How can prove or disprove that bounded smooth functions have a bounded derivative?
Can the graph of a bounded function ever have an unbounded derivative?
A positive integer $n$ is called a balancing number if  $$1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r) \tag{1}$$  for some positive integer $r$.     Problem: Find the general solution (closed-form expression) of equation $(1)$, in other words, find $f_1(m)=n, f_2(m)=r$ where $f_1, f_2$ are solution of $(1)$ for any integer $m \in \mathbb Z$.   I have asked a  with a constrain involving Pell's equation. A film or TV series about classic monsters getting together I'm a US passport holder. What documents do I need when I fly from the U.S. to Germany?  Assume I'm going for a few weeks of vacation.  Obviously I need my passport.  What about a "visa"?  Anything else? Solving a cubic polynomial equation. Overview I have tried finding a solution to this problem myself and I have flailed. Its just a challenge for me. could you please tell me how far am I in solving this question? My approach for quadratic equation : I am still a student, and when I first saw the quadratic formula I wanted to derive it myself and I did it. This was my logic: the equation $x^2 + bx + c=0$ is obviously supposed to contain 2 roots (real or imaginary). So any 2 points in an axes can be represented as: $n+k$ or $n-k$. so (1) : putting $x = n+k$, (2): putting $x=n-k$ we get two equations; on subtracting $(1) - (2)$ we get an equation in $n$ and $k$ as, $4nk+2kb=0$ which gives $n=-b/2$.  Now substituting this value of $n$ in $(1)$ and $(2)$ and adding them we get $k= (+or-) \sqrt{b^2 -4c}/2 $. My approaches for trying to solve cubic equations: Let $x^3+bx^2+cx+d = 0$ be the equation. My first trials were focused on extracting the roots by various substations. Let me explain some of them. Originally, when I derived the quadratic equation, I wanted to literary &quot;represent the solution&quot; by $n+k$ and $n-k$. most methods that I have thought of for cubic equation similar to that of the quadratic (like assigning 3 unambiguous  variables to uniquely represent 3 roots- $n$ and $k$ represents unique values), by considering 3 unique values, but they end up being ambiguous. Then I was suddenly struck with the idea that I could define certain random operators to &quot;transform&quot; the equation in some way changing the co-coefficients of $x^2, x, x^0$. I have defined the first transformation as: $x^3+bx^2+cx+d=0 \to x^3+ b'x^2 + c'x+d' =0   $ then $b' = b$, $c' =b^2+c$, $d' = bc-d$. Applying this operation changes the modified equation's zeros as follows: $\alpha'=(\alpha+\beta)/2, \beta' = (\beta+\gamma)/2, \gamma' = (\alpha+\gamma)/2 $ if the initial zeros are $\alpha, \beta, \gamma$. My latest assumption is that creating $n$ number of transformations and applying them can help us obtain the roots in some combination. The other operation I have defined is &quot;increment roots by by $\theta$ operation&quot;. in this case, $x^3+ bx^2+ cx+ d \to x^3+b'x^2+ c'x+d=0$. here, $b' = b + 3\theta, c' = 3\theta^2+ 2\theta b+ c, d' = \theta^3 +b\theta^2 + c\theta + d$. This modification transforms the original roots $\alpha, \beta, \gamma$ to $\alpha' = \alpha + \theta, \beta' = \beta + \theta, \gamma' = \gamma + \theta$ Now for my questions: How far am I from solving the cubic polynomial problem myself without any external help? More over, I give up. How to find the roots of a cubic equation? Could my latest idea for solving cubic equations actually be applied (the transformation method)? Is it possible to find the roots with my transformation method? (my second transformation definition was successful in converting the coefficient b or c to zero by transforming the roots linearly) Book covering differential geometry and topology for physics I'm interested in learning how to use geometry and topology in physics. Could anyone recommend a book that covers these topics, preferably with some proofs, physical applications, and emphasis on geometrical intuition? I've taken an introductory course in real analysis but no other higher math. How to truncate a file in C? Brand new disk brakes are barely stopping the bike Short description of the scoping rules? So from what I remember it was on cable TV. The main hero was a TV actor zapped to a alien planet. It had a evil overlord with a virgin priestesses. She wants to get laid but he refuses because the next priestesses is only three. Regardless it's a parody things include a man who was maimed by the evil overlord wants revenge the hero eventually screws the virgin priestesses it cuts to scenes of a train going into a tunnel and a volcano erupting. Extracting numbers from string field using Python Parser of ArcMap Field Calculator? I am trying to calculate field test2 with only the numeric values from a string field test. The entire script is like:    def makestr(test)      list=[]      for s in test:          if s.isdigit():              list.append(s)      for a in list:          str=''.join(list)     But I keep getting an error:    Any ideas? Blocking comment spam without using captcha Why do I get TypeError: can't multiply sequence by non-int of type 'float'? I am typing to get a sale amount (by input) to be multiplied by a defined sales tax (0.08) and then have it print the total amount (sales tax times sale amount).  