$\lim_{x\to \infty} f(x)= \text{Does not exist}$
Why do non-constant periodic functions have no limit at infinity?
If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$ If $I$ and $J$ are ideals in a ring $R$ with 1 which are  co-maximal, i.e $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n$ in $\mathbb{N}$  Work done:     Should I proceed using    By taking $$K=I^m+J^m \neq R$$  Let $$S=\{I,J \in R \mid I^m+J^n \neq R \}$$ be the required partially order set ordered by inclusion then  $$I+J \subseteq I^n+J \subseteq I^n+J^m \subseteq \ldots $$  Then, I think this chain doesn't have an upper bound...as $I^m+J^n \neq R$ for all $ m,n \in \mathbb N$. Am I right??  Thanks for any help!! If all derivatives of $f$ are uniformly bounded by a common constant and $f(1/n) = 0$ for all $n$, is $f$ identically zero? $$ f: \Re \longrightarrow \Re \ \in C^{\infty} \\ \exists \ L&gt;0: \ \forall x \in \Re, \forall n\in N \\ |f^{(n)} (x)| \le L \\ f(\frac{1}{n})=0 \ \forall n\in N \\ f(x) \equiv 0  $$  Good morning,  Can you give me an help to make this proof?  This is my thought:   $\lim_{n \rightarrow +\infty } \dfrac{1}{n}=0=\theta$ $f(\theta)=0$ $f^{(n)} (\theta) =0\\f(x)=\sum_{k=0}^n \dfrac{f^{(k)}(x-\theta)^k}{k!}$  $f$ is analytic, the rest of Taylor's expansion is 0, because the derivatives are bounded, $f$ is null.   I have thought this reasoning, but I have difficulties to demonstrate part 3. Can you help me?  Is there any mistake?  Can I use an easier way to make the proof?  Thanks. I am watching Susskinds's lecture , in which he introduced square angular momentum operator $L^2$, which can be written as $$ L^2=L_z^2+L_-L_++L_z $$ where $$ L_\pm=L_x\pm iL_y $$  Besides, he argued that the spectrum of $L$ can only be $m,m-1,\cdots,-m+1,-m$ in which $m$ is an integer or half-integer. If we denote $|m\rangle$ the eigenvector corresponds to eigenvalue $m$, we have $$ L_\pm|l\rangle=|l\pm 1\rangle\;\text{or}\;0 $$  In the lecture he showed that $L^2|m\rangle=m(m+1)|m\rangle$ for $m$ the highest eigenvalue, by $$ \begin{align*} L^2|m\rangle&amp;=L_z^2|m\rangle+L_-L_+|m\rangle+L_z|m\rangle\\ &amp;=m^2|m\rangle+0+m|m\rangle\\ &amp;=m(m+1)|m\rangle \end{align*} $$  Then he left an exercise to show that for any $-m\leqslant l&lt; m$, we also have $|l\rangle$ is an eigenvector of $L^2$ with eigenvalue $m(m+1)$.  In the lecture, he hinted by showing that it suffices to show that $L^2$ commutes with $L_i$ for $i=x,y,z$, which gives us $$ L^2|m-1\rangle=L^2L_-|m\rangle=L_-L^2|m\rangle=L_-m(m+1)|m\rangle=m(m+1)L_-|m\rangle=m(m-1)|m-1\rangle $$  I believe it is true since I find the   online.   However, my question is that, why the following proof is wrong.  I tried to mimic the proof for $|m\rangle$ to have that  $$ \begin{align*} L^2|k\rangle&amp;=L_z^2|k\rangle+L_-L_+|k\rangle+L_z|k\rangle\\ &amp;=k^2|k\rangle+L_-L_+|k+1\rangle+k|k\rangle\\ &amp;=k^2|k\rangle+|k\rangle+k|k\rangle\\ &amp;=(k^2+k+1)|k\rangle \end{align*} $$  which shows that $|k\rangle$ does not have eigenvalue $m(m+1)$ but $k^2+k+1$ for $k&lt;m$. CHECK constraint in MySQL is not working What are path and contour integrals? I am very confused on what path and contour integrals actually represent. I tried looking for an answer on Google but the only examples were related to Quantum Mechanics. For instance, real Riemann integrals can be thought as &quot;the area under the curve&quot;. What about path/contour integrals? Also, is there a difference between path and contour integrals, or are they the same thing? I've been thinking of contour integrals as path integrals where the path is a &quot;closed loop&quot;, but I'm not sure if that's the case. It's all so confusing, and not being able to visualise these concepts makes it feel like I'm just memorising formulas and theorems without any reason. I have been thinking about path integrals as integrating over a curve in three dimensions. For example, in an $xyz$ coordinate system (with $y$ pointing “up”), we have a path on the $x$-&quot;$z$&quot; plane and integrate over that path. That is, instead of integrating over and along &quot;$y=0$&quot;, we integrate over a certain path. But I don't know if this is a correct way to think of path integrals. This issue has been frustrating me a lot because I'm now studying theorems that involve path/contour integrals and I just can't understand what they really do or mean. By path integral I mean $$\int_{\gamma\vert_{[a,b]}}f(z)\hspace{0.2em}dz = \int_{a}^{b}f(\gamma(t))\cdot\gamma'(t)\hspace{0.2em}dt$$ where $f:U\subseteq\mathbb{C}\rightarrow\mathbb{C}$ is a function that satisfies all conditions for complex integration and $\gamma:[a,b]\rightarrow U\subseteq\mathbb{C}$ is at least a piece-wise regular path. Any help would be appreciated. I read this set of stories in the 80's.  