Cycles Gradient between two materials
How can you smooth the edge between two materials?
The set of limit points of the sequence $1,\frac12,\frac14,\frac34,\frac18,\frac38,\frac58,\frac78,\frac1{16},\frac3{16},\ldots$ Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix What should be the intuition when working with compactness? I have a question that may be regarded by many as duplicate since there's a  one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in $\mathbb{R}^n$ the compact sets are those that are closed and bounded, however the guy who answered this question and had his answer accepted says that compactness is some analogue of finiteness. That's the first problem: In my intuitive view of finiteness, only boundedness would suffice to say that a certain subset of $\mathbb{R}^n$ is in some sense &quot;finite&quot;. On the other hand there's the other definition of compactness (in terms of covers) which is the one I really need to work with and I cannot see how that definition implies this intuition on finiteness. Also, I feel it's pretty strange the covers people use when they want to deal with compact sets. To prove a set is compact I know they must show that for every open cover there's a finite subcover; the problem is that I can't see intuitively how one could show this for every cover. Also when trying to disprove compactness the books I've read start presenting strange covers that I would have never thought about. I think my real problem is that I didn't yet get the intuition on compactness. So, what intuition should we have about compact sets in general and how should we really put this definition to use? Can someone provide some reference that shows how to understand the process of proving (and disproving) compactness? Yesterday, I was browsing the mobile Stack Overflow site. I noticed one thing: the textareas where you write your comments/answers are very buggy. I'm using a Samsung Galaxy S2, with the standard Android browser (Android version 4.1.2). There are a number of things that aren't working when typing your comment/answer.   Automatic spacing with Swype is disabled (however this might be because of Swype, not the site). Text navigation buttons (step left, step right, select text, copy text etc.) doesn't work with the on-screen keyboard, and selecting text by holding your finger over text is extremely difficult. When you paste something it always seems to be pasted at the top of your text, no matter where you try to paste it. Replying to another comment ("@Example User") does not show suggested people to reply to, like the full site does.   Is there anything I can do to solve these issues/bugs, or is it in the hands of Stack Overflow itself, and if so, should it be looked into?  Edit: I was using the mobile version of the site when noticing these bugs. update line based on previous line Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$ Is there a nice presentation for the group $\text{PSL}(2,\mathbb{F}_q)$, for every prime $q$?  (I don't have any particular definition in mind for "nice", other than, say, a small number of generators and relations.) My homework has this problem:     Prove that there exist irrational numbers $x$ and $y$ such that $x +  y$ is rational.   There is an easy solution that I found on mathbitsnotebook.com:     \begin{align} \left(2+6\sqrt{7}\right) + \left(-6\sqrt{7}\right) = 2.\end{align}   Can we prove the same without using subtraction? How to install and run CPUSim 4.0.11 on Ubuntu 18.04 LTS? I'm going according to the steps mentioned on the cs colby .  First, I installed java on my system by typing the following commands-  sudo add-apt-repository ppa:linuxuprising/java sudo apt-get update sudo apt-get install oracle-java10-set-default sudo apt install oracle-java10-set-default     Then, according to the instructions, I downloaded the .zip file, extracted it and then in the terminal typed   java -cp .:richtextfx-fat-0.6.10.jar:reactfx-2.0-MR.jar -jar CPUSim-4.0.11.jar   But this didn't work.  of command 1 terminal.  Here is a screenshot of all the files that are there in the CPUSim folder,   On observing the names of the folders present, I thought that the previous command was wrong. So I tried the following ones,  java -cp .:richtextfx-fat-0.6.10.jar:reactfx-2.0-M4.jar -jar CPUSim-4.0.11.jar java -cp .:richtextfx-0.6.10.jar:reactfx-2.0-M4.jar -jar CPUSim-4.0.11.jar   But none of them seems to work. Now when I click on the CPUSim-4.0.11.jar file, nothing happens. I have just switched to Ubuntu from Windows 10, and have a very little knowledge about how things work here. Please tell me how I could install and run CPUSim 4.0.11 on Ubuntu 18.04 LTS. Why are there holes in the icing? Why is it that the classical path gives the dominant contribution in the quantum mechanical path integral? How do we understand this? Continuing example in Beamer Evaluate $\int_0^{\infty}\frac{\log x}{1+e^x}\,dx$ Trackpad not working after update to Ubuntu 14.10 on Dell Inspiron 15R Order of an element in a finite cyclic group Word for two collections that do not have any elements in common I'm looking for a word for when you have a collection A and a collection B and they have no overlap.  In mathematical terms: the relation between two sets where the intersection is empty. Like in this Venn diagram:    The word that keeps popping up in my mind is disparate sets, but I don't feel that completely covers the meaning when I look at the definition, because that doesn't seem to preclude overlapping sets that are different:     disparate   1. distinct in kind; essentially different; dissimilar:   disparate ideas. removing event listener Are Playstation 4 games locked to a certain region? This question has been asked before, but it was for PS3 and Xbox 360 and was asked August 2010. This time around my question goes specifically to PS4, since I'm moving from Mexico to Europe and I was wondering if the European games will run in my console or if I'll have to buy a new one. I never thought of this before (which is why I never paid attention if it was mentioned in reports or specs of the console) and the research I did wasn't too helpful. "DataBinding: 'index+NewsItem' does not contain a property with the name 'Link'", but property exists (Not a Typo) I've been given a proof that shows the following:  If $f:[a,b]\to \mathbb C$ is a continuous function and $f(t)=u(t)+iv(t)$ then $$\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$$ The proof begins by letting $\theta$ be the principle argument of the complex number $\int_a^b f(t)dt$ and there is one step in the proof I don't understand; can anyone explain to me why $$\int_a^b e^{-i\theta}f(t) dt=\Re\bigg(\int_a^be^{-i\theta}f(t)dt\bigg)$$  Would this not imply that the imaginary part of the left hand side is equal to $0$? I'm not sure why the LHS and RHS would be equal here. "Pick up something" or "pick something up"?