I was preparing for the Maclaurin Olympiad and I came across this question:  Show that: $$ \frac{1}{a} + \frac{1}{b} = \frac{5}{11}$$ has no solutions for positive integers $a,b$.  Is there any general method to solve equations like this:  $$11a + 11b = 5ab$$  Thank you
How many integer solutions does $xy+9(x+y)=2006$ have? Here $x$ and $y$ are both integers.  My trying:        I have tried to solve this problem. But I have no idea to solve this. Please help
Capitalisation of "Section" and "Chapter" in a Ph.D. Thesis I am submitting a Ph.D. thesis fairly soon and my supervisor has flagged my use of capitalisation in "Section" and "Chapter" as possibly incorrect. I have googled about a bit and I see mixed opinions.  So my question is, when writing a computer science Ph.D. thesis, what is the correct way to capitalise "Section", "Chapter", "Appendix", "Figure", "Table", ... ?  For example, what is the correct capitalisation for the following:   "In Chapter 3, it was shown that..." "In the previous Section, a method was presented to..." "The graph in Figure 3 shows..." Correct position of "only" Which is grammatically correct?  I can only do so much in this time.  or  I can do only so much in this time. When I'm connected to two networks with separate routers using Wifi and Ethernet, how does my Mac know what connection to use when I visit a website? I understand that the architecture of Convolutional Neural Networks (CNN) and Feed forward (FNN) are quite different. And that CNNs use pooling and filters of shared weights over a patch of the image. I am not so clear on the core convolution operator (1):   If anyone could link me to an explanation, I have looked at the Colah blog and Nielsen's online book, and I understand what it is doing but don't understand the convolution operator.  Also, it looks quite similar to the the core FNN function is there and difference? (2).  (1) convolution operator is $a^1 = \sigma(b + w*a^0)$ which is equivalent to:  $a^1 = \sigma(b + \sum^4_{l=0}\sum^4_{l=0}w_{l,m}a_{j+l,k+m})$  (2) feed forward operation:  $a^1_i = \sigma(\sum^n_{j=1}w_{ij}x_j +b_i)$  Many thanks  Sources: functions taken from: What are the best ways to improve a site's position in Google? Apologies if this has been asked somewhere before, but I didn't see what I was looking for after several pages of Google results.  I was reading Jech's The Axiom of Choice and was introduced to the equivalence of the ultrafilter lemma and the compactness theorem for FOL. I gather that these are also equivalent to the completeness theorem. I am primarily interested in trying to understand the "ultrafilter lemma $\implies$ completeness/compactness" direction. Jech's own proof takes an odd route through "binary messes" and I don't really follow the last couple of steps. (It's on pp.17,18 of the 2008 Dover reprint, if one is interested).  I'm wondering if anyone knows a good reference for, or can provide, a readable proof that the ultrafilter lemma implies completeness or compactness, with a preference for completeness. Alternatively, if you have Jech's book on hand, helping me bridge the concluding paragraph of his $\textrm{(ii)}\implies\textrm{(iii)}$ step with the preceding steps would clear up much of my puzzlement. (Though in this case I can't offer any help in saying what, specifically, is missing in the way of a bridge). My book gives the following definition:     Let $(M,d)$ be a metric space, and let $\mathcal{T}$ be the collection of all subsets of $M$ that are open in the metric space sense...  $\mathcal{T}$ is called the topology generated by d   Then it later says to prove things about two different metrics generating the same topology. However I'm very confused about what all this means.   When I read "all subsets," I think of the theorem that says a set with $n$ elements will have $2^n$ subsets. (ex. all the subsets of $\{1,2,3\}$ are $\{\emptyset, \{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\} \}$). So I think "all subsets that are open" would be like, If M is the set (1,5), then an open subset would be any set $(a,b)$ where $1&lt;a&lt;b&lt;5$, and then all subsets would be all possible union combinations of these "$(a,b)$" sets.     Its seems to me that metrics have nothing to do with subsets. Clearly there is something i'm not understanding. Please Help me.   It would be very enlightening, if someone could give me some simple examples of 1 set and 2 metrics generating two different typologies. I recently discovered that my computer is 64-bit. I have backed up my Home folder, and reinstalled Ubuntu. The reinstall wasn't nearly as painful as I thought. There is one thing that I can't quite seem to figure out: how do I get the 64-bit Firefox Beta build? I always get the Beta builds, but I want to take advantage of the 64-bit architecture of my computer.  says that Mozilla has come out with a 64-bit version of Firefox, but I can't seem to find it.  