I have a laptop, which comes with an SMPS charger. The laptop charger is a very common type. I know there are some earthing issues in my home. I need to fix that. But SMPS isolate mains from the supply, but when I touch the body of the laptop, it shocks. Another day, I was charging my phone through USB from the laptop and plugged the laptop charger. I own a metal earphone, and that shocked me. I know there's some problem in my home earthing part, but still aren't SMPS meant to be isolated from mains. Then why I'm getting that shock. Why does electricity want to flow through my body if it's floating? Someone, please explain the incident, with reasoning and possible ways to avoid it.
So, I got some laptop charger from my neighbor and I was thinking about to do something with it. Then one day I feel tiny shock from the output of the charger. I thought ac to dc filtering mechanism maybe shorted. Then I had tested other remaining charger to see if there was any AC current going on. And they all do the same. Is it normal to have flow some AC through DC ? Can I do something about it ? make it safer for work ? It does not feel like comfortable.
Gravity in 2d space and inverse linear law In our three-dimensional universe, gravity obeys the inverse square law. In a four-dimensional universe, gravity would be expected to obey the inverse cube law et cetera.  In a two-dimensional universe, one would similarly expect gravity to obey an inverse linear law. I've seen it claimed that this is not actually true, when you work it out according to general relativity there would actually be no gravity at all in such a universe. Is this true? If so, is there a layman's explanation of why? Is it related to the fact that gravitational potential wells would be infinitely deep?  Would electrostatic forces follow the same law as gravity?  If there existed infinitely long strings in our universe, would they have the same gravitational effect as point particles in a 2d universe? How to calculate Network Service Areas in QGIS? Is there a limit as to how fast a black hole can grow? Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$ I want to prove the assertion:      The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \pmod{4}$.   My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and $$ p = \epsilon^{-1} (1-\zeta_p)^{p-1} $$ where $\epsilon$ is a unit.  Since $p$ is an odd prime, $(p-1)/2$ is an integer and  $$ \sqrt{\epsilon p } = (1-\zeta_p)^{(p-1)/2} $$ makes sense and belongs to $\mathbb{Z}[\zeta_p]$.    How do I deal with the $\epsilon$ under the square root?  I guess the condition on the congruence class of $p$ comes from that.  Is this even the right way to proceed?  Uniqueness is not clear to me either.  I thought about looking at the valuation $v_p$ on $\mathbb{Q}$, extending it to two possible quadratic extensions beneath $\mathbb{Q}(\zeta_p)$, then seeing how those have to extend to common valuations on $\mathbb{Q}(\zeta_p)$, but I didn't see how to make it work.  I would appreciate some help. In a sentence taken from the Oxford Dictionary:     ‘The inscription on his tombstone in Groombridge Church, where he is buried alongside his three children, bears his original name and no reference to his nom de plume.’   Is ‘buried’ an adjective, a , or a   because it follows the copular verb is?      Predicate nominatives complete only linking verbs. The linking verbs include the following: the helping verbs is: am, are, was, were, be, being, and been; the sense verbs: look, taste, smell, feel, and sound; and verbs like become, seem, appear, grow, continue, stay, and turn.   Or is it a past participle because the clause is in the passive voice?   Does “he is buried” mean currently his body lies in a coffin covered with soil, or  does it mean he has been buried by someone?  If we say ‘he was buried’ it clearly refers to an event that happened and ended sometime in the past. It might even suggest that the burial ground has since been changed. Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$? We were trying to come up with an easy way to generate a rational number in between two existing rational numbers with a fairly low numerator and denominator (the way we were doing this earlier was to find the average of the two rationals, but that results in a denominator of up to $c * d$. Does this inequality hold for all values of $a, b, c, d$? How can I efficiently check the originality of my work? Is there any way to backtrack to previous areas? Why is a nonstop flight sometimes more expensive than multiple connections? Usually I find a flight I like that uses one or more connections with the same or different companies. In these cases, if the distance between the first and the second connection is not too large, I prefer to go by bus or train and then take the flight. The problem is that sometimes it is more expensive to take the flight in the second airport than to take the flight from the first one. I mean, I'm wondering why sometimes it is more expensive to take only a flight than to take the same flight and another one.  A practical example (and real, you can check with SkyScanner). Imagine I am in Madrid (Spain) and would like to go to Rio de Janeiro (Brazil) on April 1st. The cheapest flight I have found is this:     TAM airlines: MAD -> GRU -> GIG : 507 €   Fantastic! GRU is one of São Paulo's airports. That's perfect for me! I'll go on to Rio traveling overland. But then, I look for this flight (MAD -> GRU, non-stop):     TAM airlines: MAD -> GRU : 597 €   It's the same flight, 90 € more expensive! I've found this kind of behavior a lot of times and I'm intrigued to know why it happens. Use of 'prototype' vs. 