Return SQL query in one row, fetch with PHP
mysql_fetch_array does not retrieve all rows
Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably many? Uncountably many? What are the possibilities to the quantity of counterexamples? For $n \geq 2$, prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$ For $n \geq 2$ prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  We need to use induction.   The Principle of Mathematical Induction, Theorem 4.2.1, states that $n_0 \in \mathbb{Z}$. For each integer $n \geq n_o$, let $P(n)$ be a statement about $n$. Suppose that the following two statements are true:   $P(n_o)$ $( \forall n \geq n_o)[P(n) \rightarrow P(n+1)]$   Then, for all integers $n \geq n_o$, that statement $P(n)$ is true.  Suppose,  $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$, the equation, is true for some natural number $k$.  $\frac{1}{2^2})*...*(1- \frac{1}{k^2}) = \frac{k+1}{2k}$  The induction hypothesis $P(k) \rightarrow P(k+1)$ must hold for any natural number $k, P(k),$ from the basis and inductive step.  $(\frac{1}{2^2})*...*(1- \frac{1}{k^2})*(1- \frac{1}{(k+1)^2}) = \frac{k+1}{2k}(1- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $\frac{k+1}{2k}(\frac{(k+1)^2}{(k+1)^2}- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $(\frac{(k+1)}{2k})( \frac{(k+1)^2-1}{(k+1)^2})$  $\leftrightarrow$ $( \frac{(k^2+2k+1-1}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{(k^2+2k}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{k+2}{(2)(k+1)})$  We need to demonstrate that   $(\frac{1}{2^2})*...*(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  holds true for $n \geq 2$  The question is do I just substitute a number that is greater than two after I did the induction? The 's  feature (E:RftLW, p. 58; WGtE, p. 180) states:     [...] You ignore all class, race, spell, and level requirements on attuning to or using magic items.   Class and race requirements are relatively intuitive, additionally, since an Artificer is able to cast spells they can already attune to items that require you to be a spellcaster. But there's still some questions:   Can they use a spell scroll of a spell not on their spell list? Is this one of the four requirements ignored? What are level and spell requirements; what are some examples? Does the equivalence $$\neg\neg (P \rightarrow Q) \leftrightarrow (\neg\neg P \rightarrow \neg\neg Q)$$ hold in propositional intuitionistic logic? In propositional classical logic the equivalence holds obviously since $P \leftrightarrow \neg\neg P$. However, the similar-looking implication $\neg\neg (P \vee Q) \rightarrow (\neg\neg P \vee \neg\neg Q)$ is . The validity of the backward implication comes up in  as well. Can I use two different cards with the same name in Commander? If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \dfrac{n-1}{n+1}$, find $a_n$ and the sum.  By definition,  the sum of the series is the $\lim n\to\infty$ of it's $n^{th}$ partial sum.  $$\text{ Sum = } \lim_{n\to\infty} = \dfrac{n-1}{n+1} = 1$$  I am asked to find $a_n$. How do I do this? What is the procedure? Sleeping in a batch file When writing a batch file to automate something on a Windows box, I've needed to pause its execution for several seconds (usually in a test/wait loop, waiting for a process to start). At the time, the best solution I could find uses ping (I kid you not) to achieve the desired effect. I've found a better write-up of it , which describes a callable "wait.bat", implemented as follows:  @ping 127.0.0.1 -n 2 -w 1000 &gt; nul @ping 127.0.0.1 -n %1% -w 1000&gt; nul   You can then include calls to wait.bat in your own batch file, passing in the number of seconds to sleep.   (at last!). In the meantime, for those of us still using Windows&nbsp;XP, Windows 2000 or (sadly) , is there a better way?  I modified the sleep.py script in the , so that it defaults to one second if no arguments are passed on the command line:  import time, sys  time.sleep(float(sys.argv[1]) if len(sys.argv) &gt; 1 else 1) Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$ Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$  The required condition is $a^2&gt;8b^2$.I dont know how to prove it.I tried.  Let $(h,k)$ be the point from where tangents are drawn to the parabola $y^2=4ax$.Let $m_1,m_2$ be the slopes of the tangents.Equations of tangents are $y=m_1x+\frac{a}{m_1}$ and $y=m_2x+\frac{a}{m_2}$ and equations of normal to the parabola $x^2=4by$ is $y=m_1x-2bm_1-bm^3_1$ and $y=m_2x-2bm_2-bm^3_2$.  As $y=m_1x+\frac{a}{m_1}$ and $y=m_1x-2bm_1-bm^3_1$  are the same lines.Therefore,$\frac{a}{m_1}=-2bm_1-bm^3_1$  $bm^4+2bm^2+a=0$  And I am stuck.I cannot get the desired condition.Please help me. What is a "year" in Westeros? How do I install the VirtualBox version from Oracle to install an Extension Pack? I downloaded VirtualBox OSE from the Ubuntu Software Center but it seems that that version is limited. I prefer to have the complete version and don't really care if it's closed source or not.  Can I install it via a terminal or do I have to download something from a website?  I don't know how to install software I download from the web. So far I've only used the Software Center. How do I alter a message in the message queue independent of site language? I want to programatically alter a message in the message queue. The message I want to alter is set in the message queue by function comment_form_submit() in comment.module, i.e.:  if ($comment-&gt;status == COMMENT_NOT_PUBLISHED) {   if (!user_access('administer comments')) {     drupal_set_message(t('Your comment has been queued for review by site administrators and will be published after approval.'));   } }   Instead of this, I want the call to drupal_set_message() to be this:  drupal_set_message(t('Your comment will be published when you validate your e-mail address.'));   How do I do this?  Postscript: Based on the answer below by Topsitemakers, I got this working for any language by using array_search() inside a custom MYMODULE_comment_submit to search the message array, and then compare the translated strings to identify the key of the string to alter.  This is the working code I ended up with:  $mm = drupal_get_messages('status', TRUE); $key = array_search(t('Your comment has been queued for review by site administrators and will be published after approval.'), $mm['status']); if (FALSE !