Factor a given determinant using row and column operations.
Vandermonde determinant for order 4
Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$ All the units are satisfied Pell's equation $a^2-2b^2=\pm1$  for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$.  Here is my proof:  Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies $$(a+b\sqrt{2})(c+d\sqrt{2})=1, c+d\sqrt{2}\in\mathbf{Z}\sqrt{2}$$ $$\implies \mathrm{norm}((a+b\sqrt{2})(c+d\sqrt{2}))=1$$ $$\implies \mathrm{norm}(a+b\sqrt{2})\cdot \mathrm{norm}(c+d\sqrt{2})=1$$ $$\implies (a+b\sqrt{2})(a-b\sqrt{2})(c+d\sqrt{2})(c-d\sqrt{2})=1$$ $$\implies (a^2-2b^2)(c^2-2d^2)=1$$ $$\implies \left[ a^2-2b^2=1\text{ and } {c}^2-2{d}^2=1\right]\text{ or } \left[ a^2-2{b}^2=-1\text{ and }{c}^2-2{d}^2=-1\right]$$ $$\implies {a}^2-2{b}^2=\pm1$$ What is "Go" in cribbage? I've played a fair bit of cribbage, but I've never heard of scoring one point for "Go" (as mentioned in this other question: ).  What is it, and what impact does it have on the game? How is the Schroedinger equation a wave equation? Wave equations take the form:  $$\frac{ \partial^2 f}  {\partial t^2} = c^2 \nabla ^2f$$  But the Schroedinger equation takes the form:  $$i \hbar \frac{ \partial f}  {\partial t} = - \frac{\hbar ^2}{2m}\nabla ^2f + U(x) f$$  The partials with respect to time are not the same order. How can Schroedinger's equation be regarded as a wave equation?  And why are interference patterns (e.g in the double-slit experiment) so similar for water waves and quantum wavefunctions? Prove that a power series $\sum_{n=0}^{\infty}{{a_n}{z^n}}$ which converges for any $z \in \mathbb{N}$, converges for any $z \in \mathbb{C}$. PROBLEM  Prove that a power series $\sum_{n=0}^{\infty}{{a_n}{z^n}}$ which converges for any $z \in \mathbb{N}$, converges for any $z \in \mathbb{C}$.  MY ATTEMPT  Since the power series $$\sum_{n=0}^{\infty}{{a_n}{z^n}}$$ converges for any $z \in \mathbb{N}$, then $$\sum_{n=0}^{\infty}{{a_n}{z^n}}$$ can be considered (?) to be a polynomial in $\mathbb{C}$.  Hence, $$\sum_{n=0}^{\infty}{{a_n}{z^n}}$$ is entire, and therefore continuous in $\mathbb{C}$ (since differentiability implies continuity).  It follows that $$\sum_{n=0}^{\infty}{{a_n}{z^n}}$$ converges for any $z \in \mathbb{C}$.  QUESTION     I am aware that, while I may have (some) ideas for a (general) proof, admittedly there are several grey areas (i.e., gaps) that need to be filled.  Would you be kind enough as to fill in these gaps, by providing useful hints? How do Southern people discern Night's Watch deserters from truthful brothers sent south? Night's Watch deserters are sentenced to death, as we've seen in the very first episode of Game of Thrones. A man dressed all in black is apprehended by Stark men and brought before Lord Eddard Stark. Lord Stark sees him as a deserter and sentences him to die (and then executes that death penalty).  However, there are men of the Night's Watch south of the Wall, on a legitimate mission or on some kind of allowed leave. For instance, we have Benjen Stark visiting Winterfell when the king and his party is there. There is also Yoren, who travels around Westeros in search of new recruits.  I could imagine that the Warden of the North (and other lords in the northern part of Westeros) would have a good idea of who would be allowed further south and whose appearance would raise suspicion. But I reckon the southern lords don't have as many visits from the Night's Watch, so they might not know who would be allowed to be there and who wouldn't.  How would lords who are rather unfamiliar with the specific brothers of the Night's Watch discern deserters from brothers sent out for some reason?  That ends my question, but there is a spoiler from the season 5 finale that continues the question a bit more:      Sam is sent to Oldtown to become a Maester, so he can replace Maester Aemon at Castle Black.  But he is riding together with Gilly (a woman!) and her baby (a baby!), which might raise a lot of suspicion from people who don't know what he is sent out to do.  What prevents southern lords of sentencing Sam as a deserter from the Night's Watch? Schengen area 90 days limit: Can I leave on the 91st day? Can anyone tell me if the 90 days is in fact 90 nights? Can you leave the Schengen area on the 91st day? Or are you only allowed 89 nights and you are required to leave on the 90th day? I need help with this:  If a is an element of $\mathbb Z_n$ and $\gcd(a,n) &gt; 1$ , then a is not invertible.   First you show that if $a$ is an element of $\mathbb Z_n$ and $\gcd(a,n)&gt;1$, then there  is an element $b$ of $\mathbb Z_n$ and ($b$ is not equal to the zero element) from which $ab = 0$.   The second part I cant get. i.e. show if $b$ is not the zero element  and $ab=0$ then $a$ is not invertible. Read Committed Snapshot Isolation vs Read Committed - pros and cons? Can a polymorphed creature use class features that don't require speech or hand gestures? The Polymorph spell states (in part):     The target’s game statistics, including mental ability scores, are   replaced by the statistics of the chosen beast. It retains its   alignment and personality.      The creature is limited in actions it can perform by the nature of its   new form, and it can’t speak, cast spells, or take any other action   that requires hands, or speech.   So, can a polymorphed Paladin use their Divine Smite class feature?  