i want to number the equation without the number 2 in the image please   \begin{equation}\label{N} (Nu)(x,y)=\mu(x,y)+\frac{1}{\Gamma(r_1)\cdot\Gamma(r_2)}\displaystyle{\int_{1}^{x}}\displaystyle{\int_{1}^{y}} \left(\log\frac{x}{s} \right)^{r_1-1}\left(\log\frac{y}{s} \right)^{r_2-1}\frac{f(s,t,u(s,t))}{st}dtds  \end{equation}
\documentclass[]{phstyle} \usepackage{latexsym,amsfonts,amssymb, undertilde} \usepackage[fleqn]{amsmath} \usepackage{makeidx}  %Preamble  \numberwithin{equation}{chapter} %\renewcommand{\theequation}{\thesection.\arabic{equation}}  \begin{document}  %\include{chapI}   these 2 lines are because the chapters are different tex files  %\include{chap1}  \renewcommand{\thechapter}{\Roman{chapter}}  %\setcounter{chapter}{0} %\renewcommand{\theequation}{\thechapter.\arabic{equation}}  \chapter{Introduction}  \section{Chap I Section1}  \begin{equation} this is equation I.1 \end{equation}  \begin{equation} this is equation I.2 \end{equation}  \section{Chap I Section2}  \begin{equation} this is equation I.3 \end{equation}  \chapter{Chapter 1}  \section{Chap 1 Section 1}  \begin{equation} this is equation 1.1.1 \end{equation}  \begin{equation} this is equation 1.1.2 \end{equation}  \end{document}     I have been typing Chapters 1-5 that contain sections and subsections and equations. The equations were numbered globally, i.e. if I had 3 equations in section 1 of Chap 1, they were numbered 1.1.1, 1.1.2, 1.1.3 and for Section 2, the equations were 1.2.4, 1.2.5, etc.  Recently I had to change Chap 1 and everything in it to Chapter I (roman number) and Chap 2 were renumbered to Chap 1, etc.  My problem is that LaTeX started numbering the equations in Chap I (the other chapters did not have this problem) according to the sections and not globally anymore. That is, the 3 equations in Chap I were I.1, I.2, I.3, and the equations in the second section of Chap I became I.1, I.2, etc instead of I.4, I.5, etc  I needed the prefix to be the chapter and not the sections. I don't understand why there is a change. I have never used \numberwithin in my book  Here are some of my codes  \documentclass[]{phstyle} \usepackage{latexsym,amsfonts,amssymb, undertilde} \usepackage[fleqn]{amsmath}   for the equation numbering in Chap I  \renewcommand{\thechapter}{\Roman{chapter}} \renewcommand{\theequation}{\thechapter.\arabic{equation}} **   Previously when Chap I was Chap 1, ** was  \renewcommand{\theequation}{\thesection.\arabic{equation}}   I have also tried the line below but it didn't work  \renewcommand{\theequation}{I.\arabic{equation}}   Thank you in advance
A doubt from Milnor's "Topology from a Differentiable Viewpoint". This is a doubt from Milnor's "Topology from a Differentiable Viewpoint".     For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of points in $f^{-1}(y)$. It is easy to see that $n(f^{-1}(y))$ is finite. It can also be seen that $n(f^{-1}(y))$ is locally constant; i.e. there is a neighbourhood of $y$ such that for every point $y'$ in that neighbourhood, $n(f^{-1}(y'))=n(f^{-1}(y))$.   The proof for the statement in bold is the following:     Let $x_1,x_2,\dots,x_n$ be the points of $f^{-1}(y)$, and choose disjoint neighbourhoods $U_1,U_2,\dots,U_n$ of these points, which are mapped diffeomorphically into neighbourhoods $V_1,V_2,\dots,V_n$ of $y$. The required neighbourhood is $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$.   I don't understand how $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$ is open. I know that $(V_1\cap V_2\cap\dots V_n)$ is open. However, how can one prove that $f(M-U_1-U_2-\dots U_n)$ is closed? I can see this working if $f$ is surjective, but not otherwise. Ball to Ball Collision - Detection and Handling With the help of the Stack Overflow community I've written a pretty basic-but fun physics simulator.    You click and drag the mouse to launch a ball.  It will bounce around and eventually stop on the "floor".  My next big feature I want to add in is ball to ball collision.  The ball's movement is broken up into a x and y speed vector.  I have gravity (small reduction of the y vector each step), I have friction (small reduction of both vectors each collision with a wall). The balls honestly move around in a surprisingly realistic way.  I guess my question has two parts:   What is the best method to detect ball to ball collision? Do I just have an O(n^2) loop that iterates over each ball and checks every other ball to see if it's radius overlaps? What equations do I use to handle the ball to ball collisions? Physics 101 How does it effect the two balls speed x/y vectors?  