I need to show that there is no subsets of $\mathbb{R}$ of order type $\omega_1$.  This is a very tricky one, because I was first trying to prove that there are no subsets with cardinality $\omega_1$ by showing that every uncountable subset of $\mathbb{R}$ has cardinality of $P(\omega)$ and thus not $\omega_1$, but I that was not true as there are subsets with cardinality $\omega_1$. The trick for me is to work with order type instead of cardinality.
I know that the usual ordering of $\mathbb R$ is not a well-ordering but is there an uncountable $S\subset \mathbb R$ such that S is well-ordered by $&lt;_\mathbb R$?   Intuitively I'd say there is no such set but intuitively I'd also say there is no well-ordered uncountable set at all, which is obviously wrong. I still struggle to grasp the idea of an uncountable, well-ordered set.
Ring homomorphism and characteristic How to analyze trend in non-periodic time series Suppose I have following non-periodic time series. Obviously the trend is decreasing and I would like to prove it by some test (with p-value). I am unable to use classic linear regression due to strong temporal (serial) auto-correlation among values.  library(forecast) my.ts &lt;- ts(c(10,11,11.5,10,10.1,9,11,10,8,9,9,                6,5,5,4,3,3,2,1,2,4,4,2,1,1,0.5,1),             start = 1, end = 27,frequency = 1) plot(my.ts, col = "black", type = "p",      pch = 20, cex = 1.2, ylim = c(0,13)) # line of moving averages  lines(ma(my.ts,3),col="red", lty = 2, lwd = 2)     What are my options? Bibliography per chapter with continuous numbering I would like to create a document with two chapters. After the first chapter there should be a bibliography. The next chapter should be able to reference items of the former chapter, but only show the new items in its bibliography, starting with the next number from the first bibliography.  Example:     Chapter 1 - The Fu   Fu has been done by Bar [1]. Early versions ...   Bibliography   [1] Bar et. al., Doning the Fu      Chapter 2 - Compining Fu with Herp   Doning Fu as described in [1]   neglects the advantages of Herp pointed out by Derp [2].   Consequently ....   Bibliography   [2] Derp et. al., The Advantages of Using Herp   Please not too much discussion about if that is a friendly-to-use idea, it is not my own demand :) How to return value from an asynchronous callback function? Generic constraints, where T : struct and where T : class How do you teleport a certain player to spawn after they win a 1v1 I managed a KitPvP server in Minecraft and everything works fine except when players wants to duel each other. They get sent to a room where they will be dueling. The main problem is when the winner wins, I have no idea how to teleport them back to spawn. How do I select text nodes with jQuery? I would like to get all descendant text nodes of an element, as a jQuery collection. What is the best way to do that? How are the AES S-Boxes calculated? Apostrophes for possession: nouns My best friend of the moment - 'Essential Grammar in Use' for elementary students by Raymond Murphy - says that 'we normally use -'s for people.'     I stayed at my sister's house. (not the house of my sister).   Then, bellow in the same unit (64) says that 'we use of... for things, places, etc.'     Look at the roof of that building. (not that building's roof)   After that I thought:  Finally!!! I understand the rules for apostrophes!!!  But soon I lost all that naif excitement when I saw other written examples on the internet that were not following those 'rules'.     I've learnt that we can apply apostrophes to ...     one year's pay   and, finally, amazingly enough ... to , which means ANY kind of nouns (I deduce uncountable nouns are also part of the possibilities). This demolishes what is said in the grammar. So, it seems that eventually:     That building's roof -> is valid   So, now I need your help.    1) Can you tell me the rules concerning the apostrophe to show possession in respect to its usage with nouns? Can we really apply to all kind of nouns? e.g. teeth's, vehicle's, Paula's, forest's, manager's, Ireland's, etc     Scotland's worst drivers   Metadata improves information's accessibility       2) Is there a sense that justifies what is written in the grammar: 'we normally use -'s for people'? or it does not make sense at all? Thermodynamics of supercooled water Measure of Image of Linear Map I am trying to work my way through the proof of the change of variables theorem for Lebesgue integrals. A key lemma in this context is as follows:  If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear map and $A \subset \mathbb{R}^n$ is Lebesgue measurable then $\lambda(T(A)) = |\det T \  | \cdot \lambda (A)$, where $\lambda(X)$ denotes the Lebesgue measure of $X$.  Can anyone provide a reference for a proof of this lemma that clearly references the facts from linear algebra that are necessary to effect the proof? The source I have for this lemma refers to German texts that I am incapable of reading and I have been unsuccessful at finding an alternate proof.  Added For the Benefit of Future Readers: In addition to the excellent references I received in response to this question, I have managed to find an additional reference that also provides a good proof of this fact: Aliprantis and Burkinshaw's Principles of Real Analysis, Third Edition, Lemma 40.4 pp 389-390. Will Area 51 be supported in the app? Cannot add a NOT NULL column with default value NULL in Sqlite3 When should I fudge dice rolls? When I started my first D&amp;D campaign as a DM, I accidentally rolled and confirmed a crit on a PC from a goblin. It was an instant kill as our campaign had no cleric. That time I decided to fudge the dice. Now, there's a similar problem, but this time the PCs are at 3rd level and they are dealing with a very important NPC. If they fail this one diplomacy check, the entire game world will fall apart, as the king would then send his entire army against the PC's homeland and will destroy it, plunging my entire game world into chaos, at which time drow that I wanted to save for a later adventure will have to come out, which will instigate a wizard's guild to release their secret ultra-deadly virus, and everything in my game world, including the PCs, will die. So should I fudge the dice roll? When is it good to fudge it? Differential equation for Harmonic Motion Particle undergoes simple harmonic motion.  Initially Its displacement is $1$, velocity $1$ and acceleration is $-12$   Compute displacement and acceleration when the velocity is square root of $8$.  I am not familiar with this kind of application question.  So, its displacement is $1$, does that mean that $y=1$ when $x=0$ ? Many years ago I read a book or story where a group of people discover a dark city - maybe on another planet - which appears devoid of life but is in very good condition. As they gradually explore the city one of the them falls into a trap and is cut to pieces by something mechanical. This detail I can remember as it was a pretty gruesome idea. I don't know the ending but my guess is that none of them make it out alive. Phase shift of 180 degrees of transversal wave on reflection from denser medium Can anyone please provide an intuitive explanation of why phase shift of 180 degrees occurs in the Electric Field of a EM wave, when reflected from an optically denser medium?  I tried searching for it but everywhere the result is just used.The reason behind it is never specified. Will some parents be offended when being asked, "Is it male or female?" If I ask the parent about a baby's gender, will it be impolite or not appropriate to say, "Is it male or female?"  Is there any subtle difference, in terms of politeness, among   "Is it a boy or girl?" "Is it he or she?" "is it male or female?" I've been using the Linksys USB6300 Adapter but when I switched to Ubuntu it stopped working since its only compatible with Windows. I've tried using ndiswrapper and although it says my windows driver is installed, my device isn't showing up. Anyone know how I could get this wireless network adapter working? Thanks.  lsusb  Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 012: ID 05c6:6766 Qualcomm, Inc.  Bus 001 Device 006: ID 0bda:0153 Realtek Semiconductor Corp. Mass Storage Device Bus 001 Device 005: ID 05e3:0608 Genesys Logic, Inc. USB-2.0 4-Port HUB Bus 001 Device 004: ID 04ca:007d Lite-On Technology Corp.  Bus 001 Device 003: ID 04ca:008a Lite-On Technology Corp.  Bus 001 Device 010: ID 13b1:003f Linksys  Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub deleting windows system folder