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Equation for a smooth staircase function I am looking for a smooth staircase equation $f(h,w,x)$ that is a function of the step height $h$, step width $w$ in the range $x$.   I cannot use the unit step or other similar functions since they are just one step. I have been experimenting with various sigmoid curves and while I can get a single smooth step I cannot get to realize the staircase shape. The closest staircase function I have found is given in  in equation (18) and depicted in Fig. 4 and it is a close example of what I want (i.e generate a staircase in the range $x$ for arbitrary step heights and widths) but it is not smooth at all.  Regarding smooth steps, a likely starting point I found is  but it gives a smooth function of just a single step. I have been unable to modify the equation to make it into a staircase. I would like to specify arbitrary step heights and widths and generate a smooth staircase in the range $x$ specified.    Edit (Extra info): The smooth function I mention above has the problem that the upper, horizontal line is not equal in length to the lower, horizontal line which is why I have been unable to adapt it into a staircase function   Edit 2 Including some pictures     Edit 2 Plot of $s$ with a steep slope showing a different width on the first horizontal line Python: List vs Dict for look up table I have about 10million values that I need to put in some type of look up table, so I was wondering which would be more efficient a list or dict?  I know you can do something like this for both:  if something in dict_of_stuff:     pass   and  if something in list_of_stuff:     pass   My thought is the dict will be faster and more efficient.  Thanks for your help.  EDIT 1  Little more info on what I'm trying to do.  .  I'm making a look up table to see if a value calculated has all ready been calculated.   EDIT 2  Efficiency for look up.  EDIT 3  There are no values assosiated with the value...so would a set be better? How to prevent an email from accidentally being forwarded? Is it possible to prevent an email from being forwarded, or to have sensitive information automatically removed when reply-all/forward is pressed?  For example, a Google Calendar invitation includes a link to directly accept/reject the event. However, if someone clicks reply-all, and they keep the HTML formatting, the link will be shared with everyone else who received the invitation, and they can change the user's RSVP. (Specifically, the email is sent only to you, but has a "To:" header showing all invitees, so the problem remains.)  What can be done to ensure that the recipient does not accidentally share the sensitive link? Would it be possible with some MIME trickery or with a semi-standard header? Or is the best practice to refrain from including such "direct action" links when someone might be tempted to reply-all? Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question Show that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.      Let $\phi$ be the homomorphism from $\Bbb R[x]$ onto $\Bbb C$ given by   $f(x) \rightarrow f(i)$ (that is, evaluate a polynomial in $\Bbb R[x]$   at $i$).  Then $x^2 + 1 \in \operatorname{Ker} \phi$ and is clearly a   polynomial of minimum degree in $\operatorname{Ker} \phi$.  Thus,   $\operatorname{Ker} \phi = \langle x^2 + 1 \rangle$ and $\Bbb R[x] /\langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$.   My question is:  How is $x^2 + 1$ a polynomial of minimum degree in $\operatorname{Ker} \phi$? "The package lists or status file could not be parsed or opened" Everytime I sudo apt-get update it always not complete and will say in the end  E: Read error - read (5: Input/output error) E: The package lists or status file could not be parsed or opened.   Same goes, when I try to install something and do sudo apt-get install &lt;some app&gt;  What should I do ? I admit, that my drive has bad sectors, could that be the reason? Is there anything else I can do? Let $n$ be an integer, prove that every prime factor of $12x^2+1$ has the form $6m+1$, where $m$ is an integer.  I've gone as far as knowing that all primes $p&gt;3$ are either $6m+1$ or $6m-1$, how do I use this, not sure whether to even use that.  I know about the Legendre symbol and quadratic reciprocity Raise bounty on question I have this question here:  I chose 500 then before accepting decided that was too much and halfed it, but now I just want the answer so I'd like to raise it to 500 but I can't find how to it. Also it's just such an irritating problem that I'd give all my rep away if I could to the clever person who has the answer. It would be nice to have a few minutes after creating the bounty to be able to raise it. But for now could a mod or something please raise it for me to 500. Need polite phrases expressing disagreement with the information/conclusions of another person, especially an educator Why is it that Android still can't be installed as regular OS? Are there any technical or others reasons why Android is still a firmware and not an OS which can be just installed without a lot of hassle?  