Properties of the quotient ring $\mathbb{Z}_6[x]/\langle2x+4\rangle$
Cardinality of quotient ring $\mathbb{Z_6}[X]/(2X+4)$
I have some  gained from survey questions.  In my case they are  responses (Strongly Disagree-Disagree-Neutral-Agree-Strongly Agree).  In my data they are coded as 1-5.  I  don't think means would mean much here, so what basic summary statistics are considered usefull? What will reduce my rendertime? What would reduce the render time? Should I get a better GPU or more RAM? 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? How to apply CSS to iframe? Is it possible to find power armor in the wasteland? Automatically define macros in LyX documents I'm using LyX all the time and over the last 2 years I've accumulated some very handy macros for my lecture notes. As it is today, every time I start a new document, I copy and paste the macros from one of my other documents. Is it possible, somehow, to automatically load macros for all files?  (I asked this on SO a while back and got no answers. I hope in this new home I'll get a response). Product of path connected spaces is path connected Will product of path connected topological spaces be necessarily path connected? Why or why not?  Give me some hints. Thank you. Why did the First Order do this at the cantina? In The Force Awakens, after the First Order are alerted to BB-8, Rey, and Finn's presence at the cantina, an aerial bombardment of the compound begins. They also deploy ground troops.  Wouldn't it have been better to not have an aerial bombardment and simply have the troops surround the cantina, keeping as many people as possible bottled up inside it while they searched both the interior and surrounding area for BB-8?  Instead of a surgical extraction, the strategy that they chose seemed to create chaos and even risked destroying BB-8.  Perhaps this was a sign of the inexperience and emotionally-charged state of Kylo Ren, but General Hux seemed to have more direct authority over the deployment of troops and fighters.  So what was the strategy behind these tactics? How do I find out which dlls an executable will load? How do I clone a subdirectory only of a Git repository? I have my Git repository which, at the root, has two sub directories:  /finisht /static   When this was in , /finisht was checked out in one place, while /static was checked out elsewhere, like so:  svn co svn+ssh://admin@domain.com/home/admin/repos/finisht/static static   Is there a way to do this with Git? add transistor model to LTspice I'm trying to add the mmbt3904lt1g model to LTspice. - I've copied its data from the MMBT3904LT1.LIB.txt in the standard.bjt library. - I've added an NPN to my schematic then I've changed its value to mmbt3904lt1g  - through the Edit->spice directive I've pasted the path to this library I thought it was all to use this device but no simulation is achieved.. I get only: Can't find definition of model "MMBT3904LT1g"  where I'm doing wrong ? How does Newtonian gravitation conflict with special relativity? In the Wikipedia article , it says     After Newtonian gravitation was found to be inconsistent with special relativity, . . .   I don't see how Newtonian gravitation itself is inconsistent with special relativity.  After all, Newton's Universal Law of Gravitation is completley analogous to Coulomb's Law, so it would seem that, if there were an analogous "gravitational magnetic field", one could formulate a theory of gravitation in exact analogy with Maxwell's Theory of Electromagnetism, and of course, this would automatically be consistent with Special Relativity.  What about this approach to gravitation does not work?  The only problem I could see with this is the lack of evidence for a "gravitational magnetic field".  That being said, gravity is "weak" as it is, and my guess is that it would be extremely difficult to set up an experiment in which a massive body were moving fast enough for this "magnetic field effect" to be obsevable.  EDIT:  As has been pointed out to me in the answers, Newton itself is inconsistent with SR.  Similarly, however, so is Coulomb's Law, yet, electromagnetism is still consistent with SR.  Thus, how do we know that it is not the case that Newton's Law is the special "static" case of a more general gravitomagnetic theory exactly analogous to Maxwell's Theory:  Let $\mathbf{G}$ be the gravitation field, let $\rho$ be the mass density, and $\mathbf{J}$ be the mass current density, let $\gamma _0$ be defined so that $\frac{1}{4\pi \gamma _0}=G$, the Universal Gravitation Constant, let $\nu _0$ be defined so that $\frac{1}{\sqrt{\gamma _0\nu _0}}=c$, and suppose there exists a field $\mathbf{M}$, the gravitomagnetic field, so that the following equations hold: $$ \vec{\nabla}\cdot \mathbf{G}=-\frac{\rho}{\gamma _0} $$ $$ \vec{\nabla}\cdot \mathbf{M}=0 $$ $$ \vec{\nabla}\times \mathbf{G}=-\frac{\partial \mathbf{M}}{\partial t} $$ $$ \vec{\nabla}\times \mathbf{M}=\nu _0\mathbf{J}+\gamma _0\nu _0\frac{\partial \mathbf{G}}{\partial t} $$ where theses fields would produce on a mass $m$ moving with velocity $\mathbf{v}$ in our inertial frame the Lorentz force $\mathbf{F}=m\left( \mathbf{G}+\mathbf{v}\times \mathbf{M}\right)$.  This theory would automatically be consistent with SR and would reduce to Newton's Law of Gravitation for the case of gravitostatics:  $\frac{\partial \mathbf{G}}{\partial t}=\frac{\partial \mathbf{M}}{\partial t}=\mathbf{J}=0$.  (To be clear, you can't just set the time derivatives equal to $\mathbf{0}$ as it seems I have done.  Upon doing so, you obtain the corresponding static theory, which is technically incorrect (as you can easily see because it won't be relativistically invariant), but is nevertheless often a useful approximation.)  My question can thus be phrased as:  without appealing directly to GR, what is wrong with this theory? Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$ I want to prove the following: $$\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$$ what I did is(use binominal): $$\binom{n}{k}X^k\cdot 1^{n-k} = (X+1)^n$$ $$k\binom{n}{k}X^k\cdot 1^{n-k} = k(X+1)^k-1$$ now I replace $k$ by $n$ and insert $X=1$ , I will get now $n\cdot2^{n-1}$ what you are suggesting? there is another way to do that? thanks! Has a free neutron ever been shown to absorb/emit/interact with a photon? Protons only 'interact with' very high-energy photons, whether inside a nucleus or free, right?  I'm assuming the same about neutrons....  Neutrons have a small magnetic moment and a slight electric 'moment' (dipole?), correct?  They are sometimes considered to consist of a negative charge surrounding a positive charge?  I would assume that neutrons, in a nucleus or free, interact with photons even less often than protons, and that the gamma photon has to have an even higher energy than one interacting with a proton....  Lastly, I have read about 'photodisintegration', in which a nucleon is kicked out of a nucleus by a gamma ray.  It seems to happen to neutrons sometimes...  But, I am mostly curious about experimental evidence of free neutrons interacting with photons, how often it happens compared to free protons, and whether the interacting photon's energy has to be extremely high, higher even than a photon interacting with a proton... Bridged interfaces and VLAN tags Can't use my sprint key and jump at the same time Ok, so I do a lot of sprint jumping in minecraft because I pvp a ton everyday and every once in a while it won't allow me to do both at the same time. I set ctrl to my sprint key so when I want to sprint I hold ctrl. If I hold ctrl and space it won't let me jump. If I double tap w I can sprint and jump at the same time. Does anybody know why it does this? It's not something to do with my keyboard because it's just fine. It's done this a couple of times and it's quite annoying. The force of gravity is constantly being applied to an orbiting object.  And therefore the object is constantly accelerating.  Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas?  I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it? How to populate options of h:selectOneMenu from database? Is Con modifier × level added to HP every level-up? Batch fit images into ratio with blurred background I need to resize a lot of images into a 4:3 ratio with the desired effect.  Is there a way to do this using photoshop actions? (the background can be scaled or two copies moved to the sides, either way works)     image credits: