The question We are given a set of size $m$, which we can assume to be the set $[m] = \{1,2,\dots,m\}$. Say that a sequence $(A_i,B_i,C_i)_{i=1}^k$ triple-splits the set $[m]$ if:  For each $i$, $(A_i, B_i, C_i)$ is a partition of $[m]$: $A_i,B_i,C_i$ are pairwise disjoint subsets of $[m]$, and $A_i \cup B_i \cup C_i = [m]$. For every $3$-element subset $\{x,y,z\} \subset [m]$, there is an $i$ such that $A_i \times B_i \times C_i$ contains a permutation of $\{x,y,z\}$. Iin other words, there is an $i$ such that the three elements $x,y,z$ are in different parts of the partition $(A_i, B_i,C_i)$.  What is the smallest $k$, in terms of $m$, for which we can find such a sequence? I am fine with asymptotic answers, and I am interested in both upper bounds (ideally constructive ones) and lower bounds better than $\Omega(\log m)$. This can be stated in terms of hypergraphs. What is the smallest $k$ for which we can cover the complete $3$-uniform hypergraph on $m$ vertices with $k$ tripartite $3$-uniform hypergraphs? Motivation This is based on my solution to  on Puzzling.SE. A sequence that triple-splits $[m]$ would give us a solution to the problem of finding the third-smallest element of $\{x_1,x_2,\dots,x_m\}$. Specifically, the formula would be $$    \min\Big\{\max\{\min\{x_i : i \in A_1\}, \min\{x_i : i \in B_1\}, \min\{x_i : i \in C_1\}\}, \\        \qquad\max\{\min\{x_i : i \in A_2\}, \min\{x_i : i \in B_1\},  \min\{x_i : i \in C_2\}\}, \\        \dots \\        \qquad\max\{\min\{x_i : i \in A_k\}, \min\{x_i : i \in B_k\}, \min\{x_i : i \in C_k\}\}\Big\}. $$ Each max is a max of three distinct elements from $\{x_1, \dots, x_m\}$, so each max is at least the third-smallest element. On the other hand, if the sequence triple-splits $[m]$, and $x_a,x_b,x_c$ are the three smallest elements, then there has to be some $i$ for which $A_i, B_i, C_i$ each contain one of $a,b,c$. For that $i$, the max we'll get will be $\max\{x_a,x_b,x_c\}$: exactly the third-smallest element.
Given a set $\mathcal{S}$ of size $n$, let a $k$-partition of $\mathcal{S}$ be a grouping into $k$ disjoint classes $$(S_1 ,S_2,...,S_k)$$   Where the $S_i$ do not necessarily contain the same number of elements.  Let $C(S_1, S_2, ... S_k)$ be the set of combinations which can be made by taking exactly one element of each class, (i.e. $C(S_1, S_2, ... S_k)$ is the Cartesian Product $S_1 \times S_2 \times ...\times S_k$). For a given $(n,k)$, what is the minimum number of k-partitions $N$ such that  $\bigcup\limits_{i=1}^{N} C(S_1^{(i)}, S_2^{(i)}, ...,S_3^{(i)}) = $ All Possible Combinations  Example:  for $n=4$ and $k=2$, $\mathcal{S} = \{1,2,3,4\}$ we can choose:  $$ S^{(1)} = \left\{\left\{ 1, 2\right\}, \left\{ 3, 4\right\}\right\}$$ $$ S^{(2)} = \left\{\left\{ 1, 3\right\}, \left\{ 2, 4\right\}\right\}$$  Then $$C_{1} = \left\{\left\{ 1, 3\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 2, 4\right\}\right\}$$ $$C_{2} = \left\{\left\{ 1, 2\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 3, 4\right\}\right\}$$  and $C_1 \cup C_2 =$ All possible combinations and $N=2$.
