I have the following identity which I want to prove:  $$\frac{d}{dt} det(A+tB)|_{t=0} = Tr( Cof(A)^TB)$$  where $Cof(A)$ is the cofactor matrix of $A$, and $A$ is an $n\times n$ complex matrix. The reference is exercise 2 in section 1 of chapter 1 of Basic theory, volume I of PDE by Michael Taylor's book (2011 edition).  I tried the definition, i.e:  $$\frac{d}{dt}\det(A+tB)|_{t=0} = \frac{d}{dt}|_{t=0} \sum_{\sigma \in S_n} sgn(\sigma) \prod_{i=1}^n (a_{i\sigma(i)}+tb_{i\sigma(i)})=$$ $$= \sum_{\sigma \in S_n } sgn(\sigma) \bigg[ b_{1\sigma(1)}\prod_{i=2}^n a_{i\sigma(i)}+\ldots + b_{n\sigma(n)} \prod_{i=1}^{n-1} a_{i\sigma(i)}\bigg]$$  How to proceed, I am not sure how does the cofactor of $A$ will look like in the notation with permutations, otherwise what is the missing step in the proof?  Thanks in advance.
Let $\mathbb{R^{ n×n}}$ be the space of n × n real matrices. Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ is the sum of the determinants of $n$ matrices, in which the $i$th matrix is $A(t)$ except that the $i$th row is differentiated. Use this result to prove that the directional derivative of the determinant function at the matrix $A \in \mathbb{R ^{n×n}}$ along the direction $B \in \mathbb{R^{ n×n}}$ is given by $$(det)^{'} (A; B) = tr(Adj(A)B) = \langle Adj(A)^{T} , B \rangle $$ where $Adj(A)$ is the adjoint of $A$, and where the inner product on $\mathbb{R^{n×n}}$ is the trace inner product given by $\langle X, Y \rangle = tr(X^{T} Y )$ Conclude that $$D(det)(A) = Adj(A) ^T $$.
Placing nodes midway on paths using |- or -| in tikz I make frequent use of |- and -| with computed points in diagrams, like so:  \coordinate (a) at (0,0); \foreach \x in (1,...,10) {     \coordinate (b) at ({whateverFunction(\x)},{whateverOtherFunction(\x)});     \draw (a) |- (b); %Straight line segment     \draw (b) -| (a); %Another straight line segment     \coordinate (a) at (b); }   To get plots like:   I would like to be able to label the midpoint of each line segment, but doing something like node[midway,above] {a} does not act like expected:    I want something like:    Why does midway not work, and what is a good workaround here? Proof by Induction: Prove that $2^n > n^2$, for all natural numbers greater than or equal to $5$ Command for forcing a pointer ungrab (captured mouse release) Sometimes Firefox doesn't release the mouse after dragging, so I need to kill the application to force it to release its pointer grab.  Is there any command to force an application to ungrab the pointer without killing it? "thanks for flagging" popup gets messed up when the comment is automatically deleted For some comments, the comment is automatically deleted when you raise a flag. This includes certain keywords and short +1 comments. But when you raise that flag with the new comment flag interface, the "thanks for flagging" shows up in the top right when the comment is deleted:      As opposed to what happens when you raise a flag on a comment that doesn't auto-delete: Using the expression "the same" for a previously mentioned item Is there any pair of 128-bit strings M and K such that AES$_K$(M) = M?  If yes, how do I go about (efficiently) finding such a pair? How to Upgrade Ubuntu 12.04.2, 12.04.3 to 12.04.4, the next Hardware Enablement Stack? I am currently using Ubuntu 12.04.2 32bit. I installed it using LiveCD.  Tomorrow, 23rd August 2013,Ubuntu 12.04.3 is released. I want to upgrade from Ubuntu 12.04.2 to Ubuntu 12.04.3 without using any LiveCD. Is it possible? If so please suggest me how can I do.  Actually while using Ubuntu 12.04.2 I have troubled with graphics. My graphics card is Intel i845G 64 MB. When Ubuntu starts it works fine and smooth without any lagging, but after sometime it hangs for few seconds (1 or 2 seconds) with a garbage screen and becomes sluggish. All windows and browsers start lagging and also it is not possible to play any video in any player (VLC, Movie Player, Xnoise, SMPlayer etc..).  I think Upgrading to Ubuntu 12.04.3 could fix my problem.  Any help will be greatly appreciated..    I've posted an answer to upgrade Ubuntu 12.04.x to 12.04.4. Matplotlib make tick labels font size smaller Can I get smaller tires without changing my rims? I currently ride a hybrid that came with 700x32c tires. Can I swap out a narrower tire without changing the rims, or do rims only fit a specific size of tire?   How small can I go before I have to buy new rims?  I hear having narrower tires makes you go quite a bit faster.  Does it really help that much? Why does printf() not print anything before sleep()? If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation:  $$z^2w'' + \alpha zw' + \beta w = 0$$  where $w$ is a function of $z$ and $\alpha, \beta$ are constants. How would the change of variables $t =\ln(z)$ transform this equation? Updating Location of Point Features using XY Fields? I have changed the coordinates of couple points in the attribute table and I want to update the location. I have found the script that is working in field calculator.   Does anybody know how to write it using a Python script in ArcGIS Desktop (not in Field Calculator)?  