Let $A$ be a ring with $0\neq1$ having $2^n-1$ invertible elements and at most $2^n-1$ non-invertible elements. Then $A$ must be a field? How many elements does the ring need to have? Because $2^n-1$ is odd, the characteristic of the ring is $2$. Thank you!
Let $A$ be a ring with $0 \neq 1 $, which has $2^n-1$ invertible elements and less non-invertible elements. Prove that $A$ is a field.
population and subsamples : mean equality? I want to perform a two sample test, to see whether theirs means are equals. One of the sample is rather big (#60000 values) and represents the scores of a whole population. This population is partitioned in small classes of less that 1000 people. The other sample of the test is a subsamble of the former. It corresponds to people of the same class. So, each time, there is less than 1000 values in it. I have no idea about the variances, so I'd think of using Welch's t-test.  But I'm concerned regarding the independence hypothesis: can it be valid, considering that the second sample is negligible when compared to the overall population? How to do power series expansion What is the coefficient of $x^{11}$ in the power series expansion of $\frac 1{1-x-x^4}$?    How do I do power series expansions? Nature of the quantum degeneracy pressure describes the first-order vs. second-order logic as follows:     First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals.   It gives $\forall P\,\forall x (x \in P \lor x \notin P)$ as an SO-logic formula, which makes perfect sense to me.  However, in a , the poster claimed that $\forall x\forall y(x=y\leftrightarrow\forall z(z\in x\leftrightarrow z\in y))$ is an FO formula. I think this must not be the case, since in the above formula, $x$ and $y$ are sets of individuals, while $z$ is an individual (and therefore this must be an SO formula).  I mentioned this as a comment, but two users commented that ZF can be entirely described by FO logic, and therefore $\forall x\forall y(x=y\leftrightarrow\forall z(z\in x\leftrightarrow z\in y))$ is an FO formula.  I'm confused. Could someone explain this please? I wonder if someone knows any general rules of thumb regarding the number of bootstrap samples one should use, based on characteristics of the data (number of observations, etc.) and/or the variables included? Coordinate descent soft-thresholding update operator for LASSO I was reading this paper () describing the coordinate descent algorithm for LASSO, and I can't quite figure out how the soft-thresholding update for each $\beta_j$ is derived for the linear regression case. Specifically, how does one go from equation (4) to equation (5) in the paper? Can anyone show me a detailed derivation of the update? Thanks!  Edit: actually this paper deals with the elastic net penalizing term, which reduces to LASSO when $\alpha=1$. I am mostly interested in LASSO, but I guess the derivation should be principally the same for elastic net. Is there a single word that describes both Witches and Wizards? In the Harry Potter world, there's the term "muggle" for people with no magical abilities, "muggle-born" for those with magical abilities who's parents are muggles and "squib" for those with no magical abilities who's parents are magical.  But is there any collective term for all magical people (as opposed to wizards for males and witches for females)?  I don't recall seeing any word that refers to all men and women and children with magical abilities, and it's the only group left without noun or collective noun to describe them. Is it possible to search for punctuation/special characters? I tried to find a related question here, but was unsuccessful.  Someone on SO asked ... I tried searching SO and was presented with the search options page. Even Google fails at this search.  So my question is, is there some way to search for an exact word (in quotes) that won't cause it to fail? How to recursively download a folder via FTP on Linux I'm trying to ftp a folder using the command line ftp client, but so far I've only been able to use 'get' to get individual files. Backgrounder:  The  (Pointer to IMPLementation) is a technique for implementation hiding in which a public class wraps a structure or class that cannot be seen outside the library the public class is part of.  This hides internal implementation details and data from the user of the library.  When implementing this idiom why would you place the public methods on the pimpl class and not the public class since the public classes method implementations would be compiled into the library and the user only has the header file?  To illustrate, this code puts the Purr() implementation on the impl class and wraps it as well.  