I run into this error. Anyone know what could be wrong or have any suggestions?   salesAmount = raw_input (["Insert sale amount here \n"]) ['Insert sale amount here \n']20.99 &gt;&gt;&gt; salesTax = 0.08 &gt;&gt;&gt; totalAmount = salesAmount * salesTax  Traceback (most recent call last):   File "&lt;pyshell#57&gt;", line 1, in &lt;module&gt;     totalAmount = salesAmount * salesTax TypeError: can't multiply sequence by non-int of type 'float' example of discontinuous function having direction derivative Is there a function (non piece-wise unlike below) which is discontinuous but has directional derivative at particular point? I have a manual that says the function has directional derivative at $(0,0)$ but is not continuous at $(0,0)$. $$f(x,y) = \begin{cases} \frac{xy^2}{x^2+y^4} &amp; \text{ if } x \neq 0\\  0 &amp; \text{ if } x= 0 \end{cases}$$  Can anyone give me few examples which is not defined piece wise as above? How do I get the coordinates of a mouse click on a canvas element? What's the simplest way to add a click event handler to a canvas element that will return the x and y coordinates of the click (relative to the canvas element)?  No legacy browser compatibility required, Safari, Opera and Firefox will do. How do I get a list of installed files from a package? I'm trying to understand the /etc/rsyslog.conf file, but I'm missing something. For example, this is a part of the file:  auth,authpriv.*         /var/log/auth.log *.*;auth,authpriv.none      -/var/log/syslog cron.*              /var/log/cron.log daemon.*            -/var/log/daemon.log kern.*              -/var/log/kern.log lpr.*               -/var/log/lpr.log mail.*              -/var/log/mail.log user.*              -/var/log/user.log   According to  page:     The facility is one of the following keywords: auth, authpriv, cron,   daemon, kern, lpr, mail, mark, news, security (same as auth), syslog,   user, uucp and local0 through local7.      The priority is one of the following keywords, in ascending order:   debug,info, notice, warning, warn (same as warning), err, error (same   as err) crit, alert, emerg, panic (same as emerg). The keywords error,   warn and panic are deprecated and should not be used anymore. The   priority defines the severity of the message.      An asterisk ("*") stands for all facilities or all priorities,   depending on where it is used (before or after the period). The   keyword none stands for no priority of the given facility.      You can specify multiple facilities with the same priority pattern in   one statement using the comma (",") operator. You may specify as much   facilities as you want. Remember that only the facility part from such   a statement is taken, a priority part would be skipped.      Multiple selectors may be specified for a single action using the   semicolon (";") separator. Remember that each selector in the   selector field is capable to overwrite the preceding ones. Using this   behavior you can exclude some priorities from the pattern.   So, this is pretty understandable but what about the file path on the right. It can be just a path, or path with | or - before it (or maybe even something else). What is the difference between the three? A grim and tragic question perhaps, but I'm curious:  What happens when a person that co-signed on a loan dies? Is the co-signer's estate potentially liable until the loan is paid off, the same as if the co-signer had borrowed the money themselves?  Or, does the responsibility for the loan die with the co-signer?  Does the bank/lender take on that risk? Producing .bbl file with Lyx and JabRef How do I produce the .bbl file required for arXiv using JabRef for bibtex and Lyx? Let $A [a_{i,j}]$  and $B =[b_{ij}]$   be two diagonal matrices in   $M_n(F)$. Let $i \ne j \in \left\{1, \ldots, n \right\}$. Using the   product rule, we obtain      $$(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$      We claim that $a_{ik}b_{kj} =0$ for all $k \in \left\{1, \ldots, n \right\}$ thus $(AB)_{ij}=0$ for all $i \ne j$ and $AB$ is diagonal. To    prove the claim, note that since $i \ne j$ , we have $i \ne k$ or $    k\ne j$. Thus either $a_{ik} = 0$ (since $A$ is diagonal) or $b_{kj} =  0$ (since $B$ is diagonal), thus in all cases $a_{ik}b_{kj}=0$ and the   claim is proved.   Where does the bit "we have $i \ne k$ or $k\ne j$" come from? I understand that since $i \ne j$, and $k$ and $j$ run over the same set of natural numbers, we also have $i \ne k$. But the "or $k \ne j$" baffles me. I understand that $k \ne j$ implies that we can have $i=k$. But that's from the other direction ($\Longleftarrow).$  Also, when it says $a_{ik}=0$ or $b_{kj}=0$ it's talking about when $i \ne k$ or $k \ne j$ right? Many thanks. The error: "mount: only root can do that" while mounting a partition I've installed Ubuntu 12.04 along with Windows 8.1. When I tried to open the C: drive in Ubuntu a window is popping out saying to run this command:  mount -t ntfs-3g -o remove_hiberfile /dev/sda2 /media/2A36EE3236EDFEA7   After running the command I'm getting the error mount: only root can do that.   What should I do?