How to solve $5x+9y=n$?
The Frobenius Coin Problem
Difference of the floor of a product and the product of floors Is there any way the following can be simplified?  $$\lfloor f(x)\cdot g(x) \rfloor - \lfloor f(x) \rfloor  \cdot \lfloor g(x) \rfloor$$ So we have something like:  \documentclass{article} \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2} \usepackage{amsmath}  \begin{document}  \tableofcontents  \section{Brushless Motor Fundamentals} \subsection{Brushless Motor Operation} \subsubsection{DC Motor Operation} Torque is generated in DC motors from the magnetic force, also known as the Lorentz force, which is produced when an electric current is passed through a coil in a magnetic field. This force is given by $F=q[E+(v\times B)]$, where $F$ is the force perpendicular to the coil, $E$ is the electric field in the coil, $B$ is the magnetic field, and $v$ is the velocity of the charged particles in the coil. From mechanics, torque is $\tau=F\times r$.  \section{Brushless Motor Fundamentals 2} \subsection{Brushless Motor Operation} \subsubsection{DC Motor Operation 2} If the electrical force is ignored and the remaining magnetic force is used, with the assumption that $v$ is perpendicular to $B$, we find that $\tau=qvBrsin\theta$.  \end{document}   How to show subsubsections and paragraphs in TOC? How to prove this limit exists: $a_{n+2}=\sqrt{a_{n+1}}+\sqrt{a_{n}}$ Question: Consider a sequence $\{a_{n}\}$ such that $a_{1},a_{2}&gt;0$, and for all $n \in \mathbb{N}$ we have:  $$a_{n+2}=\sqrt{a_{n+1}}+\sqrt{a_{n}}$$  Prove that :  $\displaystyle\lim_{n\to\infty}a_{n}$ exists and find this limit.  My work: If this limit exists, let $\displaystyle\lim_{n\to\infty}a_{n}=x&gt;0$; then we have $$x=\sqrt{x}+\sqrt{x}\Longrightarrow x=4$$ But I can't see how to prove the limit of $\{a_{n}\}$ exists. Thank you for you help! I can't use the terminal while gedit command is running I recently installed 12.04. When I try to edit a file with gedit, I can't use the terminal until I close the editing file or I have to open a new terminal. But I think I didn't have this problem with 11.04, however I'm not sure. Is there anyway to avoid this and to use same terminal while editing files. How to report issues with the mobile SE application Secure hash and salt for PHP passwords It is currently said that MD5 is partially unsafe. Taking this into consideration, I'd like to know which mechanism to use for password protection.  This question,   suggests that hashing multiple times may be a good idea, whereas  suggests using salt.  I'm using PHP. I want a safe and fast password encryption system. Hashing a password a million times may be safer, but also slower. How to achieve a good balance between speed and safety? Also, I'd prefer the result to have a constant number of characters.   The hashing mechanism must be available in PHP It must be safe It can use salt (in this case, are all salts equally good? Is there any way to generate good salts?)   Also, should I store two fields in the database (one using MD5 and another one using SHA, for example)? Would it make it safer or unsafer?  In case I wasn't clear enough, I want to know which hashing function(s) to use and how to pick a good salt in order to have a safe and fast password protection mechanism.  Related questions that don't quite cover my question: I would like to solve the beginner's standard exercise which claims that a point of $\mathrm{Spec} \ R$ is closed iff it is a maximal ideal.  The reverse implication is easy. The direct one seems much subtler. If $P = V(I)$ then, since there exist a maximal ideal $M$ containing $P$, it would follow that $\{P, M\} \subseteq V(I) = P$, whence $P=M$. In order to show the existence of $M$ I applied Zorn's lemma to the set of ideals containing $P$.  The question is the following: is there any other proof of the above implication that does not (indirectly) use the axiom of choice? My concern is that I may be using a cannon to shoot a fly. Also, it would be the first time that I see closedness of points to require the axiom of choice. If the axiom is indeed needed in general, are there "nice" classes of rings for which we could get away without it? Is a current transformer appropriate for a voltage step up flyback converter? I am designing a flyback converter, input 170V output 1200V at 2mA.  I am having trouble finding an appropriate transformer, most do not have a high enough N. Current transformers seem to have ranges in 1:500, so I could work with that.  Is it proper to use a current transformer as a conventional step up voltage transformer? Downloading files instead of opening directly I have kept files in site assets and created hyperlinks to the files (which may be PDF or word ). When i click on the link it opens up in the browser. How do i force it to download ? Why does $1+2+3+\cdots = -\frac{1}{12}$? "Everything is a file" in the UNIX World.  Above sentence is famous. When I run echo "hello programmer" &gt;&gt; /dev/tty1 , I can watch the given string on TeleType 1  , ....  What and where is file per each socket? Suppose my friend connects to my PC, and its IP is h.h.h.h , how can I access the respective file? Is it possible? How to create a rounded cone Substitute fresh garlic instead of garlic powder? I have a recipe that calls for garlic powder, but I only have fresh garlic on hand.   What ratio should I use to substitute? Number of homomorphisms between two cyclic groups. Player A taps  targeting Player B; its "deals 1 damage to target player" ability is placed on the stack.  Player B gets priority and responds with , targeting Goblin Fireslinger.  Both players pass priority. Snap resolves, and Goblin Fireslinger is returned to its owner's hand.  Does Player B still take 1 damage from Goblin Fireslinger's activated ability?  I know that if the creature had been killed/sent to the graveyard, the ability would still resolve, but I was told recently that because a player's hand is a private zone it couldn't. If anyone could confirm or deny this for me, it would be most appreciated. Why is this sentence correct?     She suggested that he go to the cinema.   I would definitely use goes instead of go. How do I save the output of a command to a file?  Is there a way without using any software? I would like to know how. If a factor variable (e.g. gender with levels M and F) is used in the glm formula, dummy variable(s) are created, and can be found in the glm model summary along with their associated coefficients (e.g. genderM)  If, instead of relying on R to split up the factor in this way, the factor is encoded in a series of numeric 0/1 variables (e.g. genderM (1 for M, 0 for F), genderF (1 for F, 0 for M) and these variables are then used as numeric variables in the glm formula, would the coefficient result be any different?  Basically the question is: does R use a different coefficient calculation when working with factor variables versus numeric variables?  Follow-up question (possibly answered by the above): besides just the efficiency of letting R create dummy variables, is there any problem with re-coding factors as a series of numeric 0,1 variables and using those in the model instead? Can I prevent/restrict standard users from installing/uninstalling software? Can i prevent/restrict standard user from installing/uninstalling software in Ubuntu 12.04 desktop? Limit involving the exponential of an expression plus a small-o I'm trying to formalize the following limit:  $$\lim_{x \to \infty} \left[1+\frac{a}{x}+o\left(\frac{1}{x}\right) \right]^x= e^a$$ however I can't see how to rigorously show this identity.  Any suggestions?