'this' in JavaScript? I found the .bashrc file and I want to know the purpose/function of it. Also how and when is it used? Can compact sets completey determine a topology? Suppose that $\tau_1$ and $\tau_2$ are two topologies on a set $X$ with the property that $K\subset X$ is compact with respect to $\tau_1$ if and only if $K$ is compact with respect to $\tau_2$. Then is this enough information to determine whether $\tau_1 = \tau_2$?  If not (which is suspect to be the case) is there a nice counterexample, and what is the minimum amount of extra information required for $\tau_1 = \tau_2$?  I feel like there would be an issue regarding 'points at infinity.' Solving ODE with negative expansion power series I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this:  $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$  $g_{ii}$ are given, from metric, but that's not important. I need to assume that the solution (since I'm looking for components of $\xi^\mu$, which is a vector with components $\xi^t,\xi^r,\xi^\phi$) is given with power series of the form:  $\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$, and this is to be seen as expansion around $1/r$ (expansion around $r=\infty$).  Now when I plug this in the ODE I get this  $\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$, where   $\xi^\mu_{n,i}$   is the derivative with the respect to i-th component.   What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my $\mathcal{O}(r)$ terms cancel each other out? Or?  I'm kinda stuck :\ Pathologies in module theory Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.  If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.  I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.  To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind. I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 &lt; \beta$. I used substitution and integration to get $$ \lim\limits_{t\to\infty}\int\limits_1^tx^\alpha\sin(x^\beta)dx \overset{u = x^\beta}{\underset{du = \beta x^{\beta-1}}{=} }\lim\limits_{t\to\infty}\left(\frac{1}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1 - \beta}{\beta}}\sin{u}du\right) $$ If $\alpha + 1 -\beta &lt; 0$, then we have a product if a function with bounded anti-derivative and a differentiable decreasing function which tend to 0, and from Dirichlet's criterion, the integral converges.   For the converse, I used integration by parts $$  = \frac{1}{\beta}\lim\limits_{t\to\infty}\left(\left.-u^{\frac{\alpha + 1-\beta}{\beta}}\cos{u}\right|_1^{t^\beta} + \frac{\alpha+1-\beta}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1-2\beta}{\beta}}\cos{u}du\right) $$ For $\alpha + 1 -\beta\geq 0$, I get that the first term indeed diverges, but this is not enough to show total divergence. How can one show it? I asked a question that has been marked as a duplicate. I read a lot of similar questions and explanations in advance but I couldn't find a solution. In the question that should answer my question the asker used a different (not working) approach. There is even an answer that would work for my case however I'd rather consider it as a workaround than a solution since I have seen examples using my approach.  So I would like your opinion in that case. Would it be ok to repost my question while explaining why other answers didn't help me? I edited my question but I think it's not getting any attention anymore. Naturally I will remove the old question. I recently renewed my passport. I also have an expired passport which has my Canada visa (visitor's) valid for another three years. I have already used my Canadian visa which is multiple entry. I am planning to visit Canada this summer.  Since my valid Canadian visa is in my old/expired passport, do I need to renew/transfer my Canada visa or can I simply travel with valid visa in old passport? Two days had passed since she had last eaten or slept.   Or should it be:     Two days have passed since she last ate or slept.   Or something else? And why? The combinatorial explosion of past simple, past perfect and present perfect is killing me. Evidence for quantum gravity from gravitational waves The rumor spreads that .  I am far from being an experimentalist, but I want to know if there is any chance that the mentioned observations will reveal any experimental evidence for quantum gravity.  Personally, I feel like this is merely impossible as gravitational waves should propagate in agreement with classical General Relativity. But I might very well not see pitfalls if there are any.  UPDATE: by "evidence for quantum gravity" I mean experimental results which don't agree with predictions of General Relativity or its classical generalizations. They might be coming from string theory or loop gravity or from a completely different approach which we haven't discovered yet. I want to understand what are the chances that new physics will show up during the upcoming  gravitational wave observations.  Also, I expect gravitational waves to be in a highly coherent state, thus making the individual gravitons unobservable. Is this true? What could cause this? f12:    Viewport:    (the f12 has fewer samples, but even when the viewport is at lower samples it looks different..)  I know about color management, but that is at the defaults..    EDIT: oops.. I know 'log' is not a default, but that is because I was fiddling with it. The image above is rendered with 'default'    The material is quite simple, only a translucent shader. (for testing) I even tried appending everything into a new file to reset all the settings, but no difference.  EDIT2: Uploaded blend