When I went to school I was specifically told that fields are material (they occupy some region in space, and they "exist" there) and continuous.   Recently, studying quantum physics I came across the notion of gauge particles which carry all kinds of interactions. From this perspective, if I judge it correctly, fields do not exist - they are essentially these particles being exchanged all the time between the interacting objects.   So, what is the current approach to the concept of fields?
There was a reason why I constantly failed physics at school and university, and that reason was, apart from the fact I was immensely lazy, that I mentally refused to &quot;believe&quot; more advanced stuff until I understand the fundamentals (which I, eventually, never did). As such, one of the most fundamental things in physics that I still don't understand (a year after dropping out from the university) is the concept of . No one cared to explain what a field actually is, they just used to throw in a bunch of formulas and everyone was content. The school textbook definition for a field (electromagnetic in this particular case, but they were similar), as I remember it, goes like:  An electromagnetic field is a special kind of substance by which charged moving particles or physical bodies with a magnetic moment interact.  A special kind of substance, are they for real? This sounds like the authors themselves didn't quite understand what a field is so they decided to throw in a bunch of buzzwords to make it sounds right. I'm fine with the second half but a special kind of substance really bugs me, so I'd like to focus on that. Is a field material? Apparently, it isn't. It doesn't consist of particles like my laptop or even the light. If it isn't material, is it real or is it just a concept that helps to explain our observations? While this is prone to speculations, I think we can agree that in scope of this discussion particles actually do exist and laws of physics don't (the latter are nothing but human ideas so I suspect Universe doesn't &quot;know&quot; a thing about them, at least if we're talking raw matter and don't take it on metalevel where human knowledge, being a part of the Universe, makes the Universe contain laws of physics). Any laws are only a product of human thinking while the stars are likely to exist without us homo sapiens messing around. Or am I wrong here too? I hope you already see why I hate physics. Is a field not material but still real? Can something &quot;not touchable&quot; by definition be considered part of our Universe by physicists? I used to imagine that a &quot;snapshot&quot; of our Universe in time would contain information about each particle and its position, and this would've been enough to &quot;de&quot; it but I guess my programmer metaphors are largely off the track. (Oh, and I know that the uncertainty principle makes such (de)serialization impossible — I only mean that I thought the Universe can be &quot;defined&quot; as the set of all material objects in it). Is such assumption false? At this point, if fields indeed are not material but are part of the Universe, I don't really see how they are different from  except for perhaps a more geeky flavor. When I talked about this with the teacher who helped me to prepare for the exams (which I did pass, by the way, it was before I dropped out), she said to me that, if I wanted hardcore definitions,  a field is a function that returns a value for a point in space.  Now this finally makes a hell lot of sense to me but I still don't understand how mathematical functions can be a part of the Universe and shape the reality.
Is $29$ the only prime of the form $p^p+2$? This is an unproven proposition I've come across in multiple places.   Suppose $A$ is a commutative Noetherian ring, and $S$ a multiplicative subset of $A$. Then $S^{-1}A$ is Noetherian.  Why is this? I thought about taking some chain of submodules $$ S^{-1}M_1\subset S^{-1}M_2\subset\cdots $$ and pulling back to a chain  $$ M_1\subset M_2\subset\cdots $$ of submodules of $A$ which must eventually stablize. Is there more to it than this? I kind of wary of assuming all submodules of $S^{-1}A$ have form $S^{-1}M$ for $M\leq A$. Expected number of siblings, given the expectation of children in a family The number of children in a family is a random variable $X$, where    $ \mathbb E(X) = 1.8 $ and $Var(X) = 0.36 $. If we randomly choose a child, which is the expected number of its siblings?      (it is given that the answer $&gt; 0.8$)   Let the number of a child's siblings be a random variable $Y$, so we are looking for $ \mathbb E(Y) $.  If found that,  $$ \mathbb E(Y) = \sum_{k=0}^{\infty}k \; \mathbb P(Y=k) = \sum_{k=0}^{\infty}k \; \mathbb P(X=k+1) = $$ $$ = \sum_{k=0}^{\infty}k \; \mathbb P(X=k)-\sum_{k=0}^{\infty}\mathbb P(X=k) +\mathbb P(X=0)= \mathbb E(X)-1+\mathbb P(X=0) $$ which is indeed greater than $0.8$, if  $\; \mathbb P(X=0)&gt;0$.  But, I don't know how to make use of the variance to find the final answer.  Thank you in advance In Star Trek TOS, we see the Defiant disappear to unknown parts  ('The Tholian Web'), and Star Trek: Enterprise revealed that the Defiant went back in time to the mirror universe ('In a Mirror Darkly', parts 1 &amp; 2).   