I am doing research on Faraday cages for school, and I want to know how it works. Faraday cages can have holes in them, and if the diameter is smaller than the wavelength of waves you want to block, the Faraday cage blocks the waves. I have found a formula that the transmission of electromagnetic waves through a hole with diameter $d (&lt; λ)$ equals $(d/λ)^4$. However, I cannot find anywhere why. It does not make sense in my head, for me it seems like a truck cannot be too long to go through a tunnel, but why is this not the same for waves? My guess is that it has to do with quantum mechanics, can someone please explain to me how this works? Thanks!
This is a follow-up to this question:   Hans Bethe wrote a paper in 1944, "Theory of Diffraction by Small Holes," Phys. Rev. 66, 163. I don't have access to the paper, but from descriptions online it sounds like he proved the following. Suppose a plane wave impinges on an absorbing sheet, and there is a hole in the sheet of diameter $d$, with $d&lt;\lambda$. Let $P_0$ be the power incident on the hole, and $P$ the power diffracted through the hole. Then the transmission is $T=P/P_0=(d/\lambda)^4$.  As a practical application, I think this explains why microwaves don't leak strongly through the metal grille in the front of a microwave oven.  Questions:   Does the thickness of the sheet matter? From references online, it sounds like the hole is treated as a waveguide...? This seems to relate to cutoff frequencies of waveguides, etc.? It seems to me that Huygens' principle would give $T=1$ for $d\ll\lambda$, since in this limit the wavelets are in phase. Why is Huygens' principle invalid here? Is this related to question #1? Is there a simple argument for the proportionality to $(d/\lambda)^4$? Or if not, how does one prove this using the gory details of Bessel functions, etc.?
What is the plural form of "Software"? Is it correct to use softwares, as in the following sentence?  There are various video conversion softwares in the market. What is the difference between "INNER JOIN" and "OUTER JOIN"? Also how do LEFT JOIN, RIGHT JOIN and FULL JOIN fit in? Apex compile error : variable does not exist I'm having issues with creating a pretty simple test Utility Class. I want to create two records: an account and a related child contact. The error I see in the developer console is:     Variable does not exist: a.ID   My code, based on the Salesforce Apex Workbook, is as follows. I've played around with various things, but it's not working. Where am I going wrong? Thanks in advance.  @isTest public class AccountSaveTestSuite {      public static Account createOneAccount(){           Account testAccount = createAcct('ABC Computing inc.');              Contact testContact = createContact();          return testAccount;             }      // Helper methods //     //        private static Account createAcct(string accountName) {          Account a = new Account(             Name=accountName);         insert a;         return a;     }      private static Contact createContact(){         Contact c = new Contact (             FirstName = 'Jane'              , LastName = 'Smith'             , Account = a.ID); // *** Error is here ***         insert c;         return c;      }  } String concatenation: concat() vs "+" operator Computing irreducible components of algebraic set How do I get from   $$B(\alpha,\beta)=\int_{0}^{1}{ t^{ \alpha -1}  (1-t)^{ \beta -1} \,dt}$$ to:   $$2^{2- \alpha - \beta } \int_{0}^{ \infty}{ \frac{ \cosh (( \alpha - \beta )x)}{( \cosh(x))^{ \alpha + \beta }} \,dx}$$  ? I have tried different change of variables such as $t=1- \frac{1}{ \cosh (x)}$ and got $\int_{0}^{ \infty}{ \frac{ \sinh (x)  ( \cosh (x)-1)^{ \alpha -1} }{( \cosh(x))^{ \alpha + \beta }} \,dx}$ but without success. I am writing an article in English, with some pages in Russian.  \documentclass[12pt]{article}  \usepackage[T1,T2A]{fontenc}  \usepackage[utf8]{inputenc}  \usepackage[russian,english]{babel}  \usepackage{newtxmath} \usepackage{newtxtext} \begin{document}  hello \textbf{hello}  \selectlanguage{russian} привет \textbf{привет}  \end{document}   Output:  hello hello привет привет  I am using \selectlanguage{} to switch between languages when needed.  However, I noticed that when I try to use \textbf{} when in Russian language mode, it does not work because of newtxtext, which I am using for math formulas.  Does anyone know how to solve this problem?  Thank you. How can i set different wallpapers for Ubuntu 15.04 login and Desktop This has been bothering me very much. i have memory that Ubuntu 12.04 had this configuration? am i wrong? finally, is it possible for the login screen to have different wallpaper from the desktop? is this possible in Ubuntu 15.04? i would appreciate a stable understandable response. thanks :)  -ben How to gracefully handle files that exceed PHP's `post_max_size`? The Cartesian product of a family $(A_i\mid i\in I)$ is defined as $$\prod\limits_{i\in I}A_i=\{f:I\to\bigcup A_i\mid f(i)\in A_i\}$$ Let $I_n=\{i \in \mathbb N \mid i \le n\}$ for $n\in \Bbb N$ and $A_i$ be countable for all $i\in I_n$. Then $\prod\limits_{i\in I_n}A_i$ is countable.   