rotate to movement direction
How can I calculate the angle and proper turn direction between two 2D vectors?
Can you give a simple intuitive explanation of IRLS method to find the MLE of a GLM? Background:   I'm trying to follow .  I understand the basics of MLE estimation: likelihood, score, observed and expected Fisher information and the Fisher scoring technique. And I know how to justify .    The question:   I can't understand even the first line of this method :(   What's the intuition behind the $z_i$ working variables defined as:  $$ z_i = \hat\eta_i + (y_i -\hat\mu_i)\frac{d\eta_i}{d\mu_i}$$  Why are they used instead of $y_i$ to estimate $\beta$?   And what's their relation with the response/link function which is the connection between $\eta$ and $\mu$  If anyone has a simple explanation or can direct me to a more basic-level text about this I would be grateful. What do the nomodeset, quiet and splash kernel parameters mean? What do Kernel command line parameters (also also known as boot options) mean?  When you go to Ubuntu (or better, GRUB) options, you have things like nomodeset, quiet and splash. What do these options mean?  I am also interested in other parameters and would be interested to see descriptions of any kernel parameter (like acpi), but these are not essential. Why are the even and odd Regge trajectories degenerate? This is an old classic which I don't think ever got a clear answer. The Gribov-Froissart projection that gives the relativistic version of Regge trajectories treats even angular momentum differently from odd angular momentum. The trajectory functions in general are separate for even and odd angular momentum.  But in QCD, the odd trajectories interpolate the even trajectories in every case--- the two trajectories are degenerate. Is there a way of understanding why the even-odd trajectories are degenerate? Is is a symmetry argument? Can you find a natural system where they are not degenerate?  A quick google search revealed this reference, which doesn't answer the question, but gives an experimental signature for the degeneracy:  , There is probably excellent data by now on this. passing values between forms (winforms) My problem is how to somehow relate the the gcd and congruence. I know that $(a,b) = ax + by$. I also know that $a \equiv b \pmod n$ means $n\mid a-b$. Any hints?  Thanks! I notice that in a pdf file compiled from Latex, "fi" such as in "field" cannot be separated as "f" and "i" when copying text out of the pdf file. I wonder why and if this can be changed? Thanks and regards! Is it correct to say "What was your name?"? The reason I am asking this is, generally the name of the person will not change. One should say "What is your name?" Formula for coordinates (lat, lon), azimuth and distance. I would like to know how to find a set of coordinates on a small circle some distance from another point.   For example, lets say A = (39.73, -104.98) #Denver  B = (39.83, -106.06) #point approximately 50 nautical miles away from A along the great circle track to C  C = (40.75, -111.88) #Salt Lake City, approximately 323 nautical miles away from A  How do I get the set of orange coordinates when I don't know the angle but I do know the cross track distance?  For concreteness, lets say the outer points near B have a cross track distance of +- 30 nautical miles and the inner points have a cross track of +- 15nm.  And the points near C have a cross track of +- 25 nm.  Also I have an initial bearing from A to C of -1.34 #radians This is a special case of , which, I think, might have drawn little attention because it was too general. In this question, I would like to consider a specific case.  Take a classical system given by the action $$ S[x] = \int dt \left[ \frac{\alpha}{2} \ddot{x}^2 + \frac{\beta}{2} \dot{x}^2 + \frac{\gamma}{2} x^2 \right]. $$  This action is Lagrangian, but it is not what we are usually dealing with in physics, because the Lagrangian contains second time derivatives.  The equations of motion are:  $$ \alpha \ddddot{x} - \beta \ddot{x} + \gamma x = 0. $$  This can be solved as usual by employing Fourier transform:  $$ x(t) = a_1 e^{i \omega_1 t} + a_1^{*} e^{-i \omega_1 t} + a_2 e^{i \omega_2 t} + a_2^{*} e^{-i \omega_2 t}, $$  where $\omega_1^2$ and $\omega_2^2$ are the two roots of $\alpha \omega^4 + \beta \omega^2 + \gamma$, given by  $$ \omega_{1,2}^2 = \frac{- \beta \pm \sqrt{\beta^2 - 4 \alpha \gamma}}{2 \alpha} .$$  Now by definition the phase space is the space of solutions of equations of motion. In this case it is parametrized by two complex co-ordinates $a_{1,2}$, meaning that it is 4-dimensional.  I would like to know if there's a natural way to associate the symplectic structure (Poisson bracket) on this phase space. I.e.,  $$ \{a_1, a_1^{*}\} = ? $$ $$ \{a_1, a_2^{*}\} = ? $$ $$ \{a_2, a_1^{*}\} = ? $$ $$ \{a_2, a_2^{*}\} = ? $$ $$ \{a_1, a_2\} = ? $$ $$ \{a_1^{*}, a_2^{*}\} = ? $$ Prove that a non-empty closed convex $S$ has an extreme point iff it does not contain any line I am wondering how to prove the following theorem: Let $S \subset \mathbf{R}^{n}$ be a non-empty closed convex set. Then $S$ has at least one extreme point iff $S$ does not contain any line. Extreme point $x \in S$ means that there is no couple of points $y, z \in S$ and $\lambda \in (0;1)$ such that $x = \lambda y + (1-\lambda)z$. I know that every non-empty closed convex set can be represented as an intersection of all closed half-spaces containing the set but I do not know if it is helpful in proving the theorem. "Unanswered" question sort order on profile pages To help users keep track of (and hopefully get answers to) unanswered questions, it would be useful to have an "unanswered" sort order on a user's profile page.  It would order the list of questions with unanswered questions first.   This would be in addition to the "recent," "views," "newest," and "votes" sort orders. What is this agglomeration of pink cells I found attached to a stone? Is there a better way to do optional function parameters in JavaScript? Calculating half derivative of $x$ using Cauchy's repeated integral formula As far as I know, Cauchy's repeated integral formula is defined as: $$(I^nf)(t)=\frac{1}{\Gamma(n)}\int_a^x (x-t)^{n-1}\space f(t)\space\space dt,\space\space n\notin Z_{\le 0}$$  I know that Gamma function is not defined for negative integers and $0$. I also know that differentiation and integration are inverse operators so I thought of finding $\frac{-1}{2}$th antiderivative of $f(x)=x$ so that I can find the half derivative of $x$. However it seems that it does not work. What is the problem with this? Am I missing something or is it not possible? I know some other ways of finding half derivative of $x$, so what I am asking is the problem about what I tried.  My attempt: $$\frac{1}{\Gamma\left(-\frac{1}{2}\right)} \int^x_0(x-t)^{-\frac{3}{2}}\space t\space dt=\frac{x}{2\sqrt{\pi}}\int^0_x u^{-\frac{3}{2}}\space (x-u)\space du$$  This definite integral is divergent and so I cannot get to a result. Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n \left(X_i-\bar{X}\right)^2}{n-1}?$$ How do you determine using stat() whether a file is a symbolic link? Merge flash site into one executable Biliniear form to inner product Let $f:V\times V\rightarrow F$ be a bilinear form in a finite inner product space V.  If $F=R$, how can I prove that there exists a single linear transformation $T:V \rightarrow V$ so that for each $v,u\in V$, $f(u,v) = (Tu, v)$ ?  Would that sill be the case if $F=C$? Canonical replacing Unity with gnome-shell in Ubuntu 18.04 what now? The default desktop environment in Ubuntu 18.04 is going to be gnome-shell instead of Unity 8.  Does this mean development of Unity is stopped? or just that we can expect it to be beta a bit longer?  source on  post by Mark Shuttleworth. phoronix also posted about it  and here about the How may I reference the script tag that loaded the currently-executing script?