Suppose m1, m2, ..., mk are positive integers &gt; 1, not necessarily pairwise relatively prime. Also Suppose a1, a2, ..., ak ∈ Z. What can be said about the solutions of the following set of linear congruence equations? x ≡ a1 (mod m1), x ≡ a2 (mod m2), x ≡ a3 (mod m3), ..., x ≡ ak (mod mk).
Suppose $m_1, m_2, m_3,\ldots, m_k &gt; 1$ and not necessarily pairwise relatively prime. Also $a_1, a_2,\ldots, a_k\in\Bbb Z$. What can we say about the solutions of the following congruence equations: $\begin{align*} x&amp;=a_1\mod m_1\\ x&amp;=a_2\mod m_2\\  &amp;\phantom{1}\vdots\\ x&amp;=a_k\mod m_k \end{align*}$
I consistently have more than one terminal open. Anywhere from two to ten, doing various bits and bobs. Now let's say I restart and open up another set of terminals. Some remember certain things, some forget.  I want a history that:   Remembers everything from every terminal Is instantly accessible from every terminal (eg if I ls in one, switch to another already-running terminal and then press up, ls shows up) Doesn't forget command if there are spaces at the front of the command.   Anything I can do to make bash work more like that? I'm a student who uses the laptop all the time and I have a problem with Wifi on the laptop. I don't really know what it is. I can describe the problem tho. For example, i closed my laptop last night, so it goes in suspend mode. I turn it on this morning and my wifi doesn't want to connect until I restart the laptop. It says that it's connecting and in Wifi settings the circle animation is spinning, turning Wifi off and on doesn't do anything. Here's what happens when I check to see my network info: sudo lshw -C network   *-network                         description: Wireless interface        product: Wireless 3160        vendor: Intel Corporation        physical id: 0        bus info: pci@0000:02:00.0        logical name: wlp2s0        version: 83        serial: 00:1e:64:e6:1c:93        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list ethernet physical wireless        configuration: broadcast=yes driver=iwlwifi driverversion=5.4.0-52-generic firmware=17.3216344376.0 ip=192.168.0.14 latency=0 link=yes multicast=yes wireless=IEEE 802.11        resources: irq:129 memory:df100000-df101fff   *-network        description: Ethernet interface        product: RTL810xE PCI Express Fast Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        logical name: enp3s0        version: 07        serial: f8:ca:b8:52:72:49        capacity: 100Mbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 firmware=rtl8106e-1_0.0.1 06/29/12 latency=0 link=no multicast=yes port=MII        resources: irq:17 ioport:d000(size=256) memory:df000000-df000fff memory:d0000000-d0003fff  Also I have to mention that my Linux is doublebooted alongside Windows 10, but I don't think that makes any difference. Help would be much appreciated, I wouldn't want to miss my classes because of this hahah Thanks! Why is orbital angular momentum not 0 when spin and linear momentum are not collinear?   Why can it be 0 when spin and linear momentum are parallel?  Like in the example of a scalar field at rest decaying into an electron and a positron via interaction $\frac{1}{2}g\phi \bar{\Psi}\Psi $   If spin is along x axis and three-momenta is along z axis, decay rate will be 0 if two spins are opposite because orbital momentum is not 0 after decaying. Why quantum mechanics? Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before.  How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try?  If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious.  Why should this be the case, or at least, why might we expect this to be the case?   Then there is the issue of measurement which is even more difficult to motivate.  In the usual formulation of quantum mechanics, we assume that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$).  How would you convince your students that this had to be the case?  I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be.  So, what do you guys think?  Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results?  Note that, at some point, you will have to make reference to experiment.  After all, this is the reason why we needed to develop quantum mechanics.  In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying.  I think we can do better.  Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory.  You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment.  Edit:  To clarify, I will try to make use of a much simpler example.  