Many years (20+?) ago in Analog SF magazine I read a short story where a culture discovered interstellar travel shortly after the start of their iron age. They could construct air-tight ships and travel essentially as far as their oxygen supply lasted. Before they met us, the most technologically advanced culture they came across had discovered balloon flight. Needless to say they thought our jets were merely improved balloons.
I read a short story (I think, but it might have been a novel), where most races discover a very easy way to travel faster than light at a certain point in their technological development.   What I remember:   The guys in wooden ships bounced around subjugating the people without FTL travel. Earth had never figured out the FTL, and so had poured its effort in a different path, achieving present day technology. The aliens were freaking out about handing the humans FTL travel. The story was told from the point of view of the aliens. I read it in the last 10 years, but there is no telling how old it actually was, although I do remember them mentioning F-18s or F-14s.
Taylor's Remainder $x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}<\log(1+x) <x-\frac{x^2}{2}+\frac{x^3}{3}$ Prove that   $\displaystyle x-\frac{x^2}{2}+\frac{x^3}{3(1+x)}&lt;\log(1+x) &lt;x-\frac{x^2}{2}+\frac{x^3}{3}$  My attempt: I can prove this by taking one side at a time and assuming $\displaystyle f(x) = \log (1+x)-x+\frac{x^2}{2}-\frac{x^3}{3}$   and then proving that it is a decreasing function and then the same for the other side.  But I am looking for a better solution using Lagrange's Mean Value Theorem or using Taylor's remainder. I remember  due to misuse and I wonder if bots have a part of it. If the bot is POSTing something to my site what are ways I can combat it? I was thinking of setting some cookies and having the cookies changed via JavaScript + timestamp and sign (so yesterday's cookies can't be used today and next week).  I'm sure most people/bots would just use another site instead of enabling JavaScript in their bot.  What else can I do? I'm thinking daily POST limit and a honeypot for generic bots who just randomly post spam. Why do we use residuals to test the assumptions on errors in regression? Suppose that we have a model $Y_i = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2} + \dots + \beta_kX_{ik} + \epsilon_i$.  Regression has a number of assumptions, such as that the errors $\epsilon_i$ should be normally distributed with mean zero and constant variance. I have been taught to check these assumptions using a normal QQ plot to test for normality of the residuals $e_i = Y_i - \hat{Y}_i$ and a residuals vs. fitted plot to check that the residuals vary around zero with constant variance.  However, these tests are all on the residuals, not the errors.  From what I understand, the errors are defined as the deviation of each observation from their 'true' mean value. So, we can write $\epsilon_i = Y_i - \mathbb{E}[Y_i]$. These errors cannot be observed by us. *  My question is this: how good of a job do the residuals do in mimicking the errors?   If the assumptions appear to be satisfied on the residuals, does this mean they are satisfied on the errors too? Are there other (better) ways to test the assumptions, like fitting the model to a testing dataset and getting the residuals from there?    * Furthermore, does this not require that the model is specified correctly? That is, that the response really does have a relationship with the predictors $X_1, X_2,$ etc. in the way specified by the model.  If we are missing some predictors (say, $X_{k+1}\ \text{to}\ X_p$), then the expectation $\mathbb{E}[Y_i] = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2} + \dots + \beta_kX_{ik}$ would not even be the true mean, and further analysis on an incorrect model seems pointless.  How do we check whether the model is a correct one? Is a +1/+1 counter I get from, say Unleash or Undying different than +1/+1 from something like  or ? In a 2-columned document, how do you get a table to stretch across the entire width? I'm working on a document that runs two columns of text down the page, like many scientific conference proceedings.  However, I'd like for a table to run across the entire width of the page, as it is far too wide to fit into one column.  How do I do this? I've completed 20 suggested edits reviews today. Then, I came across a  and I approved it.   Then, I had a doubt that whether the review count has incremented or not, oh! it's not incrementing. System works fine.  But, take a look at review page,     The edit was already reviewed by one reviewer and then I reviewed it. As I'm the Owner of the post it's enough to approve the edit. But, the review history doesn't show the detail that I had approved it. It's misleading that only one vote of a reviewer who is neither the community user nor the OP nor the moderator, is enough.  So, what should we do in this type of situation?  Is it correct to make OP to review the suggested edit after completing daily limit?  Should we need to display/hide the OP has approved the edit in the review history? Upvoting a downvoted answer will do +2 upvote? I have first downvoted an answer. Then, I decided to leave it the way it was before, meaning to undo my downvote just like it never existed, so I upvote to remove my downvote. Suprise! SO seems to consider it as two upvotes!  