Why not implement Purr directly on the public class?  // header file: class Cat {     private:         class CatImpl;  // Not defined here         CatImpl *cat_;  // Handle      public:         Cat();            // Constructor         ~Cat();           // Destructor         // Other operations...         Purr(); };   // CPP file: #include "cat.h"  class Cat::CatImpl {     Purr(); ...     // The actual implementation can be anything };  Cat::Cat() {     cat_ = new CatImpl; }  Cat::~Cat() {     delete cat_; }  Cat::Purr(){ cat_-&gt;Purr(); } CatImpl::Purr(){    printf("purrrrrr"); } Limiting floats to two decimal points I want a to be rounded to 13.95.  &gt;&gt;&gt; a 13.949999999999999 &gt;&gt;&gt; round(a, 2) 13.949999999999999   The  function does not work the way I expected. Can I clear an area of my prison for everyone BUT max security prisoners? How can I make sure that only low and medium security inmates are in a specific area? Is it possible to reverse the overlay you can set? It seems like you are only able to clear areas for low OR medium security prisoners, but sharing is not possible. The thing I want here is to make sure that max security prisoners are not in certain areas. I think some good words would be conflicted, or confused. As an example of context, think of someone who believes in freedom of speech, but, at the same time, they would support banning comedians who make jokes about certain issues that person cares about. If there are injective homomorphisms between two groups in both directions, are they isomorphic? If I have two groups, $G$ and $H$ and two injective homomorphisms $\phi:G \to H$ and $\psi: H \to G$, then by the first isomorphism theorem applied to $\phi$, we have that $G \cong \mathrm{Im} (\phi)$, a subgroup of $H$. Similarly, $H$ is isomorphic to a subgroup of $G$. For finite groups, this guarantees that $G \cong H$ but does this hold for infinite groups?  Weird things can happen for infinite groups, e.g $n\mathbb Z \subsetneq \mathbb Z$ but $n \mathbb Z \cong \mathbb Z$. I'm wondering if this kind of thing stops an isomorphism from occurring.  I think this question generalises to rings, modules, fields and so on. Can this be answered for all of these structures? Absolute zero and Heisenberg uncertainty principle I got to read Volume I of Feynmann's lectures. It said that at absolute zero, molecular motion doesn't cease at all, because if that happens, we will be able to make precise determination of position and momentum of the atom. We do know that Heisenberg uncertainty principle holds for microscopic particles in motion. But what then is wrong to consider that all molecular motion ceases at absolute zero? In other words, does the uncertainty principle not hold when there is no motion?  Need some help! Random Gaussian Variables Is there a class in the standard library of .NET that gives me the functionality to create random variables that follow Gaussian distribution? How to update libavcodec on ubuntu 14.04? On Ubuntu 14.04.5 I just upgraded firefox to version 50.0; now there is a message on some pages which reads:   libavcodec may be vulnerable or is not supported, and should be updated to play videos   So how can I update libavcodec?  A simple   sudo apt-get install libavcodec   gives the following output:  Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package libavcodec   So maybe I don't have libavcodec installed in the first place? So why do I get this message...? I am pretty much sure that for native speakers the issue I am going to bring up might look as an uncalled question as they can easily figure out which form of a verbal part of speech should be used, be it a gerund, an infinitive, or a bare infinitive. However, it can be pretty much misleading for a foreigner because the same construction works differently with different words and I don’t see a logical explanation. For example:  The purpose of this article is to analyze the issue…  My hobby is reading. One may assume, and many foreigners really come to this conclusion, that it’s OK to switch the “gerund” and “infinitive” as the meaning will stay the same, that is:  The purpose of this article is analyzing the issue… (same as 1 above)  My hobby is to read. (same as 2 above)   However, as I was told, this assumption is erroneous. Another example: His task was watching after them. (not good) His task was to watch after them. Is there any principle which governs the choice of the right form in such cases? What is the distinction between a professional and amateur photographer? Mirror changes to side of cube across all six sides