I used   \usepackage{lineno} \linenumbers   For numbering my lines. I know it has problem with double dollars and for that case I replaced double dollars with \[\] and it worked but now I used \begin{gather*}\end{gather*} to fit an equation but it caused line-numbering doesn't work for a few lines before this equation. Id there any solution for this?
The line numbers for the text of a paragraph with an equation defined by \begin{align} command do not show as expected. How to fix this?    The LaTeX codes are attached here.   \documentclass[a4paper, 12pt]{article} \usepackage[utf8]{inputenc}    % Use the UTF-8 encoding \usepackage{amsmath}           % For the use of math \usepackage{amsfonts} \usepackage{amssymb}           % For the use of AMS symbols \usepackage{indentfirst}       % For indentation \usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}  \usepackage{lineno} \linenumbers  \usepackage{titlesec} \titleformat*{\section}{\large\bfseries} \titleformat*{\subsection}{\normalsize\bfseries} \titleformat*{\subsubsection}{\normalsize}   \renewcommand{\baselinestretch}{1.5}    \usepackage{authblk}           % Use author package \renewcommand*{\Authfont}{\large} \renewcommand*{\Affilfont}{\normalsize\normalfont}  \makeatletter \g@addto@macro\normalsize{%   \setlength\abovedisplayskip{0pt}   \setlength\belowdisplayskip{0pt}   \setlength\abovedisplayshortskip{0pt}   \setlength\belowdisplayshortskip{0pt} } \makeatother  \title{\textbf{Globally cumstomize the spacing between the text and equation}}    \date{}  \begin{document}  \maketitle  \section{Air filter} The air filter is simulated by the Control Valve module (AFT in Fig. 2(a)). The pressure drop through the air filter ($ \Delta P_{af} $) is given by the following equation [11,16]. \begin{align} \Delta P = \Delta P_{d} \left( \frac{m}{m_{d}}\right)^{1.84}  \left( \frac{T}{T_{d}}\right)  \left( \frac{P}{P_{d}}\right)^{-1}  \end{align} where, $ \Delta P_{af} $ is the pressure drop, $ m $ is the mass flow rate, $ T $ is the temperature, and $ P $ is the pressure, and subscript $ d $ denotes the design condition.  \newpage \section{Air compressor (AC)} The AC operating characteristics can be described by its performance map, which relates the following dimensionless operational variables. The AC operating characteristics can be described by its performance map, which relates the following dimensionless operational variables. \begin{align} &amp; \text{Relative pressure ratio:} &amp;&amp; PR_{r} = \frac{PR-1}{PR_{d}-1}\\ &amp; \text{Relative isentropic efficiency:} &amp;&amp; \eta_{r} = \frac{\eta}{\eta_{d}} \\ &amp; \text{Relative corrected mass flow:} &amp;&amp; m_{cor,r} = \frac{\left(\frac{m_{in}}{P_{in}}\sqrt{T_{in}}\right)}{\left(\frac{m_{in,d}}{P_{in,d}}\sqrt{T_{in,d}}\right)} \\ &amp; \text{Relative corrected speed:} &amp;&amp; N_{cor,r} = \frac{\left(\frac{N}{\sqrt{T_{in}}}\right)}{\left(\frac{N_{d}}{\sqrt{T_{in,d}}}\right)} \end{align} where, $ \eta $ is the efficiency, $ N $ is the shaft speed, and $ PR = P_{in}/P_{out} $. Subscript $ cor $ denotes the corrected value, $ r $ denotes the relative value, $ in $ denotes the inlet, and $ out $ denotes the outlet.  \end{document}
Ubuntu in VirtualBox wants to install on every boot I'd like to update the label of an operator when it's called. But this :  def execute(self,context):     self.bl_label = "good night"   ... apparently doesn't work if the bl_label property has already been defined. Am I wrong ? How to delete file(s) in secure manner? Is there a way to make sure that a deleted file can not be recovered?   As you know, deleting a file by pressing shift-del or using trash doesn't mean that file is gone forever. It resides somewhere in the computer.  In day to day life, law enforcement or thieves use recovery programs to reveal our private data or something else.   They can recover all data that resides on hard disk, in RAM, and even USB.  To protect us from these people, what should one do? Which program should one use?  Note: I don't want an answer like first use a recovery program then erase that file with shred. With just one program, I want to erase all unused/deleted files on my computer. Convergence of sequence of real numbers and (sub)subsequence Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of real numbers. A well known result is:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ converges to $a\in \mathbb{R}$.   Now, I'm working on an exercise which essentialy asks to prove the following:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if for every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ there is a (sub)subsequence $(x_{n_{k_j}})_{j\in \mathbb{N}}$ which converges to $a$.   Now, one direction is trivial. If in fact $(x_n)_{n\in \mathbb{N}}$ converges to $a$, then by the result I've mentioned, every subsequence of $(x_{n})_{n\in \mathbb{N}}$ converges to $a$. Let then $(x_{n_k})_{k\in \mathbb{N}}$ be an arbitrary subsequence. In that case, we know that it is a subsequence of itself, and hence we've found a (sub)subsequence which converges to $a$.  