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How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $? If it is possible to express  $ \cos n \alpha $ in terms of $ \cos \alpha $  as a power series for integer $n$ ...  I like to see an expression for the quotient angle that obviously tallies when $ (\alpha , \beta) $ are swapped.  EDIT1:  Something like:  $$ \cos (\alpha - \beta)= \cos \alpha \cos \beta  + \sin \alpha  \sin \beta $$  EDIT2:  Like to know why $ \cos ( \alpha / \beta )$ cannot be expressed in terms of $ \cos \alpha, \cos \beta $, but  $ \cos ( \alpha + \beta) $  can be expressed in terms of  $\cos \alpha $ and $ \cos \beta. $ How to deal with hierarchical / nested data in machine learning Possible to click on an object in the viewport and the object list hierarchy automatically goes to that object also? How to pull email address and password from text file with CMD? I have a text file named file.txt Which contains,  Combo: testing123mail@gmail.com:password As Combo:  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf some text here some more text here As Combo: hadvsfhia@ahfjbadj.com:password@1 some random text here iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   I am running the below code, to filter only mail:pass from the above text file file.txt  for /f "tokens=3" %%a in ('type file.txt^|find "As Combo:"') do &gt;&gt;mail.txt echo %%a for /f "tokens=2" %%b in ('type file.txt^|find "Combo:"') do &gt;&gt;mail.txt echo %%b for /f "tokens=1" %%c in ('type file.txt^|find ":"') do &gt;&gt;mail.txt echo %%c   Expected Output in mail.txt,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   But, am getting actual output is,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password Combo: Combo: Combo: As As iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   Note: The mail id and passwords will be differ in different text file. Above given mail id and passwords are typed by me randomly for example.  Kindly fix my code and help to get expected output. Thanks in advance. Apply a modifier as a shapekey into existing shapekey or merge shapekeys? I am making a shapekey of a model shrinkwrapped to another model. The shrinkmap modifier is too inaccurate to wrap to the model alone. So, my first step was to roughly hand position the geometry around of the first model around the second, then use shrinkwrap. The rough positioning was done as a shape key. I can then apply shrinkwrap, but its as a separate shapekey. The two do not work well together. I need them to be the same shapekey. How do I apply a modifier to an existing shapekey, or merge/combine/join the two shapekeys? Continuing in the Visual Studio debugger after an exception occurs When I debug a C# program and I get an exception throwed (either thrown by code OR thrown by the framework), the IDE stops and get me to the corresponding line in my code.  Everything is fine for now.  I then press "F5" to continue. From this moment, it seams like I'm in an infinite loop. The IDE always get me back to the exception line. I have to Shift + F5 (stop debugging/terminate the program) to get out of his.  I talked with some co-workers here and they told me that this happens sometime to them too.  What's happening? Emmy Noether's theorem in simpler terms I'd like to understand  and its contents as to what it implies in a bit simpler terms. I am familiar with mathematics unto Calculus 1,2,3 and some linear algebra and group theory. I am familiar with that for each symmetry there is a conservation law.    Now is there a one-one correspondence between these two always? What about laws like conservation of charge?  And conservation of energy?  And conservation of mass?  Considering our view of nature was changed as Special Relativity was introduced and thus, what earlier the theorem would have asserted as mass and energy to be originating from different symmetries then how did we reconcile these two as just one symmetry? I would like to say something like     The boy, Adam's, favorite toy was a bike.   What is the proper way to say this? Choosing variables to include in a multiple linear regression model How can I show that if the second fundamental form of a surface is identically equal to zero, then the surface is a plane? This is my question:  Let P be a plane considered as a surface in 3-space. Show that its second fundamental form is zero. Conversely, show that if the second fundamental form of a surface is identically zero then the surface is a plane.  So far I have that the general equation of the plane is $$ax+by+cz+d=0.$$ Then $$M(u,v)=(u,v,\alpha u + \beta v + \gamma),$$ where $a,b,c,d,\alpha,\beta, \gamma$ are all constants and $c\not=0$. Then $M_u=(1,0,\alpha)$ and $M_v=(0,1,\beta)$. I know that $\vec{n} =\frac{M_u \times M_v}{|M_u \times M_v|}$ which gives $$ \vec{n}=\frac{(-\alpha,-\beta, 1)}{\sqrt{\alpha^2+\beta^2+1}}.$$ Hence, the unit norm is a constant. $$ M_{uu}=M_{uv}=M_{vv}=0$$ Hence $l=m=n=0$, so the second fundamental form is $$ ldu^2+mdudv+ndv^2=0.$$  Now I need to show the converse, but I don't know how. Please help :) Word for the satisfaction of curiosity There is a good feeling you get when your curiosity is satisfied, especially if is about a question you found interesting or important, or have spent a lot of time and energy trying to answer. Depending on how much I wanted to know, this feeling can be quite pleasing. Is there a word specifically for either the satisfaction of a curiosity, or the feeling that results? Compactness of  Multiplication Operator on $L^2$ How do I get on a Minecraft PE server when it says 'invalid name'? I am having trouble getting onto a server in Minecraft PE on iOS. When I tap on the person's server it just says invalid name:    I have tried turning off and restarting the iPad, going to home screen and logging back into the server but I have had no luck and no change. Is there a reason why this is so? When administering Linux systems I often find myself struggling to track down the culprit after a partition goes full. I normally use du / | sort -nr but on a large filesystem this takes a long time before any results are returned.  