Find all such mappings where $A=\{1,2,3,4\}$, $B=\{1,2,3,4,5,6\}$ and $f:A \to B$ is an injective mapping such that $f(i) \neq i ~\forall i=1,2,3,4$.
How many injective functions $f:[1,...,m]\to{[1,...,n]}$ has no fixed point? $(m\le n)$
How do I create a file and mount it as a filesystem? How do I create a file of size 10M, format it with ext3 filesystem and then mount it in /media/fuse?  I tried with  mkfs -t ext3 file   then to mount it  mount -t ext3 file /media/fuse   It didn't work because it said that file wasn't a block device. Can anybody help me? How to add readable text to a light and dark image? I'm trying to add text to a photo of a city but the colors in the background make it difficult to read light and dark text. I've tried several light and dark colors but they all looked overpowered by the colors in the image. What colors could I use or what can I do to the text to make it show but also make it so you can still see the background?  Here is the image I'm trying to use: When should you use generator expressions and when should you use list comprehensions in Python?  # Generator expression (x*2 for x in range(256))  # List comprehension [x*2 for x in range(256)] Train passport control from London to Paris I'm a non-EU citizen, I will be travelling from London to Paris next month, so I was wondering since most countries track entry and leave dates, if I use the train to switch these countries and they don't control it, my UK visa will seem like entry YES but no leaving. Same for the Schengen, I wouldn't have an entry date but I'll be leaving,  Will it be trouble later? Are they controlling this on trains? If not is there anything I should do before? I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question  but to no avail.)  It starts from the Wikipedia . They say this:     The likelihood of a set of parameter values, $\theta$, given   outcomes $x$, is equal to the probability of those observed outcomes   given those parameter values, that is      $$\mathcal{L}(\theta \mid x) = P(x \mid \theta)$$   Great! So in English, I read this as: "The likelihood of parameters equaling theta, given data X = x, (the left-hand-side), is equal to the probability of the data X being equal to x, given that the parameters are equal to theta". (Bold is mine for emphasis).  However, no less than 3 lines later on the same page, the Wikipedia entry then goes on to say:     Let $X$ be a random variable with a discrete probability distribution   $p$ depending on a parameter $\theta$. Then the function      $$\mathcal{L}(\theta \mid x) = p_\theta (x) = P_\theta (X=x), \, $$      considered as a function of $\theta$, is called the likelihood   function (of $\theta$, given the outcome $x$ of the random variable   $X$). Sometimes the probability of the value $x$ of $X$ for the   parameter value $\theta$ is written as $P(X=x\mid\theta)$; often   written as $P(X=x;\theta)$ to emphasize that this differs from   $\mathcal{L}(\theta \mid x) $ which is not a conditional probability,   because $\theta$ is a parameter and not a random variable.   (Bold is mine for emphasis). So, in the first quote, we are literally told about a conditional probability of $P(x\mid\theta)$, but immediately afterwards, we are told that this is actually NOT a conditional probability, and should be in fact written as $P(X = x; \theta)$?   So, which one is is? Does the likelihood actually connote a conditional probability ala the first quote? Or does it connote a simple probability ala the second quote?  EDIT:  Based on all the helpful and insightful answers I have received thus far, I have summarized my question - and my understanding thus far as so:    In English, we say that: "The likelihood is a function of parameters, GIVEN the observed data." In math, we write it as: $L(\mathbf{\Theta}= \theta \mid \mathbf{X}=x)$. The likelihood is not a probability.  The likelihood is not a probability distribution.  The likelihood is not a probability mass. The likelihood is however, in English: "A product of probability distributions, (continuous case), or a product of probability masses, (discrete case), at where $\mathbf{X} = x$, and parameterized by $\mathbf{\Theta}= \theta$." In math, we then write it as such: $L(\mathbf{\Theta}= \theta \mid \mathbf{X}=x) = f(\mathbf{X}=x ; \mathbf{\Theta}= \theta) $ (continuous case, where $f$ is a PDF), and as $L(\mathbf{\Theta}= \theta \mid \mathbf{X}=x) = P(\mathbf{X}=x ; \mathbf{\Theta}= \theta) $ (discrete case, where $P$ is a probability mass). The takeaway here is that at no point here whatsoever is a conditional probability coming into play at all.  