Let $f$ be Lebesgue integrable on $\mathbb{R}$. Show that  $ \sum_{n=1}^{\infty} f(x+n) $ converges almost everywhere.   I was thinking that maybe the finite sum could be compared to a simple function and maybe this could lead somewhere, but i've doubt this since I don't know what the level sets look like for the $f(i)'s$. The result seems a  bit obvious but I don't have a good idea.  Any helpful hints or help would be great
Assume $f:\mathbb R\rightarrow  {\mathbb R} $ is Lebesgue integrable, prove $\sum_{n\in\mathbb N } f(x+n)$ converges for almost everywhere x. This is a question discussed with friends. Thanks
When the external resistor increases in temperature, why is the calculated value for internal resistance higher? Say we had a circuit with a cell (with internal resistance)and an external resistor of $10 \Omega$ as shown:  The $10\Omega$ resistor then increases in temperature, leading to an increase in resistance in this resistor. My thinking was that since $10 \Omega$ resistor now has a value higher than $10\Omega$, more voltage drops across this resistor. Since your emf is constant, less voltage must drop across the internal resistor. Less voltage drop means less resistance. Where is the flaw in my logic? I would like to automate some services using cron jobs.  When I try to edit the crontab file at /etc/crontab, I am not able to save it as it says that it is read only.   What can I do? Should outdated answers be voted down? On Stack Overflow, there are many questions that relate to currently-developing technologies such as HTML5 and CSS3. The accepted answers to these questions may become outdated, and better answers may appear. However, because the answers are accepted, it is likely that they will continue to be upvoted by users reading the answers, and the OP will likely not notice that a better, more up-to-date answer may exist.  Should these outdated answers be voted down? If not, what should we do with them? OS: UBUNTU 18.04  I had installed Oracle Java 8 earlier. But decided to switch to open JDK 11. it's successfully installed. java -version openjdk version "11.0.2" 2019-01-15 OpenJDK Runtime Environment (build 11.0.2+9-Ubuntu-3ubuntu118.04.3) OpenJDK 64-Bit Server VM (build 11.0.2+9-Ubuntu-3ubuntu118.04.3, mixed mode, sharing)  Then I set the JAVA_HOME sudo nano /etc/environment  File changes: PATH="/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/$ JAVA_HOME="/usr/lib/jvm/java-11-openjdk-amd64/"  Then executed: source /etc/environment  echo $JAVA_HOME /usr/lib/jvm/java-11-openjdk-amd64/  But the moment I exit the terminal and then launch the terminal again: I get this output:  ▶ echo $JAVA_HOME /usr/lib/jvm/java-8-oracle  which you can see is not what is expected. Please help me out Cardinality of a set of closed intervals What is the cardinality of the set S of all closed intervals on the real number line with rational (positive) lengths?  I believe the number of intervals with a specific fixed length but varying start points is |R|, and the number of intervals with a fixed start point but varying lengths is |Q| = |N|. I think the answer to the question would then be |S| = |R x N| = |R|? I'm not quite sure how to prove the last equality though. Would I have to define a bijective function that maps elements of (R, N) to R? How would I approach doing so? How do I show for an ideal transformer $M^2=L_1L_2$? I've been stuck on this problem for about an hour.     In an ideal transformer, the same flux passes through all turns of the primary and of the secondary . Show that in this case $M^2=L_1L_2$, where $M$ is the mutual inductance of the coils, and $L_1$, $L_2$ are their individual self-inductances.   My first instinct was the following: If we isolate the response of each coil to the other one, we get the equations $N_1\Phi_1=MI_2$ and $N_2\Phi_2=MI_1$, where $\Phi$ is magnetic flux going through a single turn of the coil and $I$ is current. Likewise, if we isolate the response of each coil to itself, we get the equations $N_1\Phi_1=L_1I_1$ and $N_2\Phi_2=L_2I_2$. I would now naturally want to say "Now just multiply $N_1\Phi_1$ and $N_2\Phi_2$ both ways, and get the desired result!" i.e.,  $$(N_1\Phi_1)(N_2\Phi_2)=(MI_2)(MI_1)=(L_1I_1)(L_2I_2)\implies M^2=L_1L_2$$  But that's incorrect, because the response of the coils to themselves and to the other coil are different quantities (i.e. not necessarily equal).  Realizing this, I write down the more general equations.  $$N_1\Phi_1=MI_2+L_1I_1$$ $$N_2\Phi_2=MI_1+L_2I_2$$  Now even with the hypothesis that $\Phi_1=\Phi_2=\Phi$, I still can't see how the identity $M^2=L_1L_2$ follows. Could you help clear up my confusion?  *Taken from D. Griffiths Introduction to Electrodynamics (4th ed.), Problem 7.58. Does Class need to implement IEnumerable to use Foreach This is in C#, I have a class that I am using from some else's DLL. It does not implement IEnumerable but has 2 methods that pass back a IEnumerator. Is there a way I can use a foreach loop on these. The class I am using is sealed. What exactly is "pressure", and what's its relation to force? I've been trying to play with a rigid body simulations.  I prepared a project of a building made of domino tails. I've placed all objects very tight, but as soon as I start animation it looks like there was some tiny gap between objects, so they all fall down a little bit. For a high building it leads to a construction disaster as the whole building colapses.  Here is an example:  (I'd suggest to use 0.25x speed).  How can I fix it? I have a Linux CentOS server, the OS+packages used around 5GB. Then, I transferred 97GB data from a Windows server to two folders on this Linux server, after calculated the disk usage, I see the total size of the two folders is larger than the disk used size.  Run du -sh on each folder, one use 50GB, the other one use 47GB  But run df -h, the used space is 96GB. (50GB + 47GB + 5GB) > 96GB  Is there any problem? Those two folders contain lots of files (1 million+). Thanks. A very basic ring theory question, which I am not able to solve. How does one show that    $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$, if $a,b$ are relatively prime.   