Doubt in Dihedral group $D_4$ regarding reflections
Generator of group $D_n$
Find $\min(\operatorname{trace}(AA^T))$ for invertible $A_{n\times n}$ For invertible $A_{n\times n}$ find $\min(\operatorname{trace}(AA^T))$  (a) $0$ (b) $1$ (c) $n$ (d) $n^2$  Clearly for $A=I$, it is $n$, and I am unable to get any lower values, but how do I prove it. How do I detect unsigned integer multiply overflow? How to know if something is a primitive concept, a law, a definition or a theorem Some basic Physics books are often misguiding in the sense that they don't make clear whether something is a primitive concept, a law, a definition or a theorem. This is often a little confusing. I've asked here some moments ago if there's a definition of force, and I've been redirected to a question about whether Newton's Laws are laws or definitions of mass and force.  This is just an example, but there are many others, like for instance, the conservation of energy, which some books present as a theorem, and some places says that this is a law of Physics. Time on the other hand, is something that according to Feynman's lectures on Physics, cannot be defined, so we just say how to measure it and leave it as a primitive concept.  In this context: how in general can we identify whether something is a primitive concept, a fundamental law of Physics, a definition or a theorem? Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens) If $X$ is measurable space, $f : X\to [-\infty,+\infty]$, function such that $f^{-1}([-\infty,r))$ measurable for all rational numbers $r$,prove than $f$ is measurable  I know that $f$ is measurable iff set $\{x\in X : f(x)&lt;a\}$ is measurable for all $a\in \mathbb{R}$ but for $a\in\mathbb{Q}$ I don't know how to proceed. Texture messes up after extruding a modified plane I want to make some 3d models of pixel art. I started by adding a plane, adding a texture to it, added loop cuts to my plane, 15 loop cuts because it's a 16x16 texture, then I selected the blank faces and deleted them. As I extrude this plane, A &gt; E &gt; $0.1$, the texture messes up on the edge of the item, as shown below  Although not all faces mess up on the edge, a lot still do. I tried recalculating normals, unwrapping the object again, but I ended with the same problem, no change at all. Though I may have done it wrong, since I never did it before. I tried smart UV project and cube projection, but they just mess it up even more. Does anyone have a solution to this? When should I use mmap for file access? POSIX environments provide at least two ways of accessing files.  There's the standard system calls open(), read(), write(), and friends, but there's also the option of using mmap() to map the file into virtual memory.  When is it preferable to use one over the other?  What're their individual advantages that merit including two interfaces? Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time.  On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to , so that one frame is preferred - but which one?  So the question is: Who will be older? And why?  And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster/slower...  How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT?  And if we are back to a situation with a preferred frame: what is the difference to the classical Galilean transform? Don't we get all the problems back that seemed to be solved by RT - e.g. the speed limit of light, because if there was a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c ?!? (I know  is only a local theory but I don't understand why the global preferred frame should not 'override' the local one).  Could anyone please enlighten me (please in a not too technical way because otherwise I wouldn't understand!)  EDIT Because there are still things that are unclear to me I posted a follow-up question: I write about  that shows a big probability mass. It's a web-based interactive visualization of topics estimated using Latent Dirichlet Allocation that is built using a combination of R and D3; the default for scaling the set of inter-topic distances defaults to Principal Components, but other algorithms are also enabled.    This is the way that it finally shows with some data I found online and used to understand Latent Dirichlet Allocation topic modeling. As far as I understand, this graph is ideal to give out final names to each topic according to the frequency or relevance of each term given the topic but I've got two questions:   1. What would you simply say about topic 8 position on the graph? Meaning that I can't tell what each square represents in this graph, I can't tell in plain words what's the difference between topic 1 and 10, or topic 8 and 3, given their position on this graph?   2. What can you say about the big gray line in comparison with the red filling when lambda is 0? I think I get that when lambda is 1, it's meaning about plain frequency of the term in the whole corpus vs the frequency of the terms in that topic, but I'm not sure I get it with putting more weight in the ratio from red to gray: what is the gray line saying about that term and how is red contrasting?  You can get this for a model named lda_model$plot() found .   