'We are wearing a jacket' or 'we are wearing jackets'?
"Those who qualify will be awarded a certificate" or "those who qualify will be awarded certificates"?
Is LAN with StarCraft 2 definitely not possible? Let $M$ be an A-module and $f \in \text{End}(M)$ an idempotent endomorphism. Show that $M = \ker(f) \oplus \ker (f-\text{id}_M)$. I don't know how to approach this exercise, using only properties of modules and direct sums, without projections. Can you help me? Thanks in advance. Decrypt all RSA ciphertexts if you can break 1% of them I was given a cryptography question during an interview the other day.   Imagine you are given an efficient algorithm which for a given RSA public key (n,e) is able to decrypt 1% of the messages encrypted with that key.   1% means that the algorithm will deterministically decrypt 1% of all possible ciphertexts on the first call, and will never decrypt the other 99% of all possible ciphertexts with any number of invocations.   Figure out an algorithm that uses this one and can decrypt any message. Any ideas? Didn't understand relationship 'AccountContactRelation' in FROM part of query call I'm getting the following error message:     System.QueryException: Didn't understand relationship 'AccountContactRelation' in FROM part of query call. If you are attempting to use a custom relationship, be sure to append the '__r' after the custom relationship name. Please reference your WSDL or the describe call for the appropriate names.   Select Email, Id, (Select ContactId, AccountId From AccountContactRelation) From Contact  In AccountContactRelation object there are two lookup fields for Account and for Contact.  Please advise how to avoid such error message? String processing in windows batch files: How to pad value with leading zeros? How to restore original size and unboot my bootable USB pendrive Is the "omniscient-omnipotent-omnipresent" definition of God consistent? is commonly defined as an  (infinite knowledge),  (unlimited power),  (present everywhere) entity.  Is there any logical  in this definition?  I have seen several paradoxes like below     Does God know what he's going to do   tomorrow? If so, could he do something   else?" If God knows what will happen,   and does something else, he's not   omniscient. If he knows and can't   change it, he's not omnipotent.      "Can 'an omnipotent being' create a   stone so heavy that it cannot lift   it?"   Do they mean that the defintion of God ( as commonly held) is logically inconsistent? Proper analyses for 2x2 contingency tables So I have multiple questions about the different types of analyses that can be used on a 2x2 contingency table and when different analyses should be used.   To open, I'll note that this stems from my efforts to find the appropriate analyses to run for research using categorical data in a 2x2 contingency table. In my case I have two conditions: control and experimental and within the experimental group there are two items of interest (say, Item A and Item B). In all cases I'm investigating whether an outcome occurs or not: How many experimental participants show the outcome on Item A? How many experimental participants show the outcome on Item B? How many control participants show the outcome? So in addition to comparing Item A with the Control group (A vs. C) and Item B with the Control group (B vs. C), I also want to compare Item A with Item B (A vs. B).   My understanding is that for between-subjects comparisons (e.g., A vs. C   &amp;   B vs. C) a  is traditionally used and that for within-subjects comparisons (e.g., A vs. B) that  is a common choice.   Question 1: is this an accurate understanding of the common choices for statistical tests using 2x2 contingency tables?    My follow-up questions, and where most of my confusion lies, are related to what to do when multiple observed cell counts are low (less than 10). I've explored numerous threads here, the Wikipedia pages for different topics (Yates's correction, Fisher's exact, etc.), a few journal articles, and descriptions in R-packages, but I have found conflicting views.   Some sources indicate that you should use , others say it's out-dated because, in addition to being overly conservative, it was only preferred because the other options were computationally intense.   