Why eigenvector of Symmetric Matrix to be orhogonal for repeated eigenvalues.
Why are real symmetric matrices diagonalizable?
Why is the derivative of a circle's area its perimeter (and similarly for spheres)? When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.  Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi r^2$.  Is this just a coincidence, or is there some deep explanation for why we should expect this? Two different expansions of $\frac{z}{1-z}$ This is exercise 21 of Chapter 1 from Stein and Shakarchi's Complex Analysis.  Show that for $|z|&lt;1$ one has $$\frac{z}{1-z^2}+\frac{z^2}{1-z^4}+\cdots +\frac{z^{2^n}}{1-z^{2^{n+1}}}+\cdots =\frac{z}{1-z}$$and   $$\frac{z}{1+z}+\frac{2z^2}{1+z^2}+\cdots \frac{2^k z^{2^k}}{1+z^{2^k}}+\cdots =\frac{z}{1-z}.$$  Justify any change in the order of summation.   [Hint: Use the dyadic expansion of an integer and the fact that $2^{k+1}-1=1+2+2^2+\cdots +2^k$.]  I don't really know how to work this through. I know that $\frac{z}{1-z}=\sum_{n=1}^\infty z^n$ and each $n$ can be represented as a dyadic expansion, but I don't know how to progress from here. Any hints solutions or suggestions would be appreciated. I would like to create a virtual device in Android Studio but I can't because hardware acceleration is disabled.  I went to the BIOS and there are two option :  AMD-SVM AMD-IOMMU   Both are enabled.  I also downloaded the HAXM installer from the SDK manager and tried to run it from the   C:\Users\%user%\AppData\Local\Android\Sdk\extras\intel\Hardware_Accelerated_Execution_Manager  with intelhaxm-android file.  I got an error telling me :   This computer does not support Intel Virtualization technology VT X or it is exclusively used by Hyper V. HAXM cannot be installed. Please ensure that hyper V is disabled in windows feature   It makes sense that I can't activate Intel Virtualization technology because I have AMD Ryzen 7 2700U.  With my previous computer it worked just fine but I had an Intel processor.  Some tutorials online suggested to enable windows Hyper-V feature but it's only available for windows 10 pro.  Does anyone know a way to activate hardware acceleration with AMD processor and without Hyper-V feature? Say we have the following sample and we are trying to estimate the variance of the sample mean of the population.  X = [0, -1, 2, 10, -3]   If I take an increasing number of bootstrap (e.g. 100, 1000, 10,000, 100,0000), the bootstrap distribution of the sample mean gets narrower and narrower. In other words, I can make the bootstrap distribution arbitrarily narrow.   Given the above, what's the point of using a different number of bootstrap samples here? And wouldn't this mean that I could make the standard error of the mean arbitrarily small by just taking more and more bootstrap samples? What am I missing?  Please note that I am doing a proper bootstrap sample in the sense that each bootstrap sample has the same exact size as the original sample (input array). What is a concise function that formats a (String) decimal into a currency format in Apex? goal: write the most concise function that formats a (String) decimal into a currency format in Apex. Feel free to trainwreck the code for points, I've kept mine exploded for readability purposes. Permission denied to change gid(group) of a file I own It appears I still miss some things about the way permissions work. I am on a debian 7 system btw. just now I have this file of which I downloaded and it belongs to myuser:myuser, that is both user and group are set to me. It also resides in my $HOME directory since that is where I downloaded it to.  So far so good. Now I want to share this file with some other users of the pc and for that I want to switch the group ownership of the file to group "users". however that fails:  nass@quarx:~/xmas_carol$ chgrp -R users *  chgrp: changing group of movie.mov': Operation not permitted   And the contents of the folder are:  -rwxr-xr-x 1 nass nass 2482411461 Feb  6 03:57 movie.mov   I am fuzzy about what is going on with the permissions. Can someone explain When should I use "in" or "on"? As it is common with people from my country, I have an immense difficulty with prepositions in English, especially with the use of in and on.   When the preposition indicates the position of the object it is a bit easier to decide which one to use, but as it requires thinking from me, I still say things like "I put the bill in the fridge", "my pencil is on my bag", etc, even though if I think about it, I would know how to say it correctly.  Of course, even in those cases where the preposition indicates position the decision is not always clear, as is the case for example with "on the bus" (you are not on top of the bus, you are inside the bus) or "he is in bed" (you are not inside the bed, you are on top of your bed).  