Given  $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and $|\mu_X|\geq10\sigma_X$, $\sigma &gt; 0$, looking for:   accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic (?) approximation of same product    This is a special case $|\mu_X|\geq10\sigma_X$, $\sigma &gt; 0$  of a more .
Given  $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$,  and $\mu_X \approx 0$, looking for:   accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic (?) approximation of same product    This is a special case $\mu_X \approx 0$ of a more .
Using Nouns or Gerunds Could anyone explain when to use a noun or a gerund?  Are these sentences correct? What is the difference between usage of gerund and noun there?        I worked on the improvement of the article.   I worked on the article improving.   I worked on the improving of the article.                The investment idea was published.   Investing idea was published. Limit with inverse trignometry We have to find the value of $$\lim_{n\to \infty}\sum_{k=2}^n\arccos\left(\frac{1+\sqrt{(k-1)k(k+1)(k+2)}}{k(k+1)}\right)$$  I tried it a lot but could not find any start.   Please help me in this . Excel-Count number of visible cells containing certain string I have a sheet in which I first apply a filter to a certain column to show "Unique records only". Now I have a column which contains, for e.g., values like:  Applepie Applesauce Crabapple Banana Mango   What I need is the count of all visible cells containing the word "Apple". Note that this should exclude the hidden (filtered out) cells. Now I have found out a formula:  `=SUMPRODUCT(SUBTOTAL(3,OFFSET(Sheet1!A:A,ROW(Sheet1!A:A)-MIN(ROW(Sheet1!A:A)),,1))*(Sheet1!A:A="&lt;Text to search&gt;"))`   However the problem with this one is this formula searches for the entire text. NOT just a part of it. How do I modify this formula to return count of cells containing "Apple" i.e. for this example it should return 3 as the result. What antivirus programs are available for Ubuntu?  We previously used Symantec Endpoint Protection but it does not work in Ubuntu. Overheard on an elevator today,     I didn't realize it was him.   Corrected by the know-it-all,     He.  "I didn't realize it was he."   The know-it-all then went on a rant about how everybody is a dolt for not knowing that.  I swear I have never heard this before in my life.  Is the know-it-all correct?    If so, why is "I didn't realize it was he," grammatically correct?  What rule makes "him" incorrect here? Are the following sentences grammatical or not? If not, does substituting a to-infinite make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, or function)  1 I have decided to purchase a jet for getting to NYC and back more quickly. 2 I have decided to buy a jet for having a quicker way to get to NYC and back.  This is my only reason and aim; this will be the jet's only purpose and function.  3 I hate hairdryers, so I am buying a space heater for drying my hair more naturally, and that is my only aim &amp; purpose &amp; reason and that will be its only function 4 For having a lasting shine in your hair, buy ShineOn Shampoo. 5 My neighbor and I have decided to cooperate more for having a better life together.   Research done: ELL  (Note the  here is a restatement of Cambridge Dictionary's ) ELL  ELL  BBC English  (Currently the best site I've found.) ELU  All of which have answers that I find  (i) incomplete (ii) contradictory (iii) terribly confusing, if not (iv) wrong. To sort through the haze, I am focusing on one issue. Background Some sources say that you need to use a to-infinitive for expressing purpose, aim, function, reason, etc. I find these to be rather hazy distinctions, given that the ODO defines  as The reason for which something is done or created or for which something exists and the same work lists as : aim, reason, function, etc. It is my assertion that at least some of the following sentences demonstrate that one can use the -ing form (be it a gerund, verbal noun, participle) to express individual purpose (aim, reason, function, intention, whatever you want to call it). Do you agree, or not? Why (not)?  In shorter terms, are the sentences grammatical? or do we have to substitute a to-infinitive (including in order to + infinitive) to make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, function, etc.) Show that the Sorgenfrey line does not have a countable basis. I am trying to understand this proof from Munkres' book which shows that the Sorgenfrey line does not have a countable basis. His proof is:  Let $\beta$ be a basis for $\mathbb{R}_l$. Choose for each $x$ an element $B_x \in \beta$ such that $x \in B_x \subset [x, x+1)$. If $y$ is not equal to $x$ then $B_y \neq B_x$ since inf($B_x$) = $x$ and $y$ = inf($B_y$). Therefore $\beta$ must be uncountable.   I am new to topology so I am probably just missing a simple point but I do not see why $\beta$ must be uncountable according to this argument. I agree that $B_y \neq B_x$ but isn't it possible that $y \in B_x$ (and thus the set $B_y$ is not needed as part of the basis)? Thanks in advance for your help. I am trying to grab the capital letters of a couple of words and wrap them in span tags. I am using  for extract and wrapping purposes, but it's not outputting anything.  preg_replace("/[A-Z]/", "&lt;span class=\"initial\"&gt;$1&lt;/span&gt;", $str) Run command on another(new) terminal window How to run any command in another terminal window?  Example: I opened one terminal window and if I run command like apropos editor, then it run and out-puts on that window. But I want to run same command on another terminal window (new window) instead on present window from first terminal.   Further clarification:  I need suggest-command &lt;command&gt; that open new terminal window and run mentioned &lt;command&gt; in that (newly opened) window. (where suggest-command is example of suggestion of command.)  