Others say that you should use , but I've seen additional information suggesting that Fisher's exact test is only appropriate if you know both the row and column totals and that other options (Boschloo, Barnard, etc.) are generally more powerful. However, which of those to use is also not clear.  Finally, even others suggest that the Pearson's $\chi^{2}$ test is appropriate as long as the expected cell counts are not low.   I've seen some indication that these arguments occur for McNemar's test as well (e.g., continuity correction &amp; McNemar's exact test).  Question 2: Is the assumption regarding cell counts about expected cell counts or observed cell counts? If the observed cell counts are low, does that matter to the type of test that should be used?  Question 3: What is an exact test and when should exact tests be used?    In addition to answers to these questions, I would also greatly appreciate any suggestions as to books or articles I can read to learn more about this. Part of my struggle with all of this has been just general difficult finding clear information about these topics and how they relate to one another. How to store and preserve lots of data? I have many DVDs with movies and games, they are getting old and dusty, some of them are damaged. And I have more than 300 GB of general files in my PC. I want to store and preserve all of my data and keep them for a long long time.  I want to know what is the best and more reliable storage media that I should use and... I can buy. I will copy all of my DVDs to a new media and throw all of my DVDs away. It will take a very smaller physical space.  I think that everything that I need to store is currently 800 GB.  I do not like optical discs, because they are fragile and I have lost many CDs and DVDs. Hard drives would be good, but they can suddenly stop working, and it is very expensive to recover data from a broken hard drive.  Other questions: How often will I have to copy the files to a new storage media? Is it good idea to have another copy of each file stored in another storage media?  Edit: If I put all of the movies and games in a HDD, if the HDD stops working, I will lose everything. This is how valuable they are for me: If I lose 10% of them, I will not cry, because I can get new ones, but if I lose 90%, it will be a disaster.  Edit 2: I do not think that that online storage is reliable even if I pay for it, because they may be closed or may get bankrupt. They are not from my country and it would be difficult to get the files back. I would like to evaluate the sum  $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}$$  Here is my attempt:    Letting    $$f(z)=\frac{1}{az^2+bz+c}$$  The poles of $f(z)$ are located at   $$z_0 = \frac{-b+\sqrt{b^2-4ac}}{2a}$$  and  $$z_1 = \frac{-b-\sqrt{b^2-4ac}}{2a}$$  Then  $$ b_0=\operatorname*{Res}_{z=z_0}\,\pi \cot (\pi z)f(z)= \lim_{z \to z_0} \frac{(z-z_0)\pi\cot (\pi z)}{az^2+bz+c}= \lim_{z \to z_0} \frac{\pi\cot (\pi z)+(z_0-z)\pi^2\csc^2 (\pi z)}{2az+b} $$  Using L'Hopital's rule. Continuing, we have the limit is  $$ \lim_{z \to z_0} \frac{\pi\cot (\pi z)+(z_0-z)\pi^2\csc^2 (\pi z)}{2az+b}= \frac{\pi\cot (\pi z_0)}{2az_0+b} $$  For $z_0 \ne 0$  Similarly, we find  $$b_1=\operatorname*{Res}_{z=z_1}\,\pi \cot (\pi z)f(z)=\frac{\pi\cot (\pi z_1)}{2az_1+b}$$  Then   $$\sum_{n=-\infty}^\infty \frac{1}{an^2+bn+c} = -(b_0+b_1)=\\ -\pi\left( \frac{\cot (\pi z_0)}{2az_0+b} + \frac{\cot (\pi z_1)}{2az_1+b}\right)=  -\pi\left( \frac{\cot (\pi z_0)}{\sqrt{b^2-4ac}} + \frac{\cot (\pi z_1)}{-\sqrt{b^2-4ac}}\right)= \frac{-\pi(\cot (\pi z_0)-\cot (\pi z_1))}{\sqrt{b^2-4ac}}= \frac{\pi(\cot (\pi z_1)-\cot (\pi z_0))}{\sqrt{b^2-4ac}} $$  Then we have  $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c} = \frac{\pi(\cot (\pi z_1)-\cot (\pi z_0))}{2\sqrt{b^2-4ac}}$$    Is this correct?  I feel like I made a mistake somewhere.  Could someone correct me?  Is there an easier way to evaluate this sum? Wild Magic Surge - What does "You cast ..." mean for your spell slots? In response to  question, I started to think about what it means when a Wild Magic Surge results in a "You cast spell-X".  