Because of this, one might wonder why (in-universe) the mirror universe didn't obtain (and maintain) technological superiority over the main timeline. (Granted, the out of universe answer is that TOS depicted the two universes as having essentially the same level of tech)  So does any work (whether STO, or a pocket books work) address why the mirror universe didn't simply leap frog the main universe's technology?   (Or to put it another way, why did the mirror universe's technology appear to stagnate for a century after obtaining access to the Defiant's advanced technology?) While modeling an Audi R8 I have the following problem. The thing I want to create is seen in this image:    I want to create a sharp crease (outlined in red) which starts around the first exhaust and then goes to the second.  Is there an efficient way to do this (I also use the Subdivision Surface Modifier for this model). If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is First: I'm obviously not talking about logging in and then automatically launching Terminal.app.  In , you used to be able to type &gt;console in the username field on the login screen and go directly into a command line terminal session.  I haven't done it in years until the other day. However, now, when I try it, my computer just locks up. I can't cancel, and it never sends me to a terminal session.  Is it even possible to do this anymore? If not, does anybody know when it was removed? It seems odd that typing &gt;console would cause the machine to lock up if it wasn't recognizing it in some way. I think the answer is yes but I know a lot of people disagree. So, I would like to ask these people when exactly does it stop being me.  Let's say I want to upload my brain into a computer using the following procedure:  I have each of my neurons replaced by electronic neurons. By electronic neurons I mean chips with the necessary hardware and software to perform the exact same function that the neuron being replaced was performing. This means that given the same inputs that the biological neuron was given it would produce the same outputs. Then I connect that electronic neuron to all the neurons that former biological neuron was connected to.   At the end of this process is it still me? If not when did it stop being me? When one neuron was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  (assuming it is still me)  Then, I divide these electronic neurons into groups and for each group I perform the following:   Write software to simulate the group of electrical neurons in my computer such that given the same inputs that go into the group of electronic neurons in my brain, it will produce the same outputs. Replace each group of neurons with another chip that communicates with my computer. Each of the chips will communicate to the corresponding software module in order to determine how to transform its inputs into the necessary outputs. Connect all of the inputs and outputs of the group of electronic neurons to that new chip.   At the end of this process is it still me? If not when did it stop being me? When one group was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  (assuming it is still me) Now, what I have is a bunch of these new chips in my head, connected to each other and for each of these chips I have a small neural network software module in my computer. So, now for each pair of chip/software module I do this:   Transfer all connections going into and out of the chip into the computer version (i.e. connect the software module to the other software modules that represent the other chips that this chip is connected to) Disconnect the chip from the other chips and remove it   At the end of this process is it still me? If not when did it stop being me? When one of the new chips was replaced? When 1,000 were replaced? When 1,000,000 were replaced?  If it is still me then I have successfully moved my brain into a computer without ceasing being myself.  So, again the question is if at the end it is not me anymore when exactly does it stop being me and why does it stop being me?  EDIT: This question assumes that our universe and everything in it (including ourselves) follows the (testable) laws of physics. As it stands it also assumes that neuroscience is mostly right, although the question could also be considered without that requirement if instead of neural simulations we were to implement physics simulation of the brain. Of course this is a lot less practical. However, if your view is that our minds have a supernatural component then the question does not make any sense.  