Variation Of Saturation Current in Photoelectric Effect
Why doesn't saturation current in the photoelectric effect depend on the frequency of light absorbed ny the metal emitter?
Question 5.33 from "Topology and its Applications"  by Baesner is to compute the fundamental group of the torus ($T^2$) with $n$ points removed. I can "see" in my mind that if we remove one point we get a bouquet of two circles. Less clear is what happens when we remove two (or more) points. Any hints? What do potential postdoc advisers seek in research statement letters? I am putting together my application for a postdoctoral position in theoretical condensed matter physics. The position is not a fellowship and is a regular postdoc position. If you have gone through the process of applying for such positions a number of times, or, even better, if you have reviewed such research statement letters, what are the main points that are sought/you seek in such letters? Specifically, should I put more emphasize on my PhD research to demonstrate my "depth of knowledge" on projects I have worked on (past research), or dwell more on things that I want/am capable to do (future research)? Abelian group (Commutative group) Prove that if in a group $(ab)^2= a^2 b^2$ then the group is commutative.  I am having a hard time doing this. Here is what I have so far: Proof:  $a^2 b^2= a^1 a^1 b^1 b^1$ =$aa^{-1}bb$ =ebb Hence,$aa^{-1}=e$  I am stuck, I do not know if this is the right process in proving this For $a,b,c,d \in \mathbb{R}$, is $\sqrt{a^2+b^2} + \sqrt{c^2+d^2} \geq \sqrt{(a+c)^2 + (b+d)^2}$? I have tried to show this using the fact that $\sqrt{x+y}\leq\sqrt{x}+\sqrt{y}\; $  for non-negative $x,y$ but haven't been able to arrive at a proof. How run a .jar file with a double-click? I have a problem running .jar files by double-clicking them. When I go to the Properties dialogue I can't find Java--only Archive Manager in the list of programs. But java -jar somefile.jar opens the program. What is the electric charge of the Sun and its corona? What does the dot product of two vectors represent? I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.  The product of two numbers, $2$ and $3$, we say that it is $2$ added to itself $3$ times or something like that.  But when it comes to vectors $\vec{a} \cdot \vec{b}$, I'm not sure what to say. "It is $\vec{a}$ added to itself $\vec{b}$ times" which doesn't make much sense to me. $$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5}  = \lim_{x\rightarrow 5} (x+5)$$  I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I understand how to do this procedurally, but I'd like to know why this works. I've only been told the methodology of expanding the $x^2-25$ into $(x-5)(x+5)$, but I don't just want to understand the methodology which my teacher tells me to "just memorize", I really want to know what's going on. I've read about factoring in abstract algebra, and about irreducible polynomials (just an example...), and I'd like to get a bigger picture of the abstract algebra in order to see why we factor the limit and why the simplified is equal to the original if it's missing the $(x-5)$, which has been cancelled. I don't want to just memorize things, I would really like to understand, but I've been told that this is "just how we do it" and that I should "practice to just memorize the procedure." I really want to understand this in abstract algebra terms, please elaborate. Thank you very much. Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $ How could we prove that this inequality holds      $$ \left(1+\frac{1}{n+1}\right)^{n+1} \gt \left(1+\frac{1}{n} \right)^{n} $$   where $n \in \mathbb{N}$, I think we could use the AM-GM inequality for this but not getting how? Can a US citizen leave and enter the US using a foreign passport? Super+D key shortcut does not show desktop I recently installed Saucy Salamander on my computer. Previously in Ringtail, Super+D used to show the desktop. However for this release, the Super+D does not seem to work properly... Is it a bug? Using C# .NET 2.0, I have a composite data class that does have the [Serializable] attribute on it.  I am creating an XMLSerializer class and passing that into the constructor:  XmlSerializer serializer = new XmlSerializer(typeof(DataClass));   I am getting an exception saying:      There was an error reflecting type.   Inside the data class there is another composite object.  Does this also need to have the [Serializable] attribute, or by having it on the top object, does it recursively apply it to all objects inside? How to align two images side by side if one label has more lines I have two same sized images and want them side by side with a caption under them.  My code looks like this:   \begin{figure}[H] \centering \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{3}     \caption{label1}   \label{fig:test1} \end{minipage}% \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{4}     \caption{label2}   \label{fig:test2} \end{minipage} \end{figure}   which works fine until one of the labels has 2 lines, than it looks like this:    and I woud like to prevent it to look something like this:    How can it be done?  Thanks Recently, I found this , that allows you to generate a 3D model from a 2D image.   It works very well and you can download an OBJ file. Vertex are colored and when I import it using Blender, I can't see any Vertex color.   