When spacetime stretches, does both space and time stretch proportionally. That is, if you stretch Planck-length doesn't Planck-time also stretch proportionally? I find this to be a conundrum.
Einstein's big revelation was that time and space are inseparable components of the same fabric. Physical observation tells us that distant galaxies are moving away from us at an accelerated rate, and because of the difficulty (impossibility?) of defining a coordinate system where things have well defined coordinates while also moving away from each other without changing the metric on the space, we interpret this to mean that space itself is expanding.   Because space and time are so directly intertwined is it possible that time too is expanding? Or perhaps it could be contracting?
Is the EmDrive, or "Relativity Drive" possible? Creating Centrelines from Road Polygons/Casings using ArcGIS Desktop? I want to create center lines from road polygons.   Are there any automated tools in ArcGIS for Desktop to do this?  I am including a Screenshot of my vector file below. Make 15 lines / ticks with Photoshop, each of them rotated of 20° etags and other tags The  tag is a synonym of , and has the description     A program to generate tags for a large project. These tags are usable by the Emacs text editor.   This is inconsistent: tags in that sense are always plural, i.e. etags, never etag.  evidently refers to another concept, .  Should  remain a synonym of , and be about HTTP Etags, and should Emacs etags be tagged +? Or should the synonym be unmade, with HTTP Etags on  and etags on ? I sold an investment in a standard brokerage account at a small loss because I needed the money for a down payment.  I still think it is a good investment, can I buy the same investment in a standard IRA without running into wash sale problems?  I will probably just wait the 30 days any way to avoid any headaches, but I want to know if there is a definitive answer on this.    Being as the cost basis in the traditional IRA doesn't matter, I wouldn't think there would be a problem. Why on-shell vs. off-shell matters? The definitions between on- and off-shell are given in .   Why is it so important in QFT to distinguish these two notions? What's the right course of action for cross-site duplicates?  There are two flavors:   A question is posted which belongs on another site and ends up already having been asked there. I can't find an example right now, but it's bound to happen soon if it hasn't already. The same question is posted in two places at the same time. Take  for example, which is a duplicate of , posted within a half hour of each other by the same user.   Possible actions:   Move it to the proper site and close it once it gets there as a duplicate. Provide a way to close it right away as a cross-site duplicate, without moving it first. Flag it and let a moderator merge them. Close it as "no longer relevant" or something else.   Additionally, flavor 2 is a bit of a special case; perhaps the question really spans both sites in terms of content and intended audience. Should multiple-postings in this case be allowed? If so, should they somehow be linked to each other? If not, how do we avoid a migration war? How can I find the program I'm hiding in bash Triangle inequality for L2 norm I am posed with a problem as below.  Let $X$ be the collection of real valued functions that are measurable on $[0,1]$. For a given number $p\geq 1$ define the $L^p$ norm by $(\int_0^1|f(x)|^p)^\frac{1}{p}$.  I need to show for $p=1,2$ that this defines a semi norm on $X$. $p=1$ was simple as was all of $p=2$ except for the triangle inequality. I am having trouble proving the triangle inequality for this. Could anybody nudge me into the right direction or am I simply over thinking it? I think I can prove it for vectors but I'm not sure how to tackle it with integrals. Right now I can only use relatively basic results because this in for an introductory measure theory course.  Thanks! I know that if (p) is maximal then Z/(p) is a field which implies its an integral domain which then implies that (p) is a prime ideal and p is prime. I'm having trouble going the forward direction assuming that p is prime and then showing it's a prime ideal. Quoting in ssh $host $FOO and ssh $host "sudo su user -c $FOO" type constructs I often end up issuing complex commands over ssh; these commands involve piping to awk or perl one-lines, and as a result contain single quotes and $'s.  I have neither been able to figure out a hard and fast rule to do the quoting properly, nor found a good reference for it.  For instance, consider the following:  # what I'd run locally: CMD='pgrep -fl java | grep -i datanode | awk '{print $1}' # this works with ssh $host "$CMD": CMD='pgrep -fl java | grep -i datanode | awk '"'"'{print $1}'"'"   (Note the extra quotes in the awk statement.)  But how do I get this to work with, e.g. ssh $host "sudo su user -c '$CMD'"?  Is there a general recipe for managing quotes in such scenarios?.. Does the given matrix have a square root? Given the following matrix, I was asked to tell whether the matrix is the square of some matrix or not.  $$ \begin{pmatrix} 1 &amp; 0 &amp; 0\\ 0 &amp; -1 &amp; 0\\ 0 &amp; 0 &amp; -1 \end{pmatrix} $$  Since this matrix is diagonal, a square root is  $$ \begin{pmatrix} 1 &amp; 0 &amp; 0\\ 0 &amp; i &amp; 0\\ 0 &amp; 0 &amp; i \end{pmatrix} $$ Am I right? L1 & L2 Regularization in Light GBM This question pertains to L1 &amp; L2 regularization parameters in Light GBM. As per official documentation:  reg_alpha (float, optional (default=0.)) – L1 regularization term on weights.  reg_lambda (float, optional (default=0.)) – L2 regularization term on weights  I have seen data scientists using both of these parameters at the same time, ideally either you use L1 or L2 not both together.   While reading about tuning LGBM parameters I cam across one such case: Kaggle official GBDT Specification and Optimization Workshop in Paris where Instructors are ML experts. And these experts have used positive values of both L1 &amp; L2 params in LGBM model. Link below (Ctrl+F 'search_spaces' to directly reach parameter grid in this long kernel)      I have seen same in XGBoost implementations.  My question is why use both at the same time in LGBM/XGBoost.  Thanks. I can think of three general ways of explaining why the universe contains more matter than antimatter:  (1) Near the Planck time, the universe had zero baryon asymmetry, but at some later time, determined by some GUT energy scale, the  were satisfied, and the baryon asymmetry became nonzero.  (2) Nonconservation of baryon number occurs only at Planckian energies. Near the Planck time, the baryon asymmetry evolved from zero to some nonzero value.  (3) The Sakharov conditions have never been satisfied. The baryon asymmetry has always been nonzero, and has simply scaled as expected. (Apparently one  $\Delta n=n_B-n_\bar{B}\propto s$, where $n$ is number density and $s$ is entropy density).  It seems like most theorists are interested in #1, but is there any reason why 2 and 3 aren't possible?  2 seems pretty reasonable, since for the reasons given in , we have good reasons to think that baryon number is not conserved under Planckian conditions.  3 also seems reasonable to me, since even if baryon number is nonconserved at Planckian energies, that's only one of the three Sakharov conditions. I don't see any obvious fine-tuning objections to #3, since the scaling of baryon asymmetry with cosmological expansion isn't particularly drastic (not an exponential decay or anything). Is there something unphysical about maintaining $\Delta n\propto s$ all the way back to the Planckian era?   Some people might object to #3 on aesthetic grounds, since we "expect" the initial conditions of the universe to be symmetric, but that seems weak to me. After all, we don't object aesthetically to the fact that homogeneity was an imperfect symmetry of the early universe, and we even accept that the early universe was in a thermodynamically unlikely state. \numcases with \tag I would like to label some cases with specific tags, and was hoping for something like  \begin{numcases}{A = } \frac{r}{a} &amp; r &gt; a \tag{G} \\ \frac{a}{r} &amp; r &lt; a \tag{H} \end{numcases}   utilizing the numcasesenvironment from the cases package, and \tag from amsmath. However, the two don't seem to be compatible:     ! Package amsmath Error: \tag not allowed here.   Is there any alternatives that would do what I hope to accomplish? Why $\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$ doesn't evaluate to 1? I am trying to identify what the flaw is exactly when reasoning about a limit such as the definition of $\mathbf e$:  $$ \lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n={e} $$  Now, I know there are ways of proving this limit, such as by considering the binomial expansion of $(1+\frac{1}{n})^n$ and comparing that to the Maclaurin series of $e$. So to make this clear, I am not looking for a proof of the limit definition of $e$.  I tried searching for "limit laws/rules" but none of the rules I found described the above case. Hence, I am looking for a specific rule (or perhaps a certain perspective) that will help me realize why the above does not in fact evaluate to 1.  My train of thought is as follows: at a first glance, if I wasn't already familiar with what the limit evaluates to, I probably would evaluate the expression inside the brackets first, and then apply the limit to the power. So, $$\displaystyle\lim_{n\to\infty}\frac{1}{n}=0\quad\text{and then}$$  $$\displaystyle\lim_{n\rightarrow \infty}\left(1+0\right)^n=\lim_{n\to\infty}1^n=1$$  Why is my reasoning flawed? integral of $\int_{0}^{1}\frac{\ln(x^{2}+1)}{x+1}dx$ using contour integration? I have an interest in contour integration. I am not that good at it, but I enjoy learning what I can about it.  Here is a version of a rational log integral rarely encountered.  $\displaystyle \int_{0}^{1}\frac{\ln(x^{2}+1)}{x+1}dx=\frac{3}{4}\ln^{2}(2)-\frac{{\pi}^{2}}{48}$  I can do this using real methods (via double integral and substitution). I can post my workings if anyone would be interested.  My question is, can this be evaluated using contour integration due to the limits being $[0,1]$ instead of $[0,\infty)$?.  Contours may not be the most efficient way to go about it, but what is the course of action when the limits are 0 to 1 instead of 0 to infinity?. How to calculate a heading on the earths surface? Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system.  How do you determine the heading in degrees clockwise from true north of movement? I initially thought that dark energy must in some way violate conservation of mass and energy since the component of the energy density of space that comes from dark energy is constant, and space is expanding. Therefore, as space expands the energy in the universe that comes from dark energy would increase. I presumed the source of this energy was not coming from the conversion of other types of energy to dark energy, so it must violate conservation.  I decided to Google this and came upon this article:    It says that dark energy does NOT violate conservation and quotes Carroll, Press, and Turner (1992):  "…the patch does negative work on its surroundings, because it has negative pressure. Assuming the patch expands adiabatically [i.e. without loss or gain of heat], one may equate this negative work to the increase of mass/energy of the patch. One thereby recovers the correct equation of state for dark energy: P = – ρ c2. So the mathematics is consistent."  Is there a way to explain this in layman's terms? (The blog attempted to do this, but it was very unclear to me.) More specifically, can you explain where my initial train of thought described above fails when I erroneously concluded that dark energy violates conservation?  Thank you. Will reseting my device to factory setting unroot and remove superuser?