Will my rims fit 25mm or 28mm tires?
What is the maximum or minimum tire width I can fit on my bicycle
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and differentiable function in all $\mathbb{R}$. If $f(0)=0$ and $|f'(x)|\leq |f(x)|$ for all  $x\in\mathbb{R}$, then $f\equiv 0$.  I've been trying to prove this using the Mean Value Theorem, but I can't get to the result. Can someone help? Finite ring has only zero divisors and units If a nonzero element $a$ of a ring $\mathbb{D}$ does not have a multiplicative inverse, then $a$ must be a zero divisor.   If $\mathbb{D}$ has finitely many elements, then the statement is true. However, I don't see why, as for the infinite case, 7 isn't a zero divisor only because every element $a &gt; 0$ when multiplied by a $b &gt; 0$ gives you $ab &gt; 0$ which has nothing to do with infinity or finite amounts of elements. Which book (free or otherwise) was the most useful to you when you started learning LaTeX?  I am frequently asked this question by friends who want to learn LaTeX, and I recommend the book which got me started, , but I feel that there might be better options around.  Also see . Does OpenSSH support multihop login? Ansible uses OpenSSH as the bottomed network communicative implementation, but does it support multi-hop login?   I mean, if Ansible can login a server directly, but can it log in server_a then from server_a login server_b for config the server_b? What is "anycast" and how is it helpful? I'd never heard of anycast until a few seconds ago when I read "".  The wikipedia "" article on it is quite formal and doesn't really evoke a mental picture of how it would be used.  Can someone explain in a few informal sentences what "anycast" is, how you configure it (just in a general sense), and what its benefits are (what does it make easier)? CG series about a beast planet How align the qedsymbol after of a big formula in display mode? Description of the problem My problem is this. For the proof of a math theorem i use the proof environment of the package amsthm. It adds a \qedsymbol in the last line of the proof, but if the proof ends with a displayed formula the qedsymbol appears in the next line leaving an unwanted blank space. The documentation of  teaches to fix this problem placing the command \qedhere after of the displayed formula but inside of the appropriate environment. This works well if the formula is of one line like \[ax^2+bx+c=0.\qedhere \]  but if the proof ends in a matrix, or cases with a &quot;great height&quot; the qedsymbol is fixed in the baseline at right and it this looks higher than expected (for my taste). The result is showed in this MWE \documentclass{article} \usepackage[width=7cm]{geometry} % Only for reducing space in this MWE  \usepackage{amsmath} % For `bmatrix` \usepackage{amsthm} % For `proof` environment \begin{document} \begin{proof} And this proof ends with \[ \begin{bmatrix}     1 &amp; 0 \\     0 &amp; 1 \\     0 &amp; 0 \end{bmatrix}.\qedhere \] \end{proof} \noindent Text \end{document}  that produces  What I want I'd like something like  with the qedsymbol aligned with the bottom of the matrix. Is this possible? Prove: If a linear map, $φ: U→V$ is bijective, then $φ(x)=0$ has only the solution $x=0$. number of prime ideals in algebraic integers Calculationg base-resistor between microcontroller and transistor I'm curious, when connecting my 2N2222 transistor's base to a PWM output pin on my Arduino, I know the Arduino can deliver a maximum of 40mA current. The 2N2222 only requires around 5 to 10 mA on it's base to switch the collector and emitter. Now, when calculating the base-resistor under "normal" conditions where I'm merely switching it directly from Vcc, the following applies:   However, when putting in a 500 ohm or 1K resistor in (1K for 5mA) directly it works great. But putting it in between the Arduino and the transistor, the transistor doesn't switch.  I suspect thus, that the resistance is too high of the base-resistor. Therefore, we may very well not need a base-resistor.  But, according to spec, the Arduino can deliver 40mA. So, am I correct in assuming the folling: 1) I should calculate the perceived resistance that the Arduino is giving, if it goes high to 5Volt and only gives 40mA, and then 2) Subtract that from the 500ohm resistor needed to make the transistor operate optimally.  Example:    So, I'm theorising that I have to then put in a 375 ohm resistor in to make the total resistance between the Arduino and the base resistor 500 ohm.  Somehow this doesn't feel right. Keep in mind, I'm not an electronic engineer, so I might have the cat by the tail here :P ListPlot multiple datasets over same x-values Does Python have a ternary conditional operator? What happens to folks who get question banned, by and large? There is a steady stream of duplicates of  coming in lately. It's got me wondering. Roughly speaking, out of say 100 folks who get this ban, how many    wander away and never interact with SO again?   do the sorts of things that might lift a ban, but don't get it lifted? do the sorts of things they should, and lift the ban? post a question of their own on Meta which we close as a dupe?   