Is it preferable to brush one's teeth before or 30-60 minutes after breakfast?  I found plenty of blog posts / testimonials but no actual scientific studies. I'm looking for the latter.    Waiting at least ~30 minutes before toothbrushing has been shown to be useful in a few studies, e.g. :     It is concluded that for protection of dentin surfaces at least 30 min should elapse before toothbrushing after an erosive attack.   Here is an example of some dentist advising to toothbrush before breakfast:  :     Brush and floss your teeth after you eat. This is the most logical answer .You do not want bacteria eating the left-overs in your mouth from breakfast . Remember to floss . Take responsibility for the food that can not be brushed or removed between your teeth.    look very similar but the options they considered were before vs. what after breakfast. They didn't consider comparing before vs. 30 minutes after breakfast. As a result, it isn't a duplicate.   contains some arguments in favor of before and after:   arguments for brushing before:   Bacteria  begin to metabolize sugars into acids almost instantly.  Salivary calcium levels are lowest in the morning putting your teeth are at a much higher risk of acidic damage from bacteria and dietary acids in the morning than at any other time Better to brush with fluoride toothpaste before so you have some fluoride left in your mouth when eating. That fluoride will immediately protect your teeth from the acidous attack of consumed carbohydrates and acidic foodstuff.   arguments for brushing after (waiting >~30 minutes):   gentle brushing can help neutralize the pH in the mouth after a meal. Obviously, if you eat something acidic which will temporarily demineralize your teeth, don't start scrubbing away.
Usually in the morning I brush my teeth after breakfast. But my dentist told me that it's better not brush them right after eating food, because eating makes them soft.  So now I brush my teeth before breakfast. Is this the correct procedure?  PS. Just found , the author has the same opinion.  Just want to get a additional confirmation.
Running OS X Apps On Ubuntu 14.04 Is there any way that still works to run OS X apps on Ubuntu 14.04? If Darling still works, please tell me how to install it.  Thanks. :) extended stars-and-bars problem(where the upper limit of the variable is bounded) Composition of two functions - Bijection I keep going around in circles with this problem:  Given the functions $f:X\to Y$ and $g:Y\to Z$, if $g\circ f:X\to Z$ is bijective, then prove $f$ must be one-to-one and $g$ must be onto. I was able to prove this without difficulty. However the second part of the problem has me stumped:  a) Find functions $f,g$ where $g\circ f$ is bijective but $f$ fails to be onto.  b) Find functions $f,g$ where $g\circ f$ is bijective but $g$ fails to be one-to-one.  My attempt for (a): Let $f:\mathbb{R}\to \mathbb{R}^+$ be defined by $f(x)=e^x$ and $g:\mathbb{R}^+\to \mathbb{R}$ be defined by $g(x)=\ln(x)$. Then $g\circ f(x)=g(f(x))=g(e^x)=\ln(e^x)=x$ which is bijective, $g$ is onto and $f$ fails to be onto... or so I thought. Upon closer inspection $f$ $\underline{is}$ onto for the given codomain $\mathbb{R}^+$.  And now I'm stuck in a rut. Anyone out there have any hints? `time echo` got no output Unique (non-repeating) random numbers in O(1)? Series convergence of $\sum_{n=1}^{\infty}\frac{1}{n^3}$ "Welcome to emergency mode!" Think it is a fsck problem My computer booted to a black screen with this error message.  Welcome to emergency mode! After logging in,type "journalctl -xb" to view    system logs, "systemctl reboot" to reboot, "systemctl default" or ^D to    try again to boot into default mode.   journalctl -xb snippet (what I think is wrong):  -- Unit systemd-fsckd.service has begun starting up. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1 contains a file system with errors, check forced. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1: Inodes that were part of a corrupted orphan linked list found. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: /dev/sdb1: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: (i.e., without -a or -p options) juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: fsck failed with error code 4. juli 09 15:40:16 kim-SSD-Sationary systemd-fsck[414]: Running request emergency.target/start/replace juli 09 15:40:16 kim-SSD-Sationary systemd[1]: systemd-fsck-root.service: main process exited, code=exited, status=1/FAILURE juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Failed to start File System Check on Root Device. -- Subject: Unit systemd-fsck-root.service has failed -- Defined-By: systemd -- Support: http://lists.freedesktop.org/mailman/listinfo/systemd-devel --  -- Unit systemd-fsck-root.service has failed. --  -- The result is failed. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Unit systemd-fsck-root.service entered failed state. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: systemd-fsck-root.service failed. juli 09 15:40:16 kim-SSD-Sationary systemd[1]: Starting Remount Root and Kernel File Systems... -- Subject: Unit systemd-remount-fs.service has begun start-up -- Defined-By: systemd -- Support: http://lists.freedesktop.org/mailman/listinfo/systemd-devel   I ignored other errors like: ACPI PCC probe failed, nvidia not proprietary, etc.  I can access my PC by pressing Ctrl+D , but it is annoying. Solve $x^2$ $mod$ $23 = 7^2$ What is the procedure to solving $x^2$ $mod$ $23 = 7^2$? According to WolframAlpha, there is no integer solution but I am completely confused as to what steps was taken to determine that.  Before asking the question, I did try to solve this using brute force by plugging in some arbitrary numbers to see whether the square of that number mod $23$ gave me a remainder of $49$ but the procedure was quite tedious. Hence the reason I turned to WolframAlpha.   Going back to the question, I want to know how WoflramAlpha determined that there was no integer solution to solving for $x$. Uninstalling Ubuntu from dual boot / changing bootloader I've recently searched how I could get the application's directory in Java. I've finally found the answer but I've needed surprisingly long because searching for such a generic term isn't easy. I think it would be a good idea to compile a list of how to achieve this in multiple languages.  Feel free to up/downvote if you (don't) like the idea and please contribute if you like it.  