$\sum_{i=0}^{n} (-1)^i l(H_i)=\sum_{i=0}^{n} (-1)^i l(G_i)$
Show that the Euler characteristic of a chain complex is equal to the Euler characteristic of its homology
Creation parameters menu greyed out When adding a UV Sphere mesh in Blender 2.8, the creation parameters menu on the bottom left of the attached image for this UV Sphere seems to be greyed out. Any idea why this might be? Thanks. I want to give a client access to my server, but I want to limit those users to their home directories. I will bind-mount in any files I want them to be able to see.  I've created a user called bob and added him to a new group called sftponly. They have a home directory at /home/bob. I've changed their shell to /bin/false to stop SSH logins. Here is their /etc/passwd line:  bob:x:1001:1002::/home/bob:/bin/false   I've also changed the /etc/ssh/sshd_config to include the following:  Match Group sftponly         ChrootDirectory /home/%u         ForceCommand internal-sftp         AllowTcpForwarding no   When I try to log in as them, here's what I see   $ sftp bob@server bob@server's password:  Write failed: Broken pipe Couldn't read packet: Connection reset by peer   If I comment out the ChrootDirectory line I can SFTP in but then they have free rein over the server. I have found that ChrootDirectory /home works, but it still gives them access to any home directory. I have explicitly tried ChrootDirectory /home/bob but that doesn't work either.  What am I doing wrong? How can I limit bob to /home/bob/?  ----EDIT-----  Okay so I just had a look at /var/log/auth.log and saw this:  May  9 14:45:48 nj sshd[5074]: pam_unix(sshd:session): session opened for user bob by (uid=0) May  9 14:45:48 nj sshd[5091]: fatal: bad ownership or modes for chroot directory component "/home/bob/" May  9 14:45:48 nj sshd[5074]: pam_unix(sshd:session): session closed for user bob   I'm not entirely sure what's going on there,  but it suggests something is wrong with the user directory. Here is the ls -h /home output:  drwxr-xr-x 26 oli      oli      4096 2012-01-19 17:19 oli drwxr-xr-x  3 bob      bob      4096 2012-05-09 14:11 bob "debconf: DbDriver "config": config.dat is locked by another process: Resource temporarily unavailable" while installing packages I'm trying to install the flashybrid but when I enter the apt-get install flashybrid command, I get this error:  root@user-desktop:/home/user# apt-get install flashybrid Reading package lists... Done Building dependency tree Reading state information... Done flashybrid is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 0B of additional disk space will be used. debconf: DbDriver "config": /var/cache/debconf/config.dat is locked by another process: Resource temporarily unavailable Setting up flashybrid (0.15+nmu2) ... debconf: DbDriver "config": /var/cache/debconf/config.dat is locked by another process: Resource temporarily unavailable dpkg: error processing flashybrid (--configure): subprocess post-installation script returned error exit status 1 Errors were encountered while processing: flashybrid E: Sub-process /usr/bin/dpkg returned an error code (1)   I also tried this one to find what is already using that file:  root@user-desktop:/home/user# lsof | grep /var/cache/debconf/config.dat lsof: WARNING: can't stat() fuse.gvfs-fuse-daemon file system /home/user/.gvfs Output information may be incomplete. dpkg-prec 4672 root 4rW REG 8,6 39658 1697827 /var/cache/debconf/config.dat   But I don't know how to fix this. Find the sum to n terms as well as the sum to infinity of the series: Find the sum to $n$ terms as well as the sum to infinity of the series:  $$\frac{1}{3} + \frac{1}{4\cdot 2!} + \frac{1}{5\cdot 3!} +\cdots.$$  I was trying this question many times but I didn't get the answer. I was using geometric series formula, but it was not applicable here...I don't know where I have to start, and I don't know any hint about this series....  If anybody helps me, I would be very thankful to him. Overriding the final destination of the OAuth2.0 flow from a custom OAuth2.0 Auth provider Background:  I have created a custom Auth. Provider in Salesforce to reach out to an authorization server that isn't on the default list of Provider Types. I used  as a good starting point to extend the Auth.AuthProviderPluginClass. Once appropriately added to my org, it then became an option on the list of Provider Types. I was able to create an Auth. Provider using that new provider type and it works to initiate the OAuth2.0 flow. I am appropriately able to get the access token and refresh token from the authorization server.  The Problem:  As it currently stands, we will have users click on a button that will send them to the provided "OAuth-Only Initialization URL" link that was generated when I created the actual Auth. Provider record. The trick is that when the OAuth2.0 flow is completed, the user is redirected back to the Salesforce Setup Home page, and I don't see a good way in the documentation to override the final destination to be something else (like a success page or something).  Question:  Does anyone know of a way to override the final destination of the OAuth2.0 flow from a custom OAuth2.0 Auth provider?    UPDATE: I tried using the startUrl (and also tried retUrl for good measure) as query params, but doing so didn't change the behavior after the OAuth2.0 flow finished.  