How would you solve such an equation if it's infeasible to just start trying different $x$ values?  Example: $$x^x = 6.$$
Is there any "superlogarithm" or something to solve an equation like this:  $$x^x = 10?$$
How to prove $\gcd(a^2, b^2) = (\gcd(a, b))^2$?  My attempt:  Let $\gcd(a, b) = d$.  Then $d|a$ and $d|b$ then $d^2|a^2$ and $d^2|b^2$. i.e $d^2$ divides $a^2 ~~\&amp;~~ b^2$. With the announcement of the detection of gravitational waves, questions about the implications proliferate.  Some relate to the possible existence of gravitons.  The analogous relationship between gravitons/gravitational waves and photons/electromagnetic waves is frequently mentioned.  The detection of individual photons required experiments of very low intensity light, yet their existence was inferred (prior to their actual detection) by Planck and Einstein (among others) using the properties of experimental black body radiation and the photo-electric effect.  If the prospects for detection of gravitons requires similar study of very low intensity gravitational waves, then those prospects are very dim indeed. My question: are there similar indirect "experimental" methods for inferring the existence of gravitons? Error with square bracket in table \begin{table}[ht] \makebox[\textwidth][c]{     \begin{tabular}{ccccc}         \hline         \hline         [] &amp; [] &amp; [] &amp; [] &amp; []\\         [] &amp; [] &amp; [] &amp; [] &amp; []\\         \hline         \hline     \end{tabular} } \caption{test} \end{table}   I cannot run the above code, latex returned an error about missing number, but I can't see anything I am missing.  And the weirdest thing is that  \begin{table}[ht] \makebox[\textwidth][c]{     \begin{tabular}{ccccc}         \hline         \hline         [] &amp; [] &amp; [] &amp; [] &amp; []\\         \hline         \hline     \end{tabular} } \caption{test} \end{table}   This code works.  Anyone have any idea? wp_customize_image_control default value How can I prove this sequence is decreasing? My problem is how to prove  $$\frac{1+2}{1}  &gt; \frac{1+3+\frac{3^2}{2!}}{1+2} &gt; \frac{1+4+\frac{4^2}{2!}+\frac{4^3}{3!}}{1+3+\frac{3^2}{2!}}&gt;\cdots $$  i am sure that the sequence is decreasing as limit exp(1).  However, the reason why the sequence is decreasing is not evident.  Thanks if you consider this issue and make me relief. If electrons were positive and protons were negative, would life be different? We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit. If H is a subgroup of G ( | G:H |=n, n > 1), then how can you show that G has a factor group G/N which is isomorphic to a subgroup D of the symmetric group Sn with |D| > 1.  I've been looking at Cayley's Theorem but it's not specific to the factor groups. For one of the clients, I have spin up a modern communication site collection and on the landing page of the same, I am seeing 'Get the mobile app' showing up, I was able to hide the 'feedback' button using css but no luck with Get the mobile app button.  Can someone please share css to hide 'get mobile app' button, appreciate the help. .       The  has no sign of the user sharth The  also does not have any sign of related activity   I would like to know why the user's name is associated with the question. At least why their display name shows up as having modified the question timeline activity.  Possible reason that I can think of is that, .  Is this the reason?  Is there a way (for normal users), to look for such details, other than looking at the user activity and the question's revision history. Can you 'exit' a loop in PHP? Finding integer solutions for $6x+15y+20z=1$ Problem: Find integers $x$, $y$, and $z$ that satisfy the equation $6x+15y+20z=1$.  I noticed that $\gcd(6,15)=3$, $\gcd(15,20)=5$, and that $\gcd(6,20)=2$. And of course $\gcd(6,15,2)=1$.   Of course I know to set one of the variables (x, y, or z) to be zero. But I want to know how to get the more trivial answers. But where do I go from here to find the solutions? I'm a student who uses the laptop all the time and I have a problem with Wifi on the laptop. I don't really know what it is. I can describe the problem tho. For example, i closed my laptop last night, so it goes in suspend mode. I turn it on this morning and my wifi doesn't want to connect until I restart the laptop. It says that it's connecting and in Wifi settings the circle animation is spinning, turning Wifi off and on doesn't do anything. Here's what happens when I check to see my network info: sudo lshw -C network   *-network                         description: Wireless interface        product: Wireless 3160        vendor: Intel Corporation        physical id: 0        bus info: pci@0000:02:00.0        logical name: wlp2s0        version: 83        serial: 00:1e:64:e6:1c:93        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list ethernet physical wireless        configuration: broadcast=yes driver=iwlwifi driverversion=5.4.0-52-generic firmware=17.3216344376.0 ip=192.168.0.14 latency=0 link=yes multicast=yes wireless=IEEE 802.11        resources: irq:129 memory:df100000-df101fff   *-network        description: Ethernet interface        product: RTL810xE PCI Express Fast Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        logical name: enp3s0        version: 07        serial: f8:ca:b8:52:72:49        capacity: 100Mbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 firmware=rtl8106e-1_0.0.1 06/29/12 latency=0 link=no multicast=yes port=MII        resources: irq:17 ioport:d000(size=256) memory:df000000-df000fff memory:d0000000-d0003fff  Also I have to mention that my Linux is doublebooted alongside Windows 10, but I don't think that makes any difference. Help would be much appreciated, I wouldn't want to miss my classes because of this hahah Thanks! php - insert a variable in an echo string Determining if a game is CPU- or GPU-limited What is a reliable, but not too time-consuming way to determine if a certain game is limited by the graphics card or by the processor speed?  