I'm trying to understand how to know the numbers of solutions for x^2=a(mod m). I know that if a has roots they are solutions but how could I know the rest of them? Is there a need to use the Chinese reminder here? if so how?
Find the number of solutions of $x^2=x\mod m$ for any integer $m$.  I was thinking about bringing the $x$ over to the other side and somehow apply chinese remainder thm.
Random operators Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H (\Omega, \mathcal F,P)$ of all $H$-valued random variables. My question is: How Can we define $|A|$ and the adjoint $A^*$? Front facing camera stopped working on the iPhone 4s I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also say why pressure is scalar instead of a vector.  I am thinking pressure at a point is $P=\lim_{\delta A \to 0} \frac{\delta F}{\delta A}$. Please let me know if I am wrong.   But I do not know at all why pressure is a scalar instead of a vector. I know it has something to do with $d \mathbf{F}=-Pd \mathbf{A}$ From time to time certain chat rooms (JavaScript and probably PHP, too) attract the attention of a fellow help vampire who thinks he can start sucking from the people in there.  Often their message end up being flagged as offensive (and those flags are likely to be validated, too, because people are annoyed by such behaviour) but it's kind of an inappropriate use of those flags.  I'm proposing a new chat flag type, which is especially meant for help vampires. Flagging it would present the flag to 10k users and owners of the room where the flag originated (no matter how much rep they have). When the flag is validated it would cause messages from this user to be shown with a low opacity for maybe three hours. It could also notify the user that help vampirism is a bad thing and that he needs to improve his behaviour.  Repeated flags of this type - obviously with some time in between in case multiple message which are really part of the same conversation are flagged - should then result in a temporary ban from the room where the messages got flagged. If that doesn't help the next step would probably be a temporary ban from all chat rooms.    For clarification, I'm not talking about the quick "this line doesn't work, why?" stuff but the kind of help vampires who get cocky even though they just got e.g. a link to a SO question which has a perfectly fine and fitting answer. Because hey, you didn't give them teh codez! How to change the chapter number to display as text (One, Two, ...)? The function $f(z)=z+\frac{1}{z}$ maps $\Omega$ biholomorphically on the upper plane The question states:     Let $\Omega$ be the upper half plane minus the unit disk. Show that the map $f: z \mapsto z+1/z$ maps $\Omega$ biholomorphically on the upper plane. Describe the preimages of the lines $\mathrm{Im}(z)=\text{a constant}&gt;0$.   So how exactly does one show the mapping is biholomorphic. I'm not sure where I should start, so unfortunately I haven't made any progress What's the difference between an argument and a parameter? When verbally talking about methods, I'm never sure whether to use the word argument or parameter or something else. Either way the other people know what I mean, but what's correct, and what's the history of the terms?  I'm a C# programmer, but I also wonder whether people use different terms in different languages.  For the record I'm self-taught without a background in Computer Science. (Please don't tell me to read  because I'm asking this for the benefit of other people who don't already have a copy of 's marvellous book.)  Summary  The general consensus seems to be that it's OK to use these terms interchangeably in a team environment. Except perhaps when you're defining the precise terminology; then you can also use "formal argument/parameter" and "actual argument/parameter" to disambiguate. Accessing a property of derived class from the base class in C# Is it possible to detect a virus with taskmanager? If I would have a running virus on my system, would I be able to see the process in taskmanager? I mean, would it be possible for a running virus to circumvent the taskmanager so the process doesn't appear in the tasklist of windows7?  Or in other words. If I really now all the processes in taskmanager to be secure, I also know that my PC is clean? formatting long numbers as strings in python What is an easy way in Python to format integers into strings representing thousands with K, and millions with M, and leaving just couple digits after comma?  I'd like to show 7436313 as 7.44M, and 2345 as 2,34K.  