Closed form estimation for f(n) in the recursive formula $f(x+1)=f(x)+{1 \over f(x)}$
Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$
Difference between static class and singleton pattern? What real (i.e. practical) difference exists between a static class and a singleton pattern?  Both can be invoked without instantiation, both provide only one "Instance" and neither of them is thread-safe. Is there any other difference? Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating? What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$? What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$.  I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this?  Please help. I am trying to understand Maximum likelihood estimation but it looks like I am missing something rather elementary.          suppose we have an iid random sample $X_1, X_2,..., X_n$ for which the probability density function of each $X_i$ is $f(x_i; \theta)$ where $\theta$ is an unknown parameter. Then, the joint probability density function of $X_1, X_2,..., X_n$ is given by:          $f(X_1=x_1,X_2=x_2,\cdots,X_n=x_n)=\prod_{i=1}^{n} f(x_i, \theta)$      In Maximum Likelihood estimation, we try to maximize $f$ as a function of $\theta$.  Question: Why do we maximize probability density instead of probability? In what way does it make sense?  Edit: As explained in the answers, the probability of choosing a finite number of sample points from a continuous probability distribution is zero, so maximizing probability doesn't make sense, but how does maximizing probability density make sense? Pretty printing XML with javascript How to insert page number in Beamer navigation symbols? Is there a single word which means "a person, and all of his ancestors", or "a person, and all of his descendants"? There are a lot of words describing ancestral relationships between people, such as:  parent grandparent great-grandparent any of the above can be described as an "ancestor".  child grandchild great-grandchild any of the above can be described as a "descendant".   I am wondering if there are single words (nouns) which describe any of the following concepts:  A person and all of his/her ancestors A person or any of his/her ancestors A person and all of his/her descendants A person or any of his/her descendants   Any ideas? I am trying to create a schema page in LaTeX. It shall look like this.   How can I create schema code which look like this image? There should be also a click to go function. When the user clicks on the user_schema link he should be redirected to that page.  What classes I can use for this? Any suggestions would be appreciated.    \documentclass{article} \usepackage[utf8]{inputenc}  \begin{document} \maketitle  \section{User schema} \mySchema{Users} {     _id: ObjectId - Id do registro     name: String - Nome do usuário     date: Date - Data de Cadastro     ren_date: Date - Data de renovação do Cadastro     email: String - Lorem ipsum dolorsitamet, consecteturadipisicingelit, sed do eiusmodtemporincididunt ut labore et dolore magna aliqua. Ut enim ad minimveniam, exercitationullamcolaborisnisi ut aliquipexeacommodoconsequat. Duisautereprehenderit in voluptatevelit esse cillumdolore eu fugiatnullapariatur. Occaecatcupidatat non proident, sunt in culpa quiofficiadeseruntmollit.     access_log: Array [          {              date: String - Data da ocorrência.             user: ObjectId - Usuário que executou.             action: String - Ação Executada.         }         {              date: String - Data da ocorrência.             user: ObjectId - Usuário que executou.             action: String - Ação Executada.         }     ]     \link{group}{group_id}: ObjectId - Id do grupo. } \end{document}   Here is the image how should it appear: How To Get Notified When Hard Drives Fail In Dell PowerEdge Servers I have a Dell PowerEdge R415 and R515 and wondering how I can be alerted remotely when a hard drive fails in either server? Both are running RAID, so I need to be notified, so I can swap out the drive. Both servers only have the basic Baseboard Management Controller. Is this possible with a third party tool/software? Understanding reference counting with Cocoa and Objective-C I'm just beginning to have a look at Objective-C and Cocoa with a view to playing with the iPhone SDK. I'm reasonably comfortable with C's malloc and free concept, but Cocoa's references counting scheme has me rather confused. I'm told it's very elegant once you understand it, but I'm just not over the hump yet.  How do release, retain and autorelease work and what are the conventions about their use?  (Or failing that, what did you read which helped you get it?) How Do I improve the performance of my Drupal7 Site? For somebody who is new to Drupal,  I see that there are many resources available on Drupal Caching but I am not sure which of them to use. If the edges of the complete graph $K_{17}$ (on 17 vertices with no three collinear) are each colored one of three colours can it be proven to have two or more monochromatic triangles? Align equation left Here is my split equation:  \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation} \begin{split} a_{ijk} =&amp; \frac {Pr(M_{I} =2  \&amp;  M_J=1 \&amp; M_K =1 | I=i , J=j , K=k)}{Pr (M_I =1 \&amp; M_J =1 \&amp; M_K=1 | I=i , J=j)} \\             =&amp; \frac {\mu_{ijk211}}{\mu_{ijk111}}\\ \end{split} \end{equation} \end{document}   I want to align this equation to the left. Is this possible in an equation environment? I am trying to install Ubuntu 12.10 for quite some time, and passing hurdles one by one. Now I am in a situation as follows.  