Recent Questions - Server Fault

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• How to delete specific files older than 1hour on cronjob?
• Setting up SSL for custom port in nginx - letsencrypt
• Using Podman containers with Ansible
• ASUS router notifying a local device on device connect/disconnect?
• Kubernetes V1.19.13 - kubeapi servers not able connecting to different etcd database
• If you're seeing this Grafana has failed to load its application files
• How to delete specific files?
• After adding a new DNSBL to Sendmail, how can I resubmit an email to see if it will be rejected?
• How can I let apache server work always on my ec2 instance?
• Packets from xfrm interface won't route, but opposite works
• Rebooted Ubuntu server, nginx site no longer accessible from browser
• Any way to join a server to Active Directory Domain by IP address in hostfile and not DNS?
• Tomcat 9 Service on Centos 7 won't start up at system boot
• How can I install the ngx_http_geoip2_module module ? on Centos
• Unable to NAT TFTP traffic because iptables is not forwarding the return connection to the client despite TFTP helper creating an expectation
• Migrating Jenkins jobs from one server to another
• SSH Suddenly returning Invalid format
• Adding Tag (i.e. Source IP) to rsyslog for sending to rsyslog remote server
• Nginx configuration for Deluge
• How to show a 404 instead of a 403 on apache host
• dmidecode weird total/data width
• nginx 405's with try_files for a DELETE request instead of proxying
• Redirect IP to a domain name using htaccess
• "A new version of /boot/grub/menu.lst is available" when upgrading Ubuntu on an AWS server
• Raid 1 can not sync with new Drive . its stopping at 30 %
• Bitnami redmine error SVN
• zimbra Error when installing
• Permanently assigning IP address for an embedded device
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How to delete specific files older than 1hour on cronjob? Posted: 31 Jul 2021 10:24 PM PDT I'm using cronjob for Ubuntu 20.04. I want to auto delete files older than 1hour and only files with filename begins with master-stdout.log* How can I do this? find /root/logs/ * -mmin +60 -exec rm {} \;

Setting up SSL for custom port in nginx - letsencrypt Posted: 31 Jul 2021 10:18 PM PDT I'm trying to enable SSL on a custom port (not 443), running a webpage. From searching around, I couldn't find much info that helped. The server has unchangable ports, external: 26143, Internal: 80. To enter the server (without SSL) you would type example.com:26143, and the system would see this as a connection to port 80. How would I set up a certificate (lets encrypt) to enable SSL on this port? From testing, it seems like whatever I do, it only accesses the server on port 80, even if I set it to 26143 here is the nginx sites-enabled config: server { listen 80; listen [::]:80; root /root/html; index index.php; server_name _; location / { try_files $uri $uri/ =404; } location ~ \.php$ { include snippets/fastcgi-php.conf; # With php-fpm (or other unix sockets): fastcgi_pass unix:/var/run/php/php7.4-fpm.sock; } location ~ /\.ht { deny all; } location /.well-known { root /var/www/ssl/example.com/; } } Commands I've tried are: certbot --nginx -d example.com:26143 certbot certonly --standalone --preferred-challanges http -d example.com:26143 certbot certonly --standalone --preferred-challenges http -d example.com certbot certonly --standalone --preferred-challenges http --http-01-port 26143 -d example.com certbot certonly --nginx --preferred-challenges http --http-01-port 26143 -d example.com certbot certonly --noninteractive --agree-tos --cert-name slickstack -d example.com -m my@mail.com --webroot -w /root/html certbot certonly --noninteractive --agree-tos --cert-name slickstack -d example.com:26143 -m my@mail.com --webroot -w /root/html certbot certonly --noninteractive --agree-tos --cert-name slickstack -d example.com --http-01-port 26143 -m my@mail.com --webroot -w /root/html certbot certonly --noninteractive --agree-tos --cert-name slickstack -d example.com --preferred-challenges http --http-01-port 26143 -m my@mail.com --webroot -w /root/html Some tweaking back and fourth, most common error I got was this: IMPORTANT NOTES: - The following errors were reported by the server: Domain: example.com Type: unauthorized Detail: Invalid response from https://example.com/.well-known/acme-challenge/ho73up1dR3KU4V37awccOw2T5xsSILWUM365ZnwVEN4 [159.81.xxx.xxx]: "<!DOCTYPE HTML PUBLIC \"-//IETF//DTD HTML 2.0//EN\">\n<html><head>\n<title>404 Not Found</title>\n</head><body>\n<h1>Not Found</h1>\n<p" To fix these errors, please make sure that your domain name was entered correctly and the DNS A/AAAA record(s) for that domain contain(s) the right IP address. The 404 is Not from my system, it's from example.com:80, instead of example.com:26143. Also, I do not have access to modifying the DNS records. In my experience, lets encrypt and SSL has been kind of confusing, and together with the rate limits, I'm not able to troubleshoot enough to understand. I know it should be possible, I just don't know how and/or what I'm doing wrong. Any help would be appreciated

Using Podman containers with Ansible Posted: 31 Jul 2021 08:42 PM PDT I created an ansible role with podman to pull the nginx image and run the container which works but i would like to now copy a custom index html from the host to the container so that it overrides the default index html page do we use volumes if so how do we use it for this scenario in the yml file? Appreciate your help.

ASUS router notifying a local device on device connect/disconnect? Posted: 31 Jul 2021 07:55 PM PDT I have a Asus RT-AX88U router, which comes with the ability to work with Alexa and IFTTT for things like notification when it detects that a device is connected/disconnected. However, I don't like the idea of sending this kind of information over the internet, and rather prefers to have a local server that processes these triggers and notifies me with other means. Is there a way to hack the router/are there unofficial APIs that I can use to get the same functionality without IFTTT/Alexa?

Kubernetes V1.19.13 - kubeapi servers not able connecting to different etcd database Posted: 31 Jul 2021 07:51 PM PDT I have upgraded Kubernets cluster ( 3 master, 3 etcd servers database) from 1.18 to v1.19.13 and ETCD to 3.4.13. since than API servers are not stable, keep switching different etcd server, because of this cluster is not working properly. these cluster running on CentOS steam 8. This Cluster worked before the upgrade, after upgrade only I have seen this issue. Any help to resolve this issue? Is there know issue with this version? API server logs I0731 00:54:39.498953 1 client.go:360] parsed scheme: "passthrough" I0731 00:54:39.499025 1 passthrough.go:48] ccResolverWrapper: sending update to cc: {[{https://0.0.0.02:2379 <nil> 0 <nil>}] <nil> <nil>} I0731 00:54:39.499035 1 clientconn.go:948] ClientConn switching balancer to "pick_first" I0731 00:54:40.241615 1 client.go:360] parsed scheme: "passthrough" I0731 00:54:40.241681 1 passthrough.go:48] ccResolverWrapper: sending update to cc: {[{https://0.0.0.01:2379 <nil> 0 <nil>}] <nil> <nil>} I0731 00:54:40.241691 1 clientconn.go:948] ClientConn switching balancer to "pick_first" I0731 00:54:45.348969 1 client.go:360] parsed scheme: "passthrough" I0731 00:54:45.349030 1 passthrough.go:48] ccResolverWrapper: sending update to cc: {[{https://0.0.0.03:2379 <nil> 0 <nil>}] <nil> <nil>} I0731 00:54:45.349040 1 clientconn.go:948] ClientConn switching balancer to "pick_first" I0731 00:55:16.460379 1 client.go:360] parsed scheme: "passthrough" I0731 00:55:16.460428 1 passthrough.go:48] ccResolverWrapper: sending update to cc: {[{https://0.0.0.01:2379 <nil> 0 <nil>}] <nil> <nil>} I0731 00:55:16.460439 1 clientconn.go:948] ClientConn switching balancer to "pick_first" I0731 00:55:17.461906 1 client.go:360] parsed scheme: "passthrough" etcd looks healthy # /opt/bin/etcdctl.sh version etcdctl version: 3.4.13 API version: 3.4 /opt/bin/etcdctl.sh endpoint health 127.0.0.1:2379 is healthy: successfully committed proposal: took = 9.739533ms # /opt/bin/etcdctl.sh check perf 60 / 60 Boooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo! 100.00% 1m0s PASS: Throughput is 150 writes/s PASS: Slowest request took 0.042491s PASS: Stddev is 0.001743s PASS # /opt/bin/etcdctl.sh endpoint status --cluster -w table +----------------------------+------------------+---------+---------+-----------+------------+-----------+------------+--------------------+--------+ | ENDPOINT | ID | VERSION | DB SIZE | IS LEADER | IS LEARNER | RAFT TERM | RAFT INDEX | RAFT APPLIED INDEX | ERRORS | +----------------------------+------------------+---------+---------+-----------+------------+-----------+------------+--------------------+--------+ | https://0.0.0.02:2379 | 15cd65a732ebd5d8 | 3.4.13 | 26 MB | false | false | 9305 | 19813854 | 19813854 | | | https://0.0.0.03:2379 | add66a254676e690 | 3.4.13 | 26 MB | true | false | 9305 | 19813854 | 19813854 | | | https://0.0.0.01:2379 | e2811ed02ce71623 | 3.4.13 | 26 MB | false | false | 9305 | 19813854 | 19813854 | | +----------------------------+------------------+---------+---------+-----------+------------+-----------+------------+--------------------+--------+

If you're seeing this Grafana has failed to load its application files Posted: 31 Jul 2021 07:51 PM PDT I want to run grafana behind nginx. I followed this instruction. The only problem I have right now is whenever I try to edit a panel, I will see this error message below. It disappears in a few seconds. The grafana.ini looks like domain=localhost root_url = %(protocol)s://%(domain)s:%(http_port)s/grafana/ serve_from_sub_path = true The scrubbed nginx.conf looks like server_name 1.2.3.4 # my public id address location /grafana/ { proxy_pass http://localhost:3000/; } I don't really experience any issue other than the annoying error message every time I want to edit a panel. When I used SSH to bind port 3000 to my computer ssh -L 3000:localhost:3000 my_id@1.2.3.4, everything worked just fine. Any suggestions on how to fix the problem?

How to delete specific files? Posted: 31 Jul 2021 07:48 PM PDT I'm running Ubuntu 20.04. I have a directory with million of files named like this master-stdout.log.20210801.024908 master-stdout.log.20210801.025524 master-stdout.log.20210801.064355 How can I delete all of master-stdout.log files?

After adding a new DNSBL to Sendmail, how can I resubmit an email to see if it will be rejected? Posted: 31 Jul 2021 04:56 PM PDT TL;DR How can I fool my own sendmail into thinking an email comes from a particular ip address, so that it rejects the message due to a DNSBL match? Details: I run my own mail server, and most spam is blocked by the DNS black lists (DNSBL) that I've added to /etc/mail/sendmail.mc like so: dnl FEATURE(`dnsbl',`dnsbl.sorbs.net',`"554 Rejected " $&{client_addr} " found in dnsbl.sorbs.net"')dnl dnl FEATURE(`dnsbl',`b.barracudacentral.org',`"554 Rejected " $&{client_addr} " found in b.barracudacentral.org"')dnl Today some spam came in (passing all tests), and after checking MX Toolbox and DNSBL Information could see that adding one of several DNSBLs would have blocked this particular spam. So, I added another DNSBL, and now I want to test it by resubmitting this email to Sendmail, but therein lies the problem: it won't be coming from the right ip address, and the DNSBL won't see it as bad. Here's the command I normally would have used: formail -s /usr/sbin/sendmail -oi -t < testmail.mbox Before I try to reinvent a wheel, I thought I'd ask here first. Possible ideas: Is there a CLI option to sendmail for faking the source ip? Maybe craft a queued message file and put it in the queue directly? Perhaps set up another ip address on my machine, and send to myself with it? Would an OpenVPN or SSH tunnel be a quick fix? Possibly a shared library could be loaded to intercept system calls, à la LibFakeTime? Dtrace looks powerful, can it alter getsockopt(2) calls like this? Thanks!

