A fun kind of question...   In 2008, researchers at the UCLA's Department of Physics and Astronomy found that unrolling of scotch tape in a vacuum caused X-rays to be created.  First, energetic electrons were produced and then X-rays.   The study stated that 300,000 x-ray photons were produced and that they were bright enough to image a finger...   As Dr. Putterman stated "kind of a microscopic lighting effect"  So why does this happen???
A couple years ago I ran upon a  video demonstrating how researchers used x-rays given off by tearing tape off its spindle in hopes to miniaturized and cheapen future x-ray devices. As of late I have been wondering what interactions are causing this. I understood from the video that the electrons are being ejected due to the added energy to the system (peeling). This instinctively pointed me towards QED where interactions between electrons produce a photon for short periods. 1)Are those same photons from electron interaction the resulting x-rays? 2)Are the frequencies of the photons being limited by the $\Delta$energy$\Delta$time uncertainty, and/or $\Delta$position$\Delta$momentum uncertainty (at the vertex of the ripping tape, where high densities of electrons may accumulate) ? Note: I've also considered that the van der Waals force may come into consideration, not certain though. Looking for a QED based descriptions/answers. Any elementary references to this work is appreciated.
For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous. This is from Apostol's Mathematical Analysis.  Let $f$ be a function defined in $[0, 1]$ such that for each real number $y$ either there is no value of $x \in [0, 1]$ such that $f(x) = y$ or there are exactly two such values of $x$. Prove that a) $f$ must be discontinuous on $[0, 1]$. b) $f$ has an infinite number of discontinuities in $[0, 1]$.  Long back I had tried to solve the problem. I reasoned for a) as follows: Suppose that $f$ is continuous and let $m, M$ be its minimum and maximum values on $[0, 1]$. Clearly if $m = M$ then $f$ is constant and clearly for every $x \in [0, 1]$ we have $f(x) = m = M$ which contradicts given property of $f$. So we must have $m &lt; M$. Clearly there are now four distinct points $a, b, c, d$ in $[0, 1]$ such that $f(a) = f(b) = m, f(c) = f(d) = M$. By checking possible linear orderings of $a, b, c, d$ I could find a find a value of $y \in (m, M)$ for which there exist more than 2 values of $x$ such that $f(x) = y$. However this approach looks very clumsy and I suspect much simpler and easier proof is possible. Part b) seemed bit complex and my only guess was that probably $f$ was discontinuous on each subinterval of $[0, 1]$ but I could not prove this claim (may be the claim is wrong!). Perhaps there is a simpler solution to this part of the problem. Apostol's also asks to provide an example of such a function. Any hints on the overall problem will be appreciated.  Update: After looking at Marc's answer, I think I have found a way to show that there are infinitely many discontinuities. Earlier the solution was a part of the question itself but it made the question look quite lengthy hence I moved it to a separate answer. Is the cube root of a prime number rational? Format timedelta to string Is it considered normal to publish job offers inviting candidates to apply based on their gender and/or race in academia? I was browsing PhD position offers in Europe on different websites and I was quite surprised to find that several offers contained sentences such as     University of X value equality and diversity. We strongly encourage women and BAME applicants for this position.    where BAME stands for Black, Asian and Minority Ethnicities; or     The research group aims to increase the number of women in scientific positions. Female candidates are therefore encouraged to apply.   Is it considered normal in the academic environment to discriminate between  candidates according to their race and/or gender for "egalitarian" reasons?   How is the choice of a candidate based on their gender and/or race considered "egalitarian"? Is there an equivalent of MATLAB's Workspace window in Mathematica? Android does not download any files from untrusted CA certificate site What is going on in front of and behind a fan? Why is it that when you drop paper behind a fan, it drops, and is not blown/sucked into the fan, whereas if you drop paper in front of a fan, it is blown away? Getting a random object from NSArray without duplication Continuous function does not map closed set to closed set Can I run two 4k (3840x2160) displays at 60hz on a Macbook Pro 13" 2015 I think the title says it all. Can I run two 4k (3840x2160) displays, both at 60hz on a Macbook Pro 13' 2015? I understand they might need to be SST and each connected to a Thunderbolt 2 port. But I cannot get a confirmation whether that setup would work.    says:     Simultaneously supports full native resolution on the built-in display and up to 3840 by 2160 pixels on up to two external displays, both at millions of colors.   …   Support for up to 5120-by-2880 resolution at 60Hz on a single external display (model with AMD Radeon R9 M370X only)   Unfortunately, it does not state at what frequency the two displays can be run.    says:     With OS X Yosemite v10.10.3 and later, most single-stream 4K (3840x2160) > displays are supported at 60Hz operation on the following Mac computers: … MacBook Pro (Retina, 13-inch, Early 2015) and later. MacBook Pro (Retina, 15-inch, Mid 2014) and later   Unfortunately it does not state whether two displays can be run at 3840 by 2160 at 60hz I'm trying to solve the following problem:  Given two CRHF $H_1:2^{4n}\to2^{2n}$, $H_2:2^{2n}\to 2^n$, construct the following hash function $H^{*}=H_2(H_1(x))$ compressing from $2^{4n}\to 2^n$. We want to demonstrate that if $H_1$ and $H_2$ are collision resistant then $H^*$ must be collision resistant too.  