I am working on some basic problems involving rings and am getting confused.  If I have a ring $\mathbb{Z}_2 \times \mathbb{Z}_4$, would this just be all elements $(a,b)$ such that $a \in  \mathbb{Z}_2$ and $b \in \mathbb{Z}_4$ or am I mistaken?  I'm then trying to find zero-divisors and units in this ring.  How would I go about doing this?  Thanks
How do I exactly find the zero divisors and units of a ring in the form of $\mathbb{Z}_n \times \mathbb{Z}_m$? For instance, how does one calculate zero divisors and units of $\mathbb{Z}_6 \times \mathbb{Z}_2$? I understand that a zero divisor is a non-zero element of $R$ such that $rs = 0$ for some other non-zero element $s$ in $R$. But I don't really quite understand how to apply that to a ring of the form $\mathbb{Z}_n \times \mathbb{Z}_m$. Also, the units of $\mathbb{Z}_6 \times \mathbb{Z}_2$ would be elements that send some element $s$ in $\mathbb{Z}_n \times \mathbb{Z}_m$ to the multiplicative identity, right? (which would be 1)? How do I apply these concepts to "product" rings? Thank you.
I have found this figure of a mode of operation using a triple AES with m||IV+ctr input:   I suppose it's a custom mode of operation since I haven't seen it anywhere else. My question is if it is IND-CPA, and how to prove it in an adversarial model? Can I run Android apps on Ubuntu? I have a service (kado-demo.service) that a normal user (kado) should be able to restart without entering any password.   I created /etc/sudoers.d/user_restart and added  kado ALL=NOPASSWD: /bin/systemctl restart kado-demo.service   Permissions:  -r--r-----   1 root root   585 Mar  5 10:47 user_restart   As user k, I now should be able to execute the given command, but this happens:  /bin/systemctl restart kado-demo.service  ==== AUTHENTICATING FOR org.freedesktop.systemd1.manage-units ===  Authentication is required to restart 'kado-demo.service'.   Multiple identities can be used for authentication:   (...)   Which is the same result as before I added the file.   The /etc/sudoers contains  #includedir /etc/sudoers.d   And I even get the same result when I add content of the user_restart file directly to /etc/sudoers.  I'm running out of ideas... I'm stuck in a teleporting loop. What can I do? I set a command block to /tp. It was supposed to activate only if you had 0 time left, and to activate the command block with a hopper clock to keep giving it power.   But it is not checking if you have 0 time left, so it keeps teleporting me. I can't get close enough to the command block to break it, so I'm stuck in a neverending teleport loop.  Any ideas? How can I implement a download rate limited in Java? Posts flagged for attention but there are none The comment with the Flagged Posts that seems to be new as of today states there are posts with flags awaiting my attention but the counter shows zero and when I go into flagged posts there is nothing waiting for me. I have seen similar questions on the matter (, ) of adding vertical space and not breaking vertical lines, the solutions suggested are to add space after the \\ using \\[0.1cm] for example or changing the \arraystretch which I do not want to do. Using \\[0.1cm] allows me to add space just before a \hline but not after as well.  How can I add vertical space after a \hline without breaking vertical lines in a table?  Please find my MWE below:  \documentclass{scrreprt} \usepackage{setspace} \usepackage[left=2.5cm,right=2.5cm,top=3cm,bottom=3cm]{geometry} \usepackage{array,tabularx} \doublespacing  \begin{document} \begin{table}[htb]   \centering   \onehalfspacing   \caption{Table of info.}      \begin{tabular}{c | c c}  &amp; C/S Analyser &amp; O/N/H Analyser \\  &amp; EMIA 320 V2 &amp; EMGA 830 \\[0.1cm]    \hline    \noalign{\vskip 0.1cm} Sensitivity (ppm) &amp; 0.1 &amp; 0.001 \\ Furnace type &amp; Induction &amp; Impulse \\ Carrier gas &amp; Pure O2 &amp; Pure He \\[0.1cm]    \hline     \end{tabular}    \doublespacing   \label{tab:chem} \end{table} \end{document} Inequality $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$ Derivative (or differential) of symmetric square root of a matrix Let A be a square, symmetric, positive-definite matrix.  Let S be its symmetric square root found by a singular value decomposition.  Let vech() be the half-vectorization operator.  