How can I let apache server work always on my ec2 instance? Posted: 31 Jul 2021 04:52 PM PDT I am learning aws's EC2 server. I configured apache and php. I started the apache server with the command Sudo service httpd start But every time I stop my pc, or the next day, when I want to continue the course. I have to start again the apache server. I mean it is not on started status always. Imagine if I have a website running on that instance, it means that users won't be able to go on to my website. Or If want my website to be available every time, I don't have to logout from my aws account. Can you explain me what is the problem. I am using the right now the 12 months free offer on aws. It's an Amazon Linux 2 with a Linux 4.14 version

Packets from xfrm interface won't route, but opposite works Posted: 31 Jul 2021 07:05 PM PDT I'm working on a site-to-site vpn, where one end us a UDM and the other is Strongswan. The goal is to provide bi-directional routing into a cloud environment. I'm completely baffled why this isn't working. The good news is Strongswan connects and will pass traffic. But I have some routing issues on the Strongswan side. My Strongswan host has two interfaces, eth0 which has the public internet IP on eth0, and an internal ip of 10.132.169.74 on eth1 Lan network[s]: 10.87.0.0/24, 10.87.35.0/24, 10.87.235.0/24 Cloud network: 10.132.0.0/16 10.87.0.1 = UDM 10.132.169.74 = Strongswan eth1 and connects to the internal cloud network 10.132.0.0/16 10.87.0.33 = test host on the LAN network 10.132.40.82 = test host on the cloud network current situation: pinging from 10.87.0.33 (Lan test host) -> 10.132.169.74 (Strongswan) works pinging from 10.132.169.74 (Strongswan) -> 10.87.0.33 (Lan test host) works pinging from 10.132.40.82 (cloud test host) -> 10.87.0.33 (Lan test host) works pinging from 10.87.0.33 (Lan test host) -> 10.132.40.82 (cloud test host) Does not work, which is the most important thing outta all of this Here's the routing table of the Strongswan host 10.132.169.74: default via x.x.x.x dev eth0 proto static 10.17.0.0/16 dev eth0 proto kernel scope link src 10.17.0.21 10.19.49.0/24 dev wg0 proto kernel scope link src 10.19.49.1 10.87.0.0/16 dev ipsec0 scope link src 10.132.169.74 10.132.0.0/16 dev eth1 proto kernel scope link src 10.132.169.74 x.x.x.y/20 dev eth0 proto kernel scope link src x.x.x.z Here's the routing table on the cloud test host (10.132.40.82): default via x.x.x.x dev eth0 proto static 10.17.0.0/16 dev eth0 proto kernel scope link src 10.17.0.24 10.87.0.0/16 via 10.132.169.74 dev eth1 10.132.0.0/16 dev eth1 proto kernel scope link src 10.132.40.82 x.x.x.y/20 dev eth0 proto kernel scope link src x.x.x.z On the Strongswan host, I'm executing this: sudo ip link add ipsec0 type xfrm dev eth0 if_id 4242 sudo ip link set ipsec0 up sudo ip route add 10.87.0.0/16 dev ipsec0 src 10.132.169.74 And finally here's my swan config: sudo tee /etc/strongswan.d/charon-systemd.conf << "EOF" charon-systemd { load=pem pkcs1 x509 revocation constraints pubkey openssl random random nonce aes sha1 sha2 hmac pem pkcs1 x509 revocation curve25519 gmp curl kernel-netlink socket-default updown vici journal { default=0 # enc=1 # asn=1 } } EOF sudo tee /etc/swanctl/conf.d/xyz.conf << "EOF" connections { vpn-cloud-udm-lan { version=2 proposals=aes128gcm16-sha256-modp2048,aes128-sha256-modp2048 unique=replace encap=yes local { id=x.x.x.x auth=psk } remote { auth=psk } children { net { local_ts=10.132.0.0/16 remote_ts=10.87.0.0/16 esp_proposals=aes128gcm16-sha256-modp2048,aes128-sha256-modp2048 start_action=trap if_id_in=4242 if_id_out=4242 } } } } secrets { ike-1 { id-vpn-cloud=x.x.x.x secret="somesecret" } ike-2 { id-udm-lan=y.y.y.y secret="somesecret" } } EOF and my sysctl on the Strongswan host: net.ipv4.ip_forward=1 net.ipv4.conf.all.forwarding=1 net.ipv4.conf.all.send_redirects=0 net.ipv4.conf.default.send_redirects=0 sudo swanctl --list-sas shows active tunnels and when I ping I can see the counters go up. Furthermore, a tcpdump listening on the cloud test host shows no traffic arriving, but a tcpdump on the Strongswan host in the particular scenario DOES show the traffic, so it's be dropped there. Any help is appreciated, thank you!

Rebooted Ubuntu server, nginx site no longer accessible from browser Posted: 31 Jul 2021 04:03 PM PDT I rebooted my Ubuntu server this morning because I was having what appeared to be a low-memory error (happens occasionally, hasn't been enough of a problem to try and fix it). But now, my site (which was previously working fine) is no longer accessible from the browser. The setup: I'm running a NuxtJS site using pm2 to daemonize it, and nginx as a reverse proxy. I have a post-receive git hook so that I can push to my remote git repo, which then rebuilds the app and restarts the pm2 instance. I can only access my site from inside the server, inside a terminal window. Lynx, wget, and cURL all work, and even follow the 301 redirect to HTTPS. And they're working when I request the domain itself, not just the localhost:3000 that's getting reverse proxied. As in, curl https://my-domain.org works. If I try to curl/lynx/etc from any other terminal window, it just waits until it times out. Same with the browser – waits until it times out. Here are the things I've tried/looked at: I'm using UFW, so I checked to see if the firewall was the problem. But 80, 443, and 8080 are all set to ALLOW. I tried seeing if maybe nginx wasn't listening somehow, so I tried sudo lsof -i -P -n | grep LISTEN. Here's the output of that: nginx 2896 root 6u IPv4 668673557 0t0 TCP *:443 (LISTEN) nginx 2896 root 7u IPv4 668673558 0t0 TCP *:80 (LISTEN) nginx 2897 www-data 6u IPv4 668673557 0t0 TCP *:443 (LISTEN) nginx 2897 www-data 7u IPv4 668673558 0t0 TCP *:80 (LISTEN) nginx 2898 www-data 6u IPv4 668673557 0t0 TCP *:443 (LISTEN) nginx 2898 www-data 7u IPv4 668673558 0t0 TCP *:80 (LISTEN) I tried checking nginx's access.log. All my curl/wget/Lynx requests are showing up as normal, but none of the browser requests are appearing. I also took a look at the error.log, and got this: 2021/07/31 11:51:52 [emerg] 885#885: bind() to 0.0.0.0:443 failed (98: Address already in use) 2021/07/31 11:51:52 [emerg] 885#885: bind() to 0.0.0.0:80 failed (98: Address already in use) 2021/07/31 11:51:52 [emerg] 885#885: bind() to 0.0.0.0:443 failed (98: Address already in use) 2021/07/31 11:51:52 [emerg] 885#885: bind() to 0.0.0.0:80 failed (98: Address already in use) 2021/07/31 11:51:52 [emerg] 885#885: still could not bind() Thus far, I haven't found any solutions. I'm just baffled, because whatever changed, it changed because of a reboot. Any ideas are much appreciated. EDIT to add some output: sudo systemctl status nginx: ● nginx.service - A high performance web server and a reverse proxy server Loaded: loaded (/lib/systemd/system/nginx.service; enabled; vendor preset: enabled) Active: active (running) since Sat 2021-07-31 15:05:53 EDT; 27min ago Process: 6834 ExecStop=/sbin/start-stop-daemon --quiet --stop --retry QUIT/5 --pidfile /run/nginx.pid (code=exited, status Process: 6840 ExecStart=/usr/sbin/nginx -g daemon on; master_process on; (code=exited, status=0/SUCCESS) Process: 6837 ExecStartPre=/usr/sbin/nginx -t -q -g daemon on; master_process on; (code=exited, status=0/SUCCESS) Main PID: 6841 (nginx) CGroup: /system.slice/nginx.service ├─6841 nginx: master process /usr/sbin/nginx -g daemon on; master_process on ├─6842 nginx: worker process └─6843 nginx: worker process Jul 31 15:05:53 parrot systemd[1]: Starting A high performance web server and a reverse proxy server... Jul 31 15:05:53 parrot systemd[1]: Started A high performance web server and a reverse proxy server. Output of sudo nginx -T is long, so I made it a gist.

Any way to join a server to Active Directory Domain by IP address in hostfile and not DNS? Posted: 31 Jul 2021 08:01 PM PDT To my knowledge their is no way to join a server to AD without the server being able to resolve the AD domain via DNS. Joining requires being able to get multiple records from DNS - including SRV records. So a simple host file entry shouldn't work. With that in mind, my question is am I correct is there no other way to join a server to AD without access to a DNS server that hosts the AD records? The reason I ask is: I have some servers that are in AWS that need to join to an AD domain inside a corporate network. We have VPN tunnel from AWS back to the corporate network. This domain is not advertised on a public DNS server we can reach from AWS. We do have an internal corporate DNS server with the appropriate records. Now with some networking changes on the corporate side we could reach this DNS through our VPN tunnel; however, in AWS we use the AWS DNS service with a delegated zone to resolve server to server communication within AWS and it then reaches out to our corporate public DNS server for anything it can't resolve. We also use the AWS DNS server for health checks on AWS to trigger region failovers. If we were to point our AWS servers to our internal corporate DNS through the VPN tunnel, we would then no longer be able to resolve internally within AWS. I only see a couple of options. Find a way to join a server to AD without using DNS, which I don't think is possible for reasons I stated previously. But if anyone knows differently, please say so. Expose the AD DNS records on our external (public) DNS. Redesign our whole DNS design of cloud and corporate environments. This option will take time and maybe it will be the long term solution. But I also need a short term solution in the meantime. Options 1 & 2 are the only short term solutions I can think of and if 1 isn't possible like I think then that leaves me with only option 2. So do you agree option 1 isn't possible and/or do you have any other ideas that I haven't already listed. Thanks in advance

