Prove that there is a function $g: [0,1] \rightarrow [1,2]$ of class $C^{1}$, such that $\int_{x}^{g(x)} f(t) dt=1$.
find a $\int_{x}^{g(x)} f(t)\,dt = 1$
A novel where a woman from "future" of 1999 speaks leet-speak Behaviour of increment and decrement operators in Python Will  be supported on the Android App in the future? I visit A51 rather frequently and I would like to see it in the App. Book series about aliens who put "caps" on people to control them Infinite Series: Fibonacci/ $2^n$ I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner)  In the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each term after the first two is the sum of the two previous terms. What is the sum to infinity of the series: $$\frac{1}{2} + \frac{1}{4}+ \frac{2}{8} + \frac{3}{16} + \frac{5}{32} +\frac{8}{64} + \frac{13}{128} +\frac{21}{256} +\frac{34}{512}+ \frac{55}{1024} + \cdots$$  Now, I solved this using an infinite geometric matrix series (incorporating the matrix version of the relation $a_n= \frac{a_{n-1}}{2}+ \frac{a_{n-2}}{4}$), and my students, after much hinting on my part, googled the necessary string to stumble across Binet's formula (which allows one to split the series into two simple, if rather messy, geometrics). Both of these are good methods, but neither really seems plausible for a challenge set for 15-18 year olds under exam conditions. So how is one supposed to do it? Regex (C#): Replace \n with \r\n How can I replace lone instances of \n with \r\n (LF alone with CRLF) using a regular expression in C#?  Sorry if it's a stupid question, I'm new to Regex.  I know to do it using plan String.Replace, like:  myStr.Replace("\n", "\r\n"); myStr.Replace("\r\r\n", "\r\n");   However, this is inelegant, and would destroy any "\r+\r\n" already in the text (although they are not likely to exist). dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) nginx The Software up-to-date Option automatically download the upgrade for nginx. but While installing it's giving me the following error:  Reading package lists... Done Building dependency tree        Reading state information... Done The following NEW packages will be installed:   nginx 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/434 kB of archives. After this operation, 977 kB of additional disk space will be used. (Reading database ... 291344 files and directories currently installed.) Unpacking nginx (from .../nginx_1.4.7-1~precise_amd64.deb) ... ----------------------------------------------------------------------  Thanks for using nginx!  Please find the official documentation for nginx here: * http://nginx.org/en/docs/  Commercial subscriptions for nginx are available on: * http://nginx.com/products/  ---------------------------------------------------------------------- dpkg: error processing /var/cache/apt/archives/nginx_1.4.7-1~precise_amd64.deb (--unpack):  trying to overwrite '/etc/logrotate.d/nginx', which is also in package nginx-common 1.4.6-1+precise0 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing:  /var/cache/apt/archives/nginx_1.4.7-1~precise_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   Actually that file /etc/logrotate.d/nginx does not exist in this folder. I know I'm asking for the close-to-impossible -- as a real full-backup certainly requires root permissions. But still there are many folks around not wanting to root their devices -- afraid of voiding their warranties or bricking their devices, or just not having their devices supported by any rooting method.  I also know there were a lot of questions asked already on this topic, and I checked all of them having the backup tag -- none covers the entire topic, all just have pieces, and especially people new to Android (but other non-power-users as well) would have a hard time figuring out the best way.  So I want a detailed canonical answer to the question:  Not involving root-solutions, how to achieve a backup as close as possible to a full backup?  Please do not give simple one-liner answers like "Use app xyz [full-stop]." If mentioning an app (and I doubt a single app would be the answer here), include what parts are covered by it -- as well as ideas on how to fill the gaps it leaves. (And to make it absolutely clear: I know the first thing coming to mind is Titanium Backup -- I use it myself, but this does not apply here: This question is strictly about methods for non-rooted devices!) Moreover, the answer should be as generic as possible in terms of applying to a wide range of devices (i.e. it should not be restricted to one device or manufacturer).  I already mentioned having investigated all questions here tagged backup. These will certainly be helpful providing details for answers here, so I will list up the topmost ones concerning my question:            All these provide some details for my question (so feel free to get inspired by them) -- but I feel there are still details missing. Also it would be helpful to have a summary -- say, an answer including the full description of "the maximum possible".  A last criterium: While I'd consider cloud-based solutions legit, I'd favour local solutions. If you can provide both: all the better. But some people value privacy quite high (I'm one of them), and thus would not trust their data to some cloud service.    Thanks to Ryan, we now have a solution for a really complete backup of all apps and their data -- which I didn't dare to hope for! The only disadvantage: His solution only works for less than 10% of all Android users; namely those whose devices are running at least with Android 4.x.  So again, I ask the close-to-impossible: Any solutions applicable at least on Gingerbread (2.3.x) or, better, even Froyo (2.2.x) -- so they cover the majority of current devices? These solutions may involve multiple tools (try to keep it as simple as possible though; think of your mother to follow your instructions ;)  And keep in mind: NO ROOT!    To further motivate high-quality answers (and hopefully avoid low-quality ones), I will be a harsh one here: As I rarely do, I will down-vote answers not fitting the mentioned criteria on one end -- but on the other end, I will setup a bounty for the best answer (which cannot be done immediately, but earliest after 2 days). Some questions and answers are marked Community Wiki and are owned by a community wiki user:  (In fact, this very question is one of them)  Why have Community Wiki posts? How do Community Wiki posts work? How does a post become a Community Wiki post? How can the Community Wiki status be removed from a post? How can I find Community Wiki posts? implementing functions in a game I'm currently typing a mathematical document which counts many sum symbols. I frequently use the align environment to render equations, although the default behaviour of this environment is to render the equations in displaystyle, which in my opinion wastes vertical space. I therefore have added \textstyle at the beginning of each line of math in this environment, but I'm looking for a more convenient solution.   