link 1:   link 2:    I have found 2 solutions, the second one convinced me. But so far I cannot formally prove that the method given in the first link gives a reminder with its size function less then the size function of the divisor. And I think maybe there exist Gaussian integers $a+bi$, $c+di$, such that using the division algorithm in link 1 to divide $c+di$ by $a+bi$ gives a remainder whose size function value is greater than the size function value of $a+bi$.  Is it possible? If it is possible, could someone find such $a+bi$ and $c+di$?If the method in link 1 also allows us to always get a remainder with size function value less then the divisor, could someone formally prove that?  Thanks in advance!
I'm having some trouble proving that the Gaussian Integer's ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i've got so far.  To be a Euclidean domain means that there is a defined application (often called norm) that verifies this two conditions:   $\forall a, b \in \mathbb{Z}[i] \backslash {0} \hspace{2 mm} a \mid b \hspace{2 mm} \rightarrow N(a) \leq N (b)$ $\forall a, b \in \mathbb{Z}[i] \hspace{2 mm} b \neq 0 \rightarrow \exists c,r \in \mathbb{Z}[i] \hspace {2 mm}$ so that $\hspace{2 mm} a = bc + r \hspace{2 mm} \text{and} \hspace{2 mm} (r = 0 \hspace{2 mm} \text{or} \hspace{2 mm} r \neq 0 \hspace{2 mm} N(r) \lt N (b) )$   I have that the application meant to be the "norm" goes: $N(a +b i) = a^2 + b^2$, and I've managed to prove the first condition, given that N is a multiplicative function, but I can not find a way to prove the second condition.   I've search for a similar question but I have not found any so far, please redirect me if there's already a question about this and forgive me for my poor use of latex.
Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$ What happens to waves when they hit smaller apertures than their wavelenghts? I was wondering this for quite a long time now. Let's say you have a water wave (like ripples, not the ones you see during tsunamis) with wavelength 10 m. Imagine you put a boundary with an opening of 1 m. Will diffraction be observed? According to my research, no. But, then, what would be seen? Proving a limit of a trigonometric function: $\lim_{x \to 2/\pi}\lfloor \sin \frac{1}{x} \rfloor=0$ $$\lim_{x \to \frac{2}{\pi}}\lfloor \sin \frac{1}{x} \rfloor=0$$    I need to prove the limit of this using the $\epsilon - \delta $ way but I don't know how to find $\delta$ when I'm given a trigonometric function I know only how to do it with polynomial functions What's the most popular way to encrypt individual files or folders? Can I get a postdoc in China more than 3 years after having received my PhD? What's the difference between cards, panels, and tiles? It seems as if with some of the modern front-end development frameworks there is a blur between the different types of 'container' elements. Whether this is an attempt to abstract the design from implementation or if this is due to there not being any standardization of terminologies I am not sure, but it leads to the fact that in some design frameworks I see the term panels being used but not cards (e.g. Bootstrap 3) while in others the term cards exists but not panels (e.g. Iconic), and perhaps both exist in others?  From a design perspective, it seems that both cards and panels are just containers, and since there's nothing that defines how big a card can get and how small a panel can be, I am wondering whether there's really any difference. From an implementation perspective, it seems that cards should be used for smaller size design elements and are preferred on mobile specific applications, whereas panels are used for a wider range of design elements and are preferred on desktop specific applications.  Is there any evidence/reference/examples that will help to illustrate or highlight the (lack of) differences? For some reason I think cards and tiles are more closely related to each other, and that panels seem to be in a slightly different class/family?  If we take the argument that there are no differences, then it is hard to understand why people associate 'tiles' with Microsoft 'Metro' style home pages, while on Google and Android apps it is 'cards' and for Java and Gnome applications it is 'panels'. Perhaps there are some contextual or implementation differences?  