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Can the problem of unification of Quantum Mechanics and General Relativity be converted into a pure Mathematical problem? I mean to say, can you make someone understand the problem who only knows Mathematics, even if a Physics description is absent? I mean, describing the problem with unification using pure Mathematical formulation.  Can we make the problem seem as if it has nothing to do with the real world and describe it only as a problem of mathematics? Generalized Tube Lemma Names of close-voters are not visible in the post notice but in the revision history and timeline I am not a user with close/reopen votes privileges on Travel Stack Exchange, therefore I can't see the names of the close voters :    However, they are visible when going to the revision history:    or the timeline of the post:    This also happens when I am not logged in: I cannot see the banner (and so if the question has been closed), but from the revision history/timeline of the post I can get that information.  Shouldn't names of close-voters be completely hidden from people without close/reopen privileges (even if the revision history is a little hard to find)? Find $cot\space x_n$ if $sin(x_{n+1}-x_n)+2^{-(n+1)}sin\space x_nsin \space x_{n+1}=0$ for $n\ge1$ Also Given     Suppose $x_1=tan^{-1}2&gt;x_2&gt;x_3&gt;......$ are positive real numbers   I cannot understand as to how I should approach but I have a hunch which is a bit hand-wavy way to prove it but tried to show that  $lim_{n \to \infty}x_n =\dfrac{\pi}{4}$ (For the proof see my edits in @hypernova's answer)  But can anybody prove it and then use it to find $cot \space x_n$ please? How do I help my third-grader focus on homework? My child just entered third grade, and the homework load is much higher this year, with a lot of writing and math drills. It is all work that she can do easily, but it's causing the whole family a lot of pain, because she's have a really hard time just sitting down and doing it — asking many questions about things she clearly actually knows, and getting distracted by the smallest thing going on around her. (Setting up a special quiet area isn't helping, because the distractions could be her pencil or a sudden burst of word association based on something in the lesson.)  So, a page of writing and a couple of pages of math which could be done in fifteen minutes total is taking an hour and a half. She can easily explain the reasoning behind the math problems, but when it comes to writing down what she just said, it's like pulling teeth. She'll slowly write a word, sounding it out in slow motion as she writes each word. And she'll say that she doesn't understand a question, but when she actually gets her mind pointed at it, she can write a great answer. And she's clearly picking up the concepts in class; I have no doubt that she actually understands, which is part of our frustration.  My wife feels like she's just wanting attention, but I think it's more about our daughter's own ability to focus. We both feel that sitting down and working painfully through each question isn't really helping her. That works, but my wife feels like it's just rewarding the behavior, and I feel like it's not really helping her learn any better focus on her own, which, you know, she'll need in actual life. And, we're happy to spend child with our kid, this pretty much the opposite of quality time.  Neither rewards or consequences seem to have any effect, and nor do charts. All of those things basically just seem to add another layer of complication, distraction, and misery. What else can we do? How to play DVDs in Ubuntu 16.04? I have an associative array in the form key =&gt; value where key is a numerical value, however it is not a sequential numerical value. The key is actually an ID number and the value is a count. This is fine for most instances, however I want a function that gets the human-readable name of the array and uses that for the key, without changing the value.  I didn't see a function that does this, but I'm assuming I need to provide the old key and new key (both of which I have) and transform the array. Is there an efficient way of doing this? What makes a spell being cast considered to be a {class} spell? My understanding is that when a spell is known or prepared via one of your class features, or if you cast it via a class feature, it's considered a "class spell" (sorcerer spell, cleric spell, etc.) for you. If you cast it via some method outside your class (items, etc.), it isn't considered a spell from your class.  For clarity, some class features care about whether or not a spell is a 'class' spell (emphasis mine):    Wild Magic Surge for Wild Magic sorcerers can only trigger "[...] immediately after you cast a sorcerer spell [...]". Overchannel for School of Evocation wizards can only be used "when you cast a wizard spell [...]" Empowered Evocation for School of Evocation wizards can also only be used on "[...] any wizard evocation spell you cast". Rod of the pact keeper gives a bonus "[...] to the saving throw DCs of your warlock spells".   