We know that, for $\alpha$, $\beta$, $\gamma$ cardinals, if $\alpha \leq \beta$, then $\alpha^\gamma \leq \beta^\gamma$, and, in general, the usual operations defined on cardinals are compatible with the order relation. I have been given the task to determine if the reversed implication holds, i.e. $\alpha^\gamma \leq \beta^\gamma \Rightarrow \alpha \leq \beta$. I think the answer is no, given that $4^{\aleph_0}=2^{\aleph_0}=\mathfrak{c}$ and $4&gt;2$. But what happens if we have a strict inequality, does this implication hold? $$\alpha^\gamma &lt; \beta^\gamma \Rightarrow \alpha &lt; \beta.$$ I proved that the above implication is true for $\gamma=$ finite. P.S. I am asking this question, because I realize an interesting thing: if we replace the exponentiation operator with multiplication or addition, the implication does hold, that is, $$\alpha+\gamma &lt; \beta+\gamma \Rightarrow \alpha &lt; \beta.$$ $$\alpha\cdot\gamma &lt; \beta\cdot\gamma \Rightarrow \alpha &lt; \beta.$$ Also, there are counterexamples for division and subtraction, $\mathfrak{c}+\aleph_0=\mathfrak{c}\cdot\aleph_0=\mathfrak{c}+1=\mathfrak{c}\cdot1=\mathfrak{c}$, and $\aleph_0&gt;1$.
Let $|A|=\alpha, |B|=\beta, |C|= \gamma$ be cardinals and $\beta \leq \gamma$. Prove $\alpha ^{\beta}\le \alpha ^{\gamma}$.  So from the given we know that there's an injection $f:B\to C$ and some functions $h:B\to A, g: C\to A$. We want to prove there's an injection $l_1:A\to C$. It appears that $f$ doesn't help here.  Trying to take representatives from $A$ and show they're in $C$ and there's an injection doesn't work so maybe the function should be $l_2: h \to g$ but I don't know how to work with it.
Accessing class members on a NULL pointer \@seccntformat only for section Rails lists have .first and .second – is there a .hundredth or .sixty_nineth ? What's the name for the relativistic paradox with the train/car travelling over a gap? There is a relativistic paradox I was introduced by a friend in Ithaca (Courtenay Footman). It involves a vehicle travelling over a ravine relativistically, so it's in the general family of ladder paradoxes. There are many ways to phrase it, but here's a simple one.   A three-car train with each car of length $L$ is travelling over a bridge across a ravine. (Assume that the couplings between cars are stiff: this train does not bend.) Unfortunately, some evil genius has precisely removed a length $6L$ of the track. The train however is relativistic, travelling at a speed such that $\gamma \approx 12$.  In the reference frame of the ground, the train should appear to be 1/24th the size of the gap, which suggests that it will fall a little and should crash into the bridge opposite. But in the reference frame of the train, each car of the train is at least twice as long as the gap and aside from the very front wheel and the very back wheel, which need to be supported by rigidity, everything is constantly well-supported. This suggests that the train makes it. Who is right?  I vaguely remember that as an undergraduate knowing only a little relativity, I trusted the train's perspective because it's at rest in that perspective, and Courtenay said that this was ultimately correct after a lot of details got resolved. But I do not remember all of these details such that I could work it out for myself. So I'm looking for some discussion that's a little more well-sourced or authoritative.  What is this paradox called, and what literature has addressed the issue? jQuery animate() WinBUGS: Multiple definitions of a node So this question is about the BUGS modeling language. So you either know it or have no clue. I'm a newbie to this so it's been driving me mad. I want to define a simple two-state hidden Markov model (HMM) where the emission of each state follows a Normal distribution. I have an array data of Nxl dimension where each row is a subject and column is a continuous variable Time associated with the subject. So in WinBUGS, I would define the model as:   for(i in 1:N) { # for each subject     # Sample the initial observation     Time[i, 1] ~ dnorm(mu[State[1]], tau[State[1]])      for(j in 2:l[i]) { # for each observation         # Sample the observed variables         Time[i, j] ~ dnorm(mu[State[j]], tau[State[j]])         # Sample the hidden states         State[j] ~ dcat(P[State[j-1], ])     } }  # P is the transition matrix of the hidden Markov chain P[1, 1:2] ~ ddirch(Pinit) P[2, 1:2] ~ ddirch(Pinit)  # Set the initial states State[1] ~ dcat(Pinit)  # Sample the initial params mu[1] ~ dnorm(200, 1.0E-6) mu[2] ~ dnorm(400, 1.0E-6)  # The precision params tau[1] ~ dgamma(0.001, 0.001) tau[2] ~ dgamma(0.001, 0.001)   Initial values have also been provided via R2WinBUGS. The model is syntactically correct. But when I run, I got this error: "multiple definitions of node State[2]"  Can you please tell me why and how to solve this? I've searched around on the error but each has their own specific case and there's not a generic solution or explanation as to why this arises. I have $k$ pairs of samples from different distributions. For each pair of samples I want to check if the samples are taken from a normal distribution with the same mean and variance. I assume the normal distributions for each pair is different and independent of other pairs. I do not know the mean and variance of any of the distributions.  