Vertical align a text to the middle of a button that includes a full heigh image to send as email
Vertically align text next to an image?
First of all, I know that light does indeed travel slower in a medium like air or water, but that's because the photons are bouncing off of the medium's particles and in different directions so the light beam as a whole is slowed down (or at least that's the classical explanation I've read somewhere).  But my question is about light traveling freely in the vacuum. What if an astronaut during a spacewalk switches on a flashlight towards the opposite direction he is moving. Will that light travel slower than normal?  If not, is there another way to make light move slower? For example, can you make a beam of light move so slow that you can visually perceive it being affected by gravity? Where can I find good literature for beginners? I am trying to find some database tutorials,books and similar, since I have started this course in college, and received some material, but it is really hard to manage through all the info on the net. I searched through the questions on the site but didn't find any. Thankfull for all your input. Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit? The limit is $$\lim_{x\to0}\left(\frac{1}{\sin^2x}-\frac{1}{x^2}\right)$$ which I'm aware can be rearranged to obtain the indeterminate $\dfrac{0}{0}$, but in an attempt to avoid L'Hopital's rule (just for fun) I tried using the fact that $\sin x\approx x$ near $x=0$. However, the actual limit is $\dfrac{1}{3}$, not $0$.  In this , the approximation reasoning works out. How can I safely prototype a mains power circuit? I'm building a device that has a mains power component, which I need to prototype before building to make sure everything works. I know I need to make sure wiring—and insulation!—is thick enough to cope with the higher current, and that I need to make sure there's enough space between exposed components so there's no arcing. I'll also be stashing whatever I prototype in a nice, insulated box before applying power, and I'll be include a small-value (250mA) fuse just in case. I'll be considering everything double-insulated rather than earthed (plastic enclosure, and there's nothing I'll need to touch while it's operating).  My plan is to solder (through-hole) components directly to each other, then insulate everything (with heatshrink tubing or electrical tape) as much as possible, then use screw-type cable joiners to connect the power cable.  I know I can't use traditional prototyping tools such as breadboards and stripboard for mains voltages, but what can I use to make prototyping at mains voltages easier?  (To be clear, 'mains voltage' in my area means 230V AC +10/-6% at 50Hz; GPOs are generally rated at 10A.) How to understand the drawbacks of K-means K-means is a widely used method in cluster analysis. In my understanding, this method does NOT require ANY assumptions, i.e., give me a dataset and a pre-specified number of clusters, k, and I just apply this algorithm which minimizes the sum of squared errors (SSE), the within cluster squared error.  So k-means is essentially an optimization problem.  I read some material about the drawbacks of k-means.  Most of them say that:   k-means assumes the variance of the distribution of each attribute (variable) is spherical;  all variables have the same variance; the prior probability for all k clusters is the same, i.e., each cluster has roughly equal number of observations;   If any one of these 3 assumptions are violated, then k-means will fail.  I could not understand the logic behind this statement. I think the k-means method makes essentially no assumptions, it just minimizes the SSE, so I cannot see the link between minimizing the SSE and those 3 "assumptions". If we map every integer to a string that represents a rational number, and produce a number different from all the ones listed, we are essentially following Cantor's algorithm. But why does it not apply? Is it because we can't be certain that the number produced is a rational number? I would like to define a theorem environment with custom layout. In particular, I want the body of the theorem to be indented, like this  Text body. Text body. Text body.  Theorem 1 (Pythagoras).     Let a,b,c the sides of a rectangular triangle.     Without loss of generality, we assume that  a&lt;b&lt;c .     Then, the following equality holds:            a^2 + b^2 = c^2  More text. And even more text.   This is similar to the quote environment, I believe.  How can I do this?  