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are two consecutive numbers relatively prime? Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Can someone give a simple explanation as to why the       $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$   doesn't converge, on the other hand it grows very slowly?   I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks. How to scale a tikzpicture to \textwidth Is it possible to see a list of the settlers in a given settlement and the jobs they are assigned to? It would make it easier to pick a settler for a new job. Display Top X results for each unique value I am trying to figure out how I can output any machine that has had the latest 3 backups be larger than 10GB. My SQL Table looks similar to the below table:  ComputerID | BackupImageName | WindowsVolumeGUID | FileSize | TimeCreated 1234       | PCBackup.spi    | kjsdfai-dadad-ddd | 10400    | 05-22-20 1234       | PCBackup1.spi   | kjsdfai-dadad-ddd | 10500    | 05-23-20 1234       | PCBackup2.spi   | kjsdfai-dadad-ddd | 10300    | 05-24-20 1234       | PCBackup3.spi   | kjsdfai-dadad-ddd | 10900    | 05-25-20 1225       | PCBackup4.spi   | kjsdfa5-dadad-ddd | 10600    | 05-22-20 1225       | PCBackup5.spi   | kjsdfa5-dadad-ddd | 5000     | 05-23-20 1225       | PCBackup6.spi   | kjsdfa5-dadad-ddd | 10000    | 05-24-20 1225       | PCBackup7.spi   | kjsdfa5-dadad-ddd | 4588     | 05-25-20   I need to output all machines (computerID's) or Volumes (WindowsVolumeGUID) that have their latest 3 files in the list greater than 10000 file size. So from the above table it would output the computerID of 1234 because its latest 3 are greater than 10000 file size and even though the computerID 1225 has 2 files greater than 10000 it would not output the result because there is a break in-between the two large files. The search will need to take into account the WindowsVolumeGUID since some machines have more than one volume and I need to take into account the individual volume backups and count the latest entries based on that.  Alternatively, I can also accomplish the same thing if there is a way to output the top 3 results for each unique WindowsVolumeGUID and not just across the board.   Any help would be greatly appreciated! Simple way to understand what derivative is I know how to count derivatives, but I actually don't understand what are those? What do they show?   Wikipedia explanation: Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity. I understand nothing what is written here?  Please, if you can, explain me as simple as you can it. Thank you very much. Counter example for a result of intersection of subspaces I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated. Is "funnest" a word? We seem to be stuck at an impasse on this issue.  Is funnest a word or not?  If so, does it mean "most fun"? What is the difference between an Intel i7 and a Xeon quad core processor? I know the  processors have been around for a long time and are mostly used in servers, but I am curious, why do people not use the Xeons in a high performance desktop?    As far as I know about the best desktop processor out there now is the  line. The i7's and Xeons are both quad-core processors, what is the main difference in these? I just saw that the MacBook Pro's are using the quad core Xeons instead of the i7's. Validate date format in php Is the EmDrive, or "Relativity Drive" possible? Continuity of a convex function I'm trying to solve the following problem:  Let $f:K\rightarrow \mathbb{R} $, $f$ convex and $K \subseteq \mathbb{R}^n$ convex. Then $f$ is continuous on $K$.   I have proved the only case $n=1$, but for an arbitrary $n$?? In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only.  It is the pentagon $ABCDE$ pictured below.   I am having trouble seeing why the central angles are all $72^\circ$, though. Can anyone provide the proof?  Also, does anyone know who this construction is due to? I haven't seen it anywhere, other than in Dixon's book; is it Dixon's result?     The result appears much more impressive without all the labels (which are slightly misplaced, please excuse this); however, I provided those so that answering would be easier. Also, it makes it easy to describe the steps in the construction:  1) Draw a circle (the red one)  with center $h$.  2) Draw the perpendicular lines $\ell_1$ and $\ell_2$ through $h$. Locate the points of intersection $f$, $B$, and $g$ with the red circle.  3) Bisect the line segment $gh$. Denote the center by $a$.  4) Draw the green circle with center $a$ and radius $ah$.  5) Draw the other green circle (as in (3) and 4)).  6) Draw the line segment through $f$ and $a$.  7) Locate the points of intersection $b$ and $c$ of the line segment with the circle constructed in step 4).  8) Draw the blue arcs (both have center at $f$ and the radii are $fb$ and $fc$).  9) Locate the points of intersection $A$, $C$, $D$, and $E$.   I actually have the solution to the first question, and will post it unless a more elegant explanation is provided (which is probably likely).  However, I find this construction particularly   beautiful, and would like to know who it is attributed to (Dixon doesn't say, explicitly). I have been facing hard time understanding meaning of "random sample" as well as "iid random variable". I tried to find out the meaning from several sources, but just got more and more confused. I am posting here what I tried and got to know:  Degroot's Probability &amp; Statistics says:     Random Samples / i.i.d. / Sample Size : Consider a given probability distribution on the real line that can be represented by either a p.f. or a p.d.f. $f$. It is said that $n$ random variables $X_1 , . . . , X_n$ form a random sample from this distribution if these random variables are independent and the marginal p.f. or p.d.f. of each of them is $f$. Such random variables are also said to be independent and identically distributed, abbreviated i.i.d. We refer to the number n of random variables as the sample size.   But one of the other statistics book I have says:       In a Random Sampling, we guarantee that every individual unit in the population gets an equal chance(probability) of being selected.   So, I have a feeling that i.i.d.s are elements that construct random sample, and the procedure to have random sample is random sampling. Am I right?   P.S.: I am very confused about this topic, so I will appreciate elaborate reply. Thanks. 'sudo su -' vs 'sudo -i' vs 'sudo /bin/bash' - when does it matter which is used, or does it matter at all? Avoiding numerical cancellation question for $\sin x -\sin y$ for $x \approx y$ When trying to avoid cancellation, one tries to reformulate the equation in order to avoid subtraction between almost equal terms.   In $\sin (x) - \sin (y), x \approx y$ the suggested solution is to reformulate it to $$2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$$  But I don't understand how it is any better? The subtraction between $x, y$ remains. Is it because $\lvert\,\sin(x)-\sin(y)\,\rvert\leq \lvert\,x-y\,\rvert$, so cancellation is less likely to happen between $x,y$ than the sines? I have a few questions regarding CAN bus load  Is there any derived equation/formula to calculate the bus load?  I need transmit below frames(Assume Standard frame format) at baud rate 500 kbps on the bus 5ms  ----- 50 frames. 10ms ----- 10 frames. 100ms ---- 30 frmaes. Event messages ---- 10 frames.   How to calculate bus load, does anyone knows kindly illustrate the same?  What is the maximum bus load on CAN bus? How can I determine the height and the least number of generators of the ideal $ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $?      I tried to calculate the dimension of the vector space $I/I\mathfrak m$ with $\mathfrak m=(x,y,z)$, but I'm not able to find it.  I think the height is $2$, because the affine variety should be a curve, and also I know that the ideal is prime, so I need a chain $0\subset \mathfrak p \subset I$. Maybe $(xz-y^2)?$ Is it prime? MySQL is giving weird result Centos mysql version is 5.5 however PHPmyadmin still says version 5.1