def XYsetVALUE( shape, X_value, Y_value):    point = shape.getPart(0)    point.X = X_value    point.Y = Y_value    return point  __esri_field_calculator_splitter__ XYsetVALUE ( !SHAPE!, !X_COORD!, !Y_COORD! )   COORDINATE FIELD TYPE ------ STRING !!!! Fourier Series for $|\cos(x)|$ It would be nice to be able to put emphasis on specific bits of a fragment or of a quote (bold can help for a quote but I find that it makes things less readable sometimes). What do you think? Update: On , you can highlight particular lines by prefixing each line with @@. See also: Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.   While it's a good book, I'm not sure it's suited for self study by itself.  I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway. Convergence of sequence of real numbers and (sub)subsequence Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of real numbers. A well known result is:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ converges to $a\in \mathbb{R}$.   Now, I'm working on an exercise which essentialy asks to prove the following:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if for every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ there is a (sub)subsequence $(x_{n_{k_j}})_{j\in \mathbb{N}}$ which converges to $a$.   Now, one direction is trivial. If in fact $(x_n)_{n\in \mathbb{N}}$ converges to $a$, then by the result I've mentioned, every subsequence of $(x_{n})_{n\in \mathbb{N}}$ converges to $a$. Let then $(x_{n_k})_{k\in \mathbb{N}}$ be an arbitrary subsequence. In that case, we know that it is a subsequence of itself, and hence we've found a (sub)subsequence which converges to $a$.  The other way around seems quite complicated for me. The reason is as follows: I must assume that every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ has a (sub)subsequence which converges to $a$ and the proof that the original one converges to $a$.  My problem is exactly that "a". If it had all (sub)subsequences converging to $a$, then by the result i've mentionted, the subsequence $(x_{n_k})_{k\in \mathbb{N}}$ would converge to $a$. In that case, we would have every subsequence converging to $a$ and the result would follow.  My idea to prove the claim was then:  First remember that a subsequence is the restriction of the original one $x: \mathbb{N}\to \mathbb{R}$ to an infinite subset $\mathbb{N}'\subset \mathbb{N}$. In that case a (sub)subsequence is the restriction of $\tilde{x} = x|\mathbb{N}'$ to an infinite subset $\mathbb{N}''\subset \mathbb{N}$. But since $\tilde{x}$ is the restriction of $x$ to $\mathbb{N}'$ and since $\mathbb{N}''\subset \mathbb{N}'$, the restriction of $\tilde{x}$ to $\mathbb{N}''$ is itself the restriction of $x$ to $\mathbb{N}''$.  In other words, every (sub)subsequence can be viewed as a subsequence of the original one. Considering this, by hypothesis every subsequence has a (sub)subsequence which converges to $a$. But by this argument, this means that we can replace every subsequence with a convergent one. And then we can use the result I stated.  Is my idea correct? Is this really the way to prove the claim? My only doubt is that I need every subsequence converging to $a$ to use the result. In this case I can't see how I can argue that this implies every subsequence converges to $a$.  How can I finish this proof? Double and triple substitution in bash and zsh Follow-up to the background part in .  In bash I can use ${!FOO} for double substitution, in zsh ${(P)FOO}. In both, the old-school (hack-y) eval \$$FOO works.  So, the smartest and most logical thing for me would be ${${FOO}}, ${${${FOO}}}… for double/triple/n substitution. Why doesn’t this work as expected?  Second: What does the \ do in the eval statement? I reckon it’s an escape, making something like eval \$$$FOO impossible. How to do a triple/n substitution with that that works in every shell? Statistics and causal inference? In his 1984 paper , Paul Holland raised one of the most fundamental questions in statistics:     What can a statistical model say about   causation?   This led to his motto:     NO CAUSATION WITHOUT MANIPULATION   which emphasized the importance of restrictions around experiments that consider causation.  Andrew Gelman makes :     "To find out what happens when you change something, it is necessary to change it."...There are things you learn from perturbing a system that you'll never find out from any amount of passive observation.   His ideas are summarized in .  What considerations should be made when making a causal inference from a statistical model? Possibly two questions in here: Are these sentence constructions logical, and if they are, why are they different?     I swam across the Ocmulgee River.   I swam across the Pacific Ocean.   I swam across the Red Sea.   I swam across Lake Winnipeg.   I can't think of a way of phrasing the lake example while using an article with a proper noun. I thought at first it was because Lake Winnipeg is different; it's generic portion (the "Lake") comes first and the name of that body comes second, but I swam across Falls Lake is the same.   Yet the sentence I swam across the lake sounds right to me. ...Why? Uncertainty principle and multiple observers