Preferably, I would like to use the existing \newtheorem command, but that's not mandatory. ${{2\pi i=0}}$? I was trying to solve for $i^i$.  I got distracted and did the following: $e^{i \theta} = \cos(\theta) + i\sin(\theta)$ $e^{i 2\pi} = cos(2\pi) + i\sin(2\pi) = 1$ ||| Take ln() of two sides of equation: $\ln(e^{i2\pi}) = \ln(1)$ $i2\pi = 0$ Clearly this is false. What did I do wrong? How do I check whether a file exists without exceptions? How do I check whether a file exists or not, without using the  statement? Every few days I get notices on my MacBook that it's running or run out of hard drive space. Curiously, restarting the computer will enable me to recover gigabytes of space (this past time, it was able to recover about 2.2GB). However, I can't identify anything in my personal activity that consumed that space.  It's possible that it's a rogue iTunes podcast or a huge software update that my Mac is automatically downloading - would either of these reclaim the space upon a restart?  One possibility that I can think of is that FileVault has some sort of disk leak, allocating but not freeing files. Does this make sense? Is there a tool that I can run to determine where this space is going? Assuming it is FileVault, should I try to disable it? What's the best way to turn of FileVault on a nearly full computer? Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$? Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=k[X_1,X_2,X_3,X_4]$? Can it be generated with three elements? (Here $k$ is a field.)    Thanks for any help. Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$ Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$.  Work so far:  $$x^8 - x = x(x^7 - 1) = x(x - 1)(x^6+x^5+x^4+x^3+x^2+x+1)$$  From here, I think the Zeros of an Irreducible over a splitting field Theorem would be the method. I am unsure how to proceed Matrix $A^T A$ as sum of outer products Rigify generated rig, hand IK controller has unexpected rotation Force Messages to use SMS for specific contact rather than iMessage? Essentially, I would like to set my phone so that all my outgoing messages to a specific friend, with whom I currently text back and forth over iMessage, are sent out over SMS, never iMessage. Additionally, I would like to block any incoming iMessage texts of his and (hopefully) they'll fail and be resent via SMS.  Let me restate that: I would like to force SMS for both parties, going both ways, for a specific individual contact.   Is this possible? Blocking any usage of iMessage between myself and this other individual, so they must either send and receive any texts to or from me via SMS (or the texts fail/never arrive). Using pgfplots, why do I get "undefined control sequence" when trying to use a foreach variable as a parameter to addplot? Since I have multiple plots with error bars, I would like to shift the error bars slightly to make the overall plot more readable. However, when I try to do this using the variable of a \foreach as the amount to shift, I get an "undefined control sequence" error.  \documentclass{article}  \usepackage{tikz} \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture} \begin{axis} \foreach \x/\y in {a/-1cm, b/0cm, c/1cm} {     \addplot+ [         %every error bar/.append style={xshift=\y},         %every node/.style={xshift=\y},         error bars/y dir=both,         error bars/y explicit,     ] coordinates {         (0, 0) +- (0, 1)         (1, 0) +- (0, 1)         (2, 0) +- (0, 1)     }; } \end{axis} \end{tikzpicture}  \end{document}   (The document does not compile when I uncomment either one of the two comments.)  What's wrong here?  EDIT: Added another variable to the \foreach. In my actual use-case it would contain the filename to plot data from or the label to use for the curve. Quotient of the ring of integers of a number field by a prime ideal R stem and leaf values How to make a stem and leaf plot with the following data in "R":  1717 1719 1645 3739 3024 3664 3830 2991 2430 2730 3469 5086 2119 3021 3292 2844 3426 2067 3215 2767 3124 2573 2840 2449 2584 1505 1390 1645 2497 3466 3228 3192  The only way to get a reasonable amount of stems is to make the stem the first digit in each number and the leaves will all be 3 digit numbers. I am not able to split the data in this way, R represents the stem with 1 digit but round my numbers  so 2584 is shown as 2| 5 (3 decimals to the right) which would be 2500. How do I adjust my stem and leaf numbers? Why are my Elytra Wings are not activating? How to retrieve the content/file in JSP?