Proving a recursive relationship
Fibonacci sequence emerging from integer partitioning
Determining ethnicity according to ancestry? How much should a "button of Cure Light Wounds" cost (should it be available at all?) How to recover deleted files? Are there any tools, methods, incantations to recover recently deleted files on Ubuntu?  If it makes any difference, I want to recover a  2.x database file. But would be better to have a method/tool that works on any kind of file. Adding more than 15 digits in Excel I want to add more than 20 digits in an Excel cell.  The current format of the cell is general, it converts the number to an exponential format. I tried with a number format and accounting, but when I enter more than 15 digits it gets converted to 0's.  What are the recommended steps for stopping Excel from converting data to exponential Format for 20 digits when in the general format?     Example: 12345678901234567890   Excel converts it to 1.23457E+19 in general format.  Without using ' before the value, is there any other way to keep value same? What's the default value for an 'object' in java? I use Chrome as my main browser.  I also spend a lot of time on a particular webapp that my company uses.  I'd like to keep those two things separate.  I can run the webapp in a separate window - but I'd like to go further than that.  I'd like to have it be identified by a different task in the Win 7 taskbar; ideally a different icon; and ideally a different title in Process Monitor.  How can I run a second, independent instance of Chrome, with a different title and icon, that won't merge with my main Chrome browser. Why do the planets' orbital distances fall on an exponential curve? Background: I was recently reading a book on the planets to my son and I noticed a pattern in the distributions of the planets.  The planets' distances roughly follow an exponential distribution.  Below is a plot of scaled, log orbital distances $$ \tilde{d_n} = \frac{\log(d_n/d_1)}{\log(d_2/d_1)} $$ with the line $an+b$:   Where $d_1$ corresponds to Mercury and so on. Ceres is included, Pluto is excluded.  By linear regression, $a = 0.90, b = -1.06$.  For the statistically minded, the data has a Pearson's correlation of 0.996.  Note that this is a well known phenomenon, see  and references.  The code used to generate the plot may be provided on request.  Question: What is the mechanism that leads to this distribution?  Aside: Is there a good introductory text on solar system formation for the mathematically inclined?  Update: This is also known as Titius–Bode law. I go to change my birthdate on Roblox, but nothing shows up, the dropdowns are just blank with no Month, Day or Year:  It also shows me a pop up asking to put in the Missing Details to improve the experience, but no matter how many times I put in my age it doesn't save and asks it again. Why would Mystique's DNA allow for Sentinels that can duplicate mutant powers? What does the human body use oxygen for besides the final electron acceptor in the electron transport chain? My biology teachers never explained why animals need to breathe oxygen, just that we organisms die if we don't get oxygen for too long. Maybe one of them happened to mention that its used to make ATP. Now in my AP Biology class we finally learned the specifics of how oxygen is used in the  due to its high electronegativity. But I assume this probably isn't the only reason we need oxygen.   What other purposes does the oxygen we take in through respiration serve? Does oxygen deprivation result in death just due to the halting of ATP production, or is there some other reason as well? What percentage of the oxygen we take in through respiration is expelled later through the breath as carbon dioxide? Understanding ROC curve Why is the gravitational force always attractive? Why is the gravitational force always attractive? Is there another way to explain this without the curvature of space time?   PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve concavely? What is the difference between these two Breusch-Pagan Tests? I'm trying to install the Jupiter power settings app but, I read it is no longer compatible with the new kernel. Is there a similar program or a way to manage the power settings on 13.04? Finite order endomorphisms Why the finite order endomorphisms are diagonalizable over the complex numbers (or any algebraically closed field where the characteristic of the field does not divide the order of the endomorphism) with roots of unity on the diagonal? Support showing image in original size in overlay How do you add a certificate authority (CA) to Ubuntu? What canonical momenta are the "right" ones? I'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the conjugate momenta $\pi^\mu_{\ \ \ \nu} = \partial \mathscr{L} / \partial(\partial_\mu A^\nu)$, I can use two methods.  First method: directly apply this to $\mathscr{L}$. We get a a factor of $2$ since there are two $F$'s, and another factor of $2$ since each $F$ contains two $\partial_\mu A_\nu$ terms, giving $$\pi^\mu_{\ \ \ \nu} = -F^\mu_{\ \ \ \nu}.$$  Second method: get $\mathscr{L}$ in terms of $A$ by expanding and integrating by parts, yielding $$\mathscr{L} = \frac{1}{2}(\partial_\mu A^\mu)^2 - \frac{1}{2}(\partial_\mu A^\nu)^2.$$ Differentiating this gets factors of $2$ and gives $$\pi^\mu_{\ \ \ \nu} = \partial_\rho A^\rho \delta^\mu_\nu - \partial^\mu A_\nu.$$  These two answers are different! (They do give the same equations of motion, at least.) I guess that means doing the integration by parts changed the canonical momenta.  Is this something I should be worried about? In particular, I have another exercise that wants me to show that one of the canonical momenta vanishes -- this isn't true for the ones I get from the second method! Plus, my stress-energy tensor is changed too. When a problem asks for "the" canonical momenta, am I forbidden from integrating by parts? Using Mosfet with seperate Logic Converter and Arduino I am trying to use an IRLB3034 with an Arduino even though it is not a logic level MOSFET. I am planning on using an LM2937 12v regulator like the schematic below:    The connection going off the page goes to an Arduino digital pin pulled down with a 10k resistor. I assumed this would work but wanted to check. As a side note, if anyone could recommend a similar MOSFET that could handle 200a and 40v (in TH) that would be great. (Only if such thing excists or is readily available)  EDIT: I have not made it clear what I am planning on using this for so I will now. It is to drive an airsoft motor and I want to use the LM2937 because I didn't think I could turn the mosfet on and off with the arduinos 5v - because the IRLB3034 isn't a logic level mosfet - so that is the reason I added the 12v regulator.  EDIT (Sorry for the previous incorrect name)  EDIT: Unfortunately, I have made a mistake, the current is not 200 amps, it is infact 30-40 amps. I don't know how I made this mistake but I think it will be helpful for people trying to follow along. I am currently trying to use busbars/6awg copper wire to carry the high-ish current around. (Only 40 amps now though)  EDIT: This question is different because it is asking how to convert 5v to 12v to run a mosfet rather then how to use a mosfet with an arduino in the best way like the other post. The accepted answer on this post also answers the other question though. What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$ What is this limit   $$   \frac{1\times3\times5\times\cdots}{2\times4\times6\times8\times\cdots} = \lim_{n \rightarrow \infty}\prod_{i=1}^{n}\frac{(2i-1)}{2i} $$   I remember that it was something involving $\pi$.  How can I compute it?    In addition; how can I compute it's sum series limit?