I have some preconceptions about whether I'm mostly GPU or CPU limited, but I'd like to verify if I'm right about that. I have some ideas how I would go around and test that but performing accurate benchmarks is notoriously tricky and I don't have much experience with it.   So I'm wondering what would be an easy way to determine the bottleneck, having only one computer with a certain configuration available? What tools would I use for that purpose?  It would also be interesting to determine if the amount of VRAM on my graphics card is a limiting factor. Expectation of a function of a random variable from CDF Is it possible to calculate the expectation of a function of a random variable with only the the r.v.'s CDF? Say I have a function $g(x)$ that has the property $\int_{-\infty}^{\infty}g(x)dx \leq \infty $ and the only information I have about the random variable is the CDF.  For example, I have a scenario where there are three timers that can be modeled as exponential random variables $X_1,X_2,X_3$ with rate parameters $\lambda_1,\lambda_2,\lambda_3$ respectively. For each moment in time I earn a reward according to some reward function $g(x)$. That is, my reward for waiting until time $t$ can be written as $\int_0^tg(x)dx$. However, $g(x)$ experiences diminishing returns so the the marginal reward received from waiting one second at $t=0$ is greater than one second at say $t=27$. This 'game' ends when one of two things happens. Either both timers $X_1$ or $X_2$ must ring or timers $X_1$ or $X_3$ must ring. I'm trying to find the expected reward of playing this game.  Currently I can calculate the the CDF of the random variable modeling the time until the game ends, but I dont know how to use this information to when what I really need is reward associated with this time.  So far I have the additional random variables: $$ W_{12}=\max(X_1,X_2) \quad W_{13}=\max(X_1,X_3) \quad Z=\min(W_{12},W_{13})$$ Also let $F_i(x), i\in \{1,2,3\}$ denote the CDF of $X_i$ The CDF of $Z$, can be written as: $$F_Z(t) = F_1(t)F_2(t) + F_1(t)F_3(t) - F_1(t)F_2(t)F_3(t)$$  I know when a random variable takes on non-negative values, you can use a shortcut to calculate the expectation using the CDF. That is, $E[X] = \int_0^\infty F(X\geq x)dx$. Is there something similar I could use for a function of a random variable, or is it necessary to compute the pdf of $Z$ first to compute $\int_0^\infty g(t)f_z(t)dx$ Using while loop to ssh to multiple servers I have a file servers.txt, with list of servers:  server1.mydomain.com server2.mydomain.com server3.mydomain.com   when I read the file line by line with while and echo each line, all works as expected. All lines are printed.  $ while read HOST ; do echo $HOST ; done &lt; servers.txt server1.mydomain.com server2.mydomain.com server3.mydomain.com   However, when I want to ssh to all servers and execute a command, suddenly my while loop stops working:  $ while read HOST ; do ssh $HOST "uname -a" ; done &lt; servers.txt Linux server1 2.6.30.4-1 #1 SMP Wed Aug 12 19:55:12 EDT 2009 i686 GNU/Linux   This only connects to the first server in the list, not to all of them. I don't understand what is happening here. Can somebody please explain?  This is even stranger, since using for loop works fine:  $ for HOST in $(cat servers.txt ) ; do ssh $HOST "uname -a" ; done Linux server1 2.6.30.4-1 #1 SMP Wed Aug 12 19:55:12 EDT 2009 i686 GNU/Linux Linux server2 2.6.30.4-1 #1 SMP Wed Aug 12 19:55:12 EDT 2009 i686 GNU/Linux Linux server3 2.6.30.4-1 #1 SMP Wed Aug 12 19:55:12 EDT 2009 i686 GNU/Linux   It must be something specific to ssh, because other commands work fine, such as ping:  $ while read HOST ; do ping -c 1 $HOST ; done &lt; servers.txt I'd like to be able to introspect a C++ class for its name, contents (i.e. members and their types) etc. I'm talking native C++ here, not managed C++, which has reflection. I realise C++ supplies some limited information using RTTI. Which additional libraries (or other techniques) could supply this information? Differences between ways to express future actions I asked this question  but I haven't gotten a useful answer. Could you tell me the difference in meaning between these sentences?        Do you think you will visit them next week?     Will you visit them next week?     Are you going to visit them next week?     Are you visiting them next week?     Are you going to be visiting them next week?     Are you planning to be visiting them next week?     Do you plan to visit them next week?      Will there be any difference in the answers?        I think I will visit them next week.     I will visit them next week.     I am going to visit them next week.     I am visiting them next week.     I am going to be visiting them next week.     I am planning to be visiting them next week.     I plan to visit them next week.      Can I use "planning to" instead of "going to"?  (E.g. "Are you planning to visit them next week? — Yes, I am planning to visit them next week.") Which of these sentences show that the subject of conversation has already been discussed? For example, I talked with a friend of mine about my plans to visit somebody. So my friend is aware of my plans.  And my last question. Which of these seven sentences express that the action will certainly take place? Reset post score to 0 on migration