It expends a spell slot, but it is not a spell, and as far as we can tell does not require hands or speech.  We know (based on the 10th level Transmutation Wizard feature) that a wizard is supposed to be able to polymorph herself, and we can assume that she is also supposed to be able to maintain concentration needed to stay in the new form (otherwise it would be a pretty useless ability).  This is (as far as I can tell) not something any of the beasts eligible as polymorph targets can normally do.  So it is clear that the polymorphed creature maintains some abilities from her old form.  The question is which ones.    The Druid's Wild Shape ability explicitly says you can use class features, while Polymorph does not, suggesting that you can't use class features while polymorphed.  However, Wild Shape also explicitly says that you can continue to concentrate on a spell (see below, emphasis mine), and Polymorph does not, yet we believe that a polymorphed creature can continue to concentrate on the polymorph spell.  So the fact that Wild Shape explicitly allows something and Polymorph is not explicit about it does not necessarily mean that the polymorphed creature cannot do those things.  From Wild Shape rules:     You can't cast spells, and your ability to speak or take any action   that requires hands is limited to the capabilities of your beast form.   Transforming doesn't break your concentration on a spell you've   already cast, however, or prevent you from taking actions that are   part of a spell, such as call lightning, that you've already cast.   So, since Polymorph is silent on the question of whether or not class abilities can be used, and states only that the polymorphed creature cannot perform actions that require hands or speech, how can we best determine whether (RAI) polymorphed creatures maintain any of their old abilities beyond spell concentration? How can I get very vivid colors in night cityscape pictures? Why isn't "Column ordering" available in my list settings? I want to change the column order visible when viewing or editing a list item. I have accessed the "Column ordering" interface via the list settings page on other lists in the past, but it isn't visible in this list's settings. how to add xsl attribute What does "\" mean in math Is there a way to turn option strict off for just a single line of code?    I'm doing some maintenance work and I need to "cheat" in just one place and I don't want to lower the standard for the entire file. Benefits of using QQ-plots over histograms In , Nick Cox wrote:     Binning into classes is an ancient method. While histograms can be useful, modern statistical software makes it easy as well as advisable to fit distributions to the raw data. Binning just throws away detail that is crucial in determining which distributions are plausible.   The context of this comment suggests using QQ-plots as an alternative means to evaluate the fit. The statement sounds very plausible, but I'd like to know about a reliable reference supporting this statement. Is there some paper which does a more thorough investigation of this fact, beyond a simple “well, this sounds obvious”? Any actual systematic comparisons of results or the likes?  I'd also like to see how far this benefit of QQ-plots over histograms can be stretched, to applications other than model fitting. Answers on  agree that “a QQ-plot […] just tells you that "something is wrong"”. I am thinking about using them as a tool to identify structure in observed data as compared to a null model and wonder whether there exist any established procedures to use QQ-plots (or their underlying data) to not only detect but also describe non-random structure in the observed data. References which include this direction would therefore be particularly useful. Suppose we have a magnetic field directed into the computer screen. Assume that there is no gravitational field. Now suppose we have a stationary positively charged particle in the magnetic field. The magnetic force is given by $$\mathbf{F}=q\mathbf{v}\times\mathbf{B}$$  Since $\mathbf{v}=0, \mathbf{F}=0$.  Now pick a reference frame moving to the left with constant velocity. In this frame, the particle is moving to the right with constant velocity. So, the magnetic force is directed upwards.  From what I've learnt in physics, forces are independent of choice of reference frames as long as the frames are inertial. But in the 2 inertial reference frames I have chosen, I get 2 different magnetic forces. How do we reconcile this? Split-like environment inside cases environment Knowing which java.exe process to kill on a windows machine When a java based application starts to misbehave on a windows machine, you want to be able to kill the process in the task manager if you can't quit the application normally. Most of the time, there's more than one java based application running on my machine. Is there a better way than just randomly killing java.exe processes in hope that you'll hit the correct application eventually?  EDIT: Thank you to all the people who pointed me to Sysinternal's Process Explorer - Exactly what I'm looking for! Unable to install "": snap "" has "install-snap" change in progress $\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?  My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains "the subset $\{\emptyset\}$".