Finding a bound on an unpleasant recurrence relation
Solving the recurrence $T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$
Why do we use "./" (dot slash) to execute a file in Linux/UNIX? Why do we use ./filename to execute a file in linux?  Why not just enter it like other commands gcc, ls etc... Cell copy results in jaggy, aliased output Give a deduction of $\forall y \exists x (x=y)$  My thinking is I can prove $\exists x(x=y)$ then use generalization theorem， which is equivalent to prove $\neg \forall x \neg (x=y)$ but I got stuck proving this... thanks in advance  Sorry that I did not specify what deduction system that I want to use here, it is the deduction sequence  where ai is either Axiom or original hypothesis or obtained from MP Big Parenthesis in an Equation I have an equation contained inside \[...\], which automatically makes a \sum with sub- and superscripts turn big--so that the summation sign looks awkward inside parenthesis.  Any idea how to make the parenthesis completely enclose the whole summation?  \documentclass{article} \begin{document} We have: \[ \sum_{i=1}^n i = (\sum_{i=1}^{n-1} i) + n = \frac{(n-1)(n)}{2} + n = \frac{n(n+1)}{2} \] \end{document} When should I use a fixed or variable time step? Do opposite categories always exist? Solving $\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$ I have the following problem that I don't know how to solve:  Let $\textbf{r}(t)$ be a curve and let $r(t):= |\textbf{r}(t)|$. Let the curve $\textbf{r}(t)$ be defined by: $$\textbf{a}(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ Where $\textbf{a}(t) = \textbf{r}''(t)$ is the acceleration of the particle and $G,M \in \mathbb{R}$. Suppose that $\textbf{r}(t)$ and $\textbf{r}'(t)$ are not parallel. The objective is to determine the trajectory $\textbf{r}(t)$ in polar coordinates.  So basically we have the following equation: $$\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ The thing is that I don't know how I should proceed from now on. If we'd have just normal scalar function this would be a differential equation and I think I would be able to solve it, but like this I have no idea how to approach the problem. How can I solve this?  Edit: If $$\textbf{r}(t):=(x(t),y(t))$$ then the equation turns into the following system of equations: $$x''=\frac{GM}{(\sqrt{x^2 + y^2})^3}x$$ $$y''=\frac{GM}{(\sqrt{x^2 + y^2})^3}y$$ I don't know if this helps but I think that this turns thing easier perhaps. Are partially charged wands generally purchasable? Dictionary for TeXstudio: “No dictionary Available” I have trouble spell checking my document in TeXstudio. When I press spellcheck, I get the error No dictionary Available. Am I supposed to install a dictionary separately? How? Any suggestion? Similarity of real matrices over $\mathbb{C}$ $A  \underset{\mathbb{C}}{\sim} B \overset{\text{def}}{\iff} A=C^{-1}BC, \space C\in M_{n}(\mathbb{C})$ and similarly for $\underset{\mathbb{R}}{\sim}$.     I want to prove that $ A \underset{\mathbb{C}}{\sim} B$ for $A,B \in M_{n}(\mathbb{R})$ therefore $A \underset{\mathbb{R}}{\sim} B$.   My idea is that elementary divisors of $A,B$ over $\mathbb{C}$ are the same, and if $(x-z)^k$ is elementary divisor than $(x-\overline{z})^k$ is also elementary divisor $\implies$ $A,B$ have same elementary divisors over $\mathbb{R}$. But i think it's not clear. How can I display site logo, name and slogan? Why are Python's 'private' methods not actually private? Can Barbarians use 2 two-handed weapons in Diablo 3? Can barbarians dual wield two-handed weapons?  My barbarian character is only a level 50 now, but I'm not sure if I can dual wield two-handed weapons at a later level or not - or is there a special skill I need?  UPDATE It is sweet, that Blizzard have finally updated Diablo 3, so that the Crusader can have a 2-handed weapon in 1 hand ( passive). It is a shame that the Crusader can't dual wield a 2-handed weapon in both hands. ...on Finnis's view all distinct instances of basic goods are incommensurable — none is of more, less, or equal value with any other. &mdash;    I expect to see more or less than any other, but then equal shows up, and is followed by with. But since with any other cannot replace than any other, do you think this sentence illegitimately omits something? Find $\sum_{n=1}^\infty \frac{n^3}{n!}$. Center of a group ring? Let $R$ be a ring with unity and let $G$ be a group. Then what is the center of the group ring $R(G)$? I feel that the center $Z(R(G))$ is $R(Z(G))$. I assumed if $x=\sum_{i=1}^{n} r_i g_i$, then it should commute with every element of $R(G)$. In particular it should commute with $g=1 \cdot g \in R(G)$ Thus we get $$\big(\sum_{i=1}^{n} r_i g_i\big)g=g\big(\sum_{i=1}^{n} r_i g_i\big)$$ Thus I get $$g_i \cdot g=g_i \cdot g \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \forall g \in G$$ Hence $x \in Z(R(G)) $. Is this correct? How can I prove my publication list is mine? In some of the questions regarding author name, it is mentioned that people can use whatever they like, however, they must be consistent across their publications.  When it comes to being hired by a university for a tenure track position or by another principal investigator as a researcher, how can someone prove that they are the author of papers when their author name does not exactly match the one on their ID card? How can a principal investigator be sure that the publication list does actually belong to the person claiming them? I'm trying to install Anbox in my new Ubuntu 16.04 using instructions from  . It seemed to work, but I got no icon in my dash, so I suspect that the installation is not complete. With Synaptic  Package Manager I can see that anbox-common &amp; anbox-modules-dkms are installed, but I have no icon in my dash, so I tried running   sudo snap install --edge --devmode anbox   and got this error:  error: cannot install "anbox": Get    https://search.apps.ubuntu.com/api/v1/snaps/details/core?channel=stable&amp;fields=anon_download_url%2Carchitecture%2Cchannel%2Cdownload_sha3_384%2Csummary%2Cdescription%2Cdeltas%2Cbinary_filesize%2Cdownload_url%2Cepoch%2Cicon_url%2Clast_updated%2Cpackage_name%2Cprices%2Cpublisher%2Cratings_average%2Crevision%2Cscreenshot_urls%2Csnap_id%2Csupport_url%2Ccontact%2Ctitle%2Ccontent%2Cversion%2Corigin%2Cdeveloper_id%2Cprivate%2Cconfinement%2Cchannel_maps_list:    dial tcp: lookup search.apps.ubuntu.com on [::1]:53: read udp    [::1]:54255-&gt;[::1]:53: read: connection refused   What am I doing wrong? Why was the connection refused? According to the instructions, it seems this is the correct command to install it.  ps: I'm not that familiar with Linux, I'm more of a Windows guy, trying to learn Linux. Is it possible to turn off the Two-Factor Authentication for an Apple ID? Is it possible to turn off the Two-Factor Authentication for an Apple ID?  I have more than one device linked to this Apple ID and I know the password and security questions. Handling mouse events for tool called by button? I have a dockable window with a 2 buttons that call 2 different tools (Draw Marker and Select Features), What i want to happen is when the user clicks on the relevant button on the dockable window and then clicks on the map, a onMouseUp event is called and it runs code i need to run.   Where does the onMouseUp event need to be placed so this can be done?