Interpolation between derivatives on bounded interval
Inequality involving the sup of a function and its first and second derivatives
Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$? I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:     Find a function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for all $x \in \mathbb{R}$.   I am also wondering, can we find $f$ so that is continuous?  I was thinking of letting $f$ be a periodic function, and adding half the period to x each time. I had no success with this, and am now thinking that such a function does not exist.  Source: I was working on a Beamer presentation, and noticed that, if I open a group scope and use a \tiny, the font size will bet set, but there will be a large space between lines. However, if I issue a \par at the end of the paragraph, it works as intended:  \documentclass[t]{beamer} \newcommand{\shortlipsum} {Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna. Donec vehicula augue euneque. Pellentesque habitant morbi tristique senectus et netus et malesuada fames ac turpis egestas. Mauris ut leo. Cras viverra}  \begin{document} \begin{frame} \shortlipsum  {\tiny \shortlipsum\par   % &lt;--- removing this \par will break line spacing } \end{frame} \end{document}   Here's a comparison between not using and using \par:    Am I doing it the right way? Why is \par the right way? Why doesn't xpath work when processing an XHTML document with lxml (in python)? Which Harry Potter works are considered canon? on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$ I noticed something when I was doing a proof of the BAC-CAB rule, and wanted to check if my intuition was correct.   First, when I actually multiplied out $\vec{B} (\vec{A} \cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B}) $ I noticed that I got the same thing. That is, the two quantities are equal. This makes sense since they are dot products. But that further implies that the expression goes to zero.  Well, that means that $\vec{A} \times (\vec{B} \times \vec{C})$ would have to be zero too. And when I thought about it it occurred to me that $\vec{B} \times \vec{C}$ describes a vector perpendicular to both B and C; it's orthogonal. But $\vec{A} \times$ (any vector) will describe a vector perpendicular to (any vector) and in this case, that's $\vec{B} \times \vec{C}$. That is going to be parallel with the vector described by $\vec{B} \times \vec{C}$ itself and that further implies that the cross product of the two is equal to zero. Which is equal to the BAC-CAB quantity above.   So, what is the big honking flaw in my reasoning here? :-) I am a seafarer (marine engineer) and I have to often visit different countries. This time I received my sign-off from Iran and they stamped my passport. Would this cause a problem for getting a US C1D1 visa since my next company wants me to have this visa? I am an indian national. Ring of order $p^2$ is commutative. I would like to show that ring of order $p^2$ is commutative.   Taking $G=(R, +)$ as group, we have two possible isomorphism classes $\mathbb Z /p^2\mathbb Z$ and $\mathbb Z/ p\mathbb Z \times \mathbb Z /p\mathbb Z$.   Since characterstic must divide the size of the group then we have two possibilities $p$ and $p^2$.   Now IU don't understand how can I reason to say that the multiplication is commutative and how can I conclude for the case when characterstic is $p$? What do you call a person who does/thinks the opposite of the crowd for just that reason? Such a person does not hold in high regard the decisions taken by a group (not a team) of people. She believes that groups tend to be anarchic and don't serve their purpose.  An example would be - for outings with friends, this person will not like planning it with the group since they will mess it up.  Therefore, this person has convinced herself that when more people are involved with something, it will be worse. She distrusts crowds instinctively basically. In Ruby on Rails, how do I format a date with the "th" suffix, as in, "Sun Oct 5th"? Batch Processing Quick Query I have like 50000 records that meet my condition for batch processing. My batch process begins at say 10:01 AM and ends at 10:10 AM. I set my batch size to be 200.  Can someone tell me what would happen  a) If an user creates a record at 10:03 AM that meet my condition...will this record be also be processed or not ?  b) Will the records which are getting processed during Batch 1 are technically locked till that particular batch is processed completely ?  c) What will be the behaviour if a record in Batch N does not meet my condition anymore but that particular batch has already been started to process ?  d) Is it possible to create a parallel processing of sorts such that Salesforce (instead of running one batch at a time) it will start processing parallely with multiple batches ?  I searched online and I could find manual/instructions with respect to how to run batch apex but I am not able to get clarity with respect to these specific cases. We are asked the following:      let $f(x)$ be a continuous function on $[0, 1]$ such that   $$\int _ {0} ^ {1} \ f(y) dy = 1$$   prove that $f(x) = 1$ for some $x ∈ [0, 1]$   The question asks to take into account Rolle's Theorem and Mean Value Theorem.  I'm aware of the definitions, but I am really unsure how to answer this question and my tutors are not really responding till after easter.    Could someone please help? Why is there a "y" involved?  Thanks you Do these new insights into prime numbers affect encryption security? reports:  Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.  Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time? 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 What is the MIME equivalent of uuencode for sendmail? e.g., I've got files I want to send by sendmail but the receiving mailbox (for posting pictures to my blog) will not accept uuencode files a la   $uuencode file | sendmail user@domain  I'm guessing it'll accept MIME though. So what command can I use instead? Series about proxy space battles via implanted computers in kids The premise of the story is that there are 2-3 main coalitions on Earth and they fight over resources in the edges of the solar system with spaceships that are controlled from Earth by kids with computers in their heads.  If I recall correctly:   The main character's name is Tom, he starts out by conning people at some sort of gambling arcade before he is found and recruited by the "army" The Pentagon in the US has a super tall tower built in the courtyard called The Spire where most of the three books take place. The reason they need kids for this has something to do with brain elasticity and putting computers in adults causes them to go mad. All the cadets are super dismissive about learning how to program their computer aside from one girl who is super good at it. The best fighters from each side of this conflict were something of superstars, but everyone else was kept completely anonymous for their protection. There were at least 3 books last I knew, released circa 2011-2014 The third book ends with Tom being the first astronaut to Mars. The first book had a green cover, the second was yellow, and the third was red. Though I might have the order wrong. Which is correct: "This is her" or "This is she"? Upon answering the telephone, the person calling asks if Joan is available. If Joan is the person who answered the phone, should she say "This is her" or "This is she"? How do I remove the limit on PC keyboard button presses? Whenever I play a PC game with the directional keys, I can only press two buttons at a time.  For instance if I am playing a shooter like R-Type, I cannot hold down the fire button while moving diagonally at the same time.  The computer simply ignores the extra button press.  How do I remove this limit so I can shoot while moving diagonally?  This limitation seems to apply for every game.  I am running Windows 7 if it makes any difference.  I noticed the same problem back when I used Windows XP. Portuguese tab names on profile page A few minutes ago, when I viewed profile pages on my phone, the tabs were all in Portuguese:   On the mobile interface:   On the full site     But it didn't happen on my desktop, and now it's back to normal.  Other users report that the site is normal, but  says it's still happening, so I've decided to post this bug report even though it seems to be working for me at the moment.   Perhaps it's some sort of intermittent problem.  I'm posting this on Meta.SE, since it seemed to happen on both Japanese.SE and ELL.SE, so it doesn't seem to be specific to a particular site. The browser's "Back" button cause problems with the review buttons What is a simple way of downloading a file from a URL path?