$$\lim\limits_{n \to \infty} g(t)=L $$  $g$ is monotonic strictly increasing, continuous in $ [0,\infty)$ and positive function.  I need to find :  $$\lim\limits_{n \to \infty} \int^2_1 g (nx) \,\mathrm{d}x  $$    I have started with:   $|g(t)-L|&lt;\epsilon\&gt;$  $L-\epsilon\&gt;$$=&lt;$$g(t)$$&lt;=$$L+\epsilon\&gt;$  $\int^2_1 (L-\epsilon\&gt;) \,\mathrm{d}x$$=&lt;$$\int^2_1 g (nx) \,\mathrm{d}x$$&lt;=$$\int^2_1 (L+\epsilon\&gt;) \,\mathrm{d}x$  I'm not sure if I'm in the right direction ...
Given function $g$ continuous in line $[0,\infty)$, positive and strongly increasing that upholds $\lim_{t\to\infty}  g(t) = L$. Calculate the limit:  $$ \lim_{n\to\infty} \int\limits_1^2 g(nx) \, dx. $$  At the moment what I think that I should do is swap the $g(nx)$ with $L$ as it approaches infinity but from there on I am just not sure whats my next step is.
We were taught that an independent clause must have a complete thought/idea.     This is the only mammal that can fly.     How can "This is the only mammal" be treated as an independent clause?            It was a grammar lesson and there was no context there. In this situation how can I decide that it is an independent clause? Though for sure, there should be an independent clause in a correct sentence, and the clause starting with 'that' is not independent. But by this way, I can't fit it with the rule that every independent clause must have a complete thought. A simple sentence must have an independent clause. If the sentence has no independent clause how we can call it a sentence? Can even two dependent clauses form a sentence? Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible? I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that discusses this. Maybe it is too trivial, but I am not able to figure it out myself.   So I would like to ask you for help. Where can I find a proof for this?  Also, I am interested in the mathematical background behind plane tiling (especially periodic), If you can suggest papers on the topic it would really help. Prove that If $m'$ is a common multiple of $s$ and $t$, then $m | m'$. Here $m$ is the LCM of $s$ and $t$.    Although the statement is intuitively clear to me I don't know how to prove. What system monitoring tools are available? I am looking for system monitoring tools which are GUI and CLI or web-based which include basic functions such as:   CPU Usage Ram Usage Swap Usage Disk Usage ( Space / I/O ) Heat Monitoring   I know there are many tools I can use, but I am looking for a single tool that has these basic functions. Randomly populating procedural shape (or Image Texture) in Cycles TL;DR  How to take a shape or Image Texture and multiply it with random Location, Rotation and Scale along object:    Background  I have created my first procedural shape on my own from this function: $(x^2 + y^2 - 1)^3 - x^2y^3 = 0$   (big yay for math idiot)   Here is the result:    And I want to do something useful with it.  Research  I've tried to find something similar over the web (this is trending right now) and I'm surprised that this question wasn't asked before or I'm looking with wrong terms. I've tried random pattern, duplicating texture, populating image, random tiles, custom tiles and such. There is a lot about tiling textures but each and every Q&amp;A I've stumbled upon are about non overlapping ones. For example:            Tryout:  The only way I can think of this is to duplicate shape as a group and manually change values in Mapping Nodes:    As you can see this is... Well. Bad.  I've tried messing with Modulo, mixing Noise between vectors, using custom node groups for LocRotScale (thanks Gandalf) but results was none, thus nothing even to show.  Question  I want to populate this shape (or any other) along the object with random Location, Rotation and Scale - with control over its number/density and maybe variation strength.  Edit: I'm not looking for alternative methods of creating this setup such as Particles, Image Textures generated in 3rd party software, OSL shaders (CPU only) and such. My intention is to use this in procedural texture generation in Cycles with GPU support.  Blend file  Nothing really needed for this question but here - have this heart from me :) Can i install ubuntu on meizu mx4 android edition like ubuntu edition with iso recovery in this page "  " Integrals of Pullbacks What is the right ffmpeg output format for a v4l2-loopback device? Following the guide in  I did:  $ sudo modprobe v4l2loopback $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f v4l2 /dev/video0   The device /dev/video0 shows up as expected but the ffmpeg command fails with following output:  $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f v4l2 /dev/video0 ffmpeg version 4.0 Copyright (c) 2000-2018 the FFmpeg developers ... [x11grab @ 0x559767794cc0] Stream #0: not enough frames to estimate rate; consider increasing probesize .... [v4l2 @ 0x55976779e6c0] ioctl(VIDIOC_G_FMT): Invalid argument Could not write header for output file #0 (incorrect codec parameters ?): Invalid argument Error initializing output stream 0:0 --  Conversion failed!   