An equivalence for $\operatorname{grade}(I,M)\ge 2$. Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Show that $\operatorname{grade}(I,M)\ge 2$ iff the canonical homomorphism $M \mapsto\operatorname{Hom}_R(I,M)$ is an isomorphism.   This question is Exercise 1.2.24 in the book: Winfried Bruns and Jürgen Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1998.
On the grade of an ideal I need to prove the following statment (actually a special case of it).     Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Then   $\operatorname{grade}(I,M)\geq 2$ if and only if the homomorphism $M\rightarrow$Hom$_R(I,M)$ given by $m\mapsto(i\mapsto im)$ is an isomorphism.   This is Exercise 1.2.24 in Bruns and Herzog, Cohen-Macaulay Rings.
The forgetful functor $\mathsf{Groups} \to \mathsf{Sets}$ admits a left adjoint, namely, "forming the free group" functor.  I was wondering if this has a left adjoint.  This seems unlikely, but I don't have a proof.  Is there a functor $\mathsf{Sets} \to \mathsf{Groups}$ that admits a left adjoint at all? I have seen only the first three movies, so I don't know whether the other movies/TV series deal with this question or not. My main point is that in the movies, the terminator that travels back furthest in time is of the worst quality, and the quality increases with the time travel (meaning they travelled from a later time). What is a good explanation for this?  Note: I am not interested in the obvious not-in-universe explanation that they wanted something better and bigger for each new movie. What flash slaves can be used with a Canon Speedlite 600EX-RT? My current flash setup is a standard shoe mount but I've been thinking about getting a remote.  I've been drooling over the Speedlite 600EX-RT. It uses radio to control slaves, which is nice (longer distance, doesn't require line of sight, etc.).  One thing I don't understand: do slaves also have to be 600EX-RTs or do other flash units use radio to communicate already? Can any (cheap?) flash slaves can be used with a Canon Speedlite 600EX-RT or do I need to buy the 600EX and a ST-E3-RT transmitter in order (or multiple 600EX-RTs) to have a radio-controlled remote flash setup? How to create a shortcut that executes an xdotool command to simulate a key press? I'm trying to simulate a media key press using a custom shortcut.  So far, I've been able to achieve the simulation I need using the command:  xdotool key XF86AudioPlay   It works perfectly, it pauses or starts the music player every time it is run.  The problem comes when trying to run it as a shortcut.  First I tried to run an alias with a custom shortcut, but it did not work.  As explained on :     The commands run by keyboard bindings are not parsed via a shell. Your best bet is to create a directory bin in your homedir.   Next I created the suggested script and named it simplay:  #!/bin/sh xdotool key XF86AudioPlay   It is located inside a bin folder which is part of the PATH, this works fine and can be run from any location.  When trying to create the Custom Shortcut, I was unable to get it to execute, the configuration is the following:  Name: test Command: simplay  Shortcut: Ctrl+Alt+R   After searching a bit more I found another way to execute the shortcut:  gnome-terminal -e simplay   I tried it on a terminal and noticed that when executing it it would rapidly open a new terminal window, execute the child process, pausing the music and immediately close the window.  I changed the command on the shortcut and when hitting it I noticed the same behavior opening a terminal and immediately being closed but this time the child process did not pause the music, which makes me think that the script has been called on all my previous attempts but it is somehow not being able to execute.  How can I solve this problem according to your knowledge? $\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$ Evaluate Integral Here is a fun integral I am trying to evaluate:  $$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$$  I thought about integrating by parts $2n$ times and then using the binomial theorem for $\sin(x)$, that is, using $\dfrac{e^{ix}-e^{-ix}}{2i}$ form in the binomial series.  But, I am having a rough time getting it set up correctly. Then, again, there is probably a better approach.   $$\frac{1}{(2n)!}\int_{0}^{\infty}\frac{1}{(2i)^{2n}}\sum_{k=0}^{n}(-1)^{2n+1-k}\binom{2n}{k}\frac{d^{2n}}{dx^{2n}}(e^{i(2k-2n-1)x})\frac{dx}{x^{1-2n}}$$  or something like that. I doubt if that is anywhere close, but is my initial idea of using the binomial series for sin valid or is there a better way?.  Thanks everyone. After using the simple LM317 on breadboard variable power supply for a while, a need for a more rugged and reliable bench power supply has arose.    