"lest" in your daily life Today I heard the word &quot;lest&quot; in our daily chat with some teammates and I even thought it was a typo. As far as I understand, it means 'avoiding', but can I use it normally in my daily speaking ? I read some examples, but none of them from daily life:   Large shell collections are built up by exchanging duplicates with other collectors. Hence it is wise to take a number of specimens. A notebook should be carried in which records may be made at the time of collecting, lest important data be forgotten.
Correct usage of "lest" Which of the followings is/are the correct usage of the word "lest"? How are these different from each other?   ...hesitant to speak out lest he be fired. ...hesitant to speak out lest he'd be fired. ...hesitant to speak out lest he would/will be fired.   P.S. I took point number 1 as the reference for this question from the definition of lest from the Merriam-Webster Dictionary.
Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic I'm reading a book about abstract algebra, but I'm having trouble solving this excercise: "Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic"  Where $(\mathbb{Q}^*,\cdot)$ is the group of nonzero rational numbers under multiplication and $(\mathbb{R}^*,\cdot)$ is the group of nonzero real numbers under multiplication.    Here is my attempt for the first.  Suppose $(\mathbb{Q}^*,\cdot)$ is cyclic, then $\mathbb{Q}^*=\langle\frac{p}{q}\rangle=\{(\frac{p}{q})^n,n\in\mathbb{Z}\}$, where $p$ and $q$ are coprime.  $\frac{2p}{q}$ is also in $\mathbb{Q}^*$ so it must be equal to $(\frac{p}{q})^n$ for some $n\in\mathbb{Z}$.  To solve $\frac{2p}{q}=(\frac{p}{q})^n$, I take a logarithm of both sides and end up with $1+\log_\frac{p}{q}(2)=n$, since $n$ is an integer $\log_\frac{p}{q}(2)$ must be an integer too, but it is possible only when $\frac{p}{q}=2^{\frac{1}{k}}, k\in\mathbb{N}$, (i.e. $\frac{p}{q}$ is a k-th root of $2$), but $k$ must be $1$ for $2^\frac{1}{k}$ to be rational so $\frac{p}{q}=2$ contradicting the hypothesis of $p$ and $q$ being coprime.    However I don't know whether this is a proper proof and the same reasoning cannot be applied to $\mathbb{R}^*$, I'd like you to just give me an hint towards a proof, without telling me the whole proof, if possible. Number of suggested edits seems to include own edits On the new History beta site I have the privilege to approve tag edits already but my own edits still go into the review queue. And the review dashboard displays a different number of pending edits depending on where I am: on  I see one pending edit, on  I see none. I am pretty sure that the pending edit being displayed is my own - I edited a tag wiki and that edit hasn't been approved yet. It shouldn't be counted no matter where I am. What are the shortcomings of the Mean Absolute Percentage Error (MAPE)? The  () is a common accuracy or error measure for time series or other predictions,  $$ \text{MAPE} = \frac{100}{n}\sum_{t=1}^n\frac{|A_t-F_t|}{A_t}\%,$$  where $A_t$ are actuals and $F_t$ corresponding forecasts or predictions.  The MAPE is a percentage, so we can easily compare it between series, and people can easily understand and interpret percentages.  However, I hear that the MAPE has drawbacks. I'd like to understand these drawbacks better so I can make an informed decision about whether to use the MAPE or some alternative like the MSE (), the MAE () or the MASE (). In $\mathsf{ZF}$, for each set $A$ exists an ordinal $\alpha$ such that no surjection $A\to \alpha$ exists Textbook recommendations for self-studying high school math? I'm a high school student who is trying to figure out a complete course of self-study for each year of high school. How can I self-learn grades of math without devoting too much time? This is a complex issue for me, as other students at my competitive high school have tutors and the like. Please recommend textbooks that have detailed explanations and progressive practice problems, for self-study for each area such as:  Algebra  Geometry  Trigonometry and Analytic Algebra  Pre-calculus  BC Calculus  Other people have skipped grades of math due to help from tutors and parents. Can I cover all of geometry and trigonometry in 8 months without going insane and be able to skip a grade? How do I remove extra Skype icon or ghost icons for removed apps from GNOME shell? I have this strange problem with conky. It doesn't show properly on the desktop:    I've tried everything from editing conky widget file to reinstalling conky and conky manager (v.2.4). Who knows how to fix it? How can I visualize the file system usage on Windows? How to make terminal transparent while still having its own background image? Is there a way to give terminal specific background image and make it 50% transparent at the same time? Find the general solution of $y'= a^{x + y}$ where y is the function Which one is correct to say: "It's me" or "It's I"? I was taught at school that the following expression is not grammatically correct:  Who is there? It's me.  