Given $f(n) = n + \left \lfloor{\sqrt n}\right \rfloor$  Prove that for any natural number $n$, the sequence $n, f(n), f(f(n)),.......$ must always contain at least one perfect square.  Now obviously if we take $n$ to be a perfect square, we are done since the sequence contains $n$. So I started by looking at the behaviour of the function at $n = 2$, and within $2$ iterations we reach $4$. Then I looked at $n = 5$, and that reaches $9$ in $2$ iterations. However, $6$ is missed in the sequence as $f(5) = 7$ and this function is strictly increasing. So I looked at the sequence when $n = 6$, which reaches $16$ in $4$ iterations.   Trying to generalise this is a bit difficult, however, as I tried to let $k^2 &lt; a &lt; (k+1)^2$ for some natural number $a$.   Clearly, $f(a) = a + k$, however I am stuck here. Is this approach worth taking or is there something completely different required?
Let $m$ be a natural number. Define $f(m) = m + \lfloor\sqrt{m}\rfloor$. Prove that at least one of the number among $m, f(m), f^2(m), \ldots$ is a perfect square. Here $f^k(m)$ denotes the composition of $f$ over itself $k$ times.  I tried the question, but the greatest integer along with square root is creating trouble.
Most incarnations of The Hulk I've seen or read about state that he is as strong as he needs, basically meaning that there is no limit to how strong he can get.  Is there any canon regarding this? Have we seen the upper limit of the Hulk's strength? Use of "would" for subjunctive phrases This has been bugging me for some time; I tried to look for previous questions here but my language tools may not be sharp enough to phrase my query correctly so please forgive me if this has already been posted here.  Speakers of English as a foreign language tend to incorrectly use the word "would" when using some particular subjunctive constructs. Here is  (this time presumably by a native English speaker):     If that would happen, you won’t be alone.     (One correct version being "If that happens, you won't be alone".) I have detected this in speakers of a number of languages so I feel this is more likely to be a feature of the English language itself than a carry-over from their native language.  I have a faint intuition that this is because , and it certainly doesn't help that , but I can neither make that precise nor even really phrase a concrete question. Can someone shed some light on this?    Edit: just to provide some real-world examples of what I mean, here is one example taken from :     Publish or Perish is designed to empower individual academics to present their case for research impact to its best advantage. We would be concerned if it would be used for academic staff evaluation purposes in a mechanistic way.   This is obviously ungrammatical (to me), but I'm not sure how to characterize the mistake, or what the likely causes are (e.g. whether it follows from analogous constructions in other languages). How do I show the current node title in page.html.twig OR in node.html.twig? If I want to show the current node title in page.html.twig (by default it seems to only be available in node.html.twig) I have to create a preprocess function like so:  function theme_preprocess_page(&amp;$variables, $hook) {   // Set our page title so we can access it in page.html.twig   if ($node = \Drupal::routeMatch()-&gt;getParameter('node')) {     $variables['my_node_title'] = $node-&gt;title-&gt;value;   } }   And then I have to overwrite page-title.html.twig by copying it into my current theme folder and commenting out everything.  This doesn't seem "right".  Is there a better way? Countdowntimer in minutes and seconds How to calculate temperature in Wire? Let's say I have 1m of 2,5mm² copper wire, insulated with H07V-K. The wire carries 40A and environment temperature is 25°C. The wire is suspended in free air. How hot will the wire get? And how can I calculate it for other values?  The use case is a 12V-230V AC converter I want to hook up to my car and determine the minimum wire thickness required. Because it's only used from time to time and not buried in a wall I suspect I can use thinner wires. This is a temporary thing and doesn't need to be up to code. Using the method of characteristics, find a solution to Burgers' equation    \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 &amp; \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g &amp; \text{on } \mathbb{R} \times\{t=0\} \end{cases}   with the initial conditions    $$g(x)=\begin{cases} 0 &amp; \text{if }x &lt; 0 \\ 1 &amp; \text{if }0 \le x \le 1 \\ 0 &amp; \text{if }x &gt; 1 \end{cases}$$   First, I realized that the equation $u_t+\left(\frac{u^2}{2} \right)_x =0$ is equivalent to this form: $$u_t+uu_x =0$$  Then should I generally follow the method of solution as outlined in the answer of ?  