Short on money, i decided to build it as many other amateurs like me did. The thing is that, for my current needs i don't need a giant digitally controlled multi-channel 500 watt beast or something like that.  Instead what i really want is a low wattage positive and negative low voltage ouput -12v-0v-12v for example. However i really like that my power supply to be safe, reliable and fool-proof as far as possible. So after searching the web here and there, i found many designs that from my point of view fell in one of two divisions:-   Ones that use discrete op-amps and transistors. And ones that use integrated voltage regulators.   So -at least for me- i think that the ones with discrete op-amps and transistors are more prone to malfunctions and issues compared to the integrated ones with all of their internal over current and thermal protection circuits which would need to be discretely implemented in the bench power supplies that utilize discrete op-amps, however those designs have the advantage of ease of customization and more importantly that it is easier to have 0V output compared to 1.2V in integrated voltage regulator ones.    So i like advice to help me choose between circuits that utilize discrete op-amps like this circuit for example:-  Or ones that use integrated voltage regulators with their intrinsic robustness and low form factor.    Also as a side but related question i have seen many designs on the web that utilize negative voltage fed to the adjust or like pins to achieve 0V output from integrated voltage regulators like this single positive power supply:-     Or a positive and negative channels power supply that i really like its design because of simplicity (not because of laziness but to avoid the drawbacks that may come with complex designs especially when implemented by an amateur like me):- And the question is      Is it wrong or a system safety and reliability lowering act to feed negative voltage to adjust pin for LM317 for example?     I ask this question because the datasheet clearly states that the working output range is 1.2V-37V for LM317 for example, also no similar circuit is mentioned in the Typical Applications section except for the following one:- And even in that circuit, however the adjust pin is tied to the negative rail, it does't regulate during this interval instead it is shut until the next half wave at least that what i understand.    So at last to sum it up which alternative is best suited for my needs, its pros and cons and suggested circuits if possible. When using vagrant Stderr: VBoxManage: error: VT-x is not available (VERR_VMX_NO_VMX) I am trying to install open edx with  ubunty 16.04 and I get this error  There was an error while executing `VBoxManage`, a CLI used by Vagrant for controlling VirtualBox. The command and stderr is shown below.  Command: ["startvm", "c60f0594-d584-4e0b-b863-2801168e641e", "--type", "headless"]  Stderr: VBoxManage: error: VT-x is not available (VERR_VMX_NO_VMX) VBoxManage: error: Details: code NS_ERROR_FAILURE (0x80004005), component ConsoleWrap, interface IConsole   this occurs when using vagrant up Calculations in quotient rings How do I export and import content types (including fields) between environments? I have a really simple Java class that effectively decorates a Map with input validation, with the obvious void set() and String get() methods.  I'd like to be able to effectively call those methods and handle return values and exceptions from outside the JVM, but still on the same machine Update: the caller I have in mind is not another JVM; thanks @Dave Ray  My implementation considerations are typical    performance ease of implementation and maintenance (simplicity?) reliability flexibility (i.e. can I call from a remote machine, etc.)   Is there a 'right way?' If not, what are my options, and what are the pro/cons for each?  (Stuff people have actually done and can provide real-life feedback on would be great!) I was making this recipe: , which was from the old Lincoln Del, one of my mom's favourite places. I mistakenly doubled the sugar, putting in 1/2 a cup instead of a 1/4 of a cup. Given the 2/3 of a cup ketchup, which was Heinz, the soup turned out way, way too sweet.   Is there any way I can rescue this? What are some efficient ways to combine two structures in MATLAB? I want to combine two structures with differing fields names.  For example, starting with:  A.field1 = 1; A.field2 = 'a';  B.field3 = 2; B.field4 = 'b';   I would like to have:  C.field1 = 1; C.field2 = 'a'; C.field3 = 2; C.field4 = 'b';   Is there a more efficient way than using "fieldnames" and a for loop?  