's description of a brachistochrone curve is      the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.   I was wondering if there was a counterpart to the brachistochrone curve, which would be the curve such that, given the aformentionned conditions, a bead would travel from A to B in the longest, finite amount of time.  I assume such curve would be some kind of concave version of the brachistochrone.
Suppose a particle only influenced by gravity is going from point a to point b on an x,y plane. Is there an equation in which the particle takes the longest amount of time, yet still makes it to point b? I know the equation for brachistochrone is the parametric for an inverted cycloid, yet I'm not sure of the opposite. For reference, I am a high school student wanting to try this for a research project. I really only need to know whether this has already been done or not, but attempted solutions are obviously appreciated.
My first issue was when I tried to apt-get update or apt-get upgrade. On upgrade I get the following error:  You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies: linux-image-server : Depends: linux-image-3.2.0-27-generic but it is not installed E: Unmet dependencies. Try using -f.   I tried running apt-get install -f and this was the output(after saying yes on prompt)  (Reading database ... 186183 files and directories currently installed.) Unpacking linux-image-3.2.0-27-generic (from .../linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb) ... Done. dpkg: error processing /var/cache/apt/archives/linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb (--unpack):  failed in write on buffer copy for backend dpkg-deb during `./boot/System.map-3.2.0-27-generic': No space left on device  No apport report written because the error message indicates a disk full error                                                                           dpkg-deb:    error: subprocess paste was killed by signal (Broken pipe) Examining /etc/kernel/postrm.d . run-parts: executing /etc/kernel/postrm.d/initramfs-tools 3.2.0-27-generic   /boot/vmlinuz-3.2.0-27-generic run-parts: executing /etc/kernel/postrm.d/zz-update-grub 3.2.0-27-generic /boot/vmlinuz-3.2.0-27-generic Errors were encountered while processing: /var/cache/apt/archives/linux-image-3.2.0-27-generic_3.2.0-27.43_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   I tried running apt-get autoremove and it gives me the same error as apt-get upgrade.  When I run df, I get this for /boot:  /dev/sda1                    233191     230297         0 100% /boot   So, I read elsewhere that I should try to purge old kernels. I checked for what kernels I had with:  $ dpkg -l linux-image-\* | grep ^ii ii  linux-image-2.6.38-13-server  2.6.38-13.52  Linux kernel image for version 2.6.38 on x86_64 ii  linux-image-3.0.0-13-server   3.0.0-13.22   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-14-server   3.0.0-14.23   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-15-server   3.0.0-15.26   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-16-server   3.0.0-16.29   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.0.0-17-server   3.0.0-17.30   Linux kernel image for version 3.0.0  on x86_64 ii  linux-image-3.2.0-24-generic  3.2.0-24.39   Linux kernel image for version 3.2.0  on 64 bit x86 SMP ii  linux-image-3.2.0-25-generic  3.2.0-25.40   Linux kernel image for version 3.2.0  on 64 bit x86 SMP ii  linux-image-3.2.0-26-generic  3.2.0-26.41   Linux kernel image for version 3.2.0  on 64 bit x86 SMP   When I try to remove the oldest with this:  $ sudo apt-get purge linux-image-2.6.38-13-server Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: linux-image-server : Depends: linux-image-3.2.0-27-generic but it is not going to be     installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).   How can I free up or extend boot without messing up my installation? This happens pretty much every time I try to bake normal maps. It's quite a broad error message and I still don't know exactly what the problems I'm having are, only that if I keep clicking things for half an hour eventually it sometimes just works.  