I found a couple examples with that word:     The song feels apropos to a midnight jaunt      Clothes that are apropos to the occasion   I'm interested in can we use that following the verb of action. For instance, if I want to express something like the following:  My frined come to my house and bring some candies with him. Could I say It's apropos that you bought the candies, I just have made a tea.
Which is more correct:     Apropos of your earlier comment, I decided to....   or      Apropos your earlier comment, I decided to...   Actually, apropos is so fancy a word, even I, a word maven if I do say so myself, am not entirely sure when to use it.  If I may add to my preposition question, are there any constructions where it would be appropriate to use something like with regard to and not apropos?
Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$ Comparing nullable column values using "IS NULL" versus "= NULL" I'm dealing with nullable values in a column of a table, i'm wondering why this:  with cte as (     select 'c1'::text as c1,null::text as c2     union     select 'c2'::text as c1,null::text as c2 ) select * from cte where c2 is null   returns a result set with the two "rows" having a null value in c2; comparting to this query:  with cte as (     select 'c1'::text as c1,null::text as c2     union     select 'c2'::text as c1,null::text as c2 ) select * from cte where c2 = null   that returns an empty result set!!  so it looks like that field is null is actually different from field = null? the question is... is there "null-compatible operator" to use? so i can write the comparision like: column_name = $1 and argument $1 can be a not null or null value. IF $1 is a null value then it will semantically equals to write as IS NULL, ELSE (if $1 is not a null value) then it would be semantically equals to = $1.  P.D: This question also applies to the case is NOT NULL versus &lt;&gt; NULL Process to convert simple Python script into Windows executable There seems to be a preference for to have the    18 mm - 55 mm 55 mm - 250 mm   lenses, but no mention of why they're better than a single 18-200mm. Could someone please explain why the two separate are better than the one combined, and if anyone has quantified this difference?  (I'm curious about Canon in particular.) Why does kinetic energy increase quadratically, not linearly, with speed? As  says:     [...] the kinetic energy of a non-rotating object of mass $m$ traveling at a speed $v$ is $\frac{1}{2}mv^2$.   Why does this not increase linearly with speed? Why does it take so much more energy to go from $1\ \mathrm{m/s}$ to $2\ \mathrm{m/s}$ than it does to go from $0\ \mathrm{m/s}$ to $1\ \mathrm{m/s}$?  My intuition is wrong here, please help it out! Sneak Damage multiplier: Does it apply to magical damage too? Example:   Dagger 1 = 18 Damage, 25 Shock  Dagger 2 = 33 Damage, no magic  Using the above, when I sneak attack and have the 15x multiple then is the damage caused by Dagger 1 is (18+25)*15 or (18*15)+25 ?  Same question with bows and arrows, does the multiplier apply to potion damage or/and the magical(enchanted)?  In other words, do the multipliers apply to the base damage only or to the final damage being caused by the weapon (base+magical+skill+potion+perks etc)?  I am confused. :-? probability that there is at least one defective part Why won't my villagers breed I have created the following structure. The intention is to create fairly compact houses for villager breeding.   This is a section of the houses.   I have 2 of these structures, which would resolve to 28 doors and 4 other types of houses meaning I have 32 houses in total. healed healed 3 villager zombies and locked them inside a small house. One of them sometimes shows hearts, but they won't breed.   I am playing vanilla Minecraft 1.8. The doors are spruce.  What is wrong with these houses? Is there something wrong with spruce doors or half slab roofs? I have a mesh and in the region around each triangle, I want to compute an estimate of the principal curvature directions. I have never done this sort of thing before and  does not help a lot. Can you describe or point me to a simple algorithm that can help me compute this estimate?  Assume that I know positions and normals of all vertices. Limit $\lim\limits_{x\to0}{\frac{\ln{(e+x)}-e^x}{\cos^2{x} -e^x}}$ Without using L'Hopital's rule find: $$\lim_{x\rightarrow 0}{\frac{\ln{(e+x)}-e^x}{\cos^2{x} -e^x}}$$ I found the first derivative because I planned to use Taylor series:  \begin{align} \frac{d}{dx}{\left(\frac{\ln{(e+x)}-e^x}{\cos^2{x} -e^x}\right)} &amp;=\frac{\left(\frac{1}{x+e}-e^x\right)\left(\cos^2{x} -e^x\right)-\left(\ln(x+e)-e^x\right)\left(\sin{2x}-e^x\right)} {(\cos^2{x} -e^x)^2}\\ &amp;=\frac{\frac{1}{x+e}-e^x}{\cos^2{x}-e^x}-\frac{\left(\ln(x+e)-e^x\right)\left(\sin{2x}-e^x\right)}{(\cos^2{x} -e^x)^2} \end{align}  However, it seems I haven't gone so far. Should I start from the beginning and try a different method?  