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How to define binary operation on arbitrary set in order to create a group structure. Given a sphere of radius $r$ with two spherical caps on it defined by the radii ($a_1$ and $a_2$) of the bases of the spherical caps, given a separation of the two spherical caps by angle $\theta$, how do you calculate the surface area of that intersection?   To clarify, the area is that of the curved surface on the sphere defined by the intersection. At the extreme where both $a_1,a_2 = r$, we would be describing a spherical lune.   Alternatively define the spherical caps by the angles  $\Phi_1 = \arcsin(a_1/r)$ and $\Phi_2 = \arcsin(a_2/r)$. $S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective? $S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is  a continuous function.  I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(x) : x \in S^2\}  \: \: \mathrm{ and }\: \: T_{\mathrm{min}}:= \mathrm{inf}\{ T(x) : x \in S^2\}$$ are reached by $T$ with the argument that $S^2$ is closed and bounded and therefore after the Heine-Borel-Theorem compact, and therefore since T is continuous it assumes its max and min in $S^2$.   Now i have answer the following:     1) Is there a point $x_0 \in S^2$ with $T(x_0)=T(-x_0)$ ?      2) Is there a value $T \in(T_{\mathrm{min}}, T_{\mathrm{max}})$ which is only assumed at exactly one point of $S^2$ ?   I wanted to show that $T$ is injective to get that 1) is false and 2) is true but i didn't get anywhere. Any ideas? Thanks in advance! How can we show that the following sequence converges? $(a_n)$ is a bounded sequence with the following condition  $a_{n+1}\geq a_n-\frac{1}{2^n}$   The sequence converges, but how do we show it? I just refreshed the [iphone]-tag-site, and I'm seeing all kinds of questions with what appears to be a grey overlay (see image). What is this for/why am I seeing this? I have never seen anything like this before.. XPS 15 stuck on 800MHz I'm running Ubuntu 16.04 with CPU: i7-6700.  Even with one of the cores sitting on 100% utilisation cpufreq-info reports.  ... analyzing CPU 7:   driver: intel_pstate   CPUs which run at the same hardware frequency: 7   CPUs which need to have their frequency coordinated by software: 7   maximum transition latency: 0.97 ms.   hardware limits: 800 MHz - 3.50 GHz   available cpufreq governors: performance, powersave   current policy: frequency should be within 800 MHz and 3.50 GHz.                   The governor "performance" may decide which speed to use                   within this range.   current CPU frequency is 800 MHz.   Each CPU shows 800 MHz.  How do I get my CPU frequency to increase under load? Etiquette on sending a thank you e-mail to respondents who gave me helpful information If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$ prove that $\det(AD)=\det(BC)$ Let $A,\ B,\ C,\ D \in \mathcal{M}_n(\mathbb{C})$. If $\operatorname{rank}\left( \begin{bmatrix} A &amp;B \\ C &amp;D \end{bmatrix}\right)=n$, prove that $\det(AD)=\det(BC)$. Which side of wave-particle duality to choose in a given situation Interesting topics to research in mathematical physics for undergraduates I want it to know how to convert string field values to integer values in QGIS Dufour... there are numbers and strings in the same field, when I use toreal(string) it stops and it won´t create the field. Kronecker delta confusion I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same):  $$g^{\mu\nu}g_{\mu\nu}=\delta_{\nu}^{\nu}=\delta_{0}^{0}+\delta_{1}^{1}+\delta_{2}^{2}+\delta_{3}^{3}=1+1+1+1=4.$$  But I've also seen (where the upstairs and downstairs indices are different):  $$g^{\mu\nu}g_{\nu\lambda}=\delta_{\lambda}^{\mu}=\left(\begin{array}{cccc} 1 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 1 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; 1 \end{array}\right).$$  How can there be two different answers to (what appears to me to be) the same operation, ie multiplying the metric tensor by its inverse? Apologies if I've got this completely wrong. How to add the name of the supervisor in a @thesis field? In the biblatex doc, it says ( Version 1.7, p.11):     thesis      A thesis written for an educational institution to satisfy the   requirements for a degree. Use the type field to specify the type of   thesis.      Required fields: author, title, type, institution, year/date      Optional fields: subtitle, titleaddon, language, note, location,   month, isbn, chapter, pages, pagetotal, addendum, pubstate, doi,   eprint, eprintclass, eprinttype, url, urldate   I would like to know if there is any "canonical" way to add the name of the supervisor(s) in a field of the .bib file (no .bbl manipulation please, I want my bibliography to be usable "as it" in several documents).  The expect result could be something like  :     Meand Myself. “A brilliant result, explained”. PhD thesis**, under the   joint supervision of Pr. Toto and Pr. Tata**. Université Paris 24,   Nov. 2013.   The "under the (joint) supervision" could be a string that would be declined in four ways:   under the supervision of  under the joint supervision of and the abridged forms What websites or other resources do users of Blender have available to them to report bugs or problems with the program?  I would like to get this issue cleared in Q&amp;A format so that we can link new users with bug-report-style questions to this post. After the events on Miller's planet, it seems to come as a revelation to Brand that     Miller had only landed a couple hours ago, and died a few minutes ago.   