My understanding of the things needed:   A "Generated" blender Image that the normals are written to (In UV/Image Editor, click Image->New)   Tell blender that this is the bake output, below (this is the really confusing bit which blender seems to try to guess and fails)  A UV Map for the low poly model (you only need the geometry of the high poly and it doesn't need a UV map) Under Bake, in the rendering tab:   Bake Mode = Normals Normal Space = Tangent Selected to Active = ✓ (for the next bit)  Select the high rez object first, then the low poly object With the two objects selected, click Bake (you may have to move the mouse to see the error pops up)   Now, from what I can tell the only  (with regard to the setup and apart from the internals of baking) is how blender finds the output image:   Cycles  Have a material using an ImageTexture node which points to your normal map output. Then  says the node must be selected, but  doesn't mention selecting. Blender Render  Have a texture attached to the material that uses the image.  is pretty vague on assigning the output image. Sometimes I've found hitting tab to go to edit mode and selecting the normal map image in the UV editor to help, or at least it seemed to.   I would like to know the exact details of what blender goes looking for when you hit Bake, rather than have to guess by following tutorials. For example, it seems to look for the high and low poly objects in your selection and based off the order you select them. How does it find the normal map/image to write to for both cycles and blender render? Does the Texture need to Influence Normals or does it just need to reference the Image? etc... How do I remove repeated elements from ArrayList? I have an ArrayList&lt;String&gt;, and I want to remove repeated strings from it. How can I do this? How can I add some html to my Mail.app signature? I want to add my company logo and and some juice to my contact information in the signature part in Mail.app. How can I do that with little or no efforts? How to check if element is visible after scrolling? I'm loading elements via AJAX. Some of them are only visible if you scroll down the page. Is there any way I can know if an element is now in the visible part of the page? Sometime I see many application such as msn, windows media player etc that are single instance applications (when user executes while application is running a new application instance will not created).  In C#, I use Mutex class for this but I don't know how to do this in Java. Derivatives of $ \frac{1}{r} $ and Dirac delta function I am trying to understand the formula \begin{equation} \nabla^2\left(\frac{1}{|{\bf r}-{\bf r}'|}\right) = - 4 \pi \delta(\bf{r}-\bf{r}'), \qquad\qquad {\rm (I)} \end{equation} where ${\bf r}=(x,y,z)$. This is something heavily used in electrostatics and the steps to 'show' this is often the following: The first derivative reads \begin{equation} \nabla \frac{1}{| {\bf r} - {\bf r}' |} = - \frac{ {\bf r} - {\bf r}'}{| {\bf r} - {\bf r}'|^3} \end{equation} And taking the second derivative gives zero, except for the singularity at ${\bf r} = {\bf r'}$. Then from the divergence theorem we have \begin{equation} \int dV \,  \nabla^2 \frac{1}{| {\bf r} - {\bf r'}|} = \int dS  \,\,{\bf n} \cdot ( \nabla\frac{1}{|{\bf r} - {\bf r}'|}) = -4 \pi \end{equation} where the integration is performed over a sphere centered at ${\bf r}'$. Q1: Is there a more direct proof for equation (I)? Then my main question is about the separate second-order differentials. For instance, we can obtain, by direct computation \begin{equation} \partial_x^2 \, \frac{1}{| {\bf r} - {\bf r}'|} = \frac{ 3 (x-x')^2 }{| {\bf r} - {\bf r'} |^5} - \frac{1}{| {\bf r} - {\bf r}'|^3} \end{equation} Q2: Should there be a $\delta$ function on the r.h.s of this equation? Problem  sudo notify-send Test "Hello World"   Displays a notification as expected.  notify-send Test "Hello World"   Does not display a notification.  Further information  Ubuntu version 16.04.  The notifications appear to use notify-osd instead of notification-daemon. Running notify-send appears to launch a notify-osd process under the user that ran notify-send. I'm not sure what dbus is.   No error messages  There are no errors in the syslog. When I run the following code no error messages occur.   #include &lt;libnotify/notify.h&gt; #include &lt;stdio.h&gt; int main() {     gboolean x = notify_init ("Hello world!");     printf( "notify_init: %d\n", x );     NotifyNotification * Hello = notify_notification_new ("Hello world", "This is an example notification.", "dialog-information");     GError *err = NULL;     x = notify_notification_show (Hello, &amp;err);     printf( "notify_notification_show: %d\n", x );     if(err != NULL) {         printf("Error detected!