Tomcat 9 Service on Centos 7 won't start up at system boot Posted: 31 Jul 2021 10:03 PM PDT I'm not very experienced with Linux generally, so please forgive me if this is obvious to you. I have done a large number of searches on various combinations of keywords, but can't find a solution to this problem. I have installed command line only (core) CentOS 7 on a virtual machine. I have installed java and downloaded tomcat 9.0.14 (I know that's not the latest version). I have set up tomcat to run as a service, using systemd, that is, I have created a file: /etc/systemd/system/tomcat.service I had to revise this post about 15 times before the submission form stopped thinking this was spam, so I had to remove a lot of information I originally wanted to include like what's in my service.tomcat file. Sorry. I'd love to include more detail, but the form just won't let me. It is the only instance of Tomcat on the Linux server. It is installed in /opt/tomcat/apache-tomcat-9.0.14, but I have created a symbolic link named "latest" so it can be referenced as /opt/tomcat/latest. Tomcat starts properly when I run the startup script manually, i.e: cd /opt/tomcat/latest/bin sudo ./startup.sh When I do so, it properly responds and I can see the landing page using a browser on another computer. I can also start up tomcat as a service - manually starting that service using: sudo systemctl start tomcat If I do so, then I can see tomcat working and using: sudo systemctl status tomcat says that it's working. I can also visit the landing page when tomcat is started this way. So it will start as a service, if I manually start the service with the above command. The problem is that when I boot up the machine, tomcat doesn't start. I don't think that I've simply failed to reference the service, rather, I believe it might be failing (but I'm not 100% certain): If I reboot the computer, tomcat doesn't start. If I then go: sudo systemctl status tomcat -l I get this: tomcat.service - Apache Tomcat Web Application Container Loaded: loaded (/etc/systemd/system/tomcat.service; enabled; vendor preset: disabled) Active: failed (Result: exit-code) since Wed 2020-05-27 14:47:27 AEST; 3min 14s ago Process: 1147 ExecStart=/opt/tomcat/latest/bin/startup.sh (code=exited, status=0/SUCCESS) Main PID: 1168 (code=exited, status=1/FAILURE) May 27 14:47:26 CentOS-7-NoGUI-Virgin systemd[1]: Starting Apache Tomcat Web Application Container... May 27 14:47:27 CentOS-7-NoGUI-Virgin systemd[1]: Started Apache Tomcat Web Application Container. May 27 14:47:27 CentOS-7-NoGUI-Virgin systemd[1]: tomcat.service: main process exited, code=exited, status=1/FAILURE May 27 14:47:27 CentOS-7-NoGUI-Virgin systemd[1]: Unit tomcat.service entered failed state. May 27 14:47:27 CentOS-7-NoGUI-Virgin systemd[1]: tomcat.service failed. Based on the fact that it says tomcat failed, I don't think that the problem is that I've not told CentOS to start the service. Rather, I think it's trying to start the service, but it's not working. If I then manually do: sudo systemctl start tomcat I get no text returned, but subsequently if I do this: sudo systemctl status tomcat -l it shows tomcat is running. I'd love to include the exact output of the command, but I had to remove that too in my many revisions of this to get the submission form not to think this was spam. I believe I have properly set up the tomcat service to start at system boot time, by doing this: sudo systemctl daemon-reload sudo systemctl enable tomcat I tried using journalctl -xe to see if I could learn why the service was failing to start at boot time, but I couldn't find anything in the results of that command that explained why this was occuring. I'm happy to provide the (very long) output from that, if that's helpful. The tomcat.service file contains the following: (I had to remove the contents of this file, even though it was marked as code, because the submission form insisted my post looked like spam. Sigh) I have set the following in my user's home folder, in the .bashrc file: export CATALINA_HOME=/opt/tomcat/latest I wonder if perhaps when the service is starting up at boot time, if it's not running as my user, then perhaps it dosn't have access to this variable somehow ? The actual startup script (referenced in the tomcat.service file is /opt/tomcat/latest/bin/startup.sh . It contains the default contents, I have not modified it in any way. Again, the above script runs and will start tomcat. I can even start tomcat as a service by manually typing sudo systemctl start tomcat. It just won't start at boot time. I have done a sudo yum check-update, followed by sudo yum install update. I did this to update CentOS 7 to the latest version. I did this after I installed tomcat, as part of the troubleshooting process I have been through. This didn't seem to help. I would be most grateful if anyone can suggest a solution, or a troubleshooting step I should try next. For example, I'm not sure how to examine the startup process on a linux box specifically to look for services failing to start up and why. Kind regards, Spencer.

How can I install the ngx_http_geoip2_module module ? on Centos Posted: 31 Jul 2021 04:08 PM PDT I installed the GeoIP package using yum. I got the geoIP files in the /usr/share/GeoIP/ folder. I need to add some rules on some countries in the: /etc/nginx/nginx.conf and to do that i need to load the module, like: load_module modules/ngx_http_geoip2_module.so;, to recognize the variables. See: geoip_country /usr/share/GeoIP/GeoIP.dat; So how can i install this module? I followed this tutorial: https://github.com/leev/ngx_http_geoip2_module/blob/master/README.md#installing and the url is outdated or invalid, not sure, but i cannot download it . Also I already have nginx being installed. Any suggestions? thnx in advance!

Unable to NAT TFTP traffic because iptables is not forwarding the return connection to the client despite TFTP helper creating an expectation Posted: 31 Jul 2021 09:04 PM PDT The Problem I have a TFTP server (Machine 'S') and a TFTP client (Machine 'C') on different subnets. They are connected via a router( Machine 'R'). All 3 machines are Debian 9/Stretch. The router is running iptables and is set to masquerade connections from the client's network to the server's network. I have configured iptables to use the Netfilter TFTP helper for tftp connections going to the TFTP server. The trouble I'm having is that the TFTP helper sets up an expectation for the return tftp connection (as expected) but, despite this, only traffic from port 69 on the TFTP server is getting translated and sent back to the client. So only the regular MASQUERADE connection tracking is being used even though the conntrack table shows the expected return connection. According to RFC1350, the server is supposed to choose a random source port for its communication and direct it to the port that the client used as a source port originally (whew...). The result is the that the router NATs the connection from the client to the server, sets up a translation rule for the return connection and happily waits for a return packet from the server with source port=69 that never arrives. The Setup Addresses are made up for clarity: TFTP Server(S): 1.1.1.1 TFTP Client(C): 2.2.2.1 Router(R): 1.1.1.2 / 2.2.2.2 Iptables on the router has the following rules. All tables have default ACCEPT policy: ======== RAW Table ======== Chain PREROUTING (policy ACCEPT 464K packets, 432M bytes) pkts bytes target prot opt in out source destination 59 2504 CT udp -- * * 0.0.0.0/0 0.0.0.0/0 udp dpt:69 CT helper tftp Chain OUTPUT (policy ACCEPT 280K packets, 36M bytes) pkts bytes target prot opt in out source destination ======== NAT Table ======== Chain POSTROUTING (policy ACCEPT 398 packets, 40794 bytes) pkts bytes target prot opt in out source destination 5678 349K MASQUERADE all -- * enp1s0 0.0.0.0/0 0.0.0.0/0 Once the TFTP client is trying to connect, conntrack -L shows the following: udp 17 28 src=2.2.2.1 dst=1.1.1.1 sport=45084 dport=69 [UNREPLIED] src=1.1.1.1 dst=1.1.1.2 sport=69 dport=45084 mark=0 helper=tftp use=1 conntrack -L EXPECT: 298 proto=17 src=1.1.1.1 dst=1.1.1.2 sport=0 dport=45084 mask-src=255.255.255.255 mask-dst=255.255.255.255 sport=0 dport=65535 master-src=2.2.2.1 master-dst=1.1.1.1 sport=45084 dport=69 class=0 helper=tftp As you can see, the TFTP helper rule is working properly and is triggered once the client attempts its connection. As you can also see, the expectation created in the EXPECT table has source port 0, which I assume means "any port". But, as you'll see, the connection is only routed back to the client if the source port from the server is port 69 (regular old NAT)! Why is this? This is not the correct behaviour as far as I can tell. I won't clutter this post anymore if I can avoid it, but what's shown by tcpdump udp and host 1.1.1.1 confirms exactly what iptables and conntrack are showing me. I did this same setup on several Debian 8/Jessie setups about a year ago and the TFTP helper worked as expected and I never had any issues. Can anyone hlep me figure out if I'm doing something wrong? Is the issue with the TFTP helper? Why would its behaviour have changed from Debian 8/Jessie?

Migrating Jenkins jobs from one server to another Posted: 31 Jul 2021 04:08 PM PDT Copied Jenkins "jobs" directory from one A (VB) to server B (AWS). The jobs directory shows up in the server B with all the files in it. But those jobs doesn't populate in Jenkins UI. Please help. Thank you!

SSH Suddenly returning Invalid format Posted: 31 Jul 2021 05:07 PM PDT So a while ago I set up a server on AWS, and used their generated SSH key. I saved the key to Lastpass, and have successfully retrieved it from there before, and got it working. However, after trying that again today, I can't get it to work. -rw------- 1 itsgreg users 1674 Jun 6 12:51 key_name I've tried ssh -i key_name, ssh-keygen -f key_name, but nothing works, I always get this error message: Load key "key_name": invalid format Is there any way to fix this?

Adding Tag (i.e. Source IP) to rsyslog for sending to rsyslog remote server Posted: 31 Jul 2021 07:01 PM PDT Is there any way to adding a Tag to Logs which sent by rsyslog? I send these logs to another server, and I can detect source IP as destination, but I need to adding tag in source.

Nginx configuration for Deluge Posted: 31 Jul 2021 08:01 PM PDT I have Nginx running on a CentOS server where i installed Deluge and configured a server block for him. In my browser, mydomain.com redirects to Deluge webUI but www.mydomain.com redirects to a web page of the hoster. In my dns, i have an entry for "www" and "mydomain" to the server ip. Here's the Deluge server block in /etc/nginx/conf.d/vhosts.conf : server { listen 80; server_name mydomain.com www.mydomain.com; location / { proxy_pass http://www.localhost:8112; proxy_set_header X-Deluge-Base "/"; } } Have you any idea ? :)

How to show a 404 instead of a 403 on apache host Posted: 31 Jul 2021 05:09 PM PDT My site was hacked about 2 months ago so I closed the site down but there are over 1,000 spam links out there still directing to the domain. As there are no files on the domain visitors, including Google, receive a 403 error, so assume the page exists. How can I change the 403 to a 404? I have a 404.html file and have tried all the different rewrite, error document variations for the htaccess file I've been able to find on this site and others and nothing seems to work. For example: Options -Indexes RewriteCond %{HTTP_HOST} ^(www\.)?fineartdecor.com [NC] RewriteRule ^(.*)/$ - [R=404,NC] ------------------------------ ErrorDocument 404 /index.html?status=404 ------------------------------- RewriteEngine on RewriteCond %{REQUEST_FILENAME} !-f RewriteCond %{REQUEST_FILENAME} !-d RewriteRule ^(.*)$ / [L,QSA] -------------------------------- Further suggestions would be gratefully received.

dmidecode weird total/data width Posted: 31 Jul 2021 07:01 PM PDT I am getting strange outputs from my workstation, which has ECC RAM. Supposedly, from what I read, the data width should be at 64 bits and the total width at 72. But... data width shows as 64 and total width as 128. Is this a problem with my configuration? For reference, my motherboard is a MSI C236A WORKSTATION. Handle 0x0042, DMI type 17, 40 bytes Memory Device Array Handle: 0x0041 Error Information Handle: Not Provided Total Width: 128 bits Data Width: 64 bits Size: 8192 MB Form Factor: DIMM Set: None Locator: ChannelA-DIMM0 Bank Locator: BANK 0 Type: DDR4 Type Detail: Synchronous Speed: 2133 MHz Manufacturer: Micron Serial Number: 18221400 Asset Tag: 9876543210 Part Number: 18ASF1G72AZ-2G1B1 Rank: 2 Configured Clock Speed: 2133 MHz Minimum Voltage: Unknown Maximum Voltage: Unknown Configured Voltage: 1.2 V Thanks, Eduardo

nginx 405's with try_files for a DELETE request instead of proxying Posted: 31 Jul 2021 05:09 PM PDT I have nginx proxying to php-fpm with the following config: location / { try_files $uri $uri/ /index.php?$args; } location ~ \.php$ { fastcgi_pass 127.0.0.1:9000; fastcgi_index index.php; fastcgi_param SCRIPT_FILENAME /vol/app/www/$fastcgi_script_name; include fastcgi_params; } ``` Everything is working great until a DELETE request comes in like: DELETE /?file&path=foo When this happens nginx returns a 405 (method not allowed) and doesn't appear to proxy the request to php-fpm. What's the best way to get DELETE/PUT requests to proxy? Is there way to bypass try_files for this type of request? When hitting this URL, I see nothing in the error.log but access.log shows: 68.50.105.169 - - [20/Mar/2016:17:48:57 +0000] "DELETE /?file=client_img1.png&fileupload=e35485990e HTTP/1.1" 405 574 "http://ec2-foo.compute.amazonaws.com/jobs/" "Mozilla/5.0 (Macintosh; Intel Mac OS X 10_11_3) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/48.0.2564.116 Safari/537.36" "-" I've confirmed that I'm not hitting the proxy. My assumption is that nginx is blocking DELETE on the first "try" of try_files

