Recommended aspect ratio for android / smartphones?
How can I develop my Android game for different phone resolutions?
Do I need to explicitly call the base virtual destructor? Given that $xyz=1$, prove that $\frac{x}{1+x^4}+\frac{y}{1+y^4}+\frac{z}{1+z^4}\le \frac{3}{2}$ Given that $xyz=1$, prove that $$\frac{x}{1+x^4}+\frac{y}{1+y^4}+\frac{z}{1+z^4}\le\frac{3}{2}.$$  I proved this with , but: is there a better/more elegant way? I added a user account to the admin group and discovered I added wrong user. So I now have to remove the account from the admin group.   How can I remove an account from a group without deleting the user? So what I decided to do is to start a small case with only 3 people. There are three possible combinations that I could pair up the people, using the symbols A, B, and C to represent the persons involved: A&amp;B, A&amp;C, B&amp;C. The probability of each pair having the same birthday is $1\over365$ (the first guy gets any birthday, and the second guy only gets one birthday to choose from). What my intuition directed me to do next is to find the probability that any of the three events are true, or the union of the three events, which involved adding up the probabilities.   To put this generally, the equation for the probability SHOULD be:  ${n \choose 2} \div 365$  However, clearly this is not the case since $n \choose 2$ gets over 365 when $n = 28$. And with 28 people, there is obviously still a chance that they all have different birthdays (Person 1 has Jan. 1, Person 2 has Jan. 2. ... Person 28 has Jan. 28). Could anyone tell me what's wrong with my intuition? I don't want to know what's the right solution, I just want to know what's wrong with my solution .. it makes sense to my intuition, even though it's incorrect. Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear. A function $T:V \rightarrow W$ is additive if $T(x+y) = T(x) + T(y)$ for every $x, y \in V$. Prove that there exists an additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.  My attempt: Let $T$ be the function $T: \mathbb{R}$ (over the field $\mathbb{Q}$) $\rightarrow \mathbb{R}$ (over the field $\mathbb{R}$). The set $\{1, \sqrt{2}\} \subseteq \mathbb{R}$ is linearly independent for the vector space $\mathbb{R}$ over $\mathbb{Q}$. Then, there must exist a linearly independent set $W \subseteq \mathbb{R}$ (over $\mathbb{Q}$) such that $\{1, \sqrt{2}\} \subseteq W \subseteq \text{span}(W) = \mathbb{R}$ (over $\mathbb{Q}$).  I have been told that the function defined as $T(1) = 1$ and $T(w) = 0$ for all $w \in \text{span}(W) \setminus \{1\}$ is additive but not linear, but I cannot see why this is? I can see why it is not linear, clearly $T(\sqrt{2} \cdot 1) = 0$ but $\sqrt{2} T(1) = \sqrt{2}$. But, why is $T$ additive? For example, $T(1+1) = T(2) =0$ but $T(1) + T(1) = 1+1 = 2$? Is there a mistake somewhere? What is the equivalent of Windows System Properties or Device Manager? I am new to Ubuntu. In Windows, if I want to know my system properties/configuration, I can see it by going to Control Panel and clicking on the System icon. I can also get a detailed list of the hardware detected/installed by going to the Device Manager.  What are the equivalents of these features in Ubuntu, where a user can see his or her system properties and hardware configuration, like RAM, CPU/processor, hard disk capacity, etc.?    Example of Windows System Properties: The Medibuntu Project has come to an end - what do I do now? Clearing NSUserDefaults I'm using +[NSUserDefaults standardUserDefaults] to store application settings. This consists of roughly a dozen string values. Is it possible to delete these values permanently instead of just setting them to a default value? Where should the comma be placed in the salutation of a letter? Sometimes I see a comma after the proper name:     Hello Mr. Black,      In order to give you....   But my native language is not English and I think that the comma in this phrase should be placed before the proper name:      Hello, Mr. Black.      In order to give you....   What is the correct punctuation in English? I've been trying to prove   $$\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$$  I've tried perturbation and inversion but still nothing. I've even tried expanding the sum to try and find some pattern that could relate this to the harmonic series but this just seems surreal. I can't find any logical reasoning behind this identity. I don't understand . Is there any proof that doesn't require the integral transform? Evaluating this integral $ \small\int \frac {x^2 dx} {(x\sin x+\cos x)^2} $ The question:      Compute$$ \int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2} $$   Tried integration by parts. That didn't work.  