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Let $G$ be a group, $H\subseteq G$ a subgroup and $a\in G$ an element of the group. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$?   If $H$ has finite index or finite order, this is not possible. If my data is overdispersed when binomial but not when Poisson does that alone mean I should use Poisson GLM? Top is Poisson, bottom is Binomial Source of variation is different people observed over a week long period. 118 people to be exact.  want to make predictions about the proportion of days/week one exercises if they wake up at a given time.  &gt; summary(FMR7)  Call: glm(formula = No.Days.Ex ~ WaketimeStand, family = poisson, data = PSD)  Deviance Residuals:       Min        1Q    Median        3Q       Max   -2.99043  -0.64677  -0.03492   0.67393   1.51943    Coefficients:               Estimate Std. Error z value Pr(&gt;|z|)     (Intercept)    1.36991    0.04689  29.218   &lt;2e-16 *** WaketimeStand -0.08309    0.04806  -1.729   0.0839 .   --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for poisson family taken to be 1)      Null deviance: 124.55  on 115  degrees of freedom Residual deviance: 121.51  on 114  degrees of freedom AIC: 479.71  Number of Fisher Scoring iterations: 4  &gt; summary(FMR6)  Call: glm(formula = cbind(No.Days.Ex, No.Days.NotEx) ~ WaketimeStand,      family = quasibinomial(link = logit), data = PSD)  Deviance Residuals:      Min       1Q   Median       3Q      Max   -3.7483  -0.9585  -0.0625   1.0428   2.9739    Coefficients:               Estimate Std. Error t value Pr(&gt;|t|)   (Intercept)     0.2595     0.1009   2.571   0.0114 * WaketimeStand  -0.1869     0.1020  -1.832   0.0695 . --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for quasibinomial family taken to be 2.015388)      Null deviance: 288.42  on 115  degrees of freedom Residual deviance: 281.57  on 114  degrees of freedom AIC: NA  Number of Fisher Scoring iterations: 4  Quite clear that there is much more overdispersion on the quasibinomial than the poisson. The response variable is no.days exercise per week and so my original thought was to analyse with binomial GLM as it is the proportion of days in a week someone exercises. However, when modelled with a Poisson GLM it is much less overdispersed. Is this a sign I am using the wrong distribution. doGet and doPost in Servlets Adjusting space between array rows and columns \documentclass[english]{article} \usepackage[T1]{fontenc} \usepackage[latin9]{inputenc} \usepackage{babel} \begin{document} % ... \[ \begin{array}{ccccc} x(1) &amp; = &amp; \frac{x(0)}{1} &amp; = &amp; x(0)\\ x(2) &amp; = &amp; \frac{x(1)}{2} &amp; = &amp; \frac{x(0)}{2}\\ x(3) &amp; = &amp; \frac{x(2)}{3} &amp; = &amp; \frac{x(0)}{2.3}\\ x(4) &amp; = &amp; \frac{x(3)}{4} &amp; = &amp; \frac{x(0)}{4!}\\  &amp;  &amp;  &amp;  &amp; \vdots\\ x(n) &amp; = &amp; \frac{x(n-1)}{n} &amp; = &amp; \frac{x(0)}{n!} \end{array} \] % ... \end{document}   Because of the fractions in the array cells, array elements are vertically too close to each other, they are literally touching to each other. Also the horizontal spacing is too much as you see in the image (screen shot from the output PDF file).  How do I make these cell-spacings look better? Inaccurate shading on object Installing Autodesk Maya on Ubuntu is possible, but can be a bit challenging.  Assuming that one already has the required specs and a solid graphics card driver installed, how can it be done? Dynamic LINQ OrderBy on IEnumerable / IQueryable I found an example in the  for Dynamic LINQ that allows you to use a sql-like string (e.g. OrderBy("Name, Age DESC")) for ordering. Unfortunately, the method included only works on IQueryable&lt;T&gt;. Is there any way to get this functionality on IEnumerable&lt;T&gt;? Some things are easier to implement just by hand (code), but some are easier through WF. It looks like WF can be used to create (almost) any kind of algorithm. So (theoretically) I can do all my logic in WF, but it's probably a bad idea to do it for all projects.  In what situations is it a good idea to use WF and when will it make things harder then they have to be? What are pros and cons/cost of WF vs. coding by hand? Was just wondering if there is any solid data available as to how many people are using ubuntu. I was also interested in the attrition rate with ubuntu users, etc. Add a non-breaking space to "reopen (1)" and its ilk Import error when opening empty notebook You know, some of us read right to left...  Certain sites like  and  have an obvious need for this, as Arabic and Hebrew are Right To Left (RTL) languages. Many answers on those sites cite the Torah and the Quran respectively. Directly quoting these sources properly requires RTL text. Precise back-reference target with hyperref and backref If $A,B,C$ are the angle of a triangle, then show that $\sin A+\sin B-\cos C\le \dfrac32$ I tried substituting $C=180^\circ-(A+B)$ and got stuck. I also tried using the formula $\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$ without any progress. Please help! Constrains relationship I have the following design. Each user has it's own contacts, it's own hosts and assign those to a domain which belongs only to him.   I want to insure that the host you assign to the domain belongs to the user that domain is assigned to.  Is it something that I have to insure in my application logic?  If I set domains.user_id foreign key to reference host.user_id and user_id.id does it eliminate my issue?    