Let $[x]$ be the greatest integer function. How to find  $\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\left[\frac{2n}{k}\right]-2\left[\frac{n}{k}\right]\right)$?
Problem: Evaluate $$ L=\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left(  \left\lfloor \frac{2n}{k} \right\rfloor -2\left\lfloor \frac{n}{k}\right \rfloor \right).$$  Please help me with this one.  I tried using the sandwich theorem by first constructing the following  $$ 2 \geq \left\lfloor \frac{2n}{k} \right\rfloor -2\left\lfloor \frac{n}{k} \right\rfloor \geq -1 $$ and then summing over. But the limit of the two bounds are different so sandwich theorem doesn't help!. Is it possible to construct sharper bounds?  Also I observed that if we ignore question of continuity, etc. then we can write,  $L= \displaystyle \int ^{1}_{0}\left\lfloor \frac{2}{x} \right\rfloor -2\left\lfloor \frac{1}{x} \right\rfloor dx$  Now I am able to compute the above integral. But is the above process correct? If then how can make it rigorous? Or if any other approach is possible, please tell.
Can virtual particles be thought of as off-shell Fourier components of a field? I just found  blog post, which gives an interpretation of virtual particles I haven't seen before.  Consider a 1D system of springs and masses, where the springs are slightly nonlinear. A "real particle" is a regular $\cos(kx-\omega t)$ wavepacket moving through the line, where $\omega$ satisfies the dispersion relation $\omega = \omega(k)$. When two real particles collide, the region where they collide temporarily looks really weird, as they interact nonlinearly, pulling and pushing on each other.  Formally, we can write this weird region as the sum of a bunch of $\cos(kx-\omega t)$ waves, but there's no guarantee they'll have the right dispersion relation. Thus, each term in the resulting expansion is a off-shell "virtual particle". If you add up all the virtual particles, you get the actual intermediate field state, which is just a weird ripple in the field where the two particles are interacting.  As another example, consider the statement "a static EM field is made of virtual particles". Under this interpretation, what that really means is, "a static field (e.g. $1/r^2$) is not equal to $\cos(kx-\omega t)$, but may be expanded in terms of such sinusoids", which is much less mysterious sounding. In fact, this is exactly what we do when we consider scattering off a potential in normal QM, e.g. in the Born approximation.  The above gives some intuition for what a 'virtual particle' means in classical field theory. They are the Fourier components of the field with $(\omega, k)$ not satisfying the dispersion relation, and they are useful in classical perturbation theory. However, they are not propagating degrees of freedom, so they only appear during interactions.   The picture above is entirely classical. Does this picture generalize to quantum field theory, giving a physical intuition for virtual particles there? I'd like to create a Windows bootable USB stick, but I don't have a Windows machine with me at the moment to do so. How could I achieve the same using Ubuntu? I have a table that is very wide. Now I want to force it into one landscape page. I already set the page orientation to landscape but now I don't know how to force it into one page with scaling. Any hints? How does the referral system work? I don't understand how the referral program works even after reading everything about it. Do I only get the 500 CC if the referred person subscribes within 14 days, or if I just refer the person? And do I get 500 CC for every month that referred person subscribes, or is it just one time? Margin on child element moves parent element We're building a feature for a website and trying to think of a hypernym for clients, partners and members so that the feature remains generic enough for re-use later on other websites. Post hoc test after ANOVA with repeated measures using R Why was Count Dooku not given the title "Darth"? I don't know the exact significance of the word "Darth" in the Sith world, but I think it was a word of honor for powerful Sith. That's why Anakin was given this word in his new name.  All powerful Sith in canon had this word in their names: Darth Plagueis, Darth Bane, Darth Sidious, Darth Vader, Darth Maul, etc. As such name similarity can't be names given by their parents, I can assume they were given such words in names as a honor.   Now, look at Count Dooku... he was a powerful Sith too. So, why wasn't he given this title?    Darth word to address Dooku hasn't been used in the movies, as far as I know. If you are claiming that Darth and Lord words are same, give citations because my vocabulary rejects it. The habitual past "would" versus "simple past" tense One can use the habitual past 'would' to describe past events. For example one could say,      Jim would spend weekends at the farm. He would cut weed and water the plants.   One can also use the simple past.       Jim spent weekends at the farm. He cut weed and watered the plants.   Would the use of habitual past be considered "weak" writing or prose, as opposed to the simple past?  How does the simple past in the following sentence "He would cut weed and water the plants." without any preceding time expression e.g. "spend weekends"  make a distinction between a repeated action/habit and a one-time action of cutting weeds and watering plants? How to retrieve the content/file in JSP? I'm trying to prove that any group $G$ of infinite order has an infinite number of subgroups.  