basis for product topology
Why does the product topology allow proper subsets for only finitely many elements?
I've got to send a bulk email out to a bunch of people in reply to emails they sent me. I've got all the original emails stored in a single outlook folder. I want to extract all of the reply addresses from the emails in that folder so I can send an email to all of them.  The article  shows you how to export the FROM address of these emails, however a large amount of them came from a web service that sends emails 'on behalf of' the person I'm trying to mail to. So I need the reply-to address, and the export wizard doesn't do that.  Does anyone have a tip on how to do this? This is Outlook 2007 on Exchange 2010. Can we calculate the standard error of prediction just based on simple linear regression output? The standard error of prediction in simple linear regression is $\hat\sigma\sqrt{1/n+(x_j-\bar{x})^2/\Sigma{(x_i-\bar{x})^2}}$.  My question is to calculate the standard error of prediction for $pop=1029$ just based on the following regression output. I can get all except $\bar{x}$. And I also know how to calculate the approximate standard error of prediction based on the standard errors of intercept and coefficient of $pop$, ignoring their correlation. Proof of a formula involving Euler's totient function: $\varphi (mn) = \varphi (m) \varphi (n) \cdot \frac{d}{\varphi (d)}$ The third formula on the  states that $$\varphi (mn) = \varphi (m) \varphi (n) \cdot \dfrac{d}{\varphi (d)} $$ where $d = \gcd(m,n)$.  How is this claim justified?  Would we have to use the Chinese Remainder Theorem, as they suggest for proving that $\varphi$ is multiplicative? For example,   If this van's a-rockin', don't come a-knockin'  (song title)   (song title)   I have a few related questions:   What is the "a-" or "-a" called? Is there a function beyond a rhythmic one? Why is "a-" attached to a word at all -- and why attach to a word at all? crossfade between 2 videos using ffmpeg What is the difference between '->' (arrow operator) and '.' (dot operator) in Objective-C? Does a magic item used as an improvised weapon count as a magic weapon? I just started real analysis.  I don't have a background in proofs or logic, simply calculus.  So I'm trying to learn more about proofs--so forgive the basic question, please.  How do you go about proving this theorem: If $a$ and $b$ are any numbers, then there is one and only one number $x$ such that $a + x=b$.  This number is given by $x=b+(-a)$.  First part of the proof I understand--it's simple.  We simply use some axioms and do the following:  $a+b-a=b$, which of course is true.  This is just true from plugging in the $b+(-a)$ for $x$.  But how about the uniqueness issue?  In my text, it says:  $$(a+x)+(-a)=b+(-a) x=b+(-a)$$  How does this prove uniqueness?  Is this equation essentially saying that the ONLY possible value of $x$ is expressed by $b+(-a)$?  How would I have known to set this up as part of the proof if I was asked to prove something is unique?  Sorry for the basic question--have to start somewhere. Does a correlation matrix of two variables always have the same eigenvectors? Check if file is already open $k$ times's draw from $n$ numbers with replacement, each number at least appear once Sample $k$ times with order from $n$ distinct numbers with replacement where $k\ge n$. Here "with order" means the sample is treated like a sequence rather than a set, so results like "1,3,3" and "3,1,3" are different.  The question is what is the probability that every number is sampled at least once? My answer is $1-\frac{{\sum\limits_{i = 1}^n {C_n^i\times i! \times {{(n - i)}^k}} }}{{{n^k}}}$ where $C_n^i$ denotes $n$ chooses $i$.  ${C_n^i\times i! \times {{(n - i)}^k}}$ means first choose $i$ ($i=1,2,...,n-1$) of numbers not to appear in the sample, and then we sample from the remaining $n-i$ numbers for $k$ times.   My question is, is there more elegant solution where the numerator can be directly computed? i.e. can we directly calculate the number of possible sample sequences in which every number has at least one occurrence? Thank you!    PS: my attempt was $C_k^n\times n!\times n^{n-k}$, meaning that we first choose $n$ positions in the sequence of length $k$ to place $n$ numbers, and then permute the $n$ numbers. This is to ensure every number to appear once. The remaining $n-k$ are sampled from the $n$ numbers without restriction.   However, I later found this seems not correct, and it counts the same sequence multiple times.  PS2: as pointed out in the comments, my first answer "1- ..." is not correct either. What does Fermi level in the band gap mean? What does it mean that the  for some semiconductors lie in the band gap?  