What if we are so used to the curvature of space caused by mass and the range of its effects that we totally ignore the possibility of the existence of "opposite" curvature1, i.e. objects that bend space opposite to mass and cause repulsion by creating local space anisotropy?   What if we are so used to space curvature caused by mass that we invent a force that has no charge to explain possible effects of opposite curvature (repelling force - diamagnetism)?   What if the oscillating electric field has the ability to cause tiny local (anisotropy) fluctuations in space and its effects on objects are interpreted as "magnetic field"?  Could magnets be just an example of objects with anomalous gravitational fields due to their ability to distort the isotropy of space?  I guess my question boils down to:   How do we know that the force at the poles of a magnet is not gravitational (large but local curvature caused by anisotropy) with opposite signs, rather than what we call magnetic?  Can space curvature arise from something else different than mass?  What would change on the r.h.s of the Einstein's equation (components of stress-energy tensor) if we assume a connection with torsion?    1. Same size of the volume element $dV$, but with stretch in one element, say $dx$ and proportional compression in the other two $dy$, $dz$.
Electric monopoles do exist, but why cannot magnetic monopoles exist?    The question is closed because I need to clarify it, but I don't know how I could ask it another way. However, I've recieved many answers that were appropiate and added something to my knowledge, so I consider it answered.
Why is conservation of angular momentum considered a law? We know that angular momentum is defined as the cross product of position and linear momentum. By taking the time derivative, we can deduce that the time rate of change of the angular momentum equals the net torque.  Then why is the conservation of angular momentum considered a law, when we can easily show it is conserved when no net torque is applied, by using Newton's laws?  It truly confuses me. How to align prices in columns when both 2 digit and 4 digit variations of a price exist? When do I use active vs passive perception? So, I'm trying to figure out how to run the scene from the beginning of Lost Mine of Phandelver (5e starter set) where the party is ambushed by goblins and I'm really confused as to how Passive Wisdom (Perception) works.  The included rulebook mentions on page 6 :     When you hide, there’s a chance someone will notice you even if they aren’t searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature’s passive Wisdom (Perception) score, which equals 10 + the creature’s Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5. For example, if a 1st-level character (with a proficiency bonus of +2) has a Wisdom of 15 (a +2 modifier) and proficiency in Perception, he or she has a passive Wisdom (Perception) of 14.   So the party is being ambushed by goblins and at that point the DM guide mentions on page 7 :     Check to see who, if anyone, is surprised. [...] Make a Dexterity (Stealth) check for the goblins, rolling once for all of them. Roll a d20, add the goblins' Stealth skill modifier (+6) to the roll and compare the result to the characters' Passive Wisdom (Perception) scores.   So to me this reads like I should roll 1d20+6 for the goblins and compare that to the static Passive Wisdom (Perception) score on the character sheet of a PC. The PCs don't get to roll anything as they're not actively looking for enemies, even though they are investigating the dead horses.  I would also think that if the characters were supposed to roll, the DM guide would clearly instruct so, given that this is the starter set.  However, watching  at around 8:01 I notice that the DM is explicitly asking his party for a perception roll. And the same thing happens on a couple of other videos where people are running the Lost Mine adventure. The DM always requests a roll.  So what gives? Shall I take it that the DMs ask for the roll because they consider the party to be actively seeking for hidden foes? Or am I just misreading the rulebook and the DM guide? This question is from PDE by Evans, 1st edition, Chapter 5, Problem 14. It has been posted  previously, however, I cannot quite put all the information together from the responses there. Hopefully you can help me now. The problem is as follows:     Let $U$ be bounded with a $C^1$ boundary. Show that a ''typical''   function $u \in L^p(U) \ (1 \leq p &lt; \infty)$ does not have a trace on   $\partial U$. More precisely, prove there does not exist a bounded   linear operator      \begin{equation} T:L^p(U) \to L^p(\partial U) \end{equation}      such that $Tu = \left. u \right|_{\partial U}$ whenever $u \in C(\overline{U}) \cap L^p(U)$.   So what we're trying to find is a bounded sequence of functions that go to infinity at the boundary, correct? Because then $Tu = \left. u \right|_{\partial U}$ is undefined.  I like the idea of using the $\mathrm{dist}(x, \partial U)$ function to do this. I was thinking about defining the function  \begin{equation} u(x) = \frac{1}{\mathrm{dist}(x,\partial U)}, \end{equation}  which must be in $L^p(U)$ since $U$ is bounded (hence integral of $\varepsilon$ over $U$ is finite, right?). But this function is not continuous at the boundary, and so $u \notin C(\overline{U})$. Can this be modified somehow?  