Is there a convenient expression for the derivative (or differential) of vech(S) with respect to vech(A)?  I know the expression for an inverse, which is sort of like a matrix version of the power rule.  Would this approach work for a symmetric square root as well (i.e., (1/2)S^(-1/2))? So I was writing up a simple binary tree in Python and came across [...]  I don't believe this to be related to the Ellipsis object, more it seems to have something to do with an infinity loop (due to Python's shallow copy?). The source of this infinity loop and why it doesn't get expanded while expanding when accessed is something I'm completely lost to, however   &gt;&gt;&gt; a [[[[[], [], 8, 3], [[], [], 3, 2], 6, 3], [], 1, 4], [[], [], -4, 2], 0, 0] &gt;&gt;&gt; Keys(a)#With a+b [0, 1, 6, 8, 3, -4] &gt;&gt;&gt; Keys(a)#With [a,b] [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]] &gt;&gt;&gt; Keys(a)[1]#?? [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...], 8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]]     Version using a+b  def Keys(x,y=[]):     if len(x):y+=[x[2]]+Keys(x[0],y)+Keys(x[1],y)#Though it seems I was using y=y[:]+, this actually outputs an ugly mess     return y   version using [a,b]  def Keys(x,y=[]):     if len(x):y+=[x[2],Keys(x[0],y),Keys(x[1],y)]     return y   So what exactly is [...]? There are n letters addressed to n eople at different addresses. The n addresses are typed on n envelopes.  A disgruntled secretary shuffles the letters and puts them in the envelopes in random order, one letter per envelope.  Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.] What is the probability approximately, for large n ?    My attempt at a solution:  P(all correct) = $\frac{1}{n!}$  P(n-1 correct) = ${n \choose n-1} \frac{1}{n!} = \frac{1}{1!(n-1)!}$  ${n \choose n-1}$ ways of selecting n-1 correct letters.  P(n-2 correct) = ${n \choose n-2} \frac{1}{n!} = \frac{1}{2!(n-2)!}$  ...  P(n-n correct) = ${n \choose n-n} \frac{1}{n!} = \frac{1}{n!(n-n)!} = \frac{1}{n!}$  This all looks ok to me, but I tested this for a case of 4 people, (I think the theoretical value should be $\frac{9}{24}$ but obviously that is not what I got. Can someone correct my logic please? Also, with regards to the hint about inclusion/exclusion - where would that come in?  I read about derangements on Wikipedia but it did not make total sense to me (the derivation).  I would appreciate any advice whatsoever including references to other material. Let $f$ be a bounded measurable function on a set of finite measure $E$.  For a measurable subset $A$ of $E$, show that $\int_A f = \int_E f \cdot \chi_A$     Lemma: Let $f$ be a bounded measurable function on a set of finite measure $E$.  Suppose $A$ and $B$ are disjoint measurable subsets of $E$.  Then $\int_{A \cup B} f = \int_A f + \int_B f$.   My first approach was to view the problem like this utilizing the above lemma:  $$\int_E  f \cdot \chi_A = \int_{E\setminus A} f \cdot X_A + \int_A f \cdot \chi_A $$  I'm not really sure how the result follows from the above. NO2 CSV downloaded files shows zero values for consider date range using Google Earth Engine When I am downloading the CSV file, the results show zeros in the CSV file. How can I download exact NO2 values as shown in Chart after running the mentioned below code in Google Earth Engine? // Create a FeatureCollection from the list and print it. var BergamoGeometry = ee.FeatureCollection(BergamoGeometry); print(BergamoGeometry); Map.centerObject(BergamoGeometry); Map.addLayer(BergamoGeometry); //////////////////////////////////////  var dataset = ee.ImageCollection(&quot;COPERNICUS/S5P/NRTI/L3_NO2&quot;)                  .filterBounds(BergamoGeometry)                   .filter(ee.Filter.date('2019-03-30', '2019-04-29'));                   var NO2 = dataset.select('NO2_column_number_density'); var band_viz = {   min: 0,   max: 0.0002,   palette: ['black', 'blue', 'purple', 'cyan', 'green', 'yellow', 'red'] }; // Create and print the chart. print(ui.Chart.image.series(NO2, BergamoGeometry, ee.Reducer.mean(), 30)); How does extra damage work for critical hits? How to render text pixel-perfectly with subpixel antialising with Illustrator? How to capture the scene exactly as my eyes can see? What's the term in music, where the singer uses words without meaning? I've been searching for the word, which is used for describing parts of lyrics, where the singer sings for example the whole refrain only using words of syllables without any meaning. Maybe you could help? (here's one song with that type of refrain: ) To run programs as sudo without password, I edited /etc/sudoers with wrong content.   