The crew of the Endurance, even when they were back at NASA headquarters, had all the information they needed to figure out the time dilation and how long Miller had been on the planet. As such, they must have known that     Miller, even in the best case, would have only had a couple hours worth of data.   Why use all those resources (fuel and time) just for that? What mechanism is responsible for the creation of these dunes on Comet 67P/Churyumov-Gerasimenko?  The following high resolution picture from ESA's Rosetta mission shows the dunes:       At a distance of 7.8 km from the surface, the image scale is about   66.5 cm/pixel, so each 1024 x 1024 pixel frame is about 680 m across (although if we assume the furthest point away is an additional ~1 km   further from the centre, the image scale is about 92 cm/pixel).   So the scale of the dunes is on the order of tens of meters along the crest. And each crest are probably just a few meters apart. Windows 10, 'System' process taking massive amounts of RAM Since I upgraded to Windows 10, my system has been consuming RAM excessively    I've been reading a bit and determined it's likely a driver leaking memory. So I got myself the Windows Driver Kit and tracked memory usage with poolmon:    However, I don't really know how to proceed from here. Is the item tagged "smNp" the culprit in this issue? How do I go from there to actually identifying the driver?  I tried some stuff like "C:\Windows\System32\drivers>findstr /s smnp ." but it returned no results. I also took a look at the pooltag.txt file and this is the description I found for it:      So yeah, any help would be appreciated. Thanks in advance. This question is a bit more subjective, but I'm hoping to get some new perspective.  I'm so used to designing for certain screen size (typically 1024x768) that I find that size to not be a problem.  Expanding the size to 1280x1024 doesn't buy you enough screen real estate to make an appreciable difference, but will give me a little more breathing room.  Basically, I just expand my "grid size" and the same basic design for the slightly smaller screen still works.  However, in the last couple of projects, my clients were all using 1080p (1920x1080) screens and they wanted solutions to use as much of that real estate as possible.  1920 pixels across provides just under twice the width I am used to, and the widescreen makes some of my old go to design approaches not to work as well.  The problem I'm running into is that when presented with so much space, I'm confronted with some major problems.   How many columns should I use?  The wide-format lends itself to a 3 column split with a 2:1:1 split (i.e. the content column bigger than the other two).  However, if I go with three columns what do I do with that extra column? How do I make efficient use of the screen real estate?  There's a temptation to put everything on the screen at once, but too much information actually makes the application harder to use.  White space is important to help make sense of complex information, but too much makes related concepts look too separate. I'm usually working with web applications that have complex data, and visualization and presentation is key to making sense of the raw data.  When your user also has a large screen (at least 24"), some information is out of eyesight and you need to move the pointer a long distance. How do you make sure everything that's needed stays within the visual hot points? Simple sites like blogs actually do better when the width is constrained, which results in a lot of wasted real estate.  I kind of wonder if having the text box and the text preview side by side would be a big benefit for the admin side of that type of screen? (1:1 two-column split).   For your answers, I know almost everything in the design is "it depends".  What I'm looking for is:   General principles you use How your approach to design has changed   I'm finding that I have to retrain myself how to work with this different format.  Every bump in the resolution I've worked through to date has been about 25%: 640 to 800 (25% increase), 800 to 1024 (28% increase), and 1024 to 1280 (25% increase).  However, the jump from 1280 to 1920 is a good 50% increase in space--the equivalent from jumping from 640 straight to 1024.  There was no commonly used middle size to help learn lessons more gradually.    To help focus the question a little bit, I had a project that was somewhat similar to Atlassian JIRA, an issue management system.  There were about six different types of records the client wanted to keep, all of them could potentially be related to each other.  Gathering data wasn't the core problem, although it did play a part.  The more important side of the problem was creating a system which would suggest potential relationships between the records and helped the analysts recognize patterns in the reported incidents.  There were different types of analysts that focused on the different problem domains, and due to the nature of their exploratory type of work, they didn't know what they wanted.  They just knew they had to make sense of a lot of data, draw correlations, and characterize classes of issues.    (Originally asked here: , but told that this might be more appropriate)    For the bounty: Stretch your minds a bit.  The pat answer "Make everything bigger" has a limit to how useful the answer is.  We are talking about users who have to make sense of a lot of data and find relationships between records.  If I simply bump up the size on everything they are going to be very limited in what they can see on the screen--particularly because the vertical is so constrained. Independence of disjoint events How to mount a remote (over SSH) directory to be available like a local directory (incl. in shell, not just in Gnome)?