The product of manifolds without boundaries is always a manifold without boundary?  $\textbf{Example:}$ $S^2\times \mathbb R$ has a boundary?
Simple question but I am confused.  What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$?  What would be the general way to evaluate the boundary of a product manifold?  Thanks for the replies!
Mirror modifier is not working, no mirrored object? Why is the speed of light defined as 299792458 m/s? Why is the speed of light defined as $299792458$ $m/s$? Why did they choose that number and no other number?  Or phrased differently:  Why is a metre $1/299792458$ of the distance light travels in a sec in a vacuum? I ride regularly in the city with my a few valuables with me. My purse, phone, keys, company ID card etc. I hate having a bag on me while ride so I am looking for a solution to have these few items in a bike bag, but also have them safe if I stop somewhere to run into  a shop or a cafe for 15 minutes.   My original idea was to look for a small saddle or top tube bag that is lockable and also safely locked to the frame, and I can simply leave it there. As it turns out there is no such thing! Or I just did not find it?  My second idea was to search for a similar bag, but which is easily detachable and I can take it with myself as a waist bag for example. Same story, I couldn't find anything like this on the market.  Do you know any bag that could work for me? Or do you have an alternate suggestion for my conundrum?   Thank you very much if you take the time to answer! Ride safe! János How does the coin limit work? Why is the derivative of a circle's area its perimeter (and similarly for spheres)? Intuition for uniform continuity of a function on $\mathbb{R}$ Laptop water damage repair procedure I'd love some advice in repairing a water-damaged laptop.  My girlfriend spilled a glass of water on her Sony Vaio and now it doesn't turn on...at all...dead! What steps should I go through to work out where the problem is and which part(s) I'll need to replace?  I can't seem to find out what has happened by just looking at the circuit boards. I'm new to electronics but could buy relevant volt-meters etc. I'm just looking for a decent trouble-shooting guide. Equals(=) vs. LIKE When using SQL, are there any benefits of using = in a WHERE clause instead of LIKE?  Without any special operators, LIKE and = are the same, right? Moderncv adding date of birth to personal information I'm doing my cv with the moderncv package and I'm using the classic style. I would like to add a date of birth line directly above the adress. Does anyone know how to do this? I've tried various solutions but they all seem to be broken. Nearby objects won't move independently? I've seen a contraction of two words.  I can't see why it wouldn't've been possible to have been contracted twice.  Is it possible and how should it be punctuated?  Update:  Ok, to sum up the answers so far   This appears in spoken British and American English It is from one of the lower  Even if spoken this way sometimes, it isn't really written as a double contraction, except as written speech in fiction. And from my own googling in , it appears most written forms are old British words, often nautical like fo'c'sle. I would like to scale an image file for each dimension differently for example 50% of the x dimension or 25% of the y dimension. Thank you very much. C++: delete vs. free and performance Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually?  I know you can do this for the sum, the product and the convolution of two functions. But I haven't seen a formula for the composition of two functions. When you pick up items in Skyrim it tells you the value of each item. However when I go and sell the items the merchants are only paying 30% of this value, at best.  Discounting any perks or other effects, how high does Speechcraft have to be before merchants start paying full value for items? Will fruit/seed of a bonsai yield another bonsai plant/tree? With practice one can create a bonsai; no complicated modification of the DNA in a high-end lab required. The bonsai will even beget bonsai fruit.   What I am curious about is - if such bonsai fruit is planted again, will it give rise to a normal, or a bonsai tree? Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$ Someone had posted a question on this site as to what would be sum of digits of $999999999999^3$ (twelve $9s$ ) equal to?   I did some computation and found the pattern that sum of digits of  $9^3 = 18$,  $99^3 = 36$,  $999^3 = 54$ and so on. So, I had replied that sum of digits of  $999999999999^3 = 12 \cdot 18 = 216$.  Can anybody help me prove this, that sum of digits of  $(\underbrace{999\dots9}_{n\text{ times}})^3 = 18n.$ utf8 and ISO 8859-1 encoding references trouble I have to use one particular template ( am obliged to use it for my thesis) it uses \bibliographystyle{plain} Ihave both english and frensh text on it (for abstract and résumé) there is no problem for characters like ( é, ç...) but some of references does not appear they are replaced by [?] knowing that foe encoding \RequirePackage[latin1]{inputenc} how to fix this problem please!   \documentclass{article}   \RequirePackage[latin1]{inputenc}   \RequirePackage{fancyhdr}     blabla \cite{nayak2015comprehensive, byun2002applications, shah2004sequential, fletcher2009support, cornuejols2011apprentissage, byun2002applications}   @article{nayak2015comprehensive,    title={A comprehensive survey on support vector machine in data mining     tasks: applications \&amp; challenges},    author={Nayak, Janmenjoy and Naik, Bighnaraj and Behera, HS},    journal={International Journal of Database Theory and Application},    volume={8},    number={1},    pages={169--186},    year={2015}    }      @inproceedings{shah2004sequential,     title={Sequential k-nearest neighbor pattern recognition for usable      speech classification},     author={Shah, Jashmin K and Smolenski, Brett Y and Yantorno, Robert E      and Iyer, Ananth N},     booktitle={Signal Processing Conference, 2004 12th European},     pages={741--744},     year={2004},     organization={IEEE}     }      @misc{fletcher2009support,      title  = {Support vector machines explained},     author = {Fletcher, Tristan},     journal= {University College London, London},     year   = {2009},     howpublished = "\url{http://sutikno.blog.undip.ac.id/files/2011/11/SVM-     Explained.pdf}",     note   = {Accessed: 2017-04-30}     }      @inbook{cornuejols2011apprentissage,      title={Apprentissage artificiel: concepts et algorithmes},      author={Cornu{\'e}jols, Antoine and Miclet, Laurent},      year={2011},      pages={279--309},      publisher={Editions Eyrolles}       }      @incollection{byun2002applications,      title={Applications of support vector machines for pattern recognition:       A survey},      author={Byun, Hyeran and Lee, Seong-Whan},      booktitle={Pattern recognition with support vector machines},       pages={213--236},      year={2002},      publisher={Springer}      }    \end{document}  knowing that my references work well on other template but this one no how to fix this problm please! here is the error I got   Edited here is the link to the template  in fact I have created a littele example ( since I could not publish my thesis ...) the error has disappeared but my references still [?] Illustrator Vector Graphics Appearing Pixellated How do I recover my accidentally lost Windows partitions after installing Ubuntu? I have a Toshiba satellite A-200 laptop with a Vista OS on it with 4 NTFS partitions  (C:) Vista  (D:) Entertainment  (E:) Work  (F:) Sources   and I wanted to start using Ubuntu instead. So I tried it first from the live CD and everything was OK and all the partitions were shown and working and so I decided to install Ubuntu to replace Vista on the (C:) drive. After I did that I can no longer find my folders and files on the (D:), (E:), (F:) partitions and the only file system that is shown is one 198 GB although my HDD is 320 GB.  I can't access the lost data on the remaining 120 GB which I hope is still there and not totally lost  I am now working from the live CD but I am unable to install testdisk. Can I recover the Vista partitions by the product recovery CD to get my laptop back to the factory settings? Can I recover the NTFS partitions using a recovery program for Windows or will that make the problem worse?   I need these data badly as I don't have a backup for them.