How to find the coefficient of $x^2$ in the polynomial $$(1-x)(1+2x)(1-3x)....(1+14x)(1-15x)$$  Is there any particular way to look at such problems?
Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$   Just to begin,  $(1-x)(1+2x) = -2x^2 + x + 1$  $(1-x)(1+2x)(1-3x) = 6x^3 - 5x^2 - 2x + 1$   But expanding on like this take too long.  In the end the terms will be like:  $(1 - 15x)(a_0 + a_1x^1 + a_2x^2 + ....)$  Then just looking at the $x^2$, it will be:  $(a_2x^2 - 15a_1x^2) = x^2(a_2 - 15a_1)$  But it it still a very hard problem, any hints?
Accessing derived class property members from base class object in CSharp How can I get the HTML source of a given web address in C#? The class of cardinal numbers is well ordered I'm looking for a proof that the class of cardinal numbers is well ordered under the order relation $|A|\leq |B| \Leftrightarrow$ exists an injection $f:A \to B$.  In fact, I've found a very beautiful proof in an article, but I think it's not the standard one. How do you prove ${n \choose k}$ is maximum when $k$ is $ \lceil \tfrac n2 \rceil$ or $ \lfloor \tfrac n2\rfloor $? Is zero input impedance good or possible when considering impedance matching? For RF applications, we are usually considering of impedance matching to choose load impedance as the conjugate of the source input impedance for the maximum transferred power. But how about design the source having zero input impedance? Is it even possible or good? A naive thought is that all the power will be on the load if so. I might think it is true for DC case and for amplifier design to have a small output impedance and big input impedance. I lack the knowledge to find the missed chain to put all the knowledge together. How do you think? Thanks! Good books on complex numbers I searched on this site but I found posts asking for books on complex analysis which I guess is not what I want.   I want a book on problems concerning with roots of unity, argand plane, exponential form, inequalities etc. Some examples : , , .  I don't think these types of problems are covered in a analysis book but I don't know. I think I want a book on algebra of complex numbers.   Are these topics covered in analysis book ? If not, please suggests some books on this topic. I guess my question is bit vague, sorry for that. :).      As suggested in the comments I would like to have books similar to . Adding Grids in Project Manager of QGIS I am using QGIS 3.12. I am new to QGIS after having worked with ArcGIS Desktop for a while. I am having issues with adding a grid to my project. I am using CRS ESPG:3587 WGS 84 / Pseudo-Mercator and noticed that the grid is showing the wrong coordinates and have included the parameters I have set for the grids. Combining values in one column (or part) when values in another column (or part) match Collinearity can pose certain problems in various kinds of regression problem. In particular, it can make the parameter estimates have high variance and be unstable.   Various methods have been proposed to deal with this including ridge regression, partial least squares regression, principal components regression, dropping variables and getting more data.   One method that is controversial is standardizing or scaling the independent variables, with various experts saying it is a good (e.g. Garcia) or bad (e.g. Belsley) idea. Belsley's issue seems to be (in non-technical terms) that changing the IVs just pushes the problem under the rug.  But other experts seem not to agree. And the authors tend to get rather heated in defending their own positions.   Back when I did my dissertation (on collinearity diagnostics) I found Belsley's arguments convincing, but that was long ago (I got my degree in 1999).  I'm looking either for expert guidance or for any current review article that is unbiased. So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel.  I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.    My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie.  My first intuition was that the frame and front wheel, together, rotate about the rear axle.  But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight.  This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.  I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong.  If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate.  The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across.  If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.    So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!  edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle  edit 2:  I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle.  A third possible axis of rotation might be the lowest point of the back wheel. Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using logarithm? More simple techniques would be nice.  Here is a simple technique used to approximate square roots by :  For example: $\sqrt {78}\approx 8\frac { 14 }{ 17 } $ , where $8$ is the nearest integer root of $78$, $14 = 78 - 8^2$, $17 = 2 \times 8 + 1$.  if $n=2^k$ we can use the method above.   For example, for $k=2$ Lets calculate $\sqrt [ 4 ]{ 136 } $: $$\sqrt [ 4 ]{ 136 } =\sqrt { \sqrt { 136 }  } \approx \sqrt { 11\frac { 136-{ 11 }^{ 2 } }{ 11\times 2+1 }  } =\sqrt { 11\frac { 15 }{ 23 }  } \\ \sqrt { 11\frac { 15 }{ 23 }  } \approx 3\frac { 11\frac { 15 }{ 23 } -{ 3 }^{ 2 } }{ 3\times 2+1 } =\frac { 544 }{ 161 } =3.38\\$$ The exact result is$$ \sqrt [ 4 ]{ 136 } =3.4149\cdots$$ The method approximates well, but it is working for only $n=2^k$ as I know. Let $a$, $b$, and $c$ be integers, where a $\ne$ 0. Then $$ $$ (i) if $a$ | $b$ and $a$ | $c$, then $a$ | ($b+c$) $$ $$ (ii) if $a$ | $b$ then $a$|$bc$ for all integers $c$; $$ $$ (iii) if $a$ |$b$ and $b$|$c$, then $a$|$c$.  Prove that if $a$|$b$ and $b$|$c$ then $a$|$c$ using a column proof that has steps in the first column and the reason for the step in the second column.  My book is really vague. I'm not really sure what to do..  