I do understand  of using a 64-bit browser, but I've decided to jump right in and do it anyway. (Flash and Java are already 64-bit, and who cares about Silverlight, since it's not for Linux anyway?)  There's only one issue, and it's a big one: I can't find the 64-bit Beta PPA!!! (I really hate using .tar.gz files, but I'd be willing to do that as long as I could still access Firefox via the Launcher. Oh, speaking of which, I don't understand .tar.gz files. Once, I managed to run one (the Dropbox Beta build,) but I have no idea whatsoever on how to install them: as in, click on the icon and go.) If $X,Y$ are equivalence relations, so is $X \times Y$ A book suggests that we should not interchange "so" and "so that". "So" means "therefore", and "so that" means "in order that". However, it seems to me that in many cases they don't actually have a difference. For example:     Alice got up at 5:00 A.M., so that she could drive her son to school.   If I substitute "so that" with "so", this sentence presents the exact same causal relation to me:     Alice got up at 5:00 A.M., so she could drive her son to school.   I must have missed something here, since English is my second language. Could somebody explain the difference? According to this question:  entire ruSO site shouldn't become hot. But I just saw the following while searching MSE:        Direct link to the question on ruSO:   Perhaps the HNQ filter works on question title (in the current case there are no non-Latin symbols) instead of site origin. Hence, according to the post mentioned before this behaviour should be interpreted as a bug.   However, I suggest to fix such "bug" by taking another feature request in mind: How can I move files into subdirectories by count? I have around 100,000 files in a directory. As you can imagine, that's not an ideal situation, and I'd like to split the directory up into subdirectories. But the filenames aren't such where I can put them into subdirectories based on their names or anything, so I'd like to create, say, 10 directories and put about 1000 files in each. Is there an easy bash command or loop that would do this? I'm trying to create a command, which calls on an environment, which defines a custom column definition with one argument using \newcolumntype, and defines a second command, containing parameters for a table column.  Then, still within the first command, I begin the tabular environment and call the custom column, with the second command as its argument. I always get an undefined control sequence error, though. Any thoughts on why that might be?  I'm sure that was clear as mud, but I've cut it down to a minimal example, which follows:  \documentclass{article} \usepackage{array} \usepackage[table,dvipsnames]{xcolor} \usepackage{colortbl}  \newenvironment{tablecolor}{     \newcommand{\tlight}{\columncolor{SkyBlue!30}\color{black}}     \newcolumntype{C}[1]{&gt;{##1}c} }{}  \newcommand{\tablegoals}{ \begin{tablecolor} \begin{tabular}{l|C{\tlight}} Goal 7  &amp; \\ \end{tabular} \end{tablecolor} }  \begin{document}     \tablegoals \end{document}     Here's my second attempt. Different approach, same undefined control sequence error with \columncolor.  \documentclass{article} \usepackage{array} \usepackage[table,dvipsnames]{xcolor} \usepackage{colortbl} \usepackage{ifthen}  \newenvironment{tablecolor}{     \newcolumntype{C}[1]{&gt;{         \ifthenelse{\equal{dark}{##1}}{             \columncolor{Cyan}\color{white}             }{             \columncolor{SkyBlue!30}\color{black}         }     }c} }{ }  \newcommand{\tablegoals}[1]{     \begin{tablecolor}         \begin{tabular}{l|C{dark}}             Goal 7  &amp;   #1         \end{tabular}     \end{tablecolor} }  \begin{document}     \tablegoals{Check mark} \end{document} JComboBox Selection Change Listener? In a USB cable, is it OK to swap the D+ and D- wires? I heard that D+ and D- are differential signals, does it matter if I swap them when connecting a USB device to the computer? Is there an algorithm for N body simulations in General Relativity I am new to general relativity but have a background in computer science.  Why is it so hard to do n-body simulations in GR?  For example, there could be a program which takes the properties (mass, position, velocity, etc.) of each particle as input and numerically integrates the evolution of the system using discrete but very small time steps.  If we throw enough memory and computing power at it, we can get arbitrarily close to the 'real' solution.  I assume the entire histories of the worldlines would need to be known, in order to calculate things like gravitational waves.  I have downloaded programs which show the evolution of the Schrodinger equation, so I am surprised they are so hard to find for GR.  Some responses I've heard elsewhere are 'the equations are nonlinear,' but from what I hear numerical algorithms are great for these types of problems.  