Running stats in R
Calculate first difference by group in R
Interscholastic Mathematics League Senior B Division #2 Points $P$, $Q$, $R$ and $S$ are chosen on the sides of parallelogram $ABCD$, so that $P$ is on line $AB$, $Q$ is on line $BC$, $R$ is on line $CD$, $S$ is on line $DA$, and $AP=BQ=CR=DS=\frac{1}{3}(AB)$. Compute the ratio of the area of $PQRS$ to the area of $ABCD$. I know it might be too easy for you guys here. I'm practicing some problems in the textbook, but this one really drove me crazy. From $\gcd( a, b ) = 1$, I have $ax + by = 1$, where should I go from here? The extra $ac$ is so annoying. Any hint?    Thanks, Chan Cannot find Background Images feature to add in a photo in Blender 2.8  Where is it? Looking for a short story about a boy discovering that the government puts mind control chemicals in his cereal I would have read this around 1985-1998 because I remember being in school at the time. It may have been in a short story collection with "The Ugly Little Boy" and a story about a family dealing with people in the future coming back to enhance the intelligence of their baby with enfante terrible results. A boy, grade school age or so and very intelligent, discovers (or possibly just decides) that there are drugs in the food that prevent one from thinking critically. The details are kind of fuzzy in my head, but I remember that he at one point distills the chemical from his cereal, which he'd detected as having a high concentration, and overdoses. He's convinced that the government is trying to silence him — a scene that sticks out in my mind was him falling on a moving sidewalk and tearing his clothing as he tries to escape two "men in black" type figures that are pursuing him. The short story ended with him deciding that he was better off just consuming the food and being happy in his ignorance.  Years later, as an adult learning about mental illness, it struck me that the story was slightly ambiguous as to whether there actually were chemicals, and that the use of drugs that dulled his creativity and intelligence, and the decision to take them anyhow, was reminiscent of the feelings that manic people have toward their medications.  Anyone know what this one was? How did the word "Bollywood" originate? As we know, American film industry is referred as Hollywood. The American film industry is called so because the film industry emerged at the place called (Correct me if I am wrong). The place adopted the name Hollywood which is informally known as Tinseltown.  But my question is not about Hollywood but the . This is widely referred as Bollywood.   From the  page of Bollywood:      Bollywood is the sobriquet for India's Hindi language film industry, based in the city of Mumbai, Maharashtra. It is more formally referred to as Hindi cinema. The term "Bollywood" is often used by non-Indians as a synecdoche to refer to the whole of Indian cinema; however, Bollywood proper is only a part of the larger Indian film industry, which includes other production centres producing films in many other Indian languages. There is no place in India called Bollywood like the Hollywood.    Many people prefer calling it as Hindi film industry and against naming film industries with wood. There are other film industries in India too in other languages. Eg: Telugu film industry is named as Tollywood, Tamil industry is named Kollywood etc.,   is the list of Hollywood inspired nicknames.  My question:   Who is the creator of the word Bollywood? Since when did the word become popular? Is there a 2D generalization of the coefficient of restitution? The  characterizes a collision in one dimension by relating the initial and final speeds of the particles involved,  $$C_R = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$$   In a 2D collision, the velocity can be split into components parallel and perpendicular to the "plane of collision" (the plane tangent to the two objects' surfaces at their contact point), and the equation above applies to the perpendicular components of the velocities only. One could write  $$\begin{pmatrix}v_{2f\shortparallel} - v_{1f\shortparallel} \\ v_{2f\perp} - v_{1f\perp}\end{pmatrix} = \begin{pmatrix}1 &amp; 0 \\ 0 &amp; -C_R\end{pmatrix}\begin{pmatrix}v_{2i\shortparallel} - v_{1i\shortparallel} \\ v_{2i\perp} - v_{1i\perp}\end{pmatrix}$$   My question is, is it useful (i.e. does it produce a more accurate description of realistic 2D collisions) to generalize that matrix? Perhaps by allowing the top left element to be unequal to 1, or allowing nonzero off-diagonal elements? Or is there some nonlinear relation that works better?  As usual, if you know of any relevant published research, references would be much appreciated. Relationship between Partial Harmonic Sum and Logarithm. Equivalent of *Nix 'which' command in PowerShell? How do I ask PowerShell where something is?    For instance, "which notepad" and it returns the directory where the notepad.exe is run from according to the current paths. Quantum mechanics scattering theory Product of pairwise coprime integers divides $b$ if each integer divides $b$ Let $a_1....a_n$ be pairwise coprime.  