Is there a word to define a period of four months?
Is "tri-quarterly" a real English word meaning 3 times a year? Are there any other words that mean 3 times a year?
Prove that if the positve term series $\sum^{\infty}_{n=1}a_n$ is convergent, also $\sum^{\infty}_{n=1}\sqrt{a_na_{n+1}}$ is convergent.  Prove that if the positive term series $\sum^{\infty}_{n=1}a_n$ and $\sum^{\infty}_{n=1}b_n$ are convergent, also $\sum^{\infty}_{n=1}a_nb_n$ is convergent.  I've tried to solve it using comparison test, but no results. $\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$ What solutions does the equation 4x≡2mod10 have? (Hint, it will have more that one.) What about solutions to an equation ax≡dmodn, where d=gcd(a,n)? What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question! How to secure Elliptic Curve ElGamal encryption against known plaintext attacks? If I have an encoding function  $f(x)$ that maps a message $m$ to a point $P$ on a suitable Elliptic Curve $E$ . If I have the public key $Q$ of my recepient then I can encrypt the message as follows:  Choosing a random number $k &lt; n -1 $ where $n$ is the order of the curve E .  Calculating $C = [k] G $ where $G$ is the generator point of the curve E   Calculating $R = [k]Q $ where $Q$ is the recipient's public key   Now the message is encrypted by adding $P$ to $R$ to get          $C_e$ $( = R + P)$ which is sent along with $C$ to the recepient.  DECRYPTION:  Now for the recepient to decrypt my message they compute $R = [privatekey] C $ (where $C = [k]G $) and simply subtract $R$ from $C_e$ to get the encoded point $P$ back which is then decoded by an inverse function $f^{-1}(x)$ to recover $m$.  Now , two problems arise : If the attacker knows the plaintext (suppose if I follow a particular format of data while sending messages ) he could encode his guessed plaintext and subtract it from $C_e$ to recover $ R $ back ! . If I used that same $ R  $ to encrypt further "blocks" of my message then the security of the later parts of my message has been breached!   I realize that doing  something like a scalar multiplication over and over again for every block of data, would be a huge drawback in efficiency and speed.   Is there any way to compute a different $ R $ for every block of data, without using much resources and that too quickly ?   Does reusing the same $R$ for another message break security ?  NOTE: This question is for educational purposes only. It's for the sake of expanding my knowledge on Crypto. In what cases I should skip 's in possessive case? I have my CIFS setup in fstab and they are working as they are supposed to on boot. They mount as they should and work for a while. Out of nowhere it seems (could be after unlocking machine etc) I get "Host is down" error trying to access it. I have multiple and they are all down. They are also shared from the same server. At this time i check on a windows computer and an outdated 14.04 machine and they are up and functioning as they are supposed to. After clicking around on the shares in nautilus and getting repeat errors they will just start working again. To access a share that is "down" takes about 2-3min of randomly clicking different mounts and going back to the first one when automagically it shows the data in the mount point.  I do not have this problem on 14.04 machines that have not been updated in a while. All of those machines are fully functional and the CIFS never go "down". On 16.04 they were not a problem until more recently.  I have made sure to update every other day and have cleaned old linux headers (in hind sight i probably should have reverted). I do this because im begging for a fix to just appear but its been weeks of battling CIFS mounts without any solution. Flag pop-up hidden beneath answer box (and it's still transparent!) Is there a way to absolutely force a Log Out, Shut Down, or Restart with open tabs and unsaved documents? I could not find any relatively simple way to log out of a public use Mac quickly with many tabs open and unsaved document in the background without resorting to sudo shutdown -h now.  The reason I ask is that sometimes a user will have his/her account logged into the iMacs here at school and they would try to log off and then leave, thinking it has completed but the process is interrupted by Safari and Office apps, leaving their account, mounts, and websites still accessible to the next person who sits down.  Obviously a shortcut string would be nice instead of posting a note telling the person to type arcane things in a terminal. For example, just having them try to find Terminal.app is going to be a chore.  