I am doing a two sample t-test for each of the pairs. I want to combine the $k$ resulting p-values.   I could use Fisher's method but I am wondering about the distribution of the average t-value. If these were z-values rather than t-values I would use Stouffer's z-test. I could also approximate the t-distribution with a z-distribution and than use Stouffer's z-test, but I am not sure if that is a good approximation.  Is there a test like Stouffer's z-test for t-values?  Is there a better way to combine the t-values? Spy from Earth studies Galactic Empire nobility and prevents plot to depose Emperor Browsing the internet from the command line I have the following data structure (a list of lists)  [  ['4', '21', '1', '14', '2008-10-24 15:42:58'],   ['3', '22', '4', '2somename', '2008-10-24 15:22:03'],   ['5', '21', '3', '19', '2008-10-24 15:45:45'],   ['6', '21', '1', '1somename', '2008-10-24 15:45:49'],   ['7', '22', '3', '2somename', '2008-10-24 15:45:51'] ]   I would like to be able to   Use a function to reorder the list so that I can group by each item in the list. For example I'd like to be able to group by the second column (so that all the 21's are together)  Use a function to only display certain values from each inner list. For example i'd like to reduce this list to only contain the 4th field value of '2somename'    so the list would look like this   [      ['3', '22', '4', '2somename', '2008-10-24 15:22:03'],       ['7', '22', '3', '2somename', '2008-10-24 15:45:51'] ] How do we get the last day of the month from a given date Is there a function or method to give the last day of a month? For example if the date is 27 feb 2013, the last date should be 28 feb 2013 , it become 29 if its a leap year. if the date was 2 jan then it should return 31 Jan.  If there is no function then any ideas on how we can get this?  Thanks How to backup symbolic link paths? Next month I am traveling to country X, where I would need Y minutes of calls, Z text messages and most importantly Q megabytes of data. How can I find the cheapest/most reliable SIM card for my needs? Allow bounty to be set with reputation from another site? Here's one to get shot down: How about letting bounties to be set using the reputation gained through the use of another site? So, for instance, I'm a programmer with a high Stack Overflow rep who hates using computers. Wouldn't it be useful for me to be able to exchange some of my excess Stack Overflow credit to incentivise a question on Super User?   Or perhaps I'd like to highlight a question on Meta Stack Overflow, but I don't spend my time there. Why shouldn't I use "Hungarian Notation"? $$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5}  = \lim_{x\rightarrow 5} (x+5)$$  I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I understand how to do this procedurally, but I'd like to know why this works. I've only been told the methodology of expanding the $x^2-25$ into $(x-5)(x+5)$, but I don't just want to understand the methodology which my teacher tells me to "just memorize", I really want to know what's going on. I've read about factoring in abstract algebra, and about irreducible polynomials (just an example...), and I'd like to get a bigger picture of the abstract algebra in order to see why we factor the limit and why the simplified is equal to the original if it's missing the $(x-5)$, which has been cancelled. I don't want to just memorize things, I would really like to understand, but I've been told that this is "just how we do it" and that I should "practice to just memorize the procedure." I really want to understand this in abstract algebra terms, please elaborate. Thank you very much. With ubuntu 19.10 I have been attempting to pair my Airpods with my laptop. There seems to be a problem with pairing and system is unable to add them as Bluetooth device. This happened after an upgrade from 19.04. Changing ControllerMode in /etc/bluetooth/main.conf seems to have no affect. Any help appreciated. I understand if you travelling in or to the US, you can lock your luggage using the TSA Luggage Lock. (Immigration is still able to open your luggage using their own special keys)  How about Europe? Especially Heathrow + Charles de Gaulle?    If the TSA Lock is not acceptable in Europe, are there any other alternative precautions I could use instead?  (FYI, information about the ) number of prime ideals in algebraic integers Using residue theorem to integrate from $-\infty$ to $\infty$ I'm trying to integrate $$\int_{-\infty}^{\infty} {x^2 \over {(x^2 + 1)}^2(x^2 + 2x + 2)} $$ given that the function  $$f(z) = {z^2 \over {(z^2 + 1)}^2(z^2+2z+2)} $$ has residues $${9i - 12 \over 100},{3 - 4i \over 25}$$ at the poles $i$ and $-1+i$ respectively. From my understanding of this I have added the two residues (by the residue theorem, and because their respective poles lie in the upper half plane) and multiplied by $2\pi i$ and got an answer of $14\pi \over 100$  Have I done this right?