I need to construct a real function with is exactly C^2 (that is, it is continuous and 2 times differentiable but it is not 3 times differentiable) such that its first derivative never vanishes.  I tried   x^5 Sin(1/x) + Exp(x)  x^(5/2) + Exp(x)  |x|^3 + Exp(x)  all these functions are exactly C^2, but their first derivatives vanish in some point(s).  :(
I need to construct a real function with is exactly $C^2$ (that is, it is continuous and two times differentiable but it is not three times differentiable) such that its first derivative never vanishes.  I tried  $x^5 \sin(\frac{1}{x}) + \exp(x)$  $x^{\frac{5}{2}} + \exp(x)$  ${|x|}^3 + \exp(x)$  all these functions are exactly $C^2$, but their first derivatives vanish in some point(s). :(
Why do networks need a common ground cable? Must one have a knowledge of fine art to be a Graphic Designer? Should all Graphic Designers know how to draw? If my app is live and users are facing a critical issue, can I upload another build and ask Apple to make it live within a day or two? There is a type and search drodown which will appear after user type in the value. I need to click on the dropdown which contains the name of the drop down field.   The problem here is that there is no unique attribute present (No unique class, no unique name) &amp; the id keeps on changing after every login so cannot take id also. In an hydrogen-like atoms the orbitals are solutions to the Schrodinger equation suitable for the problem.  They describe the regions where an electron can be found.  So, why don't they have spherical symmetric?  As in this picture:    Why, in a certain distance from the nucleus, are there points where an electron can be found, and points where it can't?  EDIT: Is there any intuation about the symmetry breaking? Why put a resistor in series with signal line? A lot of times in circuits I see a resistor placed in series in a signal line and sometimes even in series with an MCU's VDD line. Is the intention of this to smooth out noise in the line? How is this different from using a small cap, like a .1µF to do the same thing? How does body fat leave your body? So I know that it’s not from tears or sweat because they are composed of mainly water and sodium, it’s not from going poo or pee as poo and pee are made of the parts of food that supply your body with no nutrients and your body also doesn’t “burn body fat” because there is a theory that no matter is ever destroyed so it’s none of those but another thing is it’s not your hair, ear wax, finger nails or anything of that sort because people can lose over 17 pounds in 3 months but you can’t build up that amount of hair , ear wax or whatever else in three months so how does body fat leave your body Finding the confidence level and number of successes? Given that the sample size is $1500$ and the confidence interval of $p$ is $(0.050, 0.083)$. Find the confidence level and the number of successes.  I have so far that $E=Z_{1-\alpha/2}∗\sqrt{(p∗({p-1}))/n}$   Where $p$ is the center of the given interval, so $p = (0.050+0.083)/2 = 0.0665$  Also, the error $E$ as the difference of the midpoint and the endpoint of the interval so $E=0.0665-0.050=0.0165$   Plugging this into the equation I got that  $0.0165=Z_{1-\alpha/2}*\sqrt{((0.0665)(1-0.0665))/1500}$   Solving for $Z_{1-\alpha/2}$ gives $Z_{1-\alpha/2}=2.56$ and since $Z_{2.96} = 0.99$   $Z_{1-\alpha/2} = 0.01$  Is what I have so far right? If so, how do I find the number of successes? Is it just to multiply 0.99 by the number of trials?   So that way I would get $1500*0.99=1485$ ? Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?  It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers. Why is my crontab not working, and how can I troubleshoot it? What could cause a toilet to leave the flapper up and keep running? I have an American Standard toilet I installed last year. I don't know much about plumbing, but I was able to install this toilet. It seems to function okay, except for what I think is a mechanical problem.  Sometimes when the toilet is flushed, the flapper doesn't close and water keeps running.  The handle feels like it's slightly stuck in the up position.  I looked inside the tank once when it was doing this, and the chain attached to the handle was slack. I don't understand why the flapper isn't closing.  I've tried putting more slack in the chain and loosening the plastic nut that holds the handle on the toilet, but it doesn't seem to be helping.    How can I fix this? A few months ago I set full disk encryption during installation of Ubuntu 17.10. Now, I've decided to upgrade. The upgrade ran to the end without any problems. However, after reboot I am not able to log into my encrypted disk.  Where can be the problem? I am 100% sure I press the "right keys" on the keyboard, but technically I don't know what I am writing because of "*" characters and maybe after upgrade my keyboard layout changed. I use some characters which might be somewhere else on the keyboard. What language is default after upgrade?   By the way, I've already tried Caps-lock, but still no luck.  