Note that this is not a duplicate question of that page. Rather I want to know if that page can be used for my problem, even though my ICs are different.  By the way, the solution printed in my book (PDE Evans, 2nd edition, page 142) is      $$u(x,t) =  \begin{cases} 0 &amp; \text{if } x &lt; 0 \\  \frac xt &amp; \text{if } 0 &lt; x &lt; t  \\ 1 &amp; \text{if } t &lt; x &lt; 1 + \frac t2  \\ 0 &amp; \text{if } x &gt; 1 + \frac t2 \tag{$0 \le t \le 2$}  \end{cases}$$ How do I invoke a private static method using reflection (Java)? Can't save my render as an image I seem to be having issues when I try and save the output file as a .png, or .jpeg for that matter. I've set up my user preferences, File, Render Output to my desktop as well as render output in the editor window. What am I missing here? I'm obviously very new to this program and your help would be appreciated. Thanks for your time Why does the classical path give the dominant contribution in the path integral? Trying to get my head around the commutator subgroup. This is an excercise from Artin's Algebra:  Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian.  Here is what I've done:  Let $xC,yC \in G/C$ then $xyx^{-1}y^{-1}C = C$ since $xyx^{-1}y^{-1}$ is a commutator hence belongs to $C$. But then $xyC = yxC$ so $xC$ and $yC$ commute in $G/C$. This can be done for any elements, so $G/C$ is abelian.  This seems somewhat surprisingly short. Is that all there is to it?   Regards What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$? Suppose we have the open ball $B_{X}(\mathbf{a},r)$ and the closed ball $\bar{B}_{X}(\mathbf{a},r)$ of radius $r$ about $\mathbf{a}\in\mathbb{R}^n=X$ with the Euclidean metric $d_2$.  What is the closure of an open ball $B_{X}(\mathbf{a},r)$ in $(\mathbb{R}^n,d_2)$?  First, I have defined the sets like this:  $$B_X(\mathbf{a},r)=\{\mathbf{y}\in\mathbb{R}^n:d_2(\mathbf{a},\mathbf{y})&lt;r\}$$ $$\bar{B}_X(\mathbf{a},r)=\{\mathbf{y}\in\mathbb{R}^n:d_2(\mathbf{a},\mathbf{y})\le r\}$$  Then, let $U=B_X(\mathbf{a},r)$ and claim the closure is $$\bar{U}=\{\mathbf{x}\in\mathbb{R}^n:\exists (\mathbf{x}_n)\in U\text{s.t}~\mathbf{x}_n\to \mathbf{x}\}=\bar{B}_X(\mathbf{a},r).$$  Since closed balls are closed then we have $$\bar{U}\subset\bar{B}_X(\mathbf{a},r).$$  But, how can I prove the other direction $\bar{B}_X(\mathbf{a},r)\subset\bar{U}$?  Since it's a containment argument I know that I have to take any element from $\bar{B}_X(\mathbf{a},r)$ and show it is in $\bar{U}=\{\mathbf{x}\in\mathbb{R}^n:\exists (\mathbf{x}_n)\in U\text{s.t}~\mathbf{x}_n\to \mathbf{x}\}$.  But I do not know how to get an expression for $\mathbf{x}_n$.  I would appreciate any help. Notify instantly when a currently viewed post is deleted At the moment SE shows us current voting results, a notification for new answers and comments, etc., instantly. However, it doesn't show us when a post is deleted.  Please notify us also when a post is deleted. Then I don't have to type a (long) comment, only to notice the post got deleted in the meantime. If I were to attack with a 4/4 indestructible creature and it's blocked by a 4/2 creature with first strike, would my indestructible creature still attack during the second combat damage step, or would it just tap, not assigning any damage? Variable substitution with an exclamation mark in bash I have 13.04 installed from a DVD that I got on eBay but the application launcher is missing. I've tried CCSM but it won't work! Prove that $\overline {A^\complement} = (A°)^\complement $   To show that $\overline {A^\complement} \subseteq  (A°)^\complement$   We first show $A^\complement \subseteq  (A°)^\complement ...(*) $  We know that since $ A°\subseteq A \qquad (*)$ is automatically follows .  The part I didn't understand is the following:     Then we conclude that since $ (A°)^\complement$ is closed    $\therefore \overline {A^\complement} \subseteq  (A°)^\complement$ Upgrading my Wubi 11.04 to 11.10 I want to upgrade my current Wubi installation (Ubuntu 11.04) to Ubuntu 11.10. Is this possible to do inside Wubi? Or do I have delete Wubi from Windows, therefore deleting my current Ubuntu installation; and then install 11.10 via the Wubi installer?  Thanks Bibtex style for "Statistics and Computing" I would like to create a bibtex style which answers the requirements of Statistics and Computing journal. How could I do that? Here is an example of how the bibliography should look like:  Journal article: Hamburger, C.: Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations. Ann. Mat. Pura Appl. 169, 321–354 (1995) Book: Geddes, K.O., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer, Boston (1992) Book chapter: Broy, M.: Software engineering — from auxiliary to key technologies. In: Broy, M., Denert, E. (eds.) Software Pioneers, pp. 10–13. Springer, Heidelberg (2002)  In the text, the references should be as follows:  Negotiation research spans many disciplines (Hamburger 1995).  