Identify a 1950's era story about a man left on empty planet who slowly becomes native
SF Story about human adapting to alien environment
Why do we require radians in calculus? I think this is just something I've grown used to but can't remember any proof.  When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then? How to evaluate potential advisers on grounds other than their research/publications? Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A&#39; \mathrel{\overset{f&#39;}{\to}} B&#39; \mathrel{\overset{g&#39;}{\to}} C&#39; \to 0 $$ and morphisms $a : A \to A&#39;, b : B \to B&#39;, c : C \to C&#39;$ making the obvious diagram commute. The snake lemma states that there is then an exact sequence $$0 \to \ker a \to \ker b \to \ker c \to \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c \to 0$$ where the morphisms between the kernels are induced by $f$ and $g$ while the maps between the cokernels are induced by $f&#39;$ and $g&#39;$.  It is not hard to show that the morphisms induced by $f, g, f&#39;, g&#39;$ exist, are unique, and that the sequence is exact at $\ker a, \ker b, \operatorname{coker} b, \operatorname{coker} c$. With the use of a somewhat large diagram shown , we can even construct the connecting morphism $d : \ker c \to \operatorname{coker} a$. However, I'm stuck showing exactness at $\ker c$ and $\operatorname{coker} a$. I thought Freyd might have had an element-free proof in his book, but it turns out he proves it by diagram chasing and invoking the Mitchell embedding theorem [pp. 98–99]. Is there a direct proof? I'm trying to make a list of ways to tell if a given degree sequence is impossible. For example $3,1,1$  is not possible because there are only 3 vertices in total so one can't have degree 3.  The list so far   vertices has degree equal to or larger than number of vertices sum of degrees is odd for n vertices if one has degree n-1 and another has degree 0 for n vertices the sum of the degrees cannot be greater than $n(n-1)$ because this would be have more edges than a complete graph If $a > b$, is $a^2 > b^2$? Given $a &gt; b$, where $a,b ∈ ℝ$, is it always true that $a^2 &gt; b^2$? Let $f_1 : [-1, 1] \rightarrow \mathbb{R};\: f_1(0) = 0 $ be a continuously differentiable function and $\lambda &gt; 1$. Consider the sequence of functions defined inductively by $f_k(x) := \lambda f_{k-1}\left(\dfrac{x}{\lambda}\right);\: k \geq 2;\: x\in [-1, 1]$. Find the pointwise limit of the sequence of functions $(f_n)$.   I figured out that $f_k(0)=0 \:\forall\: k\implies f_k\rightarrow 0 $ when $ x=0$. Also $f_k$ is continuously differentiable $\forall \:k$.   $\Bigg($Since $\quad\displaystyle\lim_{h\rightarrow0}\dfrac{f_2(x+h)-f_2(x)}{h}=\lambda\lim_{h\rightarrow0}\dfrac{f_1\left(\frac{x+h}{\lambda}\right)-f_1\left(\frac{x}{\lambda}\right)}{h}.\: $Since$ f_1 $is continuously differentiable we see $ f_2 $ is differentiable and its derivative is continuous and hence $f_k$. $\Bigg)$.  Further since $f_1(0)=0$. So I assumed $f_1(x)=x\cdot g(x)$, where $g(x)$ is continuously differentiable and $g(0)\neq0$. Assume $x\neq0,\:f_2(x)=\lambda\cdot\dfrac{x}{\lambda}g\left(\dfrac{x}{\lambda}\right)=x\cdot g\left(\dfrac{x}{\lambda}\right)\implies f_3(x)=x\cdot g\left(\dfrac{x}{\lambda^2}\right)$ $\implies f_k(x)=x\cdot g\left(\dfrac{x}{\lambda^{k-1}}\right)$.   So taking limit as $k\rightarrow \infty$, we get $\displaystyle \lim_{k\rightarrow\infty}f_k(x)=x\cdot g(0)$. Now I am not sure how to proceed or whether there is something wrong here. Please provide hints or suggestions.  $\rule{17cm}{1pt}$  To sum up (using the arguments made by $\textbf{zhw}$ and $\textbf{Paul Frost}$)  When $\displaystyle x\neq0, \lim_{n\rightarrow\infty}f_{n+1}(x) = \lim_{n\rightarrow\infty}x\,\frac{f_1(x/\lambda^n)}{x/\lambda^n} = \lim_{n\rightarrow\infty}x\,\frac{f_1(x/\lambda^n)-f_1(0)}{x/\lambda^n}=x\:f'(0).$ Thus  $$f_n\rightarrow\begin{cases}0&amp;x=0\\x\:f'_1(0) &amp;x\neq0\end{cases}=f $$ Are undead and creatures immune to charm excluded from the HP count of Sleep? From the spell description:     Sleep (...) Creatures within 20 feet of a point you choose within range are affected in ascending order of their current hit points (ignoring unconscious creatures). Starting with the creature that has the lowest current hit points,(...) Subtract each creature’s hit points from the total before moving on to the creature with the next lowest hit points. A creature’s hit points must be equal to or less than the remaining total for that creature   to be affected. Undead and creatures immune to being charmed aren’t affected by this spell. (PH p.276)   Now, to build the ascending order of their current hitpoints, the spell instructs to count everyone in area of effect, but only ignore unconscous creatures. Then later on it says that Undead and creatures immune to being charmed aren't affected.  Are those two latter kinds of creatures excluded from the roll of current HP by default? Following the "if they were it would say so" philosophy of Sage Advice, it doesn't seem so to me. IE11 making background image text blurry Word for a task which is flawed or doomed to failure but which you have to do anyway? Why is the word 'number' abbreviated to 'No.' in UK English and '#' in American English? Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected? Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon &gt;0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)&gt; \varepsilon \}. $$ Then  $U_\varepsilon$ is open but in general not connected.  But I suppose that for sufficiently small $\varepsilon &gt;0$ also  $U_\varepsilon$ is connected. How to prove its?  Edit. Let's assume that additionally $U$ is bounded. Is then $U_\varepsilon$ connected? Keep box-shadow direction consistent while rotating Why are we limited to six questions per day? Is Dirichlet function Riemann integrable? One might expect the sum of the tracert latencies to equal 267ms: Find a formula for a sequence $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},...\}$ Need help finding a good book on Riemann Geometry Questions can be protected (i.e. marked "highly active").   What does it mean for a question to be protected? Why are some questions protected? Who can protect and unprotect questions? Who can answer a protected question? How are protected questions displayed? When should I protect or unprotect a question? When does the Community user protect a question? How to Understand the Relationships Among Random Variables, Samples, and Populations? 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.