In Bayes theorem, we have: $P(\mathbf{\Theta}= \theta \mid \mathbf{X}=x) = \frac{P(\mathbf{X}=x \mid \mathbf{\Theta}= \theta) \ P(\mathbf{\Theta}= \theta)}{P(\mathbf{X}=x)}$. Colloquially, we are told that "$P(\mathbf{X}=x \mid  \mathbf{\Theta}= \theta)$ is a likelihood", however, this is not true, since $\mathbf{\Theta}$ might be an actual random variable. Therefore, what we can correctly say however, is that this term $P(\mathbf{X}=x \mid \mathbf{\Theta}= \theta)$ is simply "similar" to a likelihood. (?) [On this I am not sure.]   EDIT II:  Based on @amoebas answer, I have drawn his last comment. I think it's quite elucidating, and I think it clears up the main contention I was having. (Comments on the image).    EDIT III:   I extended @amoebas comments to the Bayesian case just now as well: What are the differences between "to talk" and "to speak"? Hazard Function - Survival Analysis I just started taking survival analysis class and I'm stumped on this question.  Let $T_1,...,T_n$ random independent continuous variables, with hazard function of $h_1(t),...,h_n(t)$. $T=min(T_1,...,T_n)$. And we need to show that the hazard function of T is $\sum_j{h_j(t)} $  Any help or direction are welcome :) My friend and his wife visited UK during the year 2015. And they came back on time we want to know one information as they same couple want to visit UK again  For how many years or months can multiple entry visa can be given for a couple who have already visited UK? Is my US visa still valid after change in name? I'm a Malawian national and I got US Visa years back.  Recently, due to some reasons, I had to get my first name partially changed on my passport. I have got an affidavit certificate stating the reason and proof.  Since the visa on my old passport has still not expired, but there is a difference in first name, will I face any problem? What UML (Unified Modelling Language) tools are available? How can I attach a file to a post? I want to attach some files to a post on one of Stack Exchange sites. What is the best way to do this? Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$? Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$  is independent on $a$? A generic question across most board games: once a player has taken their turn (taken their finger off the piece they are playing), should they be allowed to change their mind?  Similarly, if a player forgets to take a privilege, such as an occupied building in Puerto Rico giving an extra worker, or plantation, if they remember a few moves later, should they be allowed to take the privilege?  Very few rules explicitly express how these events should be dealt with, so they are generally left up to the group. However, my group's rules will change depending on the personality of the players playing (not all members are available each time we play). I would like to make this consistent, and would appreciate feedback on how others deal with this scenario. Textbook approach to modeling non-proportional hazards in the Cox model In Cox models with time varying coefficients, the effect of covariates on the hazard is allowed to change through time. In cases where a coefficient has a linear relationship with time, I am aware of at least two text books ( and an Appendix to ) that advocate for the following model, first in R code:  coxph(Surv(time, death) ~ disease + disease:time)   And now in mathematical notation:  $$ h(t|X) = h_0(t) \exp(\beta_1X + \beta_2X_T) $$ Where $h(t|X)$ is the instantaneous hazard rate at time $t$ given covariates $X$; $h_0(t)$ is the baseline hazard rate; and $T$ is the censoring or event times. $\beta_1$ represents the log hazard ratio for individuals with $T=0$ (which is sort of nonsensical since you can't have $T=0$ in the data), while $\beta_2$ is harder to interpret... $\beta_2$ shows how the log hazard ratio depends on censoring/event times, but that is circular because event times are also the response variable (this point is made in Terry Therneau’s vignette “”).  According to Therneau's vignette, to correctly specify a time varying coefficient, you need to use the following model, first in R code:  coxph(Surv(time, death) ~ disease + tt(disease), tt=function(x,t,...){x*t})   And now in mathematical notation: $$ h(t|X) = h_0(t) \exp(\beta_1X + \beta_2X_T) $$  The difference is that the time interaction is with $t$ in the model rather than with the event times $T$ in the data. The resulting parameters are easy to interpret: they are the intercept and slope of the linear relationship between the log hazard ratio of $X$ and time. This approach seems like the proper way to model a time varying effect.  