Contrast with a feature like Blessed Healer for Life Domain clerics that just use the language of "When you cast a spell [...]", and you'll see why I believe there's an important distinction as to whether or not a spell instance is a 'class' spell.  However, comments on  made me question where my understanding of what makes a spell instance a 'class' spell comes from, and I'm not sure of a concise spot that explains it. What are the rules that govern what is or is not considered a class spell when cast?   Of particular concern is whether or not a spell is, in all instances, a class spell once you've obtained it from your class. For example, if you were a wizard that had magic missile both prepared and in your spellbook, would any instance of magic missile you cast be considered a wizard spell, even if it was cast by way of, say, a wand of magic missile? Benefits & Practicality of Virtual Machines I have been living under a rock for the past 5 - 8 years, and am just now getting up to speed with virtualization, virtual machines and platforms like VMware or VirtualBox.  I think I am starting to get them, but there are two concepts about VMs that I'm just choking on for some reason; one has to do with their intrinsic benefit to the enterprise, and the other has to do with their practicality when being used.  So I understand that you can have multiple VMs running on the same physical machine, all doing their own "thing" as if they were separate machines. So let's say I want to make a nice little virtual network of servers in my home office so I can have a big sandbox to play inside of for all my projects. I set up 2 physical machines as servers, and on each of them I have, say, 4 virtual machines running (1 might be an app server, another might be a RDBMS, another might be a message broker, etc.).   Besides saving me money (as I only have to provide 2 physical machines instead of buying 8 servers), what instrinsic benefit/purpose do these VMs serve here? I don't care how clever virtual machines are, if my computer only has 8GB of RAM, and I have 4 VMs running on it, those VMs only get 8GB of RAM to share between them. Putting 4 VMs on my server doesn't endow it with 4 times that amount of RAM! So, as far as I see it, VMs save you money on hardware, but just end up bogging down that hardware as you have multiple machines competing over the same pool of resources. What am I missing here? Can VMs communicate with each other over your network using high-level protocols like TCP/IP, HTTP, FTP, etc.? Otherwise, it just doesn't make sense to implement all these VMs if you have to feed them separate and independent chunks of data/requests to process. I'm looking to build a collection of fantasy miniatures, preferably pewter - but I don't want to buy them one at a time from my local hobby store... Much too expensive, and they don't often carry anything I'm interested in buying, which makes it a long term venture as well. So I'm looking for ideas on how to obtain many miniatures at once, preferably for cheap. D&amp;D Miniatures aren't a good source for me as they're collectibles, and the prices on those are also prohibitive.  By the way - they don't have to be pewter, that's just my preference. =) Looking for illustrator script to swap all objects with another I have a map with small circles for over 200 cities. I need to change all of the circles into map pins. Is there a way to do this? I think it's a script maybe.   I know how to select all objects of the same appearance and give them a new graphic style but that will only let me change the color, stroke, etc. not replace with a new shape. Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$ I have to prove that:     $$\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$$   I tried to open up the right side with Pascal's definition that: $$ { n \choose k} = {n-1 \choose {k}} + {n-1 \choose {k-1}}$$  Here is what I came up with, and I am sure it is wrong because it does not equal the left side:  $$ {m+n+1 \choose m+1} = {m+n \choose m+1} + {m+n \choose m} = ... ={m+n \choose m+1} + {m+n-1 \choose m} + {m+n-2 \choose m-1} + ... + {m \choose m+1} = \sum_{k=0}^{n} {m+k \choose m+k-n+1 }  $$  Which, again, probably is wrong because it is not equal $\sum_{k=0}^{n} { m+k \choose m}$. Any help is appreciated Inaccurate reputation graphs - bug or as designed? I have notice two issues with the reputation graph: 1: This user has a higher score on the graph than in reality I happened to notice  where the user's graph reputation does not match their actual reputation.  The person in question was hit by a -100 penalty for an offensive answer a while back, but the graph doesn't reflect that.  I thought it just didn't update when I noticed this at the time, but the person has since gained and lost reputation and at all times the graph was +100 of their actual reputation.  It seems, based on Shadow Wizard's comment that this is because the reputation change is non-public. OK, fair enough... , which was marked as , noticed a similar phenomenon for placed DVs.  