The correct one is:  Who is there? It's I.  Can you let me know which one is accurate? Let $f$ be a real, continuous function defined on $[0,\infty)$ such that $xf(x)$ is increasing for all sufficiently large values of $x$. Show that if   $$\int_0^x f(y)\,dy \sim Ax^\alpha \quad \left(\,x\to \infty\right)$$  for some positive constants $A$ and $\alpha$, then  $$f(x)\sim \alpha Ax^{\alpha -1} \quad \left(\,x\to \infty\right).$$  Clearly, I have to use differentiation somewhere, but I don't know how to manipulate $\lim_{x\to \infty}\frac{\int_0^x f(y)\,dy}{Ax^\alpha}=1$ to get the desired result.  From suggestions given, I know that by L'hospital's rule, it's enough to show that the limit $\lim \frac{f(x)}{\alpha Ax^{\alpha -1}}$ exists, and this limit equals $ \lim \frac{xf(x)}{\alpha Ax^{\alpha}}$. From here, I'll need to use the given assumption that $xf(x)$ is eventually increasing. But how can I show the existence of the limit based on these facts?  I would greatly appreciate any solutions, hints or suggestions. The difference between "have a lunch" and "have lunch" Is there any difference between     I am not having a lunch tomorrow.   and     I am not having lunch tomorrow.   This is a follow up question of : . I recently obtained a Seagate Expansion 3TB external harddrive. I'm trying to copy over all of my stuff to prepare for reformatting and installing a new OS. A handful of my mp3s did not transfer when I was doing my music folder. I checked them after the fact and the ORIGINAL files became corrupted somehow. I am certain of at least a few that were not corrupted in anyway before doing this. Every now and then during my transfer I got  There was an error copying the file into /media/***/Seagate Expansion Drive. Error splicing file: Input/output error   Each of these files that I check after are corrupted at some point in the duration and will not play. Why did this happen and what can I do to fix it? How is the equation of motion on an ellipse derived? I would like to show that a particle orbiting another will follow the trajectory   \begin{equation} r = \frac{a(1-e^2)}{1 + e \cos(\theta)}. \end{equation}  I would like to do this with minimal assumptions. Any pointers? I've read that this is nothing but the equation for an ellipse, as defined from the focus of the ellipse. But I'm not satisfied in just taking the result, I mean why should I assume that orbits are elliptical? I guess it should be derivable from energy and angular momentum considerations... How can I embed a View inside a content of a node? How can I embed a view inside a content of a node in Drupal 8?   I found something similar, for example in the question about "" or the  module, but it is all for Drupal 7.  I tried with an Entity reference field pointing to a View but it display only a label and from the Manage Display tab I can choose only from Label and Entity ID. I need something like the "Rendered entity" option that is present on Entity reference fields pointing to other nodes. Is it possible for a familiar to attune to and then wear or wield a magic item? How can I make a "Movie Clip" node start at a certain frame of the scene? Any way to control light halos in the compositor based on emission level? I am wanting to animate a row of LED lights. I have separated the LED lights from the rest of my object with the pass index/ID mask functions; currently all of my LED lights are on pass index #1.  What I want is to have a minor halo effect around the lights, but only when they are active. Currently, when I add a halo effect to my pass index #1, all of the LED lights have a halo effect regardless if the LEDs are emitting light. All LEDs are diffuse and emission shaders, so I can animate by controlling the mix amount between diffuse and emission.   Is there any simple way to have the halo effect created through recognizing light emission values, or will I have to just animate each LED as its own layer of compositing? Can EU citizens travel to UK after Brexit with just an identity card?