Using Cycle (I prefer a internal solution if it exists) according to some answers I've found on Blender Stack Exchange, I patched some nodes : Attribute -> Diffuse BDSF -> Material Output.  Unfortunately, I can't get it works, the model stills black.    When I open the OBJ file, I see that color per vertex are there :  v 43.747554 107.000000 69.000000 0.470588 0.380392 0.309804 v 43.780163 115.000000 64.500000 0.427451 0.337255 0.266667 v 43.782527 107.000000 68.500000 0.470588 0.380392 0.309804 v 43.801902 115.000000 65.000000 0.427451 0.337255 0.266667 v 43.806461 107.000000 69.500000 0.470588 0.380392 0.309804 v 43.821703 115.000000 64.000000 0.427451 0.337255 0.266667 v 43.838352 108.000000 68.500000 0.466667 0.376471 0.305882 v 43.854670 106.000000 69.000000 0.482353 0.392157 0.321569 v 43.856176 108.000000 69.000000 0.466667 0.376471 0.305882 v 43.873876 106.000000 69.500000 0.482353 0.392157 0.321569 v 43.888855 110.000000 68.000000 0.466667 0.376471 0.305882 v 43.891471 109.000000 68.000000 0.466667 0.376471 0.305882 v 43.892733 110.000000 67.500000 0.466667 0.376471 0.305882 v 43.895170 108.000000 68.000000 0.466667 0.376471 0.305882 v 43.900904 109.000000 68.500000 0.466667 0.376471 0.305882 v 43.907584 107.000000 68.000000 0.470588 0.380392 0.309804 v 43.917898 115.000000 65.500000 0.427451 0.337255 0.266667 v 43.925000 111.000000 67.500000 0.454902 0.364706 0.294118   According to some answers here, Blender import more easily the vertex color of a PLY file so I tried to convert to PLY with two different sites :   1. meshconvert .com : Give me a 3.5 Mo file  2. greentoken.de/onlineconv : Give me a 2.3 Mo file  (Of course, if you know some other sites I could try, it interest me)  I guess that color data are lost with greentoken (not sure).   When I import the 3.5 Mo PLY file in blender, I need to be in textured display mode, then I think I see my vertex color value but in black &amp; white instead of RGB, that is weird. Anyway, I can't get it work  EDIT 1 : I opened PLY files and color vertex aren't there... I need to find a better converter   EDIT 2 : I tried www.ofoct.com/3d-model-file-for-3d-printer-converter it doens't take vertex color into account. In fact, in each PLY files I converted with the various sites here, there is a common line :      comment Created by Open Asset Import Library -    (v3.2.0)   (the version varies)  EDIT 3 : In fact it seems that model converted by Green token, keep the vertex color. That is weird because that's the smallest file :   ply format ascii 1.0 comment Created by Open Asset Import Library - http://assimp.sf.net (v3.1.187496374) element vertex 29793 property float x property float y property float z property float nx property float ny property float nz element face 59582 property list uint uint vertex_index end_header 82 103 6.37528 0.40963 0.0331735 0.911648 82 102 6.57043 0.272087 0.235193 0.933088 82.6603 103 6.5 -0.14071 0.043367 0.9891 82 103.935 6.5 0.201161 -0.136681 0.969976 81.5475 103 7 0.825371 0.0489484 0.562465 83 102 6.62244 -0.248588 0.147929 0.957247 83 103 6.56417 -0.411487 -0.0867124 0.907281 81.5819 104 7 0.787841 -0.0338904 0.614946 82 105 6.63886 0.0715144 -0.131383 0.988749 83 104 6.77321 -0.431167 -0.171652 0.885794 83.6781 102 7 -0.470137 0.061235 0.880467 83.6327 103 7 -0.543939 -0.0905203 0.834228 83 105 6.96709 -0.445324 -0.163091 0.880391 84 103 7.23724 -0.520118 -0.141377 0.842313 84 104 7.47775 -0.546517 -0.195833 0.814229 84 102 7.16782 -0.416323 -0.021798 0.908955 83.3508 101 7 -0.314426 0.156435 0.936304 84 101 7.22581 -0.280388 0.0301462 0.959413 84.9009 101 7.5 -0.339848 -0.0219793 0.940223 84.6939 102 7.5 -0.435242 -0.0925323 0.895546   As you can see, nx, ny and nz correspond to RGB values.. Are there integer solutions to $9^x - 8^y = 1$? I'm trying to install  in VirtualBox but the installation get stuck at Do you want to install boot loader GRUB    Some background:   The VM was created from the Ubuntu 32-bit (x86) VirtualBox template The VM has 1 core + 3096 MB RAM Video memory: 32 MB PAE/NX enabled Hardware virtualization: both VTx and nested pagination enabled Graphics acceleration: disabled (both 2D and 3D) Storage: 64 GB attached to IDE virtual controller (was SATA before: makes no difference)   I've already tried all the "standard" VirtualBox procedure (PAE yes/no; SATA/IDE; no soundcard; no USB; graphics accel. yes/no).  I also tried run the Live CD in VESA mode: like this it starts, but I'd still prefer to install it. Wubi with 14.10 doesn't work - failed to partition the selected disk I tried to install Ubuntu 14.10 on my computer with Wubi. The installation appeared to work in Windows, and told me to reboot.   After rebooting the Ubuntu installer continued the installation, until it stopped with an error message: Failed to partition the selected disk with the detail This probably happened because are too many primary partitions in the partition table. How can I have a horizontally shaded frametitle in beamer? This question is similar to . I want to achieve the horizontal shading effect in the frametitle and when I use  \setbeamertemplate{frametitle}[horizontal shading] [left=red, right=white]   I get compilation errors:  LaTeX Error: Missing \begin{document}. [[]   Vertical shading seems to work just fine. Can we exempt downvoted accepted answers from getting the top spot? "many a time" means "many times"?