I'm stuck on the following exercise from Herstein's "Topics in Algebra":  "Show that if $(a,n)=1$ then you can find $[b]\in J_n$ such that $[a][b]=1$"  (note: $[a]$ is the congruence class (mod $n$) of $a$, and $J_n$ is the set of congruence classes mod $n$)  I've been thinking about using the property $(a,n)=1\Rightarrow \exists h,k\in\mathbb{Z}$ such that $ha+kn=1$ and also the Euclidean algorithm by writing $a=qn+b$ $0\leq b&lt;a$ (since $J_n=\{[0],...[n-1]\})$ but I haven't been able to come up with a proof.  So, I would appreciate any comment/hint about how to prove this fact.  Best regards,  lorenzo
I need help with this:  If a is an element of $\mathbb Z_n$ and $\gcd(a,n) &gt; 1$ , then a is not invertible.   First you show that if $a$ is an element of $\mathbb Z_n$ and $\gcd(a,n)&gt;1$, then there  is an element $b$ of $\mathbb Z_n$ and ($b$ is not equal to the zero element) from which $ab = 0$.   The second part I cant get. i.e. show if $b$ is not the zero element  and $ab=0$ then $a$ is not invertible.
I thought the new Apple font was called San Francisco, but I cannot find it in the font list of the Keynote.app for my Mac. Is it called another way or it is not available for Keynote?  I am currently using OS X El Capitan. Entropy and the principle of least action Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ? Assume you cast Barkskin in humanoid form:     You touch a willing creature. Until the spell ends, the target’s skin has a rough, bark-like appearance, and the target’s AC can’t be less than 16, regardless of what kind of arm or it is wearing.   It is clear that under the rules of Wild Shape "transforming doesn’t break your concentration on a spell you’ve already cast", but does the effect of the spell carry over to your new skin? Setting up different application instances under one DNS Code examples, like in the TikZ/PGF manual I like very much the 'side-by-side' code examples in the TikZ/PGF manual...    ... and I would like to use the same thing in a manual I'm writing. I have found the source code for the codeexample environment that the TikZ/PGF manual uses for this purpose (specifically, it's located at /usr/local/texlive/2012/texmf-dist/doc/generic/pgf/macros/pgfmanual-en-macros.tex).  My (probably rather basic) question is: how can I use the same codeexample environment in my own documents? I ask because I tried compiling the example in the related question below, but the file couldn't be found. Assume  $$\lim_{n\to \infty}  (y_{2n} - y_n)=0$$  then  $$\lim_{n\to \infty} y_n$$  exists.  I know it's not true, and I can see a sequence that disprove that $(1,1,2,1,3,2,4,1,5,3,...)$ but I want a sequence that has formula.  hint/help?  thanks Find duplicate records in MySQL In $\mathsf{ZF}$, for each set $A$ exists an ordinal $\alpha$ such that no surjection $A\to \alpha$ exists I have a Dell Inspiron 1525 laptop and I am planning on purchasing a mac mini, new version , I am tight on money for a while and want to use my current laptop screen as monitor for mac mini, I just want to know if it is at all possible to do, I have windows 7 and ubuntu 11.04 installed in my laptop, and I don't really care if need to wipe them out, or purchasing a monitos is the only option left, also I am planning to use it for coding and stuff, so VNC is probably not a good choice. Declaring Text Decorations such as Underline, Strikethrough in a Style How do I include text decorations such as Underline, Strikethrough etc in a Style definition:  &lt;Style x:Key="UnderlinedLabel"&gt;      &lt;Setter Property="Control.FontFamily" Value="Trebuchet MS" /&gt;      &lt;Setter Property="Control.FontSize" Value="14" /&gt;      &lt;!-- Next line fails --&gt;      &lt;Setter Property="Control.TextDecorations" Value="Underline" /&gt; &lt;/Style&gt;   I'm familiar with using the following XAML to underline text:  &lt;TextBlock&gt;    &lt;Underline&gt;        Underlined text    &lt;/Underline&gt; &lt;/TextBlock&gt;   However text decoration is just another style, I want to be able to define it declaritively like FontWeight, FontSize etc.  [Update]  I was applying this style to a Label control.  This was my main problem.  It appears you can't underline text in a Label.  Change to a TextBlock (thanks gix) and all is well. !! Do not use the future tense after unless   [✓]  I won't go unless you go   [✘] [I won't go] unless you will go. [✘]   Why not? Even if the above is prescriptive, what may be some reasons? In my separate question , I quote: unless = if not. If not can precede future tense; so why can't unless?  Footnote:  motivated this. Dynamic pivot: sum of sales per month My table is like this:  Sitecode    Month  Amount --------    -----  ------ XX          Jan     1000 XX          Jan     3000 XX          Apr     3000 XX          Apr     1000   What I want is, to show the result something like this:  Sitecode    MonthJAN   MonthAPR --------    --------   -------- XX          4000       4000 Numerical integration of divergent function I am having trouble with the numerical integration of a divergent function. For example, \begin{equation} n= \int f(x)\,dx = \displaystyle\int \dfrac{\Theta(x-\varepsilon)\,dx}{\sqrt{x-\varepsilon}} \end{equation} is not behaving good when calculated numerically: \begin{equation} n_\text{numerical} \approx f(x_j)(x_j-\varepsilon) + \sum_{k=j+1}^N f(x_k)\Delta x \end{equation} with \begin{equation} x_i = x_\text{min} + i\Delta x \end{equation} and $x_j$ being the smallest point larger than $\varepsilon$. Due to the divergent behavior of $f(x)$ near $\varepsilon$, the numerical result has discontinuities and jumps at every $\varepsilon=x_i$ when I sweep $\varepsilon$. The smoothness of this integration is crucial in the convergence of my simulations. Can anybody suggest a simple trick to solve the problem? All I need is to have a continuous numerical result by continuously changing $\varepsilon$. Show group of order $4n + 2$ has a subgroup of index 2. Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.)  I can easily observe that $\vert G \vert = 2(2n + 1)$ so $2 \mid \vert G \vert$ and 2 is prime. We have satisfied the hypothesis of Cauchy's Theorem so we can say $G$ contains an element of order 2.  This is where I am stuck  I am confused about how the left regular representation relates to the group. So my understanding at this point is that every group is isomorphic to a subgroup of some symmetric group. My question: is the left regular representation $\varphi : G \to S_G$ an isomorphism? where $G \cong S_G$ or is $S_G$ the same thing as $S_{\vert G \vert}$ and $\varphi$ is only an injection? I'm using Dummit and Foote for definitions.  I saw an argument online that said that since we have an element of order 2, there is a $\sigma \in S_G$ of order 2, but it is a product of $2n + 1$ disjoint 2-cycles. I don't understand how they could claim this and tried working it out on my own but didn't get there. -- They then went on to use the parity mapping $\varepsilon : S_G \to \{\pm1\}$ and since we have an odd permutation $\sigma$, we have $[S_G:\text{ker }\varepsilon] = 2$. I understood their computation but not how that directly shows that $G$ has a subgroup of order 2? unless $G \cong S_G$ because of how left regular representation is defined. (but again, I'm not understanding that concept very well yet.)  So, to be clear about my questions:   What is meant by left regular representation, is it an isomorphism or just an injection? and how would it be used here. If it is an isomorphism, the argument online starts to make more sense, but how can they say that since $\sigma$ is even, it is made up of $2n + 1$ disjoint transpositions? If you have a full, proof, I'd appreciate it, but good hints are just as good!   Thanks you! How many times can American Tourist re-enter UK in same 6 month period? Determining best fitting curve fitting function out of linear, exponential, and logarithmic functions Context:  From a question on Mathematics Stack Exchange , someone has a set of $x-y$ points, and wants to fit a curve to it, linear, exponential or logarithmic.  The usual method is to start by choosing one of these (which specifies the model), and then do the statistical calculations.  But what is really wanted is to find the 'best' curve out of linear, exponential or logarithmic.  Ostensibly, one could try all three, and choose the best fitted curve of the three according to the best correlation coefficient.  But somehow I'm feeling this is not quite kosher. The generally accepted method is to pick your model first, one of those three (or some other link function), then from the data calculate the coefficients. And post facto picking the best of all is cherry picking. But to me whether you're determining a function or coefficients from the data it is still the same thing, your procedure is discovering the best...thing (let's say that which function  is -also- another coefficient o be discovered).  Questions:   Is it appropriate to choose the best fitting model out of linear, exponential, and logarithmic models, based on a comparison of fit statistics? If so, what is the most appropriate way to do this? If regression helps find parameters (coefficients) in a function, why can't there be a discrete parameter to choose which of three curve families the best would come from? What IDEs are available for Ubuntu? This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. See  for more information.   This is a community wiki for IDEs available on Ubuntu. Please post one IDE per answer (including more than just a screenshot or a link, please at least put a short description).  In your answer, tell us what the IDE is for (which language(s) or if it is RAD capable). Explaining different Standard Deviation results from same data in ArcGIS Desktop and MS Excel? I have done some interpolation in Geostatistical Analyst in ArcGIS and I got the Standard Deviation (SD) a bit different from the SD that I calculated in Excel Spreadsheet.   What algorithm does each package use? Why are the results different? How to Send Html Emails in Google Spreadsheet Here's an example of a Google Docs Spreadsheet script that sends emails using Google spreadsheets. I want the email Sent in HTML not plain text. Can someone help me out?  function sendEmails() {   var sheet = SpreadsheetApp.getActiveSheet();   var startRow = 2;  // First row of data to process   var numRows = 2;   // Number of rows to process   // Fetch the range of cells A2:B3   var dataRange = sheet.getRange(startRow, 1, numRows, 2)   // Fetch values for each row in the Range.   var data = dataRange.getValues();   for (i in data) {     var row = data[i];     var emailAddress = row[0];  // First column     var message = row[1];       // Second column     var subject = "Sending emails from a Spreadsheet";     MailApp.sendEmail(emailAddress, subject, message);   } } onChange event doesn't trigger