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Projectile/orbital motion over very long distance We know the optimum angle for greatest horizontal displacement when launching an object with projectile motion is 45 degrees.  How to solve the angle when it is real long distance around the earth where we cannot assume that gravity is constant but is given by $F=\frac {-GMm}{r^2}$. I use differetial equation to solve it and obtain 45 degrees.   First, for the vertical distance $y$, change $v=u-gt$ to $y'=u-y''t$. Then solve it to obtain a function $y(t)$. When $y'(t)=0$ we have the time for maximum vertical distance. Then insert the value to horizontal equation.   Can anyone guide me through this problem because I am not confident with my working Regarding the sum $\sum_{p \ \text{prime}} \sin p$ I'm very confident that  $$\sum_{p \ \text{prime}} \sin p $$  diverges. Of course, it suffices to show that there are arbitrarily large primes which are not in the set $\bigcup_{n \geq 1} (\pi n - \epsilon, \pi n + \epsilon)$ for sufficiently small $\epsilon$. More strongly, it seems that $\sin p$ for prime $p$ is dense in $[-1,1]$.   This problem doesn't seem that hard though. Here's something that (to me) seems harder.      If $p_n$ is the nth prime, what is $$\limsup_{n \to +\infty} \sum_{p \ \text{prime} \leq p_n} \sin p?$$   What is $$\sup_{n \in \mathbb{N}} \sum_{p \ \text{prime} \leq p_n} \sin p? $$   Of course, we can ask analogous questions for $\inf$.   I'm happy with partial answers or ideas. For example, merely an upper bound. A closed ball in a metric space is a closed set Prove that a closed ball in a metric space is a closed set   My attempt: Suppose $D(x_0, r)$ is a closed ball. We show that $X \setminus D $ is open. In other words, we need to find an open ball contained in $X \setminus D$.   Pick $$t \in X-D \implies d(t,x_0) &gt; r \implies d(t,x_0) - r &gt; 0 $$ Let $B(y, r_1)$ be an open ball, and pick $z \in B(y,r_1)$. Then, we must have $d(y,z) &lt; r_1 $. We need to choose $r_1$ so that $d(z,x_0) &gt; r$. Notice by the triangle inequality  $$ d(x_0,t) \leq d(x_0,z) + d(z,t) \implies d(z,x_0) \geq d(x_0,t) - d(z,t) &gt; d(x_0,t) - r_1.$$  Notice, if we pick $r_1 = d(t,x_0)-r$ then we are done.  Is this correct? Bottom to top explanation of the Mahalanobis distance? I'm studying pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahalanobis distance. The books give sort of intuitive explanations, but still not good enough ones for me to actually really understand what is going on. If someone would ask me "What is the Mahalanobis distance?" I could only answer: "It's this nice thing, which measures distance of some kind" :)   The definitions usually also contain eigenvectors and eigenvalues, which I have a little trouble connecting to the Mahalanobis distance. I understand the definition of eigenvectors and eigenvalues, but how are they related to the Mahalanobis distance? Does it have something to do with changing the base in Linear Algebra etc.?  I have also read these former questions on the subject:       I have also read .  The answers are good and pictures nice, but still I don't really get it...I have an idea but it's still in the dark. Can someone give a "How would you explain it to your grandma"-explanation so that I could finally wrap this up and never again wonder what the heck is a Mahalanobis distance? :) Where does it come from, what, why?   UPDATE:   Here is something which helps understanding the Mahalanobis formula: How do I connect to a WPA wifi network using the command line? So basically, I'd like to use iwconfig to connect to my wifi network when I'm not inside X. But I just don't see a way to do it in the man page. Any idea? Is there any adjustment required when computing the correlation coefficient between:   continuous and discrete variable  two discrete variables ?    Thanks ! Using ArcMap 10.0, I am working on a map that shows many villages. Each village has a name attribute that I want to use for a label, but in some areas there are so many villages that not all can be labeled. I am using the Maplex label engine and have not checked the "Never Remove (allow overlap)" box.  I am quite happy that not all village names are displayed as this would be way too much. And it is ok that Maplex chooses which labels to display, as I have no way to know which village is more important than another one. This also means that I can't base the labeling on a field such as population or something else, even though that would be less random.  However, I would also like the village points that are not labeled to similarly not be displayed. I think it unlikely that someone would want to see the point without knowing the name of the village. If all points are shown, it is also difficult to tell which point a name label goes with in crowded areas. So instead of choosing just labels not to be shown, I also want the computer to chose features not to be shown.  To give you a better idea on what I mean:    Is there a way to make these features/points not draw if their label is not shown? I've noticed that in comments, when you create `code region` with backticks, the backslash will still work as an escape character within that region (but it should not, IMHO).  To clarify:   In comments, `\\` will be rendered as \, but it should be rendered as \\ In the post body itself, `\\` will be rendered as \\ (i.e., correctly)   This is especially confusing when you enter regular expressions into a comment, since backslash-escaping is very common there, and confusion increases if you are dealing with a regex that is double-escaped for representation in a string.   At some point you lose track how many backslashes you must make so the comment renders correctly, or you don't notice fast enough (or at all) and the comment contains wrong code where the same markup in a question or answer would come out as perfectly correct.   