Redirect IP to a domain name using htaccess Posted: 31 Jul 2021 03:03 PM PDT Let's say I have this IP Address 11.12.13.14 and the domain example.com. Now what I want is to redirect the user from IP Address to domain name (but without changing the domain name to address bar). So when the user requests 11.12.13.14/test it should open exapmle.com/test but not to redirect to domain name, in the address bar it should still remain 11.12.13.14/test. I have seen this question Redirect to other domain but keep typed domain. I don't know if it works because I haven't tested it, but I suppose it does. I am using Ubuntu 14.04 with Apache, so is there any wat to achieve this? Here is what I have tried Options +FollowSymLinks -MultiViews RewriteEngine On RewriteBase / RewriteCond %{HTTP_HOST} ^ 11.12.13.14$ [NC] RewriteRule ^ http://www.example.com%{REQUEST_URI} [L,NE,P]

"A new version of /boot/grub/menu.lst is available" when upgrading Ubuntu on an AWS server Posted: 31 Jul 2021 04:25 PM PDT I just tried to do a sudo do_release_upgrade on an AWS EC2 Ubuntu 13.10 server to upgrade to 14.04. All was going well until I got the following message: A new version of /boot/grub/menu.lst is available, but the version installed currently has been locally modified. What would you like to do about menu.lst? * install the package maintainer's version * keep the local version currently installed * show the differences between the versions * show a side-by-side difference between the versions * show a 3-way difference between available versions * do a 3-way merge between available versions (experimental) * start a new shell to examine the situation <Ok> I certainly haven't modified menu.lst, so I assume the local modifications are Amazon's doing. I'm going to hit the "keep the local version currently installed" option and hope for the best. But why am I getting this message, and is this the correct way to handle it?

Raid 1 can not sync with new Drive . its stopping at 30 % Posted: 31 Jul 2021 03:03 PM PDT i had trying to add new HDD in place of Falty HDD. but new HDD can not sync with old one .sync process shown up to 30 % after that its stopped . cat /proc/mdstat Personalities : [raid1] md2 : active raid1 sda3[0] sdb3[2](S) 1458319504 blocks super 1.0 [2/1] [U_] md1 : active raid1 sda2[3] sdb2[2] 524276 blocks super 1.0 [2/2] [UU] md0 : active raid1 sda1[0] sdb1[2] 6291444 blocks super 1.0 [2/2] [UU] md0 and md1 sync successfully , but md2 can not this is detail mdadm --detail /dev/md2 /dev/md2: Version : 1.0 Creation Time : Fri May 24 11:22:21 2013 Raid Level : raid1 Array Size : 1458319504 (1390.76 GiB 1493.32 GB) Used Dev Size : 1458319504 (1390.76 GiB 1493.32 GB) Raid Devices : 2 Total Devices : 2 Persistence : Superblock is persistent Update Time : Mon Aug 4 22:08:23 2014 State : clean, degraded Active Devices : 1 Working Devices : 2 Failed Devices : 0 Spare Devices : 1 Name : rescue:2 (local to host rescue) UUID : 96b46a6c:f520938c:f94879df:27851e8a Events : 616 Number Major Minor RaidDevice State 0 8 3 0 active sync /dev/sda3 1 0 0 1 removed 2 8 19 - spare /dev/sdb3 is that any solution . i want to backup my data

Bitnami redmine error SVN Posted: 31 Jul 2021 09:04 PM PDT I'm installing the Bitnami Redmine stack (redmine + subversion). Firstly I install configure and test it locally (Ubuntu 14.04 LTS). And everything is OK. I install Bitnami stack on server (Red Hat 4.4.7-4) and configure SVN. I commit files into SVN and connect project into Redmine with SVN repository, but when I try see it Rredmine displays 404 error. In the Redmine log file I see the following errors: Started GET "/redmine/projects/web-user-panel/repository" for 127.0.0.1 at 2014-04-24 11:34:20 +0300 Processing by RepositoriesController#show as HTML Parameters: {"id"=>"web-user-panel"} Current user: user (id=13) Error parsing svn output: #<REXML::ParseException: No close tag for /lists/list> /var/www/html/redmine/ruby/lib/ruby/1.9.1/rexml/parsers/treeparser.rb:28:in `parse' /var/www/html/redmine/ruby/lib/ruby/1.9.1/rexml/document.rb:245:in `build' /var/www/html/redmine/ruby/lib/ruby/1.9.1/rexml/document.rb:43:in `initialize' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/xml_mini/rexml.rb:30:in `new' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/xml_mini/rexml.rb:30:in `parse' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/xml_mini.rb:80:in `parse' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/abstract_adapter.rb:313:in `parse_xml' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/subversion_adapter.rb:106:in `block in entries' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/abstract_adapter.rb:258:in `call' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/abstract_adapter.rb:258:in `block in shellout' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/abstract_adapter.rb:255:in `popen' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/abstract_adapter.rb:255:in `shellout' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/abstract_adapter.rb:212:in `shellout' /var/www/html/redmine/apps/redmine/htdocs/lib/redmine/scm/adapters/subversion_adapter.rb:100:in `entries' /var/www/html/redmine/apps/redmine/htdocs/app/models/repository.rb:198:in `scm_entries' /var/www/html/redmine/apps/redmine/htdocs/app/models/repository.rb:203:in `entries' /var/www/html/redmine/apps/redmine/htdocs/app/controllers/repositories_controller.rb:116:in `show' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal/implicit_render.rb:4:in `send_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/abstract_controller/base.rb:167:in `process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal/rendering.rb:10:in `process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/abstract_controller/callbacks.rb:18:in `block in process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:491:in `_run__2883861927089110970__process_action__2542827355008294621__callbacks' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:405:in `__run_callback' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:385:in `_run_process_action_callbacks' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:81:in `run_callbacks' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/abstract_controller/callbacks.rb:17:in `process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal/rescue.rb:29:in `process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal/instrumentation.rb:30:in `block in process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/notifications.rb:123:in `block in instrument' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/notifications/instrumenter.rb:20:in `instrument' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/notifications.rb:123:in `instrument' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal/instrumentation.rb:29:in `process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal/params_wrapper.rb:207:in `process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activerecord-3.2.17/lib/active_record/railties/controller_runtime.rb:18:in `process_action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/abstract_controller/base.rb:121:in `process' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/abstract_controller/rendering.rb:45:in `process' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal.rb:203:in `dispatch' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal/rack_delegation.rb:14:in `dispatch' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_controller/metal.rb:246:in `block in action' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/routing/route_set.rb:73:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/routing/route_set.rb:73:in `dispatch' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/routing/route_set.rb:36:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/journey-1.0.4/lib/journey/router.rb:68:in `block in call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/journey-1.0.4/lib/journey/router.rb:56:in `each' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/journey-1.0.4/lib/journey/router.rb:56:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/routing/route_set.rb:608:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/rack-openid-1.3.1/lib/rack/openid.rb:98:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/best_standards_support.rb:17:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/rack-1.4.5/lib/rack/etag.rb:23:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/rack-1.4.5/lib/rack/conditionalget.rb:25:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/head.rb:14:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/params_parser.rb:21:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/flash.rb:242:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/rack-1.4.5/lib/rack/session/abstract/id.rb:210:in `context' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/rack-1.4.5/lib/rack/session/abstract/id.rb:205:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/cookies.rb:341:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activerecord-3.2.17/lib/active_record/query_cache.rb:64:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activerecord-3.2.17/lib/active_record/connection_adapters/abstract/connection_pool.rb:479:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/callbacks.rb:28:in `block in call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:405:in `_run__1805290955544829105__call__1486932417638469082__callbacks' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:405:in `__run_callback' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:385:in `_run_call_callbacks' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/callbacks.rb:81:in `run_callbacks' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/callbacks.rb:27:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/remote_ip.rb:31:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/debug_exceptions.rb:16:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/show_exceptions.rb:56:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/railties-3.2.17/lib/rails/rack/logger.rb:32:in `call_app' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/railties-3.2.17/lib/rails/rack/logger.rb:16:in `block in call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/activesupport-3.2.17/lib/active_support/tagged_logging.rb:22:in `tagged' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/railties-3.2.17/lib/rails/rack/logger.rb:16:in `call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/actionpack-3.2.17/lib/action_dispatch/middleware/request_id.rb:22:in `call' 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`block in call' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/rack-1.4.5/lib/rack/urlmap.rb:49:in `each' /var/www/html/redmine/apps/redmine/htdocs/vendor/bundle/ruby/1.9.1/gems/rack-1.4.5/lib/rack/urlmap.rb:49:in `call' /var/www/html/redmine/ruby/lib/ruby/gems/1.9.1/gems/passenger-4.0.40/lib/phusion_passenger/rack/thread_handler_extension.rb:74:in `process_request' /var/www/html/redmine/ruby/lib/ruby/gems/1.9.1/gems/passenger-4.0.40/lib/phusion_passenger/request_handler/thread_handler.rb:141:in `accept_and_process_next_request' /var/www/html/redmine/ruby/lib/ruby/gems/1.9.1/gems/passenger-4.0.40/lib/phusion_passenger/request_handler/thread_handler.rb:109:in `main_loop' /var/www/html/redmine/ruby/lib/ruby/gems/1.9.1/gems/passenger-4.0.40/lib/phusion_passenger/request_handler.rb:448:in `block (3 levels) in start_threads' ... No close tag for /lists/list Line: 4 Position: 93 Last 80 unconsumed characters: Output was: <?xml version="1.0" encoding="UTF-8"?> <lists> <list path="svn://127.0.0.1/voxysuser"> Rendered common/error.html.erb within layouts/base (0.1ms) Completed 404 Not Found in 69.1ms (Views: 15.1ms | ActiveRecord: 3.0ms) How can I resolve this problem? I googled it, but similar problem fixed should be fixed 3 years ago. I'm installing the latest Bitnami Redmine 2.5.1-1 stack. UPDATE Well, I found next way. If I use the http protocol it works fine, but I should remove access for svn by web. That's why I create virtual host on localhost and get info from svn use 127.0.0.1 IP. <VirtualHost 127.0.0.1:8000> <Location /repo> DAV svn SVNPath "PATH_TO_MY_REPOSITORY" </Location> And this it work good.

zimbra Error when installing Posted: 31 Jul 2021 10:03 PM PDT Can you guys help? I have set up an ubuntu server 12.04 for Zimbra, I download zcs 8.0.2 and put it in /opt/zimbra. but when I run install.sh, even though prerequisites ARE Found and I Agree to the License and select packages to intall, the folder is removed and I get the following error: ./install.sh line 339: /opt/zimbra/libexec/zmsetup.pl: No such file or directory And when I go back to looking for that .pl I don't find anything, everyhting's wiped out ! Any thoughts? Thanks

Permanently assigning IP address for an embedded device Posted: 31 Jul 2021 09:05 PM PDT This is a follow-up to Embedded device configured with bad IP address, can I still connect? We make embedded devices that run Linux. Users can change the networking configuration of the device (static IP, DHCP client and server). Zeroconf was supposed to be the fallback when a user forgets static IP was assigned, but Zeroconf seems spotty in implementation. Connecting a Windows client frequently results in the client getting a link-local address that cannot communicate with the device. There is no hardware reset button, sadly. I know what MAC address each device has, but I don't know how to use that information because the device's networking stack rejects data unless I know its IP address. Would it be bad to statically assign a secondary IP address in the link-local range (169.254.0.0/16) to eth0:0? That way I can write a restore utility that will work when the device is directly connected to a client. (No routers involved, but possibly a switch) What happens if two of our devices are on the same network with the same link-local IP address? They will have different primary IP addresses. Some similar products hard code a private IP (i.e. 192.168.1.2) for this particular situation.

