So I ask to prove that the function $f(x):=\begin{cases}e^{-\frac{1}x},\,\text{if}\, x&gt;0\\0\,\text{otherwise}\end{cases}$ is of class $C^\infty$. Clearly $f\in C^0$ since $\lim_{x\rightarrow 0}f(x)=0$ and since in $(-\infty,0)\cup(0,+\infty)$ the function $f$ agrees with functions of class $C^\infty$ so to prove the statement we have to prove that $f^n(0)=0$ but unfortunately I can't prove this. So could someone help me, please?
The Wikipedia page () proves that $$f(x) = \begin{cases}  \exp(-1/x),  &amp; \mbox{if }x&gt;0 \\ 0, &amp; \mbox{if }x\le0 \end{cases}$$ is a non-analytic smooth function. But I don't understand why the fact that "the Taylor series of $f$ at the origin converges everywhere [where does "everywhere" mean here?] to $0$" implies the Taylor series converges to $0$ when $x&gt;0$? Can someone explain this in greater detail? I guess there should be a gap in my understanding of power series.
Relevance of cycle detection in Pollard's rho integer factorization algorithm I've recently learned about Pollard's rho factorization algorithm, and the way I understand it, it works by using a pseudorandom sequence of integers $x_{k}$ ranging from $0$ to $n-1$, where $n$ is the number to be factored, to find repetitions in an unknown sequence $y_{k} = x_{k} \bmod p$, where $p$ is an unknown, non-trivial factor of $n$. Such a repetition $y_{i}=y_{j}$ occurs when $x_{i}\equiv x_{j} \bmod p$, which can be detected by checking whether $\gcd(x_{i}-x_{j}, n)\gt 1$.  From what I could find on the Internet, Floyd's cycle detection algorithm or a variant thereof is used to find such repetitions. And the reason for this is what I don't understand.   First, Floyd's algorithm requires that each element in the sequence depends solely on the element before it, which, even if it is the case for $x_{k}$, is not necessarily the case for $y_{k}=x_{k}\bmod p$. For example, assuming we want to factorize $6$, let's take the sequence $5, 3, 2, 4, 1, 0, 5, 3, 2, 4, 1, 0,\ldots$. The cycle period is $5, 3, 2, 4, 1, 0$, where every element is contained only once. However, if we take this sequence $\bmod 2$, we get $1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0,\ldots$. Floyd's algorithm cannot be applied here to find the cycle period $1, 1, 0, 0, 1, 0$. Besides, we are not even interested in a cycle, only in repetitions. Whether we find the congruence $\bmod 2$ between $5$ and $3$, $5$ and $1$, or $4$ and $2$ in the first sequence is irrelevant.   Please correct me if my understanding of Pollard's rho algorithm as described above is wrong. But if it is not, it seems to me that one could just as well try out some random numbers between $2$ and $n-1$ and see if they share a divisor with $n$ other than $1$. So what is the benefit of using a cycle detection method like Floyd's algorithm?  I also found , but it is not answered, and the comment on the question does not really answer the question either. What is the plural form of "Software"? Is it correct to use softwares, as in the following sentence?  There are various video conversion softwares in the market. What resolution would be needed for a billboard advertisement? Why does the Moon not revolve around the Sun directly? The sun pulls on the moon with a force that is more than twice the magnitude of the force with which the earth attracts the moon. Why, then, doesn’t the sun take the moon away from the earth? My regex works on regex101 but doesn't work in python? How do I develop black and white negatives at home? Send request for "reason for down vote"? Each question in SO keeps a list of the users who voted it down. As we all know, the list is never shown to anyone. Only the down voter can see which questions he down voted.   Sometimes, down voters down vote rather hastily. Thats okay because no one is perfect (except chuck norris). But sometimes, the down voter might have a good reason to down vote but may not leave a reason behind. As a result, the poster is left wondering if his question is stupid/bad or okay. To remedy this situation, I suggest a feature below -   The poster should have a button "request down voters for reason" which can be pushed only once. The down voters in the down voter list get the notification and can choose to respond to it. I feel that this can serve as a polite reminder to those who down vote a question which would seem to be an okay (if not good) question to most people.   Example of question which I feel should have at least 0 votes -    If okay, then this feature can be combined with -    Do you agree to the proposal ?  