By the way can anyone suggest me a better design for my schema? Solving the recurrence $T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$ A former student of mine was TA-ing an algorithms class last quarter and asked students to solve this famous recurrence relation:  $$T(n) = \sqrt n T(\sqrt{n}) + n$$  There are  and this question has already been asked here.  I was talking to my student about this problem and mentioned that it's quite hard to solve and is pretty dependent on the particular choices being made. As an example, I suggested this variant of a recurrence relation that was less simple to solve:  $$T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$$  The TA and I tried solving this recurrence for about an hour and a half without making any progress. Here are a few things we tried:   My approach to solving the initial recurrence relation was to draw out a recursion tree and notice that each level of the tree contributes $n$ to the total and that there are $O(\log \log n)$ layers, so the recurrence solves to $O(n \log \log n)$. When I tried doing this here, I noticed that the work per layer was no longer constant; instead, the top layer sums to $\sqrt{n}$, the second layer to $n^{3/4}$, the third to $n^{7/8}$, etc. We got stuck working with the summation $n^{1/2} + n^{3/4} + n^{7/8} + ...$. We tried using the iteration method to unroll the recurrence. With the original recurrence relation, this works out nicely; here, we got stuck at the same summation given above.   I'm completely stuck trying to figure out how to solve this recurrence relation. There's nothing riding on it per se - I don't need to solve it for any particular reason - but the fact that we arrived at it by a straightforward modification of a common algorithms problem set question makes it all the more enticing.  Any idea how to solve this recurrence?  Thanks! Setting the hostname: FQDN or short name? I've noticed that the "preferred" method of setting the system hostname is fundamentally different between Red Hat/CentOS and Debian/Ubuntu systems.   and the  say the hostname should be the FQDN:     HOSTNAME=&lt;value&gt;, where &lt;value&gt; should be the Fully Qualified Domain   Name (FQDN), such as hostname.example.com, but can be whatever   hostname is necessary.   The  is slightly more ambiguous:     Setup prompts you to supply a host name for this computer, either as a   fully-qualified domain name (FQDN) in the format hostname.domainname   or as a short host name in the format hostname.    says the hostname should not use the FQDN:     3.5.5. The hostname      The kernel maintains the system hostname. The init script in runlevel   S which is symlinked to "/etc/init.d/hostname.sh" sets the system   hostname at boot time (using the hostname command) to the name stored   in "/etc/hostname". This file should contain only the system hostname,   not a fully qualified domain name.   I haven't seen any specific recommendations from IBM about which to use, but  seems to have a preference.  My questions:   In a heterogeneous environment, is it better to use the vendor recommendation, or choose one and be consistent across all hosts? What software have you encountered which is sensitive to whether the hostname is set to the FQDN or short name? How do I install drivers for the RTL8821CE Network Controller on Ubuntu 20.04? My Laptop (HP Laptop 14-ck1xxx) came with the RTL8821CE Network Controller. When I run lspci | grep Network I get: 03:00.0 Network controller: Realtek Semiconductor Co., Ltd. RTL8821CE 802.11ac PCIe Wireless Network Adapter  I‘m unable to get the Wi-Fi to work. What I did so far:  Download the repo at . Unzip the file and rename the folder from rtl8821ce-master to rtl8821ce. Run the following commands in the terminal:  cd Downloads/rtl8821ce/ sudo apt install bc module-assistant build-essential dkms sudo m-a prepare sudo ./dkms-install.sh    Everything seemed to install and execute successfully. Then, I rebooted my laptop and made sure secure boot is disabled. However, Ubuntu still displays: No Wi-Fi Adapter Found - make sure you have a Wi-Fi adapter plugged in and turned on  The command uname -r returns: 5.4.0-7634-generic  The command sudo lshw -C network returns: *-network                         description: Ethernet interface        product: RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:02:00.0        logical name: eno1        version: 15        serial: c4:65:16:08:52:bf        size: 1Gbit/s        capacity: 1Gbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 duplex=full firmware=rtl8168h-2_0.0.2 02/26/15 ip=192.168.178.37 latency=0 link=yes multicast=yes port=MII speed=1Gbit/s        resources: irq:16 ioport:4000(size=256) memory:a1204000-a1204fff memory:a1200000-a1203fff   *-network UNCLAIMED        description: Network controller        product: RTL8821CE 802.11ac PCIe Wireless Network Adapter        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        version: 00        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress cap_list        configuration: latency=0        resources: ioport:3000(size=256) memory:a1100000-a110ffff  What am I doing wrong? How can this be solved? Suppose that $(U,V)$ has the following joint pdf: $$f_{U,V}(u,v)=\exp(-\theta u-\theta^{-1}v)$$ , where $u\geq0$, $v\geq0$, $\theta&gt;0$. Define $X=UV$ and $Y=U/V$. Find the marginal pdf of $X$ and $Y$.  With transformation, I got $U=\sqrt{XY}$ and $V=\sqrt{X/Y}$ and the absolute value of Jacob is $1/2Y$. Then, the joint pdf of $(X,Y)$ is  $$f_{X,Y}(x,y)=\exp(-\theta \sqrt{XY}-\theta^{-1}\sqrt{X/Y})1/2Y, X,Y&gt;0.$$  But I am not sure how to get the marginal pdf?  Thank you! Get the exact time for a remote server