My lens's focus ring can be twisted past infinity and the close-focus stops without ever hitting a hard stop— is it defective?
Why am I allowed to turn my focusing ring beyond infinity?
Suppose $G$ is a semigroup and it holds both left and right cancellation. Suppose $G$ is a semigroup and it holds both left and right cancellation. Also for each $a,b\in G$, $xa=b$ has solution in $G$. Prove G is a group.   I know this question looks very "old" style. First I think it's easy for me too since I already practice probelems like "semigroup, $xa=b$ and $ax=b$ have solution , then group" Or "finite semigroup, both left and right cancellation hold, then group ". But I still get stuck... My try:  First of course, for $xa=a$ , there exists $e_a$ such that $e_aa=a$. Now we need to proof $e_a$ is left identity for all elements. Then left inverse exists hence $G$ is a group. But the key is to proof $e_ab=b$ for all $b\in G$. How to set shell to wrap words for long commands? Don't let the association bonus be given away as bounties When a user with at least 200 rep on any Stack Exchange site registers a new account on another SE site, they immediately receive 100 rep on that site as an .  This is a deliberate feature to give users who are already familiar with how the SE platform and community works a bit of an extra boost (e.g. letting them upvote and comment immediately), and it's pretty much working as intended. What's probably not intended is that, currently, this bonus rep can be given away as . In fact, because awarding bounties currently requires only 75 rep, any user with 200+ rep on at least one site can join another site, receive the 100 rep bonus, and immediately award it all as a bounty to another user, without having to earn any actual rep on the site at all.  Not only does this seem contrary to the design intent of bounties (and of the association bonus), but it also seems ripe for potential abuse. (In fact, the only thing stopping a sockpuppet ring from gaining unlimited rep this way is the fact that the bounties would likely be sooner or later noticed by mods.) It's also possible for experienced users to (more or less) accidentally bounty their reputation so low that that they lose access to basic site features that they'd normally have due to the bonus.  Arguably, of course, this is entirely the user's own fault, but it can still be .  Also, this behavior , which currently require 110 rep to answer for users with the association bonus. The fix I'd like to suggest is simple:  If a user has been awarded the association bonus, don't let them award a bounty that would take their rep below 100.  Of course, it would still be possible for users with the bonus to drop below 100 rep via other means, e.g. by having their posts downvoted.  But they wouldn't be able to just give away this baseline rep to other users any more. Why do harmonics occur when you pluck a string? When you energise a taut string, the following resonant modes of vibration occur:    Plotting on the frequency domain, you can see their corresponding frequencies:    But what is the underlying physical principle? Why does this happen?  Is there any way of explaining it that could be understood by a smart 15 year old?  EDIT: I'm going to give my best attempt so far. Here goes:   We can start with sympathetic resonance. Sounding a particular frequency, a pure sinewave.  And noticing the string resonates sympathetically at the frequency of each harmonic. Say this is explained and understood. Now imagine that plucking a string is equivalent to a burst of white noise, which contains frequencies all the way through the spectrum. This could be approached backwards, by starting with random frequencies and noticing that resultant wave produced looks like white noise.   If the above is scientifically correct, then it restricts the domain of the question.  I would really like to be able to understand it scientifically and also be able to explain it intuitively.  PS Images from Delete commits from a branch in Git Can “whose” refer to an inanimate object? Once a post has been edited, you can go into the revision list, and click "View Source".  If a post has not been edited (and I don't have the edit privilege) is there any way to see the raw Markdown?  The reason I ask I because of  - it's impossible to see the source markdown, and how it's different from the rendered Markdown.  Not exactly a huge issue, just curious! I was just awarded +100 reputation on all of my Stack Exchange accounts.  What is this bonus for? It simply says "Association Bonus" on my reputation overview.  Also:   What privileges does the association bonus grant users? Can I lose the association bonus? Does it make my reputation 101 everywhere? Can I earn the association bonus on sites I haven't joined yet? Does it matter which site I earn the 200 reputation on? Will I get an association bonus every time my reputation on a site crosses 200 points? Does the association bonus allow me to answer protected questions? Does the association bonus count towards the reputation cap? Can I get the association bonus if I only have a profile on one site? What else is there to know about the association bonus? I am trying to estimate summary statistics of points located inside a polygon. Each points in the point layer is associated with N attributes. My target is to summarize (say mean, min,..) the attributes of points located within a polygon and populate the attribute fields of the corresponding polygon.   I am looking for a solution using GeoPandas or other Python libraries.   import geopandas as gpd    gdf_points = gpd.read_file('/path_to_points.json') gdf_polygon = gpd.read_file('/path_to_polygons.json')  dfsjoin = gpd.sjoin(gdf_polygon ,gdf_points)   Now, how can I summarize the stats for each attribute in the point layer and add it to the polygon shapefile? Which function can I use?  I am looking for something that is functionally equivalent to ESRI ArcGIS  with Q: "Do you have any juice?"   A: "Yes, there's some in the fridge."   Sounds perfectly fine to me, but:     Q: "Do you have any towels?"   A: "Yes, there's some in the closet."   Does not.  I asked for towels - plural - so wouldn't "Yes, there're some in the closet," in which there are is turned into a contraction be the correct way to say it?  Spellcheck, however, doesn't like "there're", and I think I'm the only person I've ever heard use the word "there're".  Even folks who I know say "there are" shorten it to "there's" when possible.  Am I saying it wrong, or are both forms acceptable? This is a problem from Rudin's Functional Analysis:  Show that there is a sequence of functions $\{ f_n \}$ in $X$ (the vector space of all complex functions) such that $f_n \to 0$ pointwise, but if any sequence $\{ \gamma_n \}$ tends to $\infty$, $\gamma_n f_n \not\to 0$.  The hint says to use the fact that the collections of all complex sequences converging to $0$ has the same cardinality as $[0,1]$. Honestly, this confused me more than it helped me.  My main issue is through the method of showing existence: I am not sure if I should attempt to construct an explicit sequence $f_n$, or if I need to appeal to other nonconstructive means such as Zorn's Lemma, etc. I have been trying to construct clever choices of sequences for a good bit now, but it always seems like I will be able to find a sequence that diverges slowly enough to allow that $\gamma_n f_n \to 0$.   Thanks! Are the limits $\lim\limits_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|\,$ and $\lim\limits_{n\to \infty }\sqrt[n]{|a_n|}\,$ equal? Let consider the power series $$ \sum_{k=1}^\infty a_kx^k. $$ We know that it converges when  $$\lim_{n\to \infty }\sqrt[n]{|a_n|}|x|&lt; 1\quad\text{or}\quad |x|&lt; \frac{1}{\lim_{n\to \infty }\sqrt[n]{|a_n|}}$$  But D'Alembert test tells us that it converges if $$|x|&lt; \frac{1}{\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|},$$ so since the radius of convergence is unique, we should have that $$\lim_{n\to \infty }\sqrt[n]{|a_n|}=\lim_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|.$$  I tried to prove it, but this result looks strange to me. So if it doesn't work always, how can we have those to limit as radius of convergence? I guess the equality of those two limit should be true most of the time.  So under what conditions is it true? How do I go about tracking down IO usage in Linux and other *nix OS'? Occasionally, something on my system starts hammering the disk. I would like to log data that would let me know about I/O usage per process, then I could go back to it later and see what keeps doing this. How should I go about solving this problem? Bonus points for solving this for arbitrary block devices. Error to calculate date diff with DateTime class How to link to apps on the app store I am creating a free version of my iPhone game. I want to have a button inside the free version that takes people to the paid version in the app store. If I use a standard link    the iPhone opens Safari first, and then the app store. I have used other apps that open the app store directly, so I know it is possible.  Any ideas? What is the URL Scheme for the app store? How do you find the directory of a command? How to send an SMS in iOS 10 Is it better in C++ to pass by value or pass by constant reference? Is it better in C++ to pass by value or pass by constant reference?  I am wondering which is better practice.  I realize that pass by constant reference should provide for better performance in the program because you are not making a copy of the variable. I couldn't find anything about Ubuntu acting as a  receiver or sender.   Can it work at all? Are there hardware prerequisites? Is WiFi a requirement or can it work over LAN or another kind of network connection? WiFi direct seems to be a necessary requirement, is it a sufficient one? (i.e. if a system supports WiFi direct does that mean it supports Miracast?) Are there differences in support between receiving/sending? How is the latency? (compared to the competition, i.e. VNC, commercial Miracast devices, etc.) How do I actually use it, if it's difficult?   Specifically, I plan to use it together with an Android phone (4.x Jelly Bean). Can command substitution be nested in variable substitution?