Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of  $\Phi$ in $V$.   I want to show that:   The Weyl chambers are open, convex and connected.   My attempts:   A single hyperplane $H_\alpha$ separates the space $V$ into two half-spaces: positive half-space $H_\alpha^+=\{x \in V \mid (\alpha,x)&gt;0\}$ and negative half-space $H_\alpha^-=\{x \in V \mid (\alpha,x)&lt;0\}$. It is clear that $V \setminus H_\alpha = H_\alpha^+\cup H^-_\alpha.$ As the result, the Weyl chambers can be regarded as components of  $$V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}= \bigcap_{\alpha \in \Phi} \left( V \setminus H_{\alpha} \right)= \bigcap_{\alpha \in \Phi} \left( H_\alpha^+\cup H_\alpha^- \right).$$ By the discussion above, a Weyl chamber is simply a $\textbf{intersection}$ of finite half-spaces. It is the case since one may write $$\left( H_\alpha^+\cup H_\alpha^- \right) \cap \left( H_\beta^+\cup H_\beta^- \right) =\left( H_\alpha^+\cap H_\beta^+ \right)  \cup\left( H_\alpha^+\cap H_\beta^- \right)  \cup \left( H_\alpha^-\cap H_\beta^+ \right)  \cup\left( H_\alpha^-\cap H_\beta^- \right).$$ Therefore openness and convexity of the Weyl chambers are straightforward since half-spaces are open and convex, so are their intersections. Moreover, we would replace $V$ by $\Bbb R^{\dim V}$ since they are isomorphic. Now we can prove that the Weyl chamber is connected by showing that every convex set in $\Bbb R^{\dim V}$ is connected. And I'm sure that proof of this statement existed. So am I done?
Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of  $\Phi$ in $V$.   I want to show that:   The Weyl chambers are open, convex and connected.   My attempts:   A single hyperplane $H_\alpha$ separates the space $V$ into two half-spaces: positive half-space $H_\alpha^+=\{x \in V \mid (\alpha,x)&gt;0\}$ and negative half-space $H_\alpha^-=\{x \in V \mid (\alpha,x)&lt;0\}$. It is clear that $V \setminus H_\alpha = H_\alpha^+\cup H^-_\alpha.$ As the result, the Weyl chambers can be regarded as components of  $$V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}= \bigcap_{\alpha \in \Phi} \left( V \setminus H_{\alpha} \right)= \bigcap_{\alpha \in \Phi} \left( H_\alpha^+\cup H_\alpha^- \right).$$ By the discussion above, a Weyl chamber is simply a $\textbf{intersection}$ of finite half-spaces. It is the case since one may write $$\left( H_\alpha^+\cup H_\alpha^- \right) \cap \left( H_\beta^+\cup H_\beta^- \right) =\left( H_\alpha^+\cap H_\beta^+ \right)  \cup\left( H_\alpha^+\cap H_\beta^- \right)  \cup \left( H_\alpha^-\cap H_\beta^+ \right)  \cup\left( H_\alpha^-\cap H_\beta^- \right).$$ Therefore openness and convexity of the Weyl chambers are straightforward since half-spaces are open and convex, so are their intersections. Moreover, we would replace $V$ by $\Bbb R^{\dim V}$ since they are isomorphic. Now we can prove that the Weyl chamber is connected by showing that every convex set in $\Bbb R^{\dim V}$ is connected. And I'm sure that proof of this statement existed. So am I done?
Show that if m/n is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better How can I easily encrypt a file? Is there any simple (IE: right click in Nautilus) way to password protect a particular folder/file in Ubuntu? I've got a few files containing sensitive info and I'd much prefer that if/when I leave my computer alone, they aren't accidentally accessed by someone else.  The secruty does not have to be extremely tight. My only concern is that when family/friends come over, I don't really like the idea of them looking at my bank details, accounts or, you guessed it, porn collection.  A simple, effective way to let me put my machine in the hands of someone else knowing that said machine can not cause me embarresment is the sole reason why I'd like to see this in Ubuntu. Let $P(x)$ be a degree $n$ polynomial with distinct roots $r_1, r_2, \cdots r_n$. Prove that   $$\displaystyle\sum_{i=1}^{n}\frac{P''(r_i)}{P'(r_i)}=0$$  My proof: We can equivalently rewrite the polynomial as $(x-r_1)(x-r_2)\cdots(x-r_n)$ and if we plug in the roots in the polynomial, we get that all of them are equal to $0$ so their first derivatives are equal to $1$ and then their second derivatives are equal to $0$ which means that all quotients are equal to $0$ and there for the sum is as well, $Q.E.D$  Don't show the solution please, but if you have a hint, feel free. Thank you. Does Android keep the .apk files? if so where? Can you make a seismograph with an accelerometer? Actual seismograph electronics are very expensive. I'm just wondering if a seismograph can be made with a adequately semsitive accelerometer. To be exact I want to bury the accelerometer based seismograph and have it sense footsteps.  So is this possible and what method would you reccommend for my needs and why? Ability to update profile from the Stack Exchange Android App Valid US visa on expired passport. Can I travel to the US? My old passport is in my maiden name and has already expired. My B1/B2 visa to travel to United States is still valid but is stamped in my old passport with my maiden name.   My new passport has my new name (old name got changed after marriage) but has details (like old passport number, father and mother's name) related to my old passport.    Will I be allowed to travel to US as long as I carry the my old as well as new passport?  