Plants have reclaimed Earth, women are the dominant sex
What book features people living in a tree who move up it in a bubble?
I'm trying to prove that every finite Boolean ring, $R$, is isomorphic to a finite number of copies of $\mathbb{Z} / 2 \mathbb{Z}$:  \begin{align} R \cong \mathbb{Z} / 2 \mathbb{Z} \; \times \cdots \; \times \; \mathbb{Z} / 2 \mathbb{Z} \end{align}  I'm aware that there are answers to this question on this website. However, almost all of the answers seem to use facts about modules and vector spaces defined over fields. This question is posed in section 7.6 in Dummit and Foote's Abstract Algebra. This section is part of the chapters which introduces the basics of ring theory (rings, homomorphisms, ideals, Chinese Remainder Theorem etc.).  I'm not even sure how to being answering this question. I know that if $R$ were an integral domain, then we would have that $R \cong \mathbb{Z} / 2 \mathbb{Z} $.   Please only give hints, considering that that the statement has to be proved using the material only included up till chapters 9 in the textbook. A closed ball in a metric space is a closed set Why does no one notice the Joker's presence as a policeman during the mayor's speech? How to set tick label precision How to filter a dictionary according to an arbitrary condition function? Finding a distribution from the sum of two random variables in a joint trinomial distribution Let X and Y have a trinomial distribution with join probability function:  $ f(x,y) = \frac{n!}{x!y!(n-x-y)!}p^x q^y (1-p-q)^{n-x-y}; x=0,1,...n, y=0,1,...n$  Let $T=X+Y$  I can't seem to figure out where to go exactly. Here's where I am at:  $\sum_{x=0}^{n} \frac{n!}{x!y!(n-x-y)!}p^x q^y (1-p-q)^{n-x-y}\mbox{ substitute in Y=T-X}$ $\sum_{x=0}^{n} \frac{n!}{x!(t-x)!(n-t)!}p^x q^{t-x} (1-p-q)^{n-t}$   I think I could I figure out the algebra.. if I could just see what to do with the summation. I know I have to involve a t somewhere, but I'm not sure. Maybe somebody could just show me how to set up my summation correctly with the substitution? Why use apparently meaningless do-while and if-else statements in macros? In many C/C++ macros I'm seeing the code of the macro wrapped in what seems like a meaningless do while loop.  Here are examples.  #define FOO(X) do { f(X); g(X); } while (0) #define FOO(X) if (1) { f(X); g(X); } else   I can't see what the do while is doing.  Why not just write this without it?  #define FOO(X) f(X); g(X) Ok, I just bought a brand new original iPad. Sounds impossible, right? Well, we just opened the box, and it is not even charging. There is no indication that this iPad even works even though it is technically "new." Is this something that Apple will need to fix or do I just need to let it sit on the charger overnight and hope it wakes up from a 2 year sleep in its box? ASP.NET: Is there a way that I can simply pass a model into a view and render it to a string? Give a deduction of $\forall y \exists x (x=y)$  My thinking is I can prove $\exists x(x=y)$ then use generalization theorem， which is equivalent to prove $\neg \forall x \neg (x=y)$ but I got stuck proving this... thanks in advance  Sorry that I did not specify what deduction system that I want to use here, it is the deduction sequence  where ai is either Axiom or original hypothesis or obtained from MP Using Mosfet with seperate Logic Converter and Arduino I am trying to use an IRLB3034 with an Arduino even though it is not a logic level MOSFET. I am planning on using an LM2937 12v regulator like the schematic below:    The connection going off the page goes to an Arduino digital pin pulled down with a 10k resistor. I assumed this would work but wanted to check. As a side note, if anyone could recommend a similar MOSFET that could handle 200a and 40v (in TH) that would be great. (Only if such thing excists or is readily available)  EDIT: I have not made it clear what I am planning on using this for so I will now. It is to drive an airsoft motor and I want to use the LM2937 because I didn't think I could turn the mosfet on and off with the arduinos 5v - because the IRLB3034 isn't a logic level mosfet - so that is the reason I added the 12v regulator.  