How do I make a 2D race car turn smoothly?
I am looking to create realistic car movement using vectors
Fundamental group of a torus with points removed Often times I am not interested in the question so much as I am the answer. Unfortunately, the only site option for bookmarking an optimal answer is to bookmark the question. This leaves you having to remember that no the question wasn't that great, but there's an answer in here worth a million. It'd be nice to be able to favorite an answer allowing you to jump immediately to the answer that was much more valuable than the question. I understand that I can bookmark the answers link using my browser but it seems being able to favorite the answer within the site's ecosystem is useful. 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} I have this new problem since 2014 begin. I cant remotely login to this enterprise network for giving them support. Its keep showing this new error.  How to fix it please?    EDIT: $K/F$ is a field extension of finite degree $n$. How can I prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$, so the ring of $n\times n$ matrix over $F$ contains an isomorphic copy of every extension of $F$ of degree $\leq n$? I have a Microsoft XBOX 360 gamepad with the wireless adapter and I was wondering what I would have to install to use them with Ubuntu I do know that when using this on Windows I have to manually install the drivers for the wireless receiver (if that is relevant.) Matrix Ring of a Semisimple Ring I recently read the concept of semi-simplicity of a (not necessarily commutative) ring.  A ring $R$ is said to be semi-simple if $R$ as a left module over itself is a semi-simple module (This in turn means that each submodule (i.e, left ideal) of $R$ is a direct summand for $R$.)  I am grappling with the following question:     If $R$ is semi-simple, then so is the matrix ring $M_n(R)$.   It is a theorem that each left module of a semi-simple ring is a semi-simple module. Thus $M_n(R)$ is a semi-simple module of $R$. But this doesn't help me.  I need to show that each left ideal of $M_n(R)$ is a direct summand for $M_n(R)$. Is there a characterization  of the left ideals of $M_n(R)$ in terms of the left ideals of $R$?  Lastly, does the converse also hold?     If $M_n(R)$ is a semi-simple ring, then is $R$ necessarily a semi-simple ring?   Thanks. Simple image editor? Remove all attributes from html tags Hidden fields though inheritance How to get the ASCII value of a character What governor limits impact Apex Describe methods? Construct a Borel set on R such that it intersect every open interval with non-zero non-"full" measure This is from problem $8$, Chapter II of Rudin's Real and Complex Analysis.  The problem asks for a Borel set $M$ on $R$, such that for any interval $I$, $M \cap I$ has measure greater than $0$ and less than $m(I)$.  I was thinking of taking the Cantor approach: taking $R$ to be the union of $[a,b]$ with $a$ and $b$ rationals, and for each $[a,b]$ we construct Cantor sets inside it. During theconstruction of each Cantor set, in order to have positive measure on it, we need to take off smaller and smaller intervals from it, namely the proportion goes to $0$. As a result, these Cantor sets are extremely "dense" on their ends. If for an interval $I$ it intersects with the Cantor set on $[a,b]$ while $b-a&gt;&gt;m(I)$, we shall expect the measure of intersection to be rather close to $m(I)$ and then we lose control on these cases.  Is there any way to fix this or shall I consider other approaches?  Thank you Lagrangian of an effective potential If there is a system, described by an Lagrangian $\mathcal{L}$ of the form   $$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$   where $T$ is the kinetic energy and $V$ the potential energy, it is also possible to define the total energy $E$ of the system   $$ E = T + V =\frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) - \frac{k}{r}.\tag{2}$$  If the angular momentum $M$ is defined by $$ M = m r^{2} \dot{\phi},\tag{3}$$ then $E$ can be written as  $$ E = \frac{m}{2}\dot{r}^2 + \underbrace{\frac{M^2}{2mr^2}-\frac{k}{r}}_{V_\textrm{eff}\left(r\right)},\tag{4}$$  where the last two terms are written as new, "effective" potential $V_\textrm{eff}\left(r\right)$.    In addition, using the definition of $M$, the Lagrangian $\mathcal{L}$ can be written as  $$ \mathcal{L} = \frac{m}{2}\dot{r}^2 + \underbrace{\frac{M^2}{2mr^2}+\frac{k}{r}}_{-V_\textrm{eff}\left(r\right)},\tag{5}$$  where the sign of $V_\textrm{eff}\left(r\right)$ has been changed, since $\mathcal{L} = T-V$. But from this argument, it appears that there are two different possible ways to construct the same effective potential. This seems to me contradictory. Where is my mistake? I am compiling the following MWE using XeLatex within TeXnicCenter and get a garbled equation number:  \documentclass{article} \usepackage{amsmath} \usepackage{polyglossia} \setmainlanguage{hebrew} \newfontfamily{\hebrewfont}{Arial}  \begin{document} \begin{equation}\label{eq:1} 2+2=4. \end{equation} \end{document}      What is going on here? My boss gave me a task with completion date suppose 31 March, 2017and I completed on 28 February, 2017. Is it in time or before time. I am a newbie in combinatorics and still don't have enough tools to handle this kind of problems.  Assuming I have a set of integers: $ \{1,2,3,4,\dots,n\} $  In how many ways, can I choose $k$ numbers out of those $n$, such that none of them are consecutive?  For instance, for the following set $ \{1,2,3,4,5\} $  For $ k=3 $ I have only 1 option: $\{1,3,5\}$  My purpose is first of all to understand the way you would think about this  problem, not necessarily the solution itself (I have the final answer for it).  Thanks. XGBoost Loss function Approximation With Taylor Expansion Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$? Are questions about (development) devices allowed on Stack Overflow? I've seen many questions on Stack Overflow asking which devices are best for developers. This made me wonder if questions asking for hardware device recommendations (best device for programming, best device within a certain budget, etc.) are on-topic for Stack Overflow, or if they should be asked somewhere else?  For Instance check these links:         etc.