UPDATE: with the release of Bootstrap 4, it seems like cards are basically equivalent to panels (since one component has replaced the other), but with this comes the cost of having to maintain a more flexible container that will make it difficult to implement consistently. According to Bootstrap:      A flexible and extensible content container. It includes options for   headers and footers, a wide variety of content, contextual background   colors, and powerful display options.   FURTHER UPDATE: It seems as if Bootstrap 5 and Material.io have adopted the card UI component in the new world of responsive and mobile first/multi-device ecosystem design systems while "tiles" have a very specific meaning in the Microsoft Fluent Design System:      A tile is an app's representation on the Start menu. Every app has a tile. When you create a new Universal Windows Platform (UWP) app project in Microsoft Visual Studio, it includes a default tile that displays your app's name and logo   Panels don't seem to be part of the vocabulary anymore, as least in terms of a UI component, but in the Microsoft and Java world it still represents an abstract concept of organisation layouts of UI components. I keep getting this warning whenever I try to run sudo apt-get update.  W: Duplicate sources.list entry http://archive.ubuntu.com/ubuntu/ precise-updates/main i386 Packages (/var/lib/apt/lists/archive.ubuntu.com_ubuntu_dists_precise-updates_main_binary-i386_Packages) W: You may want to run apt-get update to correct these problems   Below is the output from /etc/apt/sources.list file:  deb http://archive.ubuntu.com/ubuntu precise main restricted deb-src http://archive.ubuntu.com/ubuntu precise main restricted  deb http://archive.ubuntu.com/ubuntu precise-updates main restricted deb-src http://archive.ubuntu.com/ubuntu precise-updates main restricted  deb http://archive.ubuntu.com/ubuntu precise universe deb-src http://archive.ubuntu.com/ubuntu precise universe deb http://archive.ubuntu.com/ubuntu precise-updates universe deb-src http://archive.ubuntu.com/ubuntu precise-updates universe  deb http://archive.ubuntu.com/ubuntu precise multiverse deb-src http://archive.ubuntu.com/ubuntu precise multiverse deb http://archive.ubuntu.com/ubuntu precise-updates multiverse deb-src http://archive.ubuntu.com/ubuntu precise-updates multiverse  deb http://archive.ubuntu.com/ubuntu precise-security main restricted deb-src http://archive.ubuntu.com/ubuntu precise-security main restricted deb http://archive.ubuntu.com/ubuntu precise-security universe deb-src http://archive.ubuntu.com/ubuntu precise-security universe deb http://archive.ubuntu.com/ubuntu precise-security multiverse deb-src http://archive.ubuntu.com/ubuntu precise-security multiverse   How do I fix it? Is there a standard idiom for comparing version numbers?  I can't just use a straight String compareTo because I don't know yet what the maximum number of point releases there will be.  I need to compare the versions and have the following hold true:  1.0 &lt; 1.1 1.0.1 &lt; 1.1 1.9 &lt; 1.10 Reading and searching long man pages I finally got fed up when wanting to read about bash's read and it's -s option with man bash. I found the right spot eventually (around line 4500), but it was a frustrating as usual, since both /read and even /\s-s\s searches have way too many matches.  So, the question is: How can I read long man pages efficiently, or get same information in other ways, locally? As a specific example, how to reach the relevant documentation after seeing read -s pwd in a shell script? A good answer could be a shell script snippet, or hint about some tool and how it is used, or something else entirely, as long as it helps in finding the right spot to read.  Note: I'm not tagging with  because I want the question to be about man page reading in general, even though that quite possibly is the most commonly encountered humongous man page. Trying to install CPAN Adjective for "terrified yet calm and capable" What is an adjective to describe the duality of a first responder (such as an EMT responding to a terrorist attack) who is terrified for the wounded, yet simultaneously calm, detached, and capable enough to give aid? $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ?? translation invariant of integral on $\mathbb{R}$ Problem 1: Let $f\in L^1(\mathbb{R})$, show $\int_\mathbb{R}f(t)dt=\int_{\mathbb{R}}f(x+t)dt,\forall x\in (-\infty, \infty)$.  Problem 2: Let $f\in L^1(\mathbb{R})$,show $\displaystyle \lim_{t\rightarrow 0}\int_{\mathbb{R}}|f(x+t)-f(x)|dx=0$.  