Don’t break words in the title
On sites with new responsive theme, word wrap on long titles breaks in the middle of a word
This actually triggered me in my mind from . After some playing around I notice that the relation $a^{\log_b x} = x^{\log_b a}$ is true for any valid value of $a,b$ and $x$. I am very inquisitive to see how this holds ? How do I prevent mouse movement from waking up a suspended computer? How to filter a dictionary according to an arbitrary condition function? Big floor symbols When I write  \lfloor\dfrac{1}{2}\rfloor   the floors come out too short to cover the fraction. How can I lengthen the floor symbols? Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$  How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).  Any guidelines please? In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, "because the positive energy of matter is cancelled out by the negative energy of the gravitational field".  But I can't find anything concrete to substantiate this claim. As a first check, I did a calculation to compute the gravitational potential energy of a sphere of uniform density of radius $R$ using Newton's Laws and threw in $E=mc^2$ for energy of the sphere, and it was by no means obvious that the answer is zero!  So, my questions:   What is the basis for the claim &ndash; does one require General Relativity, or can one get it from Newtonian gravity? What conditions do you require in the model, in order for this to work? Could someone please refer me to a good paper about this? How do i convert a binary int into a string? Given a Java InputStream, how can I determine the current offset in the stream? I'd like something like a generic, re-usable getPosition() method that will tell me the number of bytes read from the starting point of the stream.  Ideally, I would prefer this to work with all InputStreams, so that I don't have to wrap each and every one of them as I get them from disparate sources.  Does such a beast exist?  If not, can anyone recommend an existing implementation of a counting InputStream? Post apocalyptic journey in search of knowledge from fallen society (ours) I'm racking my memory for a post-apocalyptic novel that I borrowed from a library maybe 15 years ago, but I'm sure it was published a few years before that though (maybe early 90s?).  I remember a party of people travelling across the flooded USA after some kind of apocalypse (climate change/flooding?). Towards the end of the book the group makes their way to New York(?) and finds a partially flooded library, discovering a set of books that gives hope for the return/rebuilding of society. Are you less likely to down-vote someone whose gravatar is a photograph of them? I know this is hard to answer since, to the extent that anyone does this, they likely don't do it consciously. In the spirit of Richard Wiseman and  I just wondered what were people's thoughts on it.  Please bear in mind that that's me in the picture, and if you down-vote this, that'll be a real person hurtin', not some abstract symbol. If $I$ is a non-zero proper ideal of a commutative integral domain, is it possible for $I$ to be its own square? Why is the application of probability in  (QM) fundamentally different from its application in other areas? QM applies probability according to the same  as in other areas of physics, engineering, etc.  Why is there a difference?  Naively one would assume one of these possibilities:   It is not the same probability (theory?) It is a matter of interpretation (of the formalism?) Something else?   Many answers (which I am still studying) focus on the fact that the combined probability of two mutually-exclusive events in QM is not equal to the sum of the probabilities of each event (which holds classically by definition). This fact (appears to) makes the formulation of another probability (a quantum one) a necessity.   Yet this again breaks down to assumed independent, if this is not so, the "classic probability" is applicable (as indeed in other areas). This is one of the main points of the question. Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics.  I need an clear argument that   involves the equations of motion for just the potentials $A^0$ and $\mathbf{A}$, and not the electric and magnetic fields.   includes sources $\rho$ and $\mathbf{J}$ in the equations of motion.  This is to justify the assertion that there are degrees of freedom that decouple from the rest of system. does not critically rely on quantum field theoretic arguments (although any supplementary remarks are welcome).     To illustrate the level of clarity I expect, I provide an argument in the Coulomb gauge $\nabla\cdot \mathbf{A} = 0$:  Of the four field degrees of freedom, the gauge condition $\nabla\cdot \mathbf{A} = 0$ removes one degree of freedom (the longitudinally polarized EM waves).  To show that among the remaining three degrees of freedom only two are propagating, consider the field equations of motion in the Coulomb gauge:  \begin{align} \nabla^2 A^0 &amp;= -\rho/\epsilon_0,\\ \big[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\big]\mathbf{A}&amp;= \mu_0 \mathbf{J}-\nabla \frac{1}{c^2}\frac{\partial}{\partial t}A^0. \end{align}  The first equation is NOT a wave equation for $A^0$, and thus does not propagate.  The final equation IS a wave equation, and describes the propagation of two degrees of freedom (Gauss' law in first eq. can be solved, and then inserted into the second equation to show that $\mathbf{A}$ only couples to the Solenoidal part of the current). Seatpost Overextension I've been riding with my very short seat post extended well past its maximum extension line for quite a while now. Looking at the seat post itself I don't see any signs of deformation or damage other than some very minor surface rust.   Could my bike frame have been damaged? Could the lever force have damaged the frame? I can post some pictures of the frame if needed. My bike is all aluminum. Where are some good places to find royalty free stock images? I'm always on the hunt for free stock images to use while I design, what are some good resources for images that are under a a royalty free or creative commons licence? How to change default window background color in Windows 10? copy ssh private keys to another computer What does the bracketed capital letter indicate in the revised version below? Is it essential, per the rules of proper punctuation, to do so? Are the ellipsis points (i.e., the spacing of the dots) spot on in that sentence to indicate the omission of one or more sentences?  Original quotation  "It need hardly be said that shortness is a merit in words. There are often reasons why shortness is not possible; much less often there are occasions when length, not shortness, is desirable. But it is a general truth that the short words are not only handier to use, but more possible in effect; extra syllables reduce, not increase, vigor."  Revised quotation  "It need hardly be said that shortness is a merit in words. . . . [S]hort words are not only handier to use, but more possible in effect; extra syllables reduce, not increase, vigor."  Thank you. How to make the window full screen with Javascript (stretching all over the screen) How should I use "eponym", "eponymous" and "namesake"? I was in London and visited the Tower Bridge. Afterwards, I went to a pub called The Tower Bridge. Just to be clear, the pub is named after the bridge.   Which of the following are correct?   The bridge is the eponym of the pub. The pub is the eponym of the bridge. From the pub, I could see the eponymous bridge. After I saw the bridge, I went to the eponymous pub. The bridge is the namesake of the pub. The pub is the namesake of the bridge.