Thank you for your help! Permalinks: custom post type -> custom taxonomy -> post I am having trouble working with WordPress rewrite rules, and could use some help.  I have a custom post type called _shows_.  All shows have a single custom taxonomy category _show-category_. A _show_ will never have more than one _show-category_.  I would like my urls to route in this manner:  www.mysite.com/shows/  =&gt;  archive-shows.php  www.mysite.com/shows/%category%/ =&gt; taxonomy-show-category.php  www.mysite.com/shows/%category%/%postname%/ =&gt; single-shows.php   So as a real world example, lets say we have a _show-category_ "Foo" and a _show_ post titled "Bar" that has "Foo" as it's _show-category_. I would expect my WordPress app to look like this:  www.mysite.com/shows/foo/ =&gt; shows all posts under the foo category www.mysite.com/shows/foo/bar =&gt; shows the indivual post   I try to avoid plugins when possible, but am open to any solution. How to convince longtable to do a page break earlier using the longtable package to typeset page spanning tables, I run into a nasty behaviour: only a dummy table head and the caption are put on the following page. The contents of the table is extracted from program sources. I don't have much influence here.  \documentclass[a4paper,twoside,DIV10,BCOR1.5cm]{scrbook} \usepackage{longtable} \newlength{\tablcol} \setlength{\tablcol}{4.7cm} \newlength{\tabrcol} \setlength{\tabrcol}{8.1cm} \itemsep0ex plus0.2ex % \begin{document} \begin{longtable}{p{\tablcol}p{\tabrcol}}% % \hline \multicolumn{1}{l}{{\bfseries Feld}} &amp; \multicolumn{1}{l}{{\bfseries Beschreibung}}\\ \hline \endfirsthead % \hline \multicolumn{2}{l}% {{\small\slshape Fortsetzung der vorherigen Seite}}\\ \hline \multicolumn{1}{l}{{\bfseries Feld}} &amp; \multicolumn{1}{l}{{\bfseries Beschreibung}}\\ \hline \endhead % \hline \multicolumn{2}{l}% {{\small\slshape Fortsetzung auf der n\"achsten Seite}}\\ \hline \endfoot % \hline \caption[show user output]{Beschreibung der Output-Struktur des show user Statements} \endlastfoot % \label{tucshowxuser}% {\scshape id} &amp; Die Nummer des Repository Objektes \\ {\scshape name} &amp; Der Name des Objektes \\ {\scshape creator} &amp; Name des Benutzers der dieses Objekt angelegt hat \\ {\scshape create\_time} &amp; Datum und Uhrzeit der Erstellung \\ {\scshape changer} &amp; Name des Benutzers der dieses Objekt zuletzt ge\"andert hat \\ {\scshape change\_time} &amp; Datum und Uhrzeit der letzten \"Anderung \\ {\scshape privs} &amp; Zeichenkette die K\"urzel f\"ur die Benutzerrechte auf dieses Objekt enth\"alt \\ {\scshape commenttype} &amp; Typ des Kommentars \\ {\scshape comment} &amp; Kommentar zum Objekt, wenn vorhanden \\ {\scshape id} &amp; Die Nummer des Repository Objektes \\ {\scshape name} &amp; Der Name des Objektes \\ {\scshape creator} &amp; Name des Benutzers der dieses Objekt angelegt hat \\ {\scshape create\_time} &amp; Datum und Uhrzeit der Erstellung \\ {\scshape changer} &amp; Name des Benutzers der dieses Objekt zuletzt ge\"andert hat \\ {\scshape change\_time} &amp; Datum und Uhrzeit der letzten \"Anderung \\ {\scshape privs} &amp; Zeichenkette die K\"urzel f\"ur die Benutzerrechte auf dieses Objekt enth\"alt \\ {\scshape commenttype} &amp; Typ des Kommentars \\ {\scshape comment} &amp; Kommentar zum Objekt, wenn vorhanden \\ {\scshape id} &amp; Die Nummer des Repository Objektes \\ {\scshape name} &amp; Der Name des Objektes \\ {\scshape creator} &amp; Name des Benutzers der dieses Objekt angelegt hat \\ {\scshape create\_time} &amp; Datum und Uhrzeit der Erstellung \\ {\scshape changer} &amp; Name des Benutzers der dieses Objekt zuletzt ge\"andert hat \\ {\scshape change\_time} &amp; Datum und Uhrzeit der letzten \"Anderung \\ {\scshape privs} &amp; Zeichenkette die K\"urzel f\"ur die Benutzerrechte auf dieses Objekt enth\"alt \\ {\scshape commenttype} &amp; Typ des Kommentars \\ {\scshape comment} &amp; Kommentar zum Objekt, wenn vorhanden \\ {\scshape id} &amp; Die Nummer des Repository Objektes \\ {\scshape name} &amp; Der Name des Objektes \\ {\scshape creator} &amp; Name des Benutzers der dieses Objekt angelegt hat \\ {\scshape create\_time} &amp; Datum und Uhrzeit der Erstellung \\ \end{longtable}  \end{document}   The problem is that I don't find a method to force a page break earlier. A page that isn't filled entirely, followed by a page with the last part of a table is better than this.  I tried to add "\needspace{someLength}" at the beginning or end of the rows, but that doesn't work either. It simply makes the rows someLength high instead of forcing a page break if there's no more than someLength room left on the page.  I'd be more than grateful for some solution or even a good hint.  TIA