Among the continue-to-interact folks, is there a difference in success rate between those who post on Meta about it (motivated, but not good rules-readers, who nonetheless get customized advice) and those who do not (better rules readers, but working to some extent blind) - which do better?  Is there a reasonable chance of recovering from a ban, assuming you care enough to try? Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$ Problem  Suppose $f$ is a non-negative integrable function on a measure space $(X,\mathcal{A},\mu).$ Prove that $$\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$$  Attempt  Let $E_t=\{x:f(x)\geq t\}$ Note that $f\chi_{E_t}\leq f$. Since $f$ is integrable, the Lebesgue dominated convergence theorem tells us that $$\lim_{t\to\infty} \int f\chi_{E_t}\rightarrow \int \lim_{t\to\infty} f\chi_{E_t}=0,$$ since $\lim_{t\to\infty} f\chi_{E_t}=0$ a.e. $\ ^{(1)}$ The result follows from the fact that $$t\mu(E_t)\leq\int f\chi_{E_t}.$$  Question  I am having trouble proving $(1)$. That is, how can I show that $f\chi_{E_t}\rightarrow 0$ a.e? Independent samples t-test with unequal sample sizes I have two independent groups: from one I have drawn a sample of size less than 30 due to its small population, while the second group has sample size 100.  Is it appropriate to use an independent samples t-test in this case? Is there a database of Linux compatible hardware? I want to know if there is a centralised community site where all Hardware is listed in terms of compatability with Linux. Something like AppDB for Wine. I asked a question  but, rather stupidly, added extra information after it had been answered.  My additional question is this: I have a table with varying lengths of words. I want to encircle different groups of cells, including diagonals. If possible, it would be great if the rectangles don't overlap each other where possible.  I looked at the answer , which, whilst tedious, worked except for the overlap.    This is where I've got to so far:  \documentclass[fleqn,12pt,a4paper,landscape]{article}  \usepackage{tikz} \usepackage{tabularx}  \begin{document}  % Introduce a new counter for counting the nodes needed for circling \newcounter{nodecount} % Command for making a new node and naming it according to the nodecount     counter \newcommand\tabnode[1]{\addtocounter{nodecount}{1} \tikz \node     (\arabic{nodecount}) {#1};}  % Some options common to all the nodes and paths \tikzstyle{every picture}+=[remember picture,baseline] \tikzstyle{every node}+=[inner sep=0pt,anchor=base, minimum width=1.8cm,align=center,text depth=.25ex,outer sep=1.5pt] \tikzstyle{every path}+=[thick, rounded corners]  \begin{table}[ht]   \begin{minipage}[b]{1 \linewidth}\centering    \begin{tabular}{cccccc} % Turn the cells needed for defining the circling paths into nodes with our     custom command \tabnode{Long ago}&amp; \tabnode{A few days ago}&amp; \tabnode{Day before         yesterday}&amp; \tabnode{Yesterday} &amp;  &amp;  \\      &amp; &amp; &amp;  &amp; \tabnode{Today} &amp; \tabnode{Now}\\      \tabnode{Long time in the future}&amp; \tabnode{A few days on}&amp;     \tabnode{Day after tomorrow}&amp; \tabnode{Tomorrow} &amp;  &amp; \\      \end{tabular}    \end{minipage}   \begin{tikzpicture}[overlay] % Define the circle paths \draw [blue](1.north west) -- (1.north east) -| (7.north east) |- (7.south     west) |- (1.north west); \draw [red](8.south west) -| (2.north east) -| (2.south west) |- (8.south     west); \draw [yellow](9.south west) -| (3.north east) -| (3.south west) |- (9.south     west); \draw [green](8.south west) -| (3.north east) -| (2.south west) |- (8.south     west);   \end{tikzpicture}  \end{table} \end{document}  %%% Local Variables:  %%% mode: latex %%% TeX-master: t %%% End: I found "A sledgehammer to crack a nut" as one example. What are some others? How does HOTP keep in sync? My understanding of  is that each password is unique and based on a counter.    $$PASSWORD = HOTP_1(K,C)$$  Where $C$ is an incremental counter.  What I wish to know, is how you keep the client and server in sync?  Looking at current systems (google authenticator), there appears to be no indication if the actual 'counter' number, but you can generate successive passwords at will.  What happens if, for example, the password you have generated uses $C = 5$, but the server is expecting $C = 3$?  Also, what is the 'integrity check' for?  Google cant seem to come up with a solution. Creating label in QGIS? I am using QGIS 2.18.16 on Windows 10.    What I want is a label to appear within the boundaries of a polygon.    I select a layer, then select command Layer>Labelling, and I get:    From here, I can change "No Labels" to "Show labels for this layer".  But I can't actually create label text.  I find a clue in a prior question:    “Labelling polygon using QGIS?”    The answer was:  The layer you are trying to label must have at least one attribute field populated with data. You can then turn on labelling in Layer Properties and select which attribute field to use for your labels.  I can enter command Layer>Open Attribute Table, and what I get is:    If I need something in the attribute table in order to apply a label, how do I accomplish that?