On diagrams the transistor have usually a direction - it is assumed that (for bipolar NPN) current goes from base to emitter and it is reflected in symbol:         &ndash; Schematic created using   However when if I look how transistors are actually being built it seems that there is no difference between collector and emitter - both are N regions (for NPN transistors). Is there a difference between them I can't see or is it just a convention to allow reader get the intended direction or current?  (I described bipolar but I'm also interested in FETs)
the (surely simplistic) model of a bipolar junction transistor one is taught in foundational physics course appears to be symmetric. - So, what is the difference between the collector and the emitter of a real BJT? If the transistor were symmetric one would not make this distinction...  Also:   Do BJTs have a 0.6V voltage drop like diodes? Are BJTs conductive in both directions, i.e. E-C and C-E?   Many thanks.
MySQL Date Between While I was looking at the values of the zeta function for the first natural numbers, I noticed that the sum of the values minus $1$, converge to $1$. Better put: $$\sum_{n=2}^{\infty} \left(\zeta(n)-1\right) = 1 $$ Furthermore, if you use only the even numbers for the zeta function, the sum will converge to $\frac{3}{4}$, or $$\sum_{n=1}^{\infty} \left(\zeta(2n)-1\right) = \frac{3}{4}$$  Leaving $$\sum_{n=2}^{\infty} \left(\zeta(2n-1)-1 \right)= \frac{1}{4}$$  This is probably common knowledge among mathematicians, but I couldn't find much about it on the internet. Is there a proof of this or perhaps even a simple explanation why this is so? Temporarily deleted answer missing from asker's inbox I  and . This answer does not appear in my inbox. I never saw a notification, and as of now the latest notification from CS is  {"SiteUrl":"cs.stackexchange.com","FaviconUrl":"https://cs.stackexchange.com/favicon.ico","CreationDate":"17 hours ago","Url":"https://cs.stackexchange.com/questions/641/language-of-the-graph-of-an-affine-function/667#667","Title":"Language of the graph of an affine function","Summary":"Hint:\n\n$53 \\cdot 100000010000000000 + 4 = 5300000530000000004$.\n\nSketch:\n\n\n  ! Assume $a$ has $n$...","Type":"answers","Count":3,"IsNew":false}   This answer was posted, deleted and undeleted while I wasn't online. I suppose the deletion removed the entry from my inbox. Then the undeletion should have recreated an entry but didn't. How to find a starting point for root finding in complex plane? The function $f(x) = x^2 +1$ has zeros in the complex plane at $x = +i$ and $x = -i$. Is there a real starting point for complex Newton's method such that the iterates converge to either of these zeros? Is there a complex starting point too?  I know that Newton's method for finding roots is $x_{n+1} = x_n - \frac {f(x)}{f'(x)}$. But I don't know how to find the proper starting point. How can I give a user one password for normal usage when logging in, and another password for system administration and sudo access? I want one user to have two passwords. I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] Avoiding confusion over the term "Research Assistant" $\sum_{n=-\infty}^{\infty}\frac{1}{(u +n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$ I've already see a  by Marko Riedel which I list it follows:     The standard way to treat these sums is to integrate   $$ f(z) = \frac{1}{(z+\alpha)^2} \pi \cot(\pi z)$$   along a contour consisting of a circle of radius $R$ and with $R$ going to infinity and hence being larger than $\alpha$, where the circle does not pass through the poles on the real axis.   Now along the semicircle in the upper half plane we have   $$|f(z)| \le \frac{1}{(R-|\alpha|)^2}\pi \left|\frac{e^{i\pi R\exp(i\theta)} + e^{-i\pi R\exp(i\theta)}} {e^{i\pi R\exp(i\theta)} - e^{-i\pi R\exp(i\theta)}}\right|= \frac{1}{(R-|\alpha|)^2} \pi  \left|\frac{e^{2i\pi R\exp(i\theta)}+1}{e^{2i\pi R\exp(i\theta)}-1}\right| &lt; \frac{1}{(R-|\alpha|)^2} \pi   \frac{1+e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}} {1-e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}}$$   This last term is clearly $O(1/R^2)$ as $R$ goes to infinity as the quotient of the two exponentials goes to one since $\exp(-R)$ vanishes and there is no singularity when $\theta = 0$ or $\theta = \pi$ as $R\cos\theta = \pm R$, which is not an integer by the assumption that the circle avoids the poles and hence cannot be one.     My question: It is clear that for each $z=Re^{i \theta}$ , $f(z)=O(1/R^2)$ ，but can this guarantee that the last term on RHS is uniform bounded ? Since $\theta$ varies on a compact interval $[0, \pi]$ , I want to show that for each $ \theta \in [0, \pi]$ , and a fixed positive number $M \gt 1 $ there exist an open ball contains $\theta$ and a fixed $R_{\theta}\gt 0$ , for every $R \ge R_{\theta}$ and $x \in $ the open ball $$\frac{1+e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} {1-e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} \le M$$  For $\theta \neq 0,\pi$ , it is easy to find the desired $R_{\theta}$ and the open ball , but if $\theta = 0$ or $ \theta = \pi$ , for every open ball containing $\theta$ , it behave erratically near $\theta$ , I have no idea how to deal with this . How do I find out which process is eating up my bandwidth? I think I'm being the victim of a bug here. Sometimes while I'm working (I still don't know why), my network traffic goes up to 200 KB/s and stays that way, even tough I'm not doing anything internet-related.   