Convergence of a sequence of integrals defined recursively
Uniformly convergence of $f_{n+1}(x)= \int^x_0 f_n(t)dt$
Examples of PCA where PCs with low variance are "useful" Cycles Normal Bake does'nt bake indents that are not tapered By conservation of energy, the solid is left in a lower energy state following emission of a photon.  Clearly absorption and emission balance at thermal equilibrium, however, thermodynamic equilibrium is a statement of the mean behaviour of the system, not a statement that the internal energy is constant on arbitrarily short timescales.  The energy has to come from somewhere during emission, and go somewhere during absorption.  Energy in a solid can be stored as kinetic and potential energy of electrons and nuclei, either individually or in collective modes such as phonons and plasmons.  In thermal equilibrium energy will be stored more or less in various forms depending on the temperature and material.  However, even if most of the thermal energy in a particular solid at temperature $T$ is stored in the form of phonons, it could be that phonons primarily interact with light indirectly via electrons, e.g. a phonon excites an electron in a phonon-electron interaction, which can interact with light via the EM field.  Given that light is an EM field, it makes sense to me that it is emitted and absorbed by charged particles.  The electron-photon interaction is probably dominant for visible and ultraviolet light, given that metals are opaque, while semiconductors and insulators are transparent to (visible and UV) light with energy lower than their bandgap. However, once you get into energies in the IR and below, or X-rays and above, other mechanisms apparently take over.  For example, on the high-energy end of the spectrum I've heard that gamma rays can interact directly with nuclear degrees of freedom, which is reasonable considering that gamma rays are emitted during a lot of nuclear reactions.  A review of  might give clues to important light-matter interactions over a broad range of wavelengths.  Whether all of the these processes are involved in blackbody emission is a somewhat different question.  What physical processes mediate energy transfer during blackbody emission, and in which energy ranges are the various processes dominant? Prove or give a counterexample:  $$(A ∪ B = A ∪ C) \land (A ∩ B = A ∩ C) \implies B = C.$$  I think this is true, but I am not sure how to show it. I don't know if there are any manipulations with unions and intersections that allows me to move things around. Thanks in advance for any hints. Very thin white line in colortbl Whenever I use the colortbl package to format tables, a thin white space between the cells sometimes shows up on screen (not in print for as far as I can tell). For example in this table:    Is there an easy way to enlarge each rectangle by a tiny amount so that the white space is not visible anymore?  (My guess is that the white space appears due to rounding in my pdf viewer, evince. I don't know if other pdf viewers have the same issue. Regardless, I would like to have my pdf file look good in common pdf readers) How to count number of words from String using shell I have something like this:  &lt;div class="content"&gt;     &lt;a href="#"&gt;A&lt;/a&gt; &lt;/div&gt; &lt;div class="content"&gt;     &lt;a href="#"&gt;B&lt;/a&gt; &lt;/div&gt; &lt;div class="content"&gt;     &lt;a href="#"&gt;C&lt;/a&gt; &lt;/div&gt;   When one of these links is clicked, I want to perform the .hide() function on the links that are not clicked. I understand jQuery has the :not selector, but I can't figure out how to use it in this case because it is necessary that I select the links using $(".content a")  I want to do something like  $(".content a").click(function() {     $(".content a:not(this)").hide("slow"); });   but I can't figure out how to use the :not selector properly in this case. In the course of creating the content for a site, we've ended up with orphaned content nodes that are not linked to from anywhere. I'd like to find and examine these. Surprisingly, I can't find a module for this.  Tiny hypothetical example: I'm building a brochureware site. Someone wrote a "Contact" page, someone else wrote an "About Us", and someone else wrote "Directions". The person writing "Directions" forgot to put it in a menu link or add a link from "About Us" to "Directions". So "Directions" is orphaned: there are no links to it. I'd like to find all such nodes so I can figure out what to do with them. Is it possible to make an underfull tabulary spread out? With this MWE  \documentclass{article} \usepackage{tabulary}  \begin{document}  \hrule  \begin{center}   \begin{tabulary}{\textwidth}{|L|L|L|}     foo&amp;bar&amp;baz   \end{tabulary} \end{center}  \hrule  \end{document}   I get a table which is narrower than the page:   But I'd like the table to be spread out to \textwidth like tabularx would do. Is this possible?  Or is there an alternative approach? If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$ Did not find this from this website...  If $$ \gcd(a,b)=1,$$ then there exists integers $x$ and $y$ such that $$xa+yb=1.$$  Now, the tip is to use particular corollary, that states:  The class $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}\Leftrightarrow \gcd(m,n)=1.$  I am totally lost with the corollary.  Let's assume that $\gcd(m,n)=1$. Then, $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}$.  OK! Then what?  