I have got a PC and 10 GB HDD which will be totally dedicated to Ubuntu so no option of Wubi and dual boot.  I was trying to install from DVD, but it is getting stuck at "Out of frequency" error. So I had to adapt for USB boot option. But my PC is USB non bootable, so workaround is "Plop Boot Manager". So I am doing the installation procedure as follows:   starting  from a CD drive which is having plop installed. opting for for USB boot in plop options. booting begins from USB. monitor eventually gives "out of frequency" error press Shift+Alt+F1 to get the terminal. open the grub with sudo nano /etc/default/grub. do necessary changes. sudo update-grub.   Now here I am getting error as follows:   /usr/sbin/grub-probe:error:failed to get canonical path of /cow.     My system is   P4 3.06 GHz, 1 GB ram , 10 GB HDD without an OS, monitor CRT lg StudioWorks (7 years old). Mobo Mercury P4 266a NDMx (865 equivalent). The whole system is perfectly in working condition under XP, but it is USB non bootable, and all other devices working perfectly.  What should I do next? NOTE: this question is sorta like a follow-on question to  I asked. In that question, the main suggestion was that I was trying to show two polys. Now, I'm asking about a SINGLE poly.  I have the following GEOGRPAHY of Los Angeles in my Sql Server 2008 :-     (or in )  It's bascially ..  -- Shape of Los Angeles.  DECLARE @WKT VARCHAR(MAX) = 'POLYGON ((-118.66815700000011 34.181235000000029, ... )) SELECT GEOGRAPHY::STGeomFromText(@WKT, 4326)   Now when I just try and render this out, It now looks like this (NOTE: this is different, but still not correct .. compared to my previous GIS Question).    Now, in the previous question people were suggesting that the previous data was a MULTIPOLYGON so there was some bad stuff I must have been doing, trying to render out holes in polys.  This time, this is a single polygon. So i'm assuming the hole theory still doesn't work. Now, I understand that by looking at this image, there's a number of holes ... but the Well Known Text data which Sql Server gives me is a POLYGON which to me means .. one single shape. So i'm not sure how I can fix this :(  This is why i'm asking the a similar question to my previous one.  Can anyone help? Can they see the data I've posted and see what happens in other systems? (I don't have any other software besides Sql Server 2008).  Maybe I need to extract all my data from Sql Server 2008 .. and clean it up (whatever that means) through another system. For example. If Sql Server is saying that the shape of Los Angeles is a single Polygon .. but when cleaned up .. it's actually 7 Polygons .. and then those shapes then get rendered correctly .. then we have a solution!   Any suggestions?  Can -anyone- render this shape onto a Google map? How can I create a video file from a set of jpg images? Update Windows 10 to 1809 manually What is the $n$ times iterate of $f(x)=\frac{x}{\sqrt{1+x^2}}$? Can virtual particles be thought of as off-shell Fourier components of a field? I just found  blog post, which gives an interpretation of virtual particles I haven't seen before.  Consider a 1D system of springs and masses, where the springs are slightly nonlinear. A "real particle" is a regular $\cos(kx-\omega t)$ wavepacket moving through the line, where $\omega$ satisfies the dispersion relation $\omega = \omega(k)$. When two real particles collide, the region where they collide temporarily looks really weird, as they interact nonlinearly, pulling and pushing on each other.  Formally, we can write this weird region as the sum of a bunch of $\cos(kx-\omega t)$ waves, but there's no guarantee they'll have the right dispersion relation. Thus, each term in the resulting expansion is a off-shell "virtual particle". If you add up all the virtual particles, you get the actual intermediate field state, which is just a weird ripple in the field where the two particles are interacting.  As another example, consider the statement "a static EM field is made of virtual particles". Under this interpretation, what that really means is, "a static field (e.g. $1/r^2$) is not equal to $\cos(kx-\omega t)$, but may be expanded in terms of such sinusoids", which is much less mysterious sounding. In fact, this is exactly what we do when we consider scattering off a potential in normal QM, e.g. in the Born approximation.  The above gives some intuition for what a 'virtual particle' means in classical field theory. They are the Fourier components of the field with $(\omega, k)$ not satisfying the dispersion relation, and they are useful in classical perturbation theory. However, they are not propagating degrees of freedom, so they only appear during interactions.   The picture above is entirely classical. Does this picture generalize to quantum field theory, giving a physical intuition for virtual particles there? Selenium 2: Open link in new tab and close tabs