Also, this is usually successful in highlighting the worst offender but I've often found myself resorting to du without the sort in more subtle cases and then had to trawl through the output.  I'd prefer a command line solution which relies on standard Linux commands since I have to administer quite a few systems and installing new software is a hassle (especially when out of disk space!) Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I have no background in Algebraic Number Theory, but if someone explains in that language, I would like to see it and learn it as much as I can at this moment. References will also be appreciated!  Let $q$ be a power of prime $p$ such that $q \equiv 1 \mod 4$ and $p \nmid 2n$.  Question. Let $\chi_{4} : \mathbb{F}_{q}^{*} \rightarrow S^{1}$ be the character of order $4$, show that $\chi_{4}(-4) = 1$.  The hint in the book says we may show that $\chi_{4}(4) = \chi_{4}(-1) = 1$ for $q \equiv 1 \mod 8$ and $\chi_{4}(4) = \chi_{4}(-1) = -1$ for $q \equiv 5 \mod 8$.  If $q \equiv 1 \mod 8$, then $\chi_{4}(4) = \chi_{4}(2)^{2} = \chi_{2}(2) = \pm 1$, where $\chi_{j}$ denotes the character of order $j$. So I think $2$ must be quadratic residue in this case, but I cannot figure out why. How can I solve the following sequence Let we have the following sequence  $$y_n= \frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\  \sqrt{n^2+n}}$$ find the limit of the sequence $y_n$ decide whether it is increasing or decreasing An Open Interval and a Half-open Interval are not Homeomorphic Use of 'const' for function parameters How far do you go with const?  Do you just make functions const when necessary or do you go the whole hog and use it everywhere?  For example, imagine a simple mutator that takes a single boolean parameter:  void SetValue(const bool b) { my_val_ = b; }   Is that const actually useful?  Personally I opt to use it extensively, including parameters, but in this case I wonder if it's worthwhile?  I was also surprised to learn that you can omit const from parameters in a function declaration but can include it in the function definition, e.g.:  .h file  void func(int n, long l);   .cpp file  void func(const int n, const long l)   Is there a reason for this? It seems a little unusual to me. I'm trying to continue the numbering of examples in enumerate environments in beamer. For example, I'd like one slide to feature   Foo Bar   and the next slide   Zip Yadda   My strategy so far has been to define a counter that saves the enumerate counter at the end of the first environment and can then be queried to set the enumerate counter at the start of the second enumerate environment. (This is essentially the proposal also found at )  That works wonderfully so long as I don't try to uncover elements of the slide sequentially. The problem, of course, is that as each of the pdf-pages that make up the slide is created, the counter is advanced. Here's a MWE.  \documentclass{beamer}  \setbeamercovered{highly dynamic}  \newcounter{saveenumi} \newcommand{\seti}{\setcounter{saveenumi}{\value{enumi}}} \newcommand{\conti}{\setcounter{enumi}{\value{saveenumi}}}   \begin{document}  \begin{frame}[&lt;+-&gt;]   \begin{enumerate}   \item foo   \item bar%     \seti   \end{enumerate} \end{frame}  \begin{frame}[&lt;+-&gt;]   \begin{enumerate}     \conti   \item zip   \item yadda%     \seti   \end{enumerate} \end{frame}  \end{document}   As the second slide is progressively uncovered (pages 3 and 4 of the resulting PDF), the counter at the beginning of the enumerate environment is advanced each time. So page 3 of the PDF has the right numbering, (3) and (4) respectively, while the fully revealed page has the wrong numbering. Pictures follow:   on one of the slides    on the following one Projecting sp objects in R I have a number of shapefiles in different CRSs (mostly WGS84 lat/lon) that I'd like to transform into a common projection (likely Albers Equal Area Conic, but I may ask for help on choosing in another question once my problem gets better-defined).  I spent a few months doing spatial stats stuff in R, but it was 5 years ago.  For the life of me, I cannot remember how to transform an sp object (e.g. SpatialPolygonsDataFrame) from one projection to another.  Example code:  P4S.latlon &lt;- CRS("+proj=longlat +datum=WGS84") hrr.shp &lt;- readShapePoly("HRR_Bdry"), verbose=TRUE, proj4string=P4S.latlon)  # Shapefile available at  #   http://www.dartmouthatlas.org/downloads/geography/hrr_bdry.zip  #   but you must rename all the filenames to have the same  #   capitalization for it to work in R   Now I have a SpatialPolygonsDataFrame with appropriate projection information, but I'd like to transform it to the desired projection.  I recall there being a somewhat unintuitively-named function for this, but I can't remember what it is.  Note that I do not want just to change the CRS but to change the coordinates to match ("reproject", "transform", etc.).  Edit  Excluding AK/HI which are annoyingly placed in Mexico for this shapefile:  library(taRifx.geo) hrr.shp &lt;-    subset(hrr.shp, !(grepl( "AK-" , hrr.shp@data$HRRCITY ) |                                      grepl( "HI-" , hrr.shp@data$HRRCITY )) ) proj4string(hrr.shp) &lt;- P4S.latlon