The other way around seems quite complicated for me. The reason is as follows: I must assume that every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ has a (sub)subsequence which converges to $a$ and the proof that the original one converges to $a$.  My problem is exactly that "a". If it had all (sub)subsequences converging to $a$, then by the result i've mentionted, the subsequence $(x_{n_k})_{k\in \mathbb{N}}$ would converge to $a$. In that case, we would have every subsequence converging to $a$ and the result would follow.  My idea to prove the claim was then:  First remember that a subsequence is the restriction of the original one $x: \mathbb{N}\to \mathbb{R}$ to an infinite subset $\mathbb{N}'\subset \mathbb{N}$. In that case a (sub)subsequence is the restriction of $\tilde{x} = x|\mathbb{N}'$ to an infinite subset $\mathbb{N}''\subset \mathbb{N}$. But since $\tilde{x}$ is the restriction of $x$ to $\mathbb{N}'$ and since $\mathbb{N}''\subset \mathbb{N}'$, the restriction of $\tilde{x}$ to $\mathbb{N}''$ is itself the restriction of $x$ to $\mathbb{N}''$.  In other words, every (sub)subsequence can be viewed as a subsequence of the original one. Considering this, by hypothesis every subsequence has a (sub)subsequence which converges to $a$. But by this argument, this means that we can replace every subsequence with a convergent one. And then we can use the result I stated.  Is my idea correct? Is this really the way to prove the claim? My only doubt is that I need every subsequence converging to $a$ to use the result. In this case I can't see how I can argue that this implies every subsequence converges to $a$.  How can I finish this proof? I posted an answer (pointing to another question already asked) but when I hit submit, I saw my answer appear as a comment to the question instead.   I tried it a couple of times and still the same result. The answer gets eaten up and is transformed into a comment.   Is this a bug? What Happens to electrons after Alpha Decay and Nuclear Fission? Where do the electrons go? In alpha decay do 2 electrons follow the alpha particle and make stable Helium or does the larger daughter nucleus become an anion?  Also what do the electrons do in the mixture of fission and alpha decay? With Beryllium-8 it decays into two alpha particles and 4 lonely electrons or do you get two Helium atoms?If someone could add an entry to Holocron it would be greatly appreciated, thanks. Intuitive reason why the Fisher Information of Binomial is inversely proportional to $p(1-p)$ It confuses/blows my mind that the Binomial has variance proportional to $p(1-p)$. Equivalently, the Fisher information is proportional to $\frac{1}{p(1-p)}$. What is the reason for this? Why is the Fisher Information minimized at $p=0.5$? That is, why is inference most difficult at $p=0.5$?  Context:  I'm working on a sample size calculator, and the formula for $N$, the sample size needed, is an increasing factor of $p(1-p)$, the result of a variance estimation in the derivation. What is "offset null" in IC 741? What is Offset null in 1st and 5th pin in IC 741 (Op-Amp)? Why it is used, though it is not used in many circuits? Give me explanation regarding the offset null! Why offset voltage was formed in IC 741? Why is $\frac{1}{2\pi i}\int_{\gamma}\frac{f'(z)}{f(z)}dz$ an integer? Let $f$ be holomorphic in an open $\Omega \subset \mathbb{C}$ and $\gamma$ a closed curve in $\text{int}(\Omega)$, along which $f$ is never zero.   Are these hypotheses enough to claim $\frac{1}{2\pi i}\int_{\gamma}\frac{f'(z)}{f(z)}dz$ is an integer? If not, what are the necessary and sufficient conditions for that?  I can prove that's an integer for some particular cases, for example when $\Omega$ is convex and  $\gamma$ is a circle. But I've seen people claming this in many other contexts, like when $\Omega$ is an annulus around $0$ and $\gamma $ is an arbitrary curve inside it. This isn't obvious to me at all.  Thanks! So I have this word problem and I’m a bit confused about it. I have the answer and explanation but I still don’t understand: The half-life of carbon-14 is approximately 5730 years, while the half-life of carbon-12 is essentially infinite. If the ratio of carbon-14 to carbon-12 in a certain sample is 25% less than the normal ratio in nature, how old is the sample?     