Electrical Generator
Can the Earth's magnetic field be used to generate electricity?
Does magnetic monopole violate $U(1)$ gauge symmetry? Does a magnetic monopole violate $U(1)$ gauge symmetry? In what sense and why? Insofar as I know, there are at least two types of magnetic monopoles. One is the Dirac monopole while the other is the monopole in Grand Unified Theories (GUT), e.g., 't Hooft-Polyakov monopole.  In the latter case, some non-Abelian gauge theory breaks down to a (compact) $U(1)$ gauge theory wherein monopoles can be observed from a large distance. The monopole field is not from the $U(1)$ degrees of freedom and nothing is singular. On the contrary, Dirac monopole is singular (topological defect?) and we must introduce Dirac string or patches of vector potential $\vec{A}(x)$. Are these no more than artifacts arising when one tries to describe the monopole field within the $U(1)$ gauge theory by assuming $\vec{B}=\nabla\times\vec{A}$? It seems true since two Maxwell's equations that magnetic monopole might affect can be cast in the differential form $\mathrm{d}F=0$ and $F=\mathrm{d}A$ in Minkowski spacetime, from which $U(1)$ gauge invariance manifests for $\mathrm{d}(A+\mathrm{d}\chi)=\mathrm{d}A$.  However, I later noticed two seminal papers on Dirac monopole  &amp; , in which the authors claimed the following   electromagnetism without monopole $\rightarrow$ connection on a trivial $U(1)$ bundle electromagnetism with monopole $\rightarrow$ connection on a nontrivial $U(1)$ bundle And the wave function of an electron around a Dirac monopole should be regarded as a section which is free from discontinuities.   I've no idea about fibre bundle. Anyway I guess $U(1)$ theory does not necessarily expel Dirac monopoles. It has something to do with the topology of some manifold in the theory? Which manifold? Can anyone shed light on this intriguing issue? Thanks in advance. Estimating partial sums $\sum_{n = 1}^m \frac{1}{\sqrt{n}}$ Apostol's Calculus, exercise number I 4.7 13.     Prove that if $n \geq 1$, then   $$ 2(\sqrt{n+1} - \sqrt{n}) &lt; \frac{1}{\sqrt{n}} &lt; 2(\sqrt{n} - \sqrt{n-1}) $$   and use this to prove that if $m \geq 2$, then   $$ 2 \sqrt{m} - 2 &lt; \sum_{n = 1}^m \frac{1}{\sqrt{n}} &lt; 2\sqrt{m} - 1 $$   In particular, when $m = 10^6$ the sum lies between $1998$ and $1999$.   After establishing the trivial case, for $n=1$, I can't think of a way to perform the inductive step:  assuming $2(\sqrt{n+1} - \sqrt{n}) &lt; \frac{1}{\sqrt{n}} &lt; 2(\sqrt{n} - \sqrt{n-1})$ and using this, deducing the same for $n+1$ is true as well:  $2(\sqrt{n+2} - \sqrt{n+1}) &lt; \frac{1}{\sqrt{n+1}} &lt; 2(\sqrt{n+1} - \sqrt{n})$  Any help is very appreciated, thanks Interesting tab on StackOverflow One day I directed to SO and was met with the awesome Interesting page, which I really like. But when I travel away from it, I have to re-enter the query string to get it back. Is there a plan to make this tab visible so I can click to it anytime? "big enough to do this" My question is what is the difference among these three sentences.   He is big enough to do such  heavy works. He is not big enough to do such heavy works. He is small enough to do such  heavy works.   I am so much confused between 2 and 3.Also, I don't get what the 3rd one means. Is there any way to get a hat wider than widehat?  Why doesn't  \widehat{abcdefghijklmnopqrstuvwxyz}   really go over all of it? What is the best book to learn probability? Question is quite straight... I'm not very good in this subject but need to understand at a good level. How can momentum but not energy be conserved in an inelastic collision? In inelastic collisions, kinetic energy changes, so the velocities of the objects also change.  So how is momentum conserved in inelastic collisions? Ubuntu 20.04 on DualBoot fail 90% of tries - Initramfs unpacking failed: Decoding failed Updating my appointment without cancelling the previous date Triangle inequality for L2 norm I am posed with a problem as below.  Let $X$ be the collection of real valued functions that are measurable on $[0,1]$. For a given number $p\geq 1$ define the $L^p$ norm by $(\int_0^1|f(x)|^p)^\frac{1}{p}$.  