Why there is no universal image which could detect appropriate drivers (kernel modules), partition NAND in some standard way and configure boot loader to boot updated OS version?  What stops Google from making a real (less firmware-like) OS (an OS for small ARM, x86 etc. devices)? (no) Properties in Java? Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$ Can I create a TIN from contours lines (shp) in QGIS? Google Chrome: Have different settings on different computers? Classifying raw LAS data as bare earth in ArcGIS for Desktop? I have a raw LAS data that has only one return (no second/ third.. return) and not classified yet. The only information I have is intensity and elevation. Is there any way that I can classify bare earth in ArcGIS? (vendor does not have more info than that). I tried classifying by selection from the profile view, but I know it is impossible to apply to all data. Good reasons to prohibit inheritance in Java? Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges? We know that $\displaystyle\zeta(2)=\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ and it converges.        Does there exists a bijective map $f:\mathbb{N} \to \mathbb{N}$ such that the sum $$\sum\limits_{n=1}^{\infty} \frac{f(n)}{n^2}$$ converges.      If our $s=2$ was not fixed, then can we have a function such that $\displaystyle \zeta(s)=\sum\limits_{n=1}^{\infty} \frac{f(n)}{n^s}$ converges Touchpad not recognized on Asus N550JK - TP500L (FocalTech) I'm installing a new laptop and the touchpad is not fully recognized. Specifically, two finger scrolling does not work.  extract from /proc/bus/input/devices  I: Bus=0011 Vendor=0002 Product=0001 Version=0000 N: Name="PS/2 FocalTech FocalTech Touchpad in mouse emulation mode" P: Phys=isa0060/serio1/input0 S: Sysfs=/devices/platform/i8042/serio1/input/input5 U: Uniq= H: Handlers=mouse0 event4 B: PROP=1 B: EV=7 B: KEY=30000 0 0 0 0 B: REL=3   Also, xinput list gives me:  ⎡ Virtual core pointer  id=2    [master pointer  (3)] ⎜   ↳ Virtual core XTEST pointer id=4   [slave  pointer  (2)] ⎜   ↳ ELAN Touchscreen id=10    [slave  pointer  (2)] ⎜   ↳ PS/2 FocalTech FocalTech Touchpad in mouse emulation mode id=14   [slave pointer  (2)] ⎣ Virtual core keyboard                     id=3    [master keyboard (2)]     ↳ Virtual core XTEST keyboard               id=5    [slave  keyboard (3)]     ↳ Power Button                              id=6    [slave  keyboard (3)]     ↳ Video Bus                                 id=7    [slave  keyboard (3)]     ↳ Video Bus                                 id=8    [slave  keyboard (3)]     ↳ Sleep Button                              id=9    [slave  keyboard (3)]     ↳ USB2.0 HD UVC WebCam                      id=11   [slave  keyboard (3)]     ↳ Asus WMI hotkeys                          id=12   [slave  keyboard (3)]     ↳ AT Translated Set 2 keyboard              id=13   [slave  keyboard (3)]   And finally, synclient tells me there is no synaptics driver active:     Couldn't find synaptics properties. No synaptics driver loaded?   So, it seems the touchpad is not recognized as a synaptics touchpad. It works, but not the synaptics specific parts.  It's in mouse emulation mode (no idea what it means). And it works together with other pointing devices, especially the touchscreen (I can confirm this works).  I have tried adding i8042.nomux=1 to the kernel parameters, but it didn't change anything... Any hint ? I am thinking about a wizard who has a rat familiar (from the  spell), and has the  spell prepared. Can he use his familiar's senses to teleport into another room (where the familiar is hiding) by looking through the eyes of the rat and casting misty step?  The wizard could use his action to switch to the familiar's senses and then use his bonus action to cast the spell. This would greatly reduce the "blind and deaf" time he experiences from doing this. Where can I download Windows 8 legally, from Microsoft? I'm really new to Mac OSX and UNIX based systems. I wanted to run a few Windows games on my Mac Mini so I started installing Wine and MacPorts. I think the installs are correct, but when I type sudo port install wine I see a not found error in my terminal. I tried other basic commands like say and clear and all of them return the same not found error. My research shows that my PATH might be incorrectly set, but it lacks steps I can implement. Specifically, I ran this command: echo $PATH and it returned:  /opt/local/bin:/opt/local/sbin:/opt/local/bin:/opt/local/sbin:/opt/local/bin  I'm a complete newbie to mac and have no idea where .bash_profile or any of those files are. My skills let me install Xcode and the Command Line tools for Xcode and I sense I need to level up on path management. At this point I need help managing my dot files and changing my path to fix these specific errors on Mountain Lion. What are my next steps?