Help will be really appreciated. I am not a fan of installing my system and all backups over and over again. Obtain the joint PDF of $\theta_{1} - \theta_{2}$ and $\theta_{1}$ where $\theta_{1}$ and $\theta_{2}$ are i.i.d $\text{Uniform}[0, 2\pi)$ I have 2 i.i.d Uniform random variables $\theta_{1}$ and $\theta_{2}$ on $[0, 2\pi)$. Here the problem pushes us to consider that all operations are made modulo $2\pi$. My question is how to obtain the joint PDF of $\theta_{1} - \theta_{2}$ and $\theta_{1}$ and then obtain the conditional PDF of $\theta_{1}$ given $\theta_{1} - \theta_{2}$??. For the second part It can be deduced from the first by writing it as the joint PDF divided by the PDF of $\theta_{1}$. But for the first part, A common idea that I saw is to derive the $\text{CDF}(x, y) = P(\theta_{1} - \theta_{2} \le x$, $\theta_{1} \le y$) and then perform derivations respect to $x$ and $y$. Is this a right approach? If yes, how to proceed with CDF especially with our uniform distribution restrictions? Add layer to a QGIS group using Python I am trying to add new layers into a specific group which I can get to work if I know the index, but how can I get Python to find the index of a group so it can load the new layer into the group wherever it is in the TOC?  qgis.utils.iface.legendInterface().moveLayer( layer, index ) Solve the equation $28^x = 19^y+87^z$ in integers Help removing subsection and subsubsection numbers from Table of Contents TOC Hello I am trying to further modify my TOC.  I got an answer  as to how I could remove the \section numbers on the left side of the TOC and now I would like to expand that.  I would like to remove the left-hand numbering for \subsection and \subsubsection as well but I do not know how.  (I also want to keep these numbers out of the document and I am using \def\sectionformat{} to do that)  Also, my TOC is slightly over one page and so I would like to adjust the spacing from 2 to 1.5,  is there any way to do this?    \documentclass[12pt, a4paper, twoside]{article}     \usepackage{baskervald}     \usepackage[margin=1in,bindingoffset=15.5mm,heightrounded]{geometry}     \usepackage[T1]{fontenc}     \usepackage[notes]{biblatex-chicago}     \usepackage{filecontents}     \usepackage{microtype}     \usepackage{url}     \usepackage{makeidx}     \usepackage{indentfirst}     \usepackage{setspace}     \usepackage{chngcntr}\counterwithin*{footnote}{section}     \newcommand*{\textsubscript}[1]{$#1{}$}     \DefineBibliographyStrings{english}{references = {Works Cited}}     \DeclareBibliographyCategory{primary}     \makeindex     \addbibresource{bibliography.bib}     \makeatletter     \let\latexl@section\l@section     \def\l@section#1#2{\begingroup\let\numberline\@gobble\latexl@section{#1}{#2}\endgroup}     \def\@seccntformat#1{\ifcsname #1format\endcsname\else\csname the#1\endcsname\quad\fi}     \def\sectionformat{}     \def\subsectionformat{}     \def\subsubsectionformat{}     \makeatother     \author{}     \title{\vspace*{36 pt}\textbf{\emph{\Huge{Big Important Title}}}\vfill}     \date{}     \begin{document}     \tableofcontents     \section{Part I}     Hello World     \end{document} I need to make a converter to go from RS422 full duplex to my Arduino. I have found lots of different chips which I think can do this, but am slightly confused by the terminology. The RS422 device is a laser distance sensor ().  It has RX+, RX-, TX+, TX-. So I have two lines for input and two lines for output.  I could use two RS485 line drivers to achieve the full duplex comms which I need.  There are a lot of chips to do that.  But then on my Arduino (clone: chipkit max32) would I need to use two UART channels? And then alternate which one I am sending each byte on?  The device uses 4 byte instructions. So if I wanted to send an instruction would I send Serial1(firstByte) then Serial2(secondByte), Serial1(thirdByte), Serial2(fourthByte)?   And then receive data the same way? I feel like I am missing something and can do this on one UART channel but am not sure how. I have a div 200 x 200 px. I want to place a 50 x 50 px image right in the middle of the div.   How can it be done?  I am able to get it centered horizontally by using text-align: center for the div. But vertical alignment is the issue.. Why Does the Transaction Log Keep Growing or Run Out of Space? This one seems to be a common question in most forums and all over the web, it is asked here in many formats that typically sound like this:  In SQL Server -  What are some reasons the transaction log grows so large? Why is my log file so big? What are some ways to prevent this problem from occurring? What do I do when I get myself on track with the underlying cause and want to put my transaction log file to a healthy size? Unable to locate package lib32bz2-1.0 I am trying to install Simplicity studio on Ubuntu 15.04 (64-bit) and I get this error  E: Unable to locate package lib32bz2-1.0 E: Couldn't find any package by regex 'lib32bz2-1.0'   I tried to install lib32bz2-1.0 package, again I get the same error. How to install package lib32bz2-1.0 on Ubuntu 15.04(64-bit)