The book I'm studying gives the answer as $\frac{1-e^{j\frac{x}{N}N}}{1- e^{j\frac{x}{N}}} = 0$ but doesn't explain how.
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:  $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr)$$  There is a slight difference in case of $\sin$, which is: $$\sum_{k=0}^{n-1}\sin (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin\biggl( \frac{2  a + (n-1)\cdot d}{2}\biggr)$$   How do we prove the above two identities?
How to fix apt: Signature by key uses weak digest algorithm (SHA1)? I started setting up by adding repositories and then went to run a sudo apt-get update again before I started installing other software, and I get the Signature key lines and it stops. So it essentially won't let me update any packages now.  d@EliteBook:~/Downloads$ sudo apt-get update Ign:1 http://dl.google.com/linux/chrome/deb stable InRelease Hit:2 http://dl.google.com/linux/chrome/deb stable Release                      Hit:4 http://security.ubuntu.com/ubuntu xenial-security InRelease               Get:5 http://ca.archive.ubuntu.com/ubuntu xenial InRelease [247 kB] Hit:6 http://ca.archive.ubuntu.com/ubuntu xenial-updates InRelease Hit:7 http://ca.archive.ubuntu.com/ubuntu xenial-backports InRelease Fetched 247 kB in 0s (256 kB/s)                    Reading package lists... Done W: http://dl.google.com/linux/chrome/deb/dists/stable/Release.gpg: Signature by  key 4CCA1EAF950CEE4AB83976DCA040830F7FAC5991 uses weak digest algorithm (SHA1) d@EliteBook:~/Downloads$   I've never seen this before whenever I setup and start installing things in Ubuntu. Is there something else I can do? What terminal command or program can list an SSD's Power On Hours? I'm looking to script using the unix shell the retrieval of SSD (and HDD) data that disk utility shows in the information window.    I'm immediately interested in the Power On Hours and Power Cycle, but if all the data comes from one data structure or tool, so much the better. The usual tools diskutil info disk0 and ioreg -l are not producing the "of interest" values listed in the screen shot above.  Is this information available from any of the built in OS X terminal tools or do I need to locate an open-source package that makes the same API calls to get this level of detail about a Mac drive? Show that $d(z,a)=|\frac{z-a}{1-\overline{a}z}|$ is a metric on the unit disk For any $z, a$ in the open unit disk, we define$$d(z,a)=\left \vert \frac{z-a}{1-\overline{a}z} \right \vert$$  Show that $d$ is a metric on the open unit disk.     It is easy to show $d(z,a)=d(a,z)$ and $d(z,a)=0$ $\rm iff$ $z=a$. However, I am having trouble showing that it satisfies triangle inequality. Any help would be appreciated! What are the requirements for a (beta) site logo? Show that $\partial_\nu T^{\mu\nu} = - j_\nu F^{\mu\nu}$ In a theoretical physics homework problem, I have to show the following: $$\partial_\nu T^{\mu\nu} = - j_\nu F^{\mu\nu}$$  Where $T$ is the Energy-Momentum-Tensor, $j$ the generalized current and $F$ the Field-Tensor. We use the $g$ for the metric tensor, I think in English the $\eta$ is more common.  I know the following relationships:   Current and magnetic potential with Lorenz gauge condition: $$\mathop\Box A^\mu = \mu_0 j^\mu$$ Energy-Momentum-Tensor: $$T^{\mu\nu} = \frac1{\mu_0} g^{\mu\alpha} F_{\alpha\beta} F^{\beta\nu} + \frac1{4\mu_0} g^{\mu\nu} F_{\kappa\lambda} F^{\kappa\lambda}$$ Field-Tensor: $$F^{\mu\nu} = 2 \partial^{[\mu} A^{\nu]} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu}$$ d'Alembert operator: $$\mathop\Box = \partial_\mu \partial^\mu$$ Bianchi identity: $$\partial^{[\mu} F^{\nu\alpha]} = 0$$   So far I have set all the definitions into the formula I have to show, but I only end up a lot of terms from antisymmetrisation and product rule. I also drew all what I have in , but I still cannot see how to tackle this problem.  Could somebody please give me a hint into the right direction? I have a site which is only for US/Canada/UK visitors. I don't want to make it visible to other visitors.  I tried to put IP ranges in .htaccess (Deny IP) but it does not work for me. May be I have inaccurate IP lists or I am doing something wrong. Pre-print service like arXiv but with private option, so I can correct my mistakes without publicly advertising my mistakes? As a student, I make mistakes all the time, and stealing my idea will not face heavy retaliation. I don't mind if someone comes up with concurrent works, but I do mind if my key idea is leaked. Many graduate students in my field therefore only use pre-print services if their papers are either publicly presented or conditionally accepted. I am looking for a website that can do most of what ArXiv does, with an additional private option or limited visibility, for example an option to