Abs and ^2 within Solve: curious speed characteristics
Why does Simplify ignore an assumption?
Must a husband and wife make separate online applications for a UK Standard Visitor visa, and so attend separate appointments for biometrics? How to lead a moving target from a moving shooter I saw this question: .  My situation is a little different though.   My target moves, and the shooter moves.  Also, the shooter's velocity is added to the bullets' velocities, i.e. bullets fired while sliding to the right will have a greater velocity toward the right.   What I'm trying to do is to get the enemy to be able to determine where they need to shoot in order to hit the player.  Using the linked SO solution, unless the player and enemy are stationary, the velocity difference will cause a miss.  How can I prevent that?      Here is the solution presented from the stack overflow answer.  It boils down to solving a quadratic equation of the form:  a * sqr(x) + b * x + c == 0   Note that by sqr I mean square, as opposed to square root. Use the following values:  a := sqr(target.velocityX) + sqr(target.velocityY) - sqr(projectile_speed) b := 2 * (target.velocityX * (target.startX - cannon.X)       + target.velocityY * (target.startY - cannon.Y)) c := sqr(target.startX - cannon.X) + sqr(target.startY - cannon.Y)   Now we can look at the discriminant to determine if we have a possible solution.  disc := sqr(b) - 4 * a * c   If the discriminant is less than 0, forget about hitting your target -- your projectile can never get there in time. Otherwise, look at two candidate solutions:  t1 := (-b + sqrt(disc)) / (2 * a) t2 := (-b - sqrt(disc)) / (2 * a)   Note that if disc == 0 then t1 and t2 are equal. Which 1970s UK fantasy kids TV show has an older brother die? I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.  Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.  Proof. Suppose $f(X)=a_{0}+a_{1}X+\cdots +a_{n}X^{n}$ is such that $a_{0}$ is a unit in $R$ and $a_{1},a_{2}, \dots,a_{r}$ are all nilpotent in $R$. Since $R$ is commutative, we get that $a_{1}X,a_{2}X^{2},\cdots,a_{n}X^{n}$ are all nilpotent and hence also their sum is nilpotent. Let $z = \sum a_{i}X^{i}$ then $a_{0}^{-1}z$ is nilpotent and so $1+a_{0}^{-1}z$ is a unit. Thus $f(X)=a_{0}+z=a_{0} \cdot (1+a_{0}^{-1}z)$ is a unit since product of two units in $R[X]$ is a unit.   I have not been able to get the converse part and would like to see the proof for the converse part. When can we interchange “who/which" and "that”? So far I thought the words who and that are usually interchangeable:   There is the man who saw me.  There is the man that saw me.    Both sentences seem fine, who can be replaced with that. But in the following examples:   This is the man to whom I spoke.  This is the man to that I spoke.    The #2 sounds odd, which is why I don’t think it might be correct. The same observation for the words which and that, while   This is the ball which we used today.  This is the ball that we used today.    Seem to be in order and which can be replaced with that. But in the following examples the #2 sounds odd again.   This is the ball with which I scored.  This is the ball with that I scored.    I wonder if there’s any rule regarding the interchangeability of the above mentioned words. Does Calculate Value (Model Only Tool) work correctly in tools run in Batch? Am I correct that if my model has a model-only tool (Calculate Value), and then I run the model as a batched process, it will not run the Calculate Value expression?   I wrote a model to upload, define, project (and clip) XY data from excel files. I had thought to use iterate tables, but it wasn't working. (I assume due to having to choose which worksheet within xlxs). So I thought to do a batch, but this seems to leave that annoying $ that arc appends to table names (and is an invalid character in most tools...)  My guess is that the Calculate value that removes my $ is not being run in the batch. Ideas? Confirmations? Why can't you use a single resistor for a number of LEDs in parallel instead of one each? Computing second partial derivative with polar coordinates Consider the polar coordinates with $x=r\cos\theta$ and $y=r\sin\theta$.  I can show using the chain rule that $$\frac{\partial}{\partial x} = \frac{x}{r} \frac{\partial}{\partial r} -\frac{y}{r^2} \frac{\partial}{\partial \theta}$$   What is the method to compute $\dfrac{\partial^2}{\partial x^2}$? I don't know how to do it. isolated non-normal surface singularity I am looking for an isolated non-normal singularity on an algebraic surface. One obvious example occurs to me: the union of two $2$-dimensional affine subspaces of $\mathbb{A}^4$ which meet in a point. But this seems like "cheating." Can someone provide an irreducible example? In Python, is there a concise way of comparing whether the contents of two text files are the same? I don't care what the differences are.  I just want to know whether the contents are different. What's the fastest connection speed / shortest elapsed time using Migration Assistant? When using Migration Assistant to move users, files, and folders from an old Mac to a new one, you have a few different choices, depending on the Macs involved. Presuming that both Macs have appropriate ports, which ports should be used in a real-world situation? Please do not provide quotations of theoretical port speeds; I'm looking for real-world experiences as to what will get the job done faster.  