Do atoms behave like waves?
Which types of particles are affected by the wave-particle duality?
How do I document a module in Python? That's it. If you want to document a function or a class, you put a string just after the definition. For instance:  def foo():     """This function does nothing."""     pass   But what about a module? How can I document what a file.py does? Assume the standard situation, that is, let $X_1, \ldots , X_n$ be independent and identically distributed with $X_k ∼ P_\theta(x; \theta)$ , where $P_\theta(x; \theta) = 2x/\theta^2$  if $0\le x \le\theta$ and $0$ otherwise. It is required to estimate $\theta$. Show that the maximum likelihood estimator for $\theta$ is $\hat{\theta} = \max{[X_1, \ldots , X_n]}$ and then show that the cumulative distribution function of $\hat{\theta}$ is $F_\theta(z) = z^{2n}/\theta^{2n}$. Here's what i did so far: Maximum Likehood estimator: $L_x(θ) = \prod_{i = 1}^{n} P_θ(x_k)$ Here we have $P_θ(x_1,...,x_n;θ) = P_θ(x_1,θ).P_θ(x_2,θ)...P_θ(x_n,θ)$ Likehood = $L_{x,θ}(θ) =P_θ(x_1,θ).P_θ(x_2,θ)...P_θ(x_n,θ)= 2x_1/θ^2. 2x_2/θ^2.. 2x_n/θ^2 = [2^n.\prod_{i = 1}^{n}x_i]/θ^{2n}$ Log-likehood: $\sum_{i = 1}^{n}log(P_θ(x_1,...x_n;θ))= \sum_{i = 1}^{n}log(2x_i/θ^2)$ Is this correct so far? Im still not sure how to get to $\hat{θ} = max{[X_1, . . . , X_n]}$ As for the the cumulative distribution part to show $F_θ(z) = z^{2n}/θ^{2n}$: $F(z) = P(max({x_k})&lt;z) = P(x_1&lt;z).P(x_2&lt;z)..P(x_n&lt;z) = 2x_1/θ^2.2x_2/θ^2...2x_n/θ^n = 2^n.\prod_{i = 1}^{n}x_i/θ^2n$ Not sure if this is correct. Would really appreciate some help. Edit: From the answers below, we can deduce the the estimator is biased. What estimator would be unbiased? How can i find it? On webmail every user gets error "ERROR: Message not sent. Server replied: 354" while sending mails.   But mails are sent properly and reaches properly.  Can you tell me where is problem, and why it is occurred?  UPDATE: I found in some resources that this is not an error but standard information message. On the other side, it's annoying. Can you tell me if it is a true, and if it is how can I avoid it? How do I get a Cron like scheduler in Python? I'm looking for a library in Python which will provide at and cron like functionality.  I'd quite like have a pure Python solution, rather than relying on tools installed on the box; this way I run on machines with no cron.  For those unfamiliar with cron: you can schedule tasks based upon an expression like:    0 2 * * 7 /usr/bin/run-backup # run the backups at 0200 on Every Sunday  0 9-17/2 * * 1-5 /usr/bin/purge-temps # run the purge temps command, every 2 hours between 9am and 5pm on Mondays to Fridays.   The cron time expression syntax is less important, but I would like to have something with this sort of flexibility.   If there isn't something that does this for me out-the-box, any suggestions for the building blocks to make something like this would be gratefully received.  Edit I'm not interested in launching processes, just "jobs" also written in Python - python functions. By necessity I think this would be a different thread, but not in a different process.  To this end, I'm looking for the expressivity of the cron time expression, but in Python.   Cron has been around for years, but I'm trying to be as portable as possible. I cannot rely on its presence. How can I verify a Google authentication API access token? How can I verify a Google authentication access token?  I need to somehow query Google and ask: Is [given access token] valid for the [example@example.com] Google account?  Short version: It's clear how an access token supplied through the  can be used to then request data from a range of Google services. It is not clear how to check if a given access token is valid for a given Google account. I'd like to know how.  Long version: I'm developing an API that uses token-based authentication. A token will be returned upon provision of a valid username+password or upon provision of a third-party token from any one of N verifiable services.  One of the third-party services will be Google, allowing a user to authenticate against my service using their Google account. This will later be extended to include Yahoo accounts, trusted OpenID providers and so on.  Schematic example of Google-based access:    The 'API' entity is under my full control. The 'public interface' entity is any web- or desktop-based app. Some public interfaces are under my control, others will not be and others still I may never even know about.  Therefore I cannot trust the token supplied to the API in step 3. This will be supplied along with the corresponding Google account email address.  I need to somehow query Google and ask: Is this access token valid for example@example.com?  In this case, example@example.com is the Google account unique identifier - the email address someone uses to log in to their Google account. This cannot be assumed to be a Gmail address - someone can have a Google account without having a Gmail account.  The Google documentation clearly states how, with an access token, data can be retrieved from a number of Google services. Nothing seems to state how you can check if a given access token is valid in the first place.  