Are there different types of infinity?
What Does it Really Mean to Have Different Kinds of Infinities?
Shouldn't the twoside option produce a wider margin at the binding/spine edge, rather than the other way around? I'm using \documentclass[twoside]{report}. Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well.   If I knew that $B(Y)$ is closed I'd have a bounded linear map $B^{-1}:B(Y)\rightarrow Y$ by the bounded inverse theorem. Therefore $A=B^{-1}BA$ is bounded. How to prove the claim if $B(Y)$ is not closed. Why isn't it $E \approx 27.642 \times mc^2$? Why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides?   Why can so many equations be expressed so neatly with small natural numbers while recycling a relatively small set of physical and mathematical constants?  For example, why is mass–energy equivalence describable by the equation $E = mc^2$ and not something like $E \approx 27.642 \times mc^2$?  Why is time dilation describable by something as neat as $t' = \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$ and not something ugly like $t' \approx 672.097 \times 10^{-4} \times \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$.  ... and so forth.    I'm not well educated on matters of physics and so I feel a bit sheepish asking this.   Likewise I'm not sure if this is a more philosophical question or one that permits a concrete answer ... or perhaps even the premise of the question itself is flawed ... so I would gratefully consider anything that sheds light on the nature of the question itself as an answer.    EDIT: I just wished to give a little more context as to where I was coming from with this question based on some of the responses:  @Jerry Schirmer comments:      You do have an ugly factor of $2.997458 \times 10^8 m/s$ in front of everything. You just hide the ugliness by calling this number c.   These are not the types of "ugly constants" I'm talking about in that this number is the speed of light. It is not just some constant needed to balance two side of an equation.  @Carl Witthoft answers:      It's all in how you define the units ...   Of course this is true, we could in theory hide all sorts of ugly constants by using different units on the right and the left. But as in the case of $E=mc^2$, I am talking about equations where the units on the left are consistent with the units on the right, irrespective of the units used. As I mentioned on a comment there:     $E=mc^2$  could be defined using units like $m$ in imperial stones ($\textsf{S}$), $c$ in cubits/fortnight ($\textsf{CF}^{−1}$) and $E$ in ... umm ... $\textsf{SC}^2\textsf{F}^{−2}$ ... so long as the units are in the same system, we still don't need a fudge factor.   When the units are consistent in this manner, there's no room for hiding fudge constants. I am very confused. Unless I am mistaken, I know "it" has to be a noun of some sort, but I am unable to figure out what noun "it" is referring to.   What is "it" in the following sentence:      It is clear that Bob likes doughnuts.   Heres another couple of examples:     It is impossible to fly. Switch off Windows Update for a particular program or feature I'm getting really tired of unchecking the same Windows Update candidates so save hundreds of MB's of dowbloads, especially for programs I never use, that are in fact not even installed, and those I don't wish to update.  Is there any way to permanently, or until another explicit change, to turn off certain updates?  There is an option called 'Hide update', but that unsettles me a bit. Changing underline color of LanguageTool in TeXstudio I've recently installed LanguageTool to check grammar and spelling mistakes in TeXstudio. However, I often miss the errors because the gray-bluish underlining makes them hard to spot. Is there a way to change the underlining to a distinctive color, such as red for example?   I checked . It says I have to do some changes to the .languagetool.cfg file, but for some reason this file doesn't appear in File Explorer. Modular exponentiation by hand ($a^b\bmod c$) How to get the last argument to a /bin/sh function What's a better way to implement print_last_arg?  #!/bin/sh  print_last_arg () {     eval "echo \${$#}"  # this hurts }  print_last_arg foo bar baz # baz   (If this were, say, #!/usr/bin/zsh instead of #!/bin/sh I'd know what to do.  My problem is finding a reasonable way to implement this for #!/bin/sh.)  EDIT: The above is only a silly example.  My goal is not to print the last argument, but rather to have a way to refer to the last argument within a shell function.    EDIT2: I apologize for such an unclearly worded question.  I hope to get it right this time.  If this were /bin/zsh instead of /bin/sh, I could write something like this  #!/bin/zsh  print_last_arg () {     local last_arg=$argv[$#]     echo $last_arg }   The expression $argv[$#] is an example of what I described in my first EDIT as a way to refer to the last argument within a shell function.  Therefore, I really should have written my original example like this:  print_last_arg () {     local last_arg=$(eval "echo \${$#}")   # but this hurts even more     echo $last_arg }   ...to make it clear that what I'm after is a less awful thing to put to the right of the assignment.  Note, however, that in all the examples, the last argument is accessed non-destructively.  IOW, the accessing of the last argument leaves the positional arguments as a whole unaffected. Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis How do I compute      $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$   What I am doing is take   $$f(z)=\frac{(\log z)^2}{1+z^2}$$  and calculating   $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log z)^2}{1+z^2}$ which came out to be $\dfrac{\pi}{2}-\dfrac{i\pi^2}{8}+\dfrac{i\pi}{2}$  Im not too sure how to move on from here. the given answer is $\dfrac{\pi^3}{8}$  Any help will be appreciated. thank you in advanced. I try to understand and visualize myself covariance matrix. Supposing I have a matrix A = [ 2 3 4; 5 5 6 ], how do I calculate its covariance matrix, and what is its practical meaning? (All I was able to understand by now is that on the diagonal of the covariance matrix, variances for particular variables are placed, and on the upper and lower fields correlations between those variables.) 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I suppose something as simple as this is only a small function you can turn off, but I can't seem to find it in the manual of my camera. Are there difference between those sentences?        Alex is getting married next month.     Alex will get married next month.      Seems that the first one is expressed in present continues, and the second on in future tense. Am I correct? Remove last character from line If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$ Unable to modify files/folders in NTFS partition find a $\int_{x}^{g(x)} f(t)\,dt = 1$ Let $f : [0, 2] → \mathbb{R}$ be a continuous, positive function and  $$\int_{0}^1 f(x)\,dx = \int_{1}^2 f(x)\,dx= 1$$ For each $x \in [0, 1]$, prove that there exists a unique function $g(x) \in [1, 2]$ such that $$\int_{x}^{g(x)} f(t)\,dt = 1$$ and $g \in C^1$. Well, I think in $g(x) = x +1$, but I'm not sure. Any hint is appreciated.  Thank you. How to *disable* automatic reboots in Windows 10? Subdividing Causes Strange Edges / Issues on Shape