I need to show for $p=1,2$ that this defines a semi norm on $X$. $p=1$ was simple as was all of $p=2$ except for the triangle inequality. I am having trouble proving the triangle inequality for this. Could anybody nudge me into the right direction or am I simply over thinking it? I think I can prove it for vectors but I'm not sure how to tackle it with integrals. Right now I can only use relatively basic results because this in for an introductory measure theory course.  Thanks! How can brackets be escaped in using string.Format? For example: String val = &quot;1,2,3&quot; String.Format(&quot; foo {{0}}&quot;, val);  This example doesn't throw an exception, but it outputs the string foo {0}. Is there a way to escape the brackets? How come the PHP private class var is not private? Multiple submit buttons in an HTML form Let's say you create a wizard in an HTML form. One button goes back, and one goes forward. Since the back button appears first in the markup when you press Enter, it will use that button to submit the form.  Example:    &lt;form&gt;   &lt;!-- Put your cursor in this field and press Enter --&gt;   &lt;input type="text" name="field1" /&gt;    &lt;!-- This is the button that will submit --&gt;   &lt;input type="submit" name="prev" value="Previous Page" /&gt;    &lt;!-- But this is the button that I WANT to submit --&gt;   &lt;input type="submit" name="next" value="Next Page" /&gt; &lt;/form&gt;     I would like to get to decide which button is used to submit the form when a user presses Enter. That way, when you press Enter the wizard will move to the next page, not the previous. Do you have to use tabindex to do this? What is the best way to remove accents (normalize) in a Python unicode string? Is it acceptable to take photos of slides at a conferences for private use? Is it accepted/appropriate to take pictures of presentation slides at conferences for private use?   I am not talking about putting those pictures out in the public afterwards, I know this is illegal without consent of the presenter.    Should it matter, I am talking about math conferences.  And I am asking because I don't see anyone taking pictures. How to start a new Linux distro? Since Naruto failed the Genin exam so many times, doesn't that mean he's older than Sasuke and Sakura? I have a server running with the timezone set to UTC. It seemed like that was generally a good practice (please correct me if I'm wrong).  Anyhow, one of the servers I connect to, in order to scp files, is running on EDT and stores files that I need to copy in the format /path/to/filename/data20120913  I looked at trying to rsync files using something like find's -mtime -1 flag for files modified in the last day, but I didn't have any luck.  I don't mind just using scp to copy the current day's file, but as of right now there is a 4-hour window where running date +%Y%m%d will give a different day on each server and that bugs me a little.  Looking through man date I see that I can have the time output as UTC, but I don't see a way to have it output as another timezone like EDT  I suppose I could also use something like the GNU date extension date -d 20100909 +%s to get the date in seconds from the epoch, apply a manual 4 * 60 * 60 second calculation, and see about rendering that as a date - but then when daylight time kicks in it will still be an hour off.  Is there a simpler way to output the date in a YYYYMMDD format for EDT on a server that is set to UTC ? Accessing a property of derived class from the base class in C# I've seen many posts regarding a combinatorial proof of the following question. But for a non-combinatorial proof would the following method work? Also... is this the easiest way to arrive at a proof? It seems to be rather verbose.  Show the formula $k {n\choose k} = n {n-1 \choose k-1}$ is true for all integers $n,k$ with $0\le k \le n$.  My answer:  Observe that  $$\begin{align*} {n\choose k}&amp;= \frac {n}{k} {n-1 \choose k-1}\\ &amp;= \frac {n}{k}\left[ {n-2 \choose k-2} + {n-2 \choose k-1}\right]\\ &amp;= \frac {n}{k}\left[ \frac{(n-2)!}{(k-2)!(n-2-(k-2))!} + \frac {(n-2)!}{(k-1)!(n-2-(k-1))!}\right]\\ &amp;= \frac {n}{k}\left[ \frac{(n-2)!}{(k-2)!(n-k)!} + \frac {(n-2)!}{(k-1)!(n-k-1)!}\right]\\ &amp;= \frac{(n-1)!}{(k-1)!(n-k)!} + \frac {(n-1)!}{(k)!(n-k-1)!}\\ &amp;= \frac{(n-1)!}{(k-1)!(n-1-(k-1))!} + \frac {(n-1)!}{(k)!(n-1-k)!}\\ &amp;={n-1 \choose k-1} + {n-1 \choose k}={n \choose k}. \end{align*}$$