Propagating uncertainties with trigonometric functions
Trigonometric Uncertainty Propagation
Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals I stumbled upon this "relation" (is the name correct?):  $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&amp;x\text{ is rational}\\ 0,&amp;x\text{ is irrational}\end{cases} $$  How is it called and why is it so? I'm really not asking for a proof since I fear it would be too complicated for me to understand, but rather for an "intuition". Take $\mathbb{R}$ as a vector space over $\mathbb{Q}$, then a basis for $\mathbb{R}$, then $|B|=2^{\aleph_0}$ I am trying to prove that Given $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and a basis for $\mathbb{R}$, then $|B|=2^{\aleph_0}$.  Proof: Suppse that $|B|&lt;2^{\aleph_0}$, then $\mathbb R = span\{ b_i ; b_i \in B \}$ So $ \mathbb R = \{ \Sigma_{i=1}^{\infty}\alpha_i b_i ; \alpha_i \in \mathbb Q, b_i \in B, i \in \mathbb N \} = \bigcup_{n=1}^{\infty}\{\Sigma_{i=1}^n \alpha_ib_i\}$. Which is a contradiction since a countable union of countable union of countable sets is countable and $\mathbb R$ is not.  What do you think?  Thank you, Shir I'm interested in the properties of zero-diagonal symmetric (or hermitian) matrices, also known as hollow symmetric (or hermitian) matrices.   The only thing I can come up with is that it cannot be positive definite (if it's not the zero matrix): The Cholesky decomposition provides for positive definite matrices $A$ the existence of a lower triangular $L$ with $A=LL^*$. However the diagonal of $LL^*$ is the inner product of each of the rows of $L$ with itself. Since the diagonal of $A$ consists of zeros, so $L$ (and thus $A$) must be the zero matrix.  The sorts of questions that interest me are:   which symmetric matrices can be transformed orthogonally into a zero-diagonal matrix? what can we say about the eigen-values of a zero-diagonal symmetric matrix?. any other interesting known properties?? Is it possible to raise stats? I have a firewall with these simple rules:  iptables -A INPUT -p tcp -s 127.0.0.1/32 --dport 6000 -j ACCEPT iptables -A INPUT -p tcp -s 192.168.16.20/32 --dport 6000 -j ACCEPT iptables -A INPUT -p tcp --dport 6000 -j REJECT   Now, suppose I am using TCPDUMP like this:  tcpdump port 6000   And I have host 192.168.16.21 trying to connect to port 6000.  Will/should tcpdump output some packets coming from 192.168.16.21? How can I group numerical data into naturally forming "brackets"? (e.g. income) The following describes what I'm trying to accomplish, but it's possible an alternative problem statement can describe my goal:  I want to    divide the following numbers into groups where the variances of the numbers within each group are not too large, and the differences between the averages of groups are not too small  compare the distribution obtained in the end with the "perfect" ones and see how "different" it is from being perfect.     Layman's explanation of goal  , and determine the "income brackets" each population is in.  The income bracket is supposed to be self-adjusting based on the input data.  My goal is to ultimately measure or calculate the difference between the income brackets.  I assume there will be many brackets, and want to see how far "apart" each tier is.  Here is a sample of hourly income for a sample set of a population of 20, and a total income of 3587:  Population= 10                   pop=2   population=5              population =3 10, 11,13,14,14,14,14,14,15,20,  40,50  ,90,91,92,93,94      999,999,900    How can I use mathematical concepts to group, sort and analyze data that acts like income distribution over a given population?  At the end of the calculation, I want to determine tiered income distribution, where a perfect distribution would look (something) like this   (each person makes $10 more per hour than the previous; total is 3587) 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279   or this:   (evenly distributed groups of people make the same per hour)  (gaps between income groups is consistent and not "too far") (income total is 3587) 99 99 99   129 129 129   159 159 159   199 199 199  229 229 229  269 269 269    Question  How should I analyze the population groups, and measure the gap in a way that will tell me how much is needed to make it more like the last two model sets listed above? I have recently been reading about Quantum Electrodynamics which I found very interesting, but even more confusing.  I understand photons mediate the electromagnetic force and interactions between charges can be described by virtual photon exchange. And that in Quantum Mechanics particles paths can be evaluated by sum over all paths...  An electron travelling from one point to another will take all paths, emitting and reabsorbing virtual photons. Does the electrons take all paths including those that involve faster than light velocities?  Is this question null, Does it mean anything to ask about properties of virtual particles?  Do I have the wrong idea about what virtual particles are? Submit Buttons on the Left or the Right? We have been puzzling over where to put the submit button, on the left or the right. In researching, we noticed that many sites put buttons on the bottom right in dialogue boxes, and on the bottom left in forms.  It makes sense: in a dialogue box it seems to denote finality, being in the endpoint of the window for left–right readers; in a form, the bottom right could be in a different position relative to the rest of the form if the window is resized.  