In an ideal gas, if you fix the volume, then the temperature is proportional to pressure.  So we may ask "Why?".  You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.".  If you choose the latter, you can then ask why that is true.  Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example).  The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment.  And in this particular case, the objective is to do this with quantum mechanics. So earlier today I came across , and I am having some trouble understanding the solution. The question is as follows:     You throw a fair six-sided die until you get 6. What is the expected   number of throws (including the throw giving 6) conditioned on the event   that all throws gave even numbers?   Quoted from Jimmy Jin in "Elchanan Mossel’s dice problem"  In the paper it goes on to state why a common wrong answer is $3$. Then afterwards explains that this problem has the same answer to,   "What is the expected number of times you can roll only $2$’s or $4$’s until you roll any other number?"  I don't understand why this is the case. If the original problem is asking for specifically a $6$, shouldn't that limit many of the possible sequences?  I also attempted to solve the problem using another method, but got an answer different from both $3$ and the correct answer of $1.5$.  I saw that possible sequences could have been something like:  $$\{6\}$$ $$\{2,6\}, \{4,6\}$$ $$\{2,2,6\}, \{2,4,6\}, \{4,2,6\}, \{4,4,6\}$$ $$\vdots$$  To which I set up the following summation and solved using :  $$\text{Expected Value} =\sum_{n=1}^\infty n\left( {\frac{1}{6}} \right)^n 2^{n-1} = 0.375$$ Obviously this is different and probably incorrect, but I can't figure out where the error in the thought process is.  Any help on understanding this would be greatly appreciated.  A blog post discussing the problem can be found When I'm using  find . -type f -name "*.htm*" -o -name "*.js*" -o -name "*.txt"  it finds all the types of file. But when I add -exec at the end:  find . -type f -name "*.htm*" -o -name "*.js*" -o -name "*.txt" -exec sh -c 'echo "$0"' {} \;  it seems it only prints .txt files. What am I doing wrong?  Note: using MINGW (Git Bash) What is dependency injection? There have been several questions already posted with specific questions about , such as when to use it and what frameworks are there for it. However,  What is dependency injection and when/why should or shouldn't it be used? Is it possible to recognise hardware (router or network card) just by its MAC address? Just wondering if that is possible. Are there any naming conventions that would allow one to deduce whether the MAC address belongs to a router (of a specific brand), or a network card, etc.. Short description of the scoping rules? What exactly are the Python scoping rules?  If I have some code:  code1 class Foo:    code2    def spam.....       code3       for code4..:        code5        x()   Where is x found?  Some possible choices include the list below:   In the enclosing source file In the class namespace In the function definition In the for loop index variable Inside the for loop   Also there is the context during execution, when the function spam is passed somewhere else. And maybe  pass a bit differently?  There must be a simple reference or algorithm somewhere.  It's a confusing world for intermediate Python programmers. How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$  I tried to prove that by proving that the following summation in in absolute value converges so the original one converges too, but that's incorrect.  Any other ideas? ARIMA equation interpretation I'm trying to replicate ARIMA (1,0,1)(1,0,1) equation in excel as a formula but I am not able to understand the interpretation of white noise residual e(t) or u(t).If could help me understand the interpretation and how to derive it from raw data it'll be really great.  Thanks icons missing from libreoffice toolbars New Ubuntu 16.04.2 installation.  Uninstalled the default libreoffice installation. Then I installed libreoffice 5.4.0.3 from this PPA:  sudo add-apt-repository ppa:libreoffice/ppa   Now all icons are missing from toolbars:    Is this a bug or a funk setting? Any thoughts on how to reset the toolbar to get the typical configuration? I already tried the basic stuff (rebooting, disable/enable toolbars, etc.) I am looking for a package to create bullet gauge graphs, but I couldn't find any.  