As an aside, do you agree that the SO question above should not get negative points ? I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? I installed the latest pgAdmin 4 (v2.1) but I found out that (), for the moment, the only way to establish a connection through an SSH tunneling is using another outside process, an  says:  I setup a tunnel as follows: ssh -f dpage(at)hostname(dot)enterprisedb(dot)com -L 5999: hostname.enterprisedb.com:5432 -N  (...) I was running on Mac, but the same procedure should work on Linux. On Windows, there's a tunnelling app in Putty iirc, but I don't know how to use it offhand.  I have found a simple . The connection is established but it seems, when I try to connect through pgAdmin it fails or is refused (using 127.0.0.1:3306 or 127.0.0.1:5432 as the tunnel's destination). Why? E: Unable to correct problems, you have held broken packages New top bar doesn't show hot threads across Stack Exchange sites There's another question on physics.SE whose , if I have understood it correctly, explains that the farther the points are in space the faster they are moving away from each other.     Actually, the apparent speed with which two of the points on the   circle in a distance $D$ of each other would move relative to each   other will be $v = H_0 D$ where $H_0$ is the speed the balloon itself   is expanding.   This means that even the ruler sitting on my desk is expanding, just that the expansion is very very slow owing to the ruler's small size relative to the cosmic scales. However, if everything is scaling then everything should apparently be of the same size to an observer in such an expanding space. The  quote from the show "The Big Bang Theory" comes to mind.   To be able to counter that, there're some phenomena that we can observe which are scale invariant. Red-shift would be one such phenomenon. But is red-shift the only way we can tell that the space is expanding? Is there a logical flaw in my thinking? I know that red-shift is a result of the Doppler effect - there's bound to be a decrease in the frequency of a signal, as perceived by an observer, emitted by a source which is moving away relative to the observer. Convert a comma delimited list of numbers to a hyphenated list or range How does one change to engineering for graduate school after undergrad in a science field? I've always had an interest in electrical engineering (actually all engineering in general) and I've decided that I'd like to pursue a graduate degree.  I guess I'm wondering if I should really try to do this or just keep it as a fun hobby I spend all my free time doing.  So, the question is,  how does one change fields between undergrad and grad school?  A specific example follows, but please provide answers that address a more general case, so that your answers will be helpful to other people with the same general question.  I already have a BS and MS in chemistry with a minor in applied math.I published three papers during my MS, one first author in the top journal in my specialty and two as coauthor in an average and a so-so journal.  I hope those will count for something.  I finished my BS with around 3.1 and my MS with a 3.9 gpa.  I'm sure that my MS advisor would write me a great letter.  My GRE scores are too old to count so I'd have to retake it or try and get a pass on it since I already have a graduate degree.  Since finishing school I've been working for 4 years as a chemist.  I can get great recommendations from my supervisor who is a PhD chemist.      I've been working for a while to cover some of the things I would have learned in a BSEE to try and make up for my lack of one.  I worked through some texts on analog/digital design, lots and lots of programming, built a balancing robot that got a bit of internet attention, and I'm currently working on an autonomous robotics project that's looking really good so far.  After that I have more project ideas than I have time or money for.    I want to apply for PhD programs (specifically in controls/robotics) but I'm worried that my application will get tossed right away because I don't have a BSEE.  Maybe I should just apply to MS programs? why does loop-cut-and-slide ignore some faces but not others? why does the loop-cut-and-slide operation ignore certain faces but happily cut others?the model is symmetrical so it is a bit strange to pick 3 faces that won't get cut. I tried applying it to see if it was just a graphics glitch causing the purple line to not display over everything, but it did end up cutting only what the line was selecting, instead of the whole mesh.    as requested,here's a paste of the blend: Compare using Thread.Sleep and Timer for delayed execution I have a method which should be delayed running for a specified amount of time.  