Integrate $\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$
how to solve $\int_{-\infty}^\infty e^{-x^2-x{\tau}} \cdot x\ dx$?
Strange OutOfMemory issue while loading an image to a Bitmap object How do I label equations without the chapter number? In a finite cyclic group, the generator has the order of the group Let $G$ be a finite cyclic group of order $n$ generated by an element $g$. I want to show that the order of $g$ is exactly the order of $G$.   My try:   We can write $G=\{g^i\;|\;i\in \mathbb Z\}$. Since $G$ is finite, we know that there will be identifications $g^i=g^j$ that will make $G$ finite. By the multiplication law on $G$ we must have $g^0=1$. Also if $i&lt;j$ and if $g^i=g^j$ then $g^{j-i}=1$. I can see all these but not see why $g^n=1$ and not $g^k=1$ for all $0&lt;k&lt;n$? Thank your for your help!! Which $n$ can be expressed as $x^2 − y^2$? Which $n$ can be expressed as $x^2 − y^2$?  Attempt: Any number which is odd or or which is a multiple of 4. From the user's perspective when switching on the phone, I see two boot animations in my  running CM10. The first animation is from LG and the second animation is from CM10. Then it settles in with a locked screen ready for my input. What happens behind the scene (screen) during a boot process when an Android device is switched on? It will be helpful if the answers can help a non geek to understand various process and jargon (like, fastboot, bootloader, recovery, if applicable, etc.) regarding boot sequence. What is the optimal path between $2$ fixed points around an invisible obstructing wall? Every day you walk from point A to point B, which are $3$ miles apart. There is a $50$% chance each walk that there is an invisible wall somewhere strictly between the two points (never at A or B). The wall extends $1$ mile in each direction perpendicular to the line segment (direct path) between A and B, and its position can be at any random location between the two points. That is, it can be any distance between A and B such as $0.1$ miles away from A, $1.5$ miles away, $2.9$ miles away.... You don't know if the wall is present or where it is until you walk into it.  You must walk around the wall if present as there is no other way to circumvent it.  Assume the wall is negligible thickness (like a force field), all ground is perfectly flat, and the y coordinates at both A and B are $0$ (although I don't think the optimal path will change much if they weren't).  What strategy minimizes the average expected walk distance between A and B?  How do we know for certain this strategy yields the shortest average walking distance?  To get the $100$ bounty points, I would like a convincing argument from you as to why you feel your solution is optimal and cannot be beaten.  For those of you using computer simulation, anything reasonably close to optimal is a candidate for the bounty.   Can anyone beat the optimal solutions submitted so far?  There are still a few days to try.  Even a slight beat (if not due to roundoff error or some other error) would help prove previous solutions are non-optimal.  You can just use the table of $31$ points (x-y coordinates) and compute that path length and then if you can find better then I will accept that answer.  I think by Sunday night I may have to award the bounty otherwise it may get auto-awarded. Does Python support short-circuiting? I noticed in this thread the OP successfully managed to connect a 30" cinema display to the new Macbook Pro USB-C, with full resolution:    I have the official Apple dual DVI adapter, going to a displayport > USB-C adapter like yours, however the screen does not fire up.  I then tried a multi adapter instead of the single displayport > USB-C, which worked but only at 1280 resolution:    Would the official Apple Dual DVI adapter not work? pylab.ion() in python 2, matplotlib 1.1.1 and updating of the plot while the program runs My significant other and I finish our master degrees at the same time and thus we are applying for PhD programmes at the same time. One major factor of accepting a potential offer would be if we could study at the same university, so I feel that mentioning a two-body problem would be advantageous for both, us and the admission committee. But where do I mention this? Is a line in the statement of purpose suitable here?     If you admit me, please also consider admitting my significant other Jane Roe.   Some additional information: We both do research in computer science. While I do research in natural language processing my partner does research in computer vision. Spider-Man is the one franchise that is on course to a third reboot in a span of 15 years.   My question is what was the reason for these numerous reboots of Spider-Man which pretty much follow the same origin story for the Marvel superhero.  Has the comic creator or studio ever spoken out about the reasoning behind the same? 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? I have two limit problems which are quite similar, so I've put them both into this post.   $((4^{10}+2^{n})^{\frac{1}{n}})$ $((3n^{2}+n)^{\frac{1}{n}})$   Attempt:  For 1, I'm not sure if this is permissible. I know that $\lim_{n \to \infty}(x^{n}+y^{n})^{\frac{1}{n}} = \max\{x,y\}$. Obviously $4^{10}$ is a constant so it's not quite in the same form, but I believe it is still okay to use this result, and so the limit is $2$.  For 2, I've got $((3n^{2}+n)^{\frac{1}{n}}) = (n(3n+1)^{\frac{1}{n}})$, but then I'm not too sure how to proceed.   I also thought of taking the log and proceeding that way, however in the notes I'm learning from they haven't covered l'hopital yet, so I'm trying to not use it.  Thanks. Why does the variance of the Random walk increase? How common is "thrice"? "What day is it today?" vs. "What day is today?" How can I find just the *.mp3 files with a recursive command like ls -R *.mp3 in a directory that contains several sub-directories and at the end copy these files in a directory that I choose.  Thanks for your support. Regression without intercept: deriving $\hat{\beta}_1$ in least squares (no matrices) What programming languages can I use in Blender? What version of Python does Blender use and are other languages available? Also, does Python work the same in Blender as when you write console applications or when working with Pygame or are there any differences? If so, what are the main differences? Why does space expansion not expand matter?