A. Less than 5730 years      B. Approximately 5730 years      C. Significantly greater than 5730 years, but less than 11460 years      D. Approximately 11460 years  Correct Answer: A Explanation: Because the half-life of carbon-12 is essentially infinite, a 25 percent decrease in the ratio of carbon-14 to carbon-12 means the same as a 25 percent decrease in the amount of carbon-14. If less than half of the carbon-14 has deteriorated, then less than one half-life has elapsed. Therefore, the sample is less than 5730 years old. Be careful with the wording here—the question states that the ratio is 25% less than the ratio in nature, not 25% of the ratio in nature, which would correspond to choice (D). How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”? Just based on the answer, it seems like the question said “25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed. I don’t see how isotopes and ratios would change this problem at all. Edit: Every answer here was very informative and helpful. It was tough to pick a best answer. I picked Cosma’s because it clicked with me. If you’re reading this and want to understand radiocarbon dating, look at every answer because there is some useful information in every answer Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated. Showing that $\gamma = -\int_0^{\infty} e^{-t} \log t \,dt$, where $\gamma$ is the Euler-Mascheroni constant. I'm trying to show that  $$\lim_{n \to \infty} \left[\sum_{k=1}^{n} \frac{1}{k} - \log n\right] = -\int_0^{\infty} e^{-t} \log t \,dt.$$  In other words, I'm trying to show that the above definitions of the Euler-Mascheroni constant $\gamma$ are equivalent.  In another post here (which I can't seem to find now) someone noted that  $$\int_0^{\infty} e^{-t} \log t \,dt = \left.\frac{d}{dx} \int_0^{\infty} t^x e^{-t} \,dt \right|_{x=0} = \Gamma&#39;(1) = \psi(1),$$  where $\psi$ is the digamma function.  This may be a good place to start on the right-hand side.  For the left-hand side I was tempted to represent the terms with integrals.  It is not hard to show that  $$\sum_{k=1}^{n} \frac{1}{k} = \int_0^1 \frac{1-x^n}{1-x} \,dx,$$  but I'm not sure this gets us anywhere.  Any help would be greatly appreciated. What is the preferred way to mention Stack Overflow from Twitter? I wanted to tweet something on the lines of "Stack&nbsp;Overflow is awesome". When I do this kind of stuff, I usually use a username (for example: ) or a hashtag ()  I found , but it seems it is from someone unrelated with Stack&nbsp;Overflow.  Is there a Twitter account I can reference, or should I use a hash-tag? Should I just point to the site?  Thanks!  (Notice: I've seen lots of questions regarding Twitter and Stack&nbsp;Overflow, but those are mostly about integration/login with Twitter for the users accounts; I'm just asking for a single account for Stack&nbsp;Overflow; probably for announcements and PR, which is different) Why do I need 50 reputation to comment? What can I do instead? Users with less than 50 reputation points cannot write comments on questions and answers that they don't own.  Why does this limit exist? But I want to contribute now. What should I do instead? But I can't write a good answer without more information! Shouldn't I be allowed to post comments if I have reputation on a different site? But I saw a new user posting a comment! How did they do it? Why do I need 50 reputation to comment on my own post?   See also:    in the Help Center If I push or hit an object in space will it rotate or move along a straight line? How to delete Windows.old in Windows 8 It seems to be a common thing that a Windows Update creates a Windows.old folder under C:\Windows.old My question is now, can it be deleted with Windows 8 and how can I do that?  If I try to manually delete it in Windows Explorer it does not let me, a permissions issue. iOS 7 Weather app expand/collapse transition Screen (not Lock Screen) Timeout Stuck at 1 Minute I recently got a new laptop (Lenovo T440p with Windows 8.1 Pro) and was having the following issue: the screen would lock (screen would go black and require a password to get back in) after 1 minute. I would prefer something more like 10 or 15 minutes. So, I tried the obvious changes (check the screensaver settings, power management settings for sleep, turning off the screen, etc.), with no success. I did, however, notice a rather odd behavior: if I changed the screensaver timeout from 1 minute to anything else (say, 15 mins), clicked OK, and re-opened the screensaver settings window, it would be back to 1 minute. Odd.  At that point, I turned to Google. I found posts ( in particular was helpful) suggesting all sorts of causes and resolutions:   If you have Cisco AnyConnect VPN client (which I did), try uninstalling it. I uninstalled, rebooted, no change. Open the User Accounts control panel and, on the left side of the screen, click Reset Security Policies. My User Accounts control panel, however, has no such link. Reboot into a repair console and run "sfc /scannow". It did indeed find and repair some issues, but didn't fix the screen timeout problem. Edit the Group Policy to force a particular timeout time; the setting showed up correctly (15 mins) in screensaver settings, power management settings, etc., but the screen would still go black and lock after 1 minute. In the registry, manually change the screensaver timeout and the "turn off the screen" timeout. No luck. Why do mathematicians use single-letter variables? How can I set conditional formatting on a particular cell that depends on another cell's value?