E: Sub-process /usr/bin/dpkg returned an error code (100) Posted: 31 Jul 2021 05:41 PM PDT I am running on xen, Debian 5.0-i386-default. I haven't touched my vps in 2 months then last night I ran the following command: myserver:/usr/bin# apt-get upgrade Reading package lists... Done Building dependency tree Reading state information... Done The following packages have been kept back: makepasswd The following packages will be upgraded: libc6 libc6-dev libc6-xen libmysqlclient15off locales mysql-client mysql-client-5.0 mysql- common mysql-server mysql-server-5.0 10 upgraded, 0 newly installed, 0 to remove and 1 not upgraded. Need to get 0B/50.1MB of archives. After this operation, 483kB of additional disk space will be used. Do you want to continue [Y/n]? y Preconfiguring packages ... E: Sub-process /usr/bin/dpkg returned an error code (100) I googled and it seems to be a permission thing for "dpkg". However, I cd into /usr/bin and there's no dpkg binary!!! Please help thanks

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GOTC 2021 深圳站首日圆满落幕 Posted: 31 Jul 2021 05:24 PM PDT 7 月 31 日，由开放原子开源基金会与 Linux 基金会亚太区联合开源中国共同举办的"全球开源技术峰会 GOTC 2021 深圳站"在深圳会展中心拉开帷幕。本次峰会汇聚国内顶级开源厂商和开源社区成员，吸引了来自全国各地的数千名开源爱好者参会，共话中国开源生态的发展趋势。 深圳市工业和信息化局局领导李郑祥为本次峰会致开...

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CNCF 宣布 Grafana 实验室升级为白金会员 Posted: 31 Jul 2021 05:35 PM PDT 可观察性方面的领导者表明了在云原生工作负载的前沿解决方案上的合作承诺 CNCF（云原生计算基金会）宣布，Grafana 实验室的会员资格从银牌会员升级到白金会员。随着承诺的扩大，该公司还将加入 CNCF 理事会。 Grafana Labs 于 2017 年作为银牌会员加入 CNCF，并一直是 CNCF 项目的大量贡献者，最著名的是 Prometheus 和 ...

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Recent Questions - Mathematics Stack Exchange

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• Understanding the proof that $A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}1&0\\0&1\end{pmatrix}}A_1\times A_2$ is an isomorphism
• Find a simpler description for each of the following rings.
• Connections of estimator and least-square fit?
• For which prime $p$, is the group units $R^*$ cyclic?
• Norm of linear map $f:R^n \rightarrow R^m$ from its matrix
• Determine the order of operator norm of Gaussian random matrix raised to the 4th power
• Attempt at a 2nd Collatz Conjecture
• Difference between weak aand distributional derivatives
• Mandelbrot set; are these trajectories chaotic?
• Solve $Ax=b$ subject to $x=Cy+d$
• Let R be a PID. Then, every nonempty set of ideals of R has a maximal element
• Find closed form of $a_{n+1} = 4a_n - 2$
• Proof of Euclidian's algorithm for non-coprime numbers
• Limit of the n-root of this sequence.
• Sequence and multiplicity of sums of integers squared $n_1^2+n_2^2+\dots+n_d^2$?
• Validity of proof $\zeta(s) \neq 0$ for $\sigma >1$.
• Homotopy groups of projective Lie groups PO(N), PSO(N), and PSpin(N)
• Can $\pi$ be defined in a p-adic context?
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• $(1+A)^n \ge (1+a_1)\cdot(1+a_2)\cdots(1+a_n) \ge (1+G)^n$
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Understanding the proof that $A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}1&0\\0&1\end{pmatrix}}A_1\times A_2$ is an isomorphism Posted: 31 Jul 2021 07:44 PM PDT Fix an abelian category. Let $A_1$ and $A_2$ be objects and let $A_1\times A_2$ be their product and $A_1+A_2$ their coproduct (i.e., sum). I am trying to understand the proof of the following theorem. Theorem 2.35 for abelian categories $A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}1&0\\0&1\end{pmatrix}}A_1\times A_2$ is an isomorphism. Proof: Let $K\to A_1+A_2$ be the kernel of $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. Then $K\longrightarrow A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}1&0\\0&1\end{pmatrix}}A_1\times A_2\mathop{\longrightarrow}\limits^{p_2}A_2\ =\ K\longrightarrow A_1+A_2\mathop{\longrightarrow}\limits^{\begin{pmatrix}0\\1\end{pmatrix}} A_2$ and $K\longrightarrow A_1+A_2$ is contained in $A_1\mathop{\longrightarrow}\limits^{u_1}A_1+A_2$. Similarly it is contained in $A_2\mathop{\longrightarrow}\limits^{u_2}A_1+A_2$, and hence it is contained in their intersection, which is zero. Thus $K=O$ and $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ is monomorphic. Dually it is epimorphic and hence an isomorphism. I don't understand the second sentence of the proof. Why are $K\longrightarrow A_1+A_2\longrightarrow A_1\times A_2\longrightarrow A_2$ and $K\longrightarrow A_1+A_2 \longrightarrow A_2$ the same, and why is $K\longrightarrow A_1+A_2$ is contained in $A_1\longrightarrow A_1+A_2$?

Find a simpler description for each of the following rings. Posted: 31 Jul 2021 07:44 PM PDT Find a simpler description for each of the following rings: a) $\mathbb{Q}[x]/(x^5+x^3)$ b) $\mathbb{Z}[x]/(x-2,x^2+3x)$ Solution of (a): $$ \dfrac{\mathbb{Q}[x]}{(x^5+x^3)} \cong \dfrac{\mathbb{Q}[x]}{(x^3(x^2+1))} \cong \dfrac{\mathbb{Q}[x]}{(x^3,x^2+1)} \cong \dfrac{\mathbb{Q}[x]/(x^2+1)}{(x^3,x^2+1)/(x^2+1)} \cong \dfrac{\mathbb{Q}[i]}{(x^3)} $$ Is this correct? Solution of (b): $$ \dfrac{\mathbb{Z}[x]}{(x-2,x^2+3x)} \cong \dfrac{\mathbb{Z}[x]}{(x-2, x, x+3)} \cong \dfrac{\mathbb{Z}[x]/(x)}{(x-2, x, x+3)/(x)} \cong \dfrac{\mathbb{Z}}{(x-2, x+3)} $$ Note that $x-2, x+3$ are pairwise relatively prime in $\mathbb{Z}[x]$. By Chinese Remainder Theorem, $$ \dfrac{\mathbb{Z}}{(x-2, x+3)} \cong \dfrac{\mathbb{Z}}{(x-2)} \times \dfrac{\mathbb{Z}}{(x+3)} $$ Can I say the last line $\cong \mathbb{Z}_2 \times \mathbb{Z}_3$?

Connections of estimator and least-square fit? Posted: 31 Jul 2021 07:42 PM PDT I don't have too much background in statistics, but I wonder if there's any connection between an estimator and a least-square fit. Can I understand the least-square fit as an option of estimator that returns a prediction of unknown results for us? Suppose I want to give a Gaussian fit to my data $\{(x_i,y_i)\}$, and there're 4 variables: $$ G(x) = B+A\exp\left[-\left(\frac{x-\mu}{\sigma}\right)^2\right] $$ Is this an estimator of the trend/distribution of my data set? If I'm particularly interested in determining the Gaussian center $\mu$ of my data, is there a way I can find an estimator of that?

For which prime $p$, is the group units $R^*$ cyclic? Posted: 31 Jul 2021 07:40 PM PDT I'm studying for my qualifying exam and found the following question in the question bank of previous years paper: Let $p$ be a prime and $\mathbb{F}_p$ the fields of $p$ elements. Set $R=\mathbb{F}_p[x]/(x^3)$. For which prime $p$, is the group of units $R^*$ cyclic? What I've got so far is that: $R$ is local ring and any element in $R$ with non-zero constant term will be a unit. I have proven it for $p=2$, but I don't know if it will be true for any other primes. Also, I have proven that if $r=a\overline{x}^2+b\overline{x}+c$ is the generator of the cyclic group $R^*$ then $b\not=0$. I tried to look at the powers of $r$ hoping that I might be able find some element in $R^*$ which can't be expressed as $r^n$, for some $n$, but it got complicated and now I'm completely lost.

Norm of linear map $f:R^n \rightarrow R^m$ from its matrix Posted: 31 Jul 2021 07:47 PM PDT Assume $f:R^n \rightarrow R^m$ is a linear map. Therefore, $f$ can be represented as a matrix. Namely, for any $x=(x_1,...,x_n)\in R^n$ \begin{align} f(x)= (x_1,...,x_n) \begin{pmatrix} a_{11} & ... & a_{1m} \\ ... & ... & ... \\ a_{n1} & ... & a_{nm} \end{pmatrix} =xA \end{align} I think the matrix $A$ contains all imformation of $f$. Define the norm of $f$ as $$ \|f\| = \sup_{x\in R^n} \frac{|f(x)|}{|x|} $$ where $|x|$ is the Euclidean norm. Therefore, there should be a way to calculate the $\|f\|$ from $A$. But I can't find what of $A$ equals to $\|f\|$. For example, the eigenvalue of $A$ is not $\|f\|$ (in fact, $A$ may not has eigenvalue, since may $n\not = m$). PS: I have a little algebra knowledge. Only a little linear algebra I know.

Determine the order of operator norm of Gaussian random matrix raised to the 4th power Posted: 31 Jul 2021 07:29 PM PDT Let $A\in \mathbb{R}^{n\times n}$ be a random matrix whose entries are i.i.d. $\mathcal{N}(0,1)$. I know that $\mathbb{E}\|A\|\lesssim \sqrt{n}$, where $\|\cdot\|$ is the operator norm. But I have no idea how to show that $\mathbb{E}\|A\|^4\lesssim n^2$. Jensen's inequality does not work here. May I have some hint/reference on this problem? Thank you!

Attempt at a 2nd Collatz Conjecture Posted: 31 Jul 2021 07:09 PM PDT I am watching fascinating videos about number theory for a while now, and I put together my own little mindworld about how prime numbers and Collatz is working. Please correct me if I'm wrong, all I have is high school maths from 15 years ago to help me. So 1) Prime numbers. As far as I understood, mathematicians think they are random, but my understanding of random is different, so what they mean by random is uncalculable/unpredicatable. Because if I look at the primes what they are? They are the empty spaces left by a multiplication table. Basically 13 is a prime, because none of the previous positive integers any variation of multiplication can results those numbers. So basically if you do a multiplication table with 2x2, 2x3, etc. with all integers, the ones missing from the results will be the primes, am I correct? 1 2 3 4 5 2 4 6 8 10 3 6 9 12 15 4 8 12 16 20 5 10 15 20 25 The numbers missing from this table are: 1, 2, 3, 5, 7, 11, 13.. basically the primes, and if do a bigger table, more primes will reveal itself. Obviously this table only capable of telling primes until the prime 5 relaibly, but this is how my thinking is. So What is really Collatz Conjecture? It basically looking for a number which is the power of 2, (or every digit is 0 in binary except the first). If it finds one, it will go towards the bottom. If it's not, it will go toward the botton for how many 0 digits there are (by deleting them) at the end of the binary value, then change the value by multiplying with 3 and adding 1 to get another number which will have 0 at the end. It keeps repeating this loop, until every digit is 0, except the first one, then it can reach the loop. It means that Collatz Conjecture is a proof that every number is the sum of multiple powers of 2. for example: 11 is 1*(2^3) + 0*(2^2) + 1*(2^1) + 1*(2^0). These are basically how we write numbers in binary: 1011. Collatz is looking for a number which is the power of 2, and if it doesn't find it, it moves to another number by multiplying 3 and adding 1. Sooner or later you will get to a power of 2 and have to the loop. So that means that other conjectures could be built, with other base system. So I took base 3, and did the same thing. The problem is, now we have 3 variant in base 3. A number ending in 0, 1 or 2. Obviously we are looking for a number which is ending only 0's, except for the first digit (1000, 2000, 10000, 20000, etc), so my rule is If the single digit sum of all digits of N are 1, 4 or 7, then N*2+1 If the single digit sum of all digits of N are 2, 5 or 8, then N*2-1 If the single digit sum of all digits of N are 3, 6 or 9, then N/3 This -I believe- will always result in the loop 3-1-3-1 My question is, do I understand these math problems right? Does anyone did another Colletz Conjecture in another base besides base 2? Sorry for my english, it is only my second language. Update: And how that relates to prime numbers? I had a train of thought about it, but it's almost 4 am now.

