In category theory, a monomorphism is a situation where     $g: A \rightarrow B \;\;$ , $\;\;h: A \rightarrow B\;\;$ and $\;\;f:B  \rightarrow C$      $f \circ g = f \circ h \;\text{ imples in } \; g = h$   I heard recently something about monomorphism to be an injective homomorphism , but it didn't make sense to me and I still couldn't prove it.   How can I prove that monomorphism is an injective homomorphism (at least for monoids)?
Let $f: M \to N$ be a mono in $\mathsf{Mon}$. I want to prove that the underlying function $U(f): U(M) \to U(N)$ is injective.   How can this be done ? Thanks in advance.
How to set systemd for automatically force fsck disks after crash (hard poweroff)?  When I used sysvinit (in Arch Linux) as /sbin/init I used the hack: in the rc.local I create /forcefsck file; in the rc.local.shutdown I remove it. At boot-up rc.sysinit enable force fsck if /forcefsck exists.  How to do same in systemd? Maybe it has built-in facilities for automatic fsck after crush? How to break multi digit number into separate digits in VHDL? I found the method .But I do not know how to perform this in VHDL. Let a fixed point number (12 downto -19) like 3456.478396 I need break this number entirely into separate numbers 3456.478396 --> 3,  4, 5,  6,  (dot) ,  4,  7,  8,  3,  9,  6...  into 11 integer type variables. How to perform this in VHDL?  signal example : sfixed(4 downto -4); --'example' contains 9.75 then it is storage as "01001.1100". --simply example ="10011100". --10011100 --&gt; integer 156   How to get back original 9.75 number from 156 to separate into single digits?  I USED,  library ieee; library ieee_proposed; use ieee.std_logic_1164.all; use ieee.numeric_std.all; use ieee_proposed.float_pkg.ALL; use ieee_proposed.fixed_pkg.ALL; use ieee_proposed.fixed_float_types.ALL; Tool to manage and/or make available a list of my publications on the web? From what I understand, the  states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon.  However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics.  Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions?  What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying  because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation! How to install the latest stable version of Chromium? Is there an easier way to install Chromium than what is described ?  Furthermore, is there an easier way to stay up to date with current Chromium releases rather than having to re-build Chromium each release? (a script which automatically grabs and installs the latest build possibly?).  Related: Strange artifacts along window borders after waking computer from sleep mode How do I store phone numbers in PostgreSQL? How to stop the find command after second match? How to make find command, that will show only first two matches and then end, something similiar to  just version for two records. Is it correct to omit preposition before date? iOS 9.2 double touch fingerprint sensor Changing the style of the first *typeset* line of a paragraph Does blocking an attack with the Shield spell still trigger Armor of Agathys? I am contemplating a Oath of Conquest Paladin / Sorcerer multiclass character, with a 1 level dip in hexblade warlock. This character will have access to the Armor of Agathys and Shield spells, which potentially allows for a pretty nice combo. The Shield spell states (PHB 275):  Casting Time: 1 reaction, which you take when you are hit by an attack or targeted by the magic missile spell... Until the start of your next turn, you have a +5 bonus to AC, including against the triggering attack,  It looks like you are indeed hit by an attack, but since you have a bonus to AC, you can potentially force that attack to miss instead. Armor of Agathys states (PHB 215),  If a creature hits you with a melee attack while you have these hit points, the creature takes 5 cold damage.  If I cast Shield and cause an attack that &quot;hit&quot; me to miss, does Armor of Agathys still deal damage? I am leaning toward yes, though this combo seems a little too strong--Shield is already a strong spell, and this character will eventually have 9 1st level spell slots and a 7th level spell slot. Not only can they freely block attacks with Shield, but they can also deal up to 35 (5d10) damage to the attacker without an attack roll or saving throw. While a Warlock would have to spend both of their spell slots to do this once, my character would be able to do this 4 times with spell slots to spare. Some notes:    ask about reducing the damage from a confirmed hit to zero. This question is asking about what happens when the Shield spell causes the attack to miss entirely. A number of online resources, such as the Roll20 5th edition compendium, seem to be missing the correct casting time for the Shield spell. Answerers should make sure they're looking at the correct version from the PHB. Own table of contents? Print out elements from an array with a comma between elements except the last word What exactly is a distribution? I know very little of Probability and Statistics, and am wishing to learn.  