Clarification:  There's a fine distinction between the directory that contains the executable file and the current working directory (given by pwd under Unix). I was originally interested in the former but feel free to post methods for determining the latter as well (clarifying which one you mean). I recently obtained a Seagate Expansion 3TB external harddrive. I'm trying to copy over all of my stuff to prepare for reformatting and installing a new OS. A handful of my mp3s did not transfer when I was doing my music folder. I checked them after the fact and the ORIGINAL files became corrupted somehow. I am certain of at least a few that were not corrupted in anyway before doing this. Every now and then during my transfer I got  There was an error copying the file into /media/***/Seagate Expansion Drive. Error splicing file: Input/output error   Each of these files that I check after are corrupted at some point in the duration and will not play. Why did this happen and what can I do to fix it? Mount ext4 partition with >4KiB block size I'm attempting to mount a partition from a NAS and the block size is 65536 bytes, this is above my distribution's precompiled kernel memory paging size of 4KiB, so I cannot mount it normally.  Which kernel compilation parameters would I need to change to get a memory paging size large enough?  Would it be possible under the x86-64 architecture?  EDIT: There doesn't seem to be an option for it in make xconfig unless I'm just missing it.  From page_types.h under arch\x86\include\asm  /* PAGE_SHIFT determines the page size */ #define PAGE_SHIFT  12 #define PAGE_SIZE   (_AC(1,UL) &lt;&lt; PAGE_SHIFT) #define PAGE_MASK   (~(PAGE_SIZE-1))`   It doesn't look to be configurable.  I could change this number directly to 16 and it would probably make pretty explosions and destroy Tokyo.  I shall probably try it later this evening. 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? :-) Every time I call \section{**Section Name**}, a number appears to the left of the name of the section.  How do I hide this number? Sentence vs proposition How to find a computer name in a LAN from the IP address? My network includes machines running Linux and others running Windows. And my machine is running Linux. Can you eradicate a disease without finding a Cure first? If you eradicate a disease in Pandemic without finding a cure first (ie remove all of one colour from the board) does it remain eradicated or can it still come back unless it has also been cured?  The rules are a bit unclear on this point ... Or I missed the rule that explicitly states this. Changing deleted partition back to NTFS without having to recover to another drive Recently, I accidentally deleted a partition off of my external hard disk. I had a copy of Minitool Power Data Recovery left over when the same drive had become RAW about a year ago. When I selected lost partition recovery, it immediately identified my deleted partition and allowed me to access and recover the files that were there.  My question is this. Is there any way to change the deleted partition back to NTFS without having to have a drive that you need to recover the files to and then move them back after formatting, as I don’t have any space left on either the computer or on my other external hard drive, so that I don’t need to buy a new one. The files on the disk were really important.  I am using Windows 7 Home Premium.  I appreciate any help in solving this crisis. Fighting a black hole: Could a strong spherical shell inside an event horizon resist falling in to the singularity? As a thought experiment imagine an incredibly strong spherical shell with a diameter a bit smaller than the event horizon of a particular large black hole.  The shell is split into two hemispheres, and the two hemispheres are released from either side of the black hole such that they will meet and join inside the event horizon, forming a stationary uniform shell around the singularity. Of course, we consider that the shell is flawless, and exactly centred around a stationary singularity - clearly this is an unstable equilibrium but we imagine it is possible.  Is is possible that the structural forces provided by such a shell could be strong enough to resist the extreme gravity conditions inside the event horizon, and hence avoid being sucked into the singularity?      (For the purposes of this question we can consider taking the shell to extreme levels of strength, e.g. consider that the shell is made from a theoretical maximally densely packed neutronium-like material that is strongly resistant to any further compression due to the Pauli exclusion principle). For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous. This is from Apostol's Mathematical Analysis.  Let $f$ be a function defined in $[0, 1]$ such that for each real number $y$ either there is no value of $x \in [0, 1]$ such that $f(x) = y$ or there are exactly two such values of $x$. Prove that a) $f$ must be discontinuous on $[0, 1]$. b) $f$ has an infinite number of discontinuities in $[0, 1]$.  Long back I had tried to solve the problem. I reasoned for a) as follows: Suppose that $f$ is continuous and let $m, M$ be its minimum and maximum values on $[0, 1]$. Clearly if $m = M$ then $f$ is constant and clearly for every $x \in [0, 1]$ we have $f(x) = m = M$ which contradicts given property of $f$. So we must have $m &lt; M$. Clearly there are now four distinct points $a, b, c, d$ in $[0, 1]$ such that $f(a) = f(b) = m, f(c) = f(d) = M$. By checking possible linear orderings of $a, b, c, d$ I could find a find a value of $y \in (m, M)$ for which there exist more than 2 values of $x$ such that $f(x) = y$. However this approach looks very clumsy and I suspect much simpler and easier proof is possible. Part b) seemed bit complex and my only guess was that probably $f$ was discontinuous on each subinterval of $[0, 1]$ but I could not prove this claim (may be the claim is wrong!). Perhaps there is a simpler solution to this part of the problem. Apostol's also asks to provide an example of such a function. Any hints on the overall problem will be appreciated.  Update: After looking at Marc's answer, I think I have found a way to show that there are infinitely many discontinuities. Earlier the solution was a part of the question itself but it made the question look quite lengthy hence I moved it to a separate answer.