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NullReferenceException in Unity Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime? This is in follow-up to .     Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the constraint, need it be prime?    Since Zorn's Lemma applies equally well to proving that either minimal prime ideals exist or that maximal ideals contained in $Z$ exist, this would provide an alternative proof of that question if true.  A naive approach to proving this is to assume $ab \in I$ with $a, b \notin I$. Since $ab \in I$, it follows that $ab$ is a zero divisor, so $abx = 0$ for some $x \neq 0$, and therefore either $a$ is a zero divisor annihilated by $bx \neq 0$ or $b$ is a zero divisor annihilated by $x$. In either case, the obvious thing to consider would be $aR+I$ or $bR+I$, but there's no obvious reason why either of these ideals should consist of zero divisors.  According to rschwieb, the answer is yes for reduced Noetherian rings. Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.   I was thinking of showing $A$ is not invertible, meaning it has an eigenvalue of $0$. Since no matter the power you give to $A$, it is still not the identity, you can deduce that at least one eigenvalue is indeed $0$. However this doesn't work if you have different eigenvalues in the Matrix, so I got stuck.  Help will be most appreciated. Intersection of two parabolae Problem: Consider two parabolae such that their axes of symmetry form a right angle. Prove that all four points of intersection lie on a common circle (it is an assumption that there exist such four points). Could you help me obtain a non-algebraic solution? (The algebraic one I did is given below.)  Algebraic solution:  There is a very simple algebraic approach, just introduce a Cartesian coordinate system such that the parabolae are given by equations:  $$y = \alpha_1 x^2 + \beta_1 $$ $$x = \alpha_2 y^2 + \beta_2 $$  and by scaling, adding and simplifying we can get the following ordinary circle equation (the right-hand side is positive because parabolae intersect): $$ \left(x-\frac{1}{2\alpha_2}\right)^2 + \left(y - \frac{1}{2\alpha_1}\right)^2 = \frac{-\beta_1}{\alpha_1} + \frac{-\beta_2}{\alpha_2} + \left(\frac{1}{2\alpha_1}\right)^2 + \left(\frac{1}{2\alpha_2}\right)^2$$  Non-algebraic solution: ?? As I understand these two statements:   An electromagnetic field gives particles charge A photon is a quantum of electromagnetic field   It must mean that a photon carries charge. But I guess it isn't true. Why not? I have a dual boot setup with Ubuntu and Windows 7. I need to shrink my root Ubuntu volume in order to get some free space to create a new partition. Are there any Windows based tools that will do this for me? I tried booting up in gparted to do this but I cannot tell which partition is which since I cannot find the size (it displays everything in blocks). CSS3's border-radius property and border-collapse:collapse don't mix. How can I use border-radius to create a collapsed table with rounded corners? packed vs unpacked vectors in system verilog Looking at some code I'm maintaining in System Verilog I see some signals that are defined like this:  node [range_hi:range_lo]x;   and others that are defined like this:  node y[range_hi:range_lo];   I understand that x is defined as packed, while y is defined as unpacked.  However, I have no idea what that means.  What is the difference between packed and unpacked vectors in System Verilog?  Edit: Responding to @Empi's answer, why should a hardware designer who's writing in SV care about the internal representation of the array?  Are there any times when I shouldn't or can't use packed signals? Kali Linux: apt update returns "Hash Sum mismatch" error I'm experiencing exactly the same issue as described in this question: . Before you mark this as a duplicate however, I have tried the solutions posted there, as well as on numerous other sites, including:  sudo apt-get clean sudo rm -rf /var/lib/apt/lists/* sudo apt-get update   Editing /etc/apt/sources.list with alternate official mirrors, such as  deb http://mirrors.ocf.berkeley.edu/kali kali-rolling main non-free contrib or deb https://http.kali.org/kali kali-rolling main non-free contrib  Everything worked after I first imported the VM. I ran sudo apt update and it found some ~650 packages to upgrade. I ran sudo apt upgrade and it encountered an error partway through. That error was solved using sudo apt --fix-broken install, but that is when this hash sum error began. Unfortunately due to hours of troubleshooting I no longer have the details of the earlier error, but I believe it was an error extracting a package due to corrupt data.  