Derivative of series and series of derivatives
Interchanging the order of differentiation and summation
Modular Arithmetic - Find the Square Root Using (or not) the word "to" in a sentence ending with "be able" Which of these is grammatically correct?   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able.   As educators, we are always looking for ways to reach all the learners in our classroom as effectively as we are able to. Universal type tranformer in Haskell How to derive $E=mc^{2}$? Is there some way to derive ? I can understand that energy in something is proportional to its mass, but the $c^{2}$ part. I have no idea. It seems like the way the units are going it would end up as $kg*(m/s)^{2}$, which I think is the unit for Newtons, not energy. And why is there no constant of proportionality $k$? Did Einstein just set it up perfectly so that there wouldn't be one? drawing different parts of a line in different colours How to draw different parts of a line in different colours. In the following code can each part of the line be drawn in different colour      \documentclass{beamer} \beamertemplatenavigationsymbolsempty \usepackage{tikz} \usepackage{pgfplots} \begin{document} \begin{frame}[fragile,t] \frametitle{} { \begin{tikzpicture} \draw [thick, green] (2.3,1.4) +(-27:2.2cm)-- +(153:2.2cm) +(42:1.88cm) -- +(-138:1.88cm); \end{tikzpicture} } \end{frame} \end{document} Since there is no word "shallowen", is there a good antonym of "deepen"? The meaning of "deepen" that I am referring to here is "to cause to become deeper." It doesn't necessarily refer to digging a hole. For example, a temporary deformation in a soft surface:     Putting the anvil on the mattress deepens the depression in it, but removing the anvil ______ it.   I would also be happy with an antonym of the intransitive version "to become deeper." Relationship between Partial Harmonic Sum and Logarithm. How can I remove a suggestion from the Chrome address bar? I've got a couple of URL autocomplete suggestions where Chrome suggests the wrong URL consistently.   How can I delete the incorrect entry from the address bar's suggestion list? The page I want shows up too but it's never the first result. tags of mundanity that are gratuitously applied to posts It's an additional unnecessary burden to the honest stacker during suggested edit duty. A question presents some code then the minor editor adds a tag like 'for-loop' or 'forms'.  These tags are so general that they only have a handful of followers. They serve no useful purpose and are the cause of many too minor edits. Why not permit culling of generic tags? I need to calculate the following determinant in order to prove the following equality:  $$\det\begin{pmatrix} a &amp; b &amp; c &amp; d \\ -b &amp; a &amp; -d &amp; c \\ -c &amp; d &amp; a &amp; -b \\ -d &amp; -c &amp; b &amp; a \end{pmatrix} = (a^2+b^2+c^2+d^2)^2.$$  I tried using Gauss-algorithm to get an easier matrix, but I'm not sure if I did it correctly.  Calling the $4$ lines $I$, $II$, $III$ and $IV$, I did:  (1) $II \cdot a$  (2) $III \cdot a$  (3) $IV \cdot a$  After this I did:  (4) $II' + I \cdot b$  (5) $III' + I \cdot c$  (6) $IV' + I \cdot d$  So finally I got the following matrix:  $$\begin{pmatrix} a &amp; b &amp; c &amp; d \\ 0 &amp; a^2+b^2 &amp; bc-ad &amp; ac+bd \\ 0 &amp; ad+bc &amp; a^2+c^2 &amp; cd-ab \\ 0 &amp; bd-ac &amp; ab-cd &amp; a^2+d^2 \end{pmatrix}.$$  I thought this would make the determinant a bit easier, unfortunately I must have done something wrong. Is multiplication with single lines allowed as I have done it?  Thank you very much. Should I use the datetime or timestamp data type in MySQL? How to set a default query timeout with JPA and Hibernate? Medium with refractive index less than unity? What is crop factor and how does it relate to focal length? WiFi problems (Atheros driver) with FreeBSD 11 With FreeBSD 10.3 my Atheros card worked fine and showed up as /dev/ath0. Now with FreeBSD 11 this is no longer the case, though the module is loaded and it seems the system recognizes the card:  root@igloo:~ # sysctl net.wlan.devices net.wlan.devices: ath0   Still, ath0 cannot be used:  root@igloo:~ # ifconfig wlan0 create wlandevice ath0 ifconfig: must specify a parent device (wlandev) when creating   What must I do to use the card? How can I apply an effect when a player left-clicks/right-clicks with an item in hand? Is there a way so that I make make it when I left click/right click with a certain item it applies an effect? I know that you can apply attributes to wearing armor and things but I want it so when you use the item it applies an effect on your player.  