(\x)^2 will plot correct parabola, while \x^2 will have negative x branch upside down. Why () are required for this formula to work?  \documentclass[10pt]{article} \usepackage[utf8]{inputenc} \usepackage{tikz} \usetikzlibrary{shapes.geometric} \usetikzlibrary{shapes.misc}  \begin{document} \begin{figure}[!htb]     \begin{tikzpicture}         \draw[very thin,color=gray] (-4.1,-4.1) grid (4.1,4.1);         \draw[-&gt;] (-4.2,0) -- (4.2,0) node[right] {$x$};         \draw[-&gt;] (0,-4.2) -- (0,4.2) node[above] {$f(x)$};          \draw[thick,purple,domain=-3:3,smooth] plot (\x,{0.5*\x^2});         \draw[thick,brown,domain=-3:3,smooth] plot (\x,{0.2*\x^3});     \end{tikzpicture}  \caption{$y=1/2 x^2, y=1/5 x^3$} \end{figure}  \end{document}
I have some files with TikZ graphics that have suddenly changed appearance since I upgraded to PGF-2.10.  Here is a minimal example:  \documentclass{minimal} \usepackage{tikz} \begin{document} \pgfversion\par \begin{tikzpicture} \draw[blue] plot[domain=-1:1] (\x,{\x^2 + \x^3}); \draw[xshift=4cm,blue] plot[domain=-1:1] (\x,{(\x)^2 + (\x)^3}); \end{tikzpicture} \end{document}   In PGF 2.00 I had this:    while in 2.10 I have this:    The one on the left in PGF 2.10 looks like the graph of y=x^3 but could also be y=x^3 + x|x|.   This has been  on sourceforge and  on the pgf-users mailing list.  Although I haven't checked myself yet, the bug remains open and some of the commenters below report that problems still exist in the CVS version.   Geoffrey Jones notes the bug is in the 2.00 CVS version:     Hi Matthew, not an answer, just an FYI   that's too large for the comment box.   My compile of your code through pgf   v2.00 produces the result you see with   v2.10, above:       I.e., the difference you observe seems   to pre-date v2.10.   There is a workaround by inserting parentheses.  But since I have so many old lecture files I'd rather find a solution that doesn't involve rewriting my TikZ code.  And I'm squeamish about using the CVS bleeding edge code even if it were fixed, since I don't want to introduce other problems.  Ideally I'd like a simple patch that I can stick at the preamble, or just fix one file in the PGF distro.
How does the Rectified Linear Unit (ReLU) activation function produce non-linear interaction of its inputs? When used as an activation function in deep neural networks The ReLU function outperforms other non-linear functions like tanh or sigmoid . In my understanding the whole purpose of an activation function is to let the weighted inputs to a neuron interact non-linearly. For example, when using $sin(z)$ as the activation, the output of a two input neuron would be:  $$ sin(w_0+w_1*x_1+w_2*x_2) $$   which would approximate the function  $$  (w_0+w_1*x_1+w_2*x_2) - {(w_0+w_1*x_1+w_2*x_2)^3 \over 6} + {(w_0+w_1*x_1+w_2*x_2)^5 \over 120} $$  and contain all kinds of combinations of different powers of the features  $x_1$ and $x_2$.   Although the ReLU is also technically a non-linear function, I don't see how it can produce non-linear terms like the $sin(), tanh()$ and other activations do.  Edit: Although my question is similar to , I'd like to know how even a cascade of ReLUs are able to approximate such non-linear terms. Merging two SQLite databases which both have junction tables How do I get the square root of a complex number? If I'm given a complex number (say $9 + 4i$), how do I calculate its square root? I'm very new to GIS concepts. Is there a way or procedure of finding out whether the co-ordinate system is in UTM, latitude/longitude or in stateplane using .prj file?   My .prj file contains the following text:      GEOGCS["WGS 84",DATUM["WGS_1984",SPHEROID["WGS   84",6378137,298.257223563,AUTHORITY["EPSG","7030"]],AUTHORITY["EPSG","6326"]],PRIMEM["Greenwich",0,AUTHORITY["EPSG","8901"]],UNIT["degree",0.01745329251994328,AUTHORITY["EPSG","9122"]],AUTHORITY["EPSG","4326"]] I've been using gksudo nautilus and sudo nautilus through Alt+F2.   What's the difference? They look very similar! Update manager doesn't offer an upgrade to the new release Ship an application with a database I'm aware that this one is far from yes or no question, but I'd like to know which techniques do you prefer in categorical data analysis - i.e. cross tabulation with two categorical variables.  