What are the technical challenges to store electricity from lightning strikes?
What kind of capacitor could build up, and maintain a charge acquired from lightning?
I want to prove the definition of limit of a function from definition (1) to definition (2).  (1) A function $f(x)$ is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for every sequence $\{x_n\}$ with limit $x^*$, the limit $\lim_{n-&gt;\infty}f(x_n)$ exists and is equal to $f^*$. (2) A function is said to have $f^*$ as the limit as $x$ tends $x^*$ iff for each $\epsilon &gt;0$, there exists a $\delta_\epsilon &gt;0$ such that $|f(x)-f^*| &lt;\epsilon \forall |x^*-x|&lt;\delta_\epsilon$  I am confused about the steps. Suppose I am given an $\epsilon &gt; 0$ and $x^*$. I have to show the existence of a $\delta_\epsilon &gt;0$ such that for all $x$ satisfying  $|x^*-x|&lt;\delta_\epsilon$ I can construct a sequence $x_n$ which converges to $x^*$ and $x_n=x$ for some $n \geq N$. Then I can say that $f(x_n)-f^*=f(x)-f^* &lt; \epsilon$. I think I may have made some mistake about the steps.  My main confusion is the following : Suppose I choose $\delta_\epsilon=\epsilon$. Then if a sequence $\{x_n\}$ converges to $x^*$ then $x_n=x$ (where $x$ satisfies $|x-x^*|&lt;\epsilon$) for some $n\geq N$. Now, $\lim_{n-&gt;\infty}f(x_n)$ exists implies there exists an $M$ such that $\forall n\geq M$, $f(x_n)-f^* &lt;\epsilon$. Now, I need to show that for some $n \geq M$ indeed $x_n=x$. How do I show that ? This $M$ and $N$ are different. How to get less spacing in math mode LaTeX commands like this:  Here again, four different simulations are run on networks of size  $C^{G} = 25 \times\ 25 = 625$, $C^{G} = 50 \times\ 50 = 2500$, $C^{G} = 250 \times\ 250 = 62500$, and $C^{G} = 500 \times\ 500 = 250000$...   When compiled generate the following:    The spacing in math mode really irritates me. There's too much white space around the equal sign, and the multiplication sign has more right padding than left padding. Is there any way to reduce this spacing? There are some kind of door devices which push the door back after we open the door and come out.  My problem is that the door area is not constructed in a way that such a device can be installed.  There is no space adjacent to the door anywhere.  Automatic closure of door is extremely important to me.   Please see following pictures and tell what choices do I have. apt-get remove with wildcard removed way more than expected. why? Assume I have a flight ticket from A->B->C and at A I checked in luggage all the way to C. However after the first flight I don't board the second flight but decide to remain in B.  As I understand procedures my luggage will need to be unloaded again from the plane in B when they notice I did not board the flight.    How and when can I retrieve my luggage at B? (Or if not, what is the fate of my luggage?) Is there any way I can advise airport/airline staff in B to not load my luggage in the first place? (I understand that asking to do so when checking in in A would be declined / met with a fare recalculation)   This question is different from  in that I am looking for a general answer, whereas in that case B happened to be the port of entry to the US where you have to collect your luggage in any case. Also it is different from   in that I do want to check in  luggage from A to B. Finally I am also aware that possible further legs of my trip would be cancelled.  is asking essentially the same but was marked duplicate to the above two questions that do not answer my question. Are repository updates secure?  As a bear of little brain from the developer side, I cannot understand why the repository list is http://security.ubuntu.com and the other http(unsecured) sites listed in /etc/apt/sources.list.  Without a certificate chain match this appears as "ask any responder for a list of packages to update" instead of "ask the ubuntu.com site..."  Can any network choose to spoof the update sites, and is this a common practice to provide a locally cached and vetted copy? I computed the homology groups of the torus, can someone tell me if this is correct? The calculation, not the result that is. Thanks!  The cells of $T^2$ are $e^0, e^1_a, e^1_b, e^2$  The chain groups are  $$ C_0(T^2) = \{ k e^0 |&#160;k \in \mathbb{Z} \} = \mathbb{Z}$$  $$ C_1(T^2) = \{&#160;k_1 e^1_a + k_2 e^1_b |&#160;k_1 , k_2 \in \mathbb{Z} \} = \mathbb{Z} \oplus \mathbb{Z}$$  $$ C_2(T^2) = \mathbb{Z}$$  $$ C_k(T^2) = \{ 0 \} , k &gt; 2$$  Now the homology groups:  $$ H_0(T^2) = \ker \partial_0 / im \partial_1 = \mathbb{Z} / 0 = \mathbb{Z}$$  where $im \partial_1 = 0$ because there is no chain in $C_1(T^2)$ whose boundary is a zero-chain in $C_0(T^2)$. (Is this reasoning correct?)  $$ H_1(T^2) = \ker \partial_1 / im \partial_2 = \mathbb{Z} \oplus \mathbb{Z}$$  where $\ker \partial_1 =  \mathbb{Z} \oplus \mathbb{Z} $, i.e. again everything maps to zero because there is no element in $C_1$ whose boundary maps to an element in $C_0$.  $im \partial_2 = 0$ again because there is no element in $C_2$ whose boundary is an element of $C_1$.  I don't want to use Hurewicz to do $H_1$.  $$ H_2(T^2) = \mathbb{Z}$$ using similar arguments as above.  Thanks for your help.  