Source in Croatian: 2.kolokvij, matematička analiza Beamer, page/frame numbers, buttons and Appendix slides Let $x$ be transcendental over $F$. Let $y=f(x)/g(x)$ be a rational function. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$ Let $F$ be a field and $x$ transcendental over $F$. Let $y=f(x)/g(x)$ be a nonconstant rational function with relatively prime polynomials $f,g\in F[x]$. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$.  My attempt: By replacing $y$ with $\frac{1}{y}$ if necessary we assume $\deg g\geq\deg f$. By the Euclid algorithm we assume $\deg g&gt;\deg f$. Then $\deg g=\max(\deg f,\deg g)$, let $n=\deg g$. Let $R(t)$ be this polynomial $$R(t)=yg(t)-f(t)=\frac{f(x)}{g(x)}g(t)-f(t)$$ Then $\deg R=\deg g=n$ because $n=\deg g&gt;\deg f$. The coefficients of $R(t)$ lie in $F(y)$ and hence $R(t)\in F(y)[t]$. Clearly $R(x)=0$.  Now I only have to prove $R(t)$ is actually the minimal polynomial of $x$ over $F(y)$. I believe the next step is to obtain a contradiction that $R(t)$ being reducible can lead to $f,g$ being not coprime, but I can't figure out how to do that.  P.S. A hint is preferred over a full answer. I have a JavaScript object like the following:  var p = {     "p1": "value1",     "p2": "value2",     "p3": "value3" };   Now I want to loop through all p elements (p1, p2, p3...) And get their keys and values. How can I do that?  I can modify the JavaScript object if necessary. My ultimate goal is to loop through some key value pairs and if possible I want to avoid using eval. Split string containing command-line parameters into string[] in C# I have a single string that contains the command-line parameters to be passed to another executable and I need to extract the string[] containing the individual parameters in the same way that C# would if the commands had been specified on the command-line. The string[] will be used when executing another assemblies entry-point via reflection.  Is there a standard function for this? Or is there a preferred method (regex?) for splitting the parameters correctly? It must handle '"' delimited strings that may contain spaces correctly, so I can't just split on ' '.  Example string:  string parameterString = @"/src:""C:\tmp\Some Folder\Sub Folder"" /users:""abcdefg@hijkl.com"" tasks:""SomeTask,Some Other Task"" -someParam foo";   Example result:  string[] parameterArray = new string[] {    @"/src:C:\tmp\Some Folder\Sub Folder",   @"/users:abcdefg@hijkl.com",   @"tasks:SomeTask,Some Other Task",   @"-someParam",   @"foo" };   I do not need a command-line parsing library, just a way to get the String[] that should be generated.  Update: I had to change the expected result to match what is actually generated by C# (removed the extra "'s in the split strings) What are the fundamental differences between the mainstream *NIX shells and what scenarios might prompt you to use one over the other?  I understand that some of it probably comes down to user preference but I've only ever used Bash and I'm interested to hear where another shell might be useful. Also, is there an impact on user-written shell scripts when running under one shell or another or is it simply a matter of changing the shell at the top of the file?  My instinct says it's not that easy. Are moderator tools, specifically flagging or voting to close going to be implemented in the Android app? If we have a dispute, we should at least have the flag option.  What about share option? If we would like to share a question, we could long press on the question title, and the question title, excerpt, and short URL copied to clipboard. Links in newsletter preview link to YouTube The problem: The links at the bottom of the Stack&nbsp;Exchange newsletter link to a YouTube video instead of the right place on the website. I´m not sure if this is an oversight or a clever hack.  Screenshot: What's the fastest way to make money online? Everything in the online world is expensive, well the best things are. We all want the luxurious apartment with 10 cars garage and the best weapons but to achieve that, we need money. Which missions or activities give the most cash ? What are some of the rule changes that Pathfinder brings in comparison to D&amp;D 3.5?  I am curious what changes were made so I can explain to my group what it has to offer. In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x)dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential). Then if we rearrange we get $dy/dx$ which could be seen as a ratio.   I wonder if the author says this because $dx$ is an independent variable, and $dy$ is a dependent variable, for $dy/dx$ to be a ratio both variables need to be independent.. maybe?