What is the resulting direction the two balls head off in?  How do I apply this to each ball?     Handling the collision detection of the "walls" and the resulting vector changes were easy but I see more complications with ball-ball collisions.  With walls I simply had to take the negative of the appropriate x or y vector and off it would go in the correct direction.  With balls I don't think it is that way.  Some quick clarifications: for simplicity I'm ok with a perfectly elastic collision for now, also all my balls have the same mass right now, but I might change that in the future.    Edit: Resources I have found useful  2d Ball physics with vectors:  2d Ball collision detection example:     Success!  I have the ball collision detection and response working great!  Relevant code:  Collision Detection:  for (int i = 0; i &lt; ballCount; i++)   {       for (int j = i + 1; j &lt; ballCount; j++)       {           if (balls[i].colliding(balls[j]))           {             balls[i].resolveCollision(balls[j]);         }     } }   This will check for collisions between every ball but skip redundant checks (if you have to check if ball 1 collides with ball 2 then you don't need to check if ball 2 collides with ball 1.  Also, it skips checking for collisions with itself).  Then, in my ball class I have my colliding() and resolveCollision() methods:  public boolean colliding(Ball ball) {     float xd = position.getX() - ball.position.getX();     float yd = position.getY() - ball.position.getY();      float sumRadius = getRadius() + ball.getRadius();     float sqrRadius = sumRadius * sumRadius;      float distSqr = (xd * xd) + (yd * yd);      if (distSqr &lt;= sqrRadius)     {         return true;     }      return false; }  public void resolveCollision(Ball ball) {     // get the mtd     Vector2d delta = (position.subtract(ball.position));     float d = delta.getLength();     // minimum translation distance to push balls apart after intersecting     Vector2d mtd = delta.multiply(((getRadius() + ball.getRadius())-d)/d);        // resolve intersection --     // inverse mass quantities     float im1 = 1 / getMass();      float im2 = 1 / ball.getMass();      // push-pull them apart based off their mass     position = position.add(mtd.multiply(im1 / (im1 + im2)));     ball.position = ball.position.subtract(mtd.multiply(im2 / (im1 + im2)));      // impact speed     Vector2d v = (this.velocity.subtract(ball.velocity));     float vn = v.dot(mtd.normalize());      // sphere intersecting but moving away from each other already     if (vn &gt; 0.0f) return;      // collision impulse     float i = (-(1.0f + Constants.restitution) * vn) / (im1 + im2);     Vector2d impulse = mtd.normalize().multiply(i);      // change in momentum     this.velocity = this.velocity.add(impulse.multiply(im1));     ball.velocity = ball.velocity.subtract(impulse.multiply(im2));  }   Source Code:     If anyone has some suggestions for how to improve this basic physics simulator let me know!  One thing I have yet to add is angular momentum so the balls will roll more realistically.  Any other suggestions?  Leave a comment! Plagiarism of Lecture Slides Original Post:  I recently attended a lecture with highly technical content.  One of the middle slides in the deck had a particular phrase that reminded me of something that I had read several years before.  It took me a few minutes, but I found the original source (another slide deck) and am shocked to find that the professor lifted 35 slides verbatim from the original source.  The University holds students accountable for plagiarism and strictly defines plagiarism such that this clearly fits.  The slides were used verbatim and no credit or reference was provided.  Further, logos were added and color schemes were changed to make make them look slightly more like original content.  What should I do?  This professor has a professional full time job and I can imagine this getting swept under the rug, but this is clearly plagiarism and is spelled out very clearly in the student conduct code.  Is this an overreaction or should I pursue this?  This is confounded by the fact that I am very disappointed by the rigor of the course and feel like the professor lacks adequate technical background to teach the course (I guess my suspicions came true).  I'm really busy at work so don't want to deal with a big todo, but believe this really discredits the academic integrity of the course.  