I am coming to Germany for holidays, and I would life to use my cell phone, however I guess a German provider may be able to offer a much better service of Internet connections. But since this is temporary trip, I don't wanna buy a full plan.  Are there SIM Cards for me to buy in Frankfurt Airport or Frankfurt area?
What SIM card would you recommend for a smartphone user for travel in Germany?  It would be probably the only or the most important connection to the Internet, so large amounts of data and relatively fast transfer is an issue.  I need to stay in touch with friends and family, so the calls to other EU countries should be as cheap as possible.   For to the first criteria, I found the T-Mobile offer interesting. The internet costs 10 Euros/month and after reaching the transfer limit it stays relatively fast. According to tests (Galileo from Pro7, AFAIR) it has also the best coverage. However, calls to other countries are very expensive.  For to second criteria, Lebara and Mobilka are interesting. The calls are very cheap, but there are problems with signal strength in many locations, where T-Mobile has no problem. The voice quality, at least in Lebara, is also limited. They both have mobile internet with unlimited transfer, but after reaching the high speed limit, the transfer is reduced to 64 kbps, which is too slow.  I've found out that O2 has foreign calls at moderate prices, but I haven't yet checked for details. There are so many other options, making it is hard to compare them all without spending the whole weekend on it. So, if someone has found a better option than having 2 SIM cards and constantly switching them, this answer would be greatly appreciated.
Enumerated description list How can a list be created, whose items are both    enumerated (a,b,c,d,...)  and allow for setting a description via \item[texthere] ?   Im am using MikTex on Windows. For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous. I was just trying to solve an exercise in Munkres' Analysis on Manifolds to get better at continuous functions and topology in $\mathbb{R}^{n}$, and I got stuck. Here it is:      Let $X = A \cup B$, $A$ and $B$ are subspaces of $X$. Let $f:X \rightarrow Y.$ Suppose that the restricted functions $f_{|A}:A \rightarrow Y$ and $f_{|B}:B \rightarrow Y$ are continuous. Show that if both $A$ and $B$ are closed in $X$, then $f$ is continuous.    Our definition of continuity is:  $f$ is continuous at $x \in X$ if for each open set $V$ of $Y$ containing $f(x)$, there is an open set $U$ of $X$ containing $x$ such that $f(U) \subset V$. $f$ is continuous if it is continuous at each point of its domain.  What I've tried so far:  Suppose $A$ and $B$ are closed in $X$. Suppose $A \cap B = \emptyset$. Then $X -A$ is open in $X$. But $X - A = B$ which is closed in $X$, which is a contradiction. Hence, $A \cap B \ne  \emptyset$. I'm not sure if this will be useful in the problem or not. If $x \in X$, then it's either in $A$ or in $B$. Suppose without loss of generality that it's in $A$. Then for all open sets $V$ of $Y$ s.t. $f(x) \in V$, $\exists$ an open set $U_{1}$ of $A$ s.t. $U_{1} \subset A, a \in U_{1}$ and $f(U_{1}) \subset V$. This is almost what we want, except that we want to find an open set $U$ of $X$ s.t. $U \subset A, a \in U$ and $f(U) \subset V$. This is because $U_{1}$ being open in $A$ does not necessarily mean that it's open in $X$ because $A$ is closed in $X$. (i know how to expand this argument out if needed) My idea now is to try to find a subset $U$ of $U_{1}$ such that $U$ is open in $X$. We have two cases:  (1) $U_{1}$ does not intersect the boundary of $A$. (2) $U_{1}$ intersects the boundary of $A$. Now I'm not sure what to do. I'm wondering if it would help to investigate the relationship between Bd(A) and Bd(B).. In any case (no pun intended), any help/advice would be greatly appreciated. :)  Sincerely,  Vien Let us look at some sample data for 5 hypothetical subjects.  library(emmeans) library(lme4)  # generate some sample data # condition (Placebo, Treatment) # type (some factor, e.g. two different medications) # value (outcome of interest) # subject  data &lt;- data.frame(condition = as.factor(rep(c(0,1), times = 10)), value = rnorm(20), type = rep(c("A","B"), each = 10), subject = as.factor(rep(rep(1:5, each = 2), times = 2)))   Now we like to conduct a mixed model and post-hoc tests with emmeans, where we test conditions per type in a pairwise fashion:  # caluculate a mixed model fit_data &lt;- lmer(value ~ condition*type + (1|subject), data = data)  # emmeans post-hoc test emmeans(fit_data, pairwise ~ condition | type)   So for each type (A or B), I have paired data (condition 0 has 5 data points and condition 1 has 5 data points). If I would do a paired t-test like so:  # paired t-test t.test(data[data$condition==0 &amp; data$type=="A","value"], data[data$condition==1 &amp; data$type=="A","value"], paired = TRUE) t.test(data[data$condition==0 &amp; data$type=="B","value"], data[data$condition==1 &amp; data$type=="B","value"], paired = TRUE)   I would get degrees of freedom of 4 for the paired t-test, but emmeans says the degrees of freedom are 12. Why is there this huge difference? If the emmeans package also would use df = 4, then the p-values would also be more comparable. GCD of the already GCD How to get the list of properties of a class? I have a problem with the interpretation of a test result in which the p-value is 8.6e-28. How should it be interpreted? What is the e for?  Specifically, I used White's Test for the homoscedasticity assumption of a linear regression:  White's general test statistic :  150.0535  Chi-sq( 9)  P-value =  8.6e-28 General formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$. If $k=1$, we have \begin{align} \sum_{n\ge1}\frac{1}{\binom{2n}{n}}&amp;=\frac12\sum_{n\ge1}n\mathrm{B}(n,n)\\ &amp;=\frac12\int _{0}^{1}\sum_{n\ge1}n(t-t^2)^{{n-1}}{\mathrm  d}t\\ &amp;=\frac12\int _{0}^{1}\frac{1}{(t-t^2-1)^2}{\mathrm  d}t\\ &amp;=\frac13+\frac{2\sqrt3\pi}{27} \end{align} I wonder if there exists a general formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$ with $k\in\Bbb N^*$. This may involve some ${}_pF_q$ functions. How can one show that for triangles of sides $a,b,c$ that   $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} &lt; 2$$  My proof is long winded, which is why I am posting the problem here.   