-ze^{a-z}}{ze^{a-z}-1}dz=\int_{|z|=1}\frac{1-z}{z-e^{z-a}}dz$$  I do not know how to proceed from here, as the $e^{z-a}$ in the denominator seems problematic. Thanks for your help! We have a org wide email addressed which has been "Verified" into our sandbox. I understant org wide email addressed is unsupported metadata, so this was achieved as pre deployment step.   Into our repo, we have a workflow that send email alert to that org wide email address. "  Workflow   ... &lt;senderAddress&gt;noreply@mycompany.com.au&lt;/senderAddress&gt; &lt;senderType&gt;OrgWideEmailAddress&lt;/senderType&gt;    The problem we are facing is that we are still getting the following error even if this org wide email address is verified:      You must specify a verified Org Wide Email Address for this sender   type.   Is that a bug in the metadata api or expected behaviour ? Ideally we would like to not have to do that as a post deployment step.   PS: We are running DX under api version 43.0 Summary table with a field that shows components of each summarized record I have a table with 2 fields the first is for community and the second for the main community that it belongs to, as in the image  I want to summarize the second field (Belongs_to) but with an extra field (multiple value field) that shows components of each main community, as in the image  Could anyone help me please? Can the system leave a deletion reason when it deletes a question? When do I clear US customs and immigration if connecting at a US airport to my final destination elsewhere in the US? Possible Duplicate:          I'm working on a website that displays distance to various locations to the nearest 1/10 mile. One of the developers (a non-native speaker) asked me an interesting question: If the distance is 1.0 mile(s), is that singular or plural? Clearly, anything other that 1.0 is plural, but is it:     1.0 miles away   or     1.0 mile away   The first option sounds more correct to my ear, but the voice of my elementary school language arts tells me that the latter option is correct.  Edit: Added the word 'away' to the example. It doesn't change the question, but might provide clarity for the answer. Also, regarding numbers between 0 and 1, after more thought, I'm not sure that the answer is as clear as I originally thought. See . Black flakes from new cast iron pan I bought a brand new Lodge 12" cast iron pan, washed it in soap and warm water and dried it with a paper towel thoroughly. Then I applied a thin layer of canola oil with a paper towel (paper towel was clean - not black after this application) and then heated the empty pan on my electric stove for 15-20 minutes. The oil on the pan smoked a bit, some parts of the pan turned slightly ashy (grey). Mostly everything looked OK.   After I turned it off and cooled the pan, I now take a clean paper towel and rub the surface and I see black flakes! What is this? Is this the oil broken down? Can I cook in the pan or wash it with soap to eliminate the black flakes? Once washed and dried should I rub canola oil again? This is my first experience with cast iron. Any pointers will help. How do I correctly clone a JavaScript object? I have an object x. I'd like to copy it as object y, such that changes to y do not modify x. I realized that copying objects derived from built-in JavaScript objects will result in extra, unwanted properties. This isn't a problem, since I'm copying one of my own literal-constructed objects.  How do I correctly clone a JavaScript object? Satisfying explanation of Aristotle's Wheel Paradox. The paradox:    We have a circle and there is another circle with smaller radius. They are co-centeric.  If circle make full turn without sliding, both smaller and bigger circle make full turn too. If we assume that the passed road is equal to the circumference of circles. We have got smaller circle's radius is equal to bigger one's.  Unsatisfying Solutions I found:    "Do not assume that smaller circle's circumference is equal to passed road since the surface that contacts to the ground is bigger one. " // Okey but it does not explain the paradox, it explains just what is the wrong assumption (even does not explain why it is a wrong assumption.) It is undeniable that every point on both smaller and bigger circle will contact exactly one and only one point on their path. Therefore we can think that this is a bijective maps and smaller circle is isomorphic to bigger one. (Okey but ....)   Question: What is the true answer? What is wrong with the definition of circumference of a circle and relationship with its taken path. Wifi not working (Intel) on HP Spectre x360 13 Admin privileges requiring for installing a package build though Packages Using osx 10.9 and packages for making package installer.  While installing a package made using Packages in /Applications location on a osx 10.9 machine, installer is asking for Admin password. Even though I had unchecked 'Require admin password for installation' while package making.  So, how to avoid this i.e a non-admin user should install this package. Package should install in /Applications location, so that non-admin user can use the app.  I check  and When doing an apt-get update/upgrade I see that warning  W: Conflicting distribution: https://downloads.plex.tv/repo/deb public InRelease (expected public but got )   I was upgrading to Ubuntu Zesty 17.04 before and do not know how to fix the above. Already have seen ? but there are no suggested fixes for that.  Maybe one of you can give some hints? Thanks! Deploying a static site on AWS EC2/nginx: what am I doing wrong? Complement of maximal multiplicative set is a prime ideal Let $R$ be a commutative ring with identity. I've been trying to prove the following:     If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$.   I have spent some time in it, but nothing is coming to my mind. Any idea/solution will be appreciated. The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ Is there any way to make a hotkey turn the screen off? How can I make a script (or make some changes) to allow me to turn the screen off in Ubuntu (12.04) whenever I want, the way thevscreen can be turned off in Cellphones?  And how can I make a combination of keys (a shortcut) to execute that script?  I want to conserve power.  I've seen this question    but there is no hotkey for it.  Moreover, in the answer, it is said that there are many ways to accomplish this so I'd like to know more. nginx support for .htaccess / rewrite rules? Differences from Apache? Adding GPX files into ArcMap? What is the smoothest route, the easiest user experience, to add  files to ArcMap?  Ideally, I'd like to interact with .gpx with the same facility as .shp, being able drag'n'drop onto the map canvas and so on. Ultimately the results will be feature classes in a file-gdb but shapefile is an acceptable intermediate.