Which of these two forces is stronger: static friction or kinetic friction?
Can the coefficient of static friction be less than that of kinetic friction?
Nested List Indices Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified? I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained  $$\begin{eqnarray} \frac{\sin z}z&amp;=&amp;\frac{\sin(x+iy)}{(x+iy)}\\ &amp;=&amp; \frac{\sin(x)\cosh(y)+i\cos(x)\sinh(y)}{x+iy}\\ &amp;=&amp; \frac{x\sin(x)\cosh(y)+y\cos(x)\sinh(y)}{x^2+y^2}+i\frac{x\cos(x)\sinh(y)-y\sin(x)\cosh(y)}{x^2+y^2}. \end{eqnarray}$$  Before calculating it, I hoped I could use this in a solution to a certain problem, but I would need to differentiate the real and imaginary parts of $\dfrac{\sin z}z$ with regard to $x$ and $y$. But what I'm getting is very complicated. I understand that it could be that that's just the way it is. But maybe there are some simplifications to be done either in the functions or in their derivatives? I've never really used hyperbolic functions before so I'm not sure I'm not missing something.  Added. The problem I'm working on is the following.     Prove that the solutions of the equation $$\tan z=z$$ are all real.   I will sketch the solution I know here.  It can be calculated that  $$\tan(x+iy)=\frac{\sin(2x)}{\cos(2x)+\cosh(2y)}+i\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}.$$  Plugging this to the equation, we get  $$\begin{eqnarray} x&amp;=&amp;\frac{\sin(2x)}{\cos(2x)+\cosh(2y)}\\ y&amp;=&amp;\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}. \end{eqnarray}$$  From this we can get   $$\sin(2x)\cdot2y=\sinh(2y)\cdot2x.$$  By plotting the functions $\sin,$ $\mathrm{id}$ and $\sinh$ (and proving the right inequalities we can see from the plots), we can see that this is only possible when $xy=0$. But when we plug $x=0$ to the second equation, we can show that the only solution is $y=0$, which ends the proof.  What strikes me in this solution is the fact that $\dfrac{\sin z}z$ is used. (Not explicitly in my statement.) This function has another connection to the problem:     The non-zero solutions of the equation $\tan z=z$ are exactly the zeros of the derivative of $\dfrac{\sin z}z.$   This is easy to prove. I wondered if there shouldn't be a solution to the problem which uses this fact, given that $\dfrac{\sin z}z$ appears in the solution I know. Alternate position of x tick labels What is the need of complex functions in wave analysis? It is commonly known that waves can be express in terms of sine or cosine function. But when I study further, I seen that for analyising the waves, it is common to use complex functions in the form $$y=y_{_0}e^{i(kx-\omega{t})}$$ where $y_{_0}$ is the amplitude, $k$ is the wave number, $\omega$ is the angular velocity and $x$ &amp; $t$ are position an time respectively. Ofcourse, I know that the function $e^{ix}$ can be written in the form $\cos{x}+i\sin{x}$ and so it is a periodic function with period $2\pi$ but my question is for what purpose we define it in terms of complex numbers? It seem to be more convenient to use real functions for real variables such as amplitude, electric and magnetic field of an electro magnetic wave, and also in quantum mechanics. What actually this interpretation means or what is the advantage of such functions? How can I set default value in ActiveRecord?  I see a post from Pratik that describes an ugly, complicated chunk of code:   class Item &lt; ActiveRecord::Base     def initialize_with_defaults(attrs = nil, &amp;block)     initialize_without_defaults(attrs) do       setter = lambda { |key, value| self.send("#{key.to_s}=", value) unless         !attrs.nil? &amp;&amp; attrs.keys.map(&amp;:to_s).include?(key.to_s) }       setter.call('scheduler_type', 'hotseat')       yield self if block_given?     end   end   alias_method_chain :initialize, :defaults end   I have seen the following examples googling around:    def initialize      super     self.status = ACTIVE unless self.status   end   and    def after_initialize      return unless new_record?     self.status = ACTIVE   end   I've also seen people put it in their migration, but I'd rather see it defined in the model code.  Is there a canonical way to set default value for fields in ActiveRecord model? how to properly save an architecture in a fpga ic forever Use /testfor to make a custom command Basically, I'm trying to recreate the Faction plugins /f map in my single-player without mods as my PC doesn't run them very well. (I know it sounds strange.) I am trying to use the /testfor to test if I have written '/f map' in chat, and if I have, it would show me a custom generated map. I don't know how to word this better xD Essentially what I want is for it to test if I have written a command in chat, and if I have, display something. What's the easiest way to download all albums from Google Photos? I have around 950 albums at Google Photos.      I want to back them up now, but organised by albums, not by date.   Whenever I create a backup from within Google Photos, the backup splits my photos by date and adds metadata here and there.  What I am doing now is to click on every album, and choose "Download all". But this will take me ages.  Is there a way through Google Drive or another app to have a easier job? Continuing enumerate Counters in Beamer I'm trying to continue the numbering of examples in enumerate environments in beamer. For example, I'd like one slide to feature   Foo Bar   and the next slide   Zip Yadda   My strategy so far has been to define a counter that saves the enumerate counter at the end of the first environment and can then be queried to set the enumerate counter at the start of the second enumerate environment. (This is essentially the proposal also found at )  That works wonderfully so long as I don't try to uncover elements of the slide sequentially. The problem, of course, is that as each of the pdf-pages that make up the slide is created, the counter is advanced. Here's a MWE.  \documentclass{beamer}  \setbeamercovered{highly dynamic}  \newcounter{saveenumi} \newcommand{\seti}{\setcounter{saveenumi}{\value{enumi}}} \newcommand{\conti}{\setcounter{enumi}{\value{saveenumi}}}   \begin{document}  \begin{frame}[&lt;+-&gt;]   \begin{enumerate}   \item foo   \item bar%     \seti   \end{enumerate} \end{frame}  \begin{frame}[&lt;+-&gt;]   \begin{enumerate}     \conti   \item zip   \item yadda%     \seti   \end{enumerate} \end{frame}  \end{document}   As the second slide is progressively uncovered (pages 3 and 4 of the resulting PDF), the counter at the beginning of the enumerate environment is advanced each time. So page 3 of the PDF has the right numbering, (3) and (4) respectively, while the fully revealed page has the wrong numbering. Pictures follow:   on one of the slides    on the following one Why don't we treat rubbish the same as spam? Can I live in the UK for a long time through repeat visits? I had recently entered the UK without a visa to be with my partner, from 21 July 2016 until 16 December 2016. I hope to return on the 9th of January. We don't meet the income requirement for a Spouse visa. I recently applied to university, and got in, and was intending to return to the UK on a student visa, but my university did not tell me about maintenance fees and my loan wasn't large enough to cover it all, so my visa would have been denied.  Now, I am hoping to attend university in the next semester, and therefore return to the US to apply for that visa. Do you think that if I try to return to the UK without a visa, I will be turned away?  I won't be working or studying during this time, however, I'm afraid that they'll consider it living in the UK illegally; I don't have a criminal record or anything like that. Are the datum and the ellipsoid the same? What is a Datum?     The Earth is shaped like a flattened sphere. This shape is called an   ellipsoid. A datum is a model of the earth that is used in mapping.   The datum consists of a series of numbers that define the shape and   size of the ellipsoid and it's orientation in space. A datum is chosen   to give the best possible fit to the true shape of the Earth.  in this   article.... How can I comment out a block of text in texmaker? I use texmaker and some time ago I found, by chance, a way to comment many lines at once, but I can't remember it now. Can somebody remind me of that? Positive operator has a positive spectrum? Definition of injective function recursion-consecutive numbers what is the number of subsets of the set {k∈N|1≤k≤n} with no two consecutive numbers? The answer says: $$a_n=a_{n-1}+a_{n-2}$$ with the starting conditions: $$a_0=1, a_1=2$$1. why does $a_1=2$? $$$$ 2. How does this recursion was created, I understand that if $a_n$ mean all the places with no consecutive numbers so, $a_{n-1}$ come from the fact that after the first place all those places satisfy this condition, but why there is need to "look" at what happen after the two places? What does "causally connected" or "causes" really mean? In a different thread, a user stated the following about events preceding or following other events:     However, if the two events are causally connected ("event A causes event B"), the causal order is preserved (i.e. "event A precedes event B") in all frames of reference.   My question is what does "causally connected" really mean? What does "causes" mean? Further, given that we know that we can have instantaneous effects in typical quantum processes (e.g. flip the polarizer, effect on another a reading light years away, even though we cannot transmit useful information with this), does that not constitute "causing" for the purpose of this statement? How to create individual chapter PDFs from included TeXs? My document structure has   one file with preamble (i.e. MainTex) and around 75 individual .tex files (i.e. Chapter_01, Chapter_02, ..., Chapter_75) representing each chapter.   I am using \include command to create a one big PDF file, the output is MainTex.pdf. Life is all good up to this point.   If I have to create individual PDF for each chapter then do I need to run 75 times with each chapter name and rename the PDF manually to match to chapter name. Because if I include only Chapter_01 and run it creates PDF named as MainTex.pdf.  Is there a way I can automate this workflow? "I left smoking", "I quit smoking", "I gave up smoking", "I stopped smoking" are these same? For various such sentences in many cases I get confused, it seems different for different cases. I want to know if someone left doing something for permanantly then what should be used? Also give some examples for different cases. Suppose... "I left my bag" means "I forgot my bag". but If I want to say the same for permanentaly "I left bringing my bag". Where should we use "stop" and "quit" then? KML to Shapefile using GDAL (LIBKML) - MultiGeometry data not transforming I am trying to convert a KML file to Shapefile using GDAL (ogr2ogr). Regular KML files convert fine but KML files containing MultiGeometry features (eg: Geometry containing both Point and Polygon) do not get transformed.   The output is simply a blank shapefile containing no shapes or attributes.  I have tried this command:  ogr2ogr -overwrite -f "ESRI Shapefile" -where OGR_GEOMETRY='MultiGeometry' &lt;ouputfolder&gt; &lt;input kml&gt;   For an example, KML file which has MultiGeometry, you can see the us_states.kml hosted by Google.    Other info: I am using 'GDAL 1.10.1, released 2013/08/26' for windows.