Converting NSString to NSDate gives nil
NSString to NSDate
I'm trying to find the derivative of $\sin^{-1}(x)$. I know the steps that lead to $\frac{1}{\sqrt{1-x^2}}$, however I don't understand why the following reasoning leads to a wrong answer.  Because  $$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} $$  If we plug in for $f(x) = \sin(x)$, and because $\frac{d}{dx}\sin(x) = \cos(x)$ we get  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\sin^{-1}(x))} $$  Since $\sin(x) = \cos(x-\frac{\pi}{2})$, we can state that $\sin^{-1}(x) = \cos^{-1}(x-\frac{\pi}{2})$. (I suspect this is what is wrong)  Thus,  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{\cos(\cos^{-1}(x-\frac{\pi}{2}))}$$  Then, by the definition of inverse function, we have  $$\frac{d}{dx}\sin^{-1}(x) = \frac{1}{x - \frac{\pi}{2}}$$ Isomorphism between $\Bbb R$ and $\Bbb R(X)$? My questions are:     $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ?      $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields?   $ $    $ $  For $1.$, here are some comments:   This is true for $\Bbb C$, see . The key point here is the fact that $\Bbb C$ is algebraically closed, of cardinality $2^{\aleph_0}$, and of characteristic $0$. The same argument doesn't work for $\Bbb R$. We can actually find a subfield $F$ of $\Bbb R$ such that $F \cong F(X) \cong F(X,Y) \cong \cdots$, see .  that $\Bbb Q(X)$ does not embed in $\Bbb Q$ (as a field).  This is equivalent to: Is there an injective ring morphism $\Bbb R[X] \hookrightarrow \Bbb R$ ? (By taking the fraction fields).   As for 2. :   They are isomorphic as abelian groups, and actually as $\Bbb Q$-vector spaces. They are not isomorphic as $\Bbb R$-algebras, because they haven't the same transcendence degree over $\Bbb R$. Let's try to see how to construct such a field isomorphism. We could try$^{[1]}$ to find a subset $B$ of $\Bbb R$ such that $\Bbb R = \Bbb Q(B)$ and $C \subsetneq B \implies \Bbb R \neq \Bbb Q(C)$.  The set $B$ must contain transcendental elements, pick such an $x_0 \in B$.  Then $\Bbb R = \Bbb Q(B) = \Bbb Q(B \setminus \{x_0\})(x_0)$ and we could try$^{[2]}$ to find a field isomorphism $\Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$, so that $\Bbb Q(B \setminus \{x_0\})(x_0) \cong \Bbb Q(B)(x_0) \cong \Bbb R(X)$.  For ${[1]}$, I tried to use Zorn's lemma on $\mathscr B = \{B \subset \Bbb R \mid \Bbb R = \Bbb Q(B)\}$ and the partial order $B_1 ≤ B_2 \iff B_1 \supset B_2$. But this is not clear to me why this should be an inductive set.  For ${[2]}$, my idea was to define $f : \Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$ by choosing a bijection $b : B \setminus \{x_0\} \to B$ (these sets should be uncountable), and then trying to extend $f\vert_{B \setminus \{x_0\}} = b$ to a field morphism.     Edit for $^{[1]}$: in  article (MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS, by LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RUZICKA) the theorem 2.4. proves that $\Bbb R$ has no minimal generating set as field (or $\Bbb Q$-algebra, i.e. as ring). The argument uses Artin-Schreier theorem. Minimal generating sets of modules are   .  I think that $B$ is a minimal generating set of a subfield $K \subset \Bbb R$ over $\Bbb Q$ if and only if any element of $B$ cannot be written as a rational fraction of other elements of $B$. Under these conditions and using (transfinite) induction, we can probably prove (and this was my first idea) that any function $f: B \to \Bbb C$ such that $b$ and $f(b)$ are conjugate over $\Bbb Q$ (i.e. have the same minimal polynomial) extends to a (unique) field morphism $\sigma : K = \Bbb Q(B) \to \Bbb C$.  Actually, the collection $\mathscr B$ defined above doesn't have to be an inductive set, at least if we replace $\Bbb R$ by $K =\Bbb Q(\sqrt 2, \sqrt[4]{2},\sqrt[8]{2},\dots)$, because the descending chain $\left(E_n = \{\sqrt[2^m]{2} \mid m \geq n\}\right)_{n \geq 1}$ satisfy $K = \Bbb Q(E_n)$, but $K \neq \Bbb Q\left(\bigcap_{n≥1} E_n\right)=\Bbb Q$, so the chain has no upper bound w.r.t. $≤$ in $\mathscr B$. I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every vector can be written as a finite sum of vectors from the basis; such bases do exist for infinite dimension vector spaces by virtue of the axiom of choice, but usually one works with orthonormal sets for which infinite sums yield all the elements of the vector space).  So my question is - how can we prove this claim? (That an orthonormal set in an infinite dimension vector space is not a vector space basis).    