For example, let's say I want something so that when I right click with a certain item (let's say an iron ingot that was given to me) I'll then have a jump boost for 10 seconds. So kinda like a potion in a way.  Is this possible with command blocks/commands/any mods? I have a DVD I was trying to rip, and handbrake froze up when scanning it.  I tried to eject the DVD by dragging and dropping it to the trash, which trashed the icon, but didn't eject the DVD.  I then forced Handbrake to quit, but without an icon on my desktop to select, how can I eject the DVD? Most common method for deciding when to stop training a neural net on a batch I have created my own neural net which is using batch gradient descent. In other words, it trains on batches of examples all at once.  My issue is trying to figure out when to stop the training of the batch. I'll try to make things as understandable as possible since there are so many options, but please comment if you need clarification.  Each batch allows me to compute a list of output errors that correspond to the error the computer has on each example. Each error is a vector with elements equal to the number of output neurons.  Let each error vector be notated by e1, e2, e3, ... so the list of output errors is [e1, e2, ...]. Let an error vector in general be notated by e.  Now there are 2 things we can do to a vector or a list:   Average: average all of the elements in a vector or list. This will be notated by the function a(). For example to find the average error of an example we would do a(e) Check: make sure that each element in the vector or list is below a certain value. If any of the elements are too high, we continue to train on the entire batch. (This function can also be applied to a single value.) This will be notated by the function c(). For example, if we wanted to make sure the average error in e was low enough, we would do c(a(e)) Sum: add up all of the elements in the vector or list. This will be notated by the function s(). For example, s(e) would be totaling the error found in error vector e.   Here are the methods I have determined for deciding when to stop gradient descent:   Iterative: just run the training a certain number of times Average of averages: Find the average error of each example, then find the average of those and make sure it's below a certain amount. This is given by the formula c(a([a(e1), a(e2), ...])) Specific error: Check to make sure the error in each output neuron is below a certain amount. This is given by the formula c(e1), c(e2), ... Specific averages: Check to make sure the average error in each example is below a certain amount. This is given by the formula c([a(e1), a(e2), ...]) Sum of sums: Total up all the error in the entire batch and then make sure it's low enough. This is given by the formula c(s([s(e1), s(e2), ...]) Sum of averages: Find the average error of each example, then add up all those and make sure it's below a certain amount. This is given by the formula c(s([a(e1), a(e2), ...])) Average of sums: Compute the average total error in each example, and then make sure it's below a certain amount. This is given by the formula c(a([s(e1), s(e2), ...])) Specific sums: Check to make sure the total of the error in each example is below a certain amount. This is given by the formula c([s(e1), s(e2), ...])   So my question is this:  Which one of these options is most commonly used when training neural nets? Why? (Is there another option that I've missed?)  (Edit: I have found a very general version of this question , where the answers seem to say there is no specifically good way. If that is still true, please say so in your answer. Note that this is not a duplicate, as I explain the specific methods I would like to use.)  Edit: as A.D pointed out, summing is pretty useless because it depends on how many values are in each list/array/batch. And if you know how many examples there are, you can just use averaging. So, the methods involving "summing" have been cut. Sequence of a product of functions in $L^p. L^q$ with $p,q$ conjugate The question is: if $f_i$ is a sequence of functions in $L^p$ converging to $f$ and $g_i$ a sequence in $L^q$ converging to $g$ show that $f_ig_i$ converges to $fg$ in $L^1$ for $p,q$ finite and $\frac{1}{p}+\frac{1}{q}=1$. Does this result hold if $p=1, q=\infty$?  So I think I showed the first part. $||f_ig_i-fg||_1=||f_ig_i-f_ig+f_ig-fg||_1\leq||f_ig_i-f_ig||_1+||f_ig-fg||_1 = ||f_i(g_i-g)||_1+||g(f_i-f)||_1\leq||f_i||_p||g_i-g||_q+||g||_q||f_i-f||_p$   (by Holder's inequality) which goes to $0$ as $i\rightarrow\infty$.  However the second part of the question throws me off because I don't see why this proof doesn't work just as well for $p=1,q=\infty$.  Am I missing something? Does the second part also hold? How To: Execute command line in C#, get STD OUT results