That is $gcd(a_i, a_k) = 1$ for distinct $i,k$, I would like to show that if each $a_i$ divides $b$ then so does the product.  I can understand intuitively why it's true - just not sure how to formulate the proof exactly. I want to say if we consider the prime factorizations of each $a_i$, then no two prime factorizations share any prime numbers.   So the product of $a_1...a_n$ must appear in the prime factorization of $b$.  Is this correct?  Or at least if if the idea is correct, any way to formulate it more clearly? Did she judge him "wrong" or "wrongly"? Which one is the correct use?  She judged him wrong. She judged him wrongly.  Or, are both correct, but have slightly different meanings? In physics sometimes we find energy that is negative. What does the negative sign indicate? What is the concept of erasure in generics in Java? What is the concept of erasure in generics in Java? Sieve of eratosthenes infinite primes I don't understand why the sieve of eratosthenes doesn't prove an infinite number of primes.  For example the number 2 eliminates 1/2 of an infinite number of numbers.  The next prime, 3, eliminates 1/3 of the remainder, not 1/3 of all the numbers and so on.  5 would eliminate 1/5 of the remaining numbers. Why doesn't that prove an infinite number of primes? Thanks How to automatically generate a stacktrace when my program crashes I am working on Linux with the GCC compiler. When my C++ program crashes I would like it to automatically generate a stacktrace.  My program is being run by many different users and it also runs on Linux, Windows and Macintosh (all versions are compiled using gcc).  I would like my program to be able to generate a stack trace when it crashes and the next time the user runs it, it will ask them if it is ok to send the stack trace to me so I can track down the problem. I can handle the sending the info to me but I don't know how to generate the trace string. Any ideas? Story about criminal being "adjusted" which stops a man's commute I'm trying to remember the name of a story, I think the author was Larry Niven, but it might possibly be from Heinlein's "Future History" series (but a check of the stories included in the Future History indicates it's not in there), about a man whose commute is stopped because a criminal has to be 'adjusted'.  In this case, the criminal was a hit-and-run driver who left his victim untreated on the roadside, and was sentenced to suffer in the same manner. So his leg was carefully broken and he was left, untreated and in pain, on the side of the road in the same place for an hour. All the time ambulances were standing by, and every effort was made to ensure that the criminal did not suffer unduly - but that he did suffer, in a manner as close to the suffering he inflicted on his victim. How can I remove duplicate rows? What is the best way to remove duplicate rows from a fairly large SQL Server table (i.e. 300,000+ rows)?  The rows, of course, will not be perfect duplicates because of the existence of the RowID identity field.  MyTable  RowID int not null identity(1,1) primary key, Col1 varchar(20) not null, Col2 varchar(2048) not null, Col3 tinyint not null In first.bat, I use  var5=C:\Program Files (x86)\Microsoft Visual Studio 10.0\VC\bin\x86_amd64\ CALL scripts\vc64.bat %var5%   And in scripts\vc64.bat, I use  SET var6=%1vcvarsx86_amd64.bat CALL %var6%   But I get : 'C:\Programvcvarsx86_amd64.bat' is not internal or external command..... error.  If the path assigned to var5 has no space, then it is fine.  I tried several combination of quotations and %1vcvarsx86_amd64.bat, but no change.  How do I make it works with path with spaces?    To be more precise, suppose it is C:\a b c\  In first.bat:       SET var6=C:\a b c\  =====&gt;  '"C:\avcvarsx86_amd64.bat"' is not.....`      SET var6="C:\a b c\" ====&gt;   There should not be a b      SET var6=C:\a\ b\ c\ ====&gt;    Can't find the specified path      SET var6=C:\a b c\ + Using "%var6" ====&gt; There should not be a b     Update: Here is an example. Change a b to ab works.  call.bat in C:\  @ECHO OFF  SET var5=C:\a b\ CALL C:\1.bat "%var5%"  pause   1.bat in C:\  @ECHO OFF  SET var6=%~1Test.bat CALL %var6%  RMDIR /S C:\NoWorry   Test.bat in C:\a b Test.bat in C:\ab  @ECHO OFF  RMDIR /S C:\ThereIsNoSuchFolder I've heard of kits that can fix your Xbox 360 if you happen to get the . Do any of these work, and do they void your warranty? Are there any other noteworthy tips or tricks? Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$ I'd like to show that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$ for all $x,y \ge 0$ and $p \in [0,1]$. I started out by defining a function $f(x,y)=(1+x)^p(1+y)^{1-p}$ and checking for critical points, but there are none in the domain I'm interested in. I think I'm probably missing a slick application of Jensen's or Holder's inequalities, but so far a proof eludes me. Any suggestions would be appreciated.