Check for the Existence of a Possible Combination of Values in an Array which Sum to a Given Value
What are bitwise shift (bit-shift) operators and how do they work?
How does the new automatic reference counting mechanism work? Why is this true?  $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$ What is the Physical Meaning of Commutation of Two Operators? I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable (self-adjoint operator) in quantum mechanics?   E.g. an operator $A$ with the Hamiltonian $H$? Sums and products of algebraic numbers How does one go about proving that the sums and products of two algebraic numbers over a field $F$ (say $a,b\in K$, where $K/F$ is a field extension) is also algebraic?   If we call $f_a$ and $f_b$ the min. poly's of $a$ and $b$, then I'm assuming the answer involves such polynomials. Perhaps looking at their roots in splitting fields for both of them? And finding a "big" splitting field, constructed from those two other ones?      In particular, I'd like a way of explicitly constructing the minimal polynomials $\ f_{ab}$ of $ab$ and $f_{a+b}$ of $a+b$.     I read somewhere that $g(x)=\Pi_j\Pi_i (x-\alpha_i\beta_j)$ works for $ab$, where the $\alpha_i$ and $\beta_j$ are the roots of $f_a$ and $f_b$, respectively, but I do not know why $g(x)\in F[x]$.  Similar remarks for $a+b$ Just like if we have any quadratic equation which has complex roots, then we are not able to factorize it easily. So we apply quadratic formula and get the roots.  Similarly if we have a cubic equation which has two complex roots (which we know conjugate of each other) and one fractional root, then we are not able to find its first root by hit &amp; trial.  So my question is like quadratic formula, is there exist any thing like cubic formula which help in solving cubic equations?  For example, I have an equation $$2x^3+9x^2+9x-7=0\tag{1}$$ and I have to find its solution which I am not able to find because it has no integral solution. Its solutions are $\dfrac {1}{2}$, $\dfrac{-5\pm \sqrt{3}i}{2} $, I know these solutions because this equation is generated by myself.  So how can I solve equations like these?  Also while typing this question, I thought about the derivation of quadratic formula, which is derived by completing the square method.  So I tried to apply ‘completing the cube’ method on the general equation $ax^3+bx^2+cx+d=0$ but it didn't help.  So please help me in finding a cubic formula or to solve the equations like given in example by an alternative method. Is it OK to reduce the charging current for a Li-ion 18650 battery? I am using NCR18650 batteries with the tp4056 charging module.  When I connect the charger at 2.8&nbsp;V (fully discharged), the tp4056 IC is getting hot at a 1&nbsp;A charge current. Is it OK to reduce this charge current by changing the charge current selecting resistor? So the tp4056 IC heat would be reduced, wouldn't it?  For my application I don't need a quick charge. With a low current, charging time will be increased, but it does not matter for my application.  Will reducing the charge current affect the battery's lifetime? Or will it damage the battery?  I am trying to reduce the heat of the tp4056 IC by reducing the charging current, maybe from 1&nbsp;A to 500&nbsp;mA or less. The 18650 battery never heats for a 1&nbsp;A charge current. I'm working on a large c++ system that is has been in development for a few years now. As part of an effort to improve the quality of the existing code we engaged on a large long-term refactoring project.  Do you know a good tool that can help me write unit tests in C++? Maybe something similar to Junit or Nunit?  Can anyone give some good advice on the methodology of writing unit tests for modules that were written without unit testing in mind? When should I fudge dice rolls? When I started my first D&amp;D campaign as a DM, I accidentally rolled and confirmed a crit on a PC from a goblin. It was an instant kill as our campaign had no cleric. That time I decided to fudge the dice. Now, there's a similar problem, but this time the PCs are at 3rd level and they are dealing with a very important NPC. If they fail this one diplomacy check, the entire game world will fall apart, as the king would then send his entire army against the PC's homeland and will destroy it, plunging my entire game world into chaos, at which time drow that I wanted to save for a later adventure will have to come out, which will instigate a wizard's guild to release their secret ultra-deadly virus, and everything in my game world, including the PCs, will die. So should I fudge the dice roll? When is it good to fudge it? Looking for intro to Conformal Bootstrap Windows registry entries for default explorer view IDW Publishing are bringing out a , Assimilation2, set in ST:TNG (presumably the TNG in the timeline prior to the events of the 2009 movie) featuring the 11th Doctor.  Is the canon of these two universes at all compatible?  Both feature a lot of time travel, so we've seen the development of humanity through many centuries in the future.  Are there direct contradictions that make it impossible for these two universes to merge, without some sort of trick like having the Doctor come from an alternate universe?  Or, given the vast spread of time, is it possible that the events of the Star Trek stories have taken place between Doctor Who stories? Find $\sum_{n=1}^\infty \frac{n^3}{n!