Is $\exists y[y=f(a)]$ valid in FOL?
Is $\exists x(x=a)$ a valid formula in first order logic with equality?
Is setting a higher nice level for a process an effective way to reduce its impact on system load/CPU time? I have an rsync cron job which is pushing the server load and triggering monitor alerts.  If I set the job to be run with a high nice level, would that effectively reduce the impact it has on system load values? Calculation of $\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$ Calculation of  $\displaystyle \lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$  $\bf{My\; Solution::}$  Let $y=\displaystyle \lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$, Now taking $\log_{e}$ on both side  $\displaystyle \log_{e}(y) = \lim_{n\to \infty}\log_{e}\left(\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}\right)=\lim_{n\to \infty}\frac{1}{n}\cdot \sum_{r=1}^{n}\log\left(1+\frac{r}{n}\right)$  Now Using Reinman Sum of Integral ...  $\displaystyle \log_{e}(y) = \lim_{n\to\infty}\frac{1}{n}\cdot \sum_{r=1}^{n}\log\left(1+\frac{r}{n}\right) = \int_{0}^{1}\ln(1+x)dx = \log_{e}(4)-\log_{e}(e) = \log_{e}\left(\frac{4}{e}\right)$  So $\displaystyle y =\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}} =\frac{4}{e}$  But how can i solve above question using Stirling Approximation,  May I ask for help?  Thank you in advance! Galois group of irreducible cubic equation In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?  I just dipped into a book, The Drunkard's Walk - How Randomness Rules Our Lives, by Leonard Mlodinow, Vintage Books, 2008. On p.107 Mlodinow says the chances are 1 in 3.   It seems obvious to me that the chances are 1 in 2. Am I correct? Is this not exactly analogous to having a bowl with an infinite number of marbles, half black and half red? Without looking I draw out a black marble. The probability of the second marble I draw being black is 1/2. Can we implement the low quality flags to delete posts after a certain number of flags? Can we implement the low quality flag to be like the abuse or spam flags. So if a certain number of low quality flags are made  on a post (whether it be a questions or an answer) the post is deleted?  This would also relieve *close vote and delete vote queues. *which is a much discussed topic.        related: How can I increment a date by one day in Java? I'm working with a date in this format: yyyy-mm-dd.  How can I increment this date by one day? How can I tell if two Minecraft mods are compatible? If I want to use two mods together, e.g.  and  (which don't state compatibility with one another), how can I tell if they will work together (and not break Minecraft - forcing me to re-download a fresh copy)? Shrinkage parameter in Adaboost? I'm unclear how the shrinkage parameter works in Adaboost.  I understand the concept of shrinkage in the theoretical sense related to ordinary least squares, but I'm not sure how to interpret this parameter in relation to Adaboost. Cardinality of cartesian product of an infinite set with N Here the problem 1.4.26 from Sohrab, Basic Real Analysis  Duplicate  This are not duplicate of this question:    the answer gives no proof  despite the title is not really connected  only finite set   Problem  Show that, if $A$ is an infinite set, then $|A\times\mathbb{N}|=|A|$. Hint: Let $\mathcal{F}$ denote the set of all bijective maps $f:S\times\mathbb{N}\to S$, where $S\subset A$. Since $|\mathbb{N}\times\mathbb{N}|=|\mathbb{N}|$, we have $\mathcal{F}\neq \varnothing$ (Why?) Show that Zorn's Lemma can be applied in $\mathcal{F}$ to produce a maximal bijection $h:B\times\mathbb{N}\to B$, with $B\subset A$, and that we must have $B=A$, by examining the cases where $S\setminus B$ is finite or infinite.  Proof  Let $\mathcal{F}$ denote the set of all bijective maps $f:S\times\mathbb{N}\to S$, where $S\subset A$. Since $|\mathbb{N}\times\mathbb{N}|=|\mathbb{N}|$ (see exercise 1.4.10(i)), we have $\mathcal{F}\neq \varnothing$ (From Exercise 1.4.22(d) we know that for a infinite set $A$, we have $|A|\geq\aleph_0$, and $|\mathbb{N}|=\aleph_0$, thus there is an injection from $g:\mathbb{N}\to A$, and, by defining $S:=g(\mathbb{N})\subset A$, we get a bijective map (composition of two bijective maps) $S\times\mathbb{N}\to\mathbb{N}\times\mathbb{N}\to\mathbb{N}\in\mathcal{F}$).  If we consider the inclusion as partial order, then every chain $S_i\subset A$, $i\in I$, has $A$ as upper bound and applying Zorn's Lemma, we get the existence of a maximal $\bar{S}$ and hence a bijection $h:\bar{S}\times\mathbb{N}\to\bar{S}$, with $\bar{S}\subset A$.  