Should I use  Thread thread = new Thread(() =&gt; {     Thread.Sleep(millisecond);     action(); }); thread.IsBackground = true; thread.Start();   Or  Timer timer = new Timer(o =&gt; action(), null, millisecond, -1);   I had read some  about using Thread.Sleep is bad design. But I don't really understand why.  But for using Timer, Timer has dispose method. Since the execution is delayed, I don't know how to dispose Timer. Do you have any suggestions?  Or if you have alternative codes for delayed execution are also appreciate. How would I prove that if an integer $n>2$, then $\bar{z}=z^{n-1}$ has $n+1$ solutions If $n$ is an integer and $n&gt;2$, then $$\bar{z}=z^{n-1}$$ has $n+1$ solutions. Does this have something to do with the rational root theorem? load() method deprecated? In Quantum Mechanics we usually consider operator-valued functions: these are functions that take in real numbers and gives back operators on the Hilbert space of the quantum system.  There are several examples of these. One of them is when we work with the Heisenberg picture, where we need to consider functions $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ such that $\alpha(t)$ is the operator at time $t$.  Another example is when we deal with exponentiation of operators, like when building the time evolution operator:  $$U(t,t_0)=\exp\left(-i\frac{H}{\hbar}(t-t_0)\right),$$  Here, the $\exp$ is usually understood as being defined via the eigenvalues of $H$.  The point is, the idea of a function $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ appears quite often in Quantum Mechanics, and sometimes one needs to differentiate these. In practice we do it formally, using all the properties we would expect, but I'm curious about how one would properly define this.  If we were dealing with bounded operators, then we could use the operator norm, which is available for this kind of operator, and define the derivative as we can usually we do when there's some norm around.  The point is that in Quantum Mechanics most of the time operators are unbounded.  So in the general case, how can one define the derivative of one operator-valued function? natbib and hyperref create by default two links to the bibentry in the reference section|chapter of the document:   \begin{filecontents}{\jobname.bib} @book{author08, title = {{A Title}}, publisher = {Alpha}, year = {2008}, author = {Author, A}, address = {London} } @book{buthor90, title = {{B Title}}, publisher = {Bravo}, year = {1990}, editor = {Buthor, B}, address = {New York} } \end{filecontents}  \documentclass{article}  \usepackage{natbib} \usepackage{hyperref} \hypersetup{colorlinks,citecolor=red}  \begin{document}  \noindent citet: \citet{author08}, \citet[see][p. 20]{author08} \\ citep: \citep{author08}, \citep[see][p. 20]{author08}  \bibliographystyle{plainnat} \bibliography{\jobname}  \end{document}     In order to avoid two links to the bibliography (one for the [author] and one for the year [year], respectively), I took  to an older question and added it to my preamble.  \usepackage{etoolbox} \makeatletter  \pretocmd{\NAT@citex}{%   \let\NAT@hyper@\NAT@hyper@citex   \def\NAT@postnote{#2}%   \setcounter{NAT@total@cites}{0}%   \setcounter{NAT@count@cites}{0}%   \forcsvlist{\stepcounter{NAT@total@cites}\@gobble}{#3}}{}{} \newcounter{NAT@total@cites} \newcounter{NAT@count@cites} \def\NAT@postnote{}  % include postnote and \citet closing bracket in hyperlink \def\NAT@hyper@citex#1{%   \stepcounter{NAT@count@cites}%   \hyper@natlinkstart{\@citeb\@extra@b@citeb}#1%   \ifnumequal{\value{NAT@count@cites}}{\value{NAT@total@cites}}     {\ifNAT@swa\else\if*\NAT@postnote*\else%      \NAT@cmt\NAT@postnote\global\def\NAT@postnote{}\fi\fi}{}%   \ifNAT@swa\else\if\relax\NAT@date\relax   \else\NAT@@close\global\let\NAT@nm\@empty\fi\fi% avoid compact citations   \hyper@natlinkend} \renewcommand\hyper@natlinkbreak[2]{#1}  % avoid extraneous postnotes, closing brackets \patchcmd{\NAT@citex}   {\ifNAT@swa\else\if*#2*\else\NAT@cmt#2\fi    \if\relax\NAT@date\relax\else\NAT@@close\fi\fi}{}{}{} \patchcmd{\NAT@citex}   {\if\relax\NAT@date\relax\NAT@def@citea\else\NAT@def@citea@close\fi}   {\if\relax\NAT@date\relax\NAT@def@citea\else\NAT@def@citea@space\fi}{}{}  \makeatother     Which gets me almost where I want to be, but leaves me with two issues I would like to be solved:   while \citet includes pre- and postnotes in the link, \citep doesn't. How can \citep be changed to include postnotes? is there a way to color the brackets and commas that are typeset in black in the unmodified first example also in black in the modified second example without removing them from the link?