Linear Function: f(ax + by) = af(x) + bf(y) vs y = ax+b.
Why is a linear transformation called linear?
The problem I am given is finding a solution to the following pair of equations:  $3x + 4y$ $\equiv$ $5$ $mod$ $13$  $2x + 5y$ $\equiv$ $7$ $mod$ $13$  By reading the methodology here:   Since the modulos in my problem are also the same, I am able to use substitution, but this leaves fractions in my expressions.  I then used matrix multiplication as follows:  $\begin{bmatrix} 3 &amp; 4\\ 2 &amp; 5\end{bmatrix}$ $\begin{bmatrix}\ x\\ y\end{bmatrix}$ = $\begin{bmatrix}\ 5\\ 7\end{bmatrix}$   And solving this also gives me a matrix with fractions, so I am a bit lost on how to solve this problem.  Any help is appreciated. Thank you very much. Counting buffer overlap in QGIS For my thesis I'm modeling the identification of recreation areas.  Therefore, there has been a survey where people were asked to specify where they recreate. They had to indicate this place on a map and give a radius. Now I have to analyse which areas were most visited, and afterwards use the data to give a weighted rating with the travel distance.  So, I use QGIS to Buffer the data points by using the given radius. Now, I want to calculate for every point in the country how many times it was visited.  Any thoughts how to do this in QGIS? Dealing with bundle/omnibus/list-of-questions question that consists of rather distinct questions I have looked at the various help pages but I couldn't find a clear guideline for the following issue: often enough someone will ask in the same post (i.e. in the same SE-question) a whole bunch of questions (in the plain English sense of the word), which while may be about the same topic are nevertheless sufficiently distinct that each may require several paragraphs of text to answer adequately.   I can see an upside for leaving related questions grouped together from the asker's perspective: "there's a thread (on SE) answering all my questions about topic T!" On the other hand, from the perspective of organizing questions by having a clear heading/title for each (actual) question and so forth, i.e. from the perspective of SE being a fairly structured database of Q&amp;As instead of being a more like a collection of forum threads "I'm confused about topic T", the omnibus question-list approach seems clearly sub-optimal.  The omnibus approach also makes the issue of accepting answers (and perhaps reviewing such actions) difficult: if two different users give replies to different parts/sub-sets of the question-list, which one is the most adequate one? Even upvoting all partial answers seems iffy because it actually penalizes the user(s) who provides(s) the most complete answer to the question-list: since a user can only give at most one upvote to another user's answer, if user X provides answers to A &amp; B in the same SE-answer, he can only get at most one upvote from user Y, whereas if these were two different SE-answers, he could get two upvotes from user Y.  So, given the above problems, which I think are real enough, is there anything in the SE system addressing list-of-questions question splitting? What is the status of pdfsync? The pdfsync homepage states that:     At the end of this summer (2007), pdfsync is slowly migrating to pdftex itself. We are working at giving pdftex all the pdfsync ability and more...   The pdfsync README file discourages people from using pdfsync:     pdfsync uses extremely sensible code. You should not use pdfsync on final documents because it can change the layout rather significantly.   From your knowledge and experience, what is the actual status of pdfsync?   Is it safe to use pdfsync? Is there an alternative for pdfsync that is built into pdftex? Is there a different tool for pdflatex that offers DVI-like functionality (forward and backward search)? The gluing lemma for closed sets states:     Given a finite closed cover $\{A_i\}$ of a topological space $X$, together with continuous maps $\{f_i : A_i \to Y\}$ into some other topological space $Y$, there exists a unique continuous map $f : X \to Y$ whose restriction to each $A_i$ is equal to $f_i$.   Question: What is a good and simple counter-example when the gluing lemma fails in the case that $\{A_i\}$ is infinite, but countably so.  My attempt: I have only been able find a silly counter-example: let $\{A_\alpha\}$ be the collection of all points of $X = [0,1] \subset \mathbb{R}$, and let $f_\alpha = 0$ for all $\alpha$ except for $\alpha = \alpha_0$, for which $A_{\alpha_0} = \{0\}$ and $f_{\alpha_0} = 1$. First quick glance at a dataset Please pardon my ignorance, but...  I keep finding myself in a situation, where I'm faced with a bunch of new data I managed to find. This data usually looks something like this:  Date     Number1  Number2  Category1  Category2 20120125      11      101        Dog      Brown 20120126      21       90        Cat      Black 20120126      31      134        Cat      Brown (...)   Usually at first glance I can't really tell if there are any trends here. The correlations between the various columns may not be very significant, but I would be delighted if I didn't have to manually create a plot for every possible combination of columns/categories.  Is there a tool out there which would accept a table of data along with information which columns should be treated as numbers, dates and categories and then proceed to plot:   correlations between each two numerical columns correlations between each two numerical columns, with separate trend lines for each category each number column as a time series, each number column as a time series, separated by category, etc.   In the end this would generate a large number of plots, most of which would show only noise. Ideally, the tool could score the plots by correlation and in the end display a slideshow starting with the highest scoring plots. This would be a very imperfect, but useful first glance at the dataset.  So? Is there a tool everyone uses for this and I just don't know about it, or is this something we need to make? Parsing date/time string with timezone abbreviated name in Python? Could gravity accelerate light? Gravity causes anything with energy to accelerate toward the source. Black holes, for example, have such strong gravity that they pull in light and don't let any escape. But can acceleration still apply to light? The speed of light is constant, of course, but why are photons affected by gravity yet aren't accelerated by it?  