How to refer to United States of America? My sentence is:     ATC experienced rapid growth between 1880 to 1911. In 1890, their five main competitors joined their consortium and by 1910, the American Tobacco Company had absorbed 250 companies, producing 80% of the tobacco products in the United States of America and had grown its equity from $25 million to $316 million.    I am conflicted on how to refer to USA (Bold part)  Is it:  in the United States of America  or  in United States of America  Also, sneaky second question if you guys don't mind :)  Do you think my sentence is too long?
Need for using the article "the" before "United States of America"? Usually, we do not use the definite article, the, before names of countries but we use the before "United States of America".   Is there a specific reason for using the article before the country "United States of America"?
Simple extension fields If I am correct simple extension fields are extensions generated by one element. I have learned that this means that elements of a simple extension can be written as powers of that element as long as the power is greater than the degree of that element. Also this means that all elements of the simple ext. Can be written in terms of decending powers of the generating element where the highest power is the degree of the generator minus 1. This concept seems to work for generating elements whose irreductable in a certain field is a descending degree polynomial. Looking at the construction of the complex numbers by starting with $\Bbb R$ (and doing $\Bbb R [x]/\langle x^2+1\rangle$), it is said that all elements are of the form $a+b\cdot \text{generator}$. But when the generator is evaluated for the irreductable, $x^2+1$, we see... $(\text{generator})^2=-1$. Thus there is no first power of gen. Can someone explain where all elements of the extension just explained are of the form $a+b\cdot \text{generator}$ comes from or explain this situation for anytime where the irreduc ( gen, F)  is not a descending degree polynomial. I hope my question is clear.  Thank you I had the following two problems:     Find a counterexample for $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$ and $ f_*(A-B) \subseteq f_*(A) -f_*(B).$ Where $f_*(X)$ is the image of $X$ under $f$ for some function $f:A \rightarrow B $ and some subset $X \subseteq A$.   I came with the following answer:  Let $ f: \mathbb{Z} \rightarrow \mathbb{Z} $, where $f: x\mapsto 0$; let $A=\{1,2\}$, and $B=\{3,4\}$.     For $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$, $$ f_*(A \cap B) = f_*\{ \varnothing\} = \varnothing \hspace{1cm} \text{and} \hspace{1cm} f_*(A) \cap f_*(B)=\{0\} \cap \{0\} = \{0\}. $$ It follows that $ \varnothing \nsupseteq \{0\} $. For $ f_*(A-B) \subseteq f_*(A) -f_*(B)$, $$f_*(A-B)=f_*(\{1,2\})=\{0\} \hspace{1cm} \text{and} \hspace{1cm} f_*(A) - f_*(B) = \{0\} - \{0\} = \varnothing$$ It follows that $ \{0\}  \nsubseteq \varnothing $.   Now (assuming my answer is correct), let's say I want to find more counterexamples or maybe a more general answer (e.g. all the conditions under which the inclusion fails). Other than by trial and error or just intuition, how would I do that?   I know this is a broad question, so I'm specifically asking for methods or advice on how to build counterexamples in elementary set theory (for example, tips on where to begin or rules that apply in general that may be unobvious). Best place to put bug reports? What websites or other resources do users of Blender have available to them to report bugs or problems with the program?  I would like to get this issue cleared in Q&amp;A format so that we can link new users with bug-report-style questions to this post. Looking for the title of a book about alien invasion, mining asteroids and using lasers to defend earth What I remember of this novel/series is that the main character creates solar laser satellites and mines asteroids to create more satellites to defend Earth from aliens. Later on, he uses the hollowed out asteroids as a base and eventually as an armored fortress to repulse the alien fleets. I last remember the main character attempting to hollow out more asteroids to defend and attack the aliens eventually. How to use early stopping properly for training deep neural network? Scheduling a team based activity with 10 different teams and X different games. Every team must meet each other ONCE but never twice, What's the most efficient way of determine individual card values for my collection? How can I stop applications and services from running? When my phone starts up there are several application/services running in the background that I would rather not.  