Difference between weak aand distributional derivatives Posted: 31 Jul 2021 07:42 PM PDT I'm studing weak and distributional derivatives and solutions and I have a few questions about it. From my understanding, one defines a weak derivative of $u \in L^{1}_{loc}(\Omega)$ such that $$ \int_{\Omega} D^{\alpha} u \varphi dx = (-1)^{|a|}\int_{\Omega} v D^{\alpha}\varphi dx$$ for all test functions $\varphi \in C^{\infty}_{c}$ and $v \in L^{1}_{loc}(\Omega)$. So $v$ is a weak derivative of u. My question is: what is the difference between this definition and the definition of distributional derivatives? From what I understood, the integral of ($\int_{\Omega} u \varphi dx$) defines a distribution so the distributional derivative is defined as seen in equation above. I understand that while working with weak derivatives, I'm differentiating functions and when one is working with distributional derivatives it's obviously differentiating distributions. But what is the difference between these two definitions? Is it necessary, for the definition of a weak derivative, that the function $u \in L^{1}_{loc}$? Thanks.

Mandelbrot set; are these trajectories chaotic? Posted: 31 Jul 2021 07:02 PM PDT I am using the Mandelbrot set, with an exponent of 2 so that the iterative equation is z = z^2 + c. The escape threshold is 4.0, and the maximum number of iterations is 5000. I find that all trajectories that belong in the set are cyclical, with the exception of some trajectories that do not repeat during the iterations. The four starting locations are: -1, -0.25 -1, 0.25 0.25, -0.5 0.25, 0.5 Are these trajectories chaotic?

Solve $Ax=b$ subject to $x=Cy+d$ Posted: 31 Jul 2021 07:37 PM PDT $Ax=b$ is a linear system of equations in which $A$ is a square invertible matrix. However, I would like to approximately solve $Ax=b$ in which the constraint $x=Cy+d$ is exactly enforced. $C$ is $m\times n$ where $m>n$. I need analytical solutions and am not interested in numerical approaches. Thank you

Let R be a PID. Then, every nonempty set of ideals of R has a maximal element Posted: 31 Jul 2021 06:58 PM PDT I came across the following proof for this proposition but there are some things I don't understand. Proof Let $S$ be the set of all proper ideals of $R$. It follows that $S$ is non-empty and it is partially ordered by inclusion. Let $I_1 \subseteq I_2 \subseteq $ ... be an arbitrary increasing chain of ideals in $S$. Let $I=\bigcup_{n} I_n$. Since the chain of $I_n$'s are nonempty, it follows that $I$ is nonempty. $I$ is an ideal. Since $R$ is a PID, $I = (a)$. We find that $a \in I=\bigcup_{n} I_n$ so $a \in I_n$ for some $n$. We get $I_n = I_{n+1} =$ .... Each chain of ideals has an upper bound. By Zorn's lemma, the nonempty set of $I_n$'s of $R$ has a maximal element,the maximal ideal containing $I$. Questions Why can we use the set of all proper ideals of $R$ for our proof when it talks about any set? Why is it true that $I_n = I_{n+1}$? I'm hoping it's a typo because what it really uses is that the previous is containned in the next. I'm also not sure why the maximal would contain $I$, I understand it's not relevant to the proof since it's only asking us to prove it exists but I got curious.

Find closed form of $a_{n+1} = 4a_n - 2$ Posted: 31 Jul 2021 07:04 PM PDT Is there a easy way to find the recurrence for $a_{n+1} = 4a_n - 2$, with $a_1 = 2$? I was learning about finding the closed form of Fibonacci-styled recurrences like $a_{n+2} = 2a_{n+1} - 3a_{n}$ or something, and then this problem appeared. The textbook did a ton of rearranging to cancel out the $-2$ term and get it into the form of a Fibonacci-styled recurrence. I'm wondering if there is a easier way.

Proof of Euclidian's algorithm for non-coprime numbers Posted: 31 Jul 2021 07:26 PM PDT This question is part of my assignment and I am really struggling with it. Let us now apply these steps to a more general situation. As before we will suppose that the Euclidean algorithm runs in $3$ steps. $$r_0 ÷ r_1 =m_1 R\;r_2$$ $$r_1 ÷ r_2 =m_2 R\;r_3$$ $$r_2 ÷ r_3 =m_3 R\;0$$ Prove that $r_3$ is factor of $r_0$ and $r_1$. They have given the example of $\gcd(81,33) = 3$. The questions is asking for a proof to show why $3$ (the GCD should be a factor of $15,33$ and $81$) You should have found that the last step of the algorithm gives 15 3 = 5R0. Use this to conclude that 3 is a factor of 15. You should have found that the second step of the algorithm gives 33 15 = 2R3. Use this to conclude that 3 is a factor of 33 and 15. You should have found that the first step of the algorithm gives 81 33 = 2R15. Use this to conclude that 3 is a factor of 81 and 33. I understand that since 15 is directly divisible by 3 , 15 has one of it's factor as 3. And intuitively also I understand that 33 and 81 has factors of 3, but I do not know how can I prove that with what they are asking me.

Limit of the n-root of this sequence. Posted: 31 Jul 2021 07:04 PM PDT I'm struggling to understand why $\lim_{n\to\infty} \sqrt[n]{\sin(\frac{1}{n^2})}=1$ while $\lim_{n\to\infty}\sin(\frac{1}{n^2})=0.$ What causes the sequence to tend to $1$ while it's under the $n$-root?

Sequence and multiplicity of sums of integers squared $n_1^2+n_2^2+\dots+n_d^2$? Posted: 31 Jul 2021 07:40 PM PDT Consider a summation over $d$ integers such as $$R_d=\sum_{n_1,n_2,\dots,n_d\in\mathbb{Z}}f\left(\sum_{i=1}^d n_i^2\right)$$ Is there a way to express this sum in terms of a specific integer sequence $S$, such that $$R_d = \sum_{n\in S}M_n f(n)$$ with term multiplicity $M_n$? What is the sequence $S$ and the multiplicity $M_n$ given by in this case?

Validity of proof $\zeta(s) \neq 0$ for $\sigma >1$. Posted: 31 Jul 2021 07:30 PM PDT The Riemann zeta function is can be expressed as an infinite series, as well as an infinite Euler product over primes $p$. $$ \zeta(s) = \sum_n 1/n^s = \prod_p(1-1/p^s)^{-1} $$ Here $s=\sigma+it$ and the function is defined for $\sigma>1$. A natural question to ask is whether there are any zeros in the region $\sigma>1$. Question: Is the following outline proof sufficient and correct? Step 1 We note that none of the factors $(1-1/p^s)^{-1}$ is ever zero, because $p^s = e^{s\ln(p)} \neq 0$ for $\sigma>1$. Step 2 The previous step is insufficient. We also need to demonstrate that the infinite product doesn't converge to zero. To do this, we make use of the following convergence criteria: if $\sum|a_n|$ converges, then $\prod(1+a_n)$ converges to a finite non-zero value. Step 3 In this case $\sum |a_n| = \sum 1/p^s$, which we know converges for $\sigma > 1$. Therefore the Euler Product converges to a non-zero finite value.

Homotopy groups of projective Lie groups PO(N), PSO(N), and PSpin(N) Posted: 31 Jul 2021 06:49 PM PDT Previously, we have learned from Homotopy groups O(N) and SO(N): $\pi_m(O(N))$ v.s. $\pi_m(SO(N))$ that: $\pi_m(SO(N))$: a table consisting of the groups $\pi_m(SO(N))$ for $1 \leq m \leq 12$ and $2 \leq N \leq 12$ can be found on the nLab page for the orthogonal group in the section on homotopy groups. $\pi_m(O(N))$: $O(N)$ consists of two connected components which are both diffeomorphic to $SO(N)$. So $\pi_0(O(N)) = \mathbb{Z}_2$, $\pi_0(SO(N)) = 0$, and for $m \geq 1$, $\pi_m(O(N)) = \pi_m(SO(N))$. $\pi_m(\operatorname{Spin}(N))$: note that $\operatorname{Spin}(N)$ is a double cover of $SO(N)$. When $N = 1$, we see that $\operatorname{Spin}(1) = \mathbb{Z}_2$ so $\pi_0(\operatorname{Spin}(1)) = \mathbb{Z}_2$ and all its other homotopy groups are trivial, while for $N = 2$ we have $\operatorname{Spin}(2) = S^1$ which has first homotopy group $\mathbb{Z}$ and all higher homotopy groups trivial. When $N \geq 3$, $\operatorname{Spin}(N)$ is the universal cover of $SO(N)$ so $\pi_1(\operatorname{Spin}(N)) = 0$ and for $m \geq 2$, $\pi_m(\operatorname{Spin}(N)) = \pi_m(SO(N))$. This post means to ask the homotopy groups of projective Lie groups $PO(N)\equiv\frac{O(N)}{Z(O(N))}$, $PSO(N)\equiv\frac{SO(N)}{Z(SO(N))}$, and $PSpin(N)\equiv\frac{Spin(N)}{Z(Spin(N))}$ where $Z(G)$ means the center subgroup of $G$. Note that (please double check): $$Z(SO(n))=\begin{cases}\mathbb{Z}_2,&n \text{ even but $n \neq 2$}\\ SO(2), & n=2.\\ 0,&n \text{ odd}\end{cases}$$ $$Z(O(n))=\mathbb{Z}_2.$$ $$Z(Spin(n))=\begin{cases}\mathbb{Z}_2,&n\equiv1,3\,\mod 4\\ \mathbb{Z}_4&n\equiv 2\,\mod 4\\ \mathbb{Z}_2\oplus\mathbb{Z}_2&n\equiv0\;\mod 4\end{cases}$$ $\pi_m(PSO(N))$: $$\pi_m(PSO(N))=?$$ $\pi_m(PO(N))$: $$\pi_m(PO(N))=?$$ $\pi_m(PSpin(N))$: $$\pi_m(PSpin(N))=?$$ It seems that we have the group isomorphism $PSO(N)\cong PO(N) \cong PSpin(N)$, so we can answer all questions at once.

Can $\pi$ be defined in a p-adic context? Posted: 31 Jul 2021 07:31 PM PDT I am not at all an expert in p-adic analysis, but I was wondering if there is any sensible (or even generally accepted) way to define the number $\pi$ in $\mathbb Q_p$ or $\mathbb C_p$. I think that circles, therefore also angles, are problematic in a p-adic context, but $\pi$ appears in many other contexts. Of course there are many known series that sum to $\pi$, some may converge p-adically, but those that converge may have different limits, and I think some more motivation would be needed to designate one as an analog of $\pi$. Maybe one could find an analog based on $e^{n\pi i} = (-1)^n$, or even $\int_{\mathbb R} e^{-x^2/2}dx = \sqrt\pi$. So my question: Is there or are there p-adic definitions of $\pi$? If not, could we sensibly define $\pi_p$, and how?

