I need to prove that if $\alpha: I\rightarrow \mathbb{R}^{2}$ is regular with curvature $\kappa$, then $\alpha$ is on a circle with radius $r&gt;0$ if and only if $|\kappa(t)|=\dfrac{1}{r}$.
I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the circunference, then its curvature is constant.  I solved the second part since $\alpha$ is the circunference, then $\alpha(t)=(r\cos\theta,r\sin\theta)$, so $|\alpha'(t)|=r$, I need help in the first part.  Thanks a lot.
Regexp that matches valid regexps It is well known that confidence intervals and testing statistical hypothesis are strongly related. My questions is focused on comparison of means for two groups based on a numerical variable. Let's assume that such hypothesis is tested using t-test. On the other side, one can compute confidence intervals for means of both groups. Is there any relation between overlapping of confidence intervals and the rejection of the null hypothesis that means are equal (in favor of the alternative that means differ - two-sided test)? For example, a test could reject the null hypothesis iff the confidence intervals do not overlap. In a case of mine a very high (compared to the video stream) resolution graphical frame (saying "access denied" (when my IP changes)) can appear instead of a video stream, which makes VLC window to resize, getting bigger than my screen is. This annoys me. Can I set up VLC to never resize its window when I don't do it myself and resize content instead? Find the slope of tangent line to the curve at the point $(1, \pi/2)$  the equation is  $$\sin(xy) = x$$  The right answer was = Slope is infinite   My answer was taking the derivative and I end up in this equation   $$ (1-y\cos(xy))/ x\cos(xy)  $$  I substitute with the points and the calculator showed an error, Does the error means infinite ? How to tell how many CPUs (cores) you have on Windows 7 Is $\exists x(x=a)$ a valid formula in first order logic with equality? Under , is $\exists x(x=a)$ true in every interpretation in first order logic (FOL) with equality? If so, could $\exists x(x=a)$ be considered a tautology? This question was motivated by  which was closed for lacking clarity. Here I have attempted to add that clarity myself so that the same question can be useful to others who are wondering a similar thing. This is a self answer question. Outlook autocleaning my line breaks and screwing up my email format I'm sending an email using the dotnet framework. Here is the template that I'm using to create the message:  Date of Hire: %HireDate% Annual Salary: %AnnualIncome% Reason for Request: %ReasonForRequest%  Name of Voluntary Employee: %FirstName% %LastName% Total Coverage Applied For:  %EECoverageAmount% Guaranteed Coverage Portion: %GICoveragePortion% Amount Subject to Medical Evident: %GIOverage%   When the messages is received in outlook, outlook tells me "Extra line breaks in this message were removed". And the message displays like this:  Date of Hire: 9/28/2001 Annual Salary: $100,000 Reason for Request: New Hire  Name of Voluntary Employee: Ronald Weasley Total Coverage Applied For:  $500,000 Guaranteed Coverage Portion: $300,000.00 Amount Subject to Medical Evident: $200,000   Note how Outlook incorrectly removes needed line breaks after the name, EECoverageAmount, etc...  It's important for the email recepients to get a correctly formatted email, and I have to assume that some of them use outlook 2003. I also can't assume they will know enough to shutoff the autoclean feature to get the message to format properly.   I have viewed these messages in other mail clients and they display correctly  some more information:   I am using UTF-8 BodyEncoding (msg.BodyEncoding = System.Text.Encoding.UTF8) The msg.Body is being read from a UTF-8 encoded text file, and each line is terminated with a crlf.   Question: How do I change the format of the message to avoid this problem? LWC: Difference between "window" and "this" I am trying to understand what exactly is the difference between window and this in the context of a LWC component.  According to this  the Locker intercepts calls to the Window and uses Secured Window instead which acts as:     Secure wrapper for the window object, which represents a window containing a DOM document.   As for this what I  was:     To add an event listener to an element that a template doesn’t own,   call addEventListener directly: this.addEventListener()   What does salesforce mean by an element that a template doesn't own? Isn't window an element that the template doesn't own? How long should I keep my tax documents, and why? How do I handle casting a spell with a cost of X when the casting cost is reduced? Probability of Obtaining the Roots in a Quadratic Equation by Throwing a Die Three Times Illogical GTK theme behaviour on Ubuntu 18.04 How do you declare an interface in C++? How do I setup a class that represents an interface?  Is this just an abstract base class? Consider the blow-up of the plane in one point. Let $E$ the exceptional divisor. We know that $(E,E)=-1$. What is the geometrical reason for which the auto-intersection of $E$ is $-1$? In general, what does it mean, geometrically, that a divisor has a negative self-intersection or that the "right" number of divisors gives rise to a negative intersection?  Thanks. I just installed the lastest version of . I works fine but in order to use it, I have to enter the following in the terminal: cd ugene-1.11.5/ then execute ./ugene -ui to launch UGENE GUI.   This takes too long. I would like to build a .desktop file for UGENE to be used in Ubuntu 12.04 so that I can simply open it from the unity launcher.  Not sure if this is helpful but the website says: By default ugene script launches the command-line version of UGENE.  Here's what I have (not sure if it's right):  [Desktop Entry] Version=1.0 Name=my Unipro UGENE GenericName=Integrated Bioinformatics Suite Comment=Unipro UGENE is a cross-platform visual environment for DNA and protein sequence analysis. MimeType=text/english;text/plain;text/x-makefile;text/x-c++hdr;text/x-c++src;text/x-chdr;text/x-csrc;text/x-java;text/x-moc;text/x-pascal;text/x-tcl;text/x-tex;application/x-shellscript;text/x-c;text/x-c++; Exec=/home/samuel/ugene-1.11.5 ./ugene -ui -c %F Icon=/usr/share/icons/hicolor/scalable/apps/ugene.svg Type=Application Terminal=true Categories=Utility;Development; StartupWMClass=UGENE   I added the launcher named my Unipro UGENE to my unity bar and tried opening it but it doesn't appear. I must be missing something. I don't know what the StartWMCClass field is suppose to be nor do I know whether I entered the correct MimeType. Using a Single system() Call to Execute Multiple Commands in C In an information security lab I'm working on, I've been tasked with executing multiple commands with a single call to "system()" (written in C, running on Fedora).  What is the syntax that will allow me to execute more than command through system()?  (The idea being you could execute arbitrary commands through a program running on a remote computer, if the program interacts with the OS through the system() call.)  I.e.:  char command[] = "????? \r\n";  system(command); Question about mosfet gate resistor I have n mosfet gate connected to  logic and Id is about 100mA. Both 4043 and mosfet has +5v. I have plan to use  mosfet  Questions are: How large gate resistor i need between 4043 and mosfet? Logic output is sometimes put on rapidly. How fast? Motherboard hdd led controls that. Do i need to place pull down resistor from logic to -0v, between 4043 and mosfet? There have been a number of cases where I would like to thank someone for providing me with a thorough response and give them feedback beyond the vote up. Often I wish people would tell me why they voted me up. Not that the thank you part is necessary but it sets it apart from the other comments related to answering the question itself.  In this type of situation is a "Thank you" acceptable in a comment?     For more information, see  in the . Power failure during upgrade During upgrade from 12.04 to 12.10 there was a power failure.  computer boots to desktop but without tools and menus to navigate.  files that can be open cannot be closed.  sidebar menu does not exist.  there is no function of the DVD drive, the USB drive nor the internet.  Possible to open GRUB but I am not familiar with it's function.  Please help. By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. I remember that there's an elegant proof, but I forget its detail. (The first step is to show $A\oplus (A\oplus B) = B$, and maybe applying this formula, we can obtain associativity.)