EDIT (Sorry for the previous incorrect name)  EDIT: Unfortunately, I have made a mistake, the current is not 200 amps, it is infact 30-40 amps. I don't know how I made this mistake but I think it will be helpful for people trying to follow along. I am currently trying to use busbars/6awg copper wire to carry the high-ish current around. (Only 40 amps now though)  EDIT: This question is different because it is asking how to convert 5v to 12v to run a mosfet rather then how to use a mosfet with an arduino in the best way like the other post. The accepted answer on this post also answers the other question though. Windows 10 Start Menu can't have multiple shortcuts with the same target Problem  If you create two shortcuts with the same target then only one of them is displayed in the Start Menu. This is true for:   search results within the Start menu manually browsing program folders within the Start menu pinned shortcuts within the Start menu   This is important because:   the shortcuts might have different settings defined in the Properties dialog, such as different compatibility settings, or be set to start in different directories. some programs install a shortcut to a command prompt which is set to open in the program's installation folder. This enables easy access to the program from the command line without having to modify %PATH%, but it means that users who are unaware of this "lose access" to CMD. Programs that do this:           Things I've tried   Renaming shortcuts   Only the shortcut that comes first alphabetically is displayed  Using folders doesn't help   Only the shortcut in the folder that comes first alphabetically is displayed  Pinning shortcuts doesn't help   the pinned shortcut updates to point to the displayed shortcut      Steps to reproduce  Fully up-to-date Windows 10 Home.   Start → search for "Command Prompt" (or "cmd"). (Optional) Right-click on Command Prompt → Pin to Start.   This makes Command Prompt available when you open the Start menu without searching for anything.  Right-click on Command Prompt → Open file location   This opens a File Explorer window in %APPDATA%\Microsoft\Windows\Start Menu\Programs\System Tools  Copy the Command Prompt shortcut and rename the copy "AAA Command Prompt"   This ensures it appears before Command Prompt when files are listed alphabetically.  (Optional) Right-click on the new AAA Command Prompt shortcut → Properties   Change something. (E.g. set the "Start in" directory to a different location, such as C:\)    Notice that it is now not possible to access Command Prompt from the Start menu - it has been replaced with AAA Command Prompt everywhere. If you rename AAA Command Prompt to ZZZ Command Prompt then you get back the ordinary version and lose the customised version. Moving the shortcuts to different folders within %APPDATA%\Microsoft\Windows\Start Menu\Programs\ doesn't help.    Any ideas? In Python, how can I parse a numeric string like "545.2222" to its corresponding float value, 545.2222? Or parse the string "31" to an integer, 31?  I just want to know how to parse a float str to a float, and (separately) an int str to an int. Why does boiling water in the microwave make a cup of tea go weird? When I boil water in the kettle, it makes a nice cup of tea.  Sometimes I need to use a microwave because a kettle isn't available.  I boil the water in the mug and it looks pretty normal, but when I drop in the teabag the water froths up and looks foamy.  I don't see what the chemical difference is here, so I assume it must be some physical difference.  I have noticed this with multiple types of tea and multiple microwaves, the results being consistent so it's not just a weird microwave or something like that.    What is the reaction here and how/why does it occur?  Here is a photo of the 'fizzy' looking tea just after dunking in the teabag. Apex PMD: Problem: Validate CRUD permission before SOQL/DML operation Background   I am using the Apex PMD plugin for VS Code and it's giving me this problem:     Validate CRUD permission before SOQL/DML operation   For this line of code:  insert contentVersion;   Which is part of this method:  private void attachReport(Id recordId) {      ContentVersion contentVersion = new ContentVersion(         versionData = Blob.valueOf(buffer.toStr()),         title = 'Import Report',         pathOnClient =  StringUtils.format('/Import-Report-{0}.txt', DateTime.now().getTime()),         FirstPublishLocationId = recordId);      insert contentVersion; }   Questions    Why am I getting the problem? What should I do to not get the problem?     Reference CreateService get error: ERROR_INVALID_ADDRESS (0x000001e7) Interpretation of ROC curve Why do you have to use copy_to_user()/copy_from_user() to access user space from the kernel? Calculate sums of inverses of binomial coefficients Help me identify this fantasy book about two near-omnipotent wizard run empires