Meaning, both of the following URLs gave me the same final destination as not having any query params at all:  https://&lt;scratch org random url&gt;.my.salesforce.com/services/auth/oauth/MyCustomAuthProvider?retUrl=lightning%2Fpage%2Fhome   https://&lt;scratch org random url&gt;.my.salesforce.com/services/auth/oauth/MyCustomAuthProvider?startUrl=lightning%2Fpage%2Fhome     Another Update: This WORKED:  https:/&lt;scratch org random url&gt;.my.salesforce.com/services/auth/oauth/MyCustomAuthProvider?startURL=%2Flightning%2Fn%2FCustomPage   Apparently, it was important to add the leading /, (of course, url encoded to be %2F) and %2Flightning%2Fpage%2Fhome gets interpreted as the setup home and sends you there.    (Parallel posted to Salesforce Dev Forum: ) Eigenvectors of a circulant matrix Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic. Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.   What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve this and showing that $|G|$ order element in $G$ exists is enough. Alt-click for edge loop Hi I am a complete newcomer to blender and am following this tutorial, where he says to alt+click to select an entire row or "edge loop". I have followed all instructions exactly up to this point (3 times to make sure!). Alt-click does nothing. Thank you for your help. p.s. If possible, please keep answers in simple layman's language. I see some similar posts about alt click and loop select but had trouble following since the explanations were too advanced. Thank you! Video starts where he demonstrates alt+click: Installing Autodesk Maya on Ubuntu is possible, but can be a bit challenging.  Assuming that one already has the required specs and a solid graphics card driver installed, how can it be done? What word/phrase can I use to describe the activity of hiding a stupid mistake/incident, such as quickly cleaning up tea spilled by accident before anyone could notice? Why are integer GCDs positive? [unit normalization of GCDs] The definition in my text reads, An integer $d$ is said to be the greatest common divisor of two non-zero integers $a$ and $b$ iff,  $d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $k|d$  Now here's the thing, if $d|a$ and $d|b$ then surely $-d|a$ and $-d|b$ as well, also $k|-d$ What I take from this? GCD is not unique! That is if $\mathrm{gcd}(12,8)= 4$ then by the definition, $\mathrm{gcd}(12,8) = -4$ as well. Yet I never ever seen a negative gcd. Someone please explain. Maybe, $4&gt;-4$, and we want the &quot;greatest common factor&quot; so...? But that still doesn't justify the definition. How can I remove the qed symbols from all lemma proofs? the following code will only make the qed not appear in the second proof. However, it it possible to make the qed symbol never appear at the end of the proofs without having to specify it locally every time?  \documentclass{article} \usepackage{amsthm} \begin{document} \begin{proof} This has the QED symbol. \end{proof} \begin{proof}\renewcommand{\qedsymbol}{} This hasn't. \end{proof} \begin{proof} And this has it again. \end{proof} \end{document} Evaluate $\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \ \mathrm{d}x$ Algorithms for computing multiplicative order modulo $n$ How to arrive at $\int_{0}^{1} f^3(x) \, dx < (\int_{0}^{1} f(x) \, dx)^2$? Define $f : [0,1] \to \mathbb{R} $ differentiable satisfying $f(0)=0$ and $0&lt;f'(x)&lt;1$. Prove:  $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$ I found the following answer,  Define $F(x)=\displaystyle\int_{0}^{x} f^3(t) \, dt - \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)^2$ We have $$F'(x)=f^3(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)f(x)=f(x)\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]$$ and $$\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]'=2f(x)\left[f'(x)-1\right]$$ from which we can arrive at the conclusion.  But I felt a little confused. In the past I usually proved the inequality by classical inequalities and something like Taylor expansion to use the condition about derivative. But this answer is little out of my expectation. Do we have other proofs or ideas to arrive at $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$? And other thoughts about solving these types of issues? I think when we talk about $\left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2 &gt;\ldots \,$ it's a little difficult to deal with. I don't quite know what tools to use. Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed? By accident, I messed up a configuration file. How do I recover the default version without reinstalling or affecting other things? In my family we have a DNA contradiction – two cousins (at about the 4th cousin level) took a Y-DNA test, but did not match and had two completely different haplogroups.   Since then, I have traced living cousins with the same surname – the only ones left living that I am aware of – and I would love to pay to have them tested to see if we can figure out where the non-paternity event (NPE) or other event happened (i.e. this new cousin could help us determine which line the NPE occurred in depending on which cousin he matches).   Does anyone have a sample letter, or know the best way to go about asking a distant cousin for DNA?   I have their address and am planning to write a letter, but I can't imagine receiving such a letter, so I am not quite sure how to ask without scaring them off. I have downloaded an ISO file recently.  How do I burn it to a CD or DVD or mount it? Cooking steak in frying pan, problem with oil splatter