I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. How to kill applications I have been playing around with Ubuntu installing new themes and and docks. Sometimes during the installation, whole screen just hangs. What is the best way in Ubuntu to Kill a process/application? Anything similar to Ctrl + Alt + Del of windows? I'm using a  freely downloadable latex class file and the style is set to \bibliographystyle{ieeetr}. But when I cite something with \cite{} I get it inside ().   For example I get it as abc et al. (1) but what I want to get is abc et al. [1].   Please tell me how to get my citations inside [ ] ? I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1}, $$ where $|x|&lt;1$. I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. Why didn't Isildur's Heir come forward sooner? In the course of creating the content for a site, we've ended up with orphaned content nodes that are not linked to from anywhere. I'd like to find and examine these. Surprisingly, I can't find a module for this.  Tiny hypothetical example: I'm building a brochureware site. Someone wrote a "Contact" page, someone else wrote an "About Us", and someone else wrote "Directions". The person writing "Directions" forgot to put it in a menu link or add a link from "About Us" to "Directions". So "Directions" is orphaned: there are no links to it. I'd like to find all such nodes so I can figure out what to do with them. Should I disconnect my MacBook Pro's power cord when the battery is fully charged? My friend told me that I need to disconnect the power cord when the battery is fully charged. Otherwise, I will shorten the life of the battery.  Other people say you should drain it once a month or point out that for long term storage, you should charge it to 50% and then periodically (3 to 6 months) bring it back to 50% and store it again.  I am going to use my laptop at home 90% of the time. So, I plan to have the power cord connected all the time.  I'd prefer to just plugging in my MacBook Pro and letting it be on charge long term and occasionally taking it away from the charger when I'm not at home or my desk.  Are these specific recommendations worth the inconvenience? What is a combo update and how is it different than the regular update? What is a combo update and how is it different from a normal update? Exporting animation created with animate package Consider the following minimal working example.   It's great to have an animation in pdf format.   But sometimes it would be good to be able to export it into an animated gif or something else (swf, video file, svg?).  How do I achieve this?   Note: I often have animations together with pgfplots.     \documentclass{article}  \usepackage{animate}  \usepackage[active,tightpage]{preview} \PreviewEnvironment{animateinline}  \begin{document}  \begin{center} \fboxsep1mm     \begin{animateinline}[autoplay,loop]{2}     a     \newframe         b     \newframe       c                      \end{animateinline} \end{center}  \end{document}   Remark   The animation is not visible in all PDF viewers. It surely works with a current Adobe Reader. How to find all files with the same extension within a directory I want to search for all .conf files in /etc/. I tried grep -r *.conf /etc, but the result isn't right. What am I doing wrong? I just saw Get Out, and there's one thing I don't understand.  Georgina turns out to be controlled by Rose's grandmother; its grandmothers brain in her body. Why then, is Georgina constantly cleaning and being treated like an actual maid.  During the diner scene, we get a glimpse into the kitchen, and we see Georgina holding up a plate in way that suggested that she was made to work and cook. The pose and the fact that she was already standing there holding the plate before Missy enters the kitchen was also quite strange and suggestive.  Also, at the "party" theres a moment that Chris sees Georgina and Missy. For a moment he thinks that Georgina is being involved in the party, but than it turns out the Georgina is just handed empty glasses to take care of.  So this I don't get. If its really grandma inside Georgina, then why is she treated in this way? Unable to get wireless netgear WNDA3100v2 to work Is every Lie subgroup of $SU(2)$ closed? Find the probability of an event determined by an inequality on exponential random variables Let X and Y be independent random variables such that $X∼Exp(1)$ and $Y∼Exp(2)$.  Find the probability that $3X+4Y≤5$.  I thought the equation for solving this was the integral from $\int_0^a \int_ 0^{(a-y)} \lambda_1 e^{-\lambda_1 x}\lambda_2 e^{-\lambda_2 y} dx dy$ where $a = 5$, so I first started with the integral from $\int_0^5 \int_0^{(5-y)} 3 \times 1e^{-1x} \times 4 \times 2e^{-2y} dx dy$ and got $23.83828927$ which is much larger than a probability can be. Any help? Let $M$ be the space of all $m\times n$ matrices. And $C=\{X\in M|\operatorname{rank}(X)\leq k\}$ where  $k\leq \min\{m,n\}$. Check whether the set $C$  is:   Closed Connected Compact Open   What are some other good properties of the set $C$,for example is it a manifold?  Clearly the set $C$ is closed  if someone is interested  a good proof can be found , hence $C$ is not open. Also as $C$ is unbounded therefore not compact. How to check whether the set $C$ is connected or not? What really goes on in a vacuum? I've been told that a vacuum isn't actually empty space, rather that it consists of antiparticle pairs spontaneously materialising then quickly annihilating, which leads me to a few questions.  Firstly, is this true? And secondly, if so, where do these particles come from?... (do the particles even have to come from anywhere?) Connection pools being reset with Error: 18056, Severity: 20, State: 46. & Perfmon Counters not showing How does lookup in $PATH work under the hood? There're way too many articles/resources on the web that teaches people HOW to set the environment variable PATH so that they can use the short hand of java or python etc instead of the absolute path in command line interface.  What I'm interested to know is that what's behind the scene when we type in the command and hit enter (similar to ).  Here is my guess:   read the command (parse/preprocess stdin to get the right arguments $@) command lookup command execution (program started, consume memory, stdout/stderr to shell) re-render the emulator by relevant environment variables (e.g. $PS#, $PROMPT, etc)   The part I want to figure out most is the command lookup. Obviously, the $PATH is consumed by some background function and separated by : / ;as delimiters, then what happened? Do we use a hash table (key: basename of the file, value: absolute dirname of the file) to store the binary files under those PATHs, or some other hooks?   NOTE: I originally thought it was hash table as I can use [ -z hash [command] ] to check if a command is available in current env, but when I use hash | grep python I get nothing from the output while which python works as anticipated. (I think the mechanism might be shell specific, but I want to get more insights into it.) How to avoid the blackening fruits once they are cut? Apples and bananas start to blacken and sour in taste soon after cutting, which spoils the presentation of dish, especially in fruit salads. How can I avoid this?  Also, does browning impact nutritional value?