How are tautologies formally constructed in propositional logic if they rely on axioms? Posted: 31 Jul 2021 07:45 PM PDT I realise that an expression like $x=x$ is not tautological in propositional logic due to the fact that "=" is not formally defined as a propositional connector. However, what I am confused about is how these propositional connectors are defined in the first place, because surely if these connectors are defined as axioms, they are assumed to be true, and so anything constructed from them would also be based on assumed axioms, including such tautologies. So whilst true statements can be constructed, I don't really understand how these can be tautologies, since they rely on axioms (defining the propositional connectors) that are assumed true, and are not then true inherently. So I was just wondering how this was got around, and how defining propositional connectors work without ending up in the same situation as the = sign in x=x. Apologies in advance if this is a silly question.

Question in generated sigma algebra Posted: 31 Jul 2021 06:54 PM PDT In measure theory, they say that, $\mathcal{F(C)}$ is a $\sigma$-algebra generated by C. And, (i) $\mathcal{F(C)} = \underset{\mathcal{C \subseteq P, \ P \ is \ a \ \sigma-algebra}}{\bigcap}$$\mathcal{P}$, (ii) $\mathcal{F(C)}$ is the smallest $\sigma$-algebra containing C. I'm having some conflicts in accepting it. Infact, for example: Let, $X = \left\{ a,b,c,d,e \right\}$ $C = \left\{ a,b,c \right\}$ $\mathcal{F(C)} = \left\{\emptyset, X, \left\{ a \right\}, \left\{ b \right\}, \left\{ c \right\}, \left\{ a,b \right\}, \left\{ a,b,c \right\}, \left\{ a,c \right\}, \left\{ d,e \right\}, \left\{ a,b,d,e \right\}, \left\{ a,c,d,e \right\},\left\{ b,d,e\right\}, \left\{ c,d,e\right\}, \left\{ a,d,e \right\} , ...\right\}$ $\mathcal{F(\left\{ d, e \right\})} = \left\{ \emptyset, X, \left\{ d\right\}, \left\{ e\right\}, \left\{ d,e\right\}, \left\{a,b,c \right\}, \left\{ a,b,c,e \right\}, \left\{ a,b,c,d\right\} \right\}$ Clearly, $\mathcal{F(\left\{ d, e \right\})}$ contains less elements to that of $\mathcal{F(C)}$. But how is that possible since $\mathcal{F(C)} = \underset{\mathcal{C \subseteq P, \ P \ is \ a \ \sigma-algebra}}{\bigcap}$$\mathcal{P}$ and $\mathcal{F(\left\{ d, e \right\})}$ is one of the $\mathcal{P}'s$? Edit based on the comments of @ArturoMagidin: $C = \left\{ \left\{a\right\},\left\{b\right\},\left\{c\right\} \right\}$ (since $C$ should be the collections of subsets of $X$) $\mathcal{F(C)} = \left\{\emptyset, X, \left\{ a \right\}, \left\{ b \right\}, \left\{ c \right\}, \left\{ a,b \right\}, \left\{ a,b,c \right\}, \left\{ a,c \right\}, \left\{ d,e \right\}, \left\{ a,b,d,e \right\}, \left\{ a,c,d,e \right\},\left\{ b,d,e\right\}, \left\{ c,d,e\right\}, \left\{ a,d,e \right\} , ...\right\}$ $\mathcal{F(\left\{ \left\{d\right\},\left\{e\right\} \right\})} = \left\{ \emptyset, X, \left\{ d\right\}, \left\{ e\right\}, \left\{ d,e\right\}, \left\{a,b,c \right\}, \left\{ a,b,c,e \right\}, \left\{ a,b,c,d\right\} \right\}$ Since $\mathcal{F(\left\{ \left\{d\right\},\left\{e\right\} \right\})}$ doesn't contain $C$, it's not one of intersecting $P's$.

Optimization problem that involve square root in the constraint? Posted: 31 Jul 2021 07:10 PM PDT I am a researcher in Telecommunication and currently I am running into a difficult optimization with a huge square root. Minimize $A$ such that the following constraints $C_1$ and $C_2$ are satisfied $\begin{gathered} {C_1}:{A_{Min}} \leqslant A \leqslant {A_{Max}} \hfill \\ {C_2}:\frac{{a{A^2}\sin (\theta )\left( {\cos (\theta )\sqrt {\frac{{{B^2}\left( {\frac{1}{{{{\sin }^2}(\theta )}}} \right)}}{{{A^2}}} - 1} + \sin (\theta )} \right) - a{B^2}}}{{{A^2} - {B^2}}} + D \leqslant \frac{{a{C^2}\sin (\theta )\left( {\cos (\theta )\sqrt {\frac{{{B^2}\left( {\frac{1}{{{{\sin }^2}(\theta )}}} \right)}}{{{C^2}}} - 1} + \sin (\theta )} \right) - a{B^2}}}{{{C^2} - {B^2}}} \hfill \\ \end{gathered}$ $\theta$ is an angle that satisfied $0 \leqslant \theta \leqslant \frac{\pi }{2}$ $a,A,B,C$ are all positive $D$ is strictly negative $A_{min}$ and $A_{max}$ are the upper bound and lower bound of $A$ respectively. From the first glance, it seems that there are a lot of symmetry in the left and right hand side of constraint $C_2$ of this problem but I do not know how to exploit it. The only effort so far is to get rid of the trigonometry function by using the Weierstrass substitution by letting $t = \tan \left( {\frac{\theta }{2}} \right)$ where $t \in \left[ {0,1} \right]$ because $0 \leqslant \theta \leqslant \frac{\pi }{2}$. After that, we have the following: $\begin{gathered} \sin (\theta ) = \frac{{2t}}{{1 + {t^2}}} \hfill \\ \cos (\theta ) = \frac{{1 - {t^2}}}{{1 + {t^2}}} \hfill \\ \end{gathered}$ With the Weierstrass substitution, I guess the optimization problem is now purely in polynomial form but nonetheless still posed a major challenge for me. I have some guts feeling that this problem might have something to do with all the square term (i.e quadratic optimization flavor). Constraint $C_2$ then becomes $\frac{{a{A^2}(2t)\left( {\frac{{{S_1}\left( {1 - {t^2}} \right)}}{{{t^2} + 1}} + \frac{{2t}}{{{t^2} + 1}}} \right)}}{{\left( {{t^2} + 1} \right)\left( {{A^2} - {B^2}} \right)}} + D \leqslant \frac{{a{C^2}(2t)\left( {\frac{{{S_2}\left( {1 - {t^2}} \right)}}{{{t^2} + 1}} + \frac{{2t}}{{{t^2} + 1}}} \right)}}{{\left( {{t^2} + 1} \right)\left( {{C^2} - {B^2}} \right)}}$ Where $\begin{gathered} {S_1}^2 = \frac{{{B^2}{{\left( {\frac{{1 + {t^2}}}{{2t}}} \right)}^2}}}{{{A^2}}} \hfill \\ {S_2}^2 = \frac{{{B^2}{{\left( {\frac{{1 + {t^2}}}{{2t}}} \right)}^2}}}{{{C^2}}} \hfill \\ \end{gathered} $ Please help me with this, thank you for your enthusiasm ! Clarification: 1/ Regarding the question of fixed and varied parameter. $A$ is the decision variable that needs to be minimized and the rest of the variable $a,B,C$ are fixed (Physical interpretation: they are some measurements value that can be captured very quickly and accurate so I considered them as fixed value). The answer of this problem should be something in the form $A$ equal to some combination of the other fixed parameters. Also,${A_{\min }}$ and ${A_{\max }}$ are fixed values depend on the transmission standard and not obtain from measure. However, in practice the angle $\theta$ will also varied as well albeit very very slowly. Therefore, I plan to solve for the case when $\theta$ is fixed first. 2/ It is not always possible to guarantee ${A^2} - {B^2}$ to be positive but if this assumption make the analysis easier I think it is possible to accept it although it limit the generality of the analysis. Note that, from my domain knowledge $0<B<1$ is always guarantee. Also, I think it is fine to assume $C^2 - B^2$ to be positive too.

$(1+A)^n \ge (1+a_1)\cdot(1+a_2)\cdots(1+a_n) \ge (1+G)^n$ Posted: 31 Jul 2021 07:29 PM PDT $A$ and $G$ are arithmetic mean and the geometric mean respectively of $n$ positive real numbers $a_1$,$a_2$,$\ldots$,$a_n$ . Prove that $(1+A)^n$$\ge$$(1+a_1)\cdot(1+a_2)\cdots(1+a_n)$$\ge$$(1+G)^n$ if $k$$\gt$$0$, $(k+A)^n$$\ge$$(k+a_1)\cdot(k+a_2)\cdots(k+a_n)$$\ge$$(k+G)^n$ My try (Ques -1): To prove $(1+a_1)\cdot(1+a_2)\cdots(1+a_n)$$\ge$$(1+G)^n$ $(1+a_1)\cdot(1+a_2)\cdots(1+a_n)=1+\sum{a_1}+\sum{a_1}\cdot{a_2}+\sum{a_1}\cdot{a_2}\cdot{a_3}+\cdots+(a_1\cdot{a_2}\cdots{a_n})$ Consider $a_1,a_2,\ldots,a_n$ be $n$ positive real numbers and then apply AM$\ge$GM $\frac{(a_1+a_2+\cdots+a_n)}{n}\ge(a_1\cdot{a_2}\cdots{a_n})^\frac{1}{n}$ imples $\sum{a_1}\ge n\cdot{G}$ because $G=(a_1\cdot{a_2}\cdots{a_n})^\frac{1}{n}$ again consider $(a_1\cdot{a_2}),(a_1\cdot{a_3}),\ldots({a_1}\cdot{a_n}),(a_2\cdot{a_3}),(a_2\cdot{a_4})\ldots,(a_2\cdot{a_n}),(a_3\cdot{a_4})\ldots(a_{n-1}\cdot{a_n})$ be $\frac{n(n-1)}{2!}$ positive real numbers. Then Applying AM$\ge$GM , we get, $\frac{\sum{a_1}\cdot{a_2}}{\frac{n(n-1)}{2!}}$ $\ge$ $\bigl(a_{1}^{n-1}\cdot{a_{2}^{n-1}}\cdots{a_{n}^{n-1}}\bigl)^\frac{2!}{n(n-1)}$ implies $\sum{a_1}\cdot{a_2}\ge \frac{n(n-1)}{2!}G^2$ Similary , $\sum{a_1}\cdot{a_2}\cdot{a_3}\ge \frac{n(n-1)(n-2)}{3!}G^3$ and so on. Therefore, $(1+a_1)\cdot(1+a_2)\cdots(1+a_n)\ge 1+nG+\frac{n(n-1)}{2!}G^2+\frac{n(n-1)(n-2)}{3!}G^3+\cdots+G^n$ Therefore, $(1+a_1)\cdot(1+a_2)\cdots(1+a_n)\ge (1+G)^n$ To prove $(1+A)^n$$\ge$$(1+a_1)\cdot(1+a_2)\cdots(1+a_n)$ Consider $(1+a_1),(1+a_2),\ldots,(1+a_n)$ be $n$ positive real numbers and then applying AM$\ge$GM $\frac{(1+a_1)+(1+a_2)+\cdots+(1+a_n)}{n} \ge [(1+a_1)\cdot(1+a_2)\cdot\cdots(1+a_n)]^\frac{1}{n}$ $\frac{n+a_1+a_2+\cdots{a_n}}{n}\ge [(1+a_1)\cdot(1+a_2)\cdot\cdots(1+a_n)]^\frac{1}{n}$ $(1+A)^n\ge(1+a_1)\cdot(1+a_2)\cdot\cdots(1+a_n)$ My try (Ques -2): To prove $(k+A)^n$$\ge$$(k+a_1)\cdot(k+a_2)\cdots(k+a_n)$ Consider $(k+a_1),(k+a_2)\ldots(k+a_n)$ be $n$ positive real numbers and applying AM$\ge$GM $\frac{(k+a_1)+(k+a_2)+\cdots(k+a_n)}{n}\ge[(k+a_1)\cdot(k+a_2)\cdots(k+a_n)]^\frac{1}{n}$ $(k+A)^n\ge (k+a_1)\cdot(k+a_2)\cdots(k+a_n)$ To prove $(k+a_1)\cdot(k+a_2)\cdots(k+a_n)$$\ge$$(k+G)^n$ $(k+a_1)\cdot(k+a_2)\cdots(k+a_n)=k^n+k^{n-1}\sum{a_1}+k^{n-2}\sum{a_1\cdot{a_2}}+\cdots+(a_1\cdot{a_2}\cdots{a_n})$ now $\sum{a_1}\ge n\cdot{G}$, $\sum{a_1}\cdot{a_2}\ge \frac{n(n-1)}{2!}G^2$ $\sum{a_1}\cdot{a_2}\cdot{a_3}\ge \frac{n(n-1)(n-2)}{3!}G^3$ and so on. Therefore $(k+a_1)\cdot(k+a_2)\cdots(k+a_n)\ge k^n+nk^{n-1}G+\frac{n(n-1)}{2!}k^{n-2}G^2+\cdots+G^n$ Therefore, $(k+a_1)\cdot(k+a_2)\cdots(k+a_n)\ge (k+G)^n$ My Ques : Have I solved the questions correctly? How to prove $\sum{a_1}\cdot{a_2}\cdot{a_3}\ge \frac{n(n-1)(n-2)}{3!}G^3$ Thanks

