Got few questions about the input/output pins of a crystal circuit.   As shown above, it's a common clock circuits. My question is: 1. If we want to measure the waveform of the clock, which pin should we probe and why? 2. In the internal circuits inside the IC, which signal (IN or OUT) is used for the clock source of the internal circuits and why?  Thanks!
I have 2 quartz crystal resonators in my pcb board: 32.768 kHz &amp; 20 MHz. They are connected to a Freescale MC12311 transceiver IC, which has an HCS08 micro-controller embedded in it. I want to test if these crystals are working properly or not.  Available tools: Oscilloscope, Frequency-Meter (Digital Counter), Digital Multimeter.  How should I use these tools to test the crystals in board?   Note: The capacitive load-effect of probes should probably be considered. If not, the measurement would not be accurate, or even worse, the crystals would not work at all.     Edit1: I used both the oscilloscope and the frequency-meter (with x10 probes), but unfortunately there was nothing monitored at all.
Types of polynomial functions. Quadratic, cubic, quartic, quintic, ...,? I would very much like to have a complete list of the types of polynomial functions. I know that theres:  Quadratic :                      (AX^2 + BX + C) Cubic     :               (AX^3 + BX^2 + CX + D) Quartic   :        (AX^4 + BX^3 + CX^2 + DX + E) Quintic   : (AX^5 + BX^4 + CX^3 + DX^2 + EX + F)   What are the names of polynomial functions to the further powers? Sexstic? Septic? Octic? I'm not sure, so if someone could enlighten me, then that would be great. Please provide a list that goes at least to the seventh power, but it would be nice if you could go further. Thanks. This lock notice about collaborative effort is partly misleading is locked  with the following notice:     This question and answer are a collaborative effort: if you see something that can be improved, edit it!    Except that while the answer can be edited because it's a Community Wiki question, the question itself  can't be, because it's locked. I don't know the reason why it was locked, but maybe the notice's text should be changed or omitted? (I have no opinion on whether the question should stay locked or not.) Prove that $x^2 = - 1$ has no solution in $\mathbb{Z}$ I have this proposition to prove: The equation $x^2 = -1$ has no solution in $\mathbb Z$. I was told that this is an opportunity for a proof by contradiction. I have already proven that for $m \in\mathbb Z$, if $m \ne 0$ then $m^2 \in\mathbb N$. I have also proven that 1 is a natural number (hence, -1 isn't one).  Here is my strategy: assume that there is a solution. Hence $x^2$ has 3 options: \begin{align*} x^2 \in\mathbb N\\ -(x^2) \in\mathbb N\\ x^2 = 0 \end{align*}  Option 1 is not possible because -1 is not a natural number. Option 2 doesn't make sense because         -$\mathbb N \in \mathbb N$ and option 3 doesn't make sense either because $0 \ne -1$. What do you think? Thank you!    Here is a simpler strategy based on the comments below.  Proof: Assume that $x^2 = -1$ has a solution in $\mathbb Z$, then $x^2$ will either be $0$ or $\in\mathbb N$. However, $-1 \notin\mathbb N$ and $-1 \ne 0$. Hence, there is a contradiction. Here is how I have proven that $-1 \notin\mathbb N$: Let $m \in\mathbb N$. Hence:  \begin{align*} -m \notin\mathbb N\\ (-1)m \notin\mathbb N \end{align*}  If $-1 \in\mathbb N$, the product of $-1$ and $m$ should $\in\mathbb N$, which is not the case here. Hence, $-1 \notin\mathbb N$. Windows 10 non-persistent installation (live ISO) We've got a Windows 10 installation running in an environment where it can occasionally lose power and restart. However, this sometimes leads to corruption of files.  I've had a look at Windows "To-Go" but this seems to be persistent. And I've also checked out virtualisation, but that would require us to run a non-persistent Windows 10, in another Windows 10 environment.  Is it possible to install Windows 10 as a "non-persistent live CD"? So that a user will have a seemingly fresh Windows 10, but doesn't require to re-install it at each startup? I'm following the , but on launching the upgrade tool I get this response:  Checking for a new ubuntu release No new release found   Am I doing something wrong? Is there a workaround? Distribution of orbital velocities in a disk galaxy for N-body simulation? I wish to travel to the US and think I am eligible to do so under the Visa Waiver Program. I found a lot of information out there but am still having some trouble understanding how this works.  What is the US Visa Waiver Program? What is an ESTA? what is the difference between IBE and ABE schemes Can somebody explain in simple words. real life example light help and limitations of these schemes etc. How do I install the latest version of gcc and g++?  