I think that if the group has an element of infinite order, then it's easy because I can take the groups generated by the powers of this element.  What if it doesn't? Every element generates a cyclic subgroup. Every element belongs to at least one cyclic subgroup (that generated by itself). So the group is the union of its cyclic subgroups. If all these are finite, we would have to have an infinite collection of subgroups anyway.  Is that correct? Is it possible to get rid of the bullets in the miniframes outer theme How do you programmatically set an attribute? Suppose I have a python object x and a string s, how do I set the attribute s on x? So:  &gt;&gt;&gt; x = SomeObject() &gt;&gt;&gt; attr = 'myAttr' &gt;&gt;&gt; # magic goes here &gt;&gt;&gt; x.myAttr 'magic'   What's the magic? The goal of this, incidentally, is to cache calls to x.__getattr__(). Correct use of "is" or "are" Possible Duplicate:          Would I use is or are in the following sentence?      The Pohutakawa Coast, Hauraki Gulf, and Waiheke Island is/are the backdrop for our day.   If I rewrote it so the subject (backdrop) starts the sentence, then it seems to me that as it is singular, the verb should be the singular is. But as it is written above, are seems less clumsy. shell: keep trailing newlines ('\n') in command substitution I want to be able to capture the exact output of a command substitution, including the trailing new line characters.  I realise that they are stripped by default, so some manipulation may be required to keep them, and I want to keep the original exit code.  For example, given a command with a variable number of trailing newlines and exit code:  f(){ for i in $(seq "$((RANDOM % 3))"); do echo; done; return $((RANDOM % 256));} export -f f   I want to run something like:  exact_output f   And have the output be:  Output: $'\n\n' Exit: 5   I'm interested in both bash and POSIX sh. Standard definition of graph says that it is an ordered pair G=(N,L) where N is set of nodes and L is set of lines which connect the nodes.   From what I've read, the set L can be empty, but can set N be empty too?  I'm mainly asking this because for my exam, I found numerous problems that go like this: Graph X is given. How many subgraphs with property Y exist? Often the number depends on the definition of graph. How to prove the set $S$ is closed under the binary operation $a*b=a+b+kab$? Let $k$ be any nonzero real number and let $S$ be all real numbers except ${-1\over k}$. Let $a,b \in S$. Define * on S by $a*b=a+b+kab$. Prove that $S$ is closed under *.  All I can think to do is set $a*b$ equal to ${-1\over k}$, but I don't know where to go from there. Is that the extent of the proof? Android app intermittently shows blank body in questions and answers (multiple sites) Note: the bug has returned in version 1.0.89 in a somewhat different form. See:      This possible bug report is similar to the bug in  that was deemed not-reproducible.  I can reproduce something similar with more recent app &amp; Android versions.   App Version: 1.0.34 (original report; is still present in 1.0.38, 1.0.42, 1.0.58) Android Version: 4.4 (up to 4.4.4) Device: Motorola Droid Maxx   In the examples below, you can see some blank answer bodies.  These are on different sites, and represent answers with both edited and non-edited text. In the first example, I clicked on the top answer tab to get to the answers, and in the second example I clicked on the bottom "View n Answers" button. (Trying to be a good little tester!)  Because I've seen this blank-answer-body intermittently over the past couple weeks (before the May 7 update), my process to test was just to go to the first site I frequent and then systematically click through the question list until I found a question with answers in which at least one answer was blank.  For all three sites shown below, it was within the first 5 or 7 questions sorted by newest.  Please note that I have seen many questions with several answers, none of which were blank. It is truly intermittent.  Example blank answer body at Workplace.SE:    Example blank answer body at Programmers.SE:    Example blank answer body at UX.SE:    And now here's the kicker, which makes me think this is some sort of content-loading latency issue rather than a rendering issue:  I returned to the left-side menu, selected the User Experience site, picked the same question as above, and then clicked the top answer tab to view answers again, and lo and behold the text is there:    I then went back to the other questions with blank answers, shown above, and the answer bodies were no longer blank.  I am happy to go through more tests/paths and do screenshots if it is helpful.  The same bug also appears in question bodies. How do I set up a supply line? The general consensus on a similar question,  is that "highly significant" is a valid, though non-specific, way to describe the strength of an association that has a p-value far below your pre-set significance threshold. However, what about describing p-values that are slightly above your threshold? I have seen some papers use terms like "somewhat significant", "nearly significant", "approaching significance", and so on. I find these terms to be a little wishy-washy, in some cases a borderline disingenuous way to pull a meaningful result out of a study with negative results. Are these terms acceptable to describe results that "just miss" your p-value cutoff?