conditioning by independent sigma-algebras
Conditional expectation on more than one sigma-algebra
Is "singular noun + and + singular noun" singular or plural? When I type "[singular noun] and [singular noun]" in Word 2010, I find some interesting things.        Bread and butter is essential. (the auto-correction warns me of subject-verb disagreement.)   Bread and butter are essential. (OK)   A knife and fork is on the table. (OK)   A knife and fork are on the table. (OK)   The needle and thread is new. (OK)   The needle and thread are new. (OK)   Cost and revenue is directly linked. (OK)   Cost and revenue are directly linked. (OK)      I wonder how a native speaker would consider this phenomenon. Is there always an equilibrium point in space for gravity? What factors influence the spring constant? For certain springs we have been taught that Hookes law, $F=-kx$ works. For such springs, what factors determine k. For instance, k is proportional to length. Can k be expressed in terms of dimensions, mass and other fundemental quantities, or is k a fundemental property of the spring itself? Unintentional Lies and Misinformation Equation left margin I have equations inside an environment which has a larger left-margin than the main text (analogous to an enumerate/itemize environment, implemented with the changepage package's adjustwidth environment). Just as equations in enumerate/itemize behave, the equations are centered with respect to the entire text, rather than just the width of the container environment. I want the latter behavior.  I'm trying to accomplish this by wrapping all display-math in minipages. Namely, I'm modifying the mathdisplay environment in amsmath.sty as follows:  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother   This is almost working, but there are some issues with spacing which I don't understand how to fix. Here's how the the normal \[ ... \] and numbered \begin{equation} ... \end{equation} look with the above modification:      My main confusion is why the numbered version is behaving so differently from the unnumbered. But all in all I want to fix this so that the spacing above and below is even as it usually is. Any help is appreciated, thanks.  Edit: Here's a compilable example:  \documentclass{amsart}  \usepackage{lipsum} \usepackage{calc} \usepackage{changepage}  \makeatletter \let\oldmd\mathdisplay \def\mathdisplay#1{%   \newline   \begin{minipage}{\textwidth-\@totalleftmargin}%     \oldmd{#1}% } \let\oldendmd\endmathdisplay \def\endmathdisplay#1{%   \oldendmd{#1}%   \end{minipage} } \makeatother  \setlength{\parindent}{0pt}  \begin{document}  \lipsum[1]  \begin{adjustwidth}{20pt}{} \lipsum[2] \[ x \mapsto \hom(-,x) \]  \lipsum[3] \begin{equation}   y \mapsto \hom(y,-) \end{equation} \lipsum[4] \end{adjustwidth}  \lipsum[5]  \end{document} How can I copy a custom Dictionary on Word 2000? When there is less than 5% battery left on a mac, the battery indicator goes red and system performance significantly drops - presumably in an attempt to make that remaining 5% last a little longer.   Is there a way to manually trigger this, so that the system performance is low (and battery life longer), even when there is plentiful battery?   This would theoretically be a way of extending the number of hours on a battery beyond the rated amount. E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages I am not sure what caused this error, but here is what the whole error says, and also this is sitting as a notification, and preventing me from updating any software using Update Manager - Please provide some assistance or tell me how to figure out what to do to fix it.  Could not calculate the upgrade  An unresolvable problem occurred while calculating the upgrade.  Please report this bug against the 'update-manager' package and include the following error message: 'E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.' Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$ I want to prove the following so I decide to rewrite the RHS  $$A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$$ $$ A\setminus(B\setminus C) = A\setminus (B \cap C^c) = A\cap (B\cap C^c)^c = A\cap(B^c\cup C) = (A\cap B^c)\cup(A\cap C) $$ there is way that $(A\cap B^c)\cup(A\cap C)\rightarrow (A\setminus B)\cup(A\cap C) = (A\setminus B)\cap (A \cup C)$?  thanks. EDIT  LHS: $$x\in A \vee x\in A\cap C \rightarrow x\in A \wedge x\in C$$ RHS: $$x\in A \wedge x\in A\cup C \rightarrow x\in A \vee x\in C$$  how it can help me? Dodecagon Area Question The distance between two opposite vertices of the dodecagon is 2. Find the area of the dodecagon.  Is there any way to do this without trigonometry?  And could you include a proof also? :O I need to find a way to import some .json models into Blender, and it seems I can't just import them the same way I can import a .obj model. So is there some way to convert a .json model to a .obj model, or another file type Blender can import? I've tried using Blockbench for this, but I think it only works with Minecraft recourse pack .json models, as those are the only models that will ever actually load in it. Thank you to anyone who can help me figure this out. How can I report bugs with or request features for the Stack Exchange mobile apps? Part of a fingernail known as the  is informally known as the "quick".  It is referenced in the saying "cut to the quick".  What is the etymology of the word quick as in reference to the fingernail? How do electrons jump orbitals? My question isn't how they receive the energy to jump, but why.  When someone views an element's emission spectrum, we see a line spectrum which proves that they don't exist outside of their orbitals (else we would see a continuous spectrum).  Electrons can be released in the form of beta decay, thus proving that they are capable of traveling outside of orbitals contrary to the statement my teacher said that they stay within orbitals. Then, to add to the confusion, the older model of rings floating around a nucleus has, from what I can tell, been outdated, which would support this model. My teacher's explanation was that the electrons made a quantum jump of some kind. How do electrons move between orbitals or do we know how they jump, excluding the reason that energy causes them to jump, and why are positrons formed sometimes instead of electrons in Beta decay?  When I'm asking "how do electrons jump" I would like to know how an electron can jump between each orbital such as how it moves and how it knows where to jump since it appears to be a jump where the electron doesn't slow into a orbital position. Specifically how they jump what is this Atomic electron transition, I understand that they jump and that they do this through absorbing and releasing energy but what is this Atomic electron transition other than what is already on the wikipedia article . Fundamental question about dimensional analysis Joris and Srikant's exchange  got me wondering (again) if my internal explanations for the difference between confidence intervals and credible intervals were the correct ones.  How you would explain the difference? Different and opposing voltage sources? &ndash; Schematic created using   The diagram above, the voltage sources are equal and wired parallel, they would increase the current(2x) flow to the load(R), but what about this case:          Are the two voltage sources in series now? How to repair a broken .EXE file association After (hopefully) scrubbing viruses out of a Windows 7 installation (after deciding not to simply run over the laptop repeatedly with my car), I've got everything almost back to normal. The only lingering issue I have is that for my non-admin users, the ".exe" file extension doesn't work. That is, clicking on the various desktop application links results in a "How do you want to open this?" dialog.  I've been through the alleged registry fixing from "winhelponline" and that had absolutely no effect.  I've tried running "assoc" for the affected users, but it reports the .exe association to be "exefile" even though it persistently does not work.  Right-clicking on a desktop icon and then choosing "start" does successfully open an application, but that's clearly a terrible situation.  For my admin user, things seem to work fine.  What do I need to do to get things working? How do I change the default session for when using auto-logins? When autologon is active, lightdm will start the Unity 3D session, not the 2D (or any other shell that I want to auto-login to).  Is there a way to start the shell I want automatically (autologon)?  Anyway, if autologon is disabled, the last used session is launched, and that's fine. I just want to choose which is default with autologon. Basically on my site I have a bunch of song lyrics and in a static page I want to show them listed by album  and album position of the song and I have two custom fields with that data, sd_album_title and `sd_album_track_n. Plus some of the songs don't have an album yet, so both values are void, and I wish to list those ones at the foot of the others, sorted by alphabetical order.  The latest thing I tried at least to sort everything by the two field is this one:  $key = 'sd_album_title'; $themeta = get_post_meta($post-&gt;ID, $key, TRUE);  add_filter( 'pre_get_posts', 'my_get_posts' ); function my_get_posts( $query ) {     if ($themeta != '') {         $query-&gt;set( 'post_type', 'lyric' );         $query-&gt;set( 'meta_key', 'sd_album_title' );         $query-&gt;set( 'orderby', 'meta_value' );         $query-&gt;set( 'order', 'ASC' );         $query-&gt;set( 'meta_query', array(               array(                     'key' =&gt; 'sd_album_title'               ),               array(                     'key' =&gt; 'sd_album_track_n',                     'value' =&gt; 'target_value',                     'compare' =&gt; '&gt;=',                     'type' =&gt; 'numeric'               )         ));     }     return $query; }  $getlist= 'my_get_posts'; $posts = get_posts( $getlist );   And my intention was to put an else at the end of it with an array ordered by title for the other songs, but it returns nothing as it is. Could something like this work with some corrections or am I on a totally wrong path?