Then I re-edited it with Mac's default text editor(course sudo cannot be used when /etc/sudoers is wrong).  I tried to run sudo agin, and get this error:   sudo: /etc/sudoers is owned by uid 501, should be 0 sudo: no valid sudoers sources found, quitting   What should I do? I cannot remember the root password (maybe I haven't ever set root password before). What exactly is the cause of RAM incompatibility? I'm looking at rather short QVL lists from manufacturers, and I'm wondering what is the underlying cause of RAM incompatibility aside from the obvious: Obvious Reasons for Incompatibility  RAM Generation (DDR, DDR2, DDR3, DDR4) Form Factor / pin count RAM Type (ECC vs Non-ECC, Registered vs not, Buffered vs not) Minimum speeds on a board that cannot adjust the memory frequency Non-standard voltage range on a board that cannot adjust the memory voltage Channel count (single vs dual vs quad) Bad or inoperable module Mismatched set  Not Obvious Reasons  Brand SKU Max Capacity Speed too high  As in, why can't motherboard manufacturers just list the specifications of the RAM they will accommodate, and support every memory chip of that category instead of listing specific SKUs from specific brands? What happens electrically/physically that causes boards to fail to POST / operate with RAM that is otherwise 'compatible' with all of the specifications of RAM that is on the QVL? What is the best class or classes in 3.5e to build a character that uses both a sword and either arcane magic or psionics which can be fluffed as arcane magic and focuses on fighting powerful individual creatures?    I have a foundness for those that can sling spell along with blade, like Dak'kon from Planescape: Torment.  So, given that, how do I make the best Gishy character I can?  Specifically, the group is looking for a character focused on delivering high damage or other debilitating effects to an individual target.  There is no need to balance sword and magic, though I would prefer to have the melee abilities remain at least relevant through at least mid-levels.  This question:  is related.  But I am specifically looking to optimize for high targeted damage output and have no need to balance between sword and spell if focusing on one is more effective.  I have considered the duskblade and that may be the best answer, but as I have limited experience with 3.5 I want to find out if I should use a pure duskblade, a duskblade which steps into other classes, or some other class I have not considered.  In this particular case, the characters will start at level 7, and equipment rules are liberal as long as the characters can afford it.  I'm not sure how long this will last so I am focused on the next few levels instead of trying to plan all the way out to 20.  It will be a party of 5.  Not all of the rest of the characters have been fully fleshed out, but we will have a cleric and a wizard.  The other two are more up in the air but they one is looking at various rogue classes and the other is looking at some sort of melee class, with this gishy rounding out the team.  The initial encounter is up in the air, but there is a definite fondness for gryphons.  We do know that we will be working up to dragon in relatively short order, which is a factor in wanting to focus on large damage output to individual creatures as opposed to being able to deal with numerous weaker creatures.  The setting is "modified Forgotten Realms".    I personally own absolutely no 3.5 books, but we have access to a fair bit and just about anything from WoTC is permitted.  I am new to 3.5, but I have a fair bit of roleplaying experience including AD&amp;D.    Thematically, I envision the character as a generally honorable, but working as a monster hunter and mercenary.  I definitely favor swords though specifically which sword is flexible.  Dak'kon from Planescape would be an excellent example of what I would like to build, as would The Witcher.  Based on Kryan's excellent answer, it seems to be leading up to using duskblade and a specialist wizard to lead into Abjurant Champion.