Let's say I have an answer with -3, then I decide to downvote it making it -4. After deeper thoughts, I prefer to remove my downvote to let it the way it was before I downvoted, meaning -3, but SO makes it -2 automatically!     Wouldn't it be supposed -3, and not -2? If someone else would have upvoted, it would have made it an upvote +1, not a +2!   Am I missing something here?  In fact, is there a way to tell SO that I only wish to undo my downvote, and not upvote the answer? I may understand why it does behave this way. I only wish to undo my downvote, though. Opening files from version 2.8 in 2.79 How to estimate camera sensor size? I am using a smartphone for capturing photos and I need to estimate the camera sensor size (mm*mm). I have the resolution (in pixels) and the focal length (in pixels). I can also measure the size of any stationary object and its distance w.r.t the lens. Basically, I don't have a pixel-to-mm metric for any of the above parameters that I can use. Is there any way I can calculate that to estimate sensor size? Is Kant's "noumenal self" argument on freedom flawed? plist or sqlite continuous function: preimage of closure of a set is closure of preimage Why is Android File Transfer necessary? Apparently newer versions of Android require the use of Android File Transfer. I have an older machine that doesn't require this, and just mounts as a USB mass storage device. I can't see a technical reason why Android File Transfer is now required when it used not to be.  What changed in order to make this necessary? Why is it necessary? Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge This is a homework problem for an undergrad topology course.  Let $l^{2}$ be the set of all real-valued sequences $(c_{n})$ where $\sum_{n=1}^{\infty}c_{n}^{2}$ converges. Let $(a_{n}),(b_{n})\in l^{2}$.  Claim: $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges.  I've done a couple of examples where $a_{n}$ and $b_{n}$ are general harmonic series. I've also written out the definition of convergence for $a_{n}$ and $b_{n} $. My problem is that thus far I do not understand on any level why my claim is true. AmpScript to select Right or Left characters from a string Is there an AmpScript function that can be used in an equivalent manner to sql left, right and mid functions?   Thanks, Mark Shortcut in Windows 7 to switch between same application's windows, like Cmd + ` in OS X Is there a shortcut in Windows 7 to switch between windows in a given application only? I very much like this feature in OS X; what is its equivalent in Windows? I am unit testing a .NET application (.exe) that uses an app.config file to load configuration properties.  The unit test application itself does not have an app.config file.    When I try to unit test a method that utilizes any of the configuration properties, they return null.  I'm assuming this is because the unit test application is not going to load in the target application's app.config.  Is there a way to override this or do I have to write a script to copy the contents of the target app.config to a local app.config?     post kind-of asks this question but the author is really looking at it from a different angle than I am.  EDIT: I should mention that I'm using VS08 Team System for my unit tests. Use of # a.k.a. read-macro Limit of a Piecewise Function defined on Rationals and Irrationals What are the restrictions on the use of Slayer's Prey? In the recent book Xanathar's Guide to Everything, there are some subclasses/archetypes added for Rangers, among them the  (p. 43), which gives some specific helpful features. Hunter's Sense  At 3rd level, you gain the ability to peer at a creature and magically discern how best to hurt it. As an action, choose one creature you can see within 60 feet of you. You immediately learn whether the creature has any damage immunities, resistances, or vulnerabilities and what they are. If the creature is hidden from divination magic, you sense that it has no damage immunities, resistances, or vulnerabilities. You can use this feature a number of times equal to your Wisdom modifier (minimum of once). You regain all expended uses of it when you finish a long rest.  The limitations on use are clearly spelled out. Slayer's Prey  Starting at 3rd level, you can focus your ire on one foe, increasing the harm you inflict on it. As a bonus action, you designate one creature you can see within 60 feet of you as the target of this feature. The first time each turn that you hit that target with a weapon attack, it takes an extra 1d6 damage from the weapon. This benefit lasts until you finish a short or long rest. It ends early if you designate a different creature.  So from this, I would interpret it as it being able to switch from creature to creature at will, but there is no guidance as to what happens if the creature under the mark dies, or if you can carry it (so to speak) from encounter to encounter. Is there a limitation on the use of Slayer's Prey in the wording that I am missing? Or is this a basically a multiple use feature? (i.e., you can designate any creature as a bonus action, with no limitations on uses per day)? The extra d6 seems a little overpowering at lower levels, and somewhat meh at higher levels. FWIW, the GM basically allowed it as multi-use for the session, and was going to also try to find clarification.