My attempt: Lemma 1: $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ and $\prod\limits_{i\in I_{n+1}}A_i$ are equinumerous for all $n \in \mathbb N$. (I presented a proof ) Lemma 2: If $A$ and $B$ are countable, then $A \times B$ is countable. (I presented a proof ) I will prove the theorem by induction on $n$. The theorem is trivially true for $n=0$. Assume that the theorem is true for $n=k$, then $\prod\limits_{i\in I_k}A_i$ is countable. Since $\prod\limits_{i\in I_k}A_i$ is countable (by inductive hypothesis) and $A_{k+1}$ is countable, then by Lemma 2 $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ is countable. Furthermore, by Lemma 1 $(\prod\limits_{i\in I_k}A_i) \times A_{k+1}$ and $\prod\limits_{i\in I_{k+1}}A_i$ are equinumerous, then $\prod\limits_{i\in I_{k+1}}A_i$ is countable too. Thus the theorem is true for $n=k+1$. This completes the proof.   Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help! Wireframe iOS design apps for iPad? Is there an app for iPad to assist in wireframe design for iPad and iPhone apps?  I don't mean actual development, just placing standard widgets and the things you usually do with photoshop and associated kits. We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit. Experience using dropdowns in-line of sentence "mad lib" style Do you have experience implementing text inputs or drop-downs in-line with other text or forms? Is this approach any better than just creating a separate field/label?  In the example below the user has the option to indicate when to be reminded again (in one, two, or three days). When they are reminded of an assignment, they simply get an email or text message. I am trying to automate these steps so that I don't need to do this on every machines manually. I need to install latest app server software (abc.tar.gz) on all the unix boxes.  I need to do this in around 12 machines - "machine1" ... "machine12". I have a master machine "machine0" which has abc.tar.gz file so I was thinking to run my script from this machine only and install abc.tar.gz software by following below steps in all those 12 machines one by one. My unix account id is david and my process runs as golden user.  This is the steps I follow if I am installing abc.tar.gz in "machine1":  david@machine1:~$ sudo scp david@machine0:/home/david/abc.tar.gz . david@machine1:~$ sudo stop test_server david@machine1:~$ sudo su - golden golden@machine1:~$ cd /opt/process golden@machine1:/opt/process$ rm -rf * golden@machine1:/opt/process$ exit david@machine1:~$ sudo cp abc.tar.gz /opt/process david@machine1:~$ sudo chown -R golden /opt/process david@machine1:~$ sudo su - golden golden@machine1:~$ cd /opt/process golden@machine1:/opt/process$ tar -xvzf abc.tar.gz golden@machine1:/opt/process$ exit david@machine1:~$ sudo start test_server   How can I automate this so that it can do the same steps in all 12 machines one bye one? I mean install abc.tar.gz in machine1 first, then install on machine2 and go on..  I want to run this script from machine0 only.  Is there any way to automate this? Which preposition in front of "line" — "on", "in", "at"? What is a word that could define someone who likes to cause conflict? What would you call someone who does things knowing specifically that his/her actions will cause pain and/or conflict or completes an action just to get someone in trouble or hurt them?   For example, in  there are two people that are in love, but the Colonel goes out of his way to tell the father of the girl that the boy is a thief and would be bad for their reputation.   What is a word that would describe him? Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$? Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H: hg \not \in H$?  I'm new to abstract algebra and this would appear to be a useful result. However, by definition, a subgroup is nearly closed, and I don't think this directly implies that for any element $g$ from $G$ not in $H$, the composition $gh \not \in H$.  Is this in fact a true statement? I'd like to be able to introspect a C++ class for its name, contents (i.e. members and their types) etc. I'm talking native C++ here, not managed C++, which has reflection. I realise C++ supplies some limited information using RTTI. Which additional libraries (or other techniques) could supply this information? It is discouraged to simply catch System.Exception. Instead, only the &quot;known&quot; exceptions should be caught. Now, this sometimes leads to unnecessary repetitive code, for example: try {     WebId = new Guid(queryString[&quot;web&quot;]); } catch (FormatException) {     WebId = Guid.Empty; } catch (OverflowException) {     WebId = Guid.Empty; }  I wonder: Is there a way to catch both exceptions and only call the WebId = Guid.Empty call once? The given example is rather simple, as it's only a . But imagine code where you modify an object multiple times, and if one of the manipulations fails expectedly, you want to &quot;reset&quot; the object. However, if there is an unexpected exception, I still want to throw that higher. How to convert a factor to integer\numeric without loss of information?