Show that there is a rational number and an irrational number between any two real numbers.
Prove that between any two different real numbers there is a rational number and an irrational number.  My attempt at a proof: (Throughout this proof it is assumed without any loss of generality that $x &lt; y$)  1.) $x$ is irrational and $y$ is irrational.  In this case the arithmetic mean $\frac{x+y}{2}$ is always an irrational number between $x$ and $y$. Also, if $y$ is irrational then it has a non-terminating decimal expansion and so we can always find a positive integer $n$ such that $x &lt; y - 10^{-n}y &lt; y$. The middle of the three numbers here will have a terminating decimal expansion, and hence it is rational. It also clearly lies between $x$ and $y$. These examples of rational and irrational numbers should also work in the cases:  2.) $x$ is irrational and $y$ is rational.  3.) $x$ is rational and $y$ is irrational.  There is one last case to consider.  4.) $x$ is rational and $y$ is rational.  In this case the arithmetic mean of $x$ and $y$ is rational. An irrational number for example could be the ratio $\frac{ex + \pi y}{e + \pi}$  I'm not sure if this proof is a good method or if it is even valid, so some feedback would be appreciated.
Convert from MySQL datetime to another format with PHP I have a datetime column in MySQL.   How can I convert it to the display as mm/dd/yy H:M (AM/PM) using PHP? Can you explain the "Axiom of choice" in simple terms? Trying to identify a short sci-fi story about a human that is "resurrected" by malignant aliens Run a command without making me wait On the CLI, sometimes a command I type takes a while to complete, and sometimes I know when that's about to happen. I'm a bit confused on "backgrounding" and such in Linux.  What is the most common (or user-friendly way) of telling the CLI that I don't want to wait, please give me back my prompt immediately. And if it could give me a progress bar or just busy-spinner, that would be great! Isomorphism equivalence implies fundamental group is abelian Who were the hooded people in The Rise of Skywalker? Near the end of Star Wars Episode IX: The Rise of Skywalker there are several hooded figures that are shown chanting stuff (in Sith?) and just hanging out with Palpatine. Who were these people? Limit of some specific "almost Riemann" sums I'm just beginning my journey with calculus, and this problem is giving me hard time.  $$\lim_{n \to \infty} \sum_{k=1}^n \frac{k}{{n^2+k}}$$  I calculated first three sums: $$\frac{1}{2}, \frac{16}{30}, \frac{117}{220}$$ From that I guessed that the limit is $$\frac{1}{2}$$. The hard part, though, is proving it.    I reckon that the squeeze theorem should be used here, but I can't seem to find anything useful. (The only sequences I was able to figure out give me limits of 1 and 0, so it's not something I can use)  Maybe you guys can give me some ideas or theorems that can be of use here? I know that this problem could be probably solved using some more advanced calculus techniques, but I'm not familiar with derivatives and integrals yet, and this problem is in the limits section of my problem book, so I'm not supposed to use anything more advanced. A thought experiment: Imagine the Sun is suddenly removed. We wouldn't notice a difference for 8 minutes, because that's how long light takes to get from the Sun's surface to Earth.   However, what about the Sun's gravitational effect? If gravity propagates at the speed of light, for 8 minutes the Earth will continue to follow an orbit around nothing. If however, gravity is due to a distortion of spacetime, this distortion will cease to exist as soon as the mass is removed, thus the Earth will leave through the orbit tangent, so we could observe the Sun's disappearance more quickly.  What is the state of the research around such a thought experiment? Can this be inferred from observation? Increasing real valued function whose image set is connected Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true:  a) $f$ is discontinuous exactly at one point.  b) $f$ is discontinuous exactly at two points.  c) $f$ is discontinuous at infinite many points.  d) $f$ is continous.  I think $f$ is continuous . We need to show that $\lim_{x \to a^+}f(x) = f(a)$ for any $a \in S$. Since $f(x)$ is increasing, $\lim_{x \to a^+}f(x) \leq f(a)$. Suppose $\lim_{x \to a^+}f(x) &lt; f(a)$. Then there is $l$ such that $\lim_{x \to a^+}f(x) &lt; l &lt;f(a)$, which contradicts connectedness of $f(S)$. Similarly we can show that $\lim_{x \to a^-}f(x) = f(a)$.  I would be thankful if anyone would check my solution and correct me if there are any errors. How can I enable Hardmode? What items do I need to make my world Hard Mode? 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. Imagine that a PC gets reduced to 0 hit points during a battle. The battle continues for a couple more rounds and they roll their death saves, one success and one failure (not sure if this part of the question is relevant).  Then the rest of the party defeats the last enemy and battle is over. Now what?    Does the downed PC keep rolling death saves until they strike 3 one way or the other? Does everyone else repeatedly roll for Medicine to stabalise the dying PC? Or (as we've been ruling it) they are just assumed to become stabilised because combat is over (I'm fairly certain this probably isn't RAW or RAI though).   I've looked through the PHB and I've only managed to find descriptions of being stabilised during combat, not after it.  