Or Do I need to again get the visa stamped on my new passport? Possible Duplicate:          So, in my calculus class (one I'm teaching, not taking), the sum $\sum_{n=1}^\infty \frac{\sin(n)}{n}$ has come up a few times.  Unfortunately, as someone not exactly an expert in the convergence of sums, it seems to resist the few techniques I know.  Certain none of the usual first year calculus tricks (integral test, alternating series test, ratio test, etc.) work, and the only more tricky technique, partial summation, I can think of doesn't seem to work either (one would need that $\sum_{n=1}^N\sin(n)$ is bounded, which I believe is false).  It seems like it should converge, since it switches sign quite often, but on the other hand, the harmonic series can mess with your intuition, so I don't have much trust in that.  So, I ask to you:     Does this series converge? There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$? There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is entire and its $0$ set has limit point in it and so $f(z)={1\over 2^{1\over z}}$ which is not analytic at $0$? Please help!  Edit: OOps! the way I defined $g(z)$ that is not entire! could any one give me hint? A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$.  The way they explained it, you could prove that there are the same amount of numbers by getting a match to each number.  For example, for any number in the range from $0$–$2$, you can find a matching number in the range $0$–$1$ by dividing the number by 2. In every case, the ending number will end up between $0$–$1$. In addition, if you multiplied any number in the range of $0$–$1$ by 2, you will always end up with a number between $0$–$2$ Therefore, there should be the same infinite amount of numbers between  $0$–$1$ as there are between $0$–$2$.  Is this principle accurate? The problem that I had with it was, if in-fact there are the same amount of numbers between 0 and 1 as there are between 0 and 2, why is 2 greater than 1? In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$?  with these parameters:    $r=\sqrt{x^2+y^2}$ $x=r\cos\theta$ $y=r\sin\theta$  Tanks. I am unable to open gnome-control-center due to an error stemming from pop repositories. Ubuntu 20.04 Make Windows 7 ignore WiFi when ethernet is available When I dock my Windows 7 laptop, I want it to prefer the wired ethernet connection over WiFi.  This is a pretty straightforward thing to do on my Mac - I just reorder my network preferences, and it "does the right thing." I just can't figure out how to achieve the same thing on my Win7 laptop.  So, when I'm docked, it connects to WiFi, and then fails to connect to servers on the local wired network. How do I fix this? Object renders black Using dconf-editor I tried enabling   org -&gt; gnome -&gt; nautilus -&gt; list-view -&gt; use-tree-view     but even after the check box is selected, plus a reboot, it still doesn't switch to the tree view on the side bar.  In previous versions of Nautilus / Ubuntu the side-pane-view setting used to be in the window-state section (which is now missing)    I'm I missing something, or is there no way to enable Tree View on the side bar in 14.04?  Is there a way to manually create the missing entry back in window-state ? Suspend broken on Ubuntu 16.04 LTS, Dell XPS 9350 Since I upgraded from Ubuntu 15.10 to Ubuntu 16.04 LTS on my Dell XPS 9350, suspend is broken. When closing my lid the laptop will suspend, but resuming fails and it reboots. I tried pm-suspend and tried resuming, resuming failed and the laptop rebooted. I got the following errors when I got back into Ubuntu. The odd thing in my opinion is that it claims the kernel is not an official ubuntu package. Either way my Linux knowledge is not sufficient to know what's going wrong here.      EDIT: I just tried installing kernel 4.4.8 which should fix some things, but it did not fix this problem. By now I also tried 4.4.9 and 4.6 RC7. None of them fixed it for me. Changing parameter values according to other parameter choice in Processing script using QGIS 3? What's the difference between faking, mocking, and stubbing? I know how I use these terms, but I'm wondering if there are accepted definitions for faking, mocking, and stubbing for unit tests?  How do you define these for your tests?  Describe situations where you might use each.  Here is how I use them:  Fake: a class that implements an interface but contains fixed data and no logic.  Simply returns "good" or "bad" data depending on the implementation.  Mock: a class that implements an interface and allows the ability to dynamically set the values to return/exceptions to throw from particular methods and provides the ability to check if particular methods have been called/not called.  Stub: Like a mock class, except that it doesn't provide the ability to verify that methods have been called/not called.  Mocks and stubs can be hand generated or generated by a mocking framework.  Fake classes are generated by hand.  I use mocks primarily to verify interactions between my class and dependent classes.  I use stubs once I have verified the interactions and am testing alternate paths through my code.  I use fake classes primarily to abstract out data dependencies or when mocks/stubs are too tedious to set up each time. My new laptop came preinstalled with Windows 8. Naively, as I am, I just formatted the harddrive and installed fine old Ubuntu. 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