It was also pointed out that this is related to the Ship of Theseus, i.e. is something still the same if all its parts have been changed. With regards to that I think it is useful to consider that most of the cells in our body (not including brain cells) are regularly replaced anyway and some people even get entire organs replaced. Yet, we don't say that they stopped being themselves. Note, however, that if you are assuming a supernatural component of the mind that this explanation is not really adequate or relevant. Which is the correct use of these two words, and in which context should one be used rather than the other? Are delphi variables initialized with a value by default? How confident is my model? Is there a way to know? I am trying to lock some boxes in a map of mine, and I used this command:   /blockdata 10 4 9 {Lock:Elyiana's Mage Key}   At first it worked, but then when I logged back on the next day it gave me this error message:    Data tag parsing failed: Expected '}' but ''' at: {Lock:Elyiana'&lt;--[HERE]     I don't know why it doesn't like the apostrophe in the key's name. I would change it, but it already worked and locked some other items. My version is 1.12, does anyone have an answer to my problem? Define $\sigma(m) = \sum$ d : d|n.  Prove that $p^j$$q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.  Attempt at Solution: I have shown that $p^k$ can never be a perfect number, and im trying to use the multiplicative property of $\sigma$ to generalize to $p^j$$q^i$ I get the following error when i do sudo apt-get update  W: GPG error: http://ppa.launchpad.net precise Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 6AF0E940624A220   I tried following ways after reading various questions here and also by googling.  I tried  sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys 6AF0E1940624A220   It did not help. I also tried  sudo gpg --keyserver hkp://keyserver.ubuntu.com:80 --recv-keys   thinking port 11371(default) may be blocked due to firewall.  I also tried to reset everything as follows  sudo rm /var/lib/apt/lists/* -vf sudo apt-get update &amp;&amp; sudo apt-get upgrade   still i get the same error or get update. Any other possible solution I can try? Can Xcode 6 and Xcode 5 coexist on the same computer? Can I run CAT5/6 cables parallel to electrical cables? I am building a new house and am planning to hardwire network cables into each room.   Can I run CAT5/6 cables parallel to electrical wires without introducing any general safety issues or losing much in connection quality?  Wiring runs for up to 25 metres from patch panel and distribution board in garage to rooms.   My research indicates that is should be fine if they are in separate conduits or if the CAT cables are shielded. VAR or VECM for a mix of stationary and nonstationary variables I have 4 time series. One of them is stationary and rest of them are not. I need to find relation between them. I will use AIC to decide lag length.   Should I use VAR or VECM to find relation between them?  Will VAR or VECM give me relation in terms of equation which can be used for forecasting?  Do I need to perform Johansen's test of cointegration?  What good would it do? Closed unit disk homeomorphic to $\mathbb{R}^2$? I've already shown the existence of a homeomorphism between the open unit disc and $\mathbb{R}^2$ and now I'm trying to work out whether the closed unit disc is homeomorphic to $\mathbb{R}^2$ or not. Clearly the only real difference is the fact it includes the points actually on the circle, but because the continuous, invertible, bijective map I defined for the open disc was $\dfrac{x}{1-|x|}$ this clearly is not defined for the points on the line.   Intuitively I'm thinking it's not possible to form a homeomorphism with the closed disk precisely because of these points on the circle but can't really form a decent argument. I was thinking along the lines of showing that the non-open, closed subset $\bar{D}(0,1) \backslash D(0,1)$ (which is just the set of points on the actual circle i.e the boundary/frontier of the closed disk) can't possibly bijectively map to a closed subset of $\mathbb{R}^2$ - which clearly would have to be the case in a homeomorphism.   Any advice would be greatly appreciated. If a $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how to find sequence $\{b_n\}$ such that $\sum |b_n|<\infty$ but $\sum |a_n||b_n|$ diverges? If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges?  I want to use this fact in another problem but don't immediately see how to prove it. Is there an easy way to see associativity or non-associativity from an operation's table? Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot &amp; a &amp; b &amp; c &amp; d &amp; e\\\hline a     &amp; e &amp; d &amp; b &amp; a &amp; c\\ b     &amp; d &amp; c &amp; e &amp; b &amp; a\\ c     &amp; b &amp; e &amp; a &amp; c &amp; d\\ d     &amp; a &amp; b &amp; c &amp; d &amp; e\\ e     &amp; c &amp; a &amp; d &amp; e &amp; b \end{array}$$ it is easy to check that it is closed (no elements occur in the table which don't occur as row or column index), commutative (the table is symmetric), has a neutral element (the row and column of $d$ are copies of the index row/column), and has an inverse element for each element (there's a $d$ in each row and column). In other words, almost all important properties can immediately be seen. The only part missing is associativity.  Therefore my question: Is there a simple way to see directly from the operation's table (i.e. without doing explicitly all the calculations) if an operation is associative?