I've tried multiple official mirrors, but I get the same error. Additionally, when I downloaded the Packages.gz file  on my Windows machine (VM host) and computed the SHA256 hash, I got the exact hash that apt printed as the expected value. This led me to believe that the error was not with the mirror but with my VM.  The next thing I tried was wget https://mirrors.ocf.berkeley.edu/kali/dists/kali-rolling/main/binary-amd64/Packages.gz followed by sha256sum Packages.gz, which provided yet another different hash output.  To be clear, I have seen 3 different hashes for the same file:   The "correct" hash shown by apt as expected, which is the one that windows also produced after downloading the file using a browser The incorrect hash calculated by apt, which led to the error A different hash calculated by sha256sum after downloading the file using wget using the same URL as for the browser download   I should also note that I have only been referencing the SHA256 hash in each step. The other hash functions are also mismatched when I run sudo apt update, but the file size is the same. I had considered that downloads might be failing due to limited disk space (it is a VM after all) but I don't think that's the case.  What am I missing? Folders Completely Hidden Except via Save As I'm using an audio program called Supercollider on Windows 10. I have saved some files in a folder called "tunes" from SC via Save As, but Windows Explorer and CMD both seem unable to recognise access or recognise the "tunes" folder.  C:\Users\HP&gt;cd "C:\Program Files (x86)\SuperCollider-3.7.2\supercollider-robin\tunes" The system cannot find the path specified.  C:\Program Files (x86)\SuperCollider-3.7.2\supercollider-robin&gt;dir/a doesn't show it either.  However, when saving from SC with Save As, it's there clear as day. I can even shift+right-click to get the path: "C:\Program Files (x86)\SuperCollider-3.7.2\supercollider-robin\tunes"  This makes no sense to me. Can anyone explain and tell me how to access my folder tunes folder please? I have created some search entries in Google Chrome using 'Edit search engines'.  How can I share some of these entries with my colleagues? Why does an atom remain uncharged after emission of an alpha particle? When an alpha particle is emitted, two protons and two neutrons leave the nucleus but the electrons remain the same in number. Why does the atom remain uncharged although it appears it should have a net charge of $-2e$? Since i upgraded to Ubuntu 15.04, if i click on the shutdown button it only logs out and i have to click again on the shutdown button (from the ubuntu login screen) to power off the pc. How can i fix this? Group files in some folders I have some files with different extensions such as *.pdf, *.mp3, *.jpg and a few others. All of them are stored in a parent directory.  How can I get a list of all extensions, create some folders based on these  extensions and then move all files into their relevant folders? Give moderators the ability to mark reviews as "bad" From what I've seen, many SO moderators trawl completed reviews and watch out for robo reviewers; and manual banning them when they see enough (>= 3?) bad reviews. This is inefficient because:   It doesn't integrate into the audit system. There's no "memory" to it. If mod A finds that user X has made two (any number less than the threshhold) bad reviews, he may make a mental note but there probably won't be any permanent/shared memory of this. If moderator B finds this user at a later date, he will have to look through the entire history, including stuff that mod A has already checked.    What I propose is that moderators get the ability to mark individual reviews as "invalid" from the completed review page. This is only to be used for reviews where any reviewer who is paying attention (and knows the rules) would choose the opposite reason (spam and other low quality stuff).   