If we launch an object on one side of the moon (the velocity being tangent to the surface of the moon), how big must the velocity be to let the object arrive at a position on the moon diametrically opposite to the launching site (in some device to catch the mass)? The mass is very, very small compared to the moon's mass.  The acceleration due to the moon's gravitation is, of course, dependent on the distance to the moon (inverse square law).  Because of this dependence on the distance of the acceleration, it's more difficult to compute a trajectory, especially this particular one that also involves the curvature of the moon. I guess integration is involved, but I don't see how. For sure the velocity has to be higher than the velocity (which is easy to calculate) to let the mass make a circle around the moon. If we didn't catch the mass on the other side, the trajectory would resemble something that looks like an ellipse. This trajectory doesn't consist of two (opposite) parabola's because the trajectory in the question isn't a parabola, again because the acceleration depends on the distance of the mass to the moon.  Is there someone who knows how to do the calculation?
We know the optimum angle for greatest horizontal displacement when launching an object with projectile motion is 45 degrees.  How to solve the angle when it is real long distance around the earth where we cannot assume that gravity is constant but is given by $F=\frac {-GMm}{r^2}$. I use differetial equation to solve it and obtain 45 degrees.   First, for the vertical distance $y$, change $v=u-gt$ to $y'=u-y''t$. Then solve it to obtain a function $y(t)$. When $y'(t)=0$ we have the time for maximum vertical distance. Then insert the value to horizontal equation.   Can anyone guide me through this problem because I am not confident with my working
The number of nonnegative integer triples with sum equal to $12$ How many triples $x,y,z$ satisfy  $x+y+z=12$, where  $x,y,z\ge 0$ are integers? Note that these are not ordered: $ (12,0,0)$ and $(0,12,0)$ are treated as same. Progress I have tried the formula ${13\choose 2}$ to select values of $x$ and $y$ from $0$ to $12$ as  $z$ would keep changing according to $x$ and $y$. (That is, $z=12-x-y$.) The problem is that that certain combinations of $x$ and $y$ would give value of $x+y$ greater than $12$. I can't seem to get around this. How do I get the UI thread's Dispatcher? Convert LaTeX to HTML How can be solved the equation $x^3+x-1=y^2$ in positive integers?    I know this equation defines an elliptic curve, but this seems to be a non-elementary way to solve this question. Is there a more elementary solution?   By the way I found three solutions: $(1,1), (2,3)$ and $(13,47)$. Is this related to the law group property of elliptic curves? Thanks for further answers. Many of the UK visa refusals we see here share a common pattern and the prevailing reasons refer to V 4.2 (a) and (c).    I do understand that, while the applicants may describe very different individual circumstances, there's a consistent pattern and, broadly, fall into specific categories:    Credibility (lifestyle, lack of ties, visit history, lies &amp; omissions) Funding (insufficient funds, provenance of funds, funds parking) Sponsorship (family, friends, employer)   Question: What most commonly triggers a UK visa refusal where V 4.2 (a) and (c) are given as the grounds?  Secondarily: Given that there is a clear pattern, to what extent are these refusals predictable? Does sponsorship make a difference? Is there a set of personal circumstances, however abstract, where a refusal is all but guaranteed? For example, why do those we see here on TSE who sought entry for the PLAB or British Army appear to be refused with a common theme? Is there a uniform shortcoming or is it just discrimination?   PLAB: What is about applications for a visa, for the purpose of sitting the Professional and Linguistic Assessments Board (PLAB) exam, that seem to invite refusals. After all, the test is given so that international medical graduates can show that they qualify to practise medicine in the UK. The first part, PLAB 1, can be taken in centres outside of the UK. However, PLAB 2 can be taken only in the UK. Why can't you can't get a visa just to sit the exam, promising that you'll leave immediately after?    British Army: Since Commonwealth citizens are eligible to apply online to join, even those who don't reside in the UK, why is it so difficult to get a visa just to attend an interview to see whether you are suitable? Even with an invitation from the Army, such visa applications seem to be unsuccessful. Isn't a career in the British Army a valid reason?     Lastly:  After such a refusal, what approaches would increase the chance of a successful application? Is there a canonical way to typeset such quantities in equations and inline math?  \documentclass{article} \begin{document} $Re$, $Nu$ are clearly not an option. $\mathit{Re}$, $\mathit{Nu}$ appear ok to me. \end{document} There is a sentence:     No sooner had he sat down than he fell asleep.   I just do not understand, is this an inversion? And if so, I still do not understand the sentence. REML or ML to compare two mixed effects models with differing fixed effects, but with the same random effect? Background:  Note: My data set and R code are included below text I wish to use AIC to compare two mixed effects models generated using the lme4 package in R.  Each model has one fixed effect and one random effect.  The fixed effect differs between models, but the random effect remains the same between models.  I've found that if I use REML=TRUE, model2 has the lower AIC score, but if I use REML=FALSE, model1 has the lower AIC score. Support for using ML: Zuur et al. (2009; p. 122) suggest that &quot;To compare models with nested fixed effects (but with the same random structure), ML estimation must be used and not REML.&quot;  This indicates to me that I ought to use ML since my random effects are the same in both models, but my fixed effects differ.  [Zuur et al. 2009.  Mixed Effect Models and Extensions in Ecology with R.  Springer.] Support for using REML: However, I notice that when I use ML, the residual variance associated with the random effects differs between the two models (model1 = 136.3; model2 = 112.9), but when I use REML, it is the same between models (model1=model2=151.5).  This implies to me that I ought instead to use REML so that the random residual variance remains the same between models with the same random variable. Question: Doesn't it make more sense to use REML than ML for comparisons of models where the fixed effects change and the random effects remain the same?  If not, can you explain why or point me to other literature that explains more? # Model2 &quot;wins&quot; if REML=TRUE: REMLmodel1 = lmer(Response ~ Fixed1 + (1|Random1),data,REML = TRUE) REMLmodel2 = lmer(Response ~ Fixed2 + (1|Random1),data,REML = TRUE) AIC(REMLmodel1,REMLmodel2) summary(REMLmodel1) summary(REMLmodel2)  # Model1 &quot;wins&quot; if REML=FALSE: MLmodel1 = lmer(Response ~ Fixed1 + (1|Random1),data,REML = FALSE) MLmodel2 = lmer(Response ~ Fixed2 + (1|Random1),data,REML = FALSE) AIC(MLmodel1,MLmodel2) summary(MLmodel1) summary(MLmodel2)  Dataset: Response    Fixed1  Fixed2  Random1 5.20    A   A   1 32.50   A   A   1 6.57    A   A   2 24.77   A   B   3 41.69   A   B   3 34.29   A   B   4 1.80    A   B   4 10.00   A   B   5 15.56   A   B   5 4.44    A   C   6 21.65   A   C   6 9.20    A   C   7 4.11    A   C   7 12.52   B   D   8 0.25    B   D   8 27.34   B   D   9 11.54   B   E   10 0.86    B   E   10 0.68    B   E   11 4.00    B   E   11 How to prevent accidental clicks in the Save button when saving a node? I need to prevent accidental clicks in the Save button when a node is being saved.  I found this   which does exactly this. But the problem is that it uses javascript (jquery). Then if the user disable javascript this module is useless.  My problem is that in my website the user get points when creating nodes, so I have to prevent this in a more secure way.  How can I do this?   What do you think of these ideas?:  1) Maybe I would forbid the node being saved more than one time    programmatically instead of disabling the Save button.   2) Or better: I would allow the user to just save a new node every 5 minutes. Is "po-TAH-to" an acceptable pronunciation for "potato"? Immortalized in the George and Ira Gershwin song "Let's Call the Whole Thing Off" is the nitpicking of pronunciation differences:      You like potato and I like potahto,   You like tomato and I like tomahto,   Potato, potahto, tomato, tomahto!   According to  and other sources, the English do pronounce tomato as tə-MAH-toh rather than tə-MAY-toh. But this doesn't seem to be the case when it comes to potato.     (From the O.E.D.)           –  Pronunciation: /təˈmɑːtəʊ/     –  Pronunciation: /pəˈteɪtəʊ/   Is "potahto" actually an acceptable way to pronounce potato? Or is the song just wrong? the slash (/) after a directory name on shell commands I have a little question here.  If I have two files, say filea and fileb, mv filea fileb would   delete fileb rename filea to fileb   Then if I have two directories, say dira and dirb, mv dira dirb would   move dira into dirb (it will become dirb/dira)   Noting that in both cases there are no notice or message, then this is pretty inconsistent to me. I think mv dira dirb should just overwrite dirb with the contents of dira (or merge the two directories under a directory named dirb).  I remember reading somewhere that a directory name with a slash (like dira/) is treated like a directory, and name with no slash (like dira) is treated like a file (to certain extents, of course). Anyway now I want to make the shell (zsh and possibly bash) respect my notation of a directory by using a slash. Is there a terminal option which enable me to enforce that?  