Is there some % string formatting operator available for that? Or that could be done only by actually dividing by 1000 in a loop and constructing result string step by step? Is this correct: "Our listeners are what make X"? Should edits with only minor code improvements be accepted. As an example, I'm reviewing an edit that changes all double quotes to single quotes and renames a variable in some javascript. With the exception of the variable rename (which is not entirely necessary), the edit is frivolous. Should I accept on the merits of a debatable variable name change? Evaluate  $$\int_0^\infty \frac{\log x \; dx}{x^{2} + 2x + 2}$$  by integrating a branch of $(\log z)^{2}/(z^{2} + 2z +2)$ along a keyhole contour.  The thing I have trouble with is why I should be examining the square of the log - I guess it has something to do with ln x in fact NOT being the real part of log z, since log z is (or can be) defined on the entire negative real axis and ln x can't. But looking at the square would definitely not have been my first plan of attack :( I am calculating the Intraclass Correlation coefficient (ICC) as per the method of Shrout-Fleiss (1979), using the (3, 1) model in which the judges are fixed as are the targets.  Some of my ICC values are negative.  Is this to be expected?  How to interpret this?  I understand that the maximum ICC is 1.0, but is there a minimum value?  How are negative values interpreted? Why didn't Sauron become invisible while wearing the One Ring? I'm curious as to why Sauron is the only being/person shown wearing the One Ring who does not become invisible?   Isildur, Gollum, Bilbo, Frodo, and Samwise all become invisible while wearing the One Ring.  Why didn't Sauron become invisible while wearing the one ring?  I became interested after reading Pearsonartphoto's question asking what effect wearing the One Ring would have had on . What should I do to avoid ear pain when flying? I always feel a pain in my ears when flying. Specially when descending. The last time I had extreme pain, feeling it inclusive around my eye balls. It felt like someone was touching the eye and pulling it from the inside. I really had to focus, close my eyes and hold to avoid screaming. I am not sure but I think both things might be related. Such a severe situation only happened once.  I just heard that with a cold the ear pain gets worse. I am bit worried that I might go through that situation again. I have a cold and I am going to fly soon.  What can I do to improve this?  I see that some consider this a duplicate of the question "best way to make my ears pop" but after reading the answers I think my question goes a bit beyond that since my pain is more extreme and I think it goes beyond just popping the ears. I also want to know if other methods, like using earplugs, might be effective. I've always managed my iPhone/iPad/iPod Manually, I recently updated to iTunes 11 and now I can see no way to get my music onto my device.  I used to be able to just drag songs from my library to my device, now there is seemingly no way to do this.  I can view the music on my device by clicking on it on the top right corner of iTunes, but I can't seem to get to a state where I can drag music from my computer onto my device.  I'm using iTunes 11.0.1.12 for Windows if that makes a difference. How do I write an Apex unit test? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, including more unit test resources, navigate to the  tag.  This canonical question is intended to address several classes of common questions by providing a quick summary and links to comprehensive resources:   How do I start writing my first unit test?  How do I unit test this code?  I need help writing this unit test.   Salesforce Stack Exchange looks for detailed, specific questions that the community can help you with, and can't write tests on your behalf. We feel that working with the resources below can help you get started, and we encourage you to make an attempt to write your test and return to SFSE with your specific questions when you encounter challenges you can't resolve. Below is experimental data for the ratio $$R=\frac{\sigma(e^+e^-\rightarrow hadrons)}{\sigma(e^+e^-\rightarrow\mu^+\mu^-)}$$ as a function of the centre of mass energy $\sqrt s$. I am interested in the peak at around $100GeV$ which corresponds to the resonance of the $Z^0$ boson. There are two ways of looking at this that I have in mind:  We can effectively ignore the electromagnetic process around this energy, and so each process should have it's own Breit-Wigner peak centred on the mass energy of the $Z^0$. The ratio of these is just a constant. Thinking in terms of Feynman rules and again ignoring the electromagnetic process around this region, each process has the same propagator and vertex factors (roughly - ignore quark mixing) and there are some extra factors due to different quarks being possible and colour degeneracy, but still the ratio should be a constant (one possible issue here is interference between Feynman diagrams which I have neglected?).  So my question is why does the peak exist in the data? I need to use a different version of circuitikz. How can I do that?