How can I prove a line can't meet an ellipse in more than two points Posted: 31 Jul 2021 07:39 PM PDT Without knowing the equation for an ellipse, but rather just with the geometric definition i.e. the ellipse is the locus of points with $PF_1 + PF_2 = k > F_1F_2$. I tried to use a classic construction: $\omega = \odot(F_1,k)$, so for $Q \in \omega$, the point $P$ of meeting of the perpendicular bisector between $Q$ and $F_2$, with $QF_1$ is in the ellipse. By the converse: given $P$ in the ellipse, the point $Q = PF_1 \cap \odot(P,PF_2)$, not in between $P$ and $F_1$, is in $\omega$. The problem is that this last operation transforms lines in weird curves, while I was hoping it would transform lines in another lines and thus, a line would meet a circle in more than two points. Notice that it is not obvious from this definition that an ellipse can be transformed in a circle by dilation on some axis.

Over skepticism in math [closed] Posted: 31 Jul 2021 07:10 PM PDT Hello I am undergraduate who is trying to self teach linear algebra. I am reading robert beezer linear algebra book. My problem is that I read the first 80 pages easily but cant move farward in the book as I get stuck revisiting old proofs making sure I understand each step well. I also sometimes find that old proofs is missing steps so I complete them with my own ,but I am scared if my completion proof is not rigorous enough or is missing something. Do any one have any advice on how not to waste time and start completing the book. Thanks in advance

What is the distribution of the dot product of a flat Dirichlet vector with a fixed vector? Posted: 31 Jul 2021 07:01 PM PDT I am using this question as a reference What is the distribution of the dot product of a Dirichlet vector with a fixed vector? I have the following problem, $S = w \cdot C$ where vector $w$ is random with components having an $N$-dimensional flat Dirichlet distribution, where $a=1$ (just uniform distribution above the $(N-1)$-simplex) How can I write directly the PDF of this function?

Expected Value Of a Random Grid That Contains Multipliers Posted: 31 Jul 2021 06:51 PM PDT I am a software developer and recently came across this problem when working on a hobby project, and my knowledge of probability is too small to solve this. I have a bag that contains 5 negative ones, 5 zeros, 10 ones, 5 twos, 5 threes. The tiles are set randomly on a 5 by 5 grid with 5 tiles left in the bag. The threes in the bag also multiply all 8 of the adjacent squares by 2 (A number is multiplied by $2^n$ adjacent 3's). What is the best way to calculate the expected value of the grid for any known bag that can contain a random amount of any numbers? Bonus points if it can handle any size (and rectangular shape) of grid. Eg. Bag ( 2 1 3 3 3 ), Grid Value: 27 $$\begin{array}{c|c|c|c|c} 3&2&1&-1&1\\ \hline 1&1&0&0&-1\\ \hline 1&0&3&2&2\\ \hline -1&0&-1&2&1\\ \hline 1&-1&1&1&0 \end{array}$$ Without the multiplier, the solution is quite simple: The number of values in the grid (25) × The sum of all of the numbers in the bag / the number of numbers in the bag. Or $(n * s) / a$ Since they do not interact, the fact they are in a grid does not even matter. For the same reason, it's not that useful for solving this problem, but I thought I would mention it as a start. A kinda related question of mine

Is there a better solution for $\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(a))}{n^{n+1}}dt}$? Posted: 31 Jul 2021 07:28 PM PDT I know there exist functions like this one for simplifying tetration based sums. There may be a way to simplify this type of sum at least using a lesser known and widely accepted functions. Here are some results as proof: graphical visualization of results and calculation of special case and integral version of special case. It was a nice idea to find the sophomore's dream, but I thought that it would be more interesting if I could find a "generalized exponential sophomore's dream". This post uses the following functions: regularized gamma functions, the exponential integral function, and tetration. There was an annoying discontinuity at n=0 hence the constant term: $$\mathrm{\int_0^b \, ^2\left(a^t\right)dt=\int_0^b a^{ta^t}dt=b+\sum_{n=1}^\infty\frac{ln^n(a)}{n!}\int_0^b t^n a^{tn}dt=\boxed{\mathrm{b+\frac1{ln(a)}\sum_{n=1}^\infty\frac{(-1)^nP\big(n+1,-n\,b\,ln(a)\big)}{n^{n+1}}}}\implies \int_{-\frac1e}^0 e^{{te}^t}dt=1+\sum_{n=1}^\infty \frac{(-1)^nP(n+1,n)}{n^{n+1}}=0.77215…}$$ $$\mathrm{\implies A(a,t)\mathop=^\text{def}\int \,^2\left(a^t\right)dt=C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(a))}{n^{n+1}}=\quad C+t-t\sum_{n=0}^\infty \frac{(t\,ln(a))^n E_{-n}(-nt\,ln(a))}{n!}}$$ This series reminds me of the Marcum Q function for non negative integers: $$\mathrm{Q_m(a,b)=1-e^{-\frac{a^2}2}\sum_{n=0}^\infty\left(\frac{a^2}{2}\right)^n\frac{P\left(m+n,\frac{b^2}2\right)}{n!}}$$ This representation is good, but is quite tedious to use as the formula requires summing an infinite amount of regularized gamma functions. I would like to find a way to get rid of the summation .I see the gamma function with powers which reminds me of the summation definition of a hypergeometric function. I would even like to see the use of a generalized hypergeometric function like Meijer G or Kampé de Fériet functions seen in the link. Please correct me and give me feedback! Result using @Arjun Vyavaharkar's function Wi(x). Perhaps it means W-Lambert Integral function: I, hopefully, found a summation form for $$\mathrm{Wi(x)=\sum_{n\ge1}n^{n-1}P(n+1,-ln(x)),|ln(x)|<\frac1e}$$Here is an interactive graph. P(a,b) is the Regularized Lower Incomplete Gamma function and the expansion used for W(x).

Delta function and $\sum_{t}\exp\{ i k t\} $ Posted: 31 Jul 2021 07:45 PM PDT I am interested in evaluating the following, which appears in an integrand, $$ \sum_{t=0}^{\infty}e^{ikt} $$ where $k$ is real. Using the following relation $$ \sum_{t=-\infty}^{\infty}e^{ikt} = 2\pi\delta(k), $$ the real part of it is \begin{eqnarray} \Re\left[\sum_{t=0}^{\infty}e^{ikt}\right]&=&\frac{1}{2}+\frac{1}{2}\Re\left[\sum_{t=-\infty}^{\infty}e^{ik}\right]\\&=&\frac{1}{2}+\pi\delta(k). \end{eqnarray} Note the addition of $\frac{1}{2}$ term! This $\frac{1}{2}$ term does not agree with the following derivation using $\epsilon$ trick! \begin{eqnarray} \lim_{\epsilon\rightarrow0_{+}}\lim_{T\rightarrow\infty}\sum_{t=0}^{T}e^{ikt}e^{-\epsilon t}&=&\lim_{\epsilon\rightarrow0_{+}}\lim_{T\rightarrow\infty}\frac{e^{\epsilon}-e^{ik\left(T+1\right)}e^{-\epsilon T}}{e^{\epsilon}-e^{ik}}\\ &=&\lim_{\epsilon\rightarrow0_{+}}\frac{1}{e^{\epsilon}-e^{ik}} \end{eqnarray} which becomes, when $k \rightarrow 0$, \begin{eqnarray} \lim_{k\rightarrow0,}\lim_{\epsilon\rightarrow0_{+}}\frac{1}{e^{\epsilon}-e^{ik}}&=&\lim_{k\rightarrow0,}\lim_{\epsilon\rightarrow0_{+}}\frac{1}{\epsilon-ik}\\ &=&\lim_{k\rightarrow0}\lim_{\epsilon\rightarrow0_{+}}\frac{i}{k+i\epsilon}\\&=&\lim_{k\rightarrow0}\left[\pi\delta\left(k\right)+i\mathcal{P}\frac{1}{k}\right] \end{eqnarray} where Sokhotsky's formula is used. Now when $k\nrightarrow0$, $$ \lim_{\epsilon\rightarrow0_{+}}\frac{1}{e^{\epsilon}-e^{ik}}=\frac{1}{1-e^{ik}} $$ Therefore, for all $k$, $$ \sum_{t=0}^{\infty}e^{ikt}=\pi\delta\left(k\right)+\mathcal{P}\frac{1}{1-e^{ik}} $$ Note that the real part of it does not have $\frac{1}{2}$ addition term, instead has $\Re\mathcal{P}\frac{1}{1-e^{ik}}$. What am I doing wrong? Building on Svyatoslav's answer which in turn supported by vitamin d, the conclusion is the following: $$ \sum_{t=0}^{\infty}e^{ikt} = \frac{1}{2} + \pi \delta(k) + \frac{i}{2} \mathcal{P} \cot (k/2) $$

Solving multivariate quadratic equations over the integers Posted: 31 Jul 2021 07:00 PM PDT I am looking for a method (if it exists) to solve over the integers the following sum of squares equation: $$ x_1^2 + x_2^2+x_3^2 + \cdots + x_n^2 = m,$$ with $m \in \mathbb{N}.$ Someone has any idea about books, articles dealing with this kind of problem? Thanks in advance!

Isometry and its inverse Posted: 31 Jul 2021 07:03 PM PDT I got this affine map: $$ f: R^3 \rightarrow R^3: \begin{pmatrix}x\\ y\\ z\\\end{pmatrix} \rightarrow A \cdot \begin{pmatrix}x\\ y\\ z\\\end{pmatrix} + \begin{pmatrix}0\\ -1\\ 1\\\end{pmatrix} $$ with $$ A = \begin{bmatrix} 1 & a_{12} & a_{22}\\ 0 & 1 & a_{21}\\ 0 & 0 & 1 \end{bmatrix}$$ Also given was this information about the inverse (which should also be an affine map?): $$ g=f^{-1}: R^3 \rightarrow R^3: \begin{pmatrix}x\\ y\\ z\\\end{pmatrix} \rightarrow B \cdot \begin{pmatrix}x\\ y\\ z\\\end{pmatrix} + \bar b $$ I have to find $\bar b$. Does anyone have an idea how to find it? I tried inputting some values but I can't seem to get there.

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