So is there an environment that is similar to align with the exception it displays textstyle math by default ? Thank you. What is the most secure hand cipher? By "hand cipher", I mean a symmetric cipher for which encryption and decryption can can both be performed with a pencil on graph paper, consuming about 10-20 seconds per character by a proficient user. Additional simple tools, e.g. a deck of cards or a multiplication table, may be used to expedite the process. No mathematical proficiency is required other than mental addition, subtraction, and maybe multiplication.  Basically, it should be a cipher that someone of average mathematical abilities can master, and it should require no digital technology or cryptographic tools that cannot be constructed out of paper.  For example, four-square in CBC mode seems secure to me because CBC is a well established mode of operation and four-square seems to be a good block cipher as it accommodates large keys and seems to produce a random-looking output. However, the fact that four-square in CBC was never widely adopted before digital encryption became a consumer technology leads me to believe that someone broke the cipher and thus discouraged people from using it.  So is there any hand cipher that the best supercomputers would take years to cryptanalyze? If not, then for which hand cipher is the best known attack the most complex? I'm testing an application which opens its own ports (acts as a server for these ports, hence listens at that ports) and where the same application connects to ports bound by other applications (acts as client for these ports).  I would like to get an overview as to which ports the application creates and to which applications and ports it connects to.  How can I do this? Give a sequence such that root test works while ratio test fails Humans don't digest fibre, and nor do any animals, but some creatures have micro-organisms (bacteria, I'm guessing) in their gut which digest fibre for them.  My question is, could humans take some sort of pill and establish our own fibre-digesting colonies in our intestines?  Has this been tried?  Supposing it could be done, would you suppose it would have any negative effects on our health?  (i.e.. would those whose gut digested fibre for them be stuck on a low-fibre diet?).    Purpose of question: just wondering.  It would be cool to be able to eat bark. How do I compare 2 rows from the same table (SQL Server)? I need to create a background job that processes a table looking for rows matching on a particular id with different statuses. It will store the row data in a string to compare the data against a row with a matching id. I know the syntax to get the row data, but I have never tried comparing 2 rows from the same table before. How is it done? Would I need to use variables to store the data from each? Or some other way? (Using SQL Server 2008) Is there a way to render an object with a subdivision surface modifier with a wireframe which is the original geometry as well as the wires that make the subdivision surface?  Example:   This wireframe here is curved. What if there was displayed the straight wireframe. The wires that make are made by the subdivision surface. Thats what I want to achieve. How to do it?  What I mean is - How to render the curved wireframe (which is the original geometry with the straight wireframe (which is the wireframe the is made by the subdivision surface)? Set theory: difference between belong/contained and includes/subset? This is a total noob question.  I am reading , and I'm having difficulty to understand something which seems to be trivial.  In the first chapter he writes:     If $x$ belongs to $A$ ($x$ is an element of $A$, $x$ is contained in $A$), we shall write      $x\in A$   I understand this.  Then, he write:     If $A$ and $B$ are sets and if every element of $A$ is an element of $B$, we say that $A$ is a subset of $B$, or $B$ includes $A$, and we write:      $A \subset B$   I understand this too.  Then he says:     The working of the definition implies that each set must be considered to be included in itself ($A \subset A$); this fact is described by saying that set inclusion is reflexive.   I understand this too.  But then:     Observe that belonging ($\in$) and inclusion ($\subset$) are conceptually very different things indeed. One important difference has already manifested itself above: inclusion is always reflexive, whereas it is not at all clear that belonging is ever reflexive. That is: $A \subset A$ is always true;  is $A\in A$ ever true? It is certainly not true of any reasonable set that anyone has ever seen.   And this is where I don't think I understand anything.  There is not more elaboration on this point in the text.  I tried to skip this but it seems it is quite fundamental for understanding what follows in the book.  Could someone explain what is meant here? I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in , it is mentioned that this is not true, and that there exist obstructions to this decomposition. Unfortunately, the level of complexity in the answers goes a bit over my head.  This type of decomposition is also addressed in .  With this loss of faith in my intuition, I have three questions:   Is it true that Lie groups do not necessarily decompose into the semidirect product of a connected group and a discrete group? If this decomposition doesn't always work, then is there a particularly nice argument as to why it doesn't? If this decomposition doesn't always work, then is there a nice example I can use to convince myself that my intuition is wrong?   Edit: As much as I love Qiaochu Yuan's answer, which is thorough and (as usual) a role model among Stack Exchange answers, this is not what I'm looking for. I'm aware of the general extension problem, and that not every short exact sequence splits, but my intuition has led me to think that each Lie group can be decomposed this way. I will certainly play around with his outline for a construction, but I would much prefer a more concrete answer. Once again, thank you!  Edit: As of yet, playing with the construction suggested by Qiaochu hasn't yielded anything interesting. This question is still open to answers! Why is $2\pi i \neq 0?$ We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$  Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's formula), then shouldn't $2\pi i=0$? What am I missing here?