I am looking for the main idea at least. If there is info I need to add please tell me. Is $(x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}}$ Differentiable in $(0,0,0)$? $$f(x,y,z)=\begin{cases} (x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}} \qquad (x,y,z) \ne (0,0,0) \\ \\ 0 \qquad (x,y,z)=(0,0,0) \end{cases} $$   At first, I study the continuity in the origin.  I apply the concept of sequential continuity: $x_n \rightarrow x_0 \Rightarrow f(x_n) \rightarrow f(x_0)$   So, $x_n=\frac{1}{n}  $ $$x_n \rightarrow 0 \Rightarrow f(x_n,x_n,x_n) \rightarrow f(0,0,0)=0$$ $$\lim_{n\rightarrow +\infty} \frac{3}{n^2} \ \sin \frac{1}{\frac{\sqrt{3}}{n}}=0 $$  How can I continue the study of differentiation?  Thanks! Is my MAC address public when browsing the Internet? Similar to '', is it also ok to add an example?   I was going to  but I noticed that one answer had most of the things I needed to say, except perhaps that the questioner didn't perhaps understand the answer and  I feel that an example may help. Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$ I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof? I am an international student in the US (important as my stay in US is tied to my student visa, issued to attend my current university).  I joined as a grad student in the MS CS program in one of the okayish public universities (ranked ~50 on QS/US news). I joined in Fall '18 and am still in my first semester. I took all the so-called "difficult" courses, but I find them very trivial (considering my undergrad was from a very good university). I am very disappointed with the curriculum and feel the coursework is not at all rigorous for a grad-school.  I might sound super arrogant, but I feel out-of-place when I talk to other students and hear them cringe about how hard the courses are, or how they don't have enough time to finish assignments within the deadlines as I find all of the coursework very trivial and feel that the assignments are a joke.   Fortunately, I am working with one of the best profs the university has to offer on a very impactful project and the research is going great. But that's the only plus point I see from sticking around.  My aim from grad school is to get research experience and join one of the Big 4s or upcoming CS unicorn startups. I cannot see myself joining one of the Big-Ns after graduating since the students from this school rarely bag a Google/Fb/unicorn startup job.  Since December is the Fall'19 admissions deadline, I am thinking about reapplying to those top 10 CS schools which I couldn't get into last year (probably because of my poor undergrad GPA).   On the other hand, since every CS company just asks leetcode-style questions in interviews, I can just be super good at it and try to nail whatever interview I get without wasting another year and then doing the same.  I'm not really interested in any kind of credit transfer either since I would like to start fresh in a more rigorous curriculum.  I'm not sure if switching schools would be a wise decision and could really use some advice from the community here.  PS :  Probably related -   Posting with a dummy account for anonymity. Is causality a formalised concept in physics? I'm using an API that requires me to pass a function pointer as a callback. I'm trying to use this API from my class but I'm getting compilation errors.   Here is what I did from my constructor:  m_cRedundencyManager-&gt;Init(this-&gt;RedundencyManagerCallBack);   This doesn't compile - I get the following error:      Error  8 error C3867: 'CLoggersInfra::RedundencyManagerCallBack': function call missing argument list; use '&amp;CLoggersInfra::RedundencyManagerCallBack' to create a pointer to member   I tried the suggestion to use &amp;CLoggersInfra::RedundencyManagerCallBack - didn't work for me.   Any suggestions/explanation for this??  I'm using VS2008.  Thanks!! Windows equivalent of the Linux command 'touch'? Can sum of a rational number and its reciprocal be an integer?   My brother asked me this question and I was unable to answer it.   The only trivial solutions which I could think of are $1$ and $-1$.  As to what I tried, I am afraid not much. I have never tried to solve such a question, and if someone could point me in the right direction, maybe I could complete it on my own.  Please don't misunderstand my question.   I am looking for a rational number $r$ where $r + \frac{1}{r}$ is an integer. How to make a disk image and restore from it later? I'm a new Linux user. I've reinstalled my Wubi from scratch at least ten times the last few weeks because while getting the system up and running (drivers, , etc.) I've broken something (X, grub, unknowns) and I can't get it back to work. Especially for a newbie like me, it's easier (and much faster) to just reinstall the whole shebang than try to troubleshoot several layers of failed &quot;fixing&quot; attempts. Coming from Windows, I expect that there is some &quot;disk image&quot; utility that I can run to make a snapshot of my Linux install (and of the boot partition!!) before I meddle with stuff. Then, after I've foobar'ed my machine, I would somehow restore my machine back to that working snapshot. What's the Linux equivalent of Windows disk imagers like  or ?  Note: I found a similar question: