Determining all homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ?
Number of Homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$
Meaning of Sun Lamp Size? In Cycles, a sun lamp is treated as infinitely far away - only the angle matters, not the position. Other lamps have their sizes defined in blender units: a point lamp with size 1 emits light from a area with size of 1. But if a sun lamp at infinity had a size of 1 blender unit, the angular arc of that sun would be nonexistent.  Clearly, when you change the size of a sun lamp, shadows get fuzzier or sharper. But what does that actually mean? What are you changing? Max Block Height for a Redstone Contraption What's the max height of blocks for a redstone contraption to still work even I'm in the surface below?  Like if I build a pillar to the max height of blocks and build a redstone contraption there, is it still going to work even I'm on the surface below? A weighted version of random.choice How does one experimentally determine chirality, helicity, spin and angular momentum of a fundamental particle? Postdoc before PhD completion I am in a situation where my PhD funding runs out in 5 months and I need to have secured a job regardless of whether I have submitted due to having a family.   Although I think I may be close to submitting by then, it may take longer. Is it worth applying for postdoc positions or would universities not consider you until you have completed the PhD? I assume that the pass rates of PhDs are very high if the candidate has got to the submission stage.   Edit: This question is aimed at the people that are are the submission stage but haven't yet defended their thesis. It is not a question asking if I can get a post-doc without a PhD. Files list file for package missing -- dpkg warning I want to install Open JDK or Oracle JDK to my machine but I got an error during installation. The error is:  Selecting previously unselected package oracle-java8-installer. dpkg: warning: files list file for package 'unity-scope-gourmet' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-guayadeque' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-gmusicbrowser' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-manpages' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-firefoxbookmarks' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-gdrive' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-home' missing; assuming package has no files currently installed E: Sub-process /usr/bin/dpkg returned an error code (2)   I searched on Google and try to something for install/reinstall this packages. Finally i try to remove Unity. I faced this error for my every tries. I am using Xmonad. How can I fix this issue? Showing that $(m)\cap (n)=(\operatorname{lcm}(m,n))$ and $(m)+(n)=(\gcd(m,n))$ for any $m,n\in\mathbb{Z}$ Characteristic polynomial of companion matrix macOS Sierra restarting download instead of resuming Consider the following figure from Faraway's Linear Models with R (2005, p. 59).    The first plot seems to indicate that the residuals and the fitted values are uncorrelated, as they should be in a homoscedastic linear model with normally distributed errors. Therefore, the second and third plots, which seem to indicate dependency between the residuals and the fitted values, suggest a different model.  But why does the second plot suggest, as Faraway notes, a heteroscedastic linear model, while the third plot suggest a non-linear model?  The second plot seems to indicate that the absolute value of the residuals is strongly positively correlated with the fitted values, whereas no such trend is evident in the third plot. So if it were the case that, theoretically speaking, in a heteroscedastic linear model with normally distributed errors  $$ \mbox{Cor}\left(\mathbf{e},\hat{\mathbf{y}}\right) = \left[\begin{array}{ccc}1 &amp; \cdots &amp; 1 \\ \vdots &amp; \ddots &amp; \vdots \\ 1 &amp; \cdots &amp; 1\end{array}\right] $$  (where the expression on the left is the variance-covariance matrix between the residuals and the fitted values) this would explain why the second and third plots agree with Faraway's interpretations.  But is this the case? If not, how else can Faraway's interpretations of the second and third plots be justified? Also, why does the third plot necessarily indicate non-linearity? Isn't it possible that it is linear, but that the errors are either not normally distributed, or else that they are normally distributed, but do not center around zero? How to download/read only the first 80KB of a file? Why are American-style options worth more than European-style options? Why are American-style options worth more than European-style options?   I realize I can exercise American-style options anytime before the  expire, but I can only exercise European-style options during their  "exercise period" (usually right when they expire, but no earlier).   So it makes sense that an American option is worth at least as much as  a European option.   But why is it worth more? If I exercise my American option early, I  may make more money than waiting until expiration, but I also may make  less. Mathematically, there's no advantage, since I'm equally likely  to make as much money by waiting for expiration.   Mathematically speaking, is there ever a good reason to exercise an  American option early?   I realize people don't always behave logically, but even the formulas  that valuate options show that American options are worth more.   EDIT (answering comments):    Many of the comments people made would apply to lookback options  (), but not to American  options.  American options can do everything European options can and more. I  understand that, and that means American options can't be worth any  LESS than European options, but I'm still not clear on how you would  mathematically calculate this extra value.  @Aaronaught: "The difference between an American and European  option is the difference between getting N chances to get it right (N  being the number of days 'til expiration) and getting just one  chance. It should be easy to see why you're more likely to profit  with the former, even if you can't accurately predict price  movement."    REPLY: I don't think you really get N chances. Once you exercise the  option, that's it, no more chances. And, if you decide not to exercise  the option today, and the underlying's price falls, you can't go back  in time and exercise it yesterday.    @jdsweet: Think of it this way, if you traveled back through time  one month - with perfect knowledge of AAPL's stock price over that  period - which happens to peak viciously then return to its old price  at the end of the period - wouldn't you pay more for an American  option?    REPLY: Well, no. I'd simply buy a cheaper European option that expires  right at the spike. Or, I'd buy a European option that expires later,  and sell it when the spike occurs. American options don't give you  precognition, so you're still guessing as to when to sell/exercise.    @jdsweet: "one of the more common reasons people exercise (as  opposed to sell) an American option before expiration is if an  unexpected dividend (larger than remaining time value of the option)  was just announced that's going to be paid before the option contract  expires"    REPLY: This makes sense. I'm actually trading FOREX options, so there  is no dividend. In that scenario, are European and American options  worth the same? I'm trying to use the \citep command in Latex and my document class is elsarticle. The problem is that it displays numbers not authoryear citations in the text which is what I'm looking for. Here are related parts of my code:      \documentclass[preprint,11pt]{elsarticle}      \usepackage{natbib}   ...      \bibliographystyle{plainnat}      \bibliography{ML}   And I'm obviously using the \citep command like this:      I once read \citep{edelman1993magnetic}   And my .bib file looks like this:  @article{wolf2009regional, title={Regional brain activation changes and abnormal functional connectivity of the         ventrolateral prefrontal cortex during working memory processing in adults with attention-deficit/hyperactivity disorder}, author={Wolf, Robert C and Plichta, Michael M and Sambataro, Fabio and Fallgatter,     Andreas J and Jacob, Christian and Lesch, Klaus-Peter and Herrmann, Martin J and     Sch{\"o}nfeldt-Lecuona, Carlos and Connemann, Bernhard J and Gr{\"o}n, Georg and others}, journal={Human brain mapping}, volume={30}, number={7}, pages={2252--2266}, year={2009}, publisher={Wiley Online Library} }   The result looks like:  I once read[1]   I'm looking for a result like this:  I once read(wolf., 2009)   Elsevier is actually looking for:  Text: All citations in the text should refer to:    Single author: the author's name (without initials, unless there is ambiguity) and the year of publication;  Two authors: both authors' names and the year of publication;  Three or more authors: first author's name followed by 'et al.' and the year of publication.    Thanks in advance Avoid footnotes from appearing before mark I have these \footnotetext and \footnotemark as illustrated in the figure below. I used these tags to enforce my own numbering (and because using only \footnote was not working the way I expected).    The code I use for generating the mark is:   \caption{Lista de notas presentes em um piano de 88 teclas,  com notação em Inglês, frequência em Hertz e comprimento de  onda em centímetros.\protect\footnotemark[1]}   Is there any way I can enforce my \footnotetext to appear in the same page as my \footnotemark? I'm looking to create a simple short-lived reservation system, and I'd like to generate confirmation numbers that are   unique random-looking alphanumeric short-ish, at least much shorter than 32 character-long strings returned by sha1   I'm only looking to have ~500 reservations, so I don't imagine high likelyhood of collissions.  One idea I had is generate an sha1 hash based on a date-time stamp and username, then truncating it to its first 10 characters.  Would something like that be reliably unique enough for the purposes of processing ~500 reservations? Choosing the best model from among different "best" models How do you choose a model from among different models chosen by different methods (e.g. backwards or forwards selection)?  Also what is a parsimonious model? Calculating extent of custom tiles? Problem  The picture is just an example of my case, but I am left with blank space on the left when I load all the tiles and I get error message in the console:   Failed to load resource: the server responded with a status of 404 (Not Found) http://path/to/the/tile//base/8/6/53.png  etc.     I could guess and limit the extent of the layer, but I would really like to know is there any way to calculate it and also be able to get rid of the error message.  What I know about the tiles?   Originally they are served as TMS They represent png images 256*256 They are in projection: EPSG:3857 In OpenLayers2 they were called like TMS layer In OpenLayers3 (v.3.8.2) they are called like XYZ layer (that includes -y):     var baseLayer = new ol.layer.Tile({   source: new ol.source.XYZ({     url: 'http://path/to/the/tiles/base/{z}/{x}/{-y}.png'   }) });   Thoughts and questions   How can I calculate extent of the tiles? Or in another words how can I find out zmin zmax xmin ymin xmax ymax or whole range of x/y?  What does x/y really represent? Are they coordinates of the left bottom corner or center of the tile? In which direction does x/y increase? How can I figure out x/y of at least one of the tile, for example left bottom one?    (I have modified the question for better understanding of the problem) 'Just' and 'only' carry a similar meaning, and while my feel for language usually helps me decide which one to use, there are times when I'm at a loss.  From my understanding, 'just' is used as a softener, to add flexibility and politeness. 'Only', on the other hand, is a relatively sharp exclusion. 'Just' also has a wider range of meanings, for instance to denote time.  While this sounds simple enough, it can get messy in practical usage. The sentence that made me stumble was "Not only airports are part of the target customer group, but also other large infrastructures".  Would the meaning of this sentence change when using 'just' instead? The way I see it, there is no difference in this case, except 'just' would make it sound a tad more colloquial. How to start applications such as xcompmgr at start up? I've been trying to make xcompmgrto autostart in Lubuntu 11.10 but it just doesn't, I have to start it manually.  I have added the next lines to the autostart script in ~/.config/openbox/  #Run a composite manager xcompmgr &amp;   But whenever I log in it doesn't do anything, I don't really need any special parameters for it I just want it to start at log in.  How can I make it run automatically at login? How can I get my graphics card to be recognized in System Info? I am new to Linux/Ubuntu. I installed Ubuntu 11.10 on my Lenovo IdeaPad y510.   System info says    Graphics: unknown.   Driver: unknown,   Experience: Standard.   What should I do to use all my laptop hardware components to work properly with Ubuntu? Screenshot: