I will see you on Monday morning. I will see you Monday morning.   Is the second version idiomatic and grammatically correct? Which version do native speakers prefer to use more?
I am a bit confused. When can we omit the preposition on before weekdays? (Monday, Tuesday etc.) Sometimes I read weekdays without the preposition on. If the preposition is left out, does it work the same way, or does it change the meaning?  Example without the preposition     The UGC issued the public notice Monday after its two previous orders   in this regard were overlooked by the DU.   Example with the preposition     The commission on Monday also issued a public notice against the FYUP   in all leading newspapers for parents and students.
Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:[0,1]\to[0,\infty)$ Find $$\lim_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$$if $f:[0,1]\rightarrow(0,\infty)$ is a continuous function.   My attempt:  Say $f(x)$ has a max. value $M$. Then $$\left(\int_0^1(f(x))^ndx\right)^\frac{1}{n}\leq\left(\int_0^1M^ndx\right)^\frac{1}{n} =M$$  I cannot figure out what to do next. From the movies we know that Obi-Wan Kenobi delivered Luke Skywalker to Owen and Beru Lars to raise on Tatooine shortly after Luke was born at the end of RotS (Episode 3).   The next think we know is Luke running into Ben Kenobi during A New Hope.   There seems to be a hint that there was at least SOME communication between Kenobi and Lars family based on him telling Luke:     Your father wanted you to have this when you were old enough, but your uncle wouldn't allow it.   (though it's possible that was referring to Kenobi offering to leave the lightsaber to Larses at the same time he dropped off Luke)  I was wondering if any details of what transpired between Kenobi and Larses in the years between RotS and ANH were noted in any canonical works (of any canon, including EU books/games/comics/etc...)? How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards? I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by profession. Knowing full well that my proof may be fallacious, erroneous, or simply lacking mathematical formality, what advice would you give me? minimal polynomial of powers of a matrix I am wondering, given the minimal polynomial of $A$, what can we say about the minimal polynomial of $A^n$?  I know that if we were looking at the roots of the characteristic polynomial ie eigenvalues then the roots of the characteristic polynomial of $A^n$ would be $nth$ powers of the eigenvalues so again these are roots of the minimal polynomial also. Characterization convex function. Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that for all $a,b\in [0,1]$ with $a&lt;b$ $$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$ How to prove that $f$ is a convex function? Thank you for your help. How do you select a particular option in a SELECT element in jQuery? Weapon damage bonuses and critical hits I am getting the following error in a Drupal 8 site.     The following module is missing from the file system   It looks as if a custom module was installed and deleted without uninstalling it.   How can I fix this error? I have a large dictionary file with 300,000+ words in it and I'm trying to find all words with the vowels aeiou in that order and have only exactly 5 vowels. My current attempt does not seem to be working and for the life of me I don't understand why.  less mywords | grep -iE [^aeiou]*a[^aeiou]*e[^aeiou]*i[^aeiou]*o[^aeiou]*u[^aeiou]*   I think it get's all the words I'm looking for but there are a few words that pop up which I can not get rid of.  abstemiousnesses ultraserious   There are a few other but they are in the same vein. Curiously, even if I add something like [^u]* to the front ultraserious keeping popping back up! Any solution would be fine however I would like it restricted to grep as we've been told it can be done using only grep. During real, everyday use, that is?  I'm looking to upgrade a drive in a MacBook Pro, and am looking at either a 750GB/7200RPM or a 1TH/5400RPM drive.  Would I care about the speed difference? Is it worth giving up the additional space on the larger drive? When is logistic regression solved in closed form? Having recently introduced an overload of a method the application started to fail. Finally tracking it down, the new method is being called where I did not expect it to be.  We had  setValue( const std::wstring&amp; name, const std::wstring&amp; value );  std::wstring avalue( func() ); setValue( L"string", avalue ); std::wstring bvalue( func2() ? L"true", L"false" ); setValue( L"bool", bvalue ); setValue( L"empty", L"" );   It was changed so that when a bool value is stored we use the same strings (internal data storage of strings)  setValue( const std::wstring&amp; name, const std::wstring&amp; value ); setValue( const std::wstring&amp; name, const bool&amp; value );  std::wstring avalue( func() ); setValue( L"string", avalue ); setValue( L"bool", func2() ); setValue( L"empty", L"" ); &lt;&lt; --- this FAILS!?!   The problem with L"" is that it is implicitly casting and previously it was happy to be a std::wstring, but not it prefers to be a bool. The MSVC compiler does not complain, or raise warning, so I'm worried that even if I "fix" the setValue( L"empty", L"" ); to be  setValue( L"empty", std::wstring() );   somebody else may come later and simply use setValue( L"empty", L"" ); and have to track down this issue again.  We thought to use explicit on the method but it is not a valid keyword for this usage. Is there some way to get the compiler to complain about this, or otherwise prevent the issue?  Otherwise I'm thinking to change the name of the method which takes a bool to ensure it can't make an incorrect guess. Functions that are sets of all function - proofs I'm going through the book Proofs and fundamentals, by Bloch, and it doesn't include a solution manual for it's examples. It doesn't have many examples on notation and proof strategy for certain cases, so I needed a little help.    Although I understand the idea, and can draw it out, I'm not sure how to write it in decent proof notation. Any help would be appreciated!   Edit: apologies, my idea is this: theres some function in $F(C,A\times B)$ such that theres some $x$ within $C$ that means $g(x)$ is within $A\times B$. Similarly, the second two imply that for some function within $F(A,C)$ there exists a value $x$ such that $g(x)=C$. and similarly for the third part from $B\rightarrow C$. It's easy to see with drawing it out that these two are the same because one will have a part within A and the other will lead to a part within $B$, so their cross will be the same. I wanted to go about proving it by setting a function $f$ within $F(C,AxB)$ and then working from there, but I really have no idea where to start or the notation. Vertical alignment of align* in enumerate Related (but not the same):         Earlier today I asked a question about aligning equations in an enumerate environment and got an answer that seemed to work for the example I gave, but it turns out I wasn't really specific enough for my problem.  I'm using align* environments, often as the only part of a list item. (If you're wondering why I'm not using more words to introduce the align*, these are mostly homework solutions where most of the work is algebra, etc.)  Here's my code and what it makes (above the middle line):  \begin{enumerate}   \item \begin{align*}     x^2 + y^2 &amp;= x^2 + (iy)^2 \\               &amp;= (x + iy) (x - iy)   \end{align*}   \item \begin{align*}     \frac{1}{n^2-4} &amp;= \frac14 \frac{4}{(n-2)(n+2)} \\                     &amp;= \frac14 \left( \frac{1}{n-2}-\frac{1}{n+2} \right)   \end{align*} \end{enumerate}     I'm trying to make the baseline of the first line of the align* environment the same as the number's baseline—I want the first line of the align* to have the same vertical position (centered horizontally in the same way) relative to its number as in the second half of my picture. (Side question: why is there so much space before the align* starts?) Unable to SSH to my virtual machine via Windows Power Shell ISE Maximum number of files in one ext3 directory while still getting acceptable performance? I have an application writing to an ext3 directory which over time has grown to roughly three million files. Needless to say, reading the file listing of this directory is unbearably slow.  I don't blame ext3. The proper solution would have been to let the application code write to sub-directories such as ./a/b/c/abc.ext rather than using only ./abc.ext.  I'm changing to such a sub-directory structure and my question is simply: roughly how many files should I expect to store in one ext3 directory while still getting acceptable performance? What's your experience?  Or in other words; assuming that I need to store three million files in the structure, how many levels deep should the ./a/b/c/abc.ext structure be?  Obviously this is a question that cannot be answered exactly, but I'm looking for a ball park estimate. Is there a name for this database schema of key values? Why can't I press the Install button when installing applications from unknown sources? Most vexing parse I have a filter using pre_get_posts to change the number of posts per page from 4 on the news landing page to 8 on all other pages (page 2, 3, etc). However I don't seem to have got the offset correct, or something, because my get_next_posts_page_link doesn't show a page 3, even though it does exist if I go to the url manually. The current offset is something I somehow managed to conjure up accidentally, I thought it worked as page 2 was correct but as I added more test posts, I realised it didn't work.  function page_2_posts_per_page_filter( $query ) {      $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;      if ( !is_admin() &amp;&amp; $query-&gt;is_paged &amp;&amp; $query-&gt;is_main_query() ) {          $offset = 4 + (($paged - 2) * 8);         $query-&gt;set( 'offset', $offset );         $query-&gt;set( 'posts_per_page', 8 );      } else if (!is_admin() &amp;&amp; $query-&gt;is_main_query()) {          $query-&gt;set( 'posts_per_page', 4 );      }     return; } add_action( 'pre_get_posts', 'page_2_posts_per_page_filter', 1 );   EDIT: SOLUTION  function page_2_posts_per_page_filter( $query ) {      $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;     $offset = 4;      if ( !is_admin() &amp;&amp; $query-&gt;is_paged &amp;&amp; $query-&gt;is_main_query() ) {          $offset = (($paged - 1) * 8) - $offset;         $query-&gt;set( 'posts_per_page', 8 );         $query-&gt;set( 'offset', $offset );      } else if (!is_admin() &amp;&amp; $query-&gt;is_main_query()) {          $query-&gt;set( 'posts_per_page', 4 );      }     return; } add_action( 'pre_get_posts', 'page_2_posts_per_page_filter');  function found_offset( $found_posts, $query ) {      $offset = 4;      if( !is_admin() &amp;&amp; $query-&gt;is_main_query() ) {         $found_posts = $found_posts + $offset;     }     return $found_posts; } add_action( 'found_posts', 'found_offset', 10, 2 );