A bullet graph looks like this:    and an ample description/specification can be found   It is available in Mathematica 9 ()  Is it possible create this kind of graphs in a easy way with TeX (like Matehmatica 9 or similar)? Inform new users if they can upvote an answer they accepted I surprisingly often stumble upon answers which were accepted but clearly not upvoted by the asker. For example, the answer has not received any upvotes at all or the asker’s vote count is zero. While I acknowledge that , most such cases happen with new users, who have only recently received the privilege to vote and who upvote after a friendly reminder.  Given the anonymity of votes and that I do not check the asker’s profile for every question with an accepted answer I stumbled upon, I assume that there is a huge dark figure for this phenomenon. Also, I naturally most often spot this on my own answers, which makes reminding the asker of their privileges somewhat awkward.  I thus propose that users which accept an answer are reminded by a pop-up that they can upvote it, if:   They haven’t already upvoted it. They have the reputation required for voting. They have cast less than a handful of votes.   I consider the latter criterion to be better than an upper reputation for which the message is displayed, since I know at least one user with several thousand reputation points, who has cast much less upvotes or votes on answers than he has accepted answers on some sites (not that his acceptance rate or general feedback on answers is spectacular). Got this problem from a friend and would like some pointers if anyone has any.  Fill an $n \times n$ matrix with the symbols $\{a,b,c,d\}$ subject to the following:     Only the symbol $a$ appears to the left or above an $a$    Only the symbol $b$ appears to the right or above a $b$    Only the symbol $c$ appears to the left or below a $c$    Only the symbol $d$ appears to the right or below a $d$     Here are some examples of a valid fillings: $$\left( \begin{matrix} a &amp; a &amp; b  &amp; b \\ a &amp; a &amp; b &amp; b \\ c &amp; c &amp; c &amp; d \\  c &amp; c &amp; c &amp; d  \end{matrix}\right) $$  $$\left( \begin{matrix} a &amp; a &amp; a  &amp; a &amp; a &amp; b  &amp; b \\ a &amp; a &amp; a  &amp; a &amp; a &amp; b  &amp; b \\ a &amp; a &amp; a &amp; d &amp; d &amp; d &amp; d \\  a &amp; a &amp; a &amp; d &amp; d &amp; d &amp; d\\a &amp; a &amp; d &amp; d &amp; d &amp; d &amp; d\\c &amp; c &amp; d &amp; d &amp; d &amp; d &amp; d\\  c &amp; c &amp; d &amp; d &amp; d &amp; d &amp; d  \end{matrix} \right) $$  What I have done: We'll it is clear to me that the $a$'s (for example) make up a sort of Young Diagram just by the property above. But there is more to the problem than that. If your YT for $a$ is just a square, then you might have a reversed (right justified) YT available to place for the $b$'s. Details like this seem to make this a tricky problem for me. I am intrigued by this problem and I just thought I'd share it here in case there is some easy way to do this counting problem that I've never heard of.  Thanks, How to add coloured line before caption? How to import a module given its name as string? Where does the Mac store account pictures? I took a picture to add to my account on Mac.   Now I want to use it on messenger, but I can't find it. On the account config, it's too small. I need to find the original.   Where are the accout pictures stored on a Mac? Using PyQGIS to transform layer of polygons into layer of holes inside bbox? I'm using python in QGIS. I have a layer of polygons (its a result of some earlier dissolving and has no overlapping polygons but some holes and islands in holes etc). I want to build a new layer which contains one polygon representing a bbox of the first layer with each polygon cut from it.  Essentially I want to "invert" the layer I have so that where the was polygon, now there is not, and where there was not polygon now there is. All inside a bounding box of the original layer.  I can build a bbox easily and I tried to "subtract" the polygon layer from my new bbox polygon layer but the process slows to a stop and then crashes my PC. Here's the algorythm  processing.runalg("qgis.difference","path/to/polygon.shp","path/to/bbox.shp","path/to/result.shp")   I dont care about attributes (obviously). I just need to cut one set of polygons from one larger polygon.  It seems so simple, I wish I could find a simple command or script to acheive this. As stated at the top I'm looking for a solution in python in QGIS. Using Elseif Conditional Statement in QGIS Field Calculator?