How to prove that $(A×B) ∩ (B×A)=(A ∩ B) × (B ∩ A)$?
Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$
If F is a finite field, then $F^F=P_F$. Integrate $e^{ax}\sin(bx)?$ Is there a general formula for finding the primitive of $$e^{ax}\sin(bx)?$$  I've done this manually with $a=9$ and $b=4$ using Euler's formulas. But it takes a bit of time. Is there a pattern here? Wire transfer to an intermediary bank account I sent money from a bank that doesn't have a space for intermediary bank, so I put the intermediary bank statement information where the beneficiary bank information is supposed to be.  I  don't know what will happen now. UBUNTU 14.04 cheese booth not detect webcam Cheese Booth shows an error:     There was an error playing video from the webcam     I test if webcam works fine or not by opening google hangout, its works fine.    Please tell, what is the issue in cheese booth.  Note: Sometimes the cheese booth works fine but most of time not Question on groups of order $pq$ Let $G$ be a group of order $pq$, $p&gt;q$  and $p$, $q$ are primes. Then prove that  If $q\mid p-1$ then there exists a non abelian group of order $pq$. Any two non-abelian groups of order $pq$ are isomorphic.   I have proved that if $q\not\mid p-1$ then $G$ is cylic . But how to prove this one I have no idea. Any kind of hint is very much welcome. This problem is in Herstein book, page 75. Differences between hash- and encryption algorithms? There is a lot of confusion between hash- and encryption algorithms; I would like to understand:   When do we use a hash algorithm and when do we use an encryption algorithm (a cipher)? How difficult it is to reverse hash- and encryption algorithms? From the theoretical/mathematical perspective, what differentiates a hash- and encryption algorithm? Both myself and my boyfriend have PS4s.  Can I just get a single PSN account and sign both consoles into the same account, or do I need to get two separate accounts? Is there a way to display a player name in /tellraw command? like the title says, I'm looking for a way to display a player name in a /tellraw command which is run by a command block. Example:  /tellraw @a {"text":"","extra":[{"text":"A player has won the game!","color":"blue"}]}   And the output is something like this:  A player has won the game!   But I want the output to be something like  JasonSPQR has won the game!   Using the @p parameter hyperlink name with biblatex authoryear (biblatex 1.4b) How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer? $\forall m,n\in\mathbb Z$ , $m\ge1$ and $n\ge1$ how to prove that  $$\frac{(mn)!}{m!(n!)^m}$$ is an integer?  Thanks in advance. I have a  and I connected it to my servo () per these details:   PIC pin 1 to positive pin 8 to negative pin 2 to servo red wire (signal) servo brown wire to negative servo orange to positive (servo has three colors: brown, red and orange wires)   I programmed PIC with this code:      #include &lt;xc.h&gt;     #define _XTAL_FREQ 20000000     #pragma config WDTE=OFF , BOREN=OFF , PWRTE=ON , MCLRE=OFF , FOSC=INTRCIO      void main(){     TRISIO = 0b0000000;     GPIO = 0b000000;         while(1==1){             GPIO = 0b000000;             __delay_ms(1);             GPIO = 0b111111;             __delay_ms(19);         }  }   But servo does not move a bit. What is the problem?  Edit: i found out it rotate servo arm very fast to its end(the point it couldn't rotate any more then it sound like tick tick tick .....)  i changed my code to this(to reverse it back):              GPIO = 0b000000;             __delay_ms(1);             GPIO = 0b111111;             __delay_ms(19);             GPIO = 0b000000;             __delay_ms(2);             GPIO = 0b111111;             __delay_ms(18);   but it behave the same. Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$ How to randomize (shuffle) a JavaScript array? Passive with "to get" In spoken English (at least in America), passives are often formed with "to get" rather than "to be;" I would wager the majority of the time:   he got hit by a car I get driven to work I've gotten called for jury duty eight times   Are there specific verbs that prefer "be" to "get," or vice-versa, or semantic differences between these forms? I cannot think of any. Assume you have a flat table that stores an ordered tree hierarchy:  Id   Name         ParentId   Order  1   'Node 1'            0      10  2   'Node 1.1'          1      10  3   'Node 2'            0      20  4   'Node 1.1.1'        2      10  5   'Node 2.1'          3      10  6   'Node 1.2'          1      20   Here's a diagram, where we have [id] Name.  Root node 0 is fictional.                          [0] ROOT                           /    \                [1] Node 1          [3] Node 2               /       \                   \     [2] Node 1.1     [6] Node 1.2      [5] Node 2.1           /            [4] Node 1.1.1   What minimalistic approach would you use to output that to HTML (or text, for that matter) as a correctly ordered, correctly indented tree?   Assume further you only have basic data structures (arrays and hashmaps), no fancy objects with parent/children references, no ORM, no framework, just your two hands. The table is represented as a result set, which can be accessed randomly.   Pseudo code or plain English is okay, this is purely a conceptional question.  Bonus question: Is there a fundamentally better way to store a tree structure like this in a RDBMS?    