I learned that the symmetric difference of sets $A, B$ is given by: $$A\triangle B=(A-B)\cup (B-A).$$  But I encountered the following expression: $A\triangle B\triangle C$, and I don’t understand what it means.    How do I apply the definition of symmetric difference when three sets are involved?  In fact I am stuck on a question   There are three set $A,B,C$.  How can I find the expression that the elements in two of these sets but are not in all three sets?  My answer is $A \cup B \cup C -[(A-B\cup C)\cup (B-A\cup C)\cup (C-A\cup B)+A\cap B\cap C]$  But the answer is $A\cup B \cup (C-A)\triangle B \triangle C$ seems more simple than mine, how to get that answer?
By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. I remember that there's an elegant proof, but I forget its detail. (The first step is to show $A\oplus (A\oplus B) = B$, and maybe applying this formula, we can obtain associativity.)
Algorithm for determining random events I'm struggling with coming up with an elegant solution to generating random events in the game that I'm working on.  Say there are 4 classes of events that can happen, with varying events in those classes that may occur.  So something along the lines of   main_events = {"common event", "not so common event", "somewhat rare event", "rare event"} sub_events[0] = {"common event 1", "common event 2",..} ...   I have a somewhat kludgey solution in place that first just has a random number generated between 0 and 100, and if the number falls within a given range then a main event will be triggered.  Then I'll do another random roll to see which sub event occurs.  Is there a better solution than something along these lines?  Like I mention, it doesn't feel very elegant, and I'd like to make it easily expandable for the addition of future events. Question about mosfet gate resistor I have n mosfet gate connected to  logic and Id is about 100mA. Both 4043 and mosfet has +5v. I have plan to use  mosfet  Questions are: How large gate resistor i need between 4043 and mosfet? Logic output is sometimes put on rapidly. How fast? Motherboard hdd led controls that. Do i need to place pull down resistor from logic to -0v, between 4043 and mosfet? CUDA Driver API vs. CUDA runtime When writing CUDA applications, you can either work at the driver level or at the runtime level as illustrated on this image (The libraries are CUFFT and CUBLAS for advanced math):   (source: )    I assume the tradeoff between the two are increased performance for the low-evel API but at the cost of increased complexity of code. What are the concrete differences and are there any significant things which you cannot do with the high-level API?  I am using CUDA.net for interop with C# and it is built as a copy of the driver API. This encourages writing a lot of rather complex code in C# while the C++ equivalent would be more simple using the runtime API. Is there anything to win by doing it this way? The one benefit I can see is that it is easier to integrate intelligent error handling with the rest of the C# code. I'm trying to follow a line in a derivation for $P(Z&gt;X+Y)$ where $X,Y,Z$ are independent continuous random variables distributed uniformly on $(0,1)$.  I've already derived the pdf of $X+Y$ using the convolution theorem, but there's a line in the answer that says:  $P(Z&gt;X+Y) = \mathbb{E}[\ P(Z&gt;X+Y\ |\  X+Y )\  ]$ where $\mathbb{E}$ is the expectation.  I'm not familiar with this result. Could anyone give a pointer to a similar result if one exists?  Thanks. Multiple domains for different products? I have a website with software applications. Is it good for SEO to choose one keyword rich domain name for each of our software products or should we stick to a single domain?  From a user's perspective I think it would be easier to remember a domain that is keyword rich as the user will instantly know what this product is for.  But I have read articles that the latest trend in SEO is to stick to one domain for all of your products and invest on this single domain website.  Is that true? What do you advise? Should I register a separate domain for each of our products or should I use only one single domain?  Should I do a 301 redirect with a .htaccess to a single domain? And what about the sitemaps? Should I register all sites in Google Webmaster Tools and post a separate sitemap for each one of them? should my main site sitemap include all pages or should separate domains have their own sitemaps? alias multi commands as one alias I made an alias for youtube-dl  with some options  like that  yd="youtube-dl -cit -f 18"   so I can download it like   yd URL   I want to modify it to show only the two lines from the output using   grep -e "Downloading video #" -e ETA  I can make it in two aliases using the |   but is there a way to make it one alias How can I get apt to use a mirror close to me, or choose a faster mirror? Total number of solutions of an equation What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 &lt; x_2 &lt; \cdots &lt; x_r &lt; N$ where $N$ is some natural number and $x_1, x_2, \cdots, x_r, m$ are integers?  Also, what would be the solution in the relaxed case where $x_1, x_2, x_3, \dots, x_r$ could be equal? I'm making a Java application with an application-logic-thread and a database-access-thread. Both of them persist for the entire lifetime of the application and both need to be running at the same time (one talks to the server, one talks to the user; when the app is fully started, I need both of them to work).  