What types of numerical algorithms are out there for GR? Dimensionality of null space when Trace is Zero This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216).  I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me.  Part (a) requires the reader to prove that $\operatorname{Tr}{(AB)} = \operatorname{Tr}{(BA)}$, which I was able to show by writing out each side of the equation using sigma notation.  Part (b) asks the reader to use part (a) to show that similar matrices have the same trace.  If $A$ and $B$ are similar, then  $\operatorname{Tr}{(A)} = \operatorname{Tr}{(S^{-1}BS)}$  $= \operatorname{Tr}(BSS^{-1})$  $= \operatorname{Tr}(B)$,  which completes part (b).  Part (c) asks the reader to show that the vector subspace of matrices with trace equal to zero have dimension $n^2 - 1$.  Curtis provides the hint that the map from $M_n(F)$ to $F$ is a linear transformation.  From this, I used the theorem that $\dim T(V) + \dim n(T) = \dim V$ to obtain the dimension of the null space.  Part (d), however, I'm stuck on.  It asks the reader to show that subspace described in part (c) is generated by matrices of the form $AB - BA$, where $A$ and $B$ are arbitrary $n \times n$ matrices.  I tried to form a basis for the subspace, but wasn't really sure what it would look like since an $n \times n$ matrix has $n^2$ entries in it, but the basis would need $n^2 - 1$ matrixes.  I also tried to think of a linear transformation whose image would have the form of $AB - BA$, but this also didn't help me.  I'm kind of stuck...  Many thanks in advance! 2018 monthly product team updates Mid-2017  started publishing monthly updates. This post will continue that effort into 2018. (.)  Updates are posted at the beginning of the month. They cover what was done last month, what is still in progress and new work that will be started in the current month.  Updates:      (CoC, nav, OpenID, responsive design, bugs, campaigns)  (CoC, comment flags, nav, OpenID)  (question wizard, nav, user survey, OpenID)  (tag watching, GDPR)  (responsive design, tags, teams)  (responsive design, OpenID, teams, sponsorship)  (responsive design, facebook auth, question lists, teams)  (newsletter, sign up, Documentation reputation, channels)  (top bar issues, hats, question template) How to monitor process/program execution in windows? We are trying to develop a small application that can monitor the programs/processes that are executing in a windows machine.  If the program/process is not supposed to run, it should be blocked. It works similar to an antivirus.  This is the basic idea.  I want to know the ways to hook into the OS to get notified about every single program/process trying to run in the machine. A random variable with neither probability density function nor probability mass function... is this example wrong? This is from Shreve's Stochastic Calculus for Finance II, Appendix A.3. Consider infinitely many independent coin tosses with outcomes $\omega_n$, $n = 1, 2, \dots$ such that $\omega_n$ can be either $H$ = heads or $T$ = tails. Define \begin{equation} Y_n\left(\omega\right) = \begin{cases}1 &amp; \text{ if }\omega_n = H \\ 0&amp; \text{ if } \omega_n = T\text{.} \end{cases} \end{equation}  Define $\displaystyle Y = \sum\limits_{n=1}^{\infty}\dfrac{2Y_n}{3^{n}}$. Shreve shows that $Y$ can only take values in the Cantor set, and since that particular set has measure zero, its Lebesgue measure is zero as well, thus it does not have a probability density function (since integrating such a density function would give a value of $0$). This part I understand, as well as my professor.  Now Shreve claims that $Y$ does not have a probability mass function.      If it did, then for some number $x \in C$ [the Cantor set] we would have $\mathbb{P}\left(Y=x\right) &gt; 0$. But $x$ has a unique base-three expansion    \begin{equation} x = \sum\limits_{n=1}^{\infty}\dfrac{x_n}{3^{n}}\text{,} \end{equation}   where each $x_n$ is either 0, 1, or 2 unless $x$ is of the form $\dfrac{k}{3^{n}}$ for some positive integers $k$ and $n$. In the latter case, $x$ has two base-three expansions... In either case, there at most two choices of $w \in \Omega_{\infty}$ [that is, the infinite coin-toss space] for which $Y(\omega) = x$. In other words, the set $\{\omega \in \Omega; Y(\omega) = x\}$ has either one or two elements. The probability of a set with one element is zero, and the probability of a set with two elements is $0 + 0 = 0$. Hence $\mathbb{P}\left\{Y = x\right\} = 0$.   My professor and I were able to follow this until the statement "The probability of a set with one element is zero." Is this a true statement? How so?   I thought it would be perhaps the fact that the Lebesgue measure of $[a, a] = \{a\}$ is zero, but I don't see any reason why Shreve would only use the Lebesgue measure, since it is not a general probability measure.