Update The token is valid for N Google services. I can't try a token against a Google service as means of verifying it as I won't know which subset of all Google's services a given user actually uses.  Furthermore, I'll never be using the Google authentication access token to access any Google services, merely as a means of verifying a supposed Google user actually is who they say they are. If there is another way of doing this I'm happy to try. How do i know which wheels will fit my bike? I run into this infinite product today. I don't have that much experience evaluating products therefore I don't have any idea of how to tackle this.   Here is the question. Evaluate (if possible) in a closed form the product:  $$\Pi = \prod_{n=1}^{\infty} \frac{1}{1+\pi^{{\large \frac{1}{2^n}}}}$$  The numerical value seems to be $\Pi= 0.534523$ which is very close to $\frac{\pi}{6}$ taking into account that $\frac{\pi}{6} \approx 0.523598$. In the mean time W|A evaluates it to $0$. I'm lost.   Can the community help?  Edit: Based on the answer we have two products. The product I asked tends to zero and the bonus product provided by @you're in my eye (thanks for that) is $\frac{\ln \pi}{\pi-1}$.   Thanks for the quick response. Diamond moderators  . In the second section from the top—entitled &quot;Ask questions, get answers, no distractions&quot;—that takes us to an opportunity to select one of a page full of questions from the site as the example question used on the about page. The selections currently available on physics are a little idiosyncratic. Is there some rhyme or reason to these? I suppose that we want a good example question to:  Be accessible to the intended audience (perhaps even to the newest and most naive person in that audience) Be judged at least decent by the community (positive score) Have more than one answer which are also judged acceptable (again positive score)  What else? How does a question end up on the list? If $x_1 = 3$, $x_{n+1} = \frac{1}{4-x_n}$ for $n \geq 1$, prove the sequence is bounded below by $0$, above by $4$. If $x_1 = 3$, $x_{n+1} = \dfrac{1}{4-x_n}$ for $n \geq 1$, prove the sequence is decreasing and bounded below by $0$, above by $4$.  Want to show: $0 &lt; x_{n+1} &lt; x_n &lt; 4$  I decided to do the bounded part first.    Base Case, $n=1$  $0 &lt; 3 &lt; 4$ [works]  IH  Suppose $0 &lt; x_k &lt; 4$, for some $k\in\mathbb{N}$  IS  Show $0 &lt; x_{k+1} &lt; 4$  We know $x_{k+1} = \dfrac{1}{4-x_k}$  We also know that $0 &lt; x_k &lt; 4$, by IH.  So we have:  $$\dfrac{1}{4} &lt; x_{k+1} &lt; \infty$$  But this doesn't show the boundness. Where have I went wrong? Linux ACL behaviour: settings the mask changes 'ls' output, but not actual permissions The mysql extension is deprecated and will be removed in the future: use mysqli or PDO instead Is there an expression to indicate the strategy of wearing someone down with numerous small irritations? I would use rope-a-dope, but it's got connotations of pretending to lose that I don't need. I'm trying to describe the behavior of someone who sends twenty detailed emails a day about various projects, and the recipient of his emails has finally gotten tired of trying to parse them, so the recipient just agrees to whatever he proposes. I'd like an expression that evokes both outward earnestness and hidden creepiness, like Uriah Heep, Dickens's "Humble servant" with veiled ambition, who overdoes it with professions of humility until he's in power. Water torture would work, but comes too close to a more racist expression that I'm not comfortable with. Carry over long sequence of missing values with Postgres I have a table like this:  create table foo (foo_label text, foo_price int, foo_date date);  insert into foo (           values           ('aaa', 100,  '2017-01-01'),           ('aaa', NULL, '2017-02-01'),           ('aaa', NULL, '2017-03-01'),           ('aaa', NULL, '2017-04-01'),           ('aaa', 140,  '2017-05-01'),           ('aaa', NULL, '2017-06-01'),           ('aaa', 180,  '2017-07-01')         );   As you can see a few values on the foo_price column are missing.  What I need is that missing values get filled up with the "previous" available value in this way:   foo_label | fixed_foo_price | foo_date -----------+-----------------+------------  aaa       | 100             | 2017-01-01  aaa       | 100             | 2017-02-01  aaa       | 100             | 2017-03-01  aaa       | 100             | 2017-04-01  aaa       | 140             | 2017-05-01  aaa       | 140             | 2017-06-01  aaa       | 180             | 2017-07-01   My attempt:  select      foo_label,      (case when foo_price is null then previous_foo_price else foo_price end) as fixed_foo_price,     foo_date from (   select        foo_label,        lag(foo_price) OVER (PARTITION BY foo_label order by foo_date::date) as previous_foo_price,        foo_price,       foo_date       from foo ) T;   As you can see from here:    It doesn't fill completely the '100' series.  