\n");         printf("Error message:%s\n", err-&gt;message);     }     else {         printf("No error detected.\n");     }     g_object_unref(G_OBJECT(Hello));     notify_uninit();     return 0; }   I do not know what I'm doing. Thanks for the help.  Python Notify  In the past, I've used a python program that has working notifications. I tested a hello world notification and it works without sudo!   #!/usr/bin/env python # -*- coding: utf-8 -*- import gi gi.require_version('Notify', '0.7') from gi.repository import Notify  Notify.init("test") n = Notify.Notification.new('test', 'test2') n.set_urgency(Notify.Urgency.CRITICAL)     n.show()   Code from   I am not sure why this works. The python Notify is just wrapping the library used in the C example. I'd prefer not to use python but will if I must.  Edit  The problem occurred again. Resolved by adding urgency critical. Showing a recursive sequence isn't bounded $a_{n+1}=a_n+\frac 1 {a_n}$ I'm used to creating Java Edition Minecraft maps, where NBT can be used to set many different block, item, and entity properties, like an item's CanPlaceOn/CanDestroy list, a chest's items, and more! To open my maps up to more players, I've started creating maps on Minecraft Bedrock Edition as well. But NBT is inaccessible from commands in BE, which I knew before, but didn't know how much of a problem I would have creating these complicated command chains. One of the things that I've always been doing is giving players items with custom item names, lores, enchantments, and other properties. But there is a problem. All this would have been done in NBT in Java Edition. In Bedrock, NBT is inaccessible from commands, meaning the commands I used in JE don't work. Are my dreams crushed? Finding Command-Line Arguments of Programs I know that unix has man pages, but is there a place I can look to discover command-line arguments of programs? Going to command prompt and typing notepad /? doesn't give anything. Mouseover events in beamer: hovering on \eqref and a comment containing the original equation popping up This idea came into my mind while I was listening to a speaker presenting his results in a seminar, he used lots of equation references in his later slides, however I can't remember which equation he was referencing to for he gave several models during an earlier stage of the presentation.   We could use hyperref to get hyperlinks on the \eqref, but clicking it would go back to the original page where the equation was labeled, and somehow interrupt the presentation.  I have several workarounds in mind: (1) exploit the \pause option as the overlay, once hitting space, the referenced equation would show. (2) use tikz to insert a minipage into the frame that contains the referenced equation.  However, both solutions require the referenced equation taking up spaces in the current slide the presenter is on, the best solution is still I said in the question title: mouse over the \eqref number, then a comment popping up showing the equation.  My google-fu leads me to a package called comment, however I didn't find any nice examples of how to use that package to produce beautiful slides.  Hence I wonder is it possible to produce some mouseover events like the title said, or any other workarounds are welcome too. Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$? Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$.   I know that   $\mathrm{char}(R) =2$  $R$ has $2^k$ elements for some $k$  (this is because, if $p\neq 2$ is a prime with $p\mid|R|$, then there exists a non-zero $x \in R$ s.t. $px=0$ and $p $ is odd $\implies x=0$, contradiction).   Does this suffice to answer my question? Arbitrary intersection of closed, connected subsets of a compact space connected? Let $(B_i)_{i\in I}$ be an indexed family of closed, connected sets in a compact space X. Suppose $I$ is ordered, sucht that $i &lt; j \implies B_i \supset B_j$.  Is $B = \bigcap_i B_i$ necessarily connected?  I can prove it, if I assume $X$ to be Hausdorff as well: If $B$ is not connected, then there are two disjoint, closed, nonempty sets $C$, $D$ in $B$, such that $C \cup D = B$. Now these sets are also closed in $X$, hence by normality there exist open disjoint neighborhoods $U$, $V$ of $C$ and $D$, respectively.  Then for all $i$: $B_i \cap U^c \cap V^c \ne \emptyset$, since $B$ is contained in $B_i$ and $B_i$ is connected. Thus we must also have  $$ B \cap U^c \cap V^c = \bigcap_i B_i \cap U^c \cap V^c \ne \emptyset $$  by compactness and the fact that the $B_i$ satisfy the finite intersection property. This is a contradiction to the choice of $U$ and $V$.  