A special solution for a functional inequality from $\mathbb{R}\times \mathbb{R} $ onto $\mathbb{R}$
A special bijection between $\mathbb{R}\times \mathbb{R}$ and $\mathbb{R}$
How to create quizzes with macqtex? Grammatically, what is "It" in the following sentence? I'm currently working at a private academy in Korea, and my boss just asked me a real head-scratcher.   In the sentence:     It doesn't have to be hot and humid for players to lose too much water from their bodies.    I'm aware that "It" is a subject pronoun replacing "The weather." Besides that, however, I'm unsure.   Is it a false subject? I wouldn't think so since it's a pronoun. Is it some crazy kind of indirect object?  Anyway, thank you all for your help! What's the smallest positive integer which is divisible by all integers $1, 2, \dots, n$?  Is there a simple way to represent the answer?  Call it $f(n)$ here.   Clearly factorial ($n!$) satisfies the condition of divisibility.  But once you get to $f(4)$ (which is $12$), factorial ($24$) is too large, because it includes as factors both $4$ and $2$, which are redundant.  Multiplying all the prime numbers up to $n$ gives you another estimate, which is this time too low, because you need to include primes multiple times when they're repeated in the factorization of the inputs.  So the answer is somewhere between factorial, and the product of primes.  Is there a simple answer? Prove that $\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}dx$ Prove that $$\frac{d}{dx}(\tan^{-1}(x))=\frac{1}{1+x^2}$$ Can anyone help me with this problem?     Let $f:[0,\infty)\longrightarrow \mathbb R$ be a continuous function such that for each $x&gt;0$, we have $\lim_{n\to \infty}f(nx)=0$. Then prove that $\lim_{x\to \infty}f(x)=0$.   Our teacher told first to prove Baire's theorem, and then show that this is a consequence of that theorem. I proved Baire's theorem, and I spend a few hours thinking on how Baire's theorem is related to this problem, but I couldn't find anything. I'd really appreciate your help. What is the difference between an int and an Integer in Java and C#? In  answer to the question "why does the moon have the same rotation and revolution periods?", we read:     The mass and speed of rotation of the Earth influence the moon in that some of its rotational energy is actually transferred to the moon. (...) Many of the moons in the solar system have also reached this point of equilibrium. In Jupiter, the moons Amalthea, Thebe, Io, Ganymede, Callista, and Europa, all have identical rotational and revolutionary periods.    My two questions are:   Can we show the mechanics of this quantitatively?  Does the system Earth-Sun experience the same phenomenon, in some measure? Will then the earth slow down its rotation speed until it will show always the same face to the sunlight? In this last case, on which timescale? Are features that allow −5 to attack to get +10 to damage mathematically sound? Some class features, monster traits, and feats allow you to take a −5 to your attack roll in order to gain a +10 to your damage roll. At first glance this looks really good, but when will it deal more damage than not using it on average?  For instance if you have a total attack bonus of +8 and you deal 1d12 damage. You have to roll a 7 to hit an AC of 15. If you take the −5 that makes it where you have to roll a 12. That's going to reduce your average damage by a lot. What is the antiderivative of $x^{x^{x^{x...}}}$? What is $\int{x^{x^{x^{x^{\dots }}}}dx}$? I managed to get it down to a complicated series after using $u$-substitution, but I want to see other methods of approach, or if my answer was right.   First I let $$u=x^{x^{x^{x^{\dots }}}}$$, and $du$ would be, in terms of $u$, $$du=  \frac{u}{\frac{1-ln(u)}{u}e^{ln(u)/u}}$$ This makes the integral of $$\int{x^{x^{x^{x^{\dots }}}}dx}=\int \frac{1 - ln(u)}{u}{e^{ln(u)/u} }du$$ Now making $w = ln(u), dw$ becomes $1/u$ and the integral becomes $$\int e^{w/e^{w}}(1-w)dw$$ From there, I distributed and made $2$ integrals of series, using the Taylor polynomial for $e^x$. From there, I integrated and back substituted. I was only trying to find the indefinite integral.   I'm still a junior in high school, so let me know if the answer is beyond my scope of understanding. Why don't Stack Exchange sites use a rich text editor? Why does the Stack Exchange network of sites use Markdown instead of some other rich text editor like FCKEditor or TinyMCE? What is an efficient way to implement a singleton pattern in Java? Relation between confidence interval and testing statistical hypothesis for t-test It is well known that confidence intervals and testing statistical hypothesis are strongly related. My questions is focused on comparison of means for two groups based on a numerical variable. Let's assume that such hypothesis is tested using t-test. On the other side, one can compute confidence intervals for means of both groups. Is there any relation between overlapping of confidence intervals and the rejection of the null hypothesis that means are equal (in favor of the alternative that means differ - two-sided test)? For example, a test could reject the null hypothesis iff the confidence intervals do not overlap. When do I clear US customs and immigration if connecting at a US airport to my final destination elsewhere in the US? I'm a British citizen travelling from London Heathrow (LHR) to the Hartford, Connecticut airport, and have a two hour stopover in Newark, New Jersey (EWR) on my itinerary. At what stage would I clear immigration and customs on arriving in the USA? Radiation– white vs black house, hot or cool? In my book's thermodynamics chapter, it says that an      "object that radiates heat faster also absorbs heat faster. This means   that an object that is a more efficient radiator comes to equilibrium   with its environment more quickly. With this in mind, is it better to   paint your house black or white?"   I am confused which it would be. The book says white because      "in summer, you house is cooler than the environment and white   reflects away the heat. In winter your house is warmer than the   environment and white radiates away the heat" Does the mass of a battery change when charged/discharged? If so, how much? Is it possible to detect it, or is it beyond any measurement?  I'd say there are two possible scenarios (depending on the battery type) and both seem interesting:   The battery reacts chemically with its environment. The battery doesn't exchange any matter with its environment except electrons. I suppose there should be some difference at least due to the principle of energy-matter equivalence, but the difference is most likely immeasurable. How to secure database passwords in PHP? How to retrieve GET parameters from JavaScript Does the Google spider render JavaScript? Removing unnecessary vertices? I have over sampled polygons that I need to simplify, without changing the shape. basically, I want to reduce the number of vertices, by eliminating the ones that are useless and have the polygons represented with the significant vertices only. I used to do that with ET GeoWizards Plug-in, but I don't have a license anymore. My table has 500.000 features, water polygons.   Any ideas? On every other site on Stack Exchange I can click on the site's title and be taken directly to the questions...     Except Meta.stackexchange.com   Here I actually have to click on the word "Questions" as there is no link active on the Meta title and logo.  Now I agree this isn't a big deal, however as I typically visit stack exchange pages up to a couple of hundred times a day, this one that doesn't fit spoils my flow. I am pretty certain it used to work.  It works for me in Firefox, and Safari on iPad, but even closing and reopening Chrome doesn't fix the issue.  Chrome is on Version 53.0.2785.116