I day ago, i've been solving some induction exercises from my textbook. But when i saw this, it seems a bit tricky and i couldn't come up with a solution. I hope someone can give clarity for this. Thanks! Let $n$, $d$ positive integers and assume $1&lt;d&lt;n$. Show that $n$ can be written in the form $$n=c_{0}+c_{1}d+...+c_{k}d^{k}$$ with $0≤c_{i}&lt;d$, and that these integers $c_{i}$ are uniquely determined. I first proved the uniqueness of $n$ using Euclidean-Algorithm with $n=qd+c_{0}$ what i am troubling now, is how will i conclude this finding to show that $n$ can be written on that form?
How can we prove that for every pair $N \in \mathbb{N}$, and natural number $\beta\in [2, \infty)$ there exists a unique set of integers $x_i \in [0, \beta -1]$, $k\in [0,\infty)$ such that:  $$N = \sum_{i=0}^{k}\beta^ix_{i}, \quad $$  It's a question if every positive integer has its unique representation in every base $\beta$. Is the validity of above statement somehow related to Horner's rule?
I think we can all agree, this sucks:  If you've been around a little while, you've probably encountered hundreds of answers like this in various forums, some of them even marked as &quot;The Answer&quot; by well-meaning1 forum admins looking to close a thread. We could try to enumerate the commonly-observed problems with it...  Link is probably broken If link isn't broken, it's spam If link isn't spam, it's to some broadly-related resource that the answerer found by pressing &quot;I'm feeling lucky&quot; when googling the question's title.  ...but the real issue here is that we don't know! Even if the link was originally pointing directly to the most awesomeest answer ever for exactly the question being asked, there's nothing in the answer itself to even hint at what direction we're being pointed in. Strip the markup, and this is what you're left with:  i think you should take The tutorial HERE! This will help you a lot  Is there even one tiny speck of information there? No. Even the title of the page being linked to would have provided something to go on, but that was entirely too much work. So let me be clear: this sort of response is not an answer. If you see this, flag it. Moderators, if you see it flagged, delete it. There's really only one valid exception to this rule, and that's when the question is:  Suggest me some tutorials where i can learn quick.  Don't get me wrong, it's still a bad answer - but when the question is kinda asking for bad answers this is to be expected. Close or flag the question and move on - this is much more efficient than hanging around to babysit it by deleting every crap answer that inevitably shows up. If you're still reading, you're probably getting really bored by now. &quot;Gee, Shog - answers that contain ZERO information are terrible, horrible, very-bad things? What shocking revelations will you share next?&quot; Yeah. It's really simple... Except, . See, the problem is that once you step away from black-and-white &quot;answers / doesn't answer&quot; decision-making, things get... Interesting. And all those other answers with links in them See, this is an answer:  And so is this:  Yes, they're both very short, and yes, they contain links. But strip the markup, and you still get at least a little bit of useful information. Does that mean these answers should forever hang around the site? No, not necessarily - if it turns out they're just not that useful, they should probably still be removed - or at very least, down-voted so that they rank below other answers. Just remember: if the text of the post contains an honest attempt at answering the question, then it is an answer - so don't flag it otherwise, and if you do, don't complain if your flag gets declined. Answers &amp; Apples If you're still confused, forget about answers. Think about apples instead:  Answers are just like that, but less tasty in pies.  1 By which I mean to say, &quot;criminally lazy&quot; Unix epoch time to Java Date object I have a string containing the , and I need to convert it to a Java Date object.  String date = "1081157732"; DateFormat df = new SimpleDateFormat(""); // This line try {   Date expiry = df.parse(date);  } catch (ParseException ex) {   ex.getStackTrace(); }   The marked line is where I'm having trouble. I can't work out what the argument to SimpleDateFormat() should be, or even if I should be using SimpleDateFormat(). Can we prove that every entire one-to-one function is linear? Draft mode for pgfplots I'm working on a fairly big document that has a lot of graphs of functions- so far it has around 200, by the end it will have around 1500. I use pgfplots for each graphic, and the functions are pretty basic: exponential, rational, polynomial, and trigonometric.  Once I am happy with each graph, I don't necessarily need to see it every time I compile, and would like to minimize compilation time for future drafts. Naturally, the document is already chopped up into chapter files, but each chapter contains around 100 figures.  