Edit (by Jonas Meyer): Knapp says in a footnote on  of Basic algebra:     In the infinite-dimensional theory the term "orthonormal basis" is used for an orthonormal set that spans $V$ when limits of finite sums are allowed, in addition to finite sums themselves; when $V$ is infinite-dimensional, an orthonormal basis is never large enough to be a vector-space basis.   Thus, without explicitly using the word, Knapp is referring only to complete infinite-dimensional inner product spaces. How to move the center of scaling Hello. I'm a beginner.  In the model shown, when I select all of the top vertices and then scale them to the center, the center of scaling is not the center of the 3d-model.   How do I move the center of scaling? How do I move the manipulator widget where I want it after selecting? Travelling from India to canada via Dubai and London. Do I need a transit visa why is the set of all binary sequences not countable? modify specific column with sed or awk I have a CSV file that looks like this:  qw12er,foo,0 bn5mgh rt8yp,foo,10 gh78jk bn852mv,foo,852 78ghjkh tgbr,foo,10 ujmyhn wsx2d,foo,0000 ui52ohn tgbr,foo,7418529 ujmyhn ikl896o,foo,22 wsxdc52   I want to modify the 3rd column and remove all numbers and space from the beginning of the 3rd column.  Then output would be as follows:  qw12er,foo,bn5mgh rt8yp,foo,gh78jk bn852mv,foo,78ghjkh tgbr,foo,ujmyhn wsx2d,foo,ui52ohn tgbr,foo,ujmyhn ikl896o,foo,wsxdc52 Change bookmark depth I want all bookmarks after "Anhang" (appendix) to be a level deeper, but I need them to be \chapter{} for the right numbering.  I write a scrbook document and that's what I got:    Is it possible to change the depth of the chapter to act like a section in the bookmarks?  UPDATE  MWE:  \documentclass[12pt,a4paper,oneside]{scrbook}  \usepackage[ngerman]{babel}     % deutsche Worttrennung \usepackage[T1]{fontenc}        % Ausgabe von Sonderzeichen und Akzenten in PDF verbessert \usepackage[utf8]{inputenc}         % Eingabe von Sonderzeichen möglich \usepackage{lipsum}  \setcounter{secnumdepth}{3}     % subsubsections nummerieren \setcounter{tocdepth}{3}        % subsubsections in TOC  \usepackage[ %     pdfproducer={Texmaker},     %      pdfcreator={pdfLaTeX},      %      hidelinks,          % versteckt die Boxen um die Links in der PDF     bookmarksnumbered,      % nummereiert Lesezeichen     ]{hyperref}         % PDF Metainformationen hinzufügen   \begin{document}  \appendix \pagenumbering{alph}  \thispagestyle{empty}  \vspace*{3em} \begin{center} \begin{Huge} \textbf{\textsf{Anhang}} \pdfbookmark{Anhang}{Anhang} \end{Huge} \end{center}  \chapter{Appendix 1}  \chapter{Appendix 2}  \section{Appendix 2.1}  \section{Appendix 2.2}  \chapter{Appendix 3}  \end{document} SF Story about human adapting to alien environment This is a story that was read on BBC radio about thirty years ago, and this is about as right as I can remember it: The human pilot of a spacecraft is doing some reconnaissance on an alien planet. He crashes but survives. For some reason, no one will know where to look for him. He sees something in the distance that looks like a building of some sort. When he gets there, it seems to be a home. Very much like a human home. There's furniture, but it's strange and uncomfortable . There's food, but it's inedible. There's a faucet but what comes out of it seems more like acid. Everything looks as though it should be right, but it's not. He leaves the house, wanders and after a few days is desperate for water. But there isn't any water anywhere. He's close to death, crawling on the ground and then something happens. There are cracks in the soil and there appears to be water bubbling up from perhaps some underground stream. The water revives him. He goes back to the house and tries the food again. It's delicious and he eats enough to gain back his strength. Then he decides to try the water again. It's pure and wonderful. He takes a shower and the last line of the story is how refreshing the water is on his skin and something about him lifting his tail. I was answering a question on Stack Overflow and required to use a trailing space in one of my code snippets.  I formatted it like so:  `&lt;a `   And it renders without the trailing space: &lt;a  Using the &lt;code&gt; tag does account for trailing spaces, though in my case it busts because of the &lt;. Here's my formatting with the &lt;code&gt; tag:  &lt;code&gt;&lt;a &lt;/code&gt;   Here's how it renders:  (Yes, you should see nothing there).  This is a pretty annoying bug. Can we get it fixed? Edge won't connect to internet, Skype for business won't screen-share, Windows Store won't update apps after Windows 10 version 1809 installed Well today as I was going home I got a strange email, in my private inbox (email associated with S.O.) asking me if I had solved a question on S.O. that I didn't ask. To be more precise I had given an answer.   