}$. Can I install a ROM made for a different device? I own a MicroMax A100. I want to install the latest version of a different ROM (Samsung or Nexus).  Is it possible? If so, can you tell me how?  If not, can I install a ROM made for a different device, but from the same manufacturer? For example, could I install a ROM for the MicroMax A110, which is the successor of A100? It's not consistent to leave an edit pending for approval (thus not making available for everyone else and locking its state) after a user who has those privileges have given his endorsement.   You could say why don't I improve the edit, but that makes no sense for something that is correct and it would not follow the "keep it simple" policy.  Update:  This is the current time line:   Someone who doesn't know about formatting makes a question. It's not legible A user without edit privileges amends the formatting. The correction is not visible, still not legible. I see the question and the pending revision and then cast my approval. It doesn't take effect yet, still not legible. Even for me if I didn't take care of copying the question source because it disappears from my own revision page!   I might agree that one's approval is indicative of both revision's quality and editor's competence. But we should retain that the main objective is to ask and answer questions. So, I propose this model:   Someone who doesn't know about formatting makes a question. It's not legible A user without edit privileges amends the formatting. The correction is not visible, still not legible. A user with edit privileges cast his approval. Revision takes effect. Although it keeps the pending approval (the Edit (1) button). If a second user with edit privileges cast a rejection, it also takes effect as a rollback. I would like to create a virtual device in Android Studio but I can't because hardware acceleration is disabled.  I went to the BIOS and there are two option :  AMD-SVM AMD-IOMMU   Both are enabled.  I also downloaded the HAXM installer from the SDK manager and tried to run it from the   C:\Users\%user%\AppData\Local\Android\Sdk\extras\intel\Hardware_Accelerated_Execution_Manager  with intelhaxm-android file.  I got an error telling me :   This computer does not support Intel Virtualization technology VT X or it is exclusively used by Hyper V. HAXM cannot be installed. Please ensure that hyper V is disabled in windows feature   It makes sense that I can't activate Intel Virtualization technology because I have AMD Ryzen 7 2700U.  With my previous computer it worked just fine but I had an Intel processor.  Some tutorials online suggested to enable windows Hyper-V feature but it's only available for windows 10 pro.  Does anyone know a way to activate hardware acceleration with AMD processor and without Hyper-V feature? Computing the $n^{\textrm{th}}$ permutation of bits. I've seen  post about the $n^{\textrm{th}}$ permutation of a set but that is not what I need. If you have a bit string (ones and zeros only) there are  to quickly permute the NEXT lexicographically ordered bit permutation. For example take $$000111 \rightarrow 001011 \rightarrow 001101\rightarrow\cdots$$ etc. If the string is long then there are going to be approximately one Bajillion of these things. And I want to know how to compute the $n^{\textrm{th}}$ guy without doing the exhaust.  Backgound: This is for a parallel computing job where I need to farm out the search space of elements of a set.  Foreground: I own a copy of Knuth's 4th volume of "The Art of Computer Programming" and I think the answer is in there but I can't seem to find it. (It's like 900 pages).  I'm posting here in the hopes that someone has knowledge of this (obviously). Even if you can point me to a source, say the part in Knuth's book where he describes this problem I would be most grateful. Is it true that The Doctor can only regenerate 12 times? Drop bar to flat bar conversion Gearing             14-Speed Shimano A070 STI Crankset            Prowheel Ounce 42/52T Double Chainring Alloy Crankset Shifters            Shimano A070 STI Front Derailleur    Shimano Claris Rear Derailleur     Shimano A070 Brakes              Tektro Dual-Pivot Caliper Brakes Brake Levers        Shimano A070 STI  Above is what I currently have. I am looking to convert from drop bar to flat bar to use the bike as a casual fitness and commuter bike. All I know is that some parts are road bike specific and may need replacing if I want the conversion done. I hear that the front derailleur, shifters/brake levers (obviously since it currently has STI) and the bar it self. Is there anything else that needs to be changed? I would like someone who has more knowledge in bikes than me to possibly tell me what parts from the above mentioned list needs to be changed. I am not looking to do any upgrades. Just looking for cheap budget parts since it will just be a leisure bike.  Thank you all  Sincerely  Tom. I have a Fujitsu laptop and all was working fine until I ended teamviewer. On the keyboard it has a numberpad on some of the letters which you press the Fn button to use. The system is acting like I have the Fn key pressed and I get numbers shown when it should be letters. If I press the Fn key the numbers go to letters. The Fn key is not stuck down. Is there a way to reset keyboard settings or a known fix. The operating system is Ubuntu 11.04 Natty. Django: AppRegistryNotReady()