EDIT: Let's assume that in the case of field name conflicts we give preference to A. How should I phrase an important question that I need to ask a professor? Why was Peeves in Hogwarts? Peeves was a poltergeist. It generally irritated people and rather then helping them as well as making fun of them.   Why was Peeves in the school? What is/was the job or work assigned to Peeves? The limit is   $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$  where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times:  $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$  For $n=1$ the limit is $\frac{1}{6}$, $n=2$, the limit is $\frac{1}{3}$ and so on.  Can we define a recurrent relation upon that given hypothesis? Also, how do I involve $n$ into calculation, because the final limit will depend on it?  Any suggestions on how to tackle this?  Thank you! Viewer-independent copyable spaces at the beginning of a line? Does the PS-Vita support PSP games Is the PS-Vita backwards compatible with PSP games?  Some games I would like to play include Final Fantasy 7: Crisis core, Jak and Daxter: The Lost Frontier, Coded Arms, Tekken: Dark Resurrection, and Secret Agent Clank. Making Map Book of pages with same area shown but different layers turned on/off In ArcMap a Map Book is best suited for a series of features that each get their own page. It allows you to pick a feature and generate a Map Book Page for each feature including any additional information you wish to add to each page via dynamic text, charts, etc. Is there an alternative for say a series of pages showing the EXACT same area but with different layers turned on and off? We have a Capital Improvements document that I have to create showing the City Boundaries on every page. One page has all police and fire stations listed, another shows proposed water systems, another shows expressways and highways and work being done near or on them... etc., ad nauseum. Is there some way to set this up so that I DON'T have to created a separate MXD for each Map Page of this document? Is there any way to include an animated GIF directly in either PDFLaTeX or XeLaTeX? I realize the animate package can include animations in a PDF, but it doesn't support animated GIFs and you have to split them up manually into EPS or PNG files as far as I can tell. Custom post type - posts list - white screen of death i am getting a weird error - white screen in the list of posts for a specific Custom post type (just for that one)   tried deactivating all the plugins tried checking for error (debugging = true)   Still nothing the page just doesnt echos anything... (nothing in source too)  I am talking about such a url in the admin:     Here is register_post_type part i am using:    function register_submodelcpt() {     $labels = array(         'name'                  =&gt; __('Sub Models', THEME_NAME),         'singular_name'         =&gt; __('Sub Models', THEME_NAME),         'add_new'               =&gt; __('New Model', THEME_NAME),         'add_new_item'          =&gt; __('Add new Model', THEME_NAME),         'edit_item'             =&gt; __('Edit Model', THEME_NAME),         'new_item'              =&gt; __('New Model', THEME_NAME),         'all_items'             =&gt; __('All Sub Models', THEME_NAME),         'view_item'             =&gt; __('Watch Model', THEME_NAME),         'search_items'          =&gt; __('Search Models', THEME_NAME),         'not_found'             =&gt;  __('No Models found', THEME_NAME),         'not_found_in_trash'    =&gt; __('No Models found in trash', THEME_NAME),          'parent_item_colon'     =&gt; '',         'menu_name'             =&gt; __('Sub Models', THEME_NAME),      );      $args = array(         'labels'                =&gt; $labels,         'public'                =&gt; true,         'publicly_queryable'    =&gt; true,         'show_ui'               =&gt; true,          'show_in_menu'          =&gt; true,          'query_var'             =&gt; true,         'rewrite'               =&gt; array('slug' =&gt; 'submodels'),         'capability_type'       =&gt; 'post',         'has_archive'           =&gt; true,          'hierarchical'          =&gt; true,         'menu_position'         =&gt; 5,         'menu_icon'             =&gt; get_stylesheet_directory_uri().'/images/cpt/subcars.png',                     'supports'              =&gt; array('title', 'thumbnail', 'revisions', 'page-attributes')     );      register_post_type('submodelscpt',$args); } add_action('init', 'register_submodelcpt');   Did anyone encounter such an issuse? can you think of a reason this might happen?   Another weird thing when i change this:    To this:     The posts list loads correctly...