And when I press the Home button while using an app, that app usually goes into the background without exiting.  How can I stop these apps from running? WOL - Wake On Lan over internet ArrayList vs List I saw this reply from Jon on :     If you want a single collection to   contain multiple unrelated types of   values, however, you will have to use   List&lt;object&gt;   I'm not comparing ArrayList vs List&lt;&gt;, but ArrayList vs List&lt;object&gt;, as both will be exposing elements of type object. What would be the benefit of using either one in this case?  EDIT: It's no concern for type safety here, since both class is exposing object as its item. One still needs to cast from object to the desired type. I'm more interested in anything other than type safety.  EDIT: Thanks Marc Gravell and Sean for the answer. Sorry, I can only pick 1 as answer, so I'll up vote both. Is there a way to delete all stack exchange accounts at the same time?  I also want to remove all traces of my posts on the site. Is that possible?   Will I have to delete all of my posts manually?  Can I use the stack exchange API for this? If $A\simeq\prod A_i$ then $\sum e_i=1$ Given $A$ commutative ring with unity. If we have rings $A_1,\dots, A_n$such that $A\simeq A_1\times \dots \times A_n$ there are $e_1,\dots,e_n$ idempotent elements of $A$ such that $e_1+\dots+e_n = 1$ and $e_i e_j =0$ if $i\neq j$.  I'm not sure how general the proof should be written...  The isomorphic relation suggest that if $f$ defines the isomorphism then there is a $(a_1,\dots, a_n)\in \prod A_i$ such that $f(a_1,\dots,a_n)=1\in A$. Could it work thinking of the idemmpotent terms as $e_i=(0,\dots,0, a_i,0, \dots,0)?$ This should satisfy $e_i e_j =0$ if $i\neq j$. But I have two problems with it (1) I don't see why $e_i$ should idempotent as no assumptions in regard to the existence of idempotent elements in $A_i$ has been made. (2) The sum $\sum e_i=1$ has no reason to be true. What does 'close invalidated' mean Recently I flagged  as a duplicate. A few days later, when I checked it again to see if it was closed, I realized it wasn't.    Searching in Meta brought me to , where a comment suggested checking the timeline. So I checked the  and saw that the close flag was set as invalidated.  Does this mean that it was disputed? When I click on it I see only "This item is no longer reviewable." Do I need more reputation (3000+ ?) to view more detailed information? How to upload string as file with jQuery or other js framework My friend and I were discussing this and we couldn't figure out how to prove it one way or another.  The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ is some product of a fraction of $\pi$.  Is $\sin(x) $ (or other trigonometric function) necessarily irrational if $ x $ is rational?  Edit: Excluding the trivial solution of 0. What is the best approach for solving the equations of the form      $p^x = x^q$ ; where $p$ and $q$ are some constants.   Basically, I wanted to solve $2^x = x^2$ to analyze algorithms with respective growth functions. Each package name is in the form: package-XubuntuY*_amd64(i386)  So my questions are:   What does the X stand for? What does the Y stand for?   for example binutils_2.22-6ubuntu1.3_amd64  What is the meaning of the numbers 6 and 1.3 Determining a patients biological age? I am looking for an algorithm that shows the exact coefficients some variables that have on a hospital patient's chronological age versus their biological age.   I understand a lot of the variables that go into determining ones biological age (i.e. Blood pressure, diet, family history of disease, obesity etc.) but not the amount of impact that it has on the actual scoring of their age.  I have found many sites that will calculate it, however I need the actual variable coefficients and influence in order to produce a proper score for thousands of patient records.   Thanks in advance for your input! How come Mortarboard is bronze and Enthusiast is silver? :   Silver  Enthusiast    Visited the site each day for 30 consecutive days   56.9k awarded Bronze  Mortarboard   Earned at least 200 reputation in a single day      15.4k awarded    It would seem that earning Enthusiast requires almost no effort while Mortarboard is highly non-trivial. This appears to be reflected in the award count (almost 4:1 in favor of enthusiasts).  So, why is the first one silver and the second one bronze?  (A similar question can be asked about Epic vs Fanatic and probably many other). The Monty Hall Problem - where does our intuition fail us?