There is also follow up, where I have to prove the converse. I am familiar with Bezout's lemma. I'm unable to uninstall MATLAB;  sandeep@sandeep-VPCEH28FN:~$ sudo rm /var/tmp/lm_TMW* [sudo] password for sandeep:  rm: cannot remove '/var/tmp/lm_TMW*': No such file or directory sandeep@sandeep-VPCEH28FN:~$ sudo apt-get purge matlab E: Could not get lock /var/lib/dpkg/lock - open (11: Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/), is another process using it? sandeep@sandeep-VPCEH28FN:~$ sudo apt-get purge matlab-support E: Could not get lock /var/lib/dpkg/lock - open (11: Resource temporarily unavailable) E: Unable to lock the administration directory (/var/lib/dpkg/), is another process using it? sandeep@sandeep-VPCEH28FN:~$ Change the default size of square brackets in latex math mode Personally I think the default size of square brackets in latex math mode is too small. For example, for an interval [a,b], I would prefer \big[\big] as opposed to plain []. My question is, what can I do to change the default size of [], so that when I input [] in math mode, they will show up automatically the same size as \big[\big]?  Right now I'm defining a new command called \b{} which takes one argument and inserts it between \big[ and \big], but the code looks ugly. Apparently \newcommand (or \def or \let) does not allow [ to be refined.  Any help is much appreciated. Thanks. The AC adapter supplied with my laptop is designed to accept voltage in the range of 100-240VAC at 50-60Hz, but the AC cable that plugs into the adapter unit itself is only rated to 125VAC 7.5A 2.5A.  I know the adapter can accept 220V power, but is it safe to use this cable with the AC adapter to plug into a 220V socket?  Edit: The part in question is a 90W HP Smart AC Adapter.  The laptop does not cause the AC adapter to draw more than about 80 watts from the socket (as measured with a Kill A Watt meter).  Edit 2: The amperage rating of the cable as marked on the attached label is 2.5A, not 7.5A. Uniform convergence of $\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$ Is $$\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$$ uniformly convergent on $(0,+\infty)$? Sadly, Weierstrass' criterion nor Cauchy's condition don't seem to work here and I just ran out of ideas. I have a couple of illustrations done in Illustrator and I am planing to animate it for a website that I am working on, I've heard of  toolkit with flash, but is it the best way to go or is there another "better" way of doing it?  Here's an example of the kind of animations I am aiming for: Using Virtual Box, how can I install an OS to a secondary, physical disk, and boot it in both (at separate times) Virtual Box, and as a typical secondary OS install? How to normalize my db schema for a quiz application? I am putting together a quiz application.   (I'm calling each quiz a "challenge")  I have a "challenge" table and a "challenge_question" table which is pretty straightforward.  challenge table: id name num_questions (I want to randomize and only display a certain number of Qs per each challenge)  challenge_question table: id challenge_id question (text of the question) required (some questions will be required to be displayed per each challenge) question_type (here I indicate what type of question: Mult. choice, T/F, Ordering, or Fill-in-the-blank)   Where I'm having trouble in normalization is with the types of question options:  Multiple choice, true/false, ordering, and fill-in-the-blank. I wonder, should I have a separate table for each type of challenge-question-option? Or is it possible to combine these into one somehow..  If I combined all possible options into a single table, it might look be something like this  challenge_options table: id challenge_question_id option (text field, can be used for both multiple choice and for fill-in-the-blank style questions) is_true (boolean field for true/false questions) is_correct (boolean, used to determine multiple choice answer) ordering_option_1  ordering_option_2 ordering_option_3 ordering_option_4 (these ordering fields would store the correct sequence of an ordering style of question)   This really strikes me as less than ideal.  Actually, I don't see how the ordering_options would work out correctly at all.  Wouldn't I need to have each of those be a separate option?  Surely there must be a better way.  Any suggestions/criticisms or alternate approaches? What was the inspiration for the design of R2-D2? Looking at R2-D2, I can't think of any robots I've seen in science fiction that vaguely resemble R2D2's design.  What was the inspiration for the design of R2-D2? Virtual Box Won't Accept .ISO To Install OS For many of my datasets I have the same imported data duplicated numerous times, each iteration displaying the data values for a different column (in my case it's fish catch weight per gridded cell per year, so lots of iterations of the same grid, with a different pattern of values per year).  For presentation purposes, it strikes me as helpful to use the same scale for each year, so I go to the original data and divide the maximum value into (arbitrarily) 10 classes and manually input the values and labels after setting the desired colour ramp and input data field:   This is extremely tedious, not just the manual excel calculation part, but also the input interface for the values &amp; labels*, hence my intention to make use of the 'save style' option. HOWEVER: when I save the style as QML and load it for the next layer, it saves everything - including the data field (in my case, year). When I change this field, QGis automatically generates the values &amp; labels based on the data for that field, ditching all of the values &amp; labels I set.   Is there any way to avoid this happening &amp; thereby duplicate class number, values and labels for each field by simply loading a style? (somewhat meta) Am I going around this the wrong way, OR, is there a different face I could use to climb this mountain?