What I recall is this:  A planet where the ruling caste was made up of humanoid women.  These women would, as almost pubescent girls, go out into the wilderness with only a spear and attempt to kill an animal in a life or death struggle.  When it happened, they would burst open a DNA packet in themselves that caused them to rapidly mature, their skin to become bronze colored, and give them some kinds of other "powers", I think over fire.  They were called "Bahronas", I think.  The population of this planet was primitive, but they were in contact with an advanced humanoid species, I think of actual homo-sapiens, who were there observing and advising them.  The advanced species wasn't allowed to intervene, just advise and watch.  The main character was a castaway boy, crash landed on their planet and stranded.  He became friendly with the ruling family of Bahronas and entered their confidence.  We find out through the course of the story that he's been genetically engineered and is not a normal boy.  At some point, the advanced species notices him and explains that he is a clone of a famous explorer and a spy, and they explain that they won't do anything in front of him.  The central part of the story is something of a blur, but, the final part of the story is told in a dramatically different style and shifts to a planet populated by catlike creatures who live in boughs of silk scarves, some of which sing.  THey are recording devices which play back the songs.  On this planet, there is another silk scarf, a white one, which doesn't play a song at all.  The cat creatures are greatly disturbed by it when they hear it's message, which they don't understand.  The narrator then goes on to say that it's a spoken word recording, made by the explorer which the boy was a clone of, explaining that he is captured and held hostage on the planet which makes the scarves, and forced to help make them (I think.) I've created a map system for a game that runs on the principle of drawing the picture of the map from tiles. There are many reasons for this which I won't go into here but if you really want to know then I'm sure you can find out how to contact me ;)  I have  so you can see exactly where the problem lies and the source. The issue is the line between the top 2 tiles and the bottom 2 tiles, I can't figure out why it's gone like this and any help would be appreciated.  In the source is a marker called "stackoverflow", if you search for "stackoverflow" when viewing source then it should take you to the table in question.  I have also uploaded an . What does one second after big bang mean? Consider the following statement:     Hadron Epoch, from $10^{-6}$ seconds to $1$ second: The temperature of the   universe cools to about a trillion degrees, cool enough to allow   quarks to combine to form hadrons (like protons and neutrons).   What does it mean to say "from $10^{-6}$ seconds to $1$ second"?   How is time being measured?  One particle might feel just $10^{-20}\ \mathrm s$ having passed and another could feel $10^{-10}\ \mathrm s$ having passed.   Is saying "1 second after the big bang" a meaningful statement? If anonymity is so important, why is it not universal? A frequent argument brought up in support of voting anonymity is that it can lead to potential grudges. Certainly the review system is of the same nature.  I cannot wrap my mind around the fact that voting is anonymous and reviewing is not. Either both should be public--and this would be fine since public reviewing has been successful--or both should be anonymous--and I assume this would be the desired option since this community holds anonymity so dear.  One could say that voting rates the posts, the actual meat of the site, while reviewing is based on clarifying the purpose of the site and cleaning it up.   answer makes the point that the community can trust that votes are a good measure of quality for the very reason that they do not know who cast them; the mentality that a high-ranking member may have supported something could blur the line between "science" and "politics/religion."  