Change easily how figures, tables, equations.. are cross-referenced based on what is recommended in style guides such as MLA, APA, IEEE
Cross-reference packages: which to use, which conflict?
How can I calculate magnification ratio I get by reversing a lens? I am looking for a way to mathematically determine the maximum magnification ratio that is possible/practical to achieve by reverse mounting a non-macro lens, prior to purchasing the lens.     See related questions: Expectation of Exit Time of Brownian Motion from Interval I am trying to solve the following:  Let $W_t$ be a brownian motion and $a,b &gt;0.$  Define $\tau$ to be the exit time of $W_t$ from $[-a,b]$, that is $$\tau = \inf\{t \ge 0\ :\ W_{t}(\omega) \notin [-a,b]\}.$$  Show that $\tau$ is integrable and compute $\mathbb{E}(\tau)$.  I have seen this stated without proof multiple places that $\mathbb{E}(\tau)=ab.$  What is the logical argument that backs up this statement?  How are we guaranteed $\tau$ is integrable? Pretty self descriptive, without being able to view the source of the noise how can we tell the difference between a sound in front of us and a sound behind us if pitch, volume and distance are all the same.   Also assume that it is not biased to one side more than the other, e.g its not to your left or right but directly in front or behind.   I'm interested in knowing if it is a part of the ear that makes this distinction, or a function of the brain. Java equivalent of unsigned long long? In C++, I enjoyed having access to a 64 bit unsigned integer, via unsigned long long int, or via uint64_t. Now, in Java longs are 64 bits, I know. However, they are signed.  Is there an unsigned long (long) available as a Java primitive? How do I use it? What is so special about speed of light in vacuum? 1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$). Legend inconsistent in QGIS map print layout After I did styling my vector data set to be classified by graduated circle and categorized by type, it appears beautifully consistent in the map canvas as shown below:   But when I create a print map layout and after adding the map, I add the legend and it come out ugly inconsistent like this:    So, I would like to ask for help to make the legend in the map print layout consistent as the legend shown in layer panel. I use the latest version QGIS 3.10.2 64bit. Finding the left and right cosets of H = {(1), (12), (34), (12) ○ (34)} in S4 I have an exercise where I am supposed to find the left and right cosets of H = {(1), (12), (34), (12) ○ (34)} in S4. But how do I generate the cosets? As I have understood it you are supposed to pick a number that is not in the set H and multiply it with every number in H. But this does not exactly give the right answer. I would really appreciate it if someone gave an easy to understand explanation of how to generate the left and right cosets. Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. I just know that ls -t and ls -f give different sorting of files and subdirectories under a directory.   What are the differences between timestamp, modification time, and created time of a file? How to get and change these kinds of information by commands? In terms of what kind of information do people say a file is "newer" than the other? What kinds of information's change will not make the file different?   For example, I saw someone wrote:     By default, the rsync program only looks to see if the files are different in size and timestamp. It doesn't care which file is newer, if it is different, it gets overwritten. You can pass the '--update' flag to rsync which will cause it to skip files on the destination if they are newer than the file on the source, but only so long as they are the same type of file. What this means is that if, for example, the source file is a regular file and the destination is a symlink, the destination file will be overwritten, regardless of timestamp.   On a side note, does the file type here mean only regular file and simlink, not the type such as pdf, jpg, htm, txt etc? How can I add some html to my Mail.app signature? I want to add my company logo and and some juice to my contact information in the signature part in Mail.app. How can I do that with little or no efforts? Annotation Scaling I made a map with annotations in QGIS 3.0.  This is how it looks like in the layout view:    If I print the map as PDF, one would expect to get the PDF to be WYSIWYG, right? However, it looks like this (it's about half the size):    Is there a way to influence the the scaling of the Annoations? I already tried to play with the Scale and Magnifier in the map view. I also tried to change the Scale in the map item.   The strange thing is, I believe it worked a few days ago. But today I installed QGIS 3.0 on a new PC and it showed the same behaviour... Why are doctors addressed as Mr. in the UK? In the US most physicians, surgeons and dentists are addressed as "doctor". Very few other professionals receive the same title.  In the UK, however, surgeons and dentists seem to prefer to be referred to as Mr. If that is true, why?  And who is addressed as Dr. in the UK?  Those who hold doctoral degrees only? How to troubleshoot booting to a black screen on VirtualBox I was using 13.04 as a guest OS on my VirtualBox 14.2.18. Host is Windows 8. After upgrade to 13.10 Ubuntu does not start. I can see a black screen and blinking cursor. I am pretty new to Ubuntu. I suspect a problem with graphics drivers. I can access tty and all files (/home) are there. Is SSIS subject to the maximum memory limitations for SQL Server 2012 Standard edition?  If I have a server with 128GB RAM and I allocate 64GB of RAM to SQL Server does SSIS utilise this memory or does it utilise the memory that is available to the OS?  I am assuming it utilises the remaining memory left to the OS since there is no settings for memory consumption and it runs as a separate service.  If anyone has any information on the available memory that should be left for SSIS to consume I would appreciate the help. Can I backup my entire Ubuntu installation to ISO DVD? Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$ I'm trying to prove $2^b-1$ does not divide $2^a + 1$ for $a,b&gt;2$. Can someone give a hint in the right direction for this? Meta sites in "Your Communities" display different reputation Will an iOS reinstall make a jailbroken iOS device "unjailbroken" and “supported by Apple” again? Will an    Work on any jailbroken-iPhone and Make a jailbroken-iPhone "unjailbroken"? Make a jailbroken-iPhone "supported by Apple"?   Similarly, does such an iOS reinstall reset the iPhone (jailbroken or not) back to a "clean, factory default" state?  My near-term purpose: I have an iPhone 4S running iOS 5.1 I'd like to jailbreak to at least install better Google Voice integration like . Move item to folder when added/edited (SP2013 or O365) How to move item to folder automatically when it has been added or edited in a library? The folder is in the same library. The correct folder will be defined by metadata. This should happen automatically when the item has been added or edited. The listitem Id should be preserved. No server-side code should be deployed.  Can this be done using workflows (SP2010/SP2013)? If not then what would be the other options?