Now we prove that $\bar{S}=A$. $R:=A\setminus \bar{S}$. $\bar{S}$ is infinite, since the example of element of $\mathcal{F}$, that we gave above is bijective to $\mathbb{N}$. We have to do this. The second is trying to follow the hint and is incomplete:   Consider that $A\neq\bar{S}$, then there is $r\in R:=A\setminus\bar{S}$. We chose an element $s\in\bar{S}$ and define the bijection $h':\bar{S}\cup\{r\}\times\mathbb{N}\to \bar{S}\cup\{r\}$ as follow  $h'(a,n)=\begin{cases}h(s,2n-1)&amp;\text{for }a=s\\ r&amp;\text{for }a=r,n=1\\ h(s,2n-2)&amp;\text{for }a=r,n\neq 1\\ h(a,n)&amp;\text{otherwise}\end{cases} $  Since $\bar{S}\subsetneq \bar{S}\cup\{r\}$ we get a contradiction to the maximality of $\bar{S}$. Now either $R$ is finite or infinite. 1) $R$ finite, let say $R=\{r_1,\cdots,r_n\}$. Then chose $n$ elements $s_1,\cdots,s_n\in\bar{S}$ and we define the bijection $h':A\times\mathbb{N}\to A$ as follow $h'(a,n)=\begin{cases}h(a,2n-1)&amp;\text{for }a\in\{s_1,\cdots,s_n\}\\ a&amp;\text{for }n=1,a\in\{r_1,\cdots,r_n\}\\ h(s_i,2n-2)&amp;\text{for }a=r_i\\ h(a,n)&amp;\text{otherwise}\end{cases}$  In case $n&gt;0$ we get a contradiction with $\bar{S}$ being the maximal element.   Thus we have only to exclude the case 2) $R$ infinite. HOW TO DO THAT?   Question  Above I tried two proofs. The first part is the same. The two second parts are given by the two bullets.  The second proof tries to follow the hint.  My questions are:   - Is my proof correct?   - In which direction should go the proof from the hint? A uncharged capacitor $C$ is connected to a battery with potential $V$. It becomes fully charged and has a charge $Q=CV$ stored on it.   Now the polarity of the battery is reversed. The capacitor will have the charge $Q$ still but with polarity reversed too.  My question is: What is the work done by the the battery?  According to me, during 1st charging it will do a work of $QV$. The energy of the capacitor is also not changing, then what is the work done to change its polarity? SQL - Query to get server's IP address Holomorphic self map on the open unit disc Does there exist a holomorphic function $f:\mathbb{D}\longrightarrow\mathbb{D}$ such that $f(1/2)=-1/2$ and $f'(1/4)=1$, where $\mathbb{D}=\{z:|z|&lt;1\}$? Does the Rogue's Assassinate feature grant criticals on multiple attacks? Does the Rogue's Assassinate feature (auto-crit on surprised targets) allow for multiple crits if you make multiple attacks that round, either through two-weapon fighting or extra attacks gained through multiclassing?  As an example: my 3rd level Human (Variant, TWF feat) Rogue-Assassin rolls high on his stealth roll, and gets up behind his target.  With two daggers, he gets 2 attacks per round if he uses his bonus action as well.  Do both of these attacks qualify for Assasinate's auto-crit feature?  That is: would I do 2 criticals for       4d4+4d6+2*DEX damage   or a single critical and a non-critical for       3d4+4d6+2*DEX damage?   Additionally, if I gained the Extra Attack feature through multiclassing, would I get crits on all 3 of my attacks, or just the first one? LCM is associative: $\ \text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))$ Definition: The least common multiple of $a.b\in\Bbb{Z}$ is the smallest  $k\in\Bbb{N}$ that is divisible by both a and b.         Proposition: $\text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))\tag{2}$    Proof: We want to show $\text{lcm}(\text{lcm}(a,b),c)\mid\text{lcm}(a,\text{lcm}(b,c))$ and $\text{lcm}(a,\text{lcm}(b,c))\mid \text{lcm}(\text{lcm}(a,b),c)$ such that $\text{lcm}(\text{lcm}(a,b),c)=\text{lcm}(a,\text{lcm}(b,c))$.  Let $k=\text{lcm}(\text{lcm}(a,b),c)$, then $c\mid k$ and $\text{lcm}(a,b)\mid k$, but if we let $l=\text{lcm}(a,b)$, then $a\mid l\,\,\text{and}\,\, b\mid l$. Therefore $c\mid k\,\,,a\mid k\,\,\text{and}\,\, b\mid k$, but then $\text{lcm}(b,c)\mid k$. Thus $\text{lcm}(a,\text{lcm}(b,c))\mid k$      Question: Does this makes sense, ifso can I do the same for $\text{lcm}(\text{lcm}(a,b),c)\mid \text{lcm}(a,\text{lcm}(b,c))$? Select from list of reasons when performing downvotes Preamble: I know that 'give a reason when downvoting' is a very common request, and I assure you that I have read these posts and the discussions in them and my request is very different. This isn't about encouraging people to comment and is more about gathering metrics.  I'd like to see a feature whereby people who are downvoted get some better feedback on why they were downvoted, especially given that people rarely leave comments in practice.  So my suggestion is that when user clicks on the downvote, they are prompted to pick from a short list (No more than 5) of common reasons. It would be a popup that replaces the existing "Please consider adding a comment" box.  The user receiving the downvote would see how many people downvoted them for each reason, but not who did it. It would significantly address the issue of people being too lazy to comment or not wanting to comment (for fear of retaliation or any other reason) while helping to ensure the user gets useful feedback.  