After reading:       and several other email related issues on S.O. I decided to let you all know about this because probably I am not the only one.   So the mail subject was:     Re: Deleted facebook account - Cannot access my apps?   which refers to   And the body (masked last name of sender):      Hi Jimmy,      Just wondering if you ever got this question answered.  If not, maybe   a brief chat with a html wizpert could help: link   .      Good luck getting this resolved!      Cheers,      Michael      ------------------------- Michael W%^%^GF Founder, CEO @ Wizpert   Now I know that my email address is not visible and I read about the MD5 regarding gravatar but I don't think this is the case. I thought about some methods and sure it's possible via metadata, github, etc for someone to get my mail, but I do not dare to think that this has reached an automated level, so that some service could easily get the email of S.O. members.   I didn't get annoyed by the message, since the CEO took the time to contact me :-p but someone else could get annoyed by these kinds of messages, since they come in the important inbox of gmail.   What is more strange is that Wizpert seems to be a nice site and also is active in the Careers 2.0 if I am not mistaken.   Another thing that crossed my mind is that since Wizpert needs unanswered questions to promote their services, possibly this is an action of Wizpert trying to attract people into their services.   Well that's it, I hope I did good to let you know (maybe I did't include the correct tags). What will happen if a plane trys to take off whilst on a treadmill? Can you create a strong blockcipher with small blocksize, given a strong blockcipher of conventional blocksize? Retrieve the position (X,Y) of an HTML element relative to the browser window What *is* an Artificial Neural Network? As we delve into  literature, we get to identify other methods with neuromorphic topologies ("Neural-Network"-like architectures). And I'm not talking about the . Examples are given below.  Then, it makes me wonder: what is the definition of an artificial Neural Network? Its topology appears to cover everything.    Examples:  One of the first identifications we make is between PCA and a linear Autoencoder with tied-weights in the encoder and decoder and thresholded activations in the bottleneck layer.  Also, a common identification is done between linear models (logistic regression in special) and a Neural Network with no hidden layer and a single output layer. This identification opens several doors.  . ? ANN. Gaussian Process? ANN (with single hidden layer with infinite hidden units).  And so, just as easily, we can incorporate arbitrary regularized versions with specialized loss functions of these algorithms into a Neural Network framework.  But the more we dig, the more similarities appear. I just stumbled into , which makes the identification of a specific ANN architecture with decision trees, allowing these to be learned by ANN methods (such as Gradient Descent backpropagation). From this we can construct Random Forests and Gradient Boosted Decision Trees from solely Neural Network topologies.  If everything can be expressed as an Artificial Neural Network, what defines an Artificial Neural Network? Please help me to find elements $a,b$ in a ring $R$ such that $a\mid b$ and $b\mid a$, but there does not exist any unit $u$ in $R$ such that $a=ub$. Looking for story title - younger gen waters down age-slowing drink of older gen, crammed into small apartment I remember a short story in which multiple generations of people being crammed into a small apartment in a city. The magical drink that either reduced or completely stopped aging was something like a soda that came in a six pack. The story focused on visual aspects of the drink at some point. The younger family members complained that their elders were living too long and apparently decided to water down the soda, I think that conclusion was ultimately left up to the reader. Many thanks for any help tracking down this story, might be one of Kurt Vonnegut's perhaps? How do I sort a dictionary by value? Which site is more reasonable for questions about R, SO or stats.stackexchange.com? I've been using SO for a little over a year now, and recently I've noticed a shift of users between SO and stats.stackexchange.com.   I'm wondering if there has been any discussion on where questions about R, a statistical environment, best fits. Would you say it's better to ask R questions on SO or would it be better to ask over at stats.stackexchange.com? What, if any, determinants would you use to decide to post between the two? What about cross posting?