This prompted me to examine graphs more carefully.  It seems that this hidden -1 has been updated, as I checked several users and did not find any others that had lower actual reputation than graph reputation, so .  (Since the number of DVs placed is public, I can be reasonable sure that these users have placed -1 rep DVs, although strictly speaking they could be 100% question DVs.) 2: Some users have a lower score on the graph than in reality I did, however, find many users where the graph says their reputation is much lower than it actually is.  Perhaps that is from deleted downvotes turning into hidden +1s, but those should be balanced by hidden -1s, so its not clear what happened.  Seems there is some error in the logic - maybe the -1s do affect the graph now as per the , but the corresponding +1s don't?  Whatever the case, I can't imagine it is intentional. Edit: I went back to find some examples today &amp; didn't find any.  I think the issue has been resolved - either it was a caching issue or a change was made to the code after this post. How to read and write into file using JavaScript? Softmax vs Sigmoid function in Logistic classifier? What decides the choice of function ( Softmax vs Sigmoid ) in a Logistic classifier ?  Suppose there are 4 output classes . Each of the above function gives the probabilities of each class being the correct output . So which one to take for a classifier ? Is it okay to have a multi-language deck? I want to buy a few booster pack from a local shop that only gets them in French. Is it generally acceptable to show up to a tournament with cards in different languages? I commonly use the word 'guys' to refer to a group of males colloquially. It's colloquial but not rude, off putting, condescending, patronizing (though I wouldn't use it with a group of men at a board meeting (hypothetically of course), unless I knew them). So, some that I would not consider as replacements would be    buddies: too old-fashioned dudes: too informal boys: too patronizing men, gentlemen: too formal or false respect   I realized that I had also been using it for any group of people, males and females, even just females. It worked for me and I didn't think of the gender implications, that women might not care for it. No one ever complained, directly or indirectly (rule of life: sometimes people don't complain about things they should).   But it occurred to me (maybe by reading something that sparked realization) that the intended hearers might not all care for it.  What might be a female gendered or non-gendered version of 'guys'?  Some that I've considered don't feel right about (though these may be reasonable answers) are:   dolls: too old-fashioned babes: too informal  ladies: too formal or I've heard from women, too creepy women: too factual girls: too patronizing (as much as 'boys') everyone, you all: too bland y'all: perfectly non-sexist but too regional (might work, but not everywhere)   Hopefully I haven't eliminated all the possibilities -- any suggestions? Set String via String.prototype function without return How do I prove this form of mean value theorem for integral? I only knew the standard mean value theorem for integrals. (i.e. $\int_a^b f(x)dx= f(c)(b-a)$ for some $c$ between $[a,b]$ where $f$ is continuous. This is directly derived by applying mean value theorem and Fundamental theorem of calculus) I'm taking numerical analysis this year and there is one theorem stated without a proof in my text. That is:  Let $f,g$ be continuous real-valued functions on $[a,b]$ such that $g$ is nonnegative. Then there exists some $c$ between $a,b$ such that $\int_a^b fg(x) dx = f(c) \int_a^b g(x) dx$  How do I prove this? Clever bounty reputation hack I say "hack" in the most favorable light. It's a clever idea and theoretically beneficial.  I noticed that an old question of mine suddenly started getting a lot of renewed attention. The question had been answered, with an answer accepted. But another user added a well reasoned and insightful answer to the list. He also opened a +100 bounty on that question.  The resulting increased attention quickly got his answer quite a lot of up-votes. Well more than the 100 points he had spent on the bounty. Therefore, he gets a net rep profit out of the transaction.  So here's the hack:   Find an interesting question Research and provide a very good answer Open up a bounty on that question to attract attention and votes to your new answer Profit!!!   2 and 3 may be out of order--I didn't see the timing of it. But either way, assuming he gets more than 10 votes, he still gets increased rep out of the deal.  I don't think this is necessarily bad for SO - it does generate some useful discussion and insightful answers. Plus you can increase your own rep very rapidly by making your answers highly visible, and therefore frequently voted upon.   However, it does slightly change the meaning and purpose of bounties away from starting a discussion and toward self-promotion. Still, interesting.