To clarify, here is my desired behaviour:   mv dira dirb results in dirb being overwritten with the contents of dira mv dira dirb/ results in dira being moved into dirb (in dirb/dira)   Has anyone thought the same way as me? Or am I just weird? How to replace the domain name in a Wordpress database? I have a Wordpress database which was installed in a development environment... thus, all references to the site itself have a fixed IP address (say 192.168.16.2). Now, I have to migrate that database to a new Wordpress installation on a hosting. The problem is that the SQL dump contains a lot of references to the IP address, and I have to replace it with: my_domain.com.  I could use sed or some other command to change the that from the command line, the problem is that there are a lot of configuration data which uses JSON. So what? Well, as you know, JSON arrays uses things like: s:4: to know how many chars an element has, and thus, if I just replace the IP with the domain name, the configuration files will get corrupted.  I used an app for Windows some years ago that allows to change values in a database and takes care of the JSON arrays. Unfortunately, I forgot the name of the app... so the question is: do you know any app that allows me to do what I want? Is it possible to activate an iPhone 4 without a SIM? So we bought an iPhone 4 from the Apple store. But we didn't get a micro-SIM as there are no PAYG tariffs on micro-SIM available.  Is there anyway to activate the device without getting hold of a SIM? I don't want to sign up to a contract just to develop with. Inverse of function is continuous or not? Disable COPY drag and drop in Windows Explorer in Windows 7 Center figure that is wider than \textwidth I have a figure that is wider than the \textwidth of my document. (I don't want to change the \textwidth.) As a result, the leftmost part of the figure is flush with the left margin, while the rightmost part spills into the right margin too much. I'd like to center the figure, so that it will exceed the \textwidth equally on both sides. Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.   The  $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative system $R\setminus I$) gives a morphism $$ f_1:\operatorname{Spec}(R_I)\to \operatorname{Spec}(R)=:X $$ The  $\widehat{R}_I$ of $R$ at $I$ gives a morphism $$ f_2:\operatorname{Spec}(\widehat{R}_I)\to X $$ For the special case $I=(a)$, the localization $R_a$ (i.e. the localization of $R$ at the multiplicative system $\{1,a,a^2,\ldots\}$) gives a morphism $$ f_3:\operatorname{Spec}(R_a)\to X $$   My question is:     What is the geometric meaning of all three constructions and how are they related?   This is what I ''know'' already or describes at least the style of answer I would appreciate. As remarked in the comments, $I$ has to be a prime ideal.   $R\to R_I$ is injective iff $R\setminus I$ contains no zero divisors which is the case if $I$ is a prime ideal. The scheme $\operatorname{Spec}(R_I)$ is the intersection of all the neightbourhoods of $I$ in $X$. This is a little contra-intuitive for me since the last statement sounds like ''$f_1$ is injective'' but the $\operatorname{Spec}$-operator should turn 'injective'' and ''surjective'' around somehow (I know this is literally not true but I only want to get a feeling like ''is contained'', ''is bigger'', ''is smaller'', etc.). $\operatorname{Spec}(R_a)$ is somehow the opposite of $\operatorname{Spec}(R_{(a)})$ (= the intersection of all the neightbourhoods of $(a)$ in $X$) because $\operatorname{Spec}(R_a)$ seems to be something like the union of all the open sets of $X$ not containing the point $(a)$. $R\to\widehat{R}_I$ is injective iff $\cap I^n=(0)$ and this holds very often (e.g. for $R$ noetherian and either an integral domain or a local ring). Hence (this is probably false intuition as remarked before), $f_2$ should be something like a ''projection'' (from something ''big'' into something ''small''). But what is geometrically the difference between localization and completion. I don't have a geometric idea of completion at all.   As remarked by Qiaochu Yuan in the comments below, one should not think of $\operatorname{Spec}$ as injective-surjective switching. Rotate group of objects over an edge How can I transition height: 0; to height: auto; using CSS? I'm trying to enable Intel SRT on my laptop. To do this, I need to change SATA controller mode from AHCI to RAID. The problem is that windows has no drivers for RAID and I can't install it while controller is in AHCI mode.   For now I have RAID driver in INF package (inf, sys, cat files). And I can load Windows recovery console with controller in RAID mode. The last thing to do is to intall this driver, but I don't know how to do it.  Google says rundll32.exe setupapi,InstallHinfSection DefaultInstall 123 &lt;filename&gt;.inf might help, but it doesn't.