Additional content in response to comments:  The University defines plagiarism as "Using another writer's words without proper citation."  There is no proper citation given here, and these are not educational materials.  This is 35 slides from someone else's tutorial presentation.  I do not believe that the professor and the author of the original content have a relationship.  I have emailed the author to enquire about any permissions or relationship.  Still, even with permission, this is still plagiarism as defined be most people and as defined by the University.  All original works in the US are implicitly copyright protected, so there is also a copyright issue, but this is not currently my concern, though it could be of interest to the original author.  I have emailed him, so this is up to him.  This is a US institution and is a top tier highly respected school.  I am also not a lay student, but a Ph.D. with a degree from a top tier university who has served as a faculty member at two top tier universities (including this one).  I have also taught undergraduate courses in the past.  I have never engaged in this type of plagiarism, have never known colleagues to do so, and am shocked that so many people suggest this is par for the course in teaching.  My doctorate is from a school that prides itself on its honor code and takes honor code violations very seriously and I have served on the honor council and deliberated on a case of plagiarism in the past.  I really am shocked by this incident.  Response from Original Author of the Slides:  I have received a response from the author of the slides and he states that the work is his original work, that there are no additional authors of the work, that no one has been given permission to present the work, and that he believes his copyright on the work has been infringed upon.  He has requested additional information to assert his claim on the copyright. Server 2012 Core (no GUI): Remote Server manager not working I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking Is there a spec that the id of elements should be made global variable? How to revert to GNOME Classic Desktop? Working with a huge file. Illustrator is super slow. What can I do to ease the load? How do I avoid rendering glitches from overlapping faces? What is the difference between "INNER JOIN" and "OUTER JOIN"? Also how do LEFT JOIN, RIGHT JOIN and FULL JOIN fit in? How to ensure only a single instance of my application runs? I am new to compact sets, and I had some hard time trying to solve this:  Let $X$ be an infinite set; $T$ the discrete topology on $X$ .Prove that $(X,T)$ is not compact.  I know that I may consider the open cover $\{\{x\},x \in X\}$ and prove that it doesn't have a finite subcover,but I don't know how to write it exactly.  Moreover, I seriously can't understant why $X$ itself cannot be considered as a finite subcover?  Any help is appreciated. Extruding a circular shape from a non-circular object I need to extrude a circular shape from a surface that's basically just a grid of quads. What's the best practice to do this without screwing up the topology too much?  I basically want to have a circular face I can extrude, in order to have a cylinder like protuberance on the object, something like this: edges with thickness and color Is there a way to vary the thickness and color of lines created by just edges and vertices? I realize that a face can be defined and then color and material applied, but is there any way to apply a color to the edges? Prove $0! = 1$ from first principles How can I prove  from first principles that $0!$ is equal to $1$? Note: This is not the same as , which is about the moderator tool. I am talking about the flag history page that is accessible through the user profile at .  I tend to have a pretty good hit rate for flags, well above 90% helpful on several sites. I also semi-regularly review the flag history page, both to see the status of any outstanding flags as well as see how flags I raised have been handled. Particularly, whenever a flag of mine is declined, I look over the post I raised the flag against with this new information in mind, to try to determine if I should change my flagging behavior in some way to better match the specific site's scope.  