I am using a variety of C++11 and C++1y features; which is why I need this. Show forms in a modal window I have a form and I am using Drupal 7. I need to show this form in a popup when a user clicks a link. The user should then be able to complete the form whilst in the pop up window. I can use anything like  or modals or anything. I just wanted to know which is the better option and what options do I have for Drupal. Fibonacci combinatorial identity: $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ Can someone explain how to prove the following identity involving Fibonacci sequence  $F_n$  $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ ? Split polygons into equal areas within each region I've a set of regions within a city. I'd like to split each region into five subregions and then get the centroid of each one of them.  I know how to make a vector grid, but the problem I have is that the generated points are not taking into account the regions, they're only related to the general map.  I have this:    And I want to split each region in five equal size areas. Something like this:    I'd like to build a visualization similar to this one: . In order to accomplish that, I need to split each region, because the original ones are pretty big. I am trying to use the  package.  I have the following counters defined (hopefully this is the right way to do it!)  \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary}   This works as I would like (i.e. Theorem 1.1, Corollary 1.2, Lemma 1.3, etc.)  My problem is \cref always returns Theorem 1.2, Theorem 1.3. I   Perhaps this is not right? (In my preamble)  \crefname{thm}{theorem}{theorem's} \crefname{lemma}{lemma}{lemma's} xUbuntu Default desktop Icons do not display? Even though I have the default icons set to display in Settings, they do not display. See the image below. Ideas???? So Black Holes Actually Merge! In 1/5th of a Second - How? I've read a lot of conflicting answers in these forums. However, today saw the awesome announcement of gravitational waves. Two black holes merged:   Not only that, they merged FAST. In 1/5th of a second revolving around each other 250 times a second. The entire event was quicker than a heartbeat. Moreover, we observed this happening as distant outsiders. So now we can say for sure:   Objects approaching the event horizon DO NOT appear to slow down Black holes CAN merge in a finite (and quick) amount of time And all this is wrt a frame of reference far, far away   To quote the NYTimes article:  One of them was 36 times as massive as the sun, the other 29. As they approached the end, at half the speed of light, they were circling each other 250 times a second.  And then the ringing stopped as the two holes coalesced into a single black hole, a trapdoor in space with the equivalent mass of 62 suns. All in a fifth of a second, Earth time.  However, everything I've read so far has let me to believe that an outside observer should never be able to measure the collision happening in a finite time. So what exactly is happening here? I must have read at lest 5 different versions of this so far everywhere in these forums over the past several years. Can someone tell me how to get a closed form for  $$\sum_{k=1}^n k^p$$  For $p = 1$, it's just the classic $\frac{n(n+1)}2$.   What is it for $p &gt; 1$? Is it possible/sensible to add a suspension fork to an old mountain bike? I have a ~1993 Giant mountain bike.  It's not great, but I am a very new and casual rider.    The bike has a solid fork and it makes for the predictable rough trail ride.  The trails are generally fairly modest (I live in North Texas, a 100' change in altitude makes our ears pop here) and the surface is generally dirt (very few rocky areas).  So, I am thinking that it would be very nice to have a suspension bike.  And seeing that a modest fork can be purchased for a modest price, I am wondering if it makes sense to add a fork to this bike.   asked about going the other direction, and there were some cautionary words about changing steering geometry.  So, on balance, is this something that I would be best off to avoid?  I suspect that if I have to replace a bunch of steering parts, the cost will quickly approach that of a new low end bike. How does the Flash breathe while utilizing his super speed? How do the Flashes breathe utilizing their super speed? The air pressure would be extreme. Do they run in a pressurized bubble? Is openssl rand command cryptographically secure? I'm wondering if the openssl rand command produces cryptographically secure random bytes. For example when in need for a random password or token:  openssl rand -hex 32   The man page unfortunately does neither state it's cryptographically secure, nor that it's not.  On the one hand, I think this is openssl, its sole purpose is to do cryptography. On the other hand, the man page states something about reading and writing seed data instead of directly querying a CSPRNG.  So it is somewhat unclear for the end user. Does someone know? Escaping HTML strings with jQuery Does anyone know of an easy way to escape HTML from strings in ?  I need to be able to pass an arbitrary string and have it properly escaped for display in an HTML page (preventing JavaScript/HTML injection attacks).  I'm sure it's possible to extend jQuery to do this, but I don't know enough about the framework at the moment to accomplish this.