Is Fermi level definition different from what is know as usual?  We define the Fermi level as the highest level of energy which is occupied by electrons.  Then what is missing here? Some kind of + (singular or plural)? I have a question about using the phrase "some kind of _".  Is it incorrect to say "some kind of + plural noun"?     For example, is it wrong to say "are these rock-like things some kind of chocolates?"   I didn't use "kinds" and that kind of adds to my question. Is "kind" always plural, like "some kinds of chocolates"? Because although I'm using "some", it feels like "kind" shouldn't be a countable noun.  Anyway, I don't think I've heard of "some kind of + plural", but I think it makes more sense then using singular, since I think chocolates are countable. Please help! , I shouldn't need to vote down or flag the answer, because the question itself is a dog.      Users should be allowed to also enable the "I'm done" button by down voting, close voting or flagging the question rather than directly taking action on the answer.    Currently, I have to click on the question's title, close or down vote it in the new tab, then go back to the review page, and click "Not sure" to continue on my adventure. My wifi connection doesnt work properly; it stays connected only for around 5 minutes after I turn on the PC. I'm using Ubuntu 16.04.  I've tried   sudo service network-manager restart   which makes no difference  Same with   killall nm-applet &amp;&amp; nm-applet &amp;   After that I tried   sudo gedit /etc/pm/config.d/config   where I insert this string  SUSPEND_MODULES = "drivername logicalchipname"   But since I don't have permission to modify that file I tried to use chmod on it, which led to  but left me still in the same situation.  $ lspci -knn | grep Net -A3; rfkill list 03:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8821AE 802.11ac PCIe Wireless Network Adapter [10ec:8821] Subsystem: XAVi Technologies Corp. RTL8821AE 802.11ac PCIe Wireless Network Adapter [1b9a:2482]  Kernel driver in use: rtl8821ae  Kernel modules: rtl8821ae  0: hci0: Bluetooth  Soft blocked: yes   Hard blocked: no 1: phy0: Wireless LAN   Soft blocked: no    Hard blocked: no   What do you suggest I do? "Effect" vs. "Affect" I know that it is defined such that there is no more than one y-value output for any given x-value input, but I"m wondering WHY it is defined that way? Why can't we apply everything we know about functions to equations that have more than one y-value output for any given x-value input?   In short, what is the benefit or reason to define a function this way?  At the same time, a little historical context would also be appreciated (i.e., when the concept originated and why).  Thanks,  Moshe Programmatically combining images in PHP I'm a big fan of Yahoo's  for speeding up websites. One of the recommendations is to combine images where possible to cut down on size and the number of requests. However, I've noticed that while it can be easy to use CSS sprites for layouts, other image uses aren't as easily combined. The primary example I'm thinking of is a blog or article list, where each blog or article also has an image associated with it. Those images can greatly affect load time and page size, especially if they aren't optimized. What I'm looking for, in concept or in practice, is a way to dynamically combine those images while running them through a loss-less compression using PHP.   A few added thoughts or concerns:   Combining the images and generating a dynamic CSS stylesheet to position the backgrounds of the images might be one way to go about it, but I also worry about accessibility and semantics. As far as I understand, CSS images should be used for layout elements and the img tag (with the alt attribute) should be used for images that are meant to convey information. I could set the image as a background to a div element and substitute a title attribute for the alt attribute, but I'm unsure about the accessibility and semantic implications of doing so. Might the GD library be a good candidate for something like this? Can you recommend other options? I simply know that a single free quark does not exist. What is the reason that we can not get a free quark?  If we can't get a free quark then what is single-top-quark? Is Switching the Neutral OK? What’s the worst that can happen if i put a switch in the neutral instead of the phase?   I just found out that 3 light bulbs sockets in my home  are connected this way, it first came to my mind to fix it but then I recalled the flow of current from positive to negative is just a convention since the electrons are charged negatively and they actually flow from N to L so i cant really find an solid excuse to mess with the wiring for the next 2 hrs