Do I use a comma when there are two items/people listed?
In what cases should I use a comma before "or"?
Eigenvalues of symmetric matrices are real without (!) complex numbers Is there any proof of the fact that the eigenvalues of symmetric matrices (i.e. $A\in\mathbb{R}^{n\times n}$ with $A^t=A$) are real without the use of the concept of complex numbers? What is the benefit/downside to using a switch statement vs. an if/else in C#.  I can't imagine there being that big of a difference, other than maybe the look of your code. Is there any reason why the resulting IL or associated runtime performance would be radically different? Related: Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable) This question is me trying to understand (again) why there can be no one-to-one correspondence between the sets of natural and real numbers.  The source of confusion is this: if we abstract completely from the properties and operations on numbers and just think of them as strings of digits, it appears to be possible to list all the real numbers between $0$ and $1$ and all positive integers having the same digits in the same (or reverse) order, for example:  $$0.1 \to 1$$  $$0.01 \to 10$$  $$0.0101 \to 1010$$  $$0.110101 \to 101011$$  I use binary expansion for simplicity.  Since irrational numbers have infinite expansions, we would have to consider some kind of 'infinite natural numbers', like:  $$101110001\dots$$  or  $$\dots101110001$$  I think the main (and only) reason why such numbers don't exist is this requirement:   We need the sets of real and natural numbers to be ordered. We need to be able to do arithmetics on real and natural numbers.   Obviously, if we consider any of two types of 'infinite natural numbers' and the usual absolute value, we find that it's impossible to order them, or do arithmetics with them.  The only way is to change the norm (to $p-$adic norm for example).  While with the real numbers it's still possible to order them, and do arithmetics with them when they have infinite amount of digits in their expansion (or at least in principle).     Is my reasoning correct? Are the two requirements (ordering and arithmetics) enough to explain why natural numbers with infinite amount of digits do not exist, while real numbers do?   This confusion is not restricted to me alone, see  for example Stop Excel from automatically converting certain text values to dates Does anyone happen to know if there is a token I can add to my csv for a certain field so Excel doesn't try to convert it to a date?  I'm trying to write a .csv file from my application and one of the values happens to look enough like a date that Excel is automatically converting it from text to a date.  I've tried putting all of my text fields (including the one that looks like a date) within double quotes, but that has no effect. UK PLAB Visa Refusal due to unexplained bank deposits I had applied for a PLAB (Doctors) Visa for the UK to appear in PLAB test. I got refused on the basis of two recent large deposits in the bank statement I provided (my mothers). In reality, these are my own savings which I had deposited in her account to make it more strong. The only objection I got was that I must provide evidence for the source of these deposits. Other than this, I have previously travelled to the UK and USA as well as other parts of the world and my travel history is fine. Can someone please advise on how to proceed and re-apply? Can 'AMD-only' RAM really be restricted to AMD chipsets, and if so, why? Every time I touch an object a full-render starts I don't know what setting I switched, or what happened, but for some reason now every time I move, scale or rotate an object in the 3d View - blender starts a render. I am not talking about the 3d View being in "Rendered" mode. I am talking about the same behavior as when you click the "Render" button (switch to image editor, begin full render). This is driving me crazy and I can't seem to figure out how to turn it off or what I did to start it. It happens whether I use the mouse or keys. If I open a new document the behavior goes away so it is probably a preference setting(?) If you can help, I will be eternally grateful.  Thanks in advance. In LWC am trying to iterate over a list and show the details from the list, this is working correctly , but as a header want to show the record number in a displayed order, below is desired outcome and simplified sample code.  Desired OutCome:   Employee Record 1: A  Employee Record 2: B  Employee Record 3: C   JavaScript:  Var empNames = ["A" , "B" , "C"];   HTML:  &lt;template for:each={empNames} for:item="employee" for:index="index"&gt; Employee Record{index} {employee} &lt;/template&gt;    Actual outcome  Employee Record 0: A  Employee Record 1: B  Employee Record 2: C   I understand as i am using {index} which indicated the position of element in an array which starts from 0 my actual outcome also starts like Employee Record 0 instead of Employee Record 1, but is there a simple way without writing much logic on JS to handle this in html and to make my headers starts from 1 instead of 0? Finding minimal form -- Velleman exercise 1.5.7a Skyrim crashes as soon as I boot up into the menu. How do I fix this? Whenever I boot up Skyrim, it crashes as soon as it reaches the main menu with no dialog or anything. I thought this was related to my using dual monitors somehow, but I disconnected the other monitor and the game still crashed. I tried running the game in different resolutions, both full-screen and windowed, at various quality settings; no dice. My computer does meet the minimum requirements to play Skyrim.  