How do I proceed? Can anyone help me with a solution? Combinatorics identity question I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out? Double Negation in C++ I just came onto a project with a pretty huge code base.   I'm mostly dealing with C++ and a lot of the code they write uses double negation for their boolean logic.    if (!!variable &amp;&amp; (!!api.lookup("some-string"))) {        do_some_stuff();  }                                      I know these guys are intelligent programmers, it's obvious they aren't doing this by accident.   I'm no seasoned C++ expert, my only guess at why they are doing this is that they want to make absolutely positive that the value being evaluated is the actual boolean representation. So they negate it, then negate that again to get it back to its actual boolean value.   Is this correct, or am I missing something? Is there a benchmark tool for Ubuntu? Is there a benchmark tool to measure computer performance for Ubuntu like SuperPI, 3DMark, or PCMark? How does one actually show from associativity that one can drop parentheses? I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?  $$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$$  Do you just say that the grouping of the parentheses now corresponds to just multiplying straight through? Thanks. OS X: Blurry Beamer presentations on Sierra / High Sierra Since OS X 10.12 (Sierra), the built-in Preview, but also many other PDF Viewers on OS X (such as Skim) show a number of problems, among them the following rendering issue:   The page is first rendered in low-res  with blurry text and graphics. About one second (!) later, the page is rendered in high fidelity. The display "wobbles", as text and graphics on the high-fidelity page is about one or two pixels up.   Example code:  \documentclass{beamer} \begin{document} \begin{frame}{Duke says}   \begin{itemize}   \item&lt;+-&gt; It don't mean a thing, if it ain't got that swing         \item&lt;+-&gt; (doo-ah, doo-ah, doo-ah, doo-ah, doo-ah, doo-ah, doo-ah, doo-ah)         \item&lt;+-&gt; It don't mean a thing all you got to do is sing         \item&lt;+-&gt; (doo-ah, doo-ah, doo-ah, doo-ah, doo-ah, doo-ah, doo-ah, doo-ah)  \end{itemize}  \end{frame}  \end{document}   While one may consider this acceptable when reading documents on screen, this is a show stopper for presenting beamer presentations with animations. The low-res display and wobbling makes the audience sick, while the 1 second rendering time makes PDF-based animations impossible.  So I am looking for a PDF Viewer for OS X that does not suffer this problems, but that a suitable for presenting beamer-based slide shows. Requirements:   page-wise full-screen/presenting mode with just the page and nothing else on the screen crisp and high-fidelity display quick rendering time   Unfortunately, the root of the problem is in Apple's own PDFKit library that is the base for most PDF viewers on OS X. So I am looking for PDF viewers that are not based on PDFKit, but fulfil the requirements above.  I am aware, of course, of Acrobat Reader DC. However, I do not particularly like its font rendering (Req 2) and I would also prefer something more light weight. I also have PDFExpert, which also is not PDFKit-based, however, does not support a presenting mode (Req 1). 101 positive integers placed on a circle A Pigeonhole Principle problem:     101 positive integers are placed on a circle whose sum is 300. Prove   that it is possible to choose from these numbers some consecutive   numbers whose sum is equal to 200.   (I don't know if the word 'consecutive' is appropriate in this case ,I mean that these numbers follow each other on that circle) How to deal with bibliography items containing long URLs? When you boot from live CD, or doing the first boot after installing Ubuntu, splash screen looks awesome. But as soon as you enable nvidia-current driver, installed with apt-get, splash screen goes all crazy.  With crazy I mean that resolution is very low, font(I assume) is very weird and it all looks like it's been broken.  This happened with 10.04 and now again with 10.10.  How do I fix splash screen after enabling nvidia drivers?