How does one prove that the unit circle is a 1-dimensional manifold?
Prove that the unit circle, $S^1:=\{(x,y)\in \mathbb{R}^2: x^2+y^2=1\}$, Is a one-dimensional manifold
Does anybody know how to move the keyboard caret in a textbox to a particular position?  For example, if a text-box (e.g. input element, not text-area) has 50 characters in it and I want to position the caret before character 20, how would I go about it?  This is in differentiation from this question:  , which requires jQuery. What methods can I use to infer a distribution if I know only three percentiles?  For example, I know that in a certain data set, the fifth percentile is 8,135, the 50th percentile is 11,259, and the 95th percentile is 23,611.  I want to be able to go from any other number to its percentile.  It's not  my data, and those are all the statistics I have.  It's clear that the distribution isn't normal.  The only other information I have is that this data represents government per-capita funding for different school districts.  I know enough about statistics to know that this problem has no definite solution, but not enough to know how to go about finding good guesses.  Would a lognormal distribution be appropriate?  What tools can I use to perform the regression (or do I need to do it myself)? How to scale and shift voltage signal? I want to scale and shift my voltage signal by below relation:  $$ V_{out} = \frac {10} {12} \cdot V_{in} - 10 $$  Indeed my primary voltage is from 0v to 24v and I want to put it from -10v to +10v.  I try to use the below circuit but its gain is always greater than 1.    Image from SLOA097 TI.  Does anyone know how I can achieve my goal? I have ±12v and 5v power lines.  EDIT: According to the innovative solution by scorpdaddy, I thought it would be valuable to edit my question:   High input impedance is required to decrease load effects. How can it be achieved? How can I achieve the more and more accuracy? zsh scripting - How do I NOT match a pattern in filename generation? 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In your opinion, what guidelines/steps are most essential for good bug reporting?  Feel free to share complete guideline recommendations from prior/current projects/organizations. Reference for the subgroup structure of $\rm{PSL}_2(q)$ This material is covered in detail in Dickson's "Linear Groups with an exposition of the Galois Field Theory", chapter XXII and Huppert's "Endliche Gruppen", chapter II, paragraph 8. Since I don't speak german and Dickson's treatment often requires deciphering, I was wondering if there is a "modern" account of this somewhere. Word for people who live in the same city What are the people who live in the same city are called? Any words for that? I want to use it in the following context:      I and my ____ are happy. I want to write a function that takes an array of letters as an argument and a number of those letters to select.   Say you provide an array of 8 letters and want to select 3 letters from that. Then you should get:  8! / ((8 - 3)! * 3!) = 56   Arrays (or words) in return consisting of 3 letters each. Mean and variance of inverse of a normal RV My random variable $X$ is normally distributed with mean $b$ and variance $p$. I defined a new RV, $Y$, such that $Y=1/X$. Does anyone know how to find the mean of $Y$ and $Y^2$? $\sigma$-algebra vs. $\sigma$-field: is there any difference? The subject says it all: is there any difference between the two concepts of $\sigma$-algebra and $\sigma$-field? In probability theory, they seem to be used more or less interchangeably. If there is no difference, is there any historical reason why some people/schools use the term $\sigma$-algebra, while others use the term $\sigma-$field? I get the point of having a minimum reputation to use the new chat feature. However, I was hoping my 2K reputation from Stack Overflow would be enough - but it looks like I need to acquire 20 rep here on Meta Stack Exchange.  So my question is - should this restriction be moved? Proof that in a field $x=0$ is equivalent to $x=-x$ How to get a file's Media Type (MIME type)? I created a partition for Ubuntu so I could dual-boot Windows Vista with it. However, Windows is no longer necessary and I want Ubuntu to take up the entire hard drive. I was wondering how to safely resize the Ubuntu (11.10) partition and take Windows out all together, without having to reinstall Ubuntu and lose all of the files. How to Generate surface from points Hi Guys,  I have a set of points in 3D. How can I convert this into a surface ? Highlight the difference between two strings in PHP What is the easiest way to highlight the difference between two strings in PHP?  I'm thinking along the lines of the Stack Overflow edit history page, where new text is in green and removed text is in red. If there are any pre-written functions or classes available, that would be ideal. High CPU usage by "System" and "System interrupts" (caused by ACPI.sys) I have a laptop, which was running Windows 8.1 x64 without any problems. Now with Windows 10 x64 installed, Task Manager constantly shows unusual CPU usage by "System" and "System interrupts". To solve this, I already tried the following, without success:   Disabling and uninstalling all non-essential drivers. Installing newer drivers than the ones that were automagically installed (if available). Disabling/enabling fast boot option. Disabling all of the non-essential services. Sysprep. Resetting BIOS to defaults and various combinations of settings. Flashing BIOS to the latest available version. Clean install from the same media that I use for other PCs. Installing all of the updates offered in Windows Update to this day. Windows Performance Recorder / Analyzer.   I'm not very familiar with Windows Performance Analyzer, so I'm hoping someone here can point me in the right direction - what exactly should I look for, to figure out which device/driver is the culprit. Or, if there's any other approach to figuring out this problem?  For the brave souls, here's my  and a screenshot of the problem: How to enable Hyper-V in Windows 8 I've got a Windows 8 (not .1, not Pro; OEM with a Lenovo Z580 laptop), which I want to use for WP8 development.  I've read  and , and a few more articles.  I have enabled in BIOS what looks like it has to do with hardware virtualisation - "Intel Virtual Technology" (is this VMX?). No signs of EPT or XD.  I've got the four Hyper-V items in the enumeration that msinfo32 shows.  However, in the 'Control Panel/Programs and Features/Turn on|off windows features' list, I don't have Hyper-V.  Is this available at all on my machine, and if not what can I do to get it working?  Thanks in advance. How to do volumetric lighting for a forest scene? Hill Cipher known plaintext attack I know a plaintext - ciphertext couple of length 6 for a hill cipher where its key is a [3x3] matrix.  Based on what I've read and learned, to attack and crack keys of [n x n], if we know a plaintext - ciphertext duo of length $n^2$ then we have our set of n equations with n variables, and this is generally solvable. However in my case there is only length of $6$ instead of $n^2 = 9$ .   My question is, how will I solve this problem and find the key? Or, since this is an obvious homework question, what should be my way of thinking? I cannot think of anything else but matrix multiplications and inverses but they do not help me at all.