EDITS AND ADDITIONS  To answer one commenter's ('s) question: A root node is not necessary, because it is never going to be displayed anyway. ParentId = 0 is the convention to express "these are top level". The Order column defines how nodes with the same parent are going to be sorted.  The "result set" I spoke of can be pictured as an array of hashmaps (to stay in that terminology). For my example was meant to be already there. Some answers go the extra mile and construct it first, but thats okay.  The tree can be arbitrarily deep. Each node can have N children. I did not exactly have a "millions of entries" tree in mind, though.  Don't mistake my choice of node naming ('Node 1.1.1') for something to rely on. The nodes could equally well be called 'Frank' or 'Bob', no naming structure is implied, this was merely to make it readable.  I have posted my own solution so you guys can pull it to pieces. Beamer -- not indenting itemize in a single environment The output in the MWE posted at the end of this post is shown below. Notice how the default indentation from itemize gives an asymmetric slide w.r.t the spacing to the left and right of the itemize environment. How can I push the bullet points to the left so that the spaces to the left and right are the same? Such a solution would ideally be:   implementable on a one-off basis, so that it is applied when necessary (since some itemize environments should be given the default indentation) independent of the margin size, so that the solution doesn't need to be changed when the margin size is changed     \documentclass{beamer}   \begin{document}  \begin{frame}{A title} \begin{itemize} \item Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod \item tempor incididunt ut labore et dolore magna aliqua. \item Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi \item ut aliquip ex ea commodo consequat. \begin{itemize} \item Duis aute irure dolor in reprehenderit in voluptate velit esse \item cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat \end{itemize} \end{itemize}    \end{frame} \end{document} Probability of Various Combinations of Weather There's a 10% chance it will rain today. There's a 15% chance it will snow today.   What percentage chance is it that it will:   Both rain and snow (it can both rain and snow at the same time)? Neither rain nor snow? -   Raining and snowing are independent of each other.  EDIT: (ADDED)   Rain or snow but not both? Print numbered paragraph after title with Biblatex My question is related with Biber and Biblatex (latest versions). I need to quote the number of a paragraph of an old Christian text which is published within a book printed nowadays.   More clearly, I'm working with this Biblatex item (it's in Italian):  @INBOOK{Clemente1984Lettera,   author = {Clemente di Roma},   title = {Lettera di Clemente Romano ai Corinti},   year = {1984},   booktitle = {I Padri apostolici},   editor = {Quaquarelli, Antonio},   location = {Roma},   pages = {41-92},   publisher = {Città Nuova} }   I quote it in a common LaTex file with the BibTex key:  \cite[42]{Clemente1985Lettera}   With a humanistic Biblatex style, the final citation is:  Clemente di Roma, Lettera di Clemente Romano ai Corinti, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Anyway, I need to quote specifically the n. 2 of the letter of Clement, so I have to print this number just after the title "Lettera di Clemente Romano ai Corinti". So, I'd like to have this citation:  Clemente di Roma, Lettera di Clemente Romano ai Corinti, n. 2, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Now, I can find no Biblatex field to store the "n. 2" text, and no way to print it after the title. Of course, I could create a specific item in the bibliographic file in which the title field, or another one which is supposed to follow it, contains the "n. 2" indication. Anyway, this solution is very bad to me, because I should fill my archive with items of specific citations, and not of single titles.  What I'm looking for is what I've already implemented in the Bibfilex citations system (), which is moreover far less powerful that Biber: the possibility to add a pipe in the postnote of a BibTex key to separate a possible indication of a number of a paragraph, to be printed just after the title, from the pages of the volume. So a key like that  \cite[2|42]{Clemente1985Lettera}   should print   Clemente di Roma, Lettera di Clemente Romano ai Corinti, n. 2, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Anybody knows how to do that with Biber and Biblatex?  Thanks. Frequently when I ask a question I have to include quite a bit of code for reference purposes.    It would be nice to be able to bold the important lines in the code.  For example this code:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     &lt;Setter Property="IsSelected" Value="False"/&gt; &lt;/DataTrigger&gt;   Is not way long, but the important part (as seen in ) is the second to last line.  However, if I try to bold it, it ends up looking like this:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     **&lt;Setter Property="IsSelected" Value="False"/&gt;** &lt;/DataTrigger&gt;   Not what I was going for.  It would be nice to be able to draw attention to specific lines of code when needed.    (One suggestion of a different color background sounds nice too (ie a yellow background to draw the eye to the relevant code.)  If there is some other way to bold in code please let me know. I have a large mesh with > 1.5 million faces. I would like to uniformly add a texture to the object without UV mapping (as blender freezes every time). The texture is quite simple and can be applied evenly throughout the object.