So, "specific beats general" gives:   You can cast a second spell when you would not normally be allowed to You can cast a spell you don't know If the surge forces you to cast a concentration spell you would lose any concentration spell that you currently had active Subject to what the specific entry says you can target it how you like Unless the specific entry prohibits it you could use metamagic on the spell   I am pretty confident in all of the above but if anyone disagrees please address in an answer.  Here comes the question.  How does this impact on your available spell slots?  "You cast ..." to me suggests that if you have an applicable slot available then you have to use it to cast the spell because that is the general rule for casting spells but if you don't then "specific beats general" and it is cast anyway.  Specific examples:     You cast fireball as a 3rd-level spell centered on   yourself   This must be a 3rd level spell:   if you have a 3rd level slot available you have to use it? if you don't, if you have 5 sorcery points and a bonus action available must you use these to get that slot? if you don't have any of that then the spell goes off anyway?      You cast confusion centered on yourself.   No level specified and confusion is a 4th level spell:   if you have a 4th level slot available you have to use it? if you have a 4th, a 6th and an 8th level slot you can choose which one to use? if you have only a 5th level slot but also 6 sorcery points and a bonus action, you could choose to use the 5th level slot or use the points to create a 4th level slot? if you don't have any 4th level or higher slots but you have 6 sorcery points and a bonus action available must you use these to get that slot? if you don't have any of that then the spell goes off anyway?   And so on ... Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$  Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}{n}}}{1-e^{\frac{i2\pi}{n}}} =0$$ by use of the sum of geometric series'. My issue is proceeding further to show that this is indeed true. Let f(t)=2/(t+8)  (a) Simplify the following difference quotient as much as possible: $$[f(t+h)−f(t)]/h$$  MY ANSWER (CORRECT): $-2 / [(t+8)(t+h+8)]$  (b) Use your result from (a) and the limit definition of the derivative to calculate $f′(t)=\lim_{h → 0} [f(t+h)−f(t)]/h=$  MY ANSWER (CORRECT): $-2 / [(t+8)^2]$  (c) Use your answer from part (b) to evaluate $f′(−3)$  MY ANSWER (CORRECT): $-2 / 25$  (d) Find the equation of the tangent line to the curve at the point $(−3,f(−3))$ y=  I have tried the point-slope formula, $y-y_1=m(x-x_1)$, but my answer was incorrect. How to expand the ext4 primary partition size in a VMware Player virtual disk? I have a VMware Player virtual machine setup, running Ubuntu 10.04 as the a guest OS.  The VM was created on a 20GB virtual drive, with a single partition for working space. Finally I got to the point where I need to expand the size of the virtual drive. Resizing the drive itself, adding some 10GB, was easy via the VMware Player menus. I now need to resize the main partition itself (I prefer this over creating a separate partition and mounting it).  According to GParted, I now have the following partitions:    My question is, how can I increase the size of the Filesystem partition to include the free 10GB?  From information I found online, it looks like one limitation is the fact that he swap partition is immediately adjacent to the filesystem, so my understanding is that first thing would be to move that partition to the end of the volume. How can this be done?  Also, some sources suggest that the system should be booted from a LiveCD, so the filesystem is not the boot partition, and thus make it easier to resize. As mentioned above, I run a VM, so I am not sure how to boot it from a LiveCD. Any ideas? Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular. Is there a function or method to give the last day of a month? For example if the date is 27 feb 2013, the last date should be 28 feb 2013 , it become 29 if its a leap year. if the date was 2 jan then it should return 31 Jan.  If there is no function then any ideas on how we can get this?  Thanks Geomagnetic reversal? Mcrypt extension is missing in 14.04 server for mysql The security guard problem How to stop the find command after second match? How to make find command, that will show only first two matches and then end, something similiar to  just version for two records.