I can neither see a counterexample for the general case, nor a proof. Any hints would be greatly appreciated!  Thanks,  S. L. On the usage of "the" article with acronyms and initialisms I've read  and  I know that  1. The article is often used with initials that are pronounced letter by letter (initialisms).  2. The article is often NOT used with initials that are pronounced as a word (acronyms).  I used 'often' because I'm not sure about possible exceptions.  In my field of study (Operations Research), initials are frequently used to represent optimization problems (e.g., the TSP). I discussed this point with my university supervisor who makes comments on my reports (English is not his native language). He responded that:  "What I learned from [...] was that whenever we use (specific) acronyms, we don't use 'the' when they are within a sentence, and we use 'the' when a sentence starts with them. For instance,  (i) The HCP is an NP-Complete problem.  (ii) We developed a new algorithm to solve HCP."  HCP stands for the Hamiltonian Cycle Problem. It is really confusing me because I do not know when to use and when not to use the article. For example, on , the author Jane Watson writes:     Use a definite article with an initialism if the spelled out term begins with “the” but is not covered in the initialism.   which implies that I should use the article with "HCP". What do you think? Automatic coloring of numbers according to size Scientific talks have their own culture, reporting the results, typically for me, in a table of the following form  1.7333    1.6472    1.5418    1.4682    1.5740 1.1525    1.2878    1.2652    1.3511    1.5102 1.2181    1.1500    1.2220    1.2425    1.5699 0.9786    1.1051    1.1472    1.3096    1.6757 0.9569    0.9190    1.0433    1.2924    1.4867   There is clearly a(n almost perfectly) monotonic pattern for rows and columns. Now for the naked eye, this is not the most quick way of capturing informing.  I would like numbers that bigger in absolute value, to appear differently (either green vs red or light vs dark) than smaller numbers, such that the pattern becomes clear instantly.  Is there a way of programming this in LaTeX? Here's the context       "Her heart leaping into her throat, she tiptoed into the salon. Such was her astonishment at seeing Mike chatting with her boss that she leaned against the wall, incapable of making another move."   Is this a phrase or what? Could it be used in other forms as well? I have a issue with my Drupal installation, for example:   when I enable or disable the modules, It redirects me to a blank page, when I create a new content type and save it, redirects me to a blank page, when I add a new view and save it, it redirects me to a blank page, when I try to clear the cache, it redirects me back to a blank page, or in similar cases.   Basically all the confirmation pages redirect me to a white screen. When I refresh it again, it shows me the page. I tried to increase the PHP memory value but it doesn't help.  Are there any other solutions for this?  The error which I'm having:      Fatal error: Allowed memory size of 100663296 bytes exhausted (tried to allocate 8192 bytes) in sites/all/modules/views/plugins/views_plugin_localization_none.inc on line 1 Why is it called an “Indian file”? Cannot see Samba configuration tool in the Dash in ubuntu 14.04 After a fresh installation of ubuntu 14.04 (my previous installation has seen to be broken somewhere reading .wmv file with VLC), I've installed Samba GUI tool from the software center.  It briefly appears in the launchpad then disappears. From this time I cannot the see in in the dash.  I've found this in askUbuntu   but the command sudo apt-get install samba samba-common gives me nothing to install.  Lecture des listes de paquets... Fait Construction de l'arbre des dépendances        Lecture des informations d'état... Fait samba est déjà la plus récente version disponible. samba passé en « installé manuellement ». samba-common est déjà la plus récente version disponible. Les paquets suivants ont été installés automatiquement et ne sont plus nécessaires :   kde-l10n-engb kde-l10n-fr Veuillez utiliser « apt-get autoremove » pour les supprimer. 0 mis à jour, 0 nouvellement installés, 0 à enlever et 8 non mis à jour.   The following guide  has no result either.  Any idea ?  Laurent06000