Consider the sequence $1,2,4,8,...,a^n = 2^n,...$ of all the powers of $2$. Prove that, given any digit $i ∈ {1,...,9}$, there exist infinitely many values of $n$ for which $a^n$ starts with that digit.  It only comes to my mind to use Poincaré's Recurrence Theorem, but I can not do that. Can anyone give a tip
Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits.
Mouseover tooltip for delete count is wrong If $|f(z)|\le |g(z)|$, can we say $f$ is constant? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? BBWC: in theory a good idea but has one ever saved your data? I'm familiar with what a BBWC (Battery-backed write cache) is intended to do - and previously used them in my servers even with good UPS. There are obvously failures it does not provide protection for. I'm curious to understand whether it actually offers any real benefit in practice.  (NB I'm specifically looking for responses from people who have BBWC and had crashes/failures and whether the BBWC helped recovery or not)  Update  After the feedback here, I'm increasingly skeptical as whether a BBWC adds any value.  To have any confidence about data integrity, the filesystem MUST know when data has been committed to non-volatile storage (not necessarily the disk - a point I'll come back to). It's worth noting that a lot of disks lie about when data has been committed to the disk (). While it seems reasonable to assume that disabling the on-disk cache might make the disks more honest, there's still no guarantee that this is the case either.  Due to the typcally large buffers in a BBWC, a barrier can require significantly more data to be commited to disk therefore causing delays on writes: the general advice is to disable barriers when using a non-volatile write back cache (and to disable on-disk caching). However this would appear to undermine the integrity of the write operation - just because more data is maintained in non-volatile storage does not mean that it will be more consistent. Indeed, arguably without demarcation between logical transactions there seems to be less opportunity to ensure consistency than otherwise.   If the BBWC were to acknowledge barriers at the point the data enters it's non-volatile storage (rather than being committed to disk) then it would appear to satisfy the data integrity requirement without a performance penalty - implying that barriers should still be enabled. However since these devices generally exhibit behaviour consistent with flushing the data to the physical device (significantly slower with barriers) and the widespread advice to disable barriers, they cannot therefore be behaving in this way. WHY NOT?  If the I/O in the OS is modelled as a series of streams then there is some scope to minimise the blocking effect of a write barrier when write caching is managed by the OS - since at this level only the logical transaction (a single stream) needs to be committed. On the other hand, a BBWC with no knowledge of which bits of data make up the transaction would have to commit its entire cache to disk. Whether the kernel/filesystems actually implement this in practice would require a lot more effort than I'm wiling to invest at the moment.  A combination of disks telling fibs about what has been committed and sudden loss of power undoubtedly leads to corruption - and with a Journalling or log structured filesystem which don't do a full fsck after an outage its unlikely that the corruption will be detected let alone an attempt made to repair it.   In terms of the modes of failure, in my experience most sudden power outages occur because of loss of mains power (easily mitigated with a UPS and managed shutdown). People pulling the wrong cable out of rack implies poor datacentre hygene (labelling and cable management). There are some types of sudden power loss event which are not prevented by a UPS - failure in the PSU or VRM a BBWC with barriers would provide data integrity in the event of a failure here, however how common are such events? Very rare judging by the lack of responses here.   Certainly moving the fault tolerance higher in the stack is significantly more expensive the a BBWC - however implementing a server as a cluster has lots of other benefits for performance and availability.  An alternative way to mitigate the impact of sudden power loss would be to implement a SAN - AoE makes this a practical proposition (I don't really see the point in iSCSI) but again there's a higher cost. I recently bought myself a pack of four YN560 IV's and a YN560-TX for my Canon camera, and apparently I got ahead of myself and didn't realise I should make sure the flashes support HSS.  I tried them out yesterday at home, and even on 1/128 and 1/250s on the camera, they were overexposing here and there. Unfortunately, I don't really have the space to move the flashes much further from my subject, so I was wondering if there is any way to trick them into HSS? I should mention that I also own a Canon 580ex II, which does support HSS. I was thinking I could perhaps use that mounted on the camera to trigger the Yongnuo's, but as far as I can tell that would render the 560-TX useless as the flashes can't be controlled wirelessly when in S1 or S2.  I guess I could buy an ND filter, but as I have lenses with differing sizes, this feels like more of a pain in the ass.  Is there any hack I can pull off here that you guys know of?  Thanks! Lock your computer after 5 minutes of inactivity How to scp in Python? How do you set password-hash for OpenLDAP? Installing QGIS Statist plugin in Mac OS X? Why do I see a saddle in this picture of a computer screen? I am not entirely sure this is an appropriate question for PSE, however since many of you have such diverse backgrounds, I'll give it a shot.  I have noticed that when one takes a picture of a computer, TV screen with a phone, you can "see" what looks like a saddle phase portrait.     Why does this happen? Does it have any relevance to differential equations or other areas of math? Why is current a scalar quantity? I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity? What is an quick way to create a new blank file in Lion? In Windows, one can create a new file using Right Click -> Add New -> Type of File. Is there an analogous action in Lion?  I just want to create a new blank file on my Desktop. Bounty-like facility for rewarding excellent questions Amongst the reasons for starting a bounty, one can select:     Reward existing answer      One or more of the answers is exemplary and worthy of an additional bounty.   Why not enable one to reward excellent questions in a similar manner? No Wifi-Adapter found on new Lenovo Yoga C930 with Ubuntu 18.10 Remove duplicate rows in MySQL Error: Account is Not Allowed To Suggest Edits I get this message while trying to edit a post on SuperUser while using the iOS mobile app. I have at least 101 reputation on SU, more than enough to be able to edit posts. This appears to be a bug. How can I share files over a Windows network? I have a media computer which I recently reformatted to Ubuntu 10.04. I am probably just missing something obvious but I can't figure out how to share folders and files, so that I can browse to them on my Windows computers in order to copy files to/from the computer.  How do I do this? How is money held in escrow for a home purchase from a IRA treated by the IRS? Linear vs nonlinear differential equation How to distinguish linear differential equations from nonlinear ones? I know, that e.g.: $$ y''-2y = \ln(x) $$ is linear, but $$ 3+ yy'= x - y $$ is nonlinear. Why? Allowed Wave Functions of System Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given by $H$. However, in situations where $H$ does not yield strict boundary conditions (e.g. the harmonic oscillator) can $\psi(x)$ be any normalised function? My intuition says no for the following reason: QM textbooks say that any valid wave function can be represented as a linear combination of the eigenstates of $H$. In the case of the harmonic oscillator, the eigenstates of $H$ are a countably infinite set. However the set of all normalised functions is uncountably infinite. It seems to me that one cannot change between one complete set of basis functions to another and reduce the "dimensionality" somehow. This makes me think that the eigenstates of $H$ are not complete, that they imply some constraints on $\psi(x)$. This is not an area of mathematics that I understand well. Can somebody help me out?