It's obvious that for every particle velocity is smooth i.e it cannot undergo sudden finite change in its position in infinitisiminal time.  Similarly any particle's velocity cannot undergo a change instantaneously (Infinite acceleration can't happen, intuitively).  Does this pattern apply to higher time derivatives of position like jerk? If yes then till how much higher derivative? 10th? 100? Infinite?
My thinking:    acceleration corresponds to a force which is instantaneous, so the acceleration of a rigid body can be rather spiky (non-smooth) velocity (angular velocity) describes the ratio of change of the distance(angle), so it is smooth in the real world.   Conclusion, it makes sense to  (e.g., simple averaging) a velocity signal (temporal velocity), but it does not make so much sense to do smoothing on acceleration signal. Am I right?
What component to use for audio signal switching? I'm looking to build a guitar pedal switcher based on a Raspberry Pi or Arduino (have not yet decided, but I have an unused Raspberry around and I lean towards that). Anyway, digital logic.  Now, I need to switch audio signal obviously and I need to do it quietly (both physical silence and electrical silence). Please note that the signal is line level throughout most of the build, except for the input which is much lower. I will probably have 2 or 3 loops before the preamp so those may not encounter line level signal.  From what I read and know, so far, mechanical relays are not really an option due to the clicking noise they make. I also read something about FETs, but that's a bit out of my league and I would appreciate any info on these.  PS: I will probably use different power rails for the digital and analog paths.  PS2: The whole thing will be mounted in a 1U rack enclosure, so please tell me if I should take into consideration any thermal issues. Why or how did Agen Kolar and Saesee Tiin seem to die so easily without any resistance? Epsilon delta proof for $\lim_\limits{x\to a} \sqrt{x} = \sqrt{a}$ where $a>0$ $f(x) = \sqrt{x}$. For right limit, $x-a &gt; 0$. Let $x-a &lt; \delta$ then $(\sqrt{x}-\sqrt{a}) \lt \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Then chose we must choose $\epsilon $ so that $\epsilon &gt; \frac{\delta}{\sqrt{x}+\sqrt{a}}$. Is this correct? How would I conclude the proof?  Edit - Follow up  From answers, I try a proof for the same limit but this time Left side:  For left side limit, $x-a &lt; 0$ so $a-x &gt; 0$. Let $a-x &lt; \delta$ then $\sqrt{a}-\sqrt{x} \lt \frac{\delta}{\sqrt{a}+\sqrt{x}} \lt \frac{\delta}{\sqrt{a}}$.   So let me begin the proof by saying to choose $\delta = \epsilon \sqrt a$ then $a-x \lt \delta $ so $(\sqrt a - \sqrt x )(\sqrt a + \sqrt x) \lt \epsilon \sqrt a$ so $\sqrt a - \sqrt x \lt \epsilon$ In the original PC release of RollerCoaster Tycoon, if playing in a scenario with a lot of grassland, I would ensure my handymen did not mow the grass as their other services seemed more beneficial to keeping the park tidy, as well as making sure the handymen did not wander off the paths. In the android release, the handymen by default do not mow the grass (that duty by default is set to off).  I know the guests will remark on the scenery and vandalism, but I have not seen anything to suggest that mowing the grass has an effect on the guests or ride construction.  My question is; does mowing the grass have any effects apart from just making the park look tidier? Does allowing it to overgrow have any negative effects? I'm trying to find a way to automatically clean/delete the temporary files from C:\Documents and Settings\user\Local Settings\Temp\ at shutdown/boot on my Windows XP Professional x64 SP2 machine.  The main reason for this is to clean Local Settings/Temp to remove old junk automatically. (I had thought I had this covered through Firefox settings, but I was only clearing temporary Internet files). Currently, there's around 3&nbsp;GB of temporary files, and I'd like to prevent this build up without having to remember to do anything on my part.  I have already looked at using  (explained in Super User question ), but it doesn't seem to touch that directory.  I'm aware of programs such as , but I would ideally like to use a solution that doesn't require additional software (so, I expect, something using a scheduled BAT/CMD script would be what I'm looking for).  One of the potential issues, I believe, is that there may be legitimate files in the Temp folder that may be needed on the next boot (for software installations or updates, for example) and the perfect solution would be able to distinguish those and leave them until they weren't needed (Note: I realise this perfect solution may not be reasonably possible).  I've already set up Disk Cleaner on a shutdown script, but when cleaning temporary files, Disk Cleaner doesn't clean this particular directory, and so I'm looking for alternative options (see Super User question  for a little more detail).  