Can anyone justify or advocate for the first approach to modeling non-proportional hazards in the Cox model? If it is really incorrect (as I suspect it is), what are the practical consequences of using this approach? Fontconfig warning: "/etc/fonts/conf.d/90-fonts-unfonts-core.conf" I am getting the following error frequently. (Example: when running zenity,firefox etc. commands / applications.)  Fontconfig warning: "/etc/fonts/conf.d/90-fonts-unfonts-core.conf", line 11: Having multiple values in &lt;test&gt; isn't supported and may not work as expected   Here is contents of file /etc/fonts/conf.d/90-fonts-unfonts-core.conf:  &lt;?xml version="1.0"?&gt; &lt;!DOCTYPE fontconfig SYSTEM "fonts.dtd"&gt; &lt;fontconfig&gt;     &lt;match target="font"&gt;         &lt;test name="family" qual="any"&gt;             &lt;string&gt;은 바탕&lt;/string&gt;&lt;string&gt;UnBatang&lt;/string&gt;                 &lt;string&gt;은 돋움&lt;/string&gt;&lt;string&gt;UnDotum&lt;/string&gt;                 &lt;string&gt;은 궁서&lt;/string&gt;&lt;string&gt;UnGungseo&lt;/string&gt;                 &lt;string&gt;은 필기&lt;/string&gt;&lt;string&gt;UnPilgi&lt;/string&gt;                 &lt;string&gt;은 그래픽&lt;/string&gt;&lt;string&gt;UnGraphic&lt;/string&gt;         &lt;/test&gt;          &lt;edit name="antialias" mode="assign" binding="strong"&gt;&lt;bool&gt;true&lt;/bool&gt;&lt;/edit&gt;         &lt;edit name="hinting"   mode="assign" binding="strong"&gt;&lt;bool&gt;false&lt;/bool&gt;&lt;/edit&gt;         &lt;edit name="autohint"  mode="assign" binding="strong"&gt;&lt;bool&gt;true&lt;/bool&gt;&lt;/edit&gt;         &lt;edit name="hintstyle" mode="assign" binding="strong"&gt;&lt;const&gt;hintslight&lt;/const&gt;&lt;/edit&gt;     &lt;/match&gt; &lt;/fontconfig&gt;   How to solve it? Is there a problem with the ClearBits data dump? I have a problem. I have several large equations within the text. And when LaTeX does not manage to put the equation on the same page and moves it to the next one, he spreads the paragraphs of the text in the first page. How could I prevent this? I know about \pagebreak and \newpage, but I do not want to force the \newpage, as the text may move (if I add something later or so on) and the pagebreaks may not be valid. I would like to make LaTeX not to spread the paragraphs over a page automatically. Is it possible?  I tried to make an example:  \documentclass[a4paper,12pt,twoside]{book} \usepackage{amsmath} \begin{document}  LaLaLa\\\  LaLaLa\\\  LaLaLa\\\  LaLaLa\\\  \begin{eqnarray} 123&amp;+&amp;123 \\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ \end{eqnarray}  LaLaLa\\\  LaLaLa\\\  \begin{eqnarray} 123&amp;+&amp;123 \\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ 123&amp;+&amp;123 \nonumber\\\ \end{eqnarray}  \end{document} Why doesn't my flash fire from the hot shoe trigger? I have a Yongnuo YN-560 III and a pair of RF-603II (C3) triggers. I have a Canon Rebel T5i.   I put one trigger on the hot shoe.  When this trigger is in TX mode I get a blue comm light and I can fire a test flash using the large test button on the trigger. But when I press the shutter button to take a picture, there's no flash.  The trigger and flash are on Channel 1. The trigger on the hot shoe is set to TRX. TX is fine, TRX gives me no flash. The flash and camera are in M mode.  I don't have a trigger on the flash because the Mark III has a built-in trigger The problem seems to be the camera.  The trigger can talk to the flash but the camera is not sending a flash command through the trigger in the hot shoe to the flash.  Grateful for anything you could suggest. Let the ring of fractions be denoted as $S^{-1}R$.       Proposition (from Robert Ash's textbook "Basic Abstract Algebra")      Define $f: R \rightarrow S^{-1}R$ by $f(a) = a/1$. Then f is a ring homomorphism. If S has no zero divisors then f is a monomorphism, and we say that R can be embedded in $S^{-1}R$. In particular:      (i) A commutative ring R cna be embedded in its complete (or *full) ring of fractions ($S^{-1}R$, where S consists of all non-divisors of zero in R).       (ii) An integral domain can be embedded in its quotient field.    Why does the proposition say "If S has no zero divisors..."? If S really had zero divisors, then this wouldn't even be properly defined, right? Because we would have a/0 for some a in R. How do I assign the output of a command into an array?