Reviewing, then, can be seen as a judgment of specific members rather than quality of edit/questions, which seems to fall precisely under the "politics/religion" umbrella--the review system is much more vulnerable to grudges due to this.  So let's discuss the differences and the implications of these disjointed anonymity policies. Number reference headline I try to number the header of my references. I succeeded to add the number to the table of contents, but I can't get it to work for the header.  To add the number to the toc I used:  \addcontentsline{toc}{section} \protect\numberline{\thesection}References}   If I try the same for \refname, I get 5References without space.   \renewcommand{\refname}{\protect\numberline{\thesection References}}   When I put the references title outside the curly brackets, it's shown in a new line. What am I missing here? The full bibliography call looks like this:  \renewcommand{\refname}{\protect\numberline{\thesection References}} \addtocounter{section}{1} \cleardoublepage \phantomsection \addcontentsline{toc}{section}{\protect\numberline{\thesection}References} \bibliographystyle{apacite} \bibliography{bachelor}   EDIT 1:  As @TeXnician stated, a tilde would solve the problem. It is not very elegant, though, as I have to use approximatly three of them to get the same space as in the other headlines.   \renewcommand{\refname}{\protect\numberline{\thesection~~~References}}   EDIT 2: @Werner suggested a shorter solution, but it throws multiple errors, when I try to integrate it into my .tex file. Here's a short reproducable example which replicates the errors.  \documentclass[12pt]{article} \usepackage[utf8]{inputenc} \usepackage[margin=1.1in]{geometry} \usepackage{listings} \usepackage{apacite} \usepackage{natbib}  \begin{filecontents}{mybib.bib}  @Article{draxler2010sample,     author    = {Draxler, Clemens},     title     = {Sample size determination for Rasch model tests},     journal   = {Psychometrika},     year      = {2010},     volume    = {75},     number    = {4},     pages     = {708--724},     publisher = {Springer}, }  \end{filecontents}   \begin{document}  \tableofcontents      \section{First Chapter}  \citep{draxler2010sample}  \let\oldsection\section% Store \section in \oldsection \renewcommand{\section}[2]{\oldsection{#2}}% Update \section to gobble * \bibliography{mybib} \bibliographystyle{apacite}  \appendix  \section{R Code}  \subsection{Greedings Function}  \begin{lstlisting}     print("Hello World") \end{lstlisting}  \end{document} Equifax credit score discrepancy in 1 month, why? I just went to Citi Bank's website (which gives free FICO scores from Equifax for users of some credit cards) and I realized that the score I saw there was a whopping ~50 points higher than the number I saw on Mint.com, despite the fact that they both claim to have received the number from Equifax. The only difference is the fact that Mint checked my score at the end of February 2015, whereas Citi checked it at the end of January 2015.  However, nothing unusual has happened to my credit history in between these two. I can confirm on Mint's credit report that nothing seems wrong, so I'm quite confused:   What can cause such a discrepancy? Whom should I contact to understand what is going on -- Mint, Citi, Equifax, or other? How should I contact them -- file a dispute, call them, or go somewhere in person?   Note that I don't feel I'm "disputing" anything, since I don't know if there is any cause for a dispute, and hence I'm not sure it makes sense to open a dispute with Equifax. I just want to know what is going on and why two third-party agencies claim to have received such wildly different numbers from the same agencies in the span of 1 month. How to disable gedit side panel? How do i disable this stupid side panel of gedit, that somehow appeared? There is nothing in the settings or plugins. Tried rm -r -f ~/.config/gedit as well, with no success. I don't have big screen and i find it totally useless and eating up what little free space for editing documents i have. See Screenshot How do I get the full url of the page I am on in C# Intermediate ring between a field and an algebraic extension. This is an exercise in some textbooks.     Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field.   The trouble is really with the inverse of $r$, where $r\in R$. How to prove that $r^{-1}\in R$, in apparent lack of a characterization of $R$.  