First I was thinking something like this: $a\mid b\Rightarrow b=as$ and  $b\mid c\Rightarrow c=bt$ and $a\mid c\Rightarrow c = au$ $$ $$ $b + c + c = b + 2c$  $$ $$ $=as + 2(bt+au)$  Yea then I just got lost. I have a laptop with win7 installed. I have now made a 60gb partition which I want to install ubuntu into. The question I have, before I do the installation, is how large each of the root, swap, and home partition should be? I have read some place that root could be as small as 8GB, but isn't that too small? Since I guess beside ubuntu all the softwares installed will reside there as well? And I think I'm going to set my swap to be 2GB large.  My main concern is how large should the root partition should be. I'm mainly going to use ubuntu for programming and browse the web. Where to place 'only' relative to prepositions? I know that questions about the placement of 'only', are often asked here; accordingly, I searched for an answer to my question before posting it.  Question  Where are focusing adverbs placed relative to prepositions?  For example,   For his presentation, he needed to talk only about tomatoes. For his presentation, he needed to talk about only tomatoes.   My Attempt to Answer the Question  In both examples, it seems to me that 'only' is placed close enough to what it is intended to modify (the topic of the subject's discourse: tomatoes) that I cannot see any justifiable misinterpretations.  One of the examples sounds better to me, but at this point in my studies, I am more interested in what the traditional rules of grammar require, than in how the wordings sound. I would like to take the full citation of a reference in my BibTeX file and write commentary in-line after it.  A \fullcitation{id} command would be perfect.  I would like to use it as:  \fullcitation{id1}   This paper talks about A with applications to B.  \fullcitation{id2}   This paper considers how C affects A.  Whenever the \fullcitation{id} command appears, I want it to be replaced with the full reference, including authors, title, journal, and date.  Is this possible? Determine the exact date of creation of an Apple ID Is it possible to determine the exact date and time (if possible) of creation of an Apple ID?   Is there any official or unofficial way to establish this if you don't have the initial sign up information recorded? If a question is in all-uppercase or all-lowercase, our quality filters should automatically reject it Proposal If a question on an English-language Stack Exchange site does not contain at least one uppercase and one lowercase letter, our quality filter should reject it. (We can make exceptions for sites which don't use a mixed-case alphabet, such as .) Justification The quality filter currently rejects neither all-uppercase nor all-lowercase posts. I've checked. The best Stack Exchange users use mixed case. If an asker is too lazy to use their Shift key, their question tends not to be of very good quality. Don't take my word for it. You can see proof. Sam Saffron helped me write  which lets you see with your very own eyes that what I am saying is true. Single-case askers tend to ask poor-quality questions such as:      In addition, posts which use only uppercase or only lowercase — even if they aren't riddled with typos or grammatical errors — tend to be hard to read. Capitalization aids readability. Why is this proposal not a duplicate? This proposal is not a duplicate of the older proposal . This is because it also deals with all-lowercase questions. All-lowercase questions are 50 times more common than all-caps questions.  Stack Overflow gets only about 100 all-uppercase questions per year; see .  Yet it gets about 5,000 all-lowercase questions per year; see . (These statistics are for Stack Overflow only; the rest of our network wasn't considered.) I suspect that the older proposal was declined for two reasons:  All-uppercase questions are rare.  Therefore, the effort required to block all-uppercase questions is too much.   This proposal is better. For just a bit more effort, the Stack Exchange developers can get a much richer reward. Conclusion What do you think of my proposal? Agree? Disagree? Thoughts? PS: This idea is modified and adapted from  by Stack Exchange user Gilles. My dear sir Gilles, I thank you. Note There's even a  which automatically detects questions with all-caps titles and alerts interested users and moderators. Gimp won't allow me to copy and paste a transparent image On the latest version of GIMP on my PC, whenever I try to copy a transparent image from the internet, GIMP will automatically fill in the transparent areas with the background color I have selected.  This is a non-issue on my Mac, but I need to do this work on my PC, so how can I make it recognize the alpha of the image and not fill it with the background color?  The image canvas that I'm working with does have an alpha channel, but that does not seem to make a difference. Helpless after exhausting daily limit for reviewing Suggested Edits Map matching links and ideas? I'm using OpenStreetMap and its vectorial road network and I'd like to implement a map matcher algorithm.   Currently I'm able, for each GPS position, to retrieve the nearest road segment and calculate the projection of this position to that segment, like on this image (Red pin is the pure GPS position, in blue the mapped segment and in Green the mapped position):    However, due to the lack of accuracy of the GPS, sometimes the mapped position jumps from segment to another and can provide some inconsistent mapped position from time to time.  My current algorithm is very basic : from the pure GPS position, I get the nearest segment and decide that the mapped matched position is on this one. I know that this can be really improved.  I can imagine that taking the vehicle direction into account will improve the map matching but do you know any other approach that would enable me to improve my map matcher ?  I seek any link, and/or open source software?