I made the following reduction  However I'm still unsure on how to calculate ”BAD event” in which $A^{H^∗}$ outputs a collision for $H_{s_1}$, in this case $x^∗=x′$ and the second part of the reduction doesn’t work. Honestly, navigation through Area 51 is a bit confusing and seems to be much different from the other SE sites.    The feature that is very lacking is the footer with all the sites you are subscribed to so that a user can easily return to a different SE site. This one really bugs me.  Can this be easily fixed?  Also, unlike all the other sites there is no "meta" site to add comments specifically about that site.    Again Area 51 seems out of sync with the other SE sites. When is the past perfect exactly needed? Let $X \ge 0$.  Show  \begin{align} \lim_{y\to\infty} y\,\Bbb{E}\left[ \frac1X \cdot1_{X&gt;y} \right] &amp;= 0 \quad\text{and} \\ \lim_{y \downarrow 0} y\,\Bbb{E}\left[ \frac1X \cdot1_{X&gt;y} \right] &amp;= 0. \end{align}  N.B.: The author doesn't assume $\Bbb{E}[1/X] &lt; \infty$.    My try:  Since the expectation appears on the RHS of Markov's inequality, I don't think this inequality can be applied to solve this question.  $$ \begin{aligned} y\,\Bbb{E}\left[ \frac1X \cdot1_{X&gt;y} \right] &amp;= \Bbb{E}\left[ \frac{y}{X} \cdot1_{X&gt;y} \right] \\ &amp;&lt; \Bbb{E}\left[ 1 \cdot1_{X&gt;y} \right] \\ &amp;= \Bbb{P}(X &gt; y) \\ &amp;\xrightarrow[y\to0]{} \Bbb{P}(X \ge 0) = 1 \end{aligned} \tag1 \label1 $$  For the first part $y \to \infty$, I apply Markov's inequality at the row $\Bbb{P}(X &gt; y)$ in \eqref{1}, so that $$\Bbb{P}(X &gt; y) \le \frac{\Bbb{E}[X]}{y} \xrightarrow[y\to\infty]{} 0?$$ But I don't have $\Bbb{E}[X] &lt; \infty$.  Thanks for any help. When I upgrade from one version of Ubuntu to the next, the upgrade tool rightly disables all my PPAs, so that software compiled for or otherwise tied to the old version of Ubuntu does not interfere with the new version that I just upgraded to.  Is there any way to automatically re-enable and update any PPAs that provide packages for the new version of Ubuntu? For example, suppose I'm currently running Maverick, and I have two PPAs enabled, "foo" and "bar". Suppose that "foo" provides packages for both Maverick and Natty, while "bar" provides packages for Natty only. Is there a tool I could run after updating to Natty that would re-enable my "foo" PPA and update it to point to the Natty packages, while leaving the "bar" PPA disabled? I get this error in the deployment step of a share-point farm based project. Most of the suggested solutions on the net are a simple Visual Studio re-start, however, a simple VS restart won't fix this.   Any ideas on what might be causing this? What happens when yellow light is passed through a prism? What happens when secondary colors like yellow,green and magenta are passed through a prism? Will it split into its component colors(for example yellow split into green and red)?If yes why we are observing secondary colors in the visible spectrum obtained when white light(composite light) is passed through a prism? Me and some friends were recently disuccing "How cool it would be to be invisible", from a military viewpoint, I said that Camouflage, when done correctly, works surprisingly well, so invisibility was unnecessary. They countered with the argument that an infrared camera could see you easily. Apart from several warm coats, are there any non-restictive ways of preventing IR radiation being emitted from your person? I try to download a file with wget and curl and it is rejected with a 403 error (forbidden).  I can view the file using the web browser on the same machine.  I try again with my browser's user agent, obtained by . I do this:  wget -U 'Mozilla/5.0 (X11; Linux x86_64; rv:30.0) Gecko/20100101 Firefox/30.0' http://...   and  curl -A 'Mozilla/5.0 (X11; Linux x86_64; rv:30.0) Gecko/20100101 Firefox/30.0' http://...   but it is still forbidden. What other reasons might there be for the 403, and what ways can I alter the wget and curl commands to overcome them?  (this is not about being able to get the file - I know I can just save it from my browser; it's about understanding why the command-line tools work differently)  update  Thanks to all the excellent answers given to this question. The specific problem I had encountered was that the server was checking the referrer. By adding this to the command-line I could get the file using curl and wget.  The server that checked the referrer bounced through a 302 to another location that performed no checks at all, so a curl or wget of that site worked cleanly.  If anyone is interested, this came about because I was reading  page to learn about embedded CSS and was trying to look at the site's css for an example. The actual URL I was getting trouble with was  and the curl I ended up with is  curl -L -H 'Referer: http://css-tricks.com/forums/topic/font-face-in-base64-is-cross-browser-compatible/' http://cloud.typography.com/610186/691184/css/fonts.css   and the wget is   wget --referer='http://css-tricks.com/forums/topic/font-face-in-base64-is-cross-browser-compatible/' http://cloud.typography.com/610186/691184/css/fonts.css   Very interesting. Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$