My attempt was to define a map, $\varphi:\mathbb{Z}[i] \to \mathbb{Z}/10\mathbb{Z}$ and show that the kernel is the ideal generated by $\langle{3-i\rangle}$. But I couldn't think of such a map. Anyhow, any ideas would be helpful. In Fallout 4 the Bobbleheads are back.  Once you pick them up they stay in your inventory with a weight of 0 and a value of 300. Will I need them later ? Do I lose the bonus if I sell them or drop them somewhere ? Same with the magazines.  Is there any disadvantage to selling them ? Why Doesn't the Adamantium Bone Covering Kill Wolverine? I've seen the  here before, as well as on many sites on the web.  I've also seen different info on what was covered.  I'm not a huge X-Men fan, but I do enjoy the characters and stories, so I know I'm probably missing out on a lot, and may have missed the obvious along the way because I didn't read a particular series or something like that.  I had a relative die from leukemia a few years ago and that reminded me of basic junior high science: that red and white blood cells are produced in the bone marrow and must be able to reach from the marrow into the blood vessels.  The leukemia my relative had was the kind where the "bad" cells would surround the marrow and prevent the new "good" blood cells from getting out to the rest of the body.  So just how much of Wolverine's skeleton is covered with Adamantium?  If it's the whole thing, there'd have to be openings for new red blood cells to get out if he is to remain alive.  I've never seen a reference to this issue, is it ever addressed?  That also raises another issue along the same line: Are hard to reach areas, such as ball and socket joints covered?  If not, then stress on other covered areas could be transferred to the weaker uncovered areas, which would lead to breakage there.  And if joints like that are covered, is there anything to say how it was done?  Are there openings or gaps in the Adamantium covering on his body?  Are there any references to any of these issues that deal with them? I own a Nikon D40X and just the kit lens and I'm considering buying either a macro lens or some extension tubes.  I don't necessarily want to do true 1:1 macro, just some close ups of bugs, fungi and flowers. Does getting disconnected from an SSH session kill your programs? How can I obfuscate (protect) JavaScript? I want to make a JavaScript application that's not open source, and thus I wish to learn how to can obfuscate my JS code? Is this possible? How to replace ${} placeholders in a text file? I want to pipe the output of a "template" file into MySQL, the file having variables like ${dbName} interspersed. What is the command line utility to replace these instances and dump the output to standard output? How do I back up AWS S3 and EBS to a non-Amazon provider? We've centered most of our operations on AWS, and love the ease of use there. However, every once in a while, the paranoid side of my brain wonders ?  Within the Amazon ecosystem, we have a pretty good backup plan: We keep most static data in S3. EBS volumes are snapshotted nightly. However, true paranoia would mean backing up to a provider outside the Amazon ecosystem. How do you handle doing this? I'm especially interested in backing up EBS snapshots and S3 contents. Bonus points for backing up RDS. How to debug EntityMalformedException? I've got fatal error EntityMalformedException: Missing bundle property on entity of type node. in entity_extract_ids() (line 7700 of .\includes\common.inc) when trying to access user/xyz.  I tried to retrieve info about the malformed node at line 7700, where the error message is built, thinks like :  if (!isset($entity-&gt;{$info['entity keys']['bundle']}) || $entity-&gt;{$info['entity keys']['bundle']} === '') {  dpm($info);// or dpm($entity);  throw new EntityMalformedException(t('Missing bundle property on entity of type @entity_type.', array('@entity_type' =&gt; $entity_type))); }   dpm($entity) returns an unexpected user object, and $info a huge qty of things.   Could somebody put me on the right way ?  I already read whatever I could find about Missing bundle property errors, but none helped solving.  dpm($entity) returns  uid (String, 2 characters ) 70 name (String, 9 characters ) John pass (String, 55 characters ) $S$DUwPuOuDPiDL4nRTYXqc7a5uOfMKey7pyhOFUEKka1XM... mail (String, 30 characters ) john@example.com theme (String, 0 characters ) signature (String, 0 characters ) signature_format (String, 13 characters ) filtered_html created (String, 10 characters ) 1396286331 access (String, 10 characters ) 1397146661 login (String, 10 characters ) 1396513460 status (String, 1 characters ) 1 timezone (NULL) language (String, 2 characters ) fr picture (NULL) init (String, 30 characters ) john@example.com data (Array, 5 elements) roles (Array, 1 element) og_user_node (Array, 0 elements) message_subscribe_email (Array, 1 element) field_bio (Array, 0 elements) field_name_first (Array, 1 element) field_name_last (Array, 1 element) field_facebook_url (Array, 0 elements) field_linkedin_url (Array, 0 elements) field_twitter_url (Array, 0 elements) user_trusted_contacts (Array, 1 element) group_group (Array, 1 element) group_access (Array, 1 element) metatags (Array, 0 elements) rdf_mapping (Array, 3 elements) realname (String, 13 characters ) John Doe content (Array, 13 elements) entity_view_prepared (Boolean) TRUE privatemsg_disabled (Boolean) FALSE QGIS Align Rasters how to retain pixel values? I am attempting to align and then merge a number of rasters of the same resolution and extent, but where the pixels are slightly misaligned (creating small gaps when merged). This works fine using the Align Rasters tool, except that the pixel values are not retained and instead given a value of something like -3.23879e+8. I mostly want to keep the nodata pixels. Is this possible?