Other cases are even less clear, and I believe there is no real rule to decide what should be said; the person should just know it. Two examples are "in a bad mood", "you are on my mind".  So my first question is whether there is really no rule to help me. The second question refers to two of the cases where I still don't know whether to use in or on:   good luck on/in your exam in/on the list   The last one is especially important to me, as I have asked a number of native speakers, and each time I get a different answer. The only consensus seems to be the case where you say "on top of the list". Apart from that, some people will say that I should say "his name is in the list" whereas others think I should say "his name is on the list". And yeah, a couple of people did tell me that I can use either.  Can someone help me? Returning an unevaluated expression with values substituted in What is the difference between logistic regression and Fractional response regression? As far as I know, the difference between logistic model and fractional response model (frm) is that the dependent variable (Y) in which frm is [0,1], but logistic is {0, 1}. Further, frm uses the quasi-likelihood estimator to determine its parameters. Normally, we can use glm to obtain the logistic models by glm(y ~ x1+x2, data = dat, family = binomial(logit)). For frm, we change family = binomial(logit) to family = quasibinomial(logit). I noticed we can also use family = binomial(logit) to obtain frm's parameter since it gives the same estimated values. See the following example library(foreign) mydata &lt;- read.dta(&quot;k401.dta&quot;)   glm.bin &lt;- glm(prate ~ mrate + age + sole + totemp,                 data = mydata, family = binomial('logit')) summary(glm.bin)  return,  Call: glm(formula = prate ~ mrate + age + sole + totemp,      family = binomial(&quot;logit&quot;),      data = mydata)  Deviance Residuals:      Min       1Q   Median       3Q      Max   -3.1214  -0.1979   0.2059   0.4486   0.9146    Coefficients:               Estimate Std. Error z value Pr(&gt;|z|)     (Intercept)  1.074e+00  8.869e-02  12.110  &lt; 2e-16 *** mrate        5.734e-01  9.011e-02   6.364 1.97e-10 *** age          3.089e-02  5.832e-03   5.297 1.17e-07 *** sole         3.636e-01  9.491e-02   3.831 0.000128 *** totemp      -5.780e-06  2.207e-06  -2.619 0.008814 **  --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for binomial family taken to be 1)      Null deviance: 1166.6  on 4733  degrees of freedom Residual deviance: 1023.7  on 4729  degrees of freedom AIC: 1997.6  Number of Fisher Scoring iterations: 6  And for family = quasibinomial('logit'), glm.quasi &lt;- glm(prate ~ mrate + age + sole + totemp,   data = mydata ,family = quasibinomial('logit')) summary(glm.quasi)  return,  Call: glm(formula = prate ~ mrate + age + sole + totemp,      family = quasibinomial(&quot;logit&quot;),      data = mydata)  Deviance Residuals:      Min       1Q   Median       3Q      Max   -3.1214  -0.1979   0.2059   0.4486   0.9146    Coefficients:               Estimate Std. Error t value Pr(&gt;|t|)     (Intercept)  1.074e+00  4.788e-02  22.435  &lt; 2e-16 *** mrate        5.734e-01  4.864e-02  11.789  &lt; 2e-16 *** age          3.089e-02  3.148e-03   9.814  &lt; 2e-16 *** sole         3.636e-01  5.123e-02   7.097 1.46e-12 *** totemp      -5.780e-06  1.191e-06  -4.852 1.26e-06 *** --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for quasibinomial family taken to be 0.2913876)      Null deviance: 1166.6  on 4733  degrees of freedom Residual deviance: 1023.7  on 4729  degrees of freedom AIC: NA  Number of Fisher Scoring iterations: 6  The estimated Beta from both `family` are the same, but the difference is the SE values.  However, to obtain the correct SE, we have to use `library(sandwich)` as in this [post][1].  Now, my questions:  1. What is the difference between these two codes? 2. Is frm about to obtain robust SE?  If my understanding is not correct, please give some suggestions.      [1]: https://stackoverflow.com/questions/37584715/fractional-response-regression-in-r Why does the find command blow up in /run/? I used the following command to check python versions as given by . It does work except for one error, shown below. Can any one explain why?  $ sudo find / -type f -executable -iname 'python*' -exec file -i '{}' \; \     | awk -F: '/x-executable; charset=binary/ {print $1}' \     | xargs readlink -f | sort -u | xargs -I % sh -c 'echo -n "%: "; % -V'   Error:  find: ‘/run/user/1000/gvfs’: Permission denied How do I remove duplicate "Open With" context menu items in Finder.app? For whatever reason, the "Open With" context menu in Finder is always listing every app four times (exactly).     