So, what are my options for automatically deleting the temp files from C:\Documents and Settings\user\Local Settings\Temp? I want to create posters for my poster presentation on a conference. What tools or LaTeX classes are available for preparing posters ? Vertical (rotated) text in HTML table Plurals of acronyms, letters, numbers — use an apostrophe or not? When I was in high school back in the 1970s, I was taught that to make a plural of an acronym, a letter, or a number, one should add an apostrophe and "s". Like I would have written this sentence, "... back in the 1970's ..." I would write "one CD, two CD's". Etc. I followed this rule faithfully for years until a co-worker told me it was wrong. Now I can't find any source that agrees with what I was taught.  Is this a rule that has changed over time? Was the convention in the 70s that one should use an apostrophe but this has changed and now we do not? Or were my high school English teachers just confused? Routing sound from different cards to a single output I have a setup where I am monitoring for silence on an incoming stream on 3 different audio cards.  When I detect silence on the master I switch to the secondary, and if I detect silence there I switch to the last card.  The way I have it set up I load/unload the loopback module for each card depending on who's the active card.  This has the inconvenience that I need a 3-to-1 minijack cable to be able to maintain the stream.  Can I load the loopback module and point the output to a single card (independently of the input card)?  How can I route the audio from any input to a single output? Partial Sums of Geometric Series When applying to a PhD program in the US, how does the admissions process work? If an applicant is weak in a particular area, is it possible to offset that by being strong in a different area?  Note that this question originated from this . Please feel free to edit the question to improve it. Access nested dictionary items via a list of keys? What kind of questions can I expect to see on a qualifying exam? I recently received this rather amusing message while reviewing a user's first post:  Congratulations! This was only a test, designed to make sure you were paying attention. This post has already been removed, but thanks for taking time to leave feedback for the author.   What is the purpose of this test? How does the audit work? What happens if I fail? Why was the system expecting me to negatively review a good post, or positively review a bad post? A review is obviously an audit. Should I report it? Do review audits count toward the Reviewer badge? Do robo-reviewers dream of electric sheep? How to plot an emission spectrum? I've tried to construct examples of rings that match all except one of the properties in the definition of a Dedekind domain. (This is an old number theory qual question from Berkeleys MGSA website).  The only starting point that I can think of would be  $R:=K[X_1,X_2,\ldots]$  for $K$ a field. This is clearly integrally closed as any polynomial relation is contained in $K(X_1,\ldots,X_n)$ for some large enough $n$ and $K[X_1,\ldots,X_n]$ is integrally closed being a UFD.  The problem is then to get rid of enough ideals from this ring $R$, so that its dimension would be 1, but I can't seem to figure out what to do.  Anyone have any other ideas for rings that could work as a starting point? The polynomial ring with an infinite number of variables is pretty much the only example that I know of for how to construct a non-noetherian ring that's still a domain. Why is "women" pronounced the way it is? As far as I know, it is the only word where wo is pronounced as wee. What is the reason for this? Does it have to do with the origin of the word? I have been reading the top posts here, and the recommendations for what to invest in are often similar. What I'd like to know, in as simply of terms as possible, is what the consensus is on priority of these investments.  I am new to this, but here are some of the obvious options:   Paying off credit card debt Paying off mortgage / student loans 401k up to employer match 401k past employer match IRA Emergency fund Other important investments that I have missed   Clearly this will vary person to person, but if you were asked to put an ordered recommendation on a highway billboard, what would it be?  (For example:   Pay off credit debt Emergency fund for 6 months 401k up to match ... ) Display lost/negative reputation in topbar achievements When you get reputation, you get a +X green spot on your reputation feed telling you that you gained reputation and if you drop down the menu, you get more detail.  How about we make it that when you lose more than you got since the last time you checked, you see it red with a little -X.  For example, let's say you've just looked at the reputation bar so any old rep notification is gone and you ask a new question. 3 users downvote your question and 1 upvotes it. You will see it green because you gained 5 rep but you should see it red because you are at -1 rep.   This would help people improve their questions / answer by seeing, as soon as they get on the site, that they got negative feedback and that something needs to be checked / changed.    It shows here so it could very well be applied to the notification count: let's suppose that I cast a Silent image of a wall on an airship. Let's suppose that I don't move it with concentration.  Does the image of the wall move with the airship or it will progressively go far?