This doesn't seem to make sense because the user can just quit each app and click a few dialog boxes but unlike you and I to whom repeatedly pounding Command-Q while hitting the spacebar and D keys are probably second nature. Why are two permutations conjugate iff they have the same cycle structure? Which equals operator (== vs ===) should be used in JavaScript comparisons? Electricity takes the path of least resistance? Electricity takes the path of least resistance!   Is this statement correct?    If so, why is it the case?  If there are two paths available, and one, for example, has a resistor, why would the current run through the other path only, and not both? What is the origin of elements heavier than iron? In all the discussions about how the heavy elements in the universe are forged in the guts of stars and especially during a star's death, I usually hear that once the star begins fusing lighter atoms to produce iron (Fe) that's the end of the star's life and the whole system collapses onto itself; and based on how massive the star was initially, it has different outcome - like a white dwarf, a neutron star or a black hole. I have rarely heard a detailed explanation of how the elements heavier than iron are produced. I would appreciate a convincing explanation of this process. Invariant Subspace of Two Linear Involutions I'd love some help with this practice qualifier problem:  If $A$ and $B$ are two linear operators on a finite dimensional complex vector space $V$ such that $A^2=B^2=I$ then show that $V$ has a one or two dimensional subspace invariant under $A$ and $B$.    Thanks! I've recently updated from Debian 'jessie' to Debian 'stretch', and have started having a problem with numbers being formatted backwards when using the bidi package in conjunction with the mathspec package.  Here's a MWE:  \documentclass{article} \usepackage{mathspec} \usepackage{bidi} \begin{document} Per 2.7.1 of the RFC. \end{document}   On Debian 'jessie', when built with xelatex, this produces "Per 2.7.1 of the RFC."  But on 'stretch' this gives "Per 1.7.2 of the RFC."    Any suggestions of how to fix this? I am doing some challenges. This is one them. I am trying to brute-force 4 digit pin with the password to get my desired answer. After connecting to the port It prompts me to enter the password then space then 4 digit pin. I tried to brute-force the pin using the script:  #!/bin/bash   nc localhost 30002   sleep 2    for i in {0000..9999};       if [[ $(echo 'UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ $i' &lt;/dev/stdin) = ^Wrong*]];                         then          continue       echo '[+] Pincode Cracked! Pincode = $i'       fi   done  but it seems that this doesn't input the pass and pin to stdin, before i tried doing something like this -&gt; if [[ $(echo 'UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ $i') = ^Wrong* ]]; What am I doing wrong here? UPDATE: Okay, so after researching around. I wrote this: for i in {0000..9999} do     if [ (echo &quot;UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ $i&quot; | nc localhost 30002 | grep -o Wrong) == &quot;Wrong&quot; ]    then       sleep 0.1       continue    fi    echo &quot;- - - - - - - - - - - - - - - - - - - - - - - - [$i]&quot; done  This might even work but as you can see it opens new connections in the loop which makes it really slow and exhaust the system. tables with lots of columns I have a table that lists various attributes of an individuals personal development plan; objective, time to complete objective, comments etc. The table is editable so the user can directly add comments without having to go to an intermediate form. The problem is that the client wants to add more columns and this is messing up the visual display.   The alternatives I have are:   allowing columns to expand out on click keeping essential attributes in table, rest in 'advanced edit option form' or somesuch I don't know - have you any ideas?   One more thing to add: I think the addition of extra columns is based on user feedback ...  edit: here is what I have gone for Calculate which day of the week a date falls in using modular arithmetic In  the main character (he is a mathematician) calculates the day of the week of someone's birthday (19/07/1992 is Sunday). I know (very) basic modular arithmetic but I can't figure out how to do it. Can someone point me to the right direction? It seems fun to do. Is there a place to see release dates for previous releases? How to suppress Terminated message after killing in bash? dimension of a subspace spanned by two subspaces If $M$ and $ N$ be the subspaces of a vector space $V$ then prove that  $(\dim M) + (\dim N) = (\dim M+N) + (\dim M \cap N)$.