I've read about (and tried) rebuilding the Launch Services database to restore the context menu to its initial state, but nothing seems to work. I've also used Onyx.app to clear user and system caches, but I'm still left with four repetitions of each app under the "Open With" service. I have restarted the system a few times just to be sure it was not some temporary corruption of the list.  I'm running OS X 10.6.6 on a 2011 17" MacBook Pro with a fresh install of OS X (i.e. I didn't use Migration Assistant.app or a Time Machine backup). I did, however, sync all my preferences and other files using MobileMe like I always do, so I imagine it's feasible that a preference file somewhere is causing problems? I didn't notice whether this was happening before or after my MobileMe sync.  How can I fix this so one app shows? Why does the zoom sometimes stop at a point? Show that the function $f: S \to \mathbb R$ given by $$f(x,s,t):=-\ln(st - ||x||^2)$$ is convex on $$S := \left\{(x,s,t) \in \mathbb R^n \times \mathbb  R \times \mathbb R: \frac{\|x\|^2}{s}&lt;t, s&gt;0, t&gt;0 \right\}$$   $-\ln(x)$ is a convex monotonic decreasing function, hence we can use composition by showing the inner part is concave. Also we know $-||x||^2$ is concave and $st$ is neither convex nor concave on $S$. Also in $S$ we have $st&gt;||x||^2$ but I was unable to show the convexity of the function as a whole How to show that topological groups are automatically Hausdorff? On page 146, James Munkres' textbook Topology(2ed),  Show that $G$ (a topological group) is Hausdorff. In fact, show that if $x \neq y$, there is a neighborhood $V$ of $e$ such that $V \cdot x$ and $V \cdot y$ are disjoint.  Noticeably, the definition of topological group in Munkres's textbook differs from that in .  A topological group $G$ is a group that is also a topological space satisfying the $T_1$ axiom, such that the map of $G \times G$ into $G$ sending $x \times y$ into $x \cdot y$ and the map of $G$ into $G$ sending $x$ into $x^{-1}$, are continuous maps. How to install Python 3.4 on Ubuntu? I'm using Drupal 7.  Is it possible to add a CC to a webform email?  I was using   function theme_webform_mail_headers($variables) {   $headers = array(     'X-Mailer' =&gt; 'Drupal Webform (PHP/' . phpversion() . ')',   );   return $headers; }   changing it with      function MYTHEME_webform_mail_headers($variables) {       $headers = array( 'CC' =&gt; 'ccmeail@domain.com', 'X-Mailer' =&gt; 'Drupal Webform (PHP/'. phpversion() .')',  );       return $headers;     }   It does not work, I've cleared all the chaches and it does not add the CC. How could I add the CC?  Thanks in advance. Make terminal use pageup/pagedown instead of shift-pageup/shift-pagedown? I've grown quite weary of shift-pgup to page up in gnome terminal, the command line, terminator, and guake, etc.  Any way to easily make these apps use the straight page-up and page-down keys for paging around? If $f \colon [a,b] \rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$ If $f \colon [a,b]\rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$.  I think that I have to proceed with the Mean Value Theorem of Integrals... Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$ I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q} $ Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having a nice "basis.  The splitting field is given by $\mathbb{Q}(\zeta_n,\alpha)$ , where $\zeta_n$ is a primitive root of unity , and $\alpha$ is some number such that $\alpha^n = a $. Well first of all, I want a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different (in this case the computation of the galois group is very simple).  For example one easy case, it's when $a&gt;0$, then $(a)^{\frac{1}{n}} \in \mathbb{R}$ , so clearly the minimal polynomial of $(a)^{\frac{1}{n}} , \zeta_n$ are distincs, and I'm done. If $n$ is odd then , it's also easy, since one root it's also real, for example $x^3-3 $, the real root is $ \root 3 \of { - 3}  =  - \root 3 \of 3  $ , so I can consider the splitting field as $\mathbb{Q}(-\root 3 \of 3 , \zeta_3 )=\mathbb{Q}(\root 3 \of 3 , \zeta_3 )$. The difficult case is when $n$ is even and $a&lt;0$ , for example  $x^8+20$ or $x^4+20$ in some cases as in the second, there are particular cases since there exist algorithms for the Galois group of quartics, but in general. It can be done?  $\underline{Remark}$ I'm searching a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different since in this case the computation of the galois group is very simple. I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$