These bad review flags are treated like audits for a user -- if the number of failed audits and bad reviews is enough, the user gets autobanned. The nice thing is that this system should (probably) easily be able to , fixing the "Why was I manual banned?" problem.  This could also be integrated well with . How to deselect the checkbox when max row selection is set to 1 in LWC? When a new tag is created, should there be some kind of notification to the user?  This could serve a dual purpose:   To make the user check that this tag is really required.  Is it spelt right? Might there be a more appropriate tag to use instead? - See  To promote the creation of tag wikis.   These are both beneficial to the site.  Making users think twice about their tags reduces the amount of retagging edits that are needed and reduces noise.  Promoting tag wikis means that more new tags would come with explainations, making them more user-friendly for others.  Creating a new tag is a privilidge earned at 1500 reputation.  Hopefully the user has gained a feel for the ethos of SO by this point, and is either enjoying being a contributer or is mainly asking good, well recieved questions.  The user is probably also wanting to contribute to the site further (even if just to gain reputation), and so it makes sense to me that there should be a gentle push towards taking a little bit of time to write a short wiki for the tag you just created.  By no means should this be mandatory, and the precise format of this gentle push is more in the scope of UX guys to think up, but I wanted to see how people felt about some kind of discreet notification.  I thought of two forms this notification could take:  Option 1 - at the time of tagging      Option 2 - after posting      Update: Raphael has suggested in the comments that this could be implemented as a 10K user tool.  While I think that's a good idea on its own, I don't think it should be used instead.  I think that encouraging users with less than 10K to complete wikis for their own tags is likely to get some wikis written because there is a sense of ownership over your own tag, and by giving it a wiki you are giving it a better start in life. As with a child, users will want to equip their tag with the tools it will need to survive in the big wide world. In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$. Is $I$ a Prime ideal of $\mathbb{Z}\left [ x \right ]$?  Let $f_{1}\left ( x \right ),f_{2}\left ( x \right ) \in \mathbb{Z}\left [ x \right ].$ Let $f_{1}\left ( x \right )f_{2}\left ( x \right ) \in I$. Clearly, I is a proper ideal of $\mathbb{Z}\left [ x \right ]$ since the elements in $\mathbb{Z}\left [ x \right ]$ evaluated at 0 can produce an odd integer but this cannot be true in I. Now, $f_{1}\left ( 0 \right )f_{2}\left ( 0 \right )$ is an even integer in I. Hence, either $f_{1}\left ( 0 \right )$ has an even constant integer or $f_{2}\left ( 0 \right )$ has an even constant integer. Thus, I is a prime ideal by definition. Is this correct? I would now like to show if I is a maximal ideal. Again, clearly, I is a proper ideal by the above argument. Let B be an ideal of $\mathbb{Z}\left [ x \right ]$ and I $\subseteq B\subseteq \mathbb{Z}\left [ x \right ]$. How should B be defined so that I would conclude either $B=I$ or $B=\mathbb{Z}[x]$? Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$ given a random sample from a normal distribution ... Suppose $Y_1, \ldots, Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Let $S^2$ be the sample variance, which is unbiased for $\sigma^2$.  GOAL: Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$.  I realize that $(n-1)S^2/ \sigma ^2$ is chi-square with $(n-1)$ degrees of freedom.  I am given that $$E(S) = E\left[\frac{\sigma}{\sqrt{n-1}} \left[\frac{(n-1)S^2}{\sigma^2}\right]^{1/2}\right] = \frac{\sigma}{\sqrt{n-1}} \int_0^\infty v^{1/2} \frac{1}{\Gamma[(n-1)/2] 2^{(n-1)/2}} v^{(n-1)/2} e^{-v/2} \, dv$$  Can anyone help me understand why they have $v^{1/2}$ and $v^{(n-1)/2}$ (combining them $v^{n/2}$ as I think it should be $v^{1/2}$ and $v^{\frac{(n-1)}{2}-1}$ (combining both terms give $v^{(n-2)/2}$ since the pdf for chi-square distribution is $$ f(v) = \frac{v^{(x/2)-1} e^{-v/2}}{2^{x/2}\Gamma(x/2)},$$ where $x$ is degree of freedom?  Could anyone please tell me which one is the correct answer with explanation?  Thank you How do I get my Python program to sleep for 50 milliseconds?