I already tried to follow the suggestions below the answer to no avail. I noticed that v4l2 is listed as demux only in ffmpeg -formats. Does that mean I can't use it for output to /dev/video0? If so, what should I use instead? I tried h264 but that crashes after ~1000 frames with following error message:  $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f h264 /dev/video0 ffmpeg version 4.0 Copyright (c) 2000-2018 the FFmpeg developers ... [x11grab @ 0x55da61e29cc0] Stream #0: not enough frames to estimate rate; consider increasing probesize ... File '/dev/video0' already exists. Overwrite ? [y/N] y .... Error writing trailer of /dev/video0: Invalid argument .... Conversion failed! Sum of independent Gamma distributions is a Gamma distribution If $X\sim \Gamma(a_1,b)$ and $Y \sim \Gamma(a_2,b)$, I need to prove $X+Y\sim\Gamma(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and just trying to multiply the gamma function but stuck ? I am trying to calculate the final velocities of two equal mass 2-dimensional circles after an elastic collision. I have tried to figure it out using formulas I know from high school physics, but nothing seems to work. My known variables are:   initial x and y velocities for each circle x and y distances between the centres of the circles at the time of the collision   The variables I would like to solve for are:   final x and y velocities for each circle   Here are diagrams showing all the variables: Socket Tester all indicators light up Does the scientific community consider the Loschmidt paradox resolved? If so what is the resolution? Is there a way to mix LIKE operator with IN ? Something Like  SELECT Id FROM Account WHERE Name LIKE IN:nameList   Where nameList is list of String? Is this is possible ? (I can make a dynamic query by doing OR with all the elements, but was looking for a more elegant way to do this) Is there an easy way to get to www.stackexchange.com from any of the SE sites? One thing I've noticed about the relatively new sitewide layout is that I can't find a quick way to get back to www.stackexchange.com without finding StackExchange in the sites list. As a result, I am typing stackexchange in my toolbar a whole lot more often now. In particular, I tend to enjoy browsing the hot questions list and my own personal feed quite often so visit these pages on a daily basis. Is there an easy link to them that I'm missing?  (On the android app, there is a feed in the menu that I really appreciate, something like that on the desktop would be ideal!) This is a  about Active Directory domain naming.   After experimenting with Windows domains and domain controllers in a virtual environment, I've realized that having an active directory domain named identically to a DNS domain is bad idea (Meaning that having example.com as an Active Directory name is no good when we have the example.com domain name registered for use as our website).   , but I'm still not sure about what other rules there are around naming Active Directory domains.  Are there any best practices on what an Active Directory name should or shouldn't be? When performing machine learning, the performance of the machine learning method is dependent on the original function $f$ that we are trying to learn (let's forget for a moment the non-deterministic error and focus on the properties of $f$).  I have a conceptual question. What would make a function easy to learn? For instance, a linear function can be easily learnt by linear methods such as linear regression and SVM. But in the general case, what would make a function easier to learn for decision trees or a neural network?   I believe feature engineering helps us move to a feature space in which the function we want to approximate is easier to represent. But what properties exactly are we trying to achieve for this function? Union of a finite set and a countably infinite set is countably infinite Ok, here is the problem statement:     Prove that if $S$ is any finite set of real numbers, then the union of $S$ and the integers is countably infinite.   This seems pretty obvious to me, knowing that 2 countable sets are countable.  But is there some step by step way to prove this?  Like do I need to prove bijectivity or something?  Thanks! Why do I get TypeError: can't multiply sequence by non-int of type 'float'? I am typing to get a sale amount (by input) to be multiplied by a defined sales tax (0.08) and then have it print the total amount (sales tax times sale amount).  I run into this error. Anyone know what could be wrong or have any suggestions?   salesAmount = raw_input (["Insert sale amount here \n"]) ['Insert sale amount here \n']20.99 &gt;&gt;&gt; salesTax = 0.08 &gt;&gt;&gt; totalAmount = salesAmount * salesTax  Traceback (most recent call last):   File "&lt;pyshell#57&gt;", line 1, in &lt;module&gt;     totalAmount = salesAmount * salesTax TypeError: can't multiply sequence by non-int of type 'float' I am running on a dual core AMD E450 CPU. When I tried to run a 64-bit OS that requires hardware virtualization using VirtualBox, it showed me an error message that "AMD-V is not enabled".    My AMD processor is supposed provide AMD-V support, but I can find no option for enabling AMD-V in BIOS.    How can I solve this problem? How can I enable AMD-V for my CPU?  Thanks in advance  lscpu :-  Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit Byte Order:            Little Endian CPU(s):                2 On-line CPU(s) list:   0,1 Thread(s) per core:    1 Core(s) per socket:    2 Socket(s):             1 NUMA node(s):          1 Vendor ID:             AuthenticAMD CPU family:            20 Model:                 2 Stepping:              0 CPU MHz:               1650.000 BogoMIPS:              3291.72 Virtualization:        AMD-V L1d cache:             32K L1i cache:             32K L2 cache:              512K NUMA node0 CPU(s):     0,1   EDITED:-  Error message from VirtualBox:-  Failed to open a session for the virtual machine XXX.  AMD-V is disabled in the BIOS. (VERR_SVM_DISABLED).  Result Code: NS_ERROR_FAILURE (0x80004005) Component: Console Interface: IConsole {1968b7d3-e3bf-4ceb-99e0-cb7c913317bb} How do I determine if an array is initialized in VB6? Passing an undimensioned array to the VB6's Ubound function will cause an error, so I want to check if it has been dimensioned yet before attempting to check its upper bound.  How do I do this?