It seems to be a convention, but I don't see anyone talking about this specific phenomenon in the following questions, nor the articles referenced therein:        So, is this level of button position optimization worth it if it creates an inconsistency in button placement? Is this a good way to go? I know of 2 basic kinds of default web icons for websites:   : place an .ico formatted file at the root of your site named favicon.ico and most desktop browsers will find it and display it in various contexts (bookmarks, tabs). Can also be added in code, which allows you to specify other names and other graphics types to the file, though there are :  &lt;link rel="shortcut icon" href="/favicon.ico" /&gt; &lt;link rel="icon" href="/favicon.ico" /&gt;  : place a png formatted file called either apple-touch-icon.png or apple-touch-icon-precomposed.png at the root of your site and iPhone OS/iOS versions of Safari will pick those up and store them when the site is added to the Home Screen. Can also be added with code like:  &lt;link rel="apple-touch-icon" href="/custom_icon.png" /&gt;    Are there any other default icon graphics for a website a webmaster can include? Why are so many different types of objects white, yet appear gray when they are wet? There are many things with different textures that appear white – salt, beer foam, paper, shaving cream, snow, talcum powder, white paint, etc. The most common answer is all of the frequencies must be reflected in order for us to perceive white, which I already know. There must be more underlining details why such a huge variety of things are white. Furthermore, if you wet paper, snow or talcum powder, they appear off-white or gray. Why is this? The Royal Order of Adverbs I know that the pattern manner-place-time shouldn't be taken too seriously if one wants to speak natural English.   In real life, people rarely use a string of adverbs. Speakers will easily break the pattern for emphatic or stylistic reasons (usually placing the adverb at the beginning of the sentence). Other factors such as length and specificity will also override the arrangement.  But I'm constructing illustrative sentences that highlight (and stick to) the Royal Order of Adverbs.   What all the online sources are doing is mixing adverbs and prepositional phrases, but I can't have the concept of prepositional phrases in my examples just yet. This means I should include only one-word or two-word adverbs (without prepositions).   So, I have these:    He sat quietly upstairs all day. She waited impatiently outside this morning. They gather noisily downtown every Saturday night.   Are the sentences above still correct, as in natural-sounding? Somehow they started sounding a bit off to me. I guess I just need to test them with a second pair of ears. Selenium + FF : How to fix error, Unable to connect to host 127.0.0.1 on port 7055 after 45000 ms? How to work around Eclipse Luna/GTK bug 339174 (change to not use oxygen) The bug is described here:  For me it fails with:  java: /build/buildd/oxygen-gtk3-1.0.2/src/animations/oxygencomboboxdata.cpp:87: void Oxygen::ComboBoxData::setButton(GtkWidget*): Assertion `!_button._widget' failed.   Problem is if I go to GTK+ Apperance in Control Module and change to anything other than oxygen Eclipse will look totally awful.  Regardless what theme I use (tried QtCurve and Raleigh) it has the same look with huge padding around all controls etc.  Is it possible to have a sensible look without oxygen GTK?  I have kubuntu-desktop installed over the stock LTS distro.   $dpkg -l libgtk* | grep -e '^i' | grep -e 'libgtk-*[0-9]'     ii  libgtk-3-0                             3.4.2-0ubuntu0.8                          GTK+ graphical user interface library     ii  libgtk-3-bin                           3.4.2-0ubuntu0.8                          programs for the GTK+ graphical user interface library     ii  libgtk-3-common                        3.4.2-0ubuntu0.8                          common files for the GTK+ graphical user interface library     ii  libgtk2.0-0                            2.24.10-0ubuntu6.1                        GTK+ graphical user interface library     ii  libgtk2.0-bin                          2.24.10-0ubuntu6.1                        programs for the GTK+ graphical user interface library     ii  libgtk2.0-common                       2.24.10-0ubuntu6.1                        common files for the GTK+ graphical user interface library How can I efficiently check the originality of my work? Install Microsoft Office Pro 2010 on Ubuntu 13.04 Find records from one table which don't exist in another I've got the following two tables (in MySQL): Phone_book +----+------+--------------+ | id | name | phone_number | +----+------+--------------+ | 1  | John | 111111111111 | +----+------+--------------+ | 2  | Jane | 222222222222 | +----+------+--------------+  Call +----+------+--------------+ | id | date | phone_number | +----+------+--------------+ | 1  | 0945 | 111111111111 | +----+------+--------------+ | 2  | 0950 | 222222222222 | +----+------+--------------+ | 3  | 1045 | 333333333333 | +----+------+--------------+  How do I find out which calls were made by people whose phone_number is not in the Phone_book? The desired output would be: Call +----+------+--------------+ | id | date | phone_number | +----+------+--------------+ | 3  | 1045 | 333333333333 | +----+------+--------------+ How do I unset an element in an array in javascript? I have browsed several ASIC manufacturer's webs, but I haven't found an actual number. I assume there would be a fixed cost associated with creating masks and such and then there will be a cost per unit.     Note: that I don't actually want to have an ASIC made, I'm just curious. Quotient of a local regular ring How can I prove this:     Let $A$ be a local regular ring with maximal ideal $\mathfrak m$ and $x \in \mathfrak m-\mathfrak m^2$. Then $A/(x)$ is a regular ring. Prove also that if $x\in\mathfrak m^2$, $x\ne 0$, the result does not hold anymore.   I don't know how to begin! Thanks to everyone who helps!  But if $d=\infty$? Java Command line arguments