I've come up with:    &chi;2 test - well, this is quite self-explanatory  Fisher's exact test - when n &lt; 40, Yates' continuity correction - when n > 40,  Cramer's V - measure of association for tables which have more than 2 x 2 cells, &Phi; coefficient - measure of association for 2 x 2 tables, contingency coefficient (C) - measure of association for n x n tables, odds ratio - independence of two categorical variables, McNemar marginal homogeniety test,   And my question here is: Which statistical techniques for cross-tabulated data (two categorical variables) do you consider relevant (and why)? I'm having difficulty with exercise 1.43 of Lang's Algebra. The question states     Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroup that is isomorphic to $G/H$.   Thinking about this for a bit, the only reasonable approach I could think of was to construct some surjective homomorphism $\phi\colon G\to K$ for $K\leq G$, and $\ker\phi=H$, and then just use the isomorphism theorems to get the result.  After a while of trying, I've failed to come up with a good map, since $H$ seems so arbitrary. I'm curious, how can one construct the desired homomorphism? This is just the approach I thought of, if there's a better one, I wouldn't mind seeing that either/instead. Thank you. Why is unity feedback used? Why is the unity feedback system used? This block diagram is for a 2 DOF ball balancing using a PID controller. Sudden increase in timing when summing over 250 entries What is the name of a small unluxurious restaurant? Is there a name for this kind of restaurant? Searching Google, I can't find any synonym of restaurant. I've tried to search for images of unluxurious restaurant, small restaurant or mundane restaurant, with no satisfactory result. The restaurant I'm looking for is small, not so clean and definitely not a place that high class people go to eat.  It can be a house:    Or lie on a side of the street: I have recently submitted a paper to arXiv, but the arXiv moderators put my paper in status "on hold", because they have to make a decision on the right classification for the paper.   To those who have experienced a similar situation, how long did it take before they made a decision and published the paper on the arXiv? I just want to have an idea of how long I will have to wait. My code follows:  \documentclass{book} \usepackage{color,colortbl} \begin{document}  \begin{tabular}{ll&gt;{\color{red}}p{20pc}} a &amp;1 &amp;Test \end{tabular}  \end{document}   The output was:    Some unnecessary vertical space came before the last column. If I remove the tag \color{red} then everything works fine, how can I fix this?  Above said issue was fixed based on the suggestion given by David Carlisle and CarLaTeX, but the same problem was there if I use the supertabular environment, I used supertabular package for auto breaking table and the code follows:  \documentclass{book} \usepackage{color,colortbl,supertabular} \begin{document}  \begin{supertabular}{ll&gt;{\leavevmode\color{red}}p{4pc}&gt;{\leavevmode\color{black}}l&gt;{\leavevmode\color{black}}l&gt;{\leavevmode\color{red}}p{4pc}} a &amp;1 &amp;Test &amp;a &amp;1 &amp;Test\\ a &amp;1 &amp;Test &amp;a &amp;1 &amp;Test\\ \end{supertabular}  \end{document} For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$ I've been messing around with Unity and broke something, how do I "start over"? Prove there are no other solutions of functional equation $f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$ I have the following functional equation. Find all continuous functions $f:(-1,1) \to \mathbb R$ such that $$ f(x+y)=\frac{f(x) + f(y)}{1 - f(x)f(y)} $$ The first obvious solution is $f(x) \equiv 0$. Another one I guessed, it is $f(x) = \pm \tan x$. I suspect, that there are no more solutions. The problem is that I don't know how to prove that.  Since we have a rational equation, I have no idea how to make any substitutions in order to get expression for $\tan x$ (as it is not a rational expression).  P.S. I can also show that it is true that (a) $f(0) = 0$ (take $x=y=0$), and (b) $f(-x) = -f(x)$ (take $y=-x$).  Update: there is a solution of this problem also here: Check whether an array is a subset of another Any idea on how to check whether that list is a subset of another?  Specifically, I have  List&lt;double&gt; t1 = new List&lt;double&gt; { 1, 3, 5 }; List&lt;double&gt; t2 = new List&lt;double&gt; { 1, 5 };   How to check that t2 is a subset of t1, using LINQ? Best approach to Windows Forms rolling log output in TextBox How does the Higgs mechanism work?