Given any continuous map between topological spaces $f: X \rightarrow Y$, we can define the mapping cone $C_{f}:=((X \times[0,1]) \sqcup Y) / \sim,$ with the equivalence relation generated by $(x, 0) \sim f(x)$ for any $x \in X$ and $\left(x_{1}, 1\right) \sim\left(x_{2}, 1\right)$ for any $x_{1}, x_{2} \in X .$ Now let $f: X \rightarrow Y$ and $g: X \rightarrow Y$ be two homotopic maps. Show that then, the mapping cones $C_{f}$ and $C_{g}$ are homotopy equivalent.  I’ve made a search and have seen a few other similar questions. However, I’d like to ask some questions below, and hopefully can come up with the solution myself. Here’s the picture of $C_f$ and $C_g$. My task would be to find maps $\alpha, \beta$ such that $\beta \circ \alpha \simeq id_{C_f}$ and $\alpha \circ \beta \simeq id_{C_g}$.  Say we want to define $\alpha$ first. My guess is that there are three kinds of point that I need to consider: $\bullet p \in X \times (0,1]$ $\bullet p \in Y \setminus (f(X) \cup g(X))$, and $\bullet p \in X \times \{0\}$ I’m thinking of letting $\alpha(p) := p$ for the first two kinds. Doing the same for $\beta$ and we’d have $\beta \circ \alpha = id_{C_f}$ and $\alpha \circ \beta = id_{C_g}$, with the domains restricted to the first two kinds of points. As for the third kind, I’m not really sure. What would happen if I also use the identity map there? If that doesn’t work, any hints on how to use the homotopy between $f$ and $g$ here?
Let $f,g:X\to Y$ be maps of spaces such that $f\simeq g$. Is it true that the mapping cones $\operatorname{cone}(f)$ and $\operatorname{cone}(g)$ are homotopy equivalent?  Can we write down an explicit homotopy equivalence? It ought not to be too hard to get a map $\operatorname{cone}(f)\to \operatorname{cone}(g)$ from the existence of a homotopy $H:X\times I\to Y$, but I haven't been able to find a sensible candidate, despite trying for quite a while.
What is the purpose of the thin wire at the top of power transmission lines? I know a power transmission line carries usually two sets of 3 phase conductors. But I have noticed a relatively thin wire at the center-top of the power transmission line. What is the purpose of this thin wire? For which positive integer values of $n$ does $a|mn-b?$ I am taking a number theory class, and I was not able to solve this homework problem.   Let $a,$ $b,$ and $m$ be constant positive integers. For which positive integer values of $n$ does $a|mn-b?$   I tried this out for a couple of values, and the values of $n$ satisfying this expression seem to occur in a cycle.  (For example, let $m=2019,$ $a=20,$ $b=19.$ The values of $n$ that work are $1,21,41,61,....$) However, I am not able to show that this always happens. How can I visualize or simulate the effect of different focal lengths? I have seen many camera review sites illustrate the effect of using different focal lengths on the same photograph using frames. There are also similar illustrations to compare different sensors (). I find this kind of visualization to be very useful in comparing the actual effect of the longer focal length\different sensor size. Of course the ability to do a wider focal length visualization would also be equally useful, but that's not possible.  Is there any ready-made software or plugin that enables this? If not, is there any simple technique to visualize the effect of longer focal lengths on an image (one would of course have to use the zoom information in the image EXIF data to accurately represent this)? All I can think of is to use some trigonometry to do the necessary calculations for the frame sizes for cropping. Why can't we use entanglement to defy Heisenberg's Uncertainty Principle? In principle, it is possible to entangle any property of two particles, including speed and momentum. Surely then, this could be used to defy the Uncertainty Principle, which states that the momentum and position of a particle cannot be together measured with accuracy, but the more accurate one gets, the less accurate the other becomes:  $$\Delta x\Delta p=\dfrac{\hbar}{2}.$$  If one was to set up an experiment where two particles are entangled, one could then measure the position of one particle with great accurary at the same time as measuring the momentum of the other particle with great accuracy, which would mean that, as the two particles are entangled, one would then know the momentum of the particle they measured the position of.  Therefore, the Uncertainty Principle would have been defied, as both momentum and position have been measured with a high accuracy.  However, this can't be true, as someone would have thought of this before.  So why wouldn't this work? Where are some places to get reference images for modeling? I'm a newbie.  I've gone through a number of tutorials and want to get some actual experience with modeling. But, I don't know where to go to get good reference images for modeling with front/sides views.  