make the paper unindexed by search engines. This way, my paper cannot be seen for at least a few months. By doing so, I can at least claim that I am one of the first few researchers who got the idea right. And most importantly, I can privately correct my mistakes, if any. Publishing mistakes is embarrassing and sometimes harms one's career. Many mistakes will be revealed during the peer-review processes and I have a chance to correct them before publicizing. Please kindly think twice before giving ethically and &quot;politically correct&quot; comments like no one will scoop me, everyone will cite me, and the scoopers will be punished. At least in my field, scooping and idea-stealing is commonly observable, and big-names don't care about publicizing pre-prints because they have resources to punish the scoopers. I do understand that keeping my work private might harm the community as the information becomes less transparent. However, it could also be possible that my un-reviewed flawed works misguide the community, and a paper reviewed will benefit the society more. I personally wasted six months because of someone else's mistake in a published work.  I heard that in molecular biology there is a widely-used service to upload your research data, like protein structure and sequencing, to get a timestamp. The uploaded materials can be set to be private for a period, and fully disclosed later. A lot of people do use this function. This is not exactly what I am looking for but shares some similar ideas. If I remember correctly, there is an interesting paper studying the scooping patterns with the aid of this database.  Solutions found!  and some other university-run pre-print services do have an embargo function. Moreover,  is a widely used repository, that, unlike arxiv, only displays the newest version to the general public, although still keeps a tag on the first submitted date! I want to draw a perfect rectangle. How can I do that with QGIS v.3+?  .  Is there a similar solution for the latest QGIS version? Exporting Data From Salesforce - REST or SOAP? What would be the associated wavelength of the particle if its velocity is zero? SQL Server replication subscriptions marked as inactive Is there any way to force SQL Server NOT to mark subscriptions as inactive, ever?  It happens sporadically when there are connection issues and I don't want to have to reinitialize the subscription every time.  Note, I'm not talking about the subscriptions being marked as expired...just as inactive.  We have retention set to 0 (infinite). I'd rather have a big distribution database than have to reinitialize a 15GB database to a subscriber over the WAN. Anyway, the real issue here is that they are getting marked as inactive within just a few hours sometimes...or sometimes a few days...it seems sporadic. Is there any way to tell WHY SQL decided in its wisdom to mark them inactive? Clicking links in pdfs opened with Okular opens Abiword (instead of the default browser) I work with Xubuntu 12.04 and use Okular (version 0.14.3) to view pdf files. If I click a web link in a pdf file (created with "pdflatex" using the "hyperref" package), "Abiword" (It's a word-processing software, like Libreoffice writer) is opened instead of my default browser google-chrome. How can I change this behavior? The settings in Okular do not seem to provide a solution.   (Note that I set google-chrome as the preferred application for web browsing under Settings -> Preferred Applications, so that's not the problem). Navigation bar overlapped by status bar Identical coefficients estimated in Poisson vs Quasi-Poisson model How to completely disable Siri on Sierra? jQuery autocomplete tagging plug-in like StackOverflow's input tags? What solutions accomplish the same auto-completion that SO uses for entering tags?  There are plugins that can handle one word but I haven't seen any that handle multiple words. How long should I keep my tax documents, and why? The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation easier, or is it an essential ingredient, without which it would be impossible to derive some results?   I can identify two different settings where the answer may be different:   It doesn't look like it is mandatory for results in Newtonian mechanics, General Relativity and classical electrodynamics. Can we say the same thing about quantum mechanics either way for sure?   Could this be a difference in quantum mechanics over the classical picture? How to create script with auto-complete? How to run Unix shell script from Java code? It is quite simple to run a Unix command from Java.   Runtime.getRuntime().exec(myCommand);   But is it possible to run a Unix shell script from Java code? If yes, would it be a good practice to run a shell script from within Java code?