== $key) {   $mm['status'][$key] = t('Your comment will be published when you validate your e-mail address.'); } foreach ($mm['status'] as $msg) {   drupal_set_message($msg, 'status'); } Are the decimal places in a CSS width respected? How to parse a string into a nullable int I'm wanting to parse a string into a nullable int in C#.  ie. I want to get back either the int value of the string or null if it can't be parsed.  I was kind of hoping that this would work  int? val = stringVal as int?;   But that won't work, so the way I'm doing it now is I've written this extension method  public static int? ParseNullableInt(this string value) {     if (value == null || value.Trim() == string.Empty)     {         return null;     }     else     {         try         {             return int.Parse(value);         }         catch         {             return null;         }     } }      Is there a better way of doing this?  EDIT:  Thanks for the TryParse suggestions, I did know about that, but it worked out about the same.  I'm more interested in knowing if there is a built-in framework method that will parse directly into a nullable int? Multi-OS Portable TeX system I'm looking to treat myself to a totally portable TeX system and was wondering: is it possible to devise an installation scheme that would run on Windows, Linux and MacOS at the "same" time?  I think TeXLive is the best choice here, but I'd much rather avoid having to maintain three separate TeXLive installations. I'd like the three installations to share the package databases for instance.  I'll be using a 32Gb USB stick, in case it matters.  PS: I think I'm onto something here... will post back when I have a working prototype.  Edit: after reading  I managed to format a USB stick with UDF and have it read and written in Linux and Windows (need to perform some tests on Mac yet). This is the first step towards interoperability.  Edit 2: well, as it turns out, the previous note was useless (the partition table breaks under Windows), I've found  and it works like a charm. The only remaining step is writing a couple of scripts to set the environmental variables and we're done ;) In comics does Batman have a rule against killing? Inspired by  question, I have to admit I don't know a lot about the Comic Book Batman, but I know a fair amount about his other incarnations.   I know for example, that the 1989 movie had him kill the Joker. He vehemently wanted the Joker dead and had no compunctions about taking a life. I also know that he had a rule against killing in The Animated Series that he never broke, and that he very nearly broke it in Batman Beyond.  But as far as the comic books go, when did Batman develop a rule against killing his foes? Does he ever kill them rather than deliver them to the proper authorities? This is a follow-up to an earlier question regarding .  Having achieved the left-alignment goal, I would now like the caption of the subfigure to be bottom-aligned, while having the subfigure itself be top-aligned. As an example, in the figure below, subfigures (a) and (b) would share an uppermost edge, and their captions would share a lowermost edge, and similarly for subfigures (c) and (d). This is most appropriate for the overall layout style of my document and also the content of the subfigures themselves.    \documentclass{article} \usepackage[font=footnotesize]{subcaption} \usepackage[draft]{graphicx}  \begin{document} \begin{figure}[!ht] \captionsetup[subfigure]{justification=justified,singlelinecheck=false} \centering \begin{subfigure}[b]{0.3\textwidth} \includegraphics[scale=0.8]{fig/workflow-S} \caption{workflow $S$} \end{subfigure} \begin{subfigure}[b]{0.5\textwidth} \includegraphics[scale=0.8]{fig/workflow-S-run} \caption{run of $S$} \end{subfigure} \vskip 8pt \begin{subfigure}[b]{0.3\textwidth} \includegraphics[scale=0.8]{fig/workflow-S'} \caption{workflow $S'$} \end{subfigure} \begin{subfigure}[b]{0.5\textwidth} \includegraphics[scale=0.8]{fig/workflow-S'-run} \caption{run of $S'$} \end{subfigure} \caption{Here's a figure} \end{figure} \end{document}   I've experimented with the minipage settings for subfigure, specifically using [t] instead of [b], but it seems to have no effect. I'm unsure as to whether a position of [t] would produce the right effect anyway, or whether it would shift the captions up as well.  Why is [t] not working here, and what's the right way to produce the layout I need?   The solution should generalise to an arbitrary number of subfigures per row, subfigures which are text or tables rather than images, and should not require any hard-coded sizes. Let $A = \mathbb{Z}$, $B = [−1, \pi]$, $C = (2, 7)$. List all elements of $A \cap (B^c \cap C)$. After working it out on a number line, I got: $\{4, 5, 6\}$.  As it stands, the expression contains the integers that do not belong to the set $B$ that cross into $C$. This would result in $4, 5, 6$. Number $7$ would not be included because of the soft brackets surrounding $C = (2, 7)$. Is this correct? Possible Duplicate:          I know that   $$\lim_{n\rightarrow\infty}\sqrt[n]n=1$$  and I can imagine that $n$ grows linearly while $n$th root compresses it exponentially and therefore the result is $1$, but how do I calculate it? Is there any noun in English which changes the first letter in the plural? Plenty of nouns change the second letter to become plural (man->men, goose->geese) but does anything change its first letter.  I've hunted high and low over the internet, and spent ages browsing the questions at  but I can't find anything. Default currency symbol I'm looking for a way to get the Default currency symbol (in a none-multi-currency org) of the salesforce org locale (to have it displayed on a visualforce page in a field that can't be &lt;apex:inputField&gt; or &lt;apex:outputField&gt;) preferably query-able from apex (via the database, a describe or whatever) and then again the actual symbol, meaning € for Euro, $ for dollar, and so on...  There must surely be some sort of mapping hidden in the code somewhere salesforce does this via the input/outputfields if bound to an sobject of type currency  Can anyone point me in the right direction?