faithful representation related to the center
$\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ .
Does the derivative of a continuous function goes to zero if the function converges to it? Why doesn't Cantor's diagonal argument also apply to natural numbers? In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string.  My question is: why can't we begin by representing each natural number as an infinite bit string? So that 0 = 00000000000..., 9 = 1001000000..., 255 = 111111110000000...., and so on.  If we could, then the diagonal argument would imply that there is a natural number not in the natural numbers, which is a contradiction. How to differentiate read and unread messages in android ListView? Find a function value given 2 points Given $f(x)$, which is differentiable at every point such that: $f'(x) \ge -5$ for every $ x \in R$   $f(2) = -13$, $f(9) = -48$  Prove that:$ f(3) = -18$  Now it's quite obvious that $ f(3) = -18$ since that if we know that the lowest value of the derivative is -5, the difference between $f(2)$ and $f(9)$ is $-48-(-13)=-35$  that means that that in the area $[2,9]$, $f'(x)=-5$ right? that's why I told that $f(3) = -18$, but I feel that my proof is insufficient, probably because I'm missing some theorems or such. can someone help me structure this proof into something RIGHT :)?  Thanks a lot! I'm trying to write a page that calls PHP that's stored in a MySQL database. The page that is stored in the MySQL database contains PHP (and HTML) code which I want to run on page load.  How could I go about doing this? Suppose that sequences of real numbers satisfy:  \begin{align*}   a_n &amp;= \sum\limits_{r=0}^n \binom{n}{r} b_r \\ \end{align*}  Prove that:  \begin{align*}   (-1)^n b_n &amp;= \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s \\ \end{align*}  My work:  The $n=0$ case:  \begin{align*}   a_0 &amp;= b_0 \\   b_0 &amp;= a_0 \\ \end{align*}  Then $n=1$ case:  \begin{align*}   a_1 &amp;= \binom{1}{0} b_0 + \binom{1}{1} b_1 = b_0 + b_1 \\   -b_1 &amp;= \binom{1}{0} a_0 - \binom{1}{1} a_1 = a_0 - a_1 \\   b_1 &amp;= a_1 - a_0 = b_0 + b_1 - a_0 = b_1 \\ \end{align*}  The inductive step:  \begin{align*}   (-1)^{n+1} b_{n+1} &amp;= \sum\limits_{s=0}^{n+1} \binom{n+1}{s} (-1)^s a_s \\ \end{align*}  I'm not sure where to go from here. I tried substituting $a_s$ into that last equation and it didn't help. Any ideas? Which languages was Gandalf proficient in? How soon can I re-enter the USA having stayed for 90 days under the Visa Waiver Program? I am currently visiting the USA from the UK (I am a UK citizen) and am intending on staying here for the full 90 days under the Visa Waiver Program.     How long would I have to stay out of the US at the end of this period before returning again?    Would I have to return to the UK, or could I stay with friends in another country (such as Canada) for a while? Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$? Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any such matrix $A$, the set $K[A]$ of polynomials in $A$ is a commutative subalgebra of $M_n(K)$; the question is whether for any pair of commuting matrices $X,Y$ at least one such commutative subalgebra can be found that contains both $X$ and $Y$.  I was asking myself this in connection with frequently recurring requests to completely characterise commuting pairs of matrices, like . While providing a useful characterisation seems impossible, a positive anwer to the current question would at least provide some answer.  Note that in many rather likely situations one can in fact take $A$ to be one of the matrices $X,Y$, for instance when one of the matrices , or more generally if its  (so coincides with the characteristic polynomial). However this is not always possible, as can be easily seen for instance for diagonal matrices $X=\operatorname{diag}(0,0,1)$ and $Y=\operatorname{diag}(0,1,1)$. However in that case both will be polynomials in $A=\operatorname{diag}(x,y,z)$ for any distinct values $x,y,z$ (then $K[A]$ consists of all diagonal matrices); although in the example in  the matrices are not both diagonalisable, an appropriate $A$ can be found there as well.   