However, on startup, I need to make sure that initially the app thread waits until the db thread is ready (currently determined by polling a custom method dbthread.isReady()). I wouldn't mind if app thread blocks until the db thread was ready.  Thread.join() doesn't look like a solution - the db thread only exits at app shutdown.  while (!dbthread.isReady()) {} kind of works, but the empty loop consumes a lot of processor cycles.  Any other ideas? Thanks. For a graph $G$, let $W$ be the (random) vertex occupied at the ﬁrst time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the following remarkable fact:      For the   random walk on an $n$-cycle, $W$ is uniformly distributed over all vertices different   from the starting vertex. Record Animation In BGE How to Identify Overfitting in Convolutional Neural network? I understand that dropout is used to reduce over fitting in the network. This is a generalization technique. In convolutional neural network how can I identify overfitting? One situation that I can think of is when I get training accuracy too high compared to testing or validation accuracy. In that case the model tries to overfit to the training samples and performs poorly to the testing samples. Is this the only way that indicates whether to apply dropout or should dropout be blindly added to the model hoping that it increases testing or validation accuracy Geomagnetic reversal? I have always been told that the geomagnetic field acts "as if" there were a bar magnet inside the Earth. I now this is not true, but I feel that knowing what is actually going on would help me understand the answer to the question that I'm about to ask:  Why does geomagnetic reversal happen? Is it something gradual or relatively fast?  (What is the origin of the geomagnetic field? Can we answer my questions starting from this mechanism? Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$ Does a closed form exist for      $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$   in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the sum is just the infinite series for $e$. I would guess that any closed form must involve the exponential function, but am at a loss to prove it. Is Python/ArcPy capable of delimiting strings with commas into different fields, and running a calculation based on those separate columns Using ArcGIS ArcMap 10.6: I have a text field in an attribute table that has comma separated values (example: 12,24, 64, 87,29). For this example, I will name this field VALUES. Please note that VALUES includes data collected from field work so the spacing between commas will be varied. My current workflow is as followed:  Export the ID and VALUES fields in the attribute table into a .csv file using Table to Table in ArcMap 10.6 Open the .csv in Excel and run the 'text to columns' function under the Data ribbon Separate the VALUES field into multiple columns based on commas Run the calculation: Copy Column I values and ID columns into a new .csv file Save and close the new .csv file and join the shapefile to the new .csv file based on ID Export the newly joined shapefile into a new file and delete the duplicate ID field in the new shapefile  I am wondering if there is a way to automate this Python/ArcPy so I can create a custom script tool to do this without leaving the ArcGIS environment. Is there a way to separate values in a text field based on commas into different columns, run the above calculation, and have the values from that calculation entered in a new field named VALUES_CALCULATED? So far I've proved the following:   $2n \geq \ln(2n) &gt; \ln(n+2)$   for $n \geq 3 $  The proof must be without the use of calculus but I have no idea on how to proceed. How to prove the property of the Lemoine point of a triangle? It is well-known that empty text files have zero bytes:    However, each of them contains , which according to my research, is stored in , and do use space.   Given this, it seems logical to me that it is possible to fill a disk by purely creating empty text files. Is this correct? If so, how many empty text files would I need to fill in a disk of, say, 1GB?    To do some checks, I run df -i but this apparently shows the % of inodes being used(?) rather than how much they weigh.  Filesystem             Inodes  IUsed    IFree IUse% Mounted on udev                   947470    556   946914    1% /dev tmpfs                  952593    805   951788    1% /run /dev/sda2            28786688 667980 28118708    3% / tmpfs                  952593     25   952568    1% /dev/shm tmpfs                  952593      5   952588    1% /run/lock tmpfs                  952593     16   952577    1% /sys/fs/cgroup /dev/sda1                   0      0        0     - /boot/efi tmpfs                  952593     25   952568    1% /run/user/1000 /home/lucho/.Private 28786688 667980 28118708    3% /home/lucho Annuity notation in LaTeX Quoting backslashes in Python string literals I have a string that contains both double-quotes and backslashes that I want to set to a variable in Python. However, whenever I try to set it, the quotes or slashes are either removed or escaped. Here's an example:  &gt;&gt;&gt; foo = 'baz "\"' &gt;&gt;&gt; foo 'baz ""'   So instead of baz "\" like I want I'm getting baz "". If I then try to escape the backslash, it doesn't help either:  &gt;&gt;&gt; foo = 'baz "\\"' &gt;&gt;&gt; foo 'baz "\\"'   Which now matches what I put in but wasn't what I originally wanted. How do you get around this problem?