Step 1: let $a=x+y$, $b=y+z$, $c=x+z$, and let $x+y+z=1$ to get   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}&lt;2$  Step 2: consider the function $f(x)=\frac{1-x}{1+x}$, and note that it is convex on the interval (0,1), so the minimum of   $\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}$  is reached when the function takes the extreme points. i.e. $x=y=0, z=1$.    But going back to the fact that this is a triangle, we note that $x=y=0 \implies a=0$ which is not possible, so the inequality is strict. I've started trying to make a 3D model that I will end up exporting to an augmented reality app but I've hit a problem in terms of the particle system. The app wants the 3D file in .dae(collada) format which blender allows, but it doesn't seem to export particle systems. I've read that blender doesn't support exporting particle systems at all so I'm now focused on converting the particle system to lots of meshes which will (hopefully)animate using key frames. Then I can export all the meshes and the particles will 'work'.  Currently I'm just using a standard particle system emitting small icospheres. I've tried a few different techniques, such as 'make duplicates real' and using a particle instance instead but they don't seem to animate. Instead it just freezes the current frame into an object(which exports fine but is not what I want). I've also tried recording the animation in the blender game engine but particle systems don't work with it so that didn't work either.  Any help would be appreciated! Thanks Is there BitTorrent software that runs in a terminal? I want to deploy it on my VPS server to download resources quickly.  This could save me a lot of time since some resources are in a foreign country, How to ignore spaces before footnotes How to get Url Hash (#) from server side I know on client side (javascript) you can use windows.location.hash but could not find anyway to access from the server side. I'm using asp.net. What's the best way of replacing a plug or switch in a house with aluminum wiring? can we have a parallel earth made of dark matter? Evaluate  $$\int\frac{e^xdx}{(e^{2x}-1)(e^x+1)}$$  I tried to decompose the integrand into A and B partial fractions.  No success. Can magnets rotate infinitely? There are many videos on youtube in which people arranged magnets in circle and rotated one placing in middle of that circle on a shaft, and the magnet (magnet motor) starts madly and continues its movement. Do they really rotate infinitely? If so, do they from where they get that extra energy to move so fast and infinitely?    EDIT: This question is about real world scenario  UPDATE: I am unable to find that particular video about which i talked about -- I added them to my favourites, but they are all now deleted :( -- there are hundreds of that that kind of videos. Some example videos links How would I go about finding $\prod_{n=1}^\infty (1+\frac{1}{n^2})$ ? Setting $a_k = \prod_{n=1}^k (1+\frac{1}{n^2})$, the first few partial products are $a_1 = 2$, $a_1 = 2.5$, $a_3 \approx 2.78$, $a_4 \approx 2.591$, $a_{20} \approx 3.50$, and they eventually appear to converge at around $3.639$. (These values were found with a quick python program. So much for my initial conjecture that the product converged to $e$.)   I tried writing $1+\frac{1}{n^2}$ as $\frac{n^2 + 1}{n^2}$, but it wasn't immediately obvious how to prove convergence / find the limit because the product doesn't really factor or telescope. Is there a way to calculate the limit? Brokerage accounts are seen as relatively tax-inefficient, since you invest with post-tax dollars, and growth is taxed upon realizing gains. The traditional IRA allows you to deduct contributions and defer taxes until you take distributions in retirement ...  Unless you are covered by a retirement plan from work and make over   $71,000/year if single, or $118,000/year if married filing jointly.   The numbers above come from . Above those thresholds, you can still make contributions to a traditional IRA, but those contributions will be post-tax. Distributions are taxed as ordinary income, excluding post-tax principal.  Similarly, eligibility to make direct Roth IRA contributions phases out to zero according to a :   $131,000 if single, or $193,000 if married filing jointly.   Looks to me that someone who cannot make direct Roth IRA contributions also cannot deduct traditional IRA deposits from income.  Assume   investment period is 20 years, is some garden variety ETF (VTI/VXUS/etc), those investments return 5%, the current marginal tax rate is 28%, the expected marginal tax rate at retirement is 25%, the long term capital gains rate is 15%, the entity is not eligible to make Roth IRA contributions, and 401k contributions are already maxed out.   A $5000 of gross income contribution into a traditional IRA at retirement might grow to an after-tax sum of:  (5000) * (1 - 0.28) * (1 + (1.05 ** 20 - 1) * (1 - 0.25)) = $8064   Instead, since contributions were going to be taxed anyway, why not put it in a brokerage account? Then you could claim the favorable long-term capital gains rate.  (5000) * (1 - 0.28) * (1 + (1.05 ** 20 - 1) * (1 - 0.15)) = $8659   All I found was , and the accepted answer says to still make a traditional IRA contribution, but I still don't see why that would be a good idea. What am I missing here? Does Stack Exchange have an Economics section? Why are Only Real Things Measurable? Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with operators in quantum mechanics, why must measurable quantities be Hermitian, and consequently real?