Any idea how can I get the wanted result? Connecting to sql server database mdf file without installing sql server on client machine? Short story ID: Bouncing-Ball Time Travel A little help please. The story I'm looking for is probably late-50's to mid-60's, and may well have occurred in an SF magazine.  A professor is demonstrating his new time machine to a scientific audience. The machine looks like two large spheres touching each other. In operation, one is sent back into deep time (let's say, although my memory has dropped this particular datum, 10 billion years). It will then return to the present, then bounce off its twin and jump back half that temporal distance (5 billion years), return, bounce half again (2.5 billion years), etc. Each trip will take half the time (in present-day units) as the previous, so the whole thing will be over in a finite time.  Before anyone can stop him, he activates it. He begins lecturing the audience, asserting that whatever has happened has always happened. At decreasing intervals the sphere reappears and bounces with a resonant "Boing!". At each jump, the narrative establishes that the appearance of the sphere did change history, and these changes are reflected in the professor's speech.  So it becomes something like "(Boing!) And those who say that this is dangerous might as well be claiming that the day is not 26 hours long (Boing!) or that there are three moons, not seven (Boing!)", etc. And when the final (Boi-Bo-B-B-Boing!) fades away he raises his tentacles in triumph and cries, "See? Nothing has changed!" Complete classification of the groups for which converse of Lagrange's Theorem holds (Up to / until / by) 50% off I wonder if someone could tell me which choice works in the following sentence and why the others do not work: (From my point of view, all of these choices work in this context.)  They have great promotion till January 5th......50% off. a) up to; b) until; c) by Adjoint pairs, triplets and quadruplets Often we have adjoint pairs $(A, B)$ (meaning $A$ is left adjoint to $B$). Sometimes we have adjoint triplets $(A,B,C)$ (meaning $A$ is left adjoint to $B$ and $B$ is left adjoint to $C$. No adjoint relation between $A$ and $C$ obviously, since they have the same source and target).  So the first question is: can we have quadruplets $(A,B,C,D)$ ? (meaning that $C$ is also left adjoint to $D$).  Second question: It would be then possible that we have $(A,D)$ besides $(A,B)$. But I think not, because I think that it is not possible to have a functor $A$ with two different right adjoints, $B$ and $D$. Is this correct? Serial protocol delimiting/synchronization techniques As asynchronous serial communication is widely spread among electronic devices even nowadays, I believe many of us have encountered such a question from time to time. Consider an electronic device D and a computer PC connected with serial line (RS-232 or similar) and required to exchange information continuously. I.e. PC is sending a command frame each X ms, and D is replying with status report/telemetry frame each Y ms (The report can be sent as response to requests or independently - doesn't really matter here). The communication frames can contain any arbitrary binary data. Assuming the communication frames are fixed-length packets. The problem: As the protocol is continuous, the receiving side might loose the synchronization or just &quot;join&quot; in the middle of an ongoing sent frame, so it just won't know where the start of frame (SOF) is. A the data has different meaning based on its position relatively to the SOF, the received data will become corrupted, potentially forever. The required solution Reliable delimiting/synchronization scheme to detect the SOF with short recovery time (i.e. it shouldn't take more than, say 1 frame to resynchronize). The existing techniques I am aware (and using some) of: 1) Header / checksum - SOF as predefined byte value. Checksum in the end of frame.  Pros: Simple. Cons: Not reliable. Unknown recovery time.  2) Byte stuffing:  Pros: Reliable, fast recovery, can be used with any hardware Cons: Not that suitable for fixed-size frame-based communication  3) 9th bit marking - prepend each byte with additional bit, while SOF marked with 1 and the data bytes are marked with 0:  Pros: Reliable, fast recovery Cons: Requires hardware support. Not directly supported by most of PC hardware and software.  4) 8th bit marking - kind of emulation of the above, while using the 8th bit instead of 9th, which is leaving only 7bits for each data word.  Pros: Reliable, fast recovery, can be used with any hardware. Cons: Requires an encoding/decoding scheme from/to the conventional 8-bit representation to/from 7-bit representation. Somewhat wasteful.  5) Timeout based - assume the SOF as the first byte coming after some defined idle time.  Pros: No data overhead, simple. Cons: Not that reliable. Won't work well with poor timing systems like, say, Windows PC. Potential throughput overhead.  Question: What are the other possible techniques/solutions exist to address the problem? Can you point to the cons in the above list which can be easily worked around, thus removing them? How do you (or would you) design your systems protocol? How can I read a file even when getting an "in use by another process" exception?