How do I fix this problem? Evaluate the integral $\int_0^{2a}\int_0^\sqrt{2ax-x^2}\frac{\phi'(y)(x^2+y^2)x dxdy}{\sqrt{4a^2x^2-(x^2+y^2)^2}}$ How do I address unchecked cast warnings? Checking if a worksheet-based checkbox is checked Proving the Lorentz invariance of the Lorentz invariant phase space element I have been looking around for a satisfactory answer to prove that  $$\frac{d^3\vec{p}}{2E_{\vec{p}}}$$  where $E_{\vec{p}}=+\sqrt{(|\vec{p}|c)^2+(mc^2)^2}$, is Lorentz invariant. The standard answer seems to be that the above measure is equal to  $$d^4p \, \delta (p^2-m^2)\theta(p_0)$$  where $p^2=p_0^2-|\vec{p}|^2$, and $\theta(x)$ is the step function. I understand these two are equivalent, but I don't understand why the second has to be Lorentz invariant, in particular why the Dirac delta has to be Lorentz invariant. I have found a  (section 2.1) that proves that $\delta^{(4)}(p-p')$ is Lorentz invariant, but I can't find a way to extend their method successfully here. In fact, all I can seem to get tells me the above isn't Lorentz invariant, and that in fact it should transform to   $$\frac{d^3\vec{p}'}{2\gamma E_{\vec{p}'}}$$  which makes sense from $d^4p$ being Lorentz invaraint and $dp_0$ transforming proportional to $\gamma$, but it isn't what everyone else says. What's the problem here?    An alternative way to "derive" the factor of $1/\gamma$:  $$\delta (p^2-m^2)\theta(p_0)=\frac{1}{E_{\vec p}}\delta(p_0-E_{\vec{p}})$$  Now, as $dp_0$ transforms to $\gamma dp_0$, the $\delta(p_0-E_{\vec{p}})$ should transform to $\gamma^{-1}\delta(p_0-E_{\vec{p}})$, as can be shown with an analogous argument to the one shown in the . The $E_{\vec{p}}$ also transforms to $\gamma(E_{\vec{p}}-v p_x)$ for an $x$-boost, but this doesn't seem to solve it. Adjective, if adjective: eg "pointed, if fatherly" They were pointed, if fatherly, remarks that echoed the themes he has stressed in his papacy but ones that resonated all the more in a newly renovated cathedral surrounded by the luxurious shops of Fifth Avenue. —   I'm not sure the meaning of the phrase &quot;if fatherly&quot; in the above sentence. I'm not familiar with the syntax using the structure 'adjective, if adjective'. Why does `ls -d *[!e]*` display all files instead of omitting all files that contain an e? ls -d *e* prints out all files containing an "e" somewhere in it. But the opposite, ls -d *[!e]* does not seem to work. Why exactly is that?  is it because ls takes * as its own argument and expands all the files?  I understand I can do: ls -d !(*e*) (in Bash with extglob set, or in ksh). "Whom" or "who" where the referent is both subject and object? I understand there has been so much on this topic but I am still confused. I get that if the person is the subject it is who and anything else is whom. However, I'm really struggling to work out this rule for the following sentence:     The scope of which his account of morality applies to is therefore restricted to those whom/who the ‘notion of justification to a being of that kind makes sense.’   Should it be who or whom? I can't work out the subject and object. I am a new user, who has just been told to run / execute a command. The answer has some text with a grey background like this text here or  sometimes this text here.   However, I don't know what "run" or "command" means in relation to Ubuntu.  How do I do this?    This is intended for use as a Q&amp;A you can link to when you tell someone to run a command. Proving that the Intersection of any Collection of Compact Sets is Compact I was trying to do this problem this way:  Let $\mathcal{B}=\{B_i\}$ be a collection of compact sets. By Heine-Borel, each of the $B_i$'s are closed and bounded. We already know that the intersection of a collection of closed sets is once again closed, so $\cap \mathcal{B}$ is closed.  If I can prove $\cap\mathcal{B}$ is bounded as well, I can use Heine-Borel to say it is compact. So I was not sure how to show it was bounded, but a friend showed me his way of doing it. He did it the following way:  Let $b,a\in\mathbb{R}$ be an upper and lower bound for $B_1$, respectively. Since $\cap\mathcal{B}\subset B_1, a$ is a lower bound for $\cap \mathcal{B}$ and $b$ is an upper bound for $\cap\mathcal{B}$. Therefore, $\cap\mathcal{B}$ is bounded and then by Heine-Borel, it is compact.  My question is on the paragraph right above. I don't understand why what he did means that $a$ and $b$ are lower and upper bounds for $\cap\mathcal{B}.$ Can anyone explain why this is? Thanks. Ahlfors "Prove the formula of Gauss" He says:     Prove the formula of Gauss:   $$ (2\pi)^\frac{n-1}{2} \Gamma(z) = n^{z - \frac{1}{2}}\Gamma(z/n)\Gamma(\frac{z+1}{n})\cdots\Gamma(\frac{z+n-1}{n}) $$   This is an exercise out of Ahlfors.  By taking the logarithmic derivative, it's easy to show the left &amp; right hand sides are the the same up to a multiplicative constant.  After that I'm lost. It's easy using another identity when $n$ is even to use induction. But when $n$ is odd I am lost.  It's obvious when $n$ is a power of 2.