Having studied the pgfplots documentation, I haven't found a draft mode, but I'm interested in knowing if one could be created.    I'm envisaging something similar to the draft mode of the graphicx package, in which graphics are not included, and blank boxes are put in their place.  Some things I've considered:   the  documentclass; but with about 1500 figures, wouldn't this mean over-modulization?  tikzexternalize, but this still requires an initial 'monster' compilation to get the images externalized before the draft option could be invoked   Here's a MWE to play with  \documentclass{article} \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture}     \begin{axis}         \addplot {x^2};     \end{axis} \end{tikzpicture}  \end{document} I have been invited to write a book chapter in an edited volume, relating to my field of work. I have a paper ready that I have not published and was planning to submit it soon. So, I am wondering is it worth to use that article for the book chapter, as both are on the same topic, or is it in a sense a waste of a journal paper, as book chapters might not be seen to have the same academic value? What should I do? Should I, instead, send the ready paper to a journal for a review and write a book chapter, based on a couple papers I have published, for instance?  Generally speaking, where does a book chapter stand? Is it something between a conference paper and a full-fledged journal article? Or, does does it have the same value as either of the above-mentioned publication types, in your opinion? Something along the lines of 'hero of the occasion', but specifically for birthday? If there isn't, how would you otherwise say that?  ('the subject of birthday party', 'the hero of this birthday party', 'birthday's hero' all sound stupid somehow). I just did my first install of any Linux OS, and I accidentally selected "Desktop GUI" in the install, but I want to build everything myself. Is there any way by which I can remove the GUI environment without re-installing OS? How can I protect my save data from casual hacking? What options are there for saving game data in a secure manner? I'm interested in solutions specifically tailored for C++.   I'm looking for something that is fast and easy to use. I'm only concerned about storing simple information such as   Which levels are and are not unlocked The user's score for each level   I'm curious again to know what's out there to use, any good libraries to use that give me nice, secure game data files that the average player can't mess with.   I just found  here which looks very nice, but it would be great to get some opinions on potential other libraries/options out there. Convert a line drawing from raster to vector I have a raster image, basically a scanned drawing of the outlines (and some detail) of various objects, and I wish to convert these lines to vectors, using either Illustrator or Inkscape.  My present solution is to have the raster image as a background layer, and in a new layer above it I use the pen to follow the lines as close as possible.  Later I tweak these lines to fit the raster image better. That has worked fine for simple drawings. However for some of my extremely complex drawings, this is very tedious to do.  Is there any tool or filter of some sort that is capable of recognizing the lines in line drawings and creating vector paths from them? How do I get started with 'painting with light' photography? Tunnels being naturally dark may not be the most intuitive choice for a place to take pictures. But I love the '' work that artists such as '' are putting out and they're clearly seeking out places like tunnels, caves, garages... Dark places in other words, to create their works of art. Which brings two questions to mind:   How do I go about finding places that I can accomplish this sort of 'tunnel photography.' What are some techniques that I can use to 'paint with light' in my photography once I have found some locations? Possible Duplicate:          I am a little bit confused about asking a question:     Can you please tell me my next work?   or     Could you please tell me my next work?   Which is more correct? How do I show that $x$ is a root of $p(x)$ in $F[x]/\langle p(x)\rangle$? Let $F$ be a field, and $p(x)$ an irreducible polynomial in $F[X]$. Define $K=F[X]/\langle p(x)\rangle$. For every $a\in F$, denote $\bar{a}=\left&lt;p(x)\right&gt;+a$. How do I show that $\bar{x}\in K$ is a root of $p(x)$?. How do I search a specific user's answers? What is the meaning of "would", and how to use it? What is the meaning of would, and which tense is it used in the following sentences?  