It occurred to me to use the smallest field containing $R$ ($R$ is easily shown to be an integral domain), that's the field of quotients, and proving that it's $R$ itself, but I don't really know how to proceed.  A not-too-weak, not-too-strong hint will be much appreciated.   Beware $ $ Readers seeking only hints should  beware that there is now a complete answer. BeanFactory vs ApplicationContext I'm pretty new to the Spring Framework, I've been playing around with it and putting a few samples apps together for the purposes of evaluating Spring MVC for use in an upcoming company project. So far I really like what I see in Spring MVC, seems very easy to use and encourages you to write classes that are very unit test-friendly.  Just as an exercise, I'm writing a main method for one of my sample/test projects. One thing I'm unclear about is the exact differences between BeanFactory and ApplicationContext - which is appropriate to use in which conditions?  I understand that ApplicationContext extends BeanFactory, but if I'm just writing a simple main method, do I need the extra functionality that ApplicationContext provides? And just exactly what kind of extra functionality does ApplicationContext provide?  In addition to answering "which should I use in a main() method", are there any standards or guidelines as far as which implementation I should use in such a scenario? Should my main() method be written to depend on the bean/application configuration to be in XML format - is that a safe assumption, or am I locking the user into something specific?  And does this answer change in a web environment - if any of my classes needed to be aware of Spring, are they more likely to need ApplicationContext?  Thanks for any help. I know a lot of these questions are probably answered in the reference manual, but I'm having a hard time finding a clear breakdown of these two interfaces and the pros/cons of each without reading thru the manual with a fine-tooth comb. Head or tail sound fine to my ESL ears.  What's the reasoning behind the plural usage?  I looked it up on etymonline but didn't find anything interesting. All my college years I have been using public, and would like to know the difference between public, private, and protected?  Also what does static do as opposed to having nothing? Delete all but the most recent n file for each group of files that share the same prefix in a directory My question is a bit different from some older questions simply asking for "deleting all but the most recent n files in a directory".  I have a directory that contains different 'groups' of files where each group of files share some arbitrary prefix and each group has at least one file. I do not know these prefixes in advance and I do not know how many groups there are.  EDIT: actually, I know something about the file names, that is they all follow the pattern prefix-some_digits-some_digits.tar.bz2. The only thing matters here is the prefix part, and we can assume that within each prefix there is no digit or dash.  I want to do the following in a bash script:   Go through the given directory, identify all existing 'groups', and for each group of files, delete all but the most recent n files of the group only. If there are less than n files for a group, do nothing for that group, i.e. do not delete any file for that group.   What is a robust and safe way of doing the above in bash? Could you please explain the commands step-by-step? Deny access to a webpage using web.config Evidence for quantum gravity from gravitational waves The rumor spreads that .  I am far from being an experimentalist, but I want to know if there is any chance that the mentioned observations will reveal any experimental evidence for quantum gravity.  Personally, I feel like this is merely impossible as gravitational waves should propagate in agreement with classical General Relativity. But I might very well not see pitfalls if there are any.  UPDATE: by "evidence for quantum gravity" I mean experimental results which don't agree with predictions of General Relativity or its classical generalizations. They might be coming from string theory or loop gravity or from a completely different approach which we haven't discovered yet. I want to understand what are the chances that new physics will show up during the upcoming  gravitational wave observations.  Also, I expect gravitational waves to be in a highly coherent state, thus making the individual gravitons unobservable. Is this true?