How to do that? How do JavaScript closures work? How would you explain JavaScript closures to someone with a knowledge of the concepts they consist of (for example functions, variables and the like), but does not understand closures themselves?  I have seen  given on Wikipedia, but unfortunately it did not help. Get list of all tables in Oracle? How do I query an Oracle database to display the names of all tables in it? Converting Older United States Dollar Bills what I want to do is this. I have an external hard disk drive, a SEAGATE expansion 1TB, you can take it a look here:  As you can see this is not a flash usb hard drive, it is a USB hard disk drive. What I want to do is installing Ubuntu in this external hard disk drive... BUT NOT PLACING THE GRUB ON MY C: DRIVE. In other words, I don't want to see the GRUB splash Window asking me every single time if I want to load Windows or Linux. I want my PC turn on or restart and go directly to Windows... BUT... I want to have Linux in my external USB hard disk drive How do I want to load Ubuntu then? Easy, when I restart my computer or just start it... in ASUS mother boards I have the possibility to select the boot device by pressing F8, and I get the boot menu. I want to turn on my computer, press F8, my mother board ask me what drive I want to use to load the Operating Sytem, I select the USB external hard disk drive where do I have Ubuntu, and bingo, I run Ubuntu. When I finish to use Ubuntu, I close it and turn off my computer. Later, I turn on my computer, and if I don't press F8, automatically load the hard disk drive C: and load Windows. My question about this, is what are the necessary steps to install the latest version from Ubuntu, doing exactly this. Not placing the GRUB in my C: hard disk drive, and installing it entirely in the external USB hard disk drive (normal one, not flash one). How do I add .e00 data into my QGIS project? Finding the 1st three derivatives of a function and plotting all four curves Use physical harddisk in Virtual Box Using Virtual Box, how can I install an OS to a secondary, physical disk, and boot it in both (at separate times) Virtual Box, and as a typical secondary OS install? I have Java 8 and Ubuntu 18.04. I'm using NetBeans and when tried to make some program, it couldn't be done because JavaFX is missing.  I installed Java JDK with NetBeans from Oracle's site and thought everything is included but obviously I didn't check it well.   What should I do, what should I install? I'm not so good at this and don't know if problem is with Ubuntu or Java. :( Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.   The  $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative system $R\setminus I$) gives a morphism $$ f_1:\operatorname{Spec}(R_I)\to \operatorname{Spec}(R)=:X $$ The  $\widehat{R}_I$ of $R$ at $I$ gives a morphism $$ f_2:\operatorname{Spec}(\widehat{R}_I)\to X $$ For the special case $I=(a)$, the localization $R_a$ (i.e. the localization of $R$ at the multiplicative system $\{1,a,a^2,\ldots\}$) gives a morphism $$ f_3:\operatorname{Spec}(R_a)\to X $$   My question is:     What is the geometric meaning of all three constructions and how are they related?   This is what I ''know'' already or describes at least the style of answer I would appreciate. As remarked in the comments, $I$ has to be a prime ideal.   $R\to R_I$ is injective iff $R\setminus I$ contains no zero divisors which is the case if $I$ is a prime ideal. The scheme $\operatorname{Spec}(R_I)$ is the intersection of all the neightbourhoods of $I$ in $X$. This is a little contra-intuitive for me since the last statement sounds like ''$f_1$ is injective'' but the $\operatorname{Spec}$-operator should turn 'injective'' and ''surjective'' around somehow (I know this is literally not true but I only want to get a feeling like ''is contained'', ''is bigger'', ''is smaller'', etc.). $\operatorname{Spec}(R_a)$ is somehow the opposite of $\operatorname{Spec}(R_{(a)})$ (= the intersection of all the neightbourhoods of $(a)$ in $X$) because $\operatorname{Spec}(R_a)$ seems to be something like the union of all the open sets of $X$ not containing the point $(a)$. $R\to\widehat{R}_I$ is injective iff $\cap I^n=(0)$ and this holds very often (e.g. for $R$ noetherian and either an integral domain or a local ring). Hence (this is probably false intuition as remarked before), $f_2$ should be something like a ''projection'' (from something ''big'' into something ''small''). But what is geometrically the difference between localization and completion. I don't have a geometric idea of completion at all.   As remarked by Qiaochu Yuan in the comments below, one should not think of $\operatorname{Spec}$ as injective-surjective switching. How do I run a program in the background of a shell, with the ability to close the shell while leaving the program running? Lets say my UI is having problems or for some reason, I need to boot up a program from the terminal window, say, nm-applet:  nm-applet   When it's started, it occupies the foreground of the terminal window.  Is there any simple way to run the program in the background without needing to leave the terminal open or have it occupy the whole terminal?  On that note, I did find a way to run programs from the terminal and have it allow for other inputs, by appending an ampersand (&amp;) to the command as such:  nm-applet &amp;   But this isn't much use as any processes started in the terminal are killed once the terminal is closed. Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura Let $I$ and $J$ be ideals of a Noetherian ring $A$. If $JA_P\subseteq IA_P$ for every $P\in \operatorname{Ass}_A(A/I)$, then $J\subseteq I$.   I'm reading Matsumura's Commutative Ring Theory book on my own. This is exercise $6.4$ in that book, and I fail to prove it. Could you help me with this? Thanks in advance.