Another option is to use that the boundary is $C^1$. So for each $x^0 \in \partial U$ there exists an $r &gt; 0$ and a $C^1$ function $\gamma : \mathbb{R}^{n-1} \to \mathbb{R}$ such that $U \cap B(x^0,r) = \{ x \in B(x^0,r) \ | \ x_n &gt; \gamma(x_1,\ldots, x_n) \}$. So around every point on the boundary we can find a ball where the n'th coordinate is greater than the $C^1$ function $\gamma$. How can we use this? Add Point to Plot3D How do you make a torus 'telescope' open? So I have a torus, and I need it to "telescope" open -- kind of like Pacman's death animation, but with a torus instead of a sphere.  I'm very new to 3D modeling/animating so I really have no clue how to tackle this problem.  Any advice to help me get pointed in the right direction would be most appreciated :) How to use java.util.Scanner to correctly read user input from System.in and act on it? Grand Prix Race Driver A has boon leading archrival B for a while by a steady 3 miles. Only 2 miles from the finish, driver A ran out of gas and decelerated thereafter at ta rate proportional to the square of his remaining speed. One mile later,driver A's speed was exactly halved.If driver B's speed remained constant,who won the race?  i have tried the set up the relation$d^2x\over dt^2$=$K ({dx\over dt})^2$ and integrate it but dont know how to do. ssh into an account which has no home directory I have a DisplayPort based monitor, and a laptop with a DP out.  I recently got a second laptop with HDMI out, and went looking for a HDMI -> DP converter.  I was very surprised to find that such a thing effectively doesn't seem to exist.   and the  seem to imply that it's just not possible to go HDMI -> DP, without a very expensive converter box (not just an adaptor).  However, I can't find anything that explains why this is the case.  Is DP designed to be backwards compatible but can't interpret HDMI?  Is it a case of HDMI-quality signal not being high enough quality to be convertible?  Some other reason?  Note that this question is not looking for an adaptor - I'm looking for the underlying cause of the lack of them. How do I get the form id of my form? I've just begun to make my modules and I'm at a standstill. I got this code from someone else and am trying to customize it:  function hook_form_alter(&amp;$form, &amp;$form_state, $form_id) {   if ($form_id == 'the form id for the node form') {     $form['#submit'][] = 'my_custom_submit_handler';   } } How can I get LaTeX to warn about unreferenced figures and unused labels in general? When writing documents, I consider it to be necessary that all figures and tables are referenced (\ref) in the text, whether or not I've remembered to \label them. LaTeX does not currently warn if a figure or table is not referenced. Is there a way to ensure that this is flagged up as a warning?  Note that I am specifically concerned about figures and tables, not chapter or section headings; it will be no good looking just at \labels that don't have an associated \ref.  In other words, it should report boo and not report bla in the following MWE:  \documentclass{article} \begin{document} \section{BLA}\label{bla} \section{BOO}\label{boo} In Section~\ref{bla} we do something. \end{document} What item bonuses apply to my follower? There are a lot of different side effects an item can have, and I assume that things like magic find have no influence on any of "our" found items at all.  However, I'm not sure about other things. For example:   Do followers get life-steal on weapons, too? Does increased orb healing make any difference? What about +experience after kills? Will my follower level up faster?   and probably a lot more.  Is there some general rule as to what effects of an item apply to followers, and which don't? Function $f(z)$ such that $|f(z)| > |z|$ for every $z\in\mathbb{C}, 1/f(z)$ $f(z)$ is a function such that $|f(z)| &gt; |z|$ for all $z\in\mathbb{C}$  How would you prove the function $\frac{1}{f(z)}$ is bounded. I think you have to use the $|f(z)|\leq M$ but I'm not sure. Find all the values of $w$ ∈ C that satisfy the equation. I can use ls -ld */ to list all the directory entries in the current directory. Is there a similarly easy way to just list all the regular files in the current directory? I know I can use find  find . -maxdepth 1 -type f   or stat  stat -c "%F %n" * | grep "regular file" | cut -d' ' -f 3-   but these do not strike me as being overly elegant. Is there a nice short way to list only the regular files (I don't care about devices, pipes, etc.) but not the sub-directories of the current directory? Listing symbolic links as well would be a plus, but is not a necessity. Is it possible to log a program's network activity in Windows 7? Does anyone know of a program or way to log all network activity to a specific program on one's computer (program and/or service)?  I know I can see this "live" in Resource Monitor, but I'm wondering if anyone knows of a program or way to track and log all network activity to/from a specific process and/or service.  I'm finding a lot of "live"-type monitors, like Windows Resource Monitor, but nothing that can log. Graph theory: Tree holds two sub-tree with an amount of vertices What is a "good fit" Brier score and Harrell's C Index Can you teleport farther away with misty step if you have a double from Invoke Duplicity? The Channel Divinity feature says:     you can cast spells as though you were in the illusion's space, but   you must use your own senses. (PHB 63)   If you were where your duplicate is, you could step to a space 30ft from its position. So if it is 60ft from you, you could teleport to a space 90ft from your actual position. But the spell says that:     you teleport up to 30 feet (PHB 260)   This would put a hard limit on the distance you can travel.  Which rule takes precedence here?