Can anyone recommend the best places to find such images? Is it possible to add a user and restrict him to run special programs? For example after this user logs in, he can only open Firefox to use Internet and no other programs can be run by this user. For example in a terminal, commands will not be accessible when you delete some environment variables like $HOME. But is it possible to avoid running programs in a graphical shell, like Gnome? If yes, how? Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$ I found the following question about compactness:     A set $S\subseteq \mathbb{R}^n$ has the following property:   if a sequence of $S$ has exactly one accumulation point in $S$, then it converges in $S$. Prove that $S$ is compact.    We know that in $\mathbb{R}^n$ compact sets correspond to closed and bounded sets.   (1) I am trying to prove that $S$ is closed. The condition in the question says if a sequence of $S$ has an accumulation point in $S$ then it converges in $S$. I think instead the condition should be if a sequence of $S$ has an accumulation point( in $\mathbb{R}^n$, not necessarily in $S$ )then it converges in $S$.  Am I correct?  (2) How to prove that $S$ is bounded?  P.S. I have made an edit (there is exactly one accumulation point).  Source of the question:  of this problem set. Why can't two series-connected diodes act as a BJT? Another form of the question is: will joining two diodes with wires (pn-np) make a transistor-equivalent?  I read that they are not equivalent, but why? Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective Puppet client not connection to master - SSL connect Error I installed the puppet master and agent on the same machine. When the client is started, I got the following error message.  puppet agent --server=agent.com --no-daemonize --debug    err: Could not retrieve catalog from remote server: SSL_connect returned=1 errno=0 state=SSLv3 read server certificate B: certificate verify failed.  This is often because the time is out of sync on the server or client Do eyeballs exhibit chromatic aberration? Fairly straightforward question. If not, why not?  I suspect that if they do, it is not perceived due to the regions of highest dispersion being in one's region of lowest visual acuity. With arrays, why is it the case that a[5] == 5[a]? As Joel points out in , in  (aka: K &amp; R), there is mention of this property of arrays in C: a[5] == 5[a]  Joel says that it's because of pointer arithmetic but I still don't understand. Why does a[5] == 5[a]? Interference between strokeopacity and fillopacity I have a few circles in a drawing, and want to highlight some of them.  \begin{tikzpicture}  \coordinate (A1) at (0,0); \coordinate (A2) at (1,0); \coordinate (A3) at (1,1); \coordinate (B)  at (1,0);  \draw (A1) -- (A2); \draw (A1) -- (A3); \draw (B)  -- (A3);  \foreach \point in {A1,A2,A3,B} {    \fill [black,draw=white](\point) circle (3pt); }  \pgfsetstrokeopacity{.3} \pgfsetfillopacity{.3} \fill [red,draw,rounded corners,line width=9pt] (A1) -- (A2) -- (A3) -- cycle; \pgfsetstrokeopacity{1} \pgfsetfillopacity{1} \end{tikzpicture}   I would like the whole area surrounding the A points to be of the same color, but the strokeopaticy and the fillopacity interfere.  Is there some ay I can prevent this?  Note: in my actuall use scenario, there are too many coordinates for manual adjusting to be an option (otherwise I would just do with a fill with rounded corners, with points calculated manually). I have a function with the same name, but with different signature in a base and derived classes. When I am trying to use the base class's function in another class that inherits from the derived, I receive an error. See the following code:  class A {     public:     void foo(string s){}; };  class B : public A {     public:     int foo(int i){}; };  class C : public B {     public:     void bar()     {         string s;         foo(s);     } };   I receive the following error from the gcc compiler:  In member function `void C::bar()': no matching function for call to `C::foo(std::string&amp;)' candidates are: int B::foo(int)   If I remove int foo(int i){}; from class B, or if I rename it from foo1, everything works fine.   What's the problem with this? Record Animation In BGE If the matrix $A^3 = 0$, is $A$ singular? How do I get the Minecraft GUI scale to go back to Auto? I have been playing Minecraft, and I went into the settings to see what could be changed. I was in display and I saw the GUI scale was set to "Auto", which was fairly large. I cycled through the scales until I got back to auto, but the GUI scale was stuck at normal size and wouldn't allow me to make it large again. I know this isn't such a big problem, but I really like big font and this is a constant bother to me. Why are SO participants so unfriendly? I have got myself a dxf which is built with units in the wrong scale. How do I upscale that in QGIS?   Note it does open in QGIS, just in the wrong map units and location.