Edit  I posted the computations as an answer below. I got two up votes but I don't know by who so I'm not yet sure I can trust them... Start Mathematica without the menu bar? How to access a variable from another script in Unity c# I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above? From my old HD (Ubuntu 16.04) I copied ~/.thunderbird and other programs to the new disk with Ubuntu 18.04 installed. All went good. I tried to install Opera. The installation was unsuccessful, a window asking something was held. After that Thunderbird Local Folders -> Inbox (only) was invaded by large icons like this screenshot  Flowers, plants, trucks and others. I restarted but those images are still there. Opera was successfully installed. What can I do to remove them? Consider the subset $B=\{(x,y):x^2+y^2\leq 1\}\ \subseteq \mathbb{R}^2$. Show that $\vert B \vert = \vert \mathbb{R}^2 \vert$.  It is easy to see tht $f:B\rightarrow\mathbb{R}^2$ defined as $f(x,y)=(x,y)$ is injective.   However, in the book I'm working out of (Book of Proof by Richard Hammack), the other injection noted is  $g:\mathbb{R}^2\rightarrow B$ defined as  $$g(x,y)=\left(\frac{x^2+y^2}{x^2+y^2+1}x,\frac{x^2+y^2}{x^2+y^2+1}y\right).$$  In the book I'm asked to verify this is indeed an injective function.  My approach was to assume $g(x,y)=g(m,n)$ and through basic algebra I was able to deduce  $x(m^2+n^2+1)(x^2+y^2)=m(x^2+y^2+1)(m^2+n^2)$  and  $y(m^2+n^2+1)(x^2+y^2)=n(x^2+y^2+1)(m^2+n^2)$  but how does this show $(x,y)=(m,n)$? Or is this the wrong approach? Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set I'm mainly trying to prove that      If $0\not \in S\subseteq R$ is a multiplicative subset of a   commutative ring $R$ with identity. Then $R-S$ contains a prime ideal.   Now, by using Zorn's lemma, one can show that $R-S$ contains a maximal ideal I. In fact, I is a prime ideal as well, so we need to show this.To do that, I assumed the opposite, i.e $\exists a,b \in R$ s.t $ab\in I$, but $a,b \not \in I$. Then looked each case. However, I stuck when $a,b \in R-[S\cup I]$. The following is what I've done for this case;  Proof:  In this case, our assumptions are $a,b\in R-[S\cup I]$ and $I + (a) = J$, where $J$ is an ideal in $R$ and $J \cap S \not = \emptyset$.   These imply that; $S^{-1}(I + (a)) = S^{-1} J$, but since $J \cap S \not = \emptyset$, by a theorem that we proved in the class, $S^{-1} J = S^{-1}R$.Therefore, it must be true that $\exists (i\in I, r\in R, s_1,s \in S)$ s.t  $$\frac{i+ra}{s_1} =\frac{1}{s}$$ but this implies $\exists s_2 \in S$ s.t  $$ s_2(is + sra -1 s_1) = 0 \quad \Rightarrow \quad (s_1 s )i + (s_1 s r) a = (s_1 s )1,$$ ,so $$(b s_1 s )i + (s_1 s r) ab = (s_1 s )b.$$ Observe that by our assumption $ab\in I$, so $(s_1 s)b \in I$.  Now, at the point, I don't know how to proceed, so I would appreciate any help or hint because I stuck at this point for over a week now.  Edit  Thanks for the great answers; however, but I'm specifically interested in how to continue from the point where I stuck. Why does Ridders' method work as well as it does? I've just read section 9.2.1 in Numerical Recipes Ed. 3 (Press et al. 2007), which describes Ridders' method of root finding. I understand that allowing for some curvature of the function by linearizing three points with an exponential will probably get you a better approximation and more rapid convergence, but I have no idea how they get that it converges quadratically, and the motivation behind choosing an exponential (rather than e.g. a polynomial) is not clear to me.  So, specifically, I'm wondering (1) why linearize with an exponential, and (2) why the method converges quadratically.  I tried looking for Ridders' original paper "A new algorithm for computing a single root of a real continuous function", but the only copy I could find was in a journal that my school's library does not give me free access to.  UPDATE:  I've realized that the fact that an exponential is strictly positive means that weighting the three points with an exponential won't change their signs. Hence, the root of the linearization will have to land inside the original bracket.  Polynomials, rationals, sinusoids, and logarithms all have positive and negative bits, so using them probably runs the risk of getting bad interpolated roots. One of these squared might be usable, but maybe non-monoticity gives some sort of undesirable effect. Since this is a complete list of all simple functions that I am aware of, exponentials seem to be the only easily available weighting function with this property.  A constant function is also technically single-signed and monotone, but it can't be used to linearize stuff. Is there an MD5 Fixed Point where md5(x) == x? Is there a fixed point in the MD5 transformation, i.e. does there exist x such that md5(x) == x? cant install or remove anything on 18.04 cause of DPKG error After an AJAX request, sometimes my application may return an empty object, like:  var a = {};   How can I check whether that's the case? How are "related" questions selected? Drawing a triangle from medians What do the brackets around processes mean? $ ps -Awwo pid,comm,args   PID COMMAND         COMMAND     1 init            /sbin/init     2 kthreadd        [kthreadd]     3 ksoftirqd/0     [ksoftirqd/0]     5 kworker/u:0     [kworker/u:0]     6 migration/0     [migration/0]     7 cpuset          [cpuset]     8 khelper         [khelper]     9 netns           [netns]    10 sync_supers     [sync_supers]    11 bdi-default     [bdi-default]    12 kintegrityd     [kintegrityd]    13 kblockd         [kblockd]    14 kacpid          [kacpid]    15 kacpi_notify    [kacpi_notify]    16 kacpi_hotplug   [kacpi_hotplug]    17 ata_sff         [ata_sff]    18 khubd           [khubd]   What do the brackets mean? Does args always return the full path to the process command (e.g. /bin/cat)?