I thought for some time that any maximal commutative subalgebra of $M_n(K)$ was of the form $K[A]$ (which would imply a positive answer) for some $A$ with minimal polynomial of degree$~n$, and that a positive answer to my question was in fact instrumental in proving this. However I was wrong on both counts: there exist (for $n\geq 4$) commutative subalgebras of dimension${}&gt;n$ (whereas $\dim_KK[A]\leq n$ for all $A\in M_n(K)$) as shown in , and I was forced to correct  in the light of this; however it seems (at least in the cases I looked at) that many (all?) pairs of matrices $X,Y$ in such a subalgebra still admit a matrix $A$ (which in general is not in the subalgebra) such that $X,Y\in K[A]$. This indicates that a positive answer to my question would not contradict the existence of such large commutative subalgebras: it would just mean that to obtain a maximal dimensional subalgebra containing $X,Y$ one should in general avoid throwing in an $A$ with $X,Y\in K[A]$. I do think these large subalgebras easily show that my question but for three commuting matrices has a negative answer.  Finally I note that  to the cited MO question mentions a result by Gerstenhaber that the dimension of the subalgebra generated two commuting matrices in $M_n(K)$ cannot exceed$~n$. This unfortunately does not settle my question (if $X,Y$ would generate a subalgebra of dimension${}&gt;n$, it would have proved a negative answer); it just might be that the mentioned result is true because of the existence of $A$ (I don't have access to a proof right now, but given the formulation it seems unlikely that it was done this way).  OK, I've tried to build up the suspense. Honesty demands that I say that I do know the answer to my question, since a colleague of mine provided a convincing one. I will however not give this answer right away, but post it once there has been some time for gathering answers here; who knows somebody will prove a different answer than the one I have (heaven forbid), or at least give the same answer with a different justification. I'm too spoiled by Windows utilities that take a digital snapshot of your entire drive, which you can then restore from in the event of a drive crash.  (e.g. like Time Machine for Mac OS X).  Is there a similar way of doing this in Linux? What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic? Evidently, $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ has order $4$, but I think it's infinite.   The four cosets are listed as $(0,0) +  \langle (2,2) \rangle$, $(0,1) +  \langle (2,2) \rangle$, $(1,0)+  \langle (2,2) \rangle$ and $(1,1) +  \langle (2,2) \rangle$. However, $(2,0)$ doesn't appear to be in any of these cosets. Maybe the answer I'm being told is wrong. Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$ Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$  I have expanded it this far:  $$\frac{k \cdot n!}{k!(n-k)!} = \frac{n \cdot (n-1)!}{(k-1)!(n-k)!} $$  but then I am stuck from where to go from there. When lasers are used to cool atoms or ions, etc where does the heat go? According to the first law of thermodynamics, sourced from wikipedia "In any process in an isolated system, the total energy remains the same."  So when lasers are used for cooling in traps, similar to the description here:  where is the heat transferred?    From what I gather of a cursory reading on traps, whether laser, magnetic, etc the general idea is to isolate the target, then transfer heat from it, thereby cooling it.  I don't understand how sending photons at a(n) atom(s) can cause that structure to shed energy, and this mechanism seems to be the key to these systems. Looking at your network profile, there is no number anywhere in your entire profile that has a sum of all your reputation across the network except for the flair page, which isn't always accurate because it's cached, and isn't exact because it rounds. Yet, there's so much empty space on the profile page.  My simple idea is to list the user's total reputation and badge counts below their Gravatar image like on any normal profile page across the network (keeping the rule of only accounts with over 200 reputation). It would make it incredibly easy to find and I wouldn't have to go search for myself on Area 51 to figure it out. Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice?  I mean : To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.  The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here). 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. Does Quicksilver's walkman speed up with him? Python/Matplotlib - Is there a way to make a discontinuous axis? How can I start a process over SSH such that it will continue to run after I disconnect? How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$