My try for Problem 1:  Consider $\displaystyle \int_{\mathbb{R}}f(t)dt=\lim_{n\rightarrow \infty}\int_{-n}^nf(t)dt=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f_n(t)dt$, where $f_n(t)=f(t)\chi_{[-n,n]}$.  And since $dt=d(x+t)$, we have $\displaystyle \int_{-n}^nf(x+t)dt=\int_{-n+x}^{n+x}f(t)dt$.  So for each $x\in (-\infty,\infty)$,  $$ \begin{split} \int_{\mathbb{R}}f(t)-f(t+x)dt &amp;=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f(t)(\chi_{[-n,n]}-\chi_{[-n+x,n+x]})dt\\ &amp;=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt\\ &amp;=\lim_{n\rightarrow \infty}\lim_{k\rightarrow \infty}\int_\mathbb{R}\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt\\ &amp;=\lim_{k\rightarrow \infty}\lim_{n\rightarrow \infty}\int_\mathbb{R}\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt \end{split} $$ where $\varphi_k(t)$ is simple function pointwise converges to $f(t)$, simple function vanish out of set of finite measure, so $\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})$ vanishes when $n$ big enough(is this right?)    Another try:  Since $f\in L^1$, $f$ is bounder a.e. so can be approximated by simple functions, and $\chi_{A+t}(x)=\chi_A(x-t)$,thus $$\int f(t)=\lim_n\int\varphi_n=\lim_n \sum_{i=1}^nc_i\mu(E_i)=\lim_n \sum_{i=1}^nc_i\mu(E_i-x)=\int f(t+x)$$  For problem 2, I guess they share the same method.   Any help or correction about these two problems? Thanks. Why does replication require primers while transcription does not? How does linear regression use the normal distribution? How to install Ubuntu on portable external Hard Drive? Ring and Subring with different Identities I'm trying to enable Intel SRT on my laptop. To do this, I need to change SATA controller mode from AHCI to RAID. The problem is that windows has no drivers for RAID and I can't install it while controller is in AHCI mode.   For now I have RAID driver in INF package (inf, sys, cat files). And I can load Windows recovery console with controller in RAID mode. The last thing to do is to intall this driver, but I don't know how to do it.  Google says rundll32.exe setupapi,InstallHinfSection DefaultInstall 123 &lt;filename&gt;.inf might help, but it doesn't. Error when using apt-get update SF novel about one-way scientific expedition to inhabited planet, where telepathic aliens "talk" sort of ++like this++ and ~*~this~*~ I'm looking for a novel I read in the mid-to-late 1990s. It was a science fiction novel about an expedition of scientists sent at sub-light-speed (maybe in cryo-sleep?) to investigate a star system. Once there they set up a permanent scientific outpost.  There was at least one inhabited planet in the system, with a sentient species that communicated telepathically. The aliens' speech was represented with various symbols instead of quotation marks, with each named character having (I think) its own characteristic set of symbols, as in the title of the question. The scientists are eventually able to communicate with these (generally friendly) aliens.  The thing I remember most clearly is the end of the book. This was a one-way trip for the scientists, so gradually they all die off from various causes. When the next-to-last member of the crew dies of old age, the survivor is looking for an epigraph for her funeral, and (ironically, considering my current plight) can't find it in his search of the electronic library. He finally discovers it under "humor" because it's by Mark Twain: "Wherever she was, there was Eden." (This quote, of course, I found immediately in  and many other sources; it's from Eve's Diary.)  Other pieces of information I'm less sure of:   I don't think the planet was hospitable to humans; it may have been something like a gas giant, with the aliens "swimming" through the gas. I think the scientists stayed in an orbital space station, though they may have made expeditions to the planet in vehicles or protective suits. There may have been a "twin" planet, or a moon in an eccentric orbit, that caused periodic "flares" of some sort on the aliens' planet. There might have been some conflict between the crew early on, maybe because there was a non-scientific (military) commander, or perhaps just because people have conflict. I think the author was of sort of middling fame; respected and known by SF enthusiasts, but not a superstar like Asimov or Clarke. Almost certainly male, and probably with a professional science background. I think the science was fairly authentic, although the first-contact bit was obviously highly speculative.