It's just that comments can't be corrected anymore after the grace period, leaving you with the choice of either deleting the comment, thus crippling the conversation, or engaging in a trial-and-error game of writing another comment with even more backslashes to explain the issue.  My question is two-fold:   Will this error be corrected? If yes, existing comments should probably be searched and "fixed" by a routine that tries to do the right thing (this is probably somewhere between hard and impossible) Or, can I rest assured that never ever the behavior of the backslash in comments is going to be changed, so I can start working around this issue by hand, knowing that they won't start displaying incorrectly at some point in the future? Q) Prove that if $0\rightarrow V_1\rightarrow V_2\rightarrow...\rightarrow V_k\rightarrow 0$ is an exact sequence of finite dimensional vector spaces, then $\sum_{i=1}^{k}(-1)^idim (V_i) = 0$.  For $k=3$, I see that the statement is true. For $k&gt;3$, am I correct in understanding that $V_2 = V_1\oplus V_3 (\implies dim(V_2)=dim(V_1)+dim(V_3))$ and $V_3 = V_2\oplus V_4(\implies dim(V_3)=dim(V_2)+dim(V_4))$ and so on?   If it is true, then for $k=4$, $\sum_{i=1}^{4}(-1)^idim (V_i) = -m_1+m_1+m_3-m_2-m_4+m_4$ and am not sure why $m_2=m_3$ for the sum to be $0$? where $m_i = dim(V_i)$. Thanks. read only access to all files in a specific sub-folder I have got a specific folder located in "/home/.../reboot/".  I have a series of users that require read-only access to the ever changing files within.  My problem is that the same users are not authorized to view any of the parent directory files.  Right now, the only thing I have in place is that their starting folder is at "/home/.../reboot/" by changing the etc/passwd file; and I need to block all access to other folders.  How could I only give them access to this directory? win 10 home how to rename my home directory and account login Choice function for collection of arbitrary finite sets. AC required? How do I login to the Gamecenter on PewDiePie's Tuber Simulator, if you have not done it when they asked? How to prove that for all natural numbers, $4^n > n^3$? This is a problem set I have, it's not a homework but it's very important practice...  Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably...   The problem is:     Prove that for all natural numbers $n$,  $4^n &gt; n^3$ Integral representation $B_T^3$ I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me? Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$. $$\lim_{x \to 0} \frac{f(x)}{|x|}=1$$ Then prove that $f$ is not differentiable at $0$. I was given this assignment and I'm not sure about my proof.. it feels like I'm assuming too much and not explaining with enough detail. Is there anything I need to add to make it sufficient? Assume toward contradiction that $f$ is differentiable at $0$. Then by definition we know that $f$ is continuous at $0$. Therefore $\lim _{x\to 0}f\left(x\right)=f\left(0\right)$ We know that $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}=1$ Therefore we can conclude $\lim _{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \left|x\right|=1\cdot 0=0$ But we can also rewrite this as $\lim _{x\to 0}\left(f\left(x\right)\right)\cdot \frac{\left|x\right|}{\left|x\right|}=f\left(0\right)\cdot 1=0$ Therefore $f(0) = 0$. Since $f$ is differentiable at $0$ we have $\lim _{x\to 0}\frac{\left(f\left(x\right)-f\left(0\right)\right)}{x-0}=\lim _{x\to 0}\frac{\left(f\left(x\right)-0\right)}{x}=\lim \:_{x\to \:0}\frac{\left(f\left(x\right)\right)}{x}\cdot \frac{\left|x\right|}{\left|x\right|}=\lim \:\:_{x\to 0}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\frac{\left|x\right|}{x}$ Since the one sided limits of $\frac{f\left(x\right)}{\left|x\right|}$ exist, and are both equal to 1, we have $\lim \:_{x\to 0^+}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^+}\frac{\left(f\left(x\right)\right)}{x}\cdot \:\:\:\frac{x}{x}=1\cdot 1=1$ $\lim \:_{x\to 0^-}\frac{\left(f\left(x\right)\right)}{\left|x\right|}\cdot \:\:\frac{\left|x\right|}{x}=\lim \:_{x\to \:0^-}\frac{\left(f\left(x\right)\right)}{-x}\cdot \:\:\:\frac{-x}{x}=1\cdot -1=-1$ Since the limit doesn't exist, we have a contradiction to the fact $f$ is differentiable at $0$. Therefore $f$ is not differentiable at 0. How do I redirect to another webpage? How can I redirect the user from one page to another using jQuery or pure JavaScript? System installation is stalled on Compaq Presario R3000 Time Dilation inside a hollow shell Assuming I have a hollow shell with total mass $M$ and radius $r$. On the surface, the gravitational time dilation would be  $$\tau=t \cdot \sqrt{1-\frac{v_{esc}^2}{c^2}}$$  where  $$v_{esc} = \sqrt{\frac{2 \cdot G \cdot M}{r}}$$  but inside the shell there would be no gravitational field (Newton's shell theorem and Birkhoff's theorem).  But still, the escape velocity required to escape to infinity would be the same as on the surface, since inside the shell you could move without any accelerating or decelerating forces acting on you until you reach the surface, from where you would get pulled backwards.  So is the time dilation inside the hollow shell relative to a field free observer at infinity    zero (I assume it's not) or  the same as on the surface (my best guess), or something completely different?   I found some not really duplicate but related threads on the interior of black holes which did not really focus on the math, but I am more behind the calculations in terms of $M$ and $r$. Why aren’t my villagers breeding? I’ve got one villager up 7 blocks surrounded with 6 doors. On ground level I have villagers who are supposed to breed. They are willing, their inventories are full of potatoes, and I’ve been sitting here for hours.  Hearts are popping up above their heads but nothing happens. I am playing on Xbox One.