Show that $\operatorname{Supp}(M)=V(\operatorname{Ann}(m))$
$\operatorname{Supp}(M)=V(\operatorname{Ann}M)$ if $M$ is finitely generated
Using WCS layer in QGIS for processing gives FAILURE: Missing destination I have a GeoServer serving raster dem data out via WCS and can add the layer easily to QGIS (3.2) but when I try to use it as the input for an operation (e.g. HillShade) I get an error.  Processing algorithm… Algorithm 'Hillshade' starting… Input parameters: { 'ALTITUDE' : 45, 'AZIMUTH' : 315, 'BAND' : 1, 'COMBINED' : False, 'COMPUTE_EDGES' : False, 'INPUT' : 'dpiMode=7&amp;identifier=training:panorama&amp;url=http://localhost:8080/geoserver/ows?version%3D1.1.1', 'MULTIDIRECTIONAL' : False, 'OPTIONS' : '', 'OUTPUT' : 'C:/Users/ishare/AppData/Local/Temp/2/processing_868cc25180f24d9f8e193bc0cfedb6d3/7cf48f64d8b54ee694c3ee93ed17e903/OUTPUT.tif', 'SCALE' : 1, 'ZEVENBERGEN' : False, 'Z_FACTOR' : 5 }  GDAL command: gdaldem hillshade dpiMode=7&amp;identifier=training:panorama&amp;url=http://localhost:8080/geoserver/ows?version%3D1.1.1 C:/Users/ishare/AppData/Local/Temp/2/processing_868cc25180f24d9f8e193bc0cfedb6d3/7cf48f64d8b54ee694c3ee93ed17e903/OUTPUT.tif -of GTiff -b 1 -z 5.0 -s 1.0 -az 315.0 -alt 45.0 GDAL command output:   FAILURE: Missing destination.   Clearly the missing destination is a red herring, and it is failing to understand the WCS layer as an input. But GDAL is built with WCS support included so shouldn't this work?  Do I need to do something else or start working out where to hack the QGIS code? Can you quickly tell me if this pseudocode makes sense or not? Create bash file to open minecraft launcher I use this command in the terminal to run the minecraft launcher with java 8:  /home/max/Java/jre1.8.0_25/bin/java -jar Minecraft.jar   I included the full java path because I installed it on /home as a normal user and i want to include it.   That command works, it opens the launcher with Java 8 but I want to make a bash file so I don't have to open the terminal. And, most important, I don't want the terminal to remain open when I click play in the Minecraft launcher and the game starts.  Now if i close manually the terminal when the game have started, the game is closed too. In Avengers: Endgame,       Thanos was able to stop and catch Mjolnir and Stormbreaker mid-air as both Captain America and Thor used/called them BEFORE he got the Infinity Stones.   Since this character is clearly not worthy and had no super powers at the time, how were they able to wield Thor's Mjolnir and Stormbreaker, when even the Hulk couldn't do it in The Avengers? How to tile textures in Cycles? Add an application to the command search path in csh Recently, I have installed an application (gnuplot) under user directory (/home/usr/bin). I want to run the application whenever I type 'gnuplot' in any directory. What could I do?  I tried to set the path in ~/.cshrc by  setenv gnuplot /home/usr/bin/gnuplot   but it doesn't work. It reports "command not found" as I type gnuplot in the terminal. How bad for TeX is omitting braces {}, even if the result is the same? Sometimes writing braces and omitting them produces the same result:  \tilde{\psi}  \mathrm{e}^{\mathrm{i} \pi}   vs  \tilde \psi   \mathrm e^{\mathrm i \pi}   Or (taken from Werner's comment below)   \frac{1}{2} vs \frac12   If that's the case, how bad for the quality of my code is omitting them? Of course, if one compiles a pair of formulas, there is no difference. But, since I ignore how TeX works, my point is: if one writes long codes with these habits,   doesn't TeX go somehow crazy because I'm not writing as it expects?   For instance, does it have any effect on how fast are long documents compiled? Any reason to keep them? We know Euler formula:  $$ e^{ix}=\cos(x)+i\sin(x). $$  Does this formula work if we replace real number $x$ with complex number $z$? How does the Java 'for each' loop work? Auto size image to page size in InDesign I've got an InDesign document with 100 A4 pages.   I've got 100 assets (a mixture of .jpg and .pdf files) and I want to place 1 per page. If an image is not the proportion of the A4 page I would like to get it as big as it can be on the page before it hits into the margin. Ex. if it's a portrait image it should be as large as it can be before hits the top/bottom margin. If it's a landscape image it should be as large as it can be before hits the left/right margin.   At the moment I'm dragging each file across and resizing them manually. Is it possible just drop them on the page and get them auto resized ? "big enough to do this" My question is what is the difference among these three sentences.   He is big enough to do such  heavy works. He is not big enough to do such heavy works. He is small enough to do such  heavy works.   I am so much confused between 2 and 3.Also, I don't get what the 3rd one means. Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1 The task is as follows:     Given: $$F(x,y,z) = 0$$    Goal:  Show    $\frac{\partial  z}{\partial y}|_x \frac{\partial  y}{\partial x}|_z \frac{\partial x}{\partial z} |_y = -1$   Here is my work so far:  (1) Differentiate with respect to y, I get:  $0 + F_2 + F_3  \frac{\partial z}{\partial y} = 0$  So $ F_3  \frac{\partial z}{\partial y} = - F_2$  (2) Differentiate with respect to x, I get:  $F_1 + F_2 \frac{\partial y}{\partial x}  + 0 = 0$  So $ F_2  \frac{\partial y}{\partial x} = - F_1$  (3) Differentiate with respect to z, I get:  $F_1  \frac{\partial x}{\partial z}    + 0  + F_3 = 0$  So $ F_1  \frac{\partial x}{\partial z} = - F_3$  (4) After some manipulations with the $F_i$, I get to the conclusion that $\frac{\partial z}{\partial y}* \frac{\partial y}{\partial x} * \frac{\partial x}{\partial z}  = -1$, so when evaluated with x, z, y respectively, conclusion is still true  My questions are:  (1) Is my proof correct?  (2) For example, when I differentiate with respect to y, I "let" $F_1$ be 0 and find partials for other coordinates.    I had a hard time trying to explain to my friend on the reason(s) why I can do such "let be 0" thing.  Although I think if I can't do that, then there is no way that I can reach the conclusion, but I somehow feel confused about the fact too.  Since my book is doing it that way, my understanding is that I can do such "let be 0" thing based on the independece of x with respect to y, when I differentiate with respect to y.  But is my thought ok?  Would someone please help me on this question? Thank you very much ^_^ Appendix without letter in TOC Adding words to links generated by hyperref \documentclass{article} \usepackage{hyperref} \usepackage{lipsum} \usepackage{polski} \begin{document} \begin{figure} . \caption{A figure} \label{fff} \end{figure} \lipsum[1]  Rysunek~\ref{fff} jest zabawny, bo na rysunku~\ref{fff} nic nie ma. \end{document}   In this sample, a clickable link is added to output PDF for 1 and 1. I want to be able to change them to Rysunek 1 and rysunku 1. I want to be able to put any other text + number of the figure into a hyperlink. The word rysunek can't be set in preamble as a default because there is a declension in Polish. The same question goes for tables, listings, equations, pages (pageref) etc (basically any reference).  HTML equivalent of my MWE:    Rysunek &lt;href="fff"&gt;1&lt;/a&gt; jest zabawny, bo na rysunku &lt;a href="fff"&gt;1&lt;/a&gt; nic nie ma.   I want it to be:    &lt;href="fff"&gt;Rysunek 1&lt;/a&gt; jest zabawny, bo na &lt;a href="fff"&gt;rysunku 1&lt;/a&gt; nic nie ma. I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day. How to promote algebraic expressions to operators in quantum mechanics? Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription   $$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $$   in the position basis. Now my questions are:    What if $x$ or $p$ appears in the denominator in a classical expression?  How to promote this to a quantum expression? What would be the meaning of division by an operator?   My expression likely contains a mixture of $x$ and $p$. For e.g., it could contain terms like $$\frac{p}{x^2}$$ or $$\frac{xp}{(x^2 + a^2)^{3/2}}.$$    How to resolve products of non-commuting operators like $x$, $p$ in a satisfactory way? reveals x2 code snippets. The first result is to  which has a lot of documentation and explanation, along with some useful discussion underneath.   However, , whilst not containing so much documentation, includes sample code for passing commands such as start, stop and restart. It also creates a PID file which can be handy for checking if the daemon is already running etc.  These samples both explain how to create the daemon. Are there any additional things that need to be considered? Is one sample better than the other, and why? Please consider gathering Meta feedback before the "ship" stage Problems installing xpatch I need to install the xpatch package to make changes to my biblatex setup. The readme file in the install package simply says to run “latex xpatch.ins.” I've moved the xpatch folder (including xpatch.ins and xpatch.dtx files) to the /usr/local/texlive/2011/../texmf-local/ folder. When I run the command above, I get the output Generating file(s) ./xpatch.sty  ! Cannot find file xpatch.dtx. \readsource ...ot find file \uptospace #1 \qStop }                                                   \else \processedLines \z@ ... l.34 ...xpatch.sty} {\from{xpatch.dtx} {package}}}   I've checked, and the .dtx file is in the folder. Any ideas what the problem is? lcm versus phi in RSA In textbook RSA, the Euler $\varphi$ function $$\varphi(pq) = (p-1)(q-1)$$ is used to define the private exponent $d$. On the other hand, real-world cryptographic specifications require the Carmichael lcm function $$\lambda(pq) = \operatorname{lcm}(p-1,q-1)$$ to define $d´$. It is clear that the $d´$ divides $d$, and therefore using $d´$ may be more efficient than using $d$. My question is: Are there any further reasons, e.g. regarding security, why one should not use $d$?