Perhaps the downvote could also be weighted based on whether or not a reason is picked.  For questions the list might look something like:   Not a question Not relevant to the subject area Trivial question with no research effort Poorly asked question Other   And for answers, this might be:   Nonconstructive or not an answer Incorrect or dangerous answer Poor explanation Doesn't answer the original question Other If the imaginary part of an entire function is never zero, the function is constant Let $f : \mathbb{C} \to \{z\in\mathbb{C}:\Im(z)\neq0\} $ entire . Show that $f$ is constant.  I took $g(z)=\frac{1}{f(z)}$ and I think that g is bounded, therefore it is constant (due to Louville's theorem) but how do I prove it? Is $|g(z)|=\frac{1}{|f(z)|}=\frac{1}{u^2+v^2}\le M$ valid? The adjective ones I have heard recently are    forward straightforward forthcoming frank   I was wondering if each of them has positive, negative or neutral meaning?  What are other similar terms that are used for positive, neutral and negative intention? How could it be that the entropy of gas is not infinite?  If we use the formula $S=k\ln(\Omega)$ we get infinity because there are an infinite number of possible situations (because there are infinite possibilities of position and momentum for each molecule). How OPTIONS are settled can be an eye opener and I hope I can get some answers The only reason why I put it here is to find out whether I am a victim or just ignorant of the way brokers deal with option positions that were exercised. Whether I am a victim or just ignorant depends on the answer that I posted here. The facts are: I had a 2 SHORT QS Option at 75 which was exercised against me on the 23 Dec 00.45 AM and I later exercised my 2 LONG QS Option at 70 on 23 Dec 9.31 AM. The broker charged me Borrow Fee for 24-27 Dec for $1102, which means instead of getting $1000 profit I turn this into a loss of $102 plus comm. My question:  Since both long and short options were exercised on the same day one at 45 mins on 23 Dec and the later at 9.31 am on the same day, how can there be a SHORT POSITION in the stocks since both are net-off in my account ?  The answer I got from my broker and this is the point that I want to clarify.  The response I got from IB was that the initial exercise at 45 mins on 23 Dec was actually on 22 Dec (not 23 as I alleged). They refused to show me how they got 22 Dec.  This is my explanation to my Broker in reply.  Even if the short call and long call was exercised on different dates 22 and 23 Dec respectively, IB can only charge me for the SHORT position for one day and not from 24-27 Dec by ignoring my long call position was not there.  Further Facts:  My Broker ignored my answer and insisted the dates are correct and therefore I was charged to borrow shares that was need to satisfy the short position on exercised. So I am lost as I do not know whether the broker is correct and they can do this by ignoring my long call position after exercised or there is something they had not tell me. That is the reason I came to this site soliciting answers from those who know more than I do or had experienced the same.  I hope I have explained this clearly now and why would I want to give an opinion when I am looking for an answer ? This is something that has long bothered me, and I have asked a few physicists and chemists and never gotten a very satisfying answer. Why are particles charged? And I'm not asking (and this is the answer you often get) why some molecule might be negatively charged because the answer to that is simply is that there was one more negatively charged particle than positively charged particle. That doesn't answer my question, however, of why there are charged particles at all?  So, on a fundamental level, what makes one particle charged and another one neutral.  For instance, protons and neutrons have almost exactly the same mass, which I'm assuming that means they have very similar internal compositions, yet one is positively charged and the other neutral. What is going on that makes these particles different?  On a similar note, and I'd really like to know this, how is it that a proton and an electron have exactly equal and opposite charge, yet an electron is about 1000 times less massive than a proton?  I have particular trouble with this question because if the answer were something along the lines of "particle X which composes protons all have partial positive charges of n amount..." I would just ask why particle X has any charge and we'd be in the same predicament.  I'm sure there is a satisfactory answer to my question, I just can't find it and don't know what resources are reliable or detailed enough to be satisfactory.