Edit: My main question is why photons aren't affected in the same way as most other particles. I'm perfectly aware that it cannot surpass lightspeed, but I want to know what makes it unaffected by acceleration while other particles are affected. How can I stop the Devel module from generating blog/forum entries for Anonymous user? I'm using Devel's generate content feature to generate Blog/Forum entries for a site I'm working on and funny enough, I get to have a blog post by an anonymous user or forum topic by an anonymous user.  How can I stop Devel from doing this? I've only given Authenticated Users/ Administrators the permission to create a blog post or forum topic but still it generate for anonymous user. How can correct this? In my textbook it is given that      'The wave function describes the position and state of the electron and its square gives the probability density of electrons.'    Can someone give me a very simple example of a wave function with explanation?  (Note: This question is not a duplicate. I have searched for other questions of this type but the answers were overwhelmingly difficult to understand.) I'm curious why PCBs have one square pad in row for 0.1" headers. What is purpose of putting one square pad? What it indicates? Can you push more amps through square pad than round? Is GTA Online cross-platform? This question is about cosmology and general relativity. I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.   Does it have infinite mass or is it dishomogeneous?  How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all? Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant. While studying for an exam in complex analysis, I came across this problem. Unfortunately I was not able to solve it. Any help would be greatly appreciated.  Let $U ⊂ \mathbb{C}$ be a domain and $f_1, . . . , f_n : U \rightarrow \mathbb{C}$ holomorphic functions, such that $\sum_{k=1}^n \left| f_k(z) \right|$ is constant on $U$. Show that $f_1, . . . , f_n$ are constant. Storing the original document class page layout with geometry Is it possible, using the geometry package, to store (and restore) the original layout specified by the document class (in my case, the class is report)?    Here is some context.  I have been editing a long document in the default page layout for the report document class.  Now I found out that I need a title page.  There is a particular title page layout, and in fact, it works perfectly if I just load the geometry package with its default and change just two settings.  But then I want the rest of the document back in the original page layout!  I can't do something like \restoregeometry, since I think geometry just overrides the class options... i.e. there doesn't seem to be anything to restore.  Similarly, it seems I can't use \savegeometry or \loadgeometry.  So how can I get back to my original layout for the rest of the document without painstakingly hard-coding in all of the specific page layout variables?    Here is a minimum working example:   \documentclass[a4paper,10pt]{report}  \RequirePackage[a4paper,dvips]{geometry} % g1 \geometry{top=2.5cm,bottom=2.3cm}        % g2  \begin{document}  \input{the_title_page}  \SomeGeometryCommandsHere  % g3  \input{the_rest}  \end{document}   If I run the above code with the lines labeled "g" commented out, then the title page does not have the correct page layout, but the rest of the document is formatted perfectly.    If I leave lines g1 and g2 uncommented, then the title page has exactly the correct layout, but no matter what I put for g3 (e.g. \loadgeometry, \restoregeometry, etc.) I can't recover the original layout that I had when g1 and g2 were commented.  Of course, I could completely specify the original layout exhaustively in line g3, and this should in principle work.  However, I spent almost one hour on this last night and when I still couldn't get it formated correctly I decided I really needed a smarter solution. Calculate the display width of a string in C#? A Java version of this question was just answered, and, well, I don't know how to do this in .net.   So how do you calculate the display width of a string in C# / .net? The limit of $((1+x)^{1/x} - e+ ex/2)/x^2$ as $x\to 0$ What is the importance of the Diophantine inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$? In Humphreys' book  Introduction to Lie algebras and Representation Theory(3rd printing, p.62), the Diophantine inequality $$\frac{1}{p} + \frac{1}{q} + \frac{1}{r} &gt; 1$$ appeared while classifying all Coxeter graphs of irreducible root systems. Under the assumption $p \geq q \geq r \geq 2 $, the possible triples turn out to be $(p,2,2), (3,3,2),(4,3,2),(5,3,2)$, which correspond to the Coxeter graph $\text{D}_{p+2}, \text{E}_6, \text{E}_7, \text{E}_8$, respectively.  It seems quite straightforward to solve the inequality in positive integers, but Humphreys said that      This inequality, by the way, has a long mathematical history.   without further comment. So I want to know if this Diophantine inequality has any important significance, or if it has appeared in other context (maybe relevant history) before. What are the main differences between performing principal component analysis (PCA) on the correlation matrix and on the covariance matrix? Do they give the same results? Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ a) Let $a&gt;0$ and the sequence $x_n$ fulfills $x_1&gt;0$ and $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ for $n \in \mathbb N$. Show that $x_n \rightarrow \sqrt a$ when $n\rightarrow \infty$. I have done it in two ways, but I guess I'm not allowed to use the first one and the second one is incomplete. Can someone please help me?  We already know $x_n \rightarrow \sqrt a$, so we do another step of the iteration and see that $x_{n+1} = \sqrt a$.  Using limit, $x_n \rightarrow x, x_{n+1} \rightarrow x$ (this is the part I think it's incomplete, don't I have to show $x_{n+1} \rightarrow x$, how?), we have that   $$x = \frac x 2 (1 + \frac a {x^2}) \Rightarrow 1 = a/x^2 \Rightarrow x = \sqrt a$$ b) Let the sequence $x_n$ be defined as $x_{n+1}= 1 + \frac 1 {x_n} (n \in \mathbb N), x_1=1$. Show that it converges and calculate its limit. &quot;Tip: Show that sequences $x_{2n}$ and $x_{2n+1}$ monotone convergent to the limit.&quot; I didn't understand the tip, how can this help me? Does it make a difference if the number is odd or even? Thanks in advance!