Ideally I'd like a rule quoted (or a tweet or something) to point to where such a thing is outlined in the rules, if it is written anywhere. How can I securely format a solid-state drive? How to see if certain representation of $SU(2)$ is symmetric/antisymmetric/mixed? I am looking at the tensor product of $n$ spin halfs (fundamental of $SU(2)$): \begin{equation} \left(\frac{1}{2}\right)^n = \frac{1}{2} \times \frac{1}{2} \times ... \times \frac{1}{2} = \frac{n}{2} + ... \end{equation} For example, for $n = 3$ \begin{equation} \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{2} \times \left(1 + 0\right) = \left(\frac{3}{2}\right)_1 + \left(\frac{1}{2}\right)_1 + \left( \frac{1}{2} \right)_0 \end{equation} where lower indexes on the right hand side specify from which tensor product representation came from. My question is how do I read off if a certain term on the right is a symmetric, antisymmetric or a mixed symmetry representation of $SU(2)$? I came up with two ways of approaching the problem.  First is Young Tableaux. As far as I understand rows of Young Tableaux correspond to symmetric indexes while columns to antisymmetric ones. So, for example, a diagram with just one row (like one that corresponds to the highest spin $\frac{3}{2}$ from the example - three boxes in a row) describes a totally symmetric representation. A diagram that has one column (maximum 2 boxes since we are dealing with $SU(2)$) describes totally antisymmetric representation. Everything that has both rows and columns is a mixed symmetry representation. Following this logic it looks like only the highest representation in the product can be totally symmetric, singlet is antisymmetric and everything in between has mixed symmetry. So, for the example \begin{equation} \left(\frac{3}{2}\right)_1: \text{symmetric rep}\\ \left(\frac{1}{2}\right)_{0,1}: \text{mixed symmetry reps}\\ \end{equation} Is this is a correct logic?  Another way to think about this is the following. Let's build representation the way we do in QM starting from the highest one, when all the spins are up. Then all components of the highest multiplet can be generated by applying lowering operators to the highest state. So, all the relative signs in the expressions are &quot;+&quot; and thus the highest representation is always symmetric as 1. also suggests. Then all the other representations must be constructed by writing down expressions orthogonal to components of the highest multiplet, so they will necessary get some &quot;-&quot; signs and as a result must be either antisymmetric or have mixed symmetry, but can not be totally symmetric.   Is this correct logic? Is it true that only the highest representation in the product can be symmetric and all the others have mixed symmetry or antisymmetric? Change multiple filenames by replacing a character I have multiple files named like that : screenshot 13:25.png Windows struggle to open these files probably because of the ":".  How can I replace it? How to measure supply currents ranging from 1μA to 10mA? I've been wondering about this for a while, since getting interested in knife sharpening.  When using a knife, it will get duller. One (the primary) reason being that you "bend" the edge. You fix this using a honing steel.  After a while though, you need to sharpen the knife again.  Another setting: If you have a brittle knife and/or mistreat your knife you will chip it.   As far as I can tell, this should mean that small pieces of metal (dust, shavings, etc.) come lose, and then likely get in the food.  Is this correct? Are there no health considerations when digesting small pieces of very sharp metal? ASCII art generator Is there a way to calculate a sorted field with sequential numbers?  I have seen  that outlines how to calculate sequential numbers, but this is always calculated on FID order, not on sorted order.  #Pre-logic Script Code: rec=0 def autoIncrement():      global rec      pStart = 1       pInterval = 1      if (rec == 0):           rec = pStart       else:           rec += pInterval       return rec  #Expression: autoIncrement()   An example of what I'm trying to do.  I've used an advanced sort to sort by year, month, day, and now want to have sequential numbers in the Seq field.  You'll see that my OBJECTID field is not in order, so the above code won't work.    Can this be done either in the Field Calculator or using an Update Cursor in arcpy? How to connect to device via SSH over direct ethernet connection I have a device that I need to connect to over SSH. The device is connected to my workstation via a direct ethernet connection. I'm attempting to assign the connected device an IP address somehow that I can SSH to, however all of the guides I'm finding have the user configure the IP from whatever device they're working with (namely Raspberry Pi's and so on). This is not something I can do with this device as I've no physical interface to work with.  I've been recommended to setup a DHCP server on my workstation so it would take care of this for me, however I've no idea how to configure it and none of the guides I have followed have been helpful. With that approach, I seem to have been able to bind an IP to the interface the device is connected on, however SSHing to that IP just brings me back to my own machine.  So my main question is: How do I assign an IP address to a device directly connected to my computer via ethernet cable without having any kind of access to the device (no interface, keyboard, monitor, etc). If the answer is to set up a DHCP server on my machine, how do I properly configure this and get an IP I can SSH to?  For reference, I am using Ubuntu 16.04 with OpenSSH installed. The device is also running some flavor of Linux and has SSH software installed, but again there's no way to interact with it except through SSH. I also do not have access to my router, so plugging the device in there and letting the router do the work is not an option.