I was reading about complexities where n is the number of elements. I do some operation different ways and which results in these two different complexity. And have a doubt that which complexity is less with bit explanation.
I'd prefer as little formal definition as possible and simple mathematics.
Powering NiChrome wire from battery I'm looking to power nichrome wire from battery. The problem I ran into is that desired length/width of the wire has very small resistance ~ 8 Ohm.  Several kinds of batteries I've tested have internal resistance much higher then that, which leads to wire staying cold and battery getting hot.  Is there any way to increase current drawn from the battery? Some kind of amplifier maybe?  Edit:    1.2V(rechargable) and 1.5V one, while not getting hot itself did not provide enough power to heat the wire. 3V did not provide enough power, even stacking 3 of them in series, only provided about 1V of voltage on the wire, and not enough amperage to heat the wire. 9V battery did provide enough amperage to heat the wire, but at the same time got very hot very quickly itself.   PS. I was wondering if  might work for this purpose? I have some troble understanding the Kernel density estimation.  If a consider the next example:  s&lt;-0 z&lt;-density(rnorm(25)) f&lt;-function(i)approx(z$x,z$y,xout=i)$y for(i in seq(min(z$x),max(z$x),length.out = 10000)) s&lt;-s+f(i) s   I expected getting that s is approximately equals to one since I am considering an approximation for the integral of the Kernel density function (that I suppose is also a density). However, I don't get the expected result. Why it doesn's sum one?  The idea behind this is that I wanted to use the Kernel as the density of my data points. I was interpolating to get my explicit function and use with other x values. Or do you know another way to get a density function estimated by fitting? Which statistical test to analyse? I am trying to find out how inflammatory markers change in response to a mechanical device change. I have daily readings for the 5 days before the event and for the 5 days after the event. I am expecting a rise in inflammatory markers in the pre readings followed by a fall after the device change. I also need to find out how the change differs in two subgroups (sample sizes of the subgroups are n=21 and n=88). The aim is to find out if the change is significant, to consider these markers as predictors of the device failing and hence needing to be changed.    I would like to know what test would be appropriate to analyse this. I need to keep things simple as I and the people reading the results are non-statisticians. Can I become leader of the Thieves' Guild without finishing the main chain and is there any benefits from being the leader anyway? I completed the quest where you get the Skeleton Key and I don't really want to continue the chain until I've done a lot of the other stuff in the game, since the key is so useful. Brynjolf said before that I would become leader of the guild after everything was over.  Can I become the leader anyway without finishing the Nightingales quest chain? Are there any bonuses for being the leader? What is the difference between a field and a property? What techniques have been used to repeat geometry or other detail to the infinite horizon? When I composed my     I would have liked to have the camera pitch up enough to show the horizon, except I didn't have enough geometry to extend the cobblestone street and the sidewalk all the way to infinity (mostly because that would require infinite stuff).  Even in that shot it is clear that I need to widen the "sidewalk" to cover up the "sky"box.    Contrast with     where the ray tracer can use the procedurally replicated geometry.   is a failed attempt at faking out the horizon, even with low detail.  It fails because lighting's interaction with the geometry gives it away.  Some techniques that prevent you from having to deal with infinity include   put the scene in a box (usually a room) put the scene in a canyon (which is just a box without a top so you can see the relatively featureless sky) route the road into a tunnel, or around a corner and hide the horizon with other geometry (which is really just a variant on the canyon). have no detail on the ground at all (flat color or gradient)   Those are all dodging the question of interest to me:  What techniques have people used in blender to fake various kinds of detail repeating into the distance (instead of just obstructing the view of infinity)? When I am writing I often come across the words "around" and "round." I was wondering what is different between them, and how they would be used in different contexts. I already know that if groups $G,H,K$ are finitely generated abelian groups, following is true.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. I prove this by uniquness of factorization of finitely generated abelian groups.  My questions are   If group $G$ is finite,$G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$? Can you give me a easiest proof and it can be proved by projection function on external direct product? If $G,H,K$ are groups.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. This statement is false. What is a counterexample?   *$\times$ is external direct product What are the uses of "using" in C#? Error message on Terminal launch I'm editing a book that includes about 25 papers written in LaTeX by different authors.  Since many of them use the same \label, to avoid collisions I redefined the \label command (and \ref and \pageref and \cite and \bibitem)  so that if the third paper has  \label{main_theorem} the aux file actually uses the tag "iii:main_theorem", and \ref{main_theorem} actually refers to the same tag.  I did a similar thing for \cite and \bibitem.  %*************** bib/cite hacks \let\TheReal@citex=\@citex \let\TheReal@bibitem=\@bibitem \let\TheReal@lbibitem=\@lbibitem \gdef\@lbibitem[#1]#2{\TheReal@lbibitem[{#1}]{#2\roman{chapter}}} \gdef\@bibitem#1{\TheReal@bibitem{#1\roman{chapter}}} \gdef\@citex[#1]#2{% \toks@={}\count@=0% \@for\@citec:=#2\do{% \advance\count@ 1 \ifnum\count@=1\edef\0{\the\toks@\expandafter\expandafter\noexpand\@citec\roman{chapter}}% \else\edef\0{\the\toks@,\expandafter\expandafter\noexpand\@citec\roman{chapter}} \fi% \toks@=\expandafter{\0}}% \TheReal@citex[#1]{\the\toks@}% } %************** label/ref/pageref hacks  %%%%%%%%%%%%% must be before ams pkgs \let\TheRealLabel=\label \let\TheRealRef=\ref \let\TheRealPageref=\pageref \gdef\label#1{\TheRealLabel{\roman{chapter}:#1}} \gdef\ref#1{\TheRealRef{\roman{chapter}:#1}} \gdef\pageref#1{\TheRealPageref{\roman{chapter}:#1}}   This all worked fine until I used hyperref, which seems to ignore the redefinition of \label, whether I load it before or after the above code.  Any suggestions?  It would be nice if the redefinition would work whether or not hyperref was loaded, so I don't want to just grab the defintion of \label out of the package and tweak it.  I'm also open to alternative methods.  Redefining \label and \ref just seemed the most straightforward (to me) solution.  Note that I also want to be able to refer to the inner labels from outside, as well.  For example, I might want to say "the main theorem of the third paper is  \ChapRef{iii}{main_theorem}  from within the introduction (a different chapter). I installed vsFTPd for running an FTP server on  (Wheezy). I checked the vsFTPd version was 2.3.5, and I configured it like so:  listen=YES local_enable=YES write_enable=YES chroot_local_user=YES pasv_min_port=15000 pasv_max_port=15200 allow_writeable_chroot=YES   I followed these articles for solving this problem:       And many others on Google and forums, but my problem was not solved.  NOTE: I have solved this problem on  (Precise Pangolin), but that solution does not work on Debian 7.3.  I really mixed up on it?! For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.   My attempt: Let  $$\begin{align*} f_n(x) &amp;= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2\right)-\ln\left(1-2a\cos x+a^2\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(\displaystyle\frac{1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2}{1-2a\cos x+a^2}\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(1+\dfrac{1}{n}\left(\displaystyle\frac{2a-2\cos x+\frac{1}{n}}{1-2a\cos x+a^2}\right)\right)}{\frac{1}{n}}. \end{align*}$$ Now it is easy to see that $f_n(x) \to \frac{2a-2\cos x}{1-2a\cos x+a^2}$ as $n \to \infty$. $|f_n(x)|\le \frac{2a+2}{(1-a)^2}$ RHS is integrable so $\lim_{n\to\infty}\int_0^\pi f_n(x)dx = \int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2} dx=I'(a)$. But  $$\int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2}=\int_0^\pi\left(1-\frac{(1-a)^2}{1-2a\cos x+a^2}\right)dx.$$ Consider $$\int_0^\pi\frac{dx}{1-2a\cos x+a^2}=\int_0^\infty\frac{\frac{dy}{1+t^2}}{1-2a\frac{1-t^2}{1+t^2}+a^2}=\int_0^\infty\frac{dt}{1+t^2-2a(1-t^2)+a^2(1+t^2)}=\int_0^\infty\frac{dt}{(1-a)^2+\left((1+a)t\right)^2}\stackrel{(*)}{=}\frac{1}{(1-a)^2}\int_0^\infty\frac{dt}{1+\left(\frac{1+a}{1-a}t\right)^2}=\frac{1}{(1-a)(1+a)}\int_0^\infty\frac{du}{1+u^2}=\frac{1}{(1-a)(1+a)}\frac{\pi}{2}.$$ So $$I'(a)=\frac{\pi}{2}\left(2-\frac{1-a}{1+a}\right)\Rightarrow I(a)=\frac{\pi}{2}\left(3a-2\ln\left(a+1\right)\right).$$  It looks too easy, is there any crucial lack?  $(*)$ — we have to check $a=1$ here by hand and actually consider $[0,1), (1,\infty)$ but result on these two intervals may differ only by constant - it may be important but in my opinion not crucial for this proof. Looking for succinct way of describing the act of solidifying an idea into reality Looking for a succinct way of describing the act of solidifying an idea into reality. "Building [idea] out" seems too generic but I'm trying to avoid a long winded, step by step explanation of the process.  Any ideas would be greatly appreciated. Thanks! How can I calculate sensor size (width and height) from focus length, vertical, and horizontal angle of view? “situation where” vs. “situation in which” In my mother tongue I can use the word where not only to describe something connected to a location, but also to substitute in which.   My question is: Is it correct to use where in a sentence like this?       Describe a situation where you had to show leadership.   Or it would be better to use in which instead? Symbols of derivatives Accelerating particles to speeds infinitesimally close to the speed of light? What is the maximum size of a web browser's cookie's key? What is the maximum size of a web browser's cookie's key?  I know the maximum size of a cookie is 4KB, but does the key have a limitation as well? How do I pass this array of c strings correctly?