Why do we define the convolution?
Meaning of convolution?
PXE Boot Server Installation Steps in Ubuntu Server VM Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$ What happens when you plug two sides of a cable to a single networking device? What is likely to happen when you plug two ends of a network cable to a single switch/router? Will this create problems on the network, or just be ignored? How To: Execute command line in C#, get STD OUT results Let A and B be sets. Show that A is a subset of B if and only if for any set C, one has A union C is a subset of B union C. Can you verify my proof if it is right?  Let $A$ and $B$ be sets.  (a) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ union $C$ is a subset of $B$ union $C$. (b) Show that $A$ is a subset of $B$ if and only if for any set $C$, one has $A$ intersect $C$ is a subset of $B$ intersect $C$.  Edited Solution :  (a)  (=>) Given A is a subset of B, this implies A U B = B. Given a set C, A U B U C = B U C. Since A is a subset of B, this implies A U C is a subset of B U C.      Solution 1 :(&lt;=) Given A U C is a subset of B U C , For A U C , x is a member of A U C . For B U C, x is a member of B U C. Suppose x is a member of A but not a member of B, meaning A is not a subset of B. Therefore , A U C is not a subset of B U C. Hence, A must be a subset of B.      Solution 2 :(&lt;=) Given A U C is a subset of B U C, For A U C, x is a member of A U C and for B U C, x is a member of B U C. Suppose C is a disjoint set. Since A U C is a subset of B U C, it follows that A is a subset of B.  (b) (=>) Since A is a subset of B , it follows that A U B = B. [A intersect C] is a subset of A and  [B intersect C ] is a subset of B by definition of intersection. Let x be member of A and member of C. Since [ A intersect C ] is a subset of A, x must be in A. Let x be a member of B and member of C. Since [ B intersect C ] is a subset of B, x must be in B. Therefore, A intersect C is a subset of B intersect C.  (&lt;=) Since A intersect C is a subset of B intersect C, suppose we let Set C be a subset of B and a subset of A as well. C is a subset of A implies A intersect C is equals to C. C is a subset of B implies B intersect C is equals to C as well. Since A intersect C is equals to B intersect C, A is a subset of B. "cron" processes spawning and piling up, but no crontab EDIT: I solved this, so see below for my own answer. While the bounty asks for a fix to the problem, I'd like to award it to someone who can explain why a cron process that ran for 6 years over multiple MacOS versions suddenly became a problem in Catalina.  First note, this is not a duplicate of  because I don't have active crontabs on my computer. Cron processes spawn, but they never finish, so they just keep piling up. I'm running MacOS Catalina 10.15.1  Cron jobs are spawning every couple of minutes and never dying. The oldest cron process always has a parent process "launchd", and I've searched for populated crontabs and can't find any. If I let this go on long enough, eventually I run into a scenario where I have hundreds of running cron processes and everything on my computer grinds to a halt, and I can't even open new apps (I get cryptic messages like ".app cannot be opened" when I try to open an app). My only solution at this point has been to keep activity monitor open and periodically killing off all active cron processes.  I've been able to create a new user account that seems to be immune from this issue. However, as soon as I logged into my apple ID on that account to set up iCloud drive, the problem started with that account, too. I logged out of my Apple ID and attempted uninstall iCloud drive, but the problem persisted. Clean installs of Catalina don't clear up the problem, I've reset SMC and PRAM multiple times, and repairing permissions (via this ). I've also ensured that cron has full disk access (via this ). I'm at a dead end with trying to troubleshoot this problem.  I'm not sure it's related, but my "system.log" file seems to get spammed the following few lines:  &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: Entered:_AMMuxedDeviceDisconnected, mux-device:63 &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: Entered:__thr_AMMuxedDeviceDisconnected, mux-device:63 &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: tid:9a27 - Mux ID not found in mapping dictionary   ANY help is appreciated. Is the Eel immortal? How to use iSight webcam without activating the green light? Recently a the Dutch government advised to put a tape on your webcam to prevent hackers from taking pictures and video using your device ().  The build in iSight camera in Apple laptops have a green light next to the camera, this light turns green when you use the camera.  Is it possible to use the camera without the activation of the green light? Personally I think the green light turns on when the sensor requires power and this is hardwired in the hardware. Am I wrong?  I have found similar questions on stackexchange, but I found non that answer this specific question. Show discontinuity of $\frac{xy}{x^2+y^2}$ How to get current CPU and RAM usage in Python? How do I write a cpp macro which expands to include newlines? Why does the following code does'nt work when i am using PDFLaTeX or just LaTeX ? I tried it in different editors as well.  This is the code   $\{f^'_n\}$   Surprisingly it works on Math.SE site Cannot refer to a non-final variable inside an inner class defined in a different method I was trying to experiment with docker, Ubuntu 14.04, 64-bit. I managed to run an image, and I gave it the path on one of my removable drives, /media/Drive1.  Now, whenever I reboot, I see there is a /media/Drive1 with only a docker directory inside it, and the actual drive which should be Drive1 ends up mounted on /media/Drive11, and this happens every reboot - even when I unmount these drives, and do a sudo rm -rfv /media/Drive1 before rebooting!  So something probably creates /media/Drive1 before it is fully mounted, and I guess that something is docker, because when I boot, I have:  $ ps axf | grep docker  2085 ?        Ssl    0:00 /usr/bin/docker -d   So, how can I disable the docker autostart, so it doesn't interfere with the mounting of external USB drives at boot? How do I find the package that provides a file? Complete Statistic: Uniform distribution Take a random sample $X_1, X_2,\ldots X_n$ from the distribution $f(x;\theta)=1/\theta$ for $0\le x\le  \theta$.  I need to show that $Y=\max(X_1,X_2,...,X_n)$ is complete.  Now, I know I should multiply the sample distribution of $Y$ and multiply it with a function of $Y$, then integrate over the range of $\theta$ and equate them to zero. But how do I get the sampling distribution of $Y$? How do some Ws on transcripts affect PhD admissions in mathematics? How can you depend on test code from another module in Maven?    Example, I have 2 modules:   Base Main   I would like a test case in Main to extend a base test class in Base.  Is this possible?  Update: Found an , which involves creating a test jar. Order of "not" with infinitive Find a value anywhere in a database