I see the word "distribution" used all over the place in different contexts.  For example, a discrete random variable has a "probability distribution."  I know what this is. A continuous random variable has a probability density function, then for $x\in\mathbb{R}$, the integral from $-\infty$ to $x$ of the probability density function is the cumulative distribution function evaluated at $x$.   And apparently just "distribution function" is synonymous with "cumulative distribution function," at least when talking about continuous random variables (question: are they always synonyms?).  Then there are many famous distributions.  $\Gamma$ distribution $\chi^2$ distribution, etc.  But what exactly is a $\Gamma$ distribution?  Is it the cumulative distribution function of a $\Gamma$ random variable?  Or the probability density function of a $\Gamma$ random variable?  But then a frequency distribution of a finite data set appears to be a histogram.  Long story short: in Probability and Statistics, what is the definition of the word "distribution"?    I know the definition of distribution in Mathematics (an element of the dual space of the collection of test functions equipped with the inductive limit topology), but not Probability and Statistics. See screenshot:    Curiously, there are older badge achievement messages further down the list which display properly. In what cases should I use a comma before "or"? Is there a converter that will convert LaTeX to HTML? I have tried several services but the LaTeX diagrams and some of the maths came out weird.  I actually converted the Latex to PDF without a problem. Then I used  to convert from PDF to HTML. The square root signs did not appear properly and some of the Cartesian coordinate plane diagrams had nothing on them except the grid.   \documentclass[12pt,a4paper]{report} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsmath} \usepackage{pstricks,pst-plot} \usepackage{pst-eucl} \addtolength{\hoffset}{0.2cm} \addtolength{\textwidth}{1.1cm} \addtolength{\textheight}{1.0cm} \addtolength{\voffset}{-0.7cm} \linespread{1.3} \begin{document}  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$     $\hspace{30 mm} $     (6)\\  (4) and (6) tell us that\\  (4)  $-1 \leq x \leq 0$  \\  (6)  If $x$ is non-positive then $x \leq -\sqrt{\frac{1}{2}}$\\  Therefore $-1 \leq x \leq -\sqrt{\frac{1}{2}}$\\  \vspace{50 mm}  \begin{align*} \psgrid[unit=2cm](0,0)(-2,-2)(2,2) \pnode(-1.8,.87){b}\uput[50](b){b} \pnode(-1.8,1.8){a}\uput[50](a){a} \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,4) \psset{linecolor=green} \psline[linewidth=1pt](-1.8, 0)(-1.8,-4) \psset{linecolor=blue} \psellipticarc(0,0)(2,2){0}{180} \psset{linecolor=red} \psline[linewidth=1pt,linestyle=dashed,dash=2pt](-2,2)(-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt](-1.41,1.41) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(4,-4) \psset{linecolor=red} \psline[linewidth=1pt]{-&gt;}(-2,2)(-4,4) \end{align*}  \end{document}   PDF output:    PDF->HTML output: Given a set $\mathcal{S}$ of size $n$, let a $k$-partition of $\mathcal{S}$ be a grouping into $k$ disjoint classes $$(S_1 ,S_2,...,S_k)$$   Where the $S_i$ do not necessarily contain the same number of elements.  Let $C(S_1, S_2, ... S_k)$ be the set of combinations which can be made by taking exactly one element of each class, (i.e. $C(S_1, S_2, ... S_k)$ is the Cartesian Product $S_1 \times S_2 \times ...\times S_k$). For a given $(n,k)$, what is the minimum number of k-partitions $N$ such that  $\bigcup\limits_{i=1}^{N} C(S_1^{(i)}, S_2^{(i)}, ...,S_3^{(i)}) = $ All Possible Combinations  Example:  for $n=4$ and $k=2$, $\mathcal{S} = \{1,2,3,4\}$ we can choose:  $$ S^{(1)} = \left\{\left\{ 1, 2\right\}, \left\{ 3, 4\right\}\right\}$$ $$ S^{(2)} = \left\{\left\{ 1, 3\right\}, \left\{ 2, 4\right\}\right\}$$  Then $$C_{1} = \left\{\left\{ 1, 3\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 2, 4\right\}\right\}$$ $$C_{2} = \left\{\left\{ 1, 2\right\} ,\left\{ 1, 4\right\},\left\{ 2, 3\right\},\left\{ 3, 4\right\}\right\}$$  and $C_1 \cup C_2 =$ All possible combinations and $N=2$. $$\frac{1}{n+1}&lt; \log(1+ 1/n)$$  Any ideas? I tried estimating the difference between $1/n$ and the logarithm and comparing with $1/n-1/(n+1)$ but I miserably failed.