However, comment flags are difficult in this regard. Most often, there is only a binary status: helpful or declined. Helpful comment flags nearly always lead to the comment being deleted, so that obviously could present a bit of a problem, but that's not really relevant because a helpful flag means you should just keep doing what you've been doing. Declined comment flags, on the other hand, usually mean the comment is kept. It's the declined comment flags that I want to learn more about. But there's no way to!  All the flag says is which post it was on, how I flagged a comment, that I was the one to raise the flag (which frankly is rather useless information when you are already looking at your own flags summary), when the flag was raised and whether it's pending, helpful or declined. In a very few cases the moderator handling the flag may leave a helpful return comment.  This can get even more confusing if I raise multiple flags in quick succession against comments on the same question including any answers to it, because they basically all get bundled together.  Even in the case of a declined flag where the comment remains, there is no way to find your way back to the specific comment you flagged! Thus it is very difficult to recognize the error of your ways. Because comment flags (with few exceptions) are handled by diamond moderators only, the lack of feedback about which comment's flag was declined means it's easy to keep wasting the moderators' time without even intending to do so.  Hence my feature request: add a link to the specific comment that the specific flag was raised against, ideally only if that comment remains. "None is" or "none are" I am drawing a 3D array like the attached picture, which actually describes a multidimensional array. However, I can not put the name of the axis and take the value from the matrix like the attached picture.  This is the code that I have:  \documentclass[margin=5mm]{standalone} \usepackage{tikz} \usetikzlibrary{matrix,calc}  \begin{document} \begin{tikzpicture}[every node/.style={anchor=north east,fill=white,minimum width=1.4cm,minimum height=7mm}] \matrix (mA) [draw,matrix of math nodes] { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mB) [draw,matrix of math nodes] at ($(mA.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mC) [draw,matrix of math nodes] at ($(mB.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \draw[dashed](mA.north east)--(mC.north east); \draw[dashed](mA.north west)--(mC.north west); \draw[dashed](mA.south east)--(mC.south east); \end{tikzpicture}   \end{document} In my custom Drupal 8 plug-in I perform such query:  $select = db_select('my_table', 'e'); $select-&gt;addField('e', 'field1');  $select-&gt;condition('field3', 'NULL', '!='); $select-&gt;condition('field4', array(11, 12, 31, 32, 33), 'IN');  $select-&gt;groupBy("e.field1"); $select-&gt;orderBy("e.field1"); $select-&gt;orderBy("e.field5"); $entries = $select-&gt;execute()-&gt;fetchAll(\PDO::FETCH_ASSOC);    And it works fine. It returns me all records grouped by field1 that meets the condition, orderered. But it returns only field1 column.  I wanted to get more columns, so I have added another $select-&gt;addField('e', 'field2'); just after first one:  $select = db_select('my_table', 'e'); $select-&gt;addField('e', 'field1'); $select-&gt;addField('e', 'field2');  $select-&gt;condition('field3', 'NULL', '!='); $select-&gt;condition('field4', array(11, 12, 31, 32, 33), 'IN');  $select-&gt;groupBy("e.field1"); $select-&gt;orderBy("e.field1"); $select-&gt;orderBy("e.field5"); $entries = $select-&gt;execute()-&gt;fetchAll(\PDO::FETCH_ASSOC);    This time I get error:     Drupal\Core\Database\DatabaseExceptionWrapper: SQLSTATE[42000]: Syntax error or access violation: 1055 Użyto 'my_database.e.field2' bez umieszczenia w group by:   Which means "my_database.e.field2 has been used without putting in into group by". But I don't want to group by every column I select.  Unfortunatelly Drupal 8 added ONLY_FULL_GROUP_BY :( Then, how can I use groupBy and select more then one column at the same time? What is the name of this connector from a fan? I am EE hobbyist and I often used recycled parts in my projects. I came across this connector to a Heat Sink Fan and would like to know the name of it.  Thanks for the help. This is probably a silly question, but how can I run ANOVA using only the mean, standard deviation, and the sample size?  i.e.  Group  mean    sd    n  1:    200      5    5    2:    100     22    5  3:    150     14    5   I am using Statsoft Statistica software.  I know how to run it using the spreadsheet with all the data, but I have never done this using only the information listed above.  Any information would be greatly appreciated.