This sometimes happens to me with the CPU usage. When it does, I just run a top command to find out which process is responsible and then kill it. Problem is: I have no way of knowing which process is responsible for my high network usage. Both the resource monitor and the top command only tell me my total network usage, neither of them tells me process specific network info.  I've found questions here about monitoring total bandwidth usage, but, as I mentioned, that's not what I need. Is there another command I can use to find out which process is getting out of hand?  The command iftop gives results that disagree entirely with the information reported by System Monitor. While the latter claims there's high network traffic, the former claims there's barely 1 KB/s.  I've already tried killing all the obvious ones (Firefox, update-manager, Pidgin, etc) with no luck. So far, restarting the machine is the only way I found of getting rid of the issue. Primary Decomposition of $\mathbb{R^n}$ under T I have a question related to the Primary Decomposition Theorem for a vector space under a linear transformation.  The following question is a direct corollary of this theorem.  However, I wish to prove it without calling upon theorem.  I am currently studying for a comprehensive exam and ran across this question.  I never really felt comfortable with the proof I saw of the Primary Decomposition Theorem so I hoping with your help with this problem I will understand this particular problem and better understand that proof as I believe their proofs will be essentially the same techniques.   Question:  Suppose we have a linear transformation $T: \mathbb{R^n}\to \mathbb{R^n}$ with minimal polynomial $m(x)=f(x)g(x)$ and $f(x),g(x)$ are relatively prime.  Show that we can write $\mathbb{R^n}=M\oplus N$ where $M$ and $N$ are $T$-invariant subspaces s.t $f(x)$ is the minimal polynomial for $T|_M$ and $g(x)$ is the minimal polynomial for $T|_N$.   So I know that I want to use $M=ker(f(T))$ and $N=ker(g(T))$ but after that I really don't have much.  Since this isn't a homework problem for a class I am essentially looking for someone to present me a proof of this simpler question and walk me through their logic so that I can hopefully understand the general theorem and its proof.  Thanks in advance for the help! Are there any concrete rules that say which words (parts of speech) in a title should start with a capital letter? What would be a correct capitalization for the title of this question? The following is an Exercise of Conway's Functional Analysis.   Prove that $X$ is a $\,Banach$ space iff whenever $\{x_n\}$ is a sequence in $X$, such that $\sum \| x_n \| &lt; \infty$, then $\sum x_n$ converges.   I easily can show that if $X$ is a Banach space then $\sum x_n$ converges. My problem is showing that $X$ is a Banach space. For this I suppose that $\{s_n\}$ is a Cauchy sequence in $X$, then I want to make a series. For this I do not have any idea. Please help me. Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $ I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand.  This problem is from some old high school IMO training courses.  I am not sure how to solve it. The only information that may be of help is $$ \sin x=\sum_{n=1}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \quad \cos x=\sum_{n=1}^\infty \frac{ x^{2n}}{(2n)!} \quad \forall \ x $$ Thanks, I am looking for a complete solution, not a description as of what to do. Could a ship warp into Borg cube and destroy it? During the Wolf-359 incident, 39 starships were destroyed, and 11,000 people were killed or assimilated.  Why didn’t one of the smaller ships warp into that one Borg cube? Doesn’t warping into it destroy anything?  Even if it just warps through without touching, one could program the navicom to self destruct while the ship is in the middle of Borg cube. MathJax isn't currently working on the Android app, which makes it totally useless for the 'technical' sites like  or .  For example, in the Android app:    Same post in a (mobile) browser:     Please add this functionality! What exactly happened with Dr. Crusher's husband and Captain Picard? Find the MLE of the unknown parameter $\theta$ when $X_1,X_2,...,X_n$ is a sample from the distribution whose density function is: $$f_X(x) = \frac12e^{-|x-\theta|}, -\infty&lt;x&lt;\infty$$ What I did so far: $$\text{maximize }(\frac12)^ne^{-|x_1-\theta|-|x_2-\theta|-...-|x_n-\theta|} \\ \text{thus we want to minimize }|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|$$ Since $|x_i-\theta|$ is positive for all $i$, we want to solve for $\theta$ so: $$|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|=0$$ What do I do from here? examples:         Gift (German) = poison   poisson (French) = fish   embarazada (Spanish) = pregnant   sauce (Spanish) = willow   triviale (Italian) = vulgar   parentes (Portuguese) = relatives   slim (Dutch) = smart How does ssh run a command? I'm using Bash on both client and server. When running a command over SSH:   ssh &lt;host&gt; 'declare' gives a list of shell variables. ssh &lt;host&gt; 'mount' gives a list of mountpoints.   However, declare is a Bash builtin, while mount is an external command. How does SSH know which to run if there is a shell builtin and an external command with the same name on the server? I was asked to evaluate the following sum: $$\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$$  I'm trying to use $$\int_{0}^{\infty} \frac{\sin(x)}{x}dx=\frac{\pi}{2}$$ However, it doesn't seem to work. Any help is greatly appreciated.