Port choices:   FireWire 400 in target disk mode FireWire 400 to 800 in target disk mode FireWire 800 in target disk mode 100 Megabit Ethernet to Gigabit Ethernet in the OS Gigabit Ethernet to Gigabit Ethernet in the OS Thunderbolt cable in target disk mode Where do I find the maximum node? A lot of people recommend to use the maximum node for glass setup, but I can't find that node. Nothing comes up when I search for it.  Where do I find it? I'm asked to show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$. I've tried to tackle this using Fermat's Little Theorem and Euler's Theorem, but I can't even get started. My main problem seems to be the "for every $a \in \mathbb{N}$" part, because if it restricted $\gcd(a, 1001) = 1$ I would be able to factor out 1001 and try to show that it's factors divide $a^{61} - a$. I'm kinda lost here. Any help is appreciated. Tools for automating document compilation A lot of people write makefiles that say something like  paper.pdf: paper.tex         pdflatex paper         bibtex paper         pdflatex paper         pdflatex paper   To handle re-running TeX to get new/changed references and so forth. Is there a better way to do this? Personalize monitor position before login I'm using multiple monitors.  When logged in everything is fine: The left monitor is positioned to the left and the right monitor is positioned to the right, but before I log in, the login-screen positions them the other way around.  This is also the position presented by bios and then the grub menu.  I don't mind where the login is displayed, but when moving the mouse, the boundary is the wrong way around and that's annoying. I am reading  that finite-dimensional subspaces of normed vector spaces have closed direct sum complements. This is the proof:      Let $\{e_1, ..., e_n\}$ be a basis for $\mathcal M$. Every $x \in   \mathcal M$ has then a unique representation $$x = \alpha_1(x)e_1 +...+  \alpha_n(x)e_n.$$ Each $\alpha_i$ is a continuous linear functional on    $\mathcal M$ (a linear map from  finite dimensional space is always    continuous) which extends to a member of $\mathcal X^*$,  by the    Hahn-Banach theorem ($\mathcal X^*$ is the dual of $\mathcal X$). Let    $\mathcal N$ be the intersection of the null spaces of these exten­sions. Then $\mathcal X = \mathcal M\oplus \mathcal N$.   I follow it until the last sentence. Clearly $\mathcal N$ is closed, as it is the intersection of closed sets (the inverse of $0$ under any continuous function, and in particular any linear functional on a vector space, is closed). Also it is easy to see that $\mathcal M \cap \mathcal N= \{0\}$, since if some $m\in\mathcal M$ is in every null space, it must have the zero coefficient for every basis vector, and so it must be zero.  My question: Why do we have $\mathcal X =\mathcal M + \mathcal N$? This seems to only hold if the extension given by Hahn-Banach is zero off of $\mathcal M$, but I see no reason why this should be the case. Razr Maxx HD, How to wipe data with broken screen My screen is totally destroyed, it's black and unresponsive. The phone turns on just fine, though. I just got my new phone and I have to send this one back now.  I want to wipe the phone before I send it back, but I'm not sure how. I don't care too much about sending back a rooted phone, it seems Asurion doesn't really care about that from what I've read, but I don't want my personal data on the phone when I send it in.  What are my options to wipe the phone without a screen? I've already read a lot of threads on other forums and here on the subject, but everyone seems to say to wipe through ADB shell. Well, I can see the phone in ADB, but I am unable to enter SU through ADB shell. I assume it's because the screen is locked, and I have no way to unlock it. Others have also said to wipe through fastboot in ADB, but I am seeing conflicting recommendations and I'm not sure how to proceed. I'm not that proficient with ABD, so I want to make sure I don't mess this up.  Any advice would be greatly appreciated. I'm new to ArcGIS 10 and am trying to do a really simple field calculator equation and it is not working. I'm just trying to find the percentage of a certain demographic trait by doing:  [asian]/[total]*100   It is populating about 10% of the column correctly and then I get an error:     There was a failure during processing, check the Geoprocessing Results   window for details   Obviously my syntax is not incorrect or else it wouldn't start populating the field correctly.   Does anyone have any idea why this is happening? Double dual mappings Given a Möbius transformation that maps one pair of concentric circles to another pair of concentric circles, why is the ratio of the radii preserved through the map?   I thought about how Möbius transformations are compositions of rotations, scaling, inversion, and translation, and that intuitively, these types of maps shouldn't change the ratio of radii between two circles.   Would it be correct to just say that if $\frac{r_1}{r_2}$ is the ratio of radii between the two circles, then   1) The radii are invariant under translation, $z \mapsto z+a$, so $\frac{r_1}{r_2}$ stays the same  2) Under scaling by a factor $z \mapsto az$, $\frac{ar_1}{ar_2} = \frac{r_1}{r_2}$   3) Under inversion, $z \mapsto \frac{1}{z}$, $\frac{1/r_1}{1/r_2} = \frac{r_2}{r_1}$  Or is there a different/better way to think about this problem?