For example,   That would be Ram calling. I'll answer it. (the meaning of would and which tense is used) The guy on the phone had southern accent. That would be Tom. (the meaning of would and which tense is used) “We saw a police helicopter overhead yesterday morning.” “Really? They would have been looking for those bank robbers.” (the meaning of would and which tense is used) You would have to say/do that. (Why is would used here? What happens to the meaning if would is removed?) John said he didn't steal any money. Well, he would, wouldn't he ? (Why is would used here?)   Why is would used in sentences 1, 2, and 3 to mean "presumption" instead of using other modals, such as: must and should? Does the meaning change if used others?   Can we use will, must, should and could in the place of would, without changing the meaning of sentences?   What kind of meanings would it make if they (will, must, should, and could) are used in the place of would?     I'd be grateful if you could answer me. $\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$ This problem comes from exercise from my lecturer  Let $A$ be a square matrix such that $\mbox{rank}(A)=1$. Prove that $\det(A+I)=\mbox{trace}(A)+1$.  ($I$ is the identy matrix)  I usually haven't any idea in solving linear algebra problem. I am developing my first 2D game in Unity and I have come across what seems an important question.  How do I handle data between scenes?  There seems to be different answers to this:   Someone mention using , while other people told me this should be used to store other things like screen brightness and so on. Someone told me that the best way was to make sure to write everything into a savegame everytime that I changed scenes, and to make sure that when the new scene loads, get the info from the savegame again. This seemed to me wasteful in performance. Was I wrong? The other solution, which is the one I have implemented so far is to have a global game object that isn't destroyed between scenes, handling all the data between scenes. So when the game starts, I load a Start Scene where this object is loaded. After this ends, it loads the first real game scene, usually a main menu.   This is my implementation:  using UnityEngine; using UnityEngine.UI; using System.Collections;  public class GameController : MonoBehaviour {      // Make global     public static GameController Instance {         get;         set;     }      void Awake () {         DontDestroyOnLoad (transform.gameObject);         Instance = this;     }      void Start() {         //Load first game scene (probably main menu)         Application.LoadLevel(2);     }      // Data persisted between scenes     public int exp = 0;     public int armor = 0;     public int weapon = 0;     //... }   This object can be handled on my other classes like this:  private GameController gameController = GameController.Instance;   While this has worked so far, it presents me with one big problem: If I want to load directly a scene, let's say for instance the final level of the game, I can't load it directly, since that scene does not contain this global game object.  Am I handling this problem the wrong way? Are there better practices for this kind of challenge? I would love to hear your opinions, thoughts and suggestions on this issue.  Thanks How to address a professor in person at US universities? At US universities, how should one address a professor while talking to him or her? In my home country, we usually address the professors as either "sir" or "madam" while talking about our queries in person.   What's the norm at US universities? High contrast theme and Firefox-Font colour issues? How to identify "deictic center" for distinguishing "come" & "go"? Read this paragraph from this    My question is:  How to identify "deictic center" for distinguishing "come" &amp; "go"?  Let say I am at home &amp; I say "I am going to the zoo this evening", so the "deictic center" in this case is my home &amp; I move away my "deictic center".  But let say my friend Tom &amp; I currently at my home &amp; Tom said "Are you Coming to the zoo to see the lion?" &amp; I said "Ok, I am coming to the zoo this evening". So, the "deictic center" in this case is "the zoo" &amp; we move towards the "deictic center" (ie the zoo).  It feels like it is very arbitrary.  Note: See this , it said the selection of "deictic center" could be due to sympathy &amp; politeness. I want to use some constants, height and width, when creating an ellipse but it won't work with \newcommand. Here is a minimal example:  \documentclass[12pt, a4paper]{article}  \usepackage{amsthm, tikz} \usetikzlibrary{arrows, shapes, trees, positioning} \begin{document}  \begin{figure} \centering \begin{tikzpicture}       \newcommand*{\width}{2}%     \newcommand*{\height}{0.5*\width}%      \draw[fill=gray!40] (0, 0) ellipse (\width and \height); \